Parameterization of Atmospheric Convection: (In 2 Volumes): Volume 1: Theoretical Background and Formulation: Volume 2: Current Issues and New Theories 1783266902, 9781783266906

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Series on the Science of Climate Change ISSN: 2045-9726 Editor: Hans-F Graf (University of Cambridge, UK)

Published Vol. 1

Parameterization of Atmospheric Convection (In 2 Volumes) Volume 1: Theoretical Background and Formulation Volume 2: Current Issues and New Theories edited by Robert S Plant and Jun-Ichi Yano

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientiic Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Plant, Robert S. author Parameterization of atmospheric convection / Robert S. Plant, Jun-Ichi Yano. pages cm -- (Series on the science of climate change ; volume 1) Includes bibliographical references and index. ISBN 978-1-78326-693-7 (hardcover, v. 1 : alk. paper) -ISBN 978-1-78326-694-4 (hardcover, v. 2 : alk. paper) -ISBN 978-1-78326-690-6 (set : alk. paper) 1. Convection (Meteorology). 2. Atmospheric physics. 3. Weather forecasting. I. Yano, Jun-Ichi. II. Title. QC880.4.C64P595 2015 551.51'5--dc23 2015014333

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Preface

In numerical modelling of weather and climate, we face an issue called subgrid-scale physical parameterization. That is the main topic of this two-volume set with a specific focus on convection parameterization. Despite its importance in atmospheric modelling, currently it is difficult to find a complete, self-contained, and up-to-date description of the convection parameterization problem in the literature. Thus, the aim of this set is to fill a gap in the literature. In order to best fill this gap, this set takes the somewhat unusual position of considering parameterization as a theoretical problem. This is in contrast to the common attitude in the wider modelling community, which usually considers it rather a purely technical problem. As a result, model parameterizations are sometimes seen as something to be tuned and manipulated, and only in a piecemeal fashion, rather than as something fundamental to be periodically rethought and redeveloped. This set takes the view that parameterization is a fundamental science problem demanding clear and rigorous thinking and physical insight. It also intends to demonstrate that such a systematic approach to parameterization development is indeed possible. Clearly we have to recognize that parameterization is in some ways a rather “messy” problem in which compromises, short-cuts, and appeals to limited data sources are sometimes needed in order to reduce the full complexity of the physical problem into a parameterization fit for practical use. Not everything can be established in a very lucid manner as a matter of fundamental physical principles. Nonetheless, some basic principles for convection parameterization can clearly be set, and much valuable guidance is available in trying to adhere to those principles as faithfully as possible when constructing a parameterization. This approach is based on the colv

vi

Preface

lective conviction of our group that a solid theoretical approach offers good prospects for making faster progress on a problem on which relatively slow progress has been made so far. There remain places where we have no such clear-cut answers, but only more or less plausible alternatives, notably for issues such as closure. For these issues, we pay great attention to some of the important technical details that come into play, especially in operational implementations. A few chapters are devoted to operational issues, partly for this reason. The set focuses mainly on the deep-convection parameterization problem, and mainly on its mass-flux formulation. The shallow convection problem is mainly discussed in rather tangential ways throughout the text, although more thorough attention is given to it in Vol. 2, Ch. 24, dedicated to the similarity approach. This omission may be considered unfortunate given the importance of non-precipitating shallow-convection parameterization in its own right, and also its interactions with deep convection. However, we also have to realize that shallow-convection parameterization along with related boundary-layer parameterizations has its own extensive development over the years, sometimes related to but sometimes independent of the development of deep-convection parameterizations. A full and comprehensive consideration of shallow-convection parameterization would, thus, amount to another book volume. We believe that the focus on deep convection is also legitimate considering the fact that this was how the parameterization problem became most acute in global atmospheric modelling in a historical context, and where many of the most acute issues remain. Various issues are examined from historical perspectives throughout the book, because we believe that it is often important to ask where we are coming from in order to know where we are going. For this reason, we often pay special attention to some important historical papers. For example, within the first few chapters we trace some historical roots of the deep-convection parameterization problem through to Riehl and Malkus’s (1958) hot-tower hypothesis. The concept of mass flux naturally arises from this hypothesis. The set is comprised of two volumes and has five parts. Volume 1 contains Parts I and II. Part I examines various basic concepts that are central for parameterization of subgrid-scale physical processes in geophysical modelling: scale separation, quasi-equilibrium, and closure. Part II is devoted to a thorough exploration of mass-flux convection parameterization. Certainly there are alternative approaches, and these are discussed within both volumes, but the mass-flux parameterization framework demands particular

Preface

vii

attention given its status as by far the most popular approach in operational models and at major climate modelling centres. Specific concepts that are heavily exploited in mass-flux parameterization are introduced and discussed one by one in Part II, and strengths and limitations are critically examined. In Volume 2, we consider the actual behaviour of convection parameterizations within the context that they are required to operate, i.e., numerical weather prediction and climate models. Issues within our numerical models and the continuing evolution of the these models pose difficult challenges to mass-flux parameterizations. Volume 2 considers the responses to those challenges. Further critical reconsiderations are required of traditional concepts, and these provoke our interest in possible new frameworks. With their increasing horizontal resolutions, a new generation of numerical models particularly urges us to abandon the traditional framework and assumptions arising from the concept of scale separation. How can we construct a parameterization without this concept? This challenging question is addressed over several chapters in Part III, alongside some closely related issues concerning the need to reconsider our strategies for analysing observations and comparing models to observations. In Part IV we consider some alternative parameterization approaches and formulations inspired by some of the treatments used by other parameterization schemes within the models, notably similarity theory for turbulent flows, the layer-cloud scheme expressed using distribution functions, and various aspects of microphysics schemes. Such considerations naturally lead to and accompany discussions of self-consistency in the formulation of our models, or the lack thereof. Finally, in Part V we explore two perspectives from theoretical physics that seem to hold some promise for parameterization development in the future. First, motivated by recent observational evidence, we consider a proposal in the recent literature that atmospheric convection may occur at self-organized criticality, a macroscopic state far from conventional statistical equilibrium. Second, we consider some aspects of symmetry classifications for partial differential equations, which can provide constraints on the acceptable forms of parameterization such that fundamental properties of the original equations remain as properties of the parameterized system. Some examples are presented for how such theoretical and mathematical principles can be exploited for developing solidly based parameterizations. In order to make the book as self-contained as possible, it also covers various related issues for deep-convection parameterization. For example,

viii

Preface

the inclusion of cloud microphysics into deep convection is inevitable considering its precipitating nature. However, this raises many questions about which microphysical processes need to be included, how they should be represented, and how the representations can be made properly compatible with the deep-convection parameterization framework. Two chapters (Ch. 18 and Ch. 23) in Volume 2 are devoted to a self-contained description of cloud microphysics. Some special considerations are also required for verifying the performance of convection parameterization, especially for the precipitation field at higher resolutions, and Ch. 21 is devoted to model validation issues. We also recognize the importance of extensively exploiting rapidly evolving satellite observations for the development and verification of convection parameterization. Chapter 16 discusses this promising but challenging issue. This set is intended to be both textbook and a reference book. As a textbook, we expect that it can be used for a specialized course at a graduate level, and some selected materials could be appropriate to supplement some undergraduate courses. For this reason, no prior knowledge of meteorology is assumed in most parts of the book. Certainly in places some basic background knowledge would be helpful, say, in order to appreciate fully a discussion about the role of convection within a certain larger-scale phenomenon. Nonetheless, it is intended that the book should be comprehensible to a relative novice in meteorology. Readers are, however, assumed to be familiar with basic physics and mathematics, and under this assumption, we try to introduce the key concepts of meteorology that are required throughout the book, although often in a compact manner. We hope that this book will prove useful not only to meteorologists who wish to develop their expertise in an important aspect of their subject, but also for some engineers and physicists interested in atmospheric modelling, who would like to understand the details of the modelling called parameterization from a theoretical perspective. We should admit that not all of the background issues are carefully introduced or explained. Some more advanced chapters in Volume 2 implicitly assume knowledge not carefully introduced in earlier chapters. We rather hope that the readers would grasp the basic concepts behind atmospheric modelling in the course of reading, or alternatively by following some of the supplementary reading materials that are suggested. As a reference book, it does not intend to replace the detailed documentation that is available for expert users or the slightly more generic documentation that is available for user communities of existing opera-

Preface

ix

tional forecast models and climate models. If we were to attempt such a goal, the materials would rapidly become obsolete. Our intention is rather to provide basic materials so that the readers will then be able to study detailed model documentation and specialist papers on convection parameterization comfortably, and even in a critical manner. As well as a single, coherent reference point, we also hope to provide a source of key questions, unresolved issues, ideas, inspirations, and even occasionally provocations for future research into convection parameterizations and related issues. For the same reasons, the emphasis is often on historical work. Very recent work is also included, but it is often included in the spirit of providing modern examples in a modern context, or else of illustrating an emerging or expanding strand of research. The research evolves rapidly, and it is beyond our capacity to judge which modern articles will still be read and considered important ten or twenty years hence. For this reason, references to the most recent work are partial at best and may simply reflect the materials most familiar to the various authors at the moment of writing, but all best efforts were made to cover what is now considered to be important historical work as much as possible. This set has evolved from a European network activity, COST Action ES0905 Basic Concepts for Convection Parameterization in Weather Forecast and Climate Models (Yano et al., 2015). EU funding for this network has provided opportunities for frequent meetings and discussions on fundamental issues of convection parameterization over a four-year period between 2009 and 2014. The concept of the book has also gradually taken shape through this COST Action activity. Many of the participants in this COST Action also agreed to contribute to the book, and many other participants are owed thanks for their contributions to the spirit of the COST Action which is well reflected in the spirit of the book. The experience of this COST Action has been unique for all of the participants, and has been characterized by many lively debates coupled with a willingness to delve into details of a very wide range of specialisms. Participants ranging from operational forecasters to numerical modellers to cloud microphysicists to plasma physicists to Lie algebra specialists have all taken an active part. Published reports on five workshops and a summer school organized under this COST Action would provide a good flavour of the whole activity (Plant et al., 2014; Yano et al., 2010b, 2012a, 2013b, 2014a,b). Two preliminary workshops may also be of interest

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Preface

(Yano et al., 2008; Yano and Bechtold, 2009). The stimulating range of contributions to the COST Action is also reflected in the range of this book.

R.S. Plant and J.-I. Yano Reading, UK and Toulouse, France September 2014

Acknowledgments

As with many other books, the genesis, development, and writing of this book has had many sources of influence, input, support, and encouragement from various people at various times. We try to acknowledge some of those general sources here, with the hope that our omissions may be forgiven as failures of our memories rather than failures of our gratitude. RSP would like to thank Lisa Bengtsson, Hans Graf, Johannes Quaas, Sandra Turner, and Till Wagner for early discussions on the scope and structure of the book, Dmitrii Mironov for originally proposing and encouraging him as an editor, Olga Petrikova and Lukas Muesele for help with maintaining a website for swapping draft materials, Michael Glinton for his work on the bibliography and index and his attention to detail, and Hiroko and Freya Plant for many reasons. JIY would like to thank Peter Bechtold, Leo Donner, and Guang Zhang for helpful communications in the drafting process. RSP and JIY would like to thank Graeme Stephens for the use of his painting on the cover, and Natalie Tourville for providing the electronic image. COST Action ES0905 has been a significant driving force of this book by providing travel funds for many meetings involving the various named authors and many others from across Europe. The discussions in those meetings were an indispensable basis for many aspects of this book. Furthermore, JIY and RSP acknowledge travel funding from CNRS and from the Royal Society, which have aided the writing and editing of the book. For Ch. 4, RSP and JIY acknowledge discussions with Laura Davies on the range of interpretations of convective quasi-equilibrium, with Mick Short on etymology, and with Paul Williams and Maarten Ambaum on slow manifolds and geostrophic adjustment. xi

xii

Acknowledgments

For Ch. 6, JIY thanks Ed Zipser for pointing out that the phrase “hot tower” does not appear in the original paper of Riehl and Malkus (1958). For Ch. 9, RSP and OMA acknowledge that part of the work was funded by the NERC project DIAMET, NE/I005234/1.

Contents

Preface

v

Acknowledgments

xi

List of Contributors

Part I

xvii

Basic parameterization concepts and issues

1

Introduction to Part I

3

1. Moist atmospheric convection: An introduction and overview ´ Horv´ A. ath

5

2. Sub-grid parameterization problem

33

J.-I. Yano 3. Scale separation

73

J.-I. Yano 4. Quasi-equilibrium

101

R.S. Plant and J.-I. Yano 5. Tropical dynamics: Large-scale convectively coupled waves ˇ Fuchs Z. xiii

147

xiv

Contents

Part II

Mass-flux parameterization

171

Introduction to Part II

173

6. Hot-tower hypothesis and mass-flux formulation

175

J.-I. Yano 7. Formulation of the mass-flux convective parameterization

195

J.-I. Yano 8. Thermodynamic effects of convection under the mass-flux formulation

227

J.-I. Yano 9. Spectral and bulk mass-flux representations

249

R.S. Plant and O. Mart´ınez-Alvarado 10. Entrainment and detrainment formulations for massflux parameterization

273

W.C. de Rooy, J.-I. Yano, P. Bechtold and S.J. B¨ oing 11. Closure

325

J.-I. Yano and R.S. Plant 12. Convective vertical velocity

403

J.-I. Yano 13. Downdraughts

419

J.-I. Yano 14. Momentum transfer J.-I. Yano

449

Contents

xv

Bibliography

481

Index

511

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List of Contributors

P. Bechtold European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading, RG2 9AX. United Kingdom. [email protected] S.J. B¨ oing Delft University of Technology, TU Delft, P.O. Box 5046, 2600 GA Delft. Netherlands. [email protected] ´ Horv´ A. ath Hungarian Meteorological Service Storm Warning Observatory, Vitorlas utca 1, 8600 Siofok. Hungary. [email protected] ˇ Fuchs Z. Department of Physics, University of Split, Teslina 12, 21000 Split. Croatia. [email protected] O. Mart´ınez-Alvarado Department of Meteorology, University of Reading, Earley Gate, PO Box 243, Reading, RG6 6BB. United Kingdom [email protected]

xvii

xviii

List of Contributors

R.S. Plant Department of Meteorology, University of Reading, Earley Gate, PO Box 243, Reading, RG6 6BB. United Kingdom [email protected] W.C. de Rooy Royal Netherlands Meteorological Institute (KNMI), PO Box 201, 3730 AE De Bilt. The Netherlands. [email protected] J.-I. Yano ´ Groupe d’Etude de l’Atmosph`ere M´et´eorologique/Centre National de Recherches M´et´eorologiques, and Centre National de la Recherche Scientifique, M´et´eo-France, 42 Av. Coriolis, 31057 Toulouse. France. [email protected]

PART I

Basic parameterization concepts and issues

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Introduction to Part I

The goal of Part I is to introduce basic concepts for atmospheric subgridscale parameterization, with the subsequent focus on deep convection in mind. In order to set the scene, moist atmospheric convection is introduced in a phenomenological manner in Ch. 1: what exactly is it, what does it look like, and why and how is it such an important aspect of the Earth’s atmosphere? A basic formulation for the subgrid-scale physical parameterization problem is introduced in a very general manner in Ch. 2. Here also there will be a discussion of why and for what purposes we need to introduce a subgrid-scale parameterization into a weather forecast or climate projection model. Special emphasis will be placed on the point that a subgrid-scale parameterization is not automatically required simply because not everything is explicitly resolved by the numerical model. We have to introduce a parameterization only if a given subgrid-scale process contributes significantly to the evolution of the variables that are explicitly computed in the model. For the case of convection this requirement is very obviously satisfied, certainly if we are interested in the tropics, and certainly if we are interested in forecasting precipitation. Nonetheless, this simple point is often neglected in model developments, and is always worth recalling when contemplating an elaboration of an existing approach. Most of the current subgrid-scale parameterizations are built upon two complementary concepts: scale separation and quasi-equilibrium. These two key concepts are discussed in detail in Chs. 3 and 4 respectively. It may be worthwhile to remark from the outset that these two concepts are not necessarily always reliable guiding principles in the real atmosphere, and that there are clearly limitations to current methods for that reason. 3

4

Introduction to Part I

Nonetheless, there are many situations encountered where such principles do genuinely work well. Moreover, they are worthy of attention if only because any future methods designed to work more generally must, for example, be able to handle equilibrium situations well in the appropriate limits. Part I is completed by Ch. 5 which studies, in a very idealized and analytically tractable way, the interactions between convection and dynamical waves in the tropics. It focuses on a particular mode of variability: the convectively coupled Kelvin wave. By demonstrating those aspects of convective behaviour that are linked most strongly with tropical dynamics, it provides a challenge to the convective parameterization approach to be developed in Part II, and to be assessed in Part III. Can the parameterizations be designed to interact appropriately with the large-scale flow?

Chapter 1

Moist atmospheric convection: An introduction and overview

´ Horv´ A. ath Editors’ introduction: This chapter introduces atmospheric convection, the process that needs to be captured in the construction of a parameterization. ´ It is written by Akos Horv´ ath, of the Hungarian Meteorological Service, and includes some of his own photographs illustrating the morphology and phenomenology of moist convection. In a parameterization context, it is common to talk about “dry” convection as that produced by thermals which are not associated with cloud and which are confined within the boundary layer; to talk about “shallow” convection as relatively weak, non-precipitating cumuli; and to talk about “deep” convection as cumulonimbus clouds. These aspects are typically treated separately in numerical models, within the boundary-layer parameterization, within the shallow-convection parameterization and within the deep-convection parameterization respectively. This chapter reminds us that such distinctions are physically meaningful as well as practically useful, with distinctive physical processes being involved for each type. At the same time, the chapter provides a valuable reminder that such distinctions are not completely simple and clear-cut, and can only be caricatures of the full range of behaviours of atmospheric convection.

1

Introduction

Atmospheric convection has some specific features that distinguish the associated phenomena from other atmospheric formations. Convection is the smallest compact weather system which can affect the entire troposphere: it is able to transport mass and energy from the surface to the tropopause. 5

6

Part I: Basic parameterization concepts and issues

A convective cell has its own circulation system with a concentrated updraught and surrounding downdraught channels. A single cumulonimbus, which is the manifestation of developed moist convection, is able to produce significant or severe weather events like windstorms, hail, or lightning. The characteristic size of a convective cell is in the range that is visible to the human eye and the humidity condensed into water droplets, like ink in a fluid, visualizes the phenomena. Considering atmospheric motions, the role of convection is pivotal in the tropical zone and minimal in the polar regions. In the zone of westerly winds, convection has a wide range of intensities determined by synoptic-scale weather systems. Convective phenomena have long been a focus of meteorological research. There were several books in the post-war era (e.g., Byers and Braham, 1949) that present a comprehensive picture of thunderstorms. Studies had been made to find the place of convection among different meteorological scales (Orlanski, 1975) and describe dynamical backgrounds (Kessler, 1985, pp. 259–276) and the possibilities for numerical prediction (Haltiner and Williams, 1980). In the last three decades hundreds of articles have been published concerned with the phenomenology of atmospheric convection from the cumulus scale to convective systems. Here, only some of the general summary works can be referred to such as Cotton and Anthes (1989, pp. 368–707) and Houze Jr. (1993) about storm and cloud dynamics, the overview of convection by Emanuel (1994) or the monograph of Doswell (2001). New results of convection-related research gradually built in the meteorological text books (Bluestein, 1992, pp. 4–6, 413–418; Bluestein, 1993, pp. 431–445, 545–563; Holton, 2004) and mesoscale-meteorology-related books (Markowski and Richardson, 2010). The aim of this chapter is to introduce moist convection. It will be shown that water vapour behaves like a fuel to deep convection and that the existence or absence of atmospheric humidity may affect the occurrence of cumulus clouds and eventually the entire circulation.

2

Convective components

Investigating the causes of convection in the atmosphere, three main aspects can be highlighted: the buoyancy force, which is responsible for free convection; convergence, which causes forced convection; and, vertical wind shear, which can play an additional role in the formation of organized thunderstorms. These aspects are referred to as convective components hereinafter.

Moist atmospheric convection: An introduction and overview

7

The first of the convective components is buoyancy, and this originates from the density difference between a rising air parcel and its environment. To demonstrate free convection let us consider a hot air balloon with an insulating and elastic wall for which the pressure in the balloon is equivalent to its environmental pressure. Inflating the balloon with hot air and releasing it, the balloon begins to rise. In the rising balloon the air temperature decreases rapidly following the dry adiabatic temperature gradient, and thus its temperature quickly drops beneath the temperature of its environment. Because the pressure of the air inside the balloon does not differ from the pressure of the surrounding atmosphere, the density of the air becomes higher, and the balloon starts to sink. The sinking balloon with the insulating wall heats up by the adiabatic rate again, and the process repeats itself, creating an oscillation. (In reality, because the wall is not fully insulating, the balloon cools and falls down.) To keep the balloon’s air sufficiently warm to create rising motion, a gas burner is applied which is turned on or off by the balloonist to hold a desired elevation. In the atmosphere a cumulus cloud may be considered like a hot air balloon and the role of the burning gas is played by the condensing water vapour. In the real atmosphere, dry convective cells are “thermals” in which the initial warm air is available from the hot near-surface layers, but because of the lack of humidity they are not able to develop into a cumulus phase and they can not rise up into higher elevations. Because buoyancy depends on density differences, the vertical change of the environmental air temperature is also a very important factor concerning free convection. The second convective component is convergence, which is responsible for forced convection, and it can be envisaged as obstacles at mountainous regions or giant pistons at cold fronts. The simplest case of convergence is when orography behaves as an obstacle to the horizontal flow and forces air masses to rise. In this case the energy for the uplift is provided by the kinetic energy of the horizontal current (wind), but the condensation may also help the raising process. As the rising air cools adiabatically, it reaches its dew point and the condensation starts to heat up the air. When the air mass has crossed the mountain its temperature remains lower (and its density remains higher) than the environment’s, and it starts to sink and warms up on the lee side. Finally, a similar oscillation begins as for the unheated balloon and wave clouds demonstrate this effect well. (In this way, wave clouds such as altocumulus lenticularis can be referred to as the result of forced convection.) Convergence can be generated by frontal systems as well, especially by

8

Part I: Basic parameterization concepts and issues

cold fronts. The efficiency of the frontal elevation depends on the bend of the front and the wind shear along the front (Weisman and Rotunno, 2004). On frontal systems with a tilting angle that is too small and motion that is too fast, convective clouds bend over the frontal surface and they are not able to grow. Generally this case applies for a cold front of the second kind (katabatic cold front; see Kurz, 1998). Slower moving fronts with a steeper angle of tilting generally produce better conditions for convection. Inside an air mass much weaker convergence lines appear and they are also able to help convection as they trigger effects within unstable air. The third convective component is vertical wind shear. This effect can be demonstrated by considering the thunderstorm as a huge chimney, where, in the updraught channel, low-level hot air rises up to the top of the troposphere, to the layer of the jet stream. The jet stream causes a non-hydrostatic pressure depression in the chimney and just like a venturi it helps the updraught in the tube. The physical background of this effect is the principle of Bernoulli (cf. Ch. 14, Sec. 8.1.1). This effect appears only for well-developed thunderstorms where the updraught channel has already been built up. Several conditions have to co-exist for the chimney effect to work properly: instability for strong thunderstorms, a large amount of water vapour, and a jet stream at a high level. However, there are some synoptic patterns where conditions are favourable and severe thunderstorms can grow supported by wind shear. The above-described components are the basic reasons for convection, and their combination shapes and influences real convective phenomena. The concept of convective components can be introduced via the vertical motion equation, which describes the rate of change of the vertical velocity (w) as a function of basic parameters such as pressure (p), density (ρ), and acceleration due to gravity (g): 1 ∂p dw = −g − . (2.1) dt ρ ∂z The operator d/dt is the total derivative operator: d ∂ ∂ ∂ ∂ = +u +v +w , (2.2) dt ∂t ∂x ∂y ∂z where t is the time and by the chain rule u = ∂x/∂t, v = ∂y/∂t, and w = ∂z/∂t are the wind components. On the right-hand side of Eq. 2.1 the variations of pressure and density are much more significant in the vertical direction than the horizontal, which allows them to be written as p = p¯(z) + p′ (x, y, z, t),

(2.3)

Moist atmospheric convection: An introduction and overview

ρ = ρ¯(z) + ρ′ (x, y, z, t),

9

(2.4)

where p¯(z) and ρ¯(z) are base-state pressure and density. “Base state” means some average-like condition that is useful to describe the state of larger-scale phenomena. For pressure, the hydrostatic condition can be considered as a base state where vertical motions are negligible. This condition requires 0 = −g −

1 ∂ p¯ . ρ¯ ∂z

(2.5)

Parameters p′ (x, y, z, t) and ρ′ (x, y, z, t) represent perturbations of pressure and density, respectively, superimposed on the base state. Mesoscale atmospheric convection is a typical perturbation process on the base state of a synoptic-scale cyclone. Removing the base-state hydrostatic balance Eq. 2.5 from the vertical motion equation, Eq. 2.1, gives ρ′ 1 dp′ dw =− g− dt ρ¯ ρ¯ dz

(2.6)

under a leading-order approximation of a Taylor expansion. The first term on the right-hand side represents the first convective component: buoyancy. In the second term, the perturbation pressure p′ can be responsible for the second and third convective components: convergence and wind shear. The terminology “moist convection”, in most cases, implies atmospheric processes that are closely coupled to buoyancy: b=−

ρ′ g. ρ¯

(2.7)

The above approach is an application of the so-called “anelastic approximation for vertical motion” (cf. Ch. 7, Sec. 3). Because density is not a directly measurable parameter, in practice the buoyancy is formulated using the temperature. The equation of state (p/ρ = RT ) reads: p¯ + p′ 1 = ρ¯ + ρ′ , ¯ T + T′ R

(2.8)

where R is the gas constant for dry air. Multiplying this equation by T¯ + T ′ , and applying the state equation again for the base state parameters ρ¯T¯ = p¯/R and finally neglecting multiplication of perturbed values, for example ρ′ T ′ , it becomes: p′ = ρ¯T ′ + ρ′ T¯. R

(2.9)

10

Part I: Basic parameterization concepts and issues

Dividing the above equation by ρ¯ and using the definition of the buoyancy, the following equation is obtained:   ′ p T′ − ¯ . (2.10) b ≈ −g p¯ T In the real atmosphere the relative pressure perturbation |p′ /¯ p| is an order of magnitude smaller than the relative temperature perturbation |T ′ /T¯ |. For example, in a moderate thunderstorm the pressure increases 2–5 hPa p = 0.005), while the temperature at 1000 hPa base-state pressure (p′ /¯ can decrease 10 K at 290 K base-state temperature (T ′ /T¯ = 0.03). That p| ≪ |T ′ /T¯| can also be proved theoretically in conditions where the |p′ /¯ maximum wind speed is definitely smaller than the speed of sound. Regarding convective phenomena, the base state means the surrounding conditions of the rising or sinking air parcel. If the temperature of the parcel is Tp , and the environmental temperature is T¯ = Te , then the perturbation is T ′ = Tp − Te , and so given the above deliberations, the buoyancy can be written as: Tp − Te . (2.11) b≈g Te Here, however, note an ambiguity in defining an environment. This issue will be discussed further in Vol. 2, Ch. 19. Humidity is always present in the atmosphere and it has an especially important role in moist convection. A common humidity descriptor is the mixing ratio qv that can be written as the ratio of the density of the water vapour (ρv ) to that of dry air (ρd ). Applying the gas law: qv =

R e e/Rv T ρv = , = ρd (p − e)/RT Rv p − e

(2.12)

where Rv is the vapour gas constant and e is the partial pressure of the vapour. Using R/Rv = 0.622 and the fact that the vapour pressure e is small compared to the dominant dry pressure (p ≫ e), the above equation can be written: 0.622e . (2.13) qv = p Considering the gas law for the moist atmosphere with a pressure of p + e and with a density of ρ(1 + qv ), the virtual temperature Tv can be defined from: p+e = RTv . (2.14) ρ(1 + qv )

Moist atmospheric convection: An introduction and overview

11

Obtaining an expression for e from Eq. 2.12 and substituting it into the left-hand side of Eq. 2.14, and further applying the gas law for p/ρ, we find that:  1 + 1.608q  p  1 + 1.608qv  p+e v = (2.15) = RT = RTv , ρ(1 + qv ) ρ 1 + qv 1 + qv

and from the above equation the virtual temperature can be expressed as:  1 + 1.608q  v ≈ T (1 + 0.61qv ). (2.16) Tv = 1 + qv The most important advantage of Tv is that the buoyancy equation can be extended to the case of a moist atmosphere by replacing T with Tv : b≈g

3

3.1

Tv p − Tv e . Tv e

(2.17)

Convective developments: From thermals to cumulonimbus First phase: the thermal

The thermal is the first phase of convection and it is driven by the first convective component: buoyancy. Using the earlier analogy, the thermal can be thought of as a hot air balloon released without turning the gas burner on. Depending on the weather conditions, the surface albedo, soil type and land use are factors that determine the appearance and strength of thermals. The heat of the “balloon” may come from the solar-irradiated hot surface. Surface albedo largely determines the amount of solar energy that heats up the surface, and that heat can supply the energy for the thermal. (The source of the initial heating is not necessarily the sun’s radiation; for example, a warm sea surface can also supply heat energy for thermals.) The energy provided by the surface to the atmosphere is divided into sensible and latent heat fluxes. The latent heat flux depends on the soil humidity content. When the energy is used to produce evaporation of soil humidity, the sensible heat flux remains moderate and the initial heating of the thermals is weaker. Glider pilots know well that on the morning after a rainy day, thermals start much later and remain weaker than during a dry period. Around dawn after a clear night, the surface and the low levels of the atmosphere have cooled down and a temperature inversion evolves as the result of outgoing

12

Part I: Basic parameterization concepts and issues

long wave radiation. The morning sun starts to warm up the surface and lower atmospheric layers. Because of warming from below, the inversion becomes thinner. The low-level inversion acts as a filter for thermals; only stronger air bubbles are able to break through, and the rising thermal causes a mass deficit below. To refill that deficit, horizontal flows are induced, towards the inversion-breaking thermal. Instead of trying to find its own way up, the low-level air streams up through the opened gap. The longevity of a thermal is also supported by the compensating subsidence, which makes the inversion stronger because the sinking air warms up dry-adiabatically. In this way the subsidence decreases the chance for development of inversionbreaking competitor thermals in the surrounding area (Fig. 1.1).

Fig. 1.1 Only a few thermals can break through the inversion and near-surface compensating flows move air towards these thermals, providing the warm air supply.

The location and timing of the inversion-breaking thermals depends on the above-mentioned surface conditions and on the near-surface turbulence. On a sunny morning, an aviation meteorologist can recognize the beginning of convection from the refreshing wind coming from a changing direction.

Moist atmospheric convection: An introduction and overview

13

The temperature in the thermal cools at the adiabatic lapse rate and quickly becomes colder than its environment. The future of the rising air bubble primarily depends on its water vapour content: with a higher water vapour mixing ratio the air becomes saturated and during the condensation the latent heat released starts to warm up the air. The appearance of liquid water makes the rising air bubble visible and the thermal develops into the cumulus phase. These processes for a rising unit mass parcel can be described using the first law of thermodynamics: dp dT −α , (3.1) Q = Cp dt dt where Q is the heating rate per unit mass, Cp is the specific heat for processes at constant pressure, and α = 1/ρ is the specific volume. It is supposed that in the thermal phase of the convection there is no heat loss or heat release so that Q is zero. Thus, parcels in the thermal rise adiabatically. Neglecting any horizontal motion in the thermal and applying the chain rule in the quasi-static state, it can be written: dT dz dT = (3.2) dt dz dt dp dp dz = . (3.3) dt dz dt The combination of the adiabatic form of Eq. 3.1 with Eqs. 3.2 and 3.3, and applying the base-state equation pα = RT yields: Cp dT dp = . (3.4) R T p The above equation can be integrated from the surface where the pressure is ps and the temperature is Ts to the altitude where the pressure is p. After integration and taking an anti-logarithm, the temperature at the higher altitude can be expressed:  R/Cp p T = Ts . (3.5) ps This equation also allows a constant characteristic for the given parcel to be defined, introducing a potential temperature θ defined in respect of the p0 reference pressure level (generally p0 = 1000 hPa):  R/Cp p0 . (3.6) θ=T p

14

Part I: Basic parameterization concepts and issues

The value of θ remains conserved (constant) during the rising or sinking of the dry adiabatic thermal. The cooling rate of the air in the rising thermal (Γd ) can be calculated from the adiabatic form of Eq. 3.1 in combination with the hydrostatic equation αdp = −gdz. Thus: g dT (3.7) =− . Γd = dz Cp Here, the condition was used that the pressure in the thermal is equivalent to the environmental pressure, which is itself hydrostatic (the wall of the balloon is flexible). Trajectories of constant dry adiabatic processes are printed on meteorological diagrams (Fig. 1.2). One such thermodynamic diagram is the emagram where the vertical axis is log p (decreasing upwards) and the horizontal axis is scaled by temperature. On the emagram, blue lines show constant dry potential temperatures: θ(p, T ) = constant. 3.2

Second phase: the cumulus

In the dry thermal phase, the free atmosphere works against dry convection because its temperature generally decreases less strongly with altitude than the adiabatically cooling thermal’s temperature. When the cooling air bubble becomes saturated and the water vapour condensation starts, cumulus humilis appear in the sky. The level at which condensation starts is named the “lifted condensation level” (LCL), or the “cumulus condensation level”. By the balloon analogy, the gas burner is turned on at the LCL (Fig. 1.2). Above the cumulus condensation level, the rising air bubble is heated by the “gas burner” of the latent heating from condensation. The saturated air bubble’s cooling rate is then less than the dry-adiabatic rate. This process can be described by the so-called pseudo-wet adiabatic process (or “pseudo-adiabatic” for short), for which the condensed moisture is assumed to fall out of the parcel, but the condensational heat is assumed to warm the air parcel. (Thus, the condensed water is assumed to precipitate, or alternatively, the water-loading effect is simply neglected.) The cooling rate of the rising cumulus, produced by a pseudo-adiabatic process, can be less than the vertical cooling rate of the free atmosphere and in this way the cumulus temperature may become warmer than that of the surrounding air. The name of the level above which the rising air parcel becomes warmer than the environment is the “level of free convection” (LFC). Reaching the LFC is an important moment in the life of the cumulus because above this

Moist atmospheric convection: An introduction and overview

15

Fig. 1.2 The trajectory of a rising parcel on the emagram. The thick black line represents the temperature profile, while the dashed black lines show profiles of the dew point. The trajectory of the rising parcel in the thermal phase is shown by the thick blue line, and in the condensing phase by the thick red line. Key: LPL, lifted parcel level; LCL, lifted condensation level; CIN, convective inhibition; LFC, level of free convection; EL, equilibrium level; LMCT, level of the maximum cloud top.

level the surrounding temperature is cooler than the cumulus’s temperature and the buoyancy helps to raise the air bubble further. Not all cumuli have enough energy to reach their LFC. Especially in developing anti-cyclonic weather situations, when the air is stable, the morning sun heats up the wet surface and fair-weather cumuli appear. These are cumulus humilis which do not reach the LFC and thus remain as thin cumulus clouds (Fig. 1.3). Reaching the LFC is the next filter for convective cells. Their success depends on the initial kinetic energy taken from the thermal phase, the vertical temperature gradient of the atmosphere, and the water vapour content of the rising air. The vertical temperature gradient depends on the actual weather situation. The higher the cooling rate, the greater the chance of a thermal becoming a cumulus and reaching the LFC. However, a stable lower layer does not necessarily mean that the convection fritters

16

Part I: Basic parameterization concepts and issues

Fig. 1.3 Cumulus humilis clouds in a stable post-frontal weather situation (picture taken by the skywatch camera of the Hungarian Meteorological Service on the top of the hill Kab-hegy).

away, but rather it causes fewer but stronger cumuli to develop (or else it may indeed prohibit all convective phenomena). The required amount of water vapour to supply cumulus development can come from the lower layers of the atmosphere along with the thermal, or it can be mixed in from the environment (this mixing process is called “entrainment”). Part of the vapour carried by the thermal ultimately originates from the soil through the latent heat flux. The soil is able to store and to release moisture to the atmosphere, and in respect of the convection, the soil works as a kind of memory of the weather: precipitation fallen yesterday can influence today’s convective activity. Once the rising air parcel has managed to reach the LFC, the surrounding free atmosphere supports its further rise. The energy the parcel requires to reach the LFC is called convective inhibition (CIN). Note that along a parcel trajectory, a Bernoulli function is conserved, and thus either a sufficient kinetic energy or else a pressure force is required in order to overcome CIN. Recall that the pseudo-wet adiabatic process assumes that the condensed water falls out of the rising parcel immediately, and so in this process T (p) can not be written directly in an explicit form as for a dry adiabatic

Moist atmospheric convection: An introduction and overview

17

process. Nevertheless, a combination of the first law of thermodynamics, the Clausius–Clapeyron equation, and the Dalton law for partial pressure allows the process to be described through a sequence of short segments. For the water vapour of a saturated air parcel, with qv denoting the mixing ratio and e the partial pressure of water vapour, the equation of the first law of thermodynamics (Eq. 3.1) can be applied: de∗ Rv T = Q, (3.8) e∗ where an asterisk denotes saturation and Cp v is the specific heat of water vapour at constant pressure. The same equation for the dry air component of the parcel is: Cp v dT −

Cp dT −

d(p − e∗ ) RT = 0. p − e∗

(3.9)

The heat source term Q in Eq. 3.8 now is not zero because the latent heat release (the “gas burner”) warms the rising parcel. The heating may be written as: Q = −Lv dq ∗ ,

(3.10)

where Lv is the latent heat of condensation and dq ∗ is a condensation rate of saturated vapour. From the above three equations it follows that: −Lv q ∗ dq ∗ = (Cp + q ∗ Cp v )dT −

∗ d(p − e∗ ) ∗ de RT − q Rv T. p − e∗ e∗

(3.11)

To modify the last term of Eq. 3.11, we substitute the Clausius– Clapeyron equation: Lv de∗ = dT Rv T 2

(3.12)

and integrate from a level (p1 , T1 ) to the level (p2 , T2 ). As a result, Eq. 3.11 reduces to: 1  p − e¯∗  1 T2 1 R, (3.13) = (Cp + q¯∗ Cp v ) ln − − ln −Lv q¯∗ T2 T1 T1 p2 − e¯∗

where q¯∗ = (q ∗ 1 + q ∗ 2 )/2 and e¯∗ = (e∗ 1 + e∗ 2 )/2 are mean values of the two levels. Equation 3.13 allows the temperature of the rising air mass with saturated humidity to be calculated (or the rising balloon temperature when the gas burner is on). Supposing that p1 and p2 are sufficiently close to each other, and that the first level is given by (p1 , T1 ), then e∗ 1 can be calculated

18

Part I: Basic parameterization concepts and issues

by a practical form of the Clausius–Clapeyron equation, for example by the August–Roche–Magnus formula:   17.625T (3.14) es (T ) = 6.1094 exp for T in ◦ C. T + 243.04 The next step is the calculation of the saturated mixing ratio q ∗ 1 using the approximate formula: 0.622e∗ . (3.15) q∗ = p (Derivation of the above equation is not detailed here; see for example North and Erukhimova, 2009.) The following step is the calculation of q¯∗ and e¯s , which are approximated by q¯∗ = q ∗ 1 and e¯s = es1 , while T2 can be determined by applying Eq. 3.13. As T2 is known, the q¯∗ 2 and e¯s2 parameters can be retrieved from Eqs. 3.14 and 3.15. Recalculating the mean values from the newly obtained q¯∗ 2 and e¯s2 , a more accurate T2 may be determined from Eq. 3.13. The iteration procedure described above allows us to follow the temperature change of the rising saturated air parcel with small steps. Going to the reference p0 = 1000 hPa pressure level the temperature obtained in this way is known as the “wet saturated potential temperature” θws , which is a constant (conservative) value for the “wet part” of the convection. Another related parameter is the “wet potential temperature” θw . This can be calculated by lifting the air parcel dry-adiabatically until it becomes saturated (i.e., to its LCL), then stepping down along the wet-adiabatic path described above to the p0 = 1000 hPa reference level. A widely used conservative parameter is the “equivalent potential temperature” (θe ). This can be calculated such that the parcel is lifted up to the condensation level, then the saturated air parcel is raised in a pseudoadiabatic manner until all the water vapour condenses out. The dried-out parcel is finally taken down to the reference level p0 = 1000 hPa along the dry adiabat. In the condensing phase of the convection, the saturated equivalent potential temperature (θs ) can be written in the explicit form:   Lv q ∗ θs ≈ θ exp , (3.16) Cp T

where θ, q ∗ and T are parameters of the saturated air parcel. (For details about the derivation and aproximations used, see Vol. 2, Ch. 22, Sec. 5.2; Ambaum, 2010, Ch. 6; or Emanuel, 1994, Ch. 4.) On the emagram (Fig. 1.2), red lines represent wet saturated potential temperatures: θws (p, T, q ∗ ) = constant, and grey dashed lines show

Moist atmospheric convection: An introduction and overview

19

constant saturated mixing ratios: q ∗ (p, T ) = constant. An actual vertical T (p) profile of the atmosphere is shown by the black lines. The dew point profile is shown by a black dashed line. The thick blue line represents the trajectory of the rising thermal from the LPL to the LCL where the parcel becomes saturated. This movement is along the dry adiabatic curve θ(pLPL , TLPL ). The lifted condensation level is set where q ∗ (pLPL , TLPL ) crosses θ(pLPL , TLPL ) and the diabatic heating starts. The parcel-buoyancy equation (Eq. 2.17) can be rewritten using the equivalent potential temperature: θe p − θ e e dw =b=g , (3.17) dt θe e where θep and θee are the parcel’s and the environment’s equivalent potential temperatures, respectively. Suppose that the parcel starts from a near surface z1 = 0 m elevation and rises to the elevation z. The environmental potential temperature can be written θe (z) = θ(0) + (dθe /dz)δz, and evidently the parcel’s potential temperature remains constant: θp = θ(0). Applying the above conditions for Eq. 3.17 yields:   dθe d g d2 δz = −g (ln θe ) δz. (δz) = − (3.18) dt2 θe (z) dz dz

Introducing the squared moist-air Brunt–V¨ais¨ al¨ a frequency d ln θe (3.19) N2 = g dz (see Vol. 2, Ch. 22, Sec. 6.4 for more details), Eq. 3.18 then takes the form of an oscillation equation for the parcel’s vertical displacement as a function of time: d2 (δz) = −N 2 δz, (3.20) dt2 which has the frequency of oscillation N . The solution of the above equation is: δz = AeiN t ,

(3.21)

where A is the amplitude of the oscillation. N 2 > 0 is the condition for stable harmonic oscillation. Lee wave clouds or linearly organized fair-weather cumuli are typical indicators of stable atmospheric stratification, when the equivalent potential temperature increases with height. In contrast, when N 2 < 0 the stratification is unstable, N is imaginary, and δz increases exponentially. In dry thermals, we might expect from the unstable condition

20

Part I: Basic parameterization concepts and issues

that dry potential temperature decreases with height (dθ/dz < 0), which is rare in the free atmosphere although dry convection is very common. This apparent contradiction is solved by the mechanism of thermals described earlier: the near-surface air heated by the surface breaks through the stable layers in concentrated vertical air flows. In the condensing phase, the vertical gradient of the wet potential temperature is the relevant indicator of instability and this can indeed be negative, especially in the summer.

3.3

Third phase: towering cumulus (cumulus congestus)

In the towering cumulus phase, the temperature of the rising air parcel is warmer than its environmental temperature. To maintain this state, the condensation has to work continuously, providing latent heat release (the “gas burner” has to work at full capacity). Neither the structure of the towering cumulus cloud nor the condensation process is homogeneous. As the condensation-heated bubble accelerates upward, the cooler ambient air can penetrate into the cloud by turbulent mixing and this cools the inner temperature of the bubble. This is the process of entrainment. Such cooling decreases the buoyancy and the bubble slows down. At the same time, this extra cooling generates extra condensation and so an additional latent heat release that tends to warm up the parcel again. Typical forms of tufts and turrets indicate these pulsating processes very clearly. In a rotating thunderstorm (supercell) there are fewer turrets because the centrifugal force works against the entrainment process, and the edges of these clouds are much sharper. The forms of a towering cumulus also depend on the wind shear. In the absence of wind shear, the cumulus is more symmetrical, whereas in a sheared environment, convective cloud could show rather deformed shapes (Fig. 1.4). When wind shear is present, the main updraught channel tilts considerably. The cloud base of the towering cumulus is rather rough compared to fair-weather cumulus clouds because of the strong turbulence. In wind shear cases, gliders often report 20 to 30 ms−1 updraught velocities in the main updraught channel of cumulus congestus clouds. During intense condensation, rain drops appear and the cloud becomes visible to weather radar. If the growing cloud has insufficient humidity to provide condensation or if the surrounding air’s cooling rate is not large enough to maintain buoyancy, then the cumulus congestus does not develop further. However,

Moist atmospheric convection: An introduction and overview

21

Fig. 1.4 Wind shear can deform the towering cumulus. The inflow channel becomes narrow and asymmetric.

if the conditions are favourable and the first lightning appears, the cumulus enters the cumulonimbus phase. 3.4

Fourth phase: cumulonimbus

Considering the thermodynamic diagram, the rising parcel is supported by buoyancy as long as its temperature is warmer than the ambient temperature. The highest point where buoyancy supports the updraught is called the “equilibrium level” (EL), as shown in Fig. 1.2. Between the level of free convection and the equilibrium level, buoyancy accelerates the rising parcel and the parcel gains kinetic energy. This kinetic energy is proportional to the area enclosed by the rising parcel trajectory and the temperature profile of the surrounding air between the LFC and EL. The area is referred to as the “positive area” (PA) when the parcel is warmer than its environment. The parcel reaching the EL does not immediately stop but penetrates into the upper layers and rises until it loses all of its kinetic energy. In the idealized case, the work of the negative buoyancy above the EL would be equal to the work represented by the PA. The level where the parcel uses up all of its energy and finally stops is

22

Part I: Basic parameterization concepts and issues

named the “level of the maximum cloud top” (LMCT). Figure 1.2 shows that the temperature of the cloud top could be cooler than the coldest point of the air column. This is the explanation for the extreme cold cloud tops of thunderstorms appearing on infrared satellite images. The glorious sights of thunderstorms are the “anvils” on the top of the cumulonimbus (Fig. 1.5). Here, it has been supposed that the entrainment effects can be neglected for the moment (the wall of the balloon is insulating and prohibits mass exchange with the environment).

Fig. 1.5

A single-cell cumulonimbus capillatus.

The cloud classification distinguishes two species of cumulonimbus clouds: the cumulonimbus capillatus which has an anvil, and the cumulonimbus calvus which has no anvil. The high-level anvil (or incus) clouds contain only ice crystals and indicate that the cell has reached its mature phase. There are several supplementary features of a cumulonimbus that allow subspecies to be defined. The most frequent features are cumulonimbus mammatus (Fig. 1.6), cumulonimbus pileus (Fig. 1.7), or the cumulonimbus arcus (Fig. 1.8) that is induced by the outflowing cold air (gust front) from the cumulonimbus.

Moist atmospheric convection: An introduction and overview

Fig. 1.6

23

Cumulonimbus mammatus.

The rising parcel between the level of free convection and the level of equilibrium is accelerated by buoyancy. To calculate the work of the buoyancy force, Eq. 2.17 may be rewritten as: Tv p − Tv e dw dz dw 1 dw2 dw = =w = ≈g . (3.22) dt dz dt dz 2 dz Tv e Integration of the above equation from zLF C to zEL yields: z EL  Tv p − Tv e  1 2 dz ≡ CAPE. (3.23) wEL = g 2 Tv e zLF C

The wEL so calculated is the maximum vertical velocity that the rising parcel has when it gets to the equilibrium level. In extreme cases wEL can reach 50 ms−1 . The integral represents the amount of energy within the air column that can be transformed into kinetic energy: the “convective available potential energy” (CAPE). The parcel uses the kinetic energy to pierce into the upper, stable layers, and the work of the negative buoyancy slows down the parcel. In the ideal case, the upper “negative area” in Fig. 1.2 that represents the work of negative buoyancy, is equal with the positive area that represents the CAPE. The LMCT is defined as the level at which these two areas are equal, and the idealized parcel stops.

24

Part I: Basic parameterization concepts and issues

Fig. 1.7

Rapidly developing cumulonimbus with pileus.

The commonly accepted most important limitation of the parcel method (cf. Ch. 12) is the entrainment. The surrounding air, mixing into the rising air parcel, decreases Tv p − Tv e and in this way the buoyancy is reduced. It can be shown that, defining the entrainment rate as λ = d ln m/dz, which is proportional to the amount of inflowing mass into a unit air mass, the kinetic energy change in a layer of thickness is: Tv p − Tv e d w2 − λw2 =g dz 2 Tv e

(3.24)

(Holton, 2004). The above equation shows that the entrainment appears as a negative feedback and decreases the CAPE. The estimation of entrainment is an important challenge of the cumulus parameterization. (See Ch. 10 for more detail.) The third convective component, wind shear, and the first convective component, buoyancy, are together responsible for rotating thunderstorms (supercells). The wind shear implies vorticity with a horizontal axis, and

Moist atmospheric convection: An introduction and overview

Fig. 1.8

25

Gust front with cumulonimbus arcus hits Lake Balaton.

the buoyancy-generated updraught tilts this axis into the vertical, making the rotating structure of the cell. The rotation adds characteristic shapes to the cloud and evolves a special circulation mechanism in the supercell (Fig. 1.9). The cumulonimbus affects the entire troposphere and it works like a cloud factory. From low levels to high levels, from stratiform to cumuliform, all kinds of clouds may appear or remain as a result of the thunderstorm cloud activities (e.g., Fig. 1.10).

4

Conditions for convection

Atmospheric convection to a great extent depends on the larger-scale weather systems (synoptic-scale phenomena) and it is also influenced by local-scale phenomena (local circulations, orography, soil state, etc.). In mid-latitude areas, synoptic-scale phenomena provide convective instability or stability allowing or prohibiting convection. The synoptic-scale impact

26

Part I: Basic parameterization concepts and issues

Fig. 1.9

Supercell above Lake Balaton.

can sometimes be so strong that thunderstorms can develop much faster and stronger in mid-latitudes than in tropical areas. This is also true in reverse: synoptic-scale conditions such as mid-tropospheric inversions may prohibit deep convection despite favourable local conditions, such as intense solar radiation, warm low-level air mass, etc. Generally speaking, in the tropics the large-scale conditions and local conditions both support convection, especially wet convection, whereas in mid-latitude both synoptic- and local-scale effects may have very different roles. Concerning the synoptic scales, the following typical weather patterns favour deep convection: • Weak pressure gradient patterns, where there is no significant convergence or divergence indicated by the pressure field and values of the mean sea level pressure are almost constant over larger areas. In such circumstances, local thunderstorms, or “air-mass” thunderstorms, can develop. Local effects may have an important role in triggering convection. A local lake circulation or even a cropfield–forest border are able to influence the appearance of the first thermal that may determine the place and strength of the first

Moist atmospheric convection: An introduction and overview

Fig. 1.10

27

Cirrus clouds drifted from the top of a distant thunderstorm.

thunderstorms. • Non-frontal convergence patterns, where there is a weak convergence line inside an air mass. This convergence could develop as a result of earlier thunderstorms within a weak pressure gradient pattern, or it could be a result of a local circulation. Near-surface convergence in the air mass encourages local thunderstorms. These thunderstorms collect humidity and make the convergence even stronger, and can also produce positive feedbacks such that afternoon thunderstorms can appear for several days. In weak pressure gradient and convergence patterns the role of high-level cold air, or at least more or less unstable vertical stratification, is quite important. In both cases, buoyancy (the first convective component) has the determining role. • High-level vorticity, and high-level cold pool patterns generally are the remains of a former cyclone (most of the time cut-off cyclones or former Mediterranean cyclones). In these cases only a weak depression can be analysed in the surface pressure field. However, at higher levels (above about the 500 hPa pressure level) the depression and the cold air core with stronger wind can still be found.

28

Part I: Basic parameterization concepts and issues

In this case thunderstorms are driven not only by buoyancy but by the third convective component, the wind shear, too. These thunderstorms still have a daily period but they are stronger than the air mass ones (Fig. 1.11).

Fig. 1.11 EUMETSAT infrared image shows a cut-off cyclone with the centre over the Adriatic Sea. Thunderstorms over the Balkan peninsula and southern Italy are indicated by high-level cloud.

• Cold front patterns are frequent convection-supporting weather phenomena. Cold fronts as long-lived convergence lines are able to concentrate large amounts of warm and wet air to provide enough fuel for the convection in the pre-frontal region. The moving cold front forces the pre-frontal air to rise and forced convection is an important added effect which makes thunderstorms more severe. The first (buoyancy) and second (forced convection) convective compo-

Moist atmospheric convection: An introduction and overview

29

nents play basic roles here. When a wave appears on the cold front, the high-level jet stream can drift above the unstable warm sector. In such a situation, the third convective component, wind shear, can also become a major effect and as a result of all three convective components acting in concert, frontal or pre-frontal squall lines may appear (Fig. 1.12).

5

Convective cloud systems: From cloud streets to squall lines

In most cases, convective cells do not develop independently from one another and they react to their environment as well. As can be seen in thermals, the most vigorous thermal cell creates convergence beneath the inversion and collects near-surface material at the expense of other thermals. When the thermal or the cumulus moves along the main flow, it drags the convergence along. In the low levels the convergence line behind the cumulus remains even after the passage of the cloud, triggering new thermals and cumulus clouds. The result is a line of organized cumulus clouds, or “cloud streets” (Fig. 1.13). Especially in highly unstable weather conditions, thunderstorm cells appear very close to each other and younger cells are able to derive energy from the older ones. This is the case of “multicell convection”. The first cell develops using mainly buoyancy. Upward streams, condensation, and convergence produce a large amount of potential energy which turns into kinetic energy via downdraughts in the decaying phase. The downdraught creates convergence at a gust front, and this provides favourable conditions for the next cell (Fig. 1.14). As a result, the second cell can be stronger than the first cell by inheriting the energy of the first cell, etc. Thunderstorm systems often form a linear structure. Thunderstorm lines can be distinguished by the movement of the cells. When storms move along the line they can produce torrential rain and flash floods especially when several cells run over the same valley or water catchment area. When individual thunderstorms move perpendicular to the line, extreme heavy and fast moving squall lines can appear, sometimes with supercells among the convective cells. A multicell convective system has a longer life span than a simple cell thunderstorm. The low-level convergence resulting from the combined effect of several thunderstorm cells is able to collect air masses from greater

30

Part I: Basic parameterization concepts and issues

Fig. 1.12 Cold front and pre-frontal squall over Hungary on 18 May 2005: (top) a radar image from the Hungarian Meteorological Service; and, (bottom) a numerical model (Weather Research and Forecasting; WRF) simulation of the wind field. The colours of the wind barbs (green, yellow, orange red, blue) represent wind speeds from light to stormy. The label SC shows rotating thunderstorms (supercells). Dashed lines indicate the pre-frontal squall lines.

Moist atmospheric convection: An introduction and overview

31

Fig. 1.13 The leading cumulus cell creates convergence which contributes to the creation of a cumulus line along the mean wind direction.

Fig. 1.14 During multicell thunderstorm development, outflow from the mature cell helps to develop the new cell by near-surface convergence.

32

Part I: Basic parameterization concepts and issues

distances than a single cell. The Coriolis force acts on these large-scale movements to produce the rotation associated with the multicell convective system. The most significant example of this scale interaction occurs in hurricanes, where a tropical cyclone develops from initial thunderstorms.

Chapter 2

Sub-grid parameterization problem

J.-I. Yano Editors’ introduction: This chapter provides both a careful introduction to and a general overview of the parameterization problem. At least for some simple problems, a parameterization can be derived analytically as an expansion in some suitable small parameter. Such problems are instructive in revealing the role of concepts such as scale separation and quasi-equilibrium, which will occupy much of the following chapters. Some possible strategies for the construction of parameterizations in more complex situations are also outlined, drawing on moment expansions for turbulence theory, the use of probability density functions for the small and/or fast scales, and the identification of dominant structures or modes within the small and/or fast scales. All of these ideas have been manifest in the development of convective parameterizations to a greater or lesser extent, and certainly they are reflected in much of the language used to discuss convective parameterization concepts. This chapter also gives a sense of the character of many parameterization studies, which often require imaginative physical ideas and assumptions, or even rather hopeful leaps of faith, but which are often most effective when combined with solid organizing principles arising from solid theoretical structures.

1

Introduction

A global infrared image from a geostationary satellite (Fig. 2.1) nicely summarizes the richness of atmospheric phenomena with the wide range of horizontal scales involved. In particular, it is interesting to focus specifically on the tropical region, where extensive convection activitiy is suggested by 33

34

Part I: Basic parameterization concepts and issues

Fig. 2.1 An example of a global infrared image from a geostationary satellite with the brightness temperature shown by false colours. An image from JMA geostationary meteorological satellite at 9:28 UTC, 2 April 2003.

the cloud distribution in the image. These convective clouds clearly cover a wide spectral range. The smallest scale corresponds to individual cumulus convective towers that are only marginally resolved in this image. Those individual convective elements are organized on the mesoscale, as suggested by reddish clusters in the image. These mesoscale elements are further organized on the planetary scale as suggested by a modulation of the clustering. A well known example of such planetary-scale coherency is the Madden–Julian oscillation (cf., Zhang, 2005). As a whole, the image suggests that this wide spectrum range of convective cloud variability involves interactions across scales (cf., Yano, 1998). Thus, it is likely that even for simulating the planetary-scale Madden–Julian oscillation, a contribution of individual convective towers must, somehow, be taken into account, as commonly assumed (cf., Ch. 5). However, these individual convective towers are far from resolved by a typical global model. Such a situation can be inferred by imagining a longitude–latitude mesh superposed on the image corresponding to a nu-

Sub-grid parameterization problem

35

merical mesh. Thus, the contribution of the convective towers to the global tropical dynamics must somehow be accounted for in an indirect parametric, manner. This is just one of many examples suggesting how parameterization of subgrid-scale physical processes is important for successfully modelling global atmospheric circulations both for forecasts and climate-change projections. This chapter provides an overview of the subgrid-scale parameterization problem from a general perspective. It begins with a formal statement of the subgrid-scale parameterization problem in the next section and from there goes on to broach various aspects of the problem. Section 2 considers a simple linear problem in order to see how parameterization works in a more concrete manner. Various basic concepts associated with the parameterization problem (scale separation, quasiequilibrium) are introduced in this very simple setting. Section 3 continues with the idea of an asymptotic expansion introduced in Sec. 2, and considers two more subgrid-scale problems with increasing complexity. The asymptotic expansion approach is presented in a more formal manner in the next chapter in discussing the concept of the scale-separation principle. In this respect, Sec. 3 and Sec. 4 may be considered a prelude for the next chapter. Section 5 reviews parameterization approaches to the turbulent kineticenergy spectrum. Section 6 lists possible approaches for subgrid-scale parameterization, and Sec. 7 provides, more specifically, an overview of the convection parameterization from a historical perspective. The last three sections (Sec. 8 to Sec. 10) offer some philosophical reflections on the subgrid-scale parameterization problem. Section 2 and Sec. 3 provide necessary base knowledge for subsequent chapters, while the remaining sections are selective stand-alone reading materials. Readers can proceed to the following chapters without them, though they offer useful perspectives on various other discussions. Readers may prefer to return to these sections later as required.

36

2

Part I: Basic parameterization concepts and issues

Formal statement of the problem

Throughout this book, we often work with a generic physical variable designated by ϕ, which is governed by an equation: 1 ∂ ∂ ϕ + ∇ · uϕ + ρwϕ = F. (2.1) ∂t ρ ∂z Here, u is the horizontal wind, w the vertical velocity, and F designates a total source term for ϕ. Note that defining the source term F is a significant task in itself, often requiring a good understanding of the physics behind it. In particular, there are extensive uncertainties associated with cloud microphysics, as will be discussed in Vol. 2, Ch. 18, around defining this term. However, for the most part, we will assume that this term is known. Unfortunately, in the literature, the process of defining this source term itself is often called a parameterization, leading to a confusion of terminology. The problem of defining F is an issue of physical representation that must clearly be distinguished from subgrid-scale parameterization that is going to be introduced in this section. This confusion is usually rectified by the fact that cloud microphysicists often insist that they are also dealing with a parameterization problem in the process of defining source terms as a macroscopic average, in contrast to a more fundamental level of description. A macroscopic description based on a statistical treatment of microphysical processes is often called “phenomenology”, and thus such efforts in cloud microphysics may be better called “phenomenological” descriptions, in order to clearly distinguish the issue from the subgrid-scale parameterization problem. The problem of parameterization for the subgrid-scale physical processes arises because atmospheric models have only a limited horizontal resolution. The easiest way of seeing the issue is to take the interpretation that a single grid-point value within a model represents a grid-box mean value ϕ, where the overbar indicates an average over a grid box. By taking such an average, the generic prognostic equation of Eq. 2.1 reduces to: ∂ ¯ · uϕ¯ + 1 ∂ ρw ϕ¯ + ∇ ¯ ϕ¯ = Q. (2.2) ∂t ρ ∂z The term on the right-hand side, Q, is called the “apparent source” by Yanai et al. (1973), which is defined in terms of the subgrid-scale variables as: 1 ∂ρw′ ϕ′ Q = −∇ · u′ ϕ′ − + F¯ , (2.3) ρ ∂z

Sub-grid parameterization problem

37

where the prime designates a deviation from the grid-box mean (e.g., ϕ′ = ϕ − ϕ). ¯ The goal of subgrid-scale parameterization is to seek a closed expression for the apparent source Q in terms of the known resolved-scale variables (grid-averaged quantities ϕ). ¯ Of course, this is a non-trivial problem, because by design, a numerical model only deals with the grid-averaged quantities. The subgrid-scale fluctuation ϕ′ is totally implicit and thus it must be estimated indirectly. Note that the apparent source term Q consists of two distinctive types of term: the first two terms describe the eddy transport, and the last term arises from the source term for a given physical variable. The reason as to why the grid box-averaged source term F¯ is also considered a part of the parameterization problem may not be immediately clear. However, a source term is often defined by a non-linear function of a dependent variable ϕ along with the other physical variables. Thus, eddy terms similar to the eddy-transport terms arise from an averaging procedure. These must clearly be “parameterized”. Most importantly, the phenomenological description of the microphysical processes given as a term F must furthermore be parameterized as a term F¯ . This very basic point is often not recognized in the literature. The parameterization problem takes a different flavour depending on the dominant term in the apparent source Q. In many turbulent problems, the dominant term is the vertical eddy transport ρw′ ϕ′ . On the other hand, in moist deep convection, diabatic heating F associated with condensation of water vapour becomes a key term to be determined. To some extent, the approach to be taken depends on the nature of the apparent source term to be parameterized. Note that in defining the parameterization problem above, by closely following a classical derivation by Yanai et al. (1973), we have talked as if a grid box literally exists within a numerical model. However, this is strictly true only if a finite volume approach (cf., LeVeque, 2002) is taken. In most of the numerical approaches, the concept of a grid box, in fact, does not exist. A more formal approach for defining the grid-box average in turbulence studies is based on filtering (cf., Leonard, 1974). An alternative, more mathematically heuristic approach is a multiscale analysis under an asymptotic expansion (cf., Majda, 2007). Note that in both approaches, the notion of a grid box becomes implicit, or even unnecessary. For this reason, the concept of a grid box is often referred to instead as the “large scale” in the following discussions.

38

Part I: Basic parameterization concepts and issues

Both approaches also help in understanding the concept of scale separation. For this reason, in introducing the scale-separation principle in the next chapter, we are going to derive the subgrid-scale parameterization based on these two approaches. More precisely, the subgrid-scale “estimation” problem consists of two parts: (1) estimation of the subgrid-scale fluctuation (distribution), ϕ′ , itself; and, (2) a procedure for evaluating the apparent source term, Q. The first part, especially when a spatial distribution of a physical variable is concerned, is often separately dealt with as downscaling (Maraun et al., 2010). The second part is a core of parameterization. However, we clearly require subgrid-scale information for the latter purpose, at least implicitly. In this respect, it is important to realize that downscaling and parameterization constitutes a single, pair problem (cf., Yano, 2010). 2.1

Eddy diffusion: Renormalization approach

In order to be more specific, let us consider a case where the forcing term is simply a diffusion: F = κ△ϕ

(2.4)

with a diffusion coefficient κ. When a model with a finite resolution is considered, the forcing term F is replaced by an apparent source Q, defined by Eq. 2.3, by adding eddy effects to the diffusion effect. Here, we may somehow anticipate that even under a finite resolution with averaging over each grid box, the same form of equation as Eq. 2.1 may still be satisfied simply by replacing the full physical variables ϕ by averaged values ϕ. ¯ Thus, the apparent source term takes the same form as the forcing (dissipation) term but the diffusion coefficient κ is replaced by a coefficient which also takes account of the eddy effects. Thus: Q = κe △ϕ

(2.5)

with κe now an eddy diffusion coefficient. Such a simple eddy parameterization is often employed in turbulence simulations with eddy diffusion coefficients estimated by varying degrees of sophistication. A full description of this approach is beyond the scope of this text, although a short review is given in Sec. 5 in order to put the parameterization problem in atmospheric modelling into perspective. However, what is more important is this general idea, rather than more technical details that are often found in specialized textbooks.

Sub-grid parameterization problem

39

The basic premise is to assume that the physical system still takes an analogous form as the original, even after grid-box averaging by modifying the constants of the system. The procedure for changing the constants of the system for a purpose of parameterization may be called “renormalization”. We will find that an idea of renormalization often works, especially in simple problems. 2.2

Observational diagnosis of apparent source

Although it is by no means a trivial problem to construct a formulation (i.e., subgrid-scale parameterization) for estimating the apparent source Q in more general cases, it is reasonably straightforward to diagnose the apparent source from observation. All that is needed is a carefully designed observational network of rawinsonde stations. Then, the apparent source (e.g., for heat and moisture) can be estimated from the large-scale values observed by using the left-hand side of Eq. 2.2. In other words, the apparent source itself can be estimated without having subgrid-scale information. Such diagnoses for heat (entropy) and moisture are usually designated as Q1 and Q2 , and the analysis is often simply called “Q1 , Q2 analysis”. Many field campaigns have been organized which are designed to provide diagnoses for the apparent source. Probably the two best known examples are GATE (GARP Atlantic Tropical Experiment: Nitta, 1977; Thompson et al. 1979) under GARP (Global Atmospheric Research Program) and TOGA-COARE (Tropical Ocean Global Atmosphere Coupled Ocean Atmosphere Response Experiment: Lin and Johnson, 1996). This observationally diagnosed apparent source is used, in turn, for verifying subgrid-scale parameterizations. See Ch. 8, Sec. 3.1 for more explicit formulae and further discussions. 2.3

Super-parameterization

If available computing power is not an issue, there is a way to get around the parameterization problem. The simplest and most naive solution would be to run a high-resolution model over a grid box so that subgrid-scale information ϕ′ would be directly obtained. Nesting (cf., Clark and Farley, 1984; Zhang et al., 1986) would be a formal approach to achieve this goal for selective grid boxes. Models which can be used for this purpose are known as “cloud-resolving models” (CRMs). The procedure can even be simplified by taking a periodic domain for the subgrid-scale (fine-scale)

40

Part I: Basic parameterization concepts and issues

explicit simulations. The scale-separation principle to be discussed in the next chapter justifies such a simplification. This is the basic idea of super-parameterization (cf., Grabowski and Smolarkiewicz, 1999; Randall et al., 2003). Of course, it is extremely computationally expensive to place a CRM over every grid box that fills up the domain. For this reason, in a standard super-parameterization implementation, a much smaller domain size, often with a two-dimensional configuration, is adopted. Although such a strategy is extremely prohibitive by itself, it provides a good starting point to think about how a subgrid-scale parameterization can be constructed. This perspective is further discussed in Sec. 6.3. 3

A simple example: Basic concepts

A parameterization problem is often considered something unsolvable analytically. In the following, it will be shown that this is not always the case. A parameterization formulation can actually be solved analytically in some relatively simple problems with the help of asymptotic expansion methods. Readers are encouraged to follow the first example closely, because it also introduces some basic concepts that are to be further discussed in subsequent chapters. The following section constitutes a further elucidation of the asymptotic expansion method. As a first example, we consider a simple linear system with two variables, x and y x˙ = −ax + by

(3.1a)

y˙ = dx − cy.

(3.1b)

Here, the dots on top of x and y designates a time derivative, while a, b, c, and d are constants, and we assume them to be positive for now. Note that this system can even be solved perfectly analytically without any approximations. Because of its extreme simplicity, it can elucidate basic concepts behind the subgrid-scale problem. Since we are dealing with a set of ordinary differential equations, the subgrid is best considered in terms of the timescale: something slow is considered that of a large scale, and something fast is considered that of a subgrid scale. Let us assume that x and y are slow and fast processes, respectively. In order to see the difference in the timescales explicitly, we divide (only) the right-hand side of Eq. 3.1b by a small parameter ǫ. Thus,

Sub-grid parameterization problem

41

the set of equations becomes: x˙ = −ax + by (3.2a) 1 (3.2b) y˙ = (dx − cy). ǫ The above shows that the temporal tendency for x is the order unity with respect to ǫ, whereas for y the order is 1/ǫ, or O(1/ǫ), which is much larger than the former. It further shows that the timescale associated with y is shorter than that for x by the factor ǫ. Under this formulation, S = by is an apparent source term that is to be parameterized, and y is a subgrid-scale variable that is not directly available in large-scale modelling. At first glance, we may simply wish to neglect such a fast process as a useless “noise”. In other words, we may neglect the fast variable by simply setting y = 0 in Eq. 3.2a. Then we readily obtain a solution: x = x0 e−at .

(3.3)

This serves as a basic reference for the subsequent considerations. A second approach would be to solve the above set of equations solely in terms of the leading order (in a very naive manner). The leading order for Eq. 3.2b is clearly O(1/ǫ) on the right-hand side, where the left-hand side is only O(1). Thus to the leading order, we may set: dx − cy = 0, which leads to a solution for y in terms of x: d x. c Eq. 3.4 can be substituted into Eq. 3.2a. Then:   d x˙ = −a + b x c or y=

(3.4)

x˙ = −a′ x

(3.5)

with a′ = a − bd/c. Its solution ′

x = x0 e−a t

(3.6)

is the same as the above in Eq. 3.3, but with the constant a modified into a′ as defined by Eq. 3.5 above, which is an example of the renormalization discussed in Sec. 2.1.

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Part I: Basic parameterization concepts and issues

This is a very simple demonstration of the fact that we should not consider a fast process y simply as “noise” as an initial naive thought may suggest, and shows why a parameterization for fast processes is needed. Eq. 3.4 is a very simple parameterization for this fast variable: the fast variable (left-hand side) is nicely expressed in terms of a slow variable (right-hand side) so that the system can be closed. This is more or less all we need for this system as long as we are concerned only with a large-scale process. As a result, the apparent source can be expressed as: x bd x= c τ0 with a characteristic adjustment timescale defined by: S = by ≃

τ0 = c/bd.

(3.7)

(3.8)

An adjustment formulation with a Newtonian relaxation is also adopted in many operational schemes as reviewed in Sec. 7 below. However, simply setting the left-hand side of Eq. 3.2b to zero to the leading order may not quite be correct: in general, there is a fast timescale process that controls the evolution of y. In order to consider such a fast process explicitly, we introduce a fast timescale as τ , and reset the time derivative as: 1 d d d = + . (3.9) dt ǫ dτ dt Here, the time is separated into two variables that describe the fast and the slow processes. The existence of two such separate scales is the basic idea behind the scale-separation principle, which is to be further discussed in the next chapter. Note that although only a timescale separation has been discussed here, the same idea can also be easily extended to a spacescale separation. Substituting Eq. 3.9 into Eqs. 3.2a,b leads to:   d 1 d + x = −ax + by (3.10a) ǫ dτ dt   d 1 d 1 + (3.10b) y = (dx − cy). ǫ dτ dt ǫ To the leading order, O(1/ǫ), these read: d x=0 dτ d y = dx − cy. dτ

(3.11a) (3.11b)

Sub-grid parameterization problem

43

Eq. 3.11a simply says that the variable x is stationary in the fast timescale. On the other hand, Eq. 3.11b can be re-written as:   d + c y = dx dτ or d cτ ye = dxecτ , dτ which can be readily solved to give:   d d y = y0 − x0 e−cτ + x, (3.12) c c assuming that x is almost constant, but nevertheless taking into account its time-dependence in writing down the final result. Here, the subscript 0 denotes the initial conditions. Note that the solution rapidly approaches Eq. 3.4 with increasing τ , regardless of the initial conditions x0 and y0 . For this reason, the solution to Eq. 3.4 may be considered an equilibrium solution of the system, and τ1 ≡ 1/c

(3.13)

is interpreted as a characteristic timescale that the system approaches to this equilibrium. Such an equilibrium may, more precisely, be called “quasiequilibrium”, because the variable y is not perfectly in equilibrium, but slowly changes with time by following the evolution of the slow variable x. Note also that the transient process towards quasi-equilibrium can be interpreted as a type of adjustment process. This is an important conclusion, because it suggests that we may not have to explicitly consider a fast process in order to develop a parameterization; a consideration only of a (quasi-) equilibrium state may be enough. In other words, we may assume a subgrid-scale process to be at a quasi-steady state. This hypothesis will be referred to as the “steady-plume hypothesis” in the context of the mass-flux formulation later, when convection is idealized as an ensemble of plumes. In the literature, the importance of considering a life cycle of convection is often argued for, or more precisely processes such as “triggering”, in order to make a parameterization more complete. However, it should be noted that such an argument is fundamentally inconsistent with the concept of quasi-equilibrium. In order to consider these transiencies, it is necessary to develop a fundamentally different type of parameterization (i.e., non-equilibrium parameterization). More importantly, the analysis

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Part I: Basic parameterization concepts and issues

just presented suggests that such fast processes may not be as important as they first appear for the purpose of parameterization: the scale-separation principle may allow us not to consider these transient behaviours explicitly for a parameterization. Here, we should carefully distinguish the two adjustment timescales introduced so far: the timescale τ0 in Eq. 3.8, through which the subgridscale process drives the system slowly under an equilibrium (Eq. 3.4) and the timescale τ1 in Eq. 3.13, through which the subgrid-scale process adjusts towards this equilibrium. Unfortunately, the literature is rather unclear on which timescale is introduced under a given adjustment scheme, although most adjustment schemes takes the form of Eq. 3.7 rather than Eq. 3.12. As far as the conventional wisdom discussed so far is concerned, τ0 is the only useful timescale to introduce. However, the situation changes if the parameter c turns out to be negative. In this case, unless the initial state is under a perfect equilibrium with y0 − dx0 /c = 0, then the system exponentially deviates away from the equilibrium state with time. Such a system is fundamentally unstable, and both the slow and the fast variables evolve rapidly with time. Note especially that at a certain point, the approximation of Eq. 3.11a ceases to be valid as y grows exponentially with time. Then, in turn, x also begins to evolve rapidly as well. Under such a situation, making an effort for a parameterization is meaningless. In a more realistic situation, the system would not grow exponentially without limit, but a certain non-linearity would catch up, and maintain a system to a bounded state. In this respect, the notion of self-organized criticality may be worth mentioning here: it refers to a system in which an equilibrium state is fundamentally unstable in the linear sense (i.e., critical). Nevertheless, the fully non-linear behaviour of the system is such that it always remains close to the unstable equilibrium state. We further discuss this concept in Vol. 2, Ch. 27. On the other hand, a system that self-regulates itself towards an equilibrium state in a manner presented by the solution in Eq. 3.12 may be called “homeostasis”, borrowing a term from physiology (cf., Ch. 4, Sec. 3.5).

4

Further examples using asymptotic expansions

The asymptotic expansion method just introduced is a powerful tool for analysing a system under a separation into two different scales. We now

Sub-grid parameterization problem

45

introduce a spatial dependence and consider two more examples. The first example is a one-dimensional heat equation system, while the second is a two-dimensional fluid flow.

4.1

Heat equation problem

This example is taken from Sec. 9.6.2 of Fritsch (1995). Here, we consider a one-dimensional heat diffusion problem by introducing a spatial coordinate x. A heat equation for a temperature θ is given by: ∂ ∂θ ∂θ = κ . ∂t ∂x ∂x

(4.1)

In this system, we assume that the heat diffusivity κ varies with a 2πperiodicity with a non-vanishing mean value < κ > = 0. The goal here is to find a uniform effective diffusivity κe that is valid for scales much longer than 2π, i.e., for a large scale. Symbolically, if we designate the scale of our interest by L, we expect that a condition L ≫ 2π is to be satisfied. This is considered a spatial version of the scale separation. Note that, different from the example of Eq. 3.1 in the last section, Eq. 4.1 is scaled for small-scale processes. In order to focus on the largescale processes, we introduce the large-scale variables by X = ǫx and T = ǫ2 t, and then the derivatives are given by: ∂ ∂ ∂ = +ǫ ∂x ∂x ∂X ∂ ∂ ∂ = + ǫ2 . ∂t ∂t ∂T

(4.2a) (4.2b)

Here, ǫ is a small parameter for an asymptotic expansion as before. Note that the space and time coordinates are stretched with different rates as expected for diffusion processes. We try to perform an asymptotic expansion more formally than in the last section also by expanding θ in ǫ: θ = θ(0) + ǫθ(1) + ǫ2 θ(2) + · · ·

(4.2c)

By substituting Eqs. 4.2a, b, c into Eq. 4.1, and seeking a balance at each

46

Part I: Basic parameterization concepts and issues

order in expansion by ǫ, as the first three terms, we obtain: ∂ ∂ (0) ∂ (0) θ = κ θ (4.3a) ∂t ∂x ∂x ∂ (0) ∂ ∂ ∂ (1) ∂ ∂ ∂ κ θ + κ θ + κ θ(0) (4.3b) O(ǫ) : θ(1) = ∂t ∂x ∂x ∂x ∂X ∂X ∂x ∂ (0) ∂ ∂ (2) ∂ ∂ (1) ∂ θ = κ θ + κ θ O(ǫ2 ) : θ(2) + ∂t ∂T ∂x ∂x ∂x ∂X ∂ ∂ (0) ∂ ∂ κ θ(1) + κ θ . + (4.3c) ∂X ∂x ∂X ∂X We solve the above series of equations at each order of ǫ. The solution for Eq. 4.3a to O(1) is considered in the same manner as in Sec. 3: this equation, which may be solved analytically, describes a transient behaviour (an adjustment process) of the system in the short timescale. We are only interested in the quasi-equilibrium state, as discussed in Sec. 3, which would be given by a spatially homogeneous state. Thus: O(1) :

∂ (0) θ = 0. ∂x Note that this is equivalent to the statement: ∂ θ = O(ǫ). ∂x Under the asymptotic expansion of Eq. 4.2c, the equation for θ(0) considers only the terms of the order unity. To O(ǫ), we may also set: ∂θ(1) =0 ∂t by realizing the fact that this term is also describing only a transient behaviour towards quasi-equilibrium. Thus: ∂ θ = O(ǫ2 ). ∂t The remaining terms can be integrated in x to give:   ∂ (0) ∂ (1) = Θ(X, T ). θ + θ κ ∂x ∂X

(4.4)

Here, Θ(X, T ) is a function depending only on the large-scale variables. We also designate an average over a 2π period by a bracket, then < ∂θ(1) /∂x >= 0, and as a result, the above equation reduces to: ∂ (0) 1 θ =< > Θ(X, T ). ∂X κ

(4.5)

Sub-grid parameterization problem

47

For O(ǫ2 ), by noting < ∂θ(2) /∂t >= 0, < (∂/∂x)κ(∂/∂x)θ(2) >= 0, the average for Eq. 4.3c gives: ∂ (0) ∂ ∂ (0) ∂ ∂ θ =

θ + < κ θ(1) > . ∂T ∂X ∂X ∂X ∂x

(4.6)

Here, note from Eqs. 4.4 and 4.5, we find: ∂ ∂ (0) ∂ ∂ ∂Θ

θ =− < κ θ(1) > + . ∂X ∂X ∂X ∂x ∂X Its substitution back to Eq. 4.6 gives: ∂ (0) ∂Θ θ = . ∂T ∂X By removing Θ with the help of Eq. 4.5, we obtain a final answer: ∂ (0) ∂ ∂ (0) θ = κe θ ∂T ∂X ∂X with the effective diffusivity given by: κe =
. κ

Here, again we find another example that a parameterization problem can be solved analytically in a closed form. Note that this problem can in all likelihood be generalized to a case that allows the diffusivity to fluctuate at small scales, provided that the small-scale fluctuations vanish after largescale averaging. Also note that this is another example of a renormalization procedure initially discussed in Sec. 2.1. Here, more emphatically, the molecular heat diffusion κ is systematically replaced by an eddy diffusion given by κe . In the case presented here, this formula is obtained by effectively removing the fast- and the small-scale processes by means of asymptotic expansion. 4.2

Two-dimensional fluid flow

The last example considered here is a fluid flow in a two-dimensional space with the second spatial coordinate designated by y. The example is taken from Sivashinsky (1985). Here, we assume a simple eddy flow defined by: u′ = sin y.

(4.7)

Note that the system is non-dimensionalized by taking the scale and the amplitude of the eddy flow.

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Part I: Basic parameterization concepts and issues

The goal is to define an equation for a large-scale flow under the prescribed eddy flow of Eq. 4.7 in a closed form. The same multiscale asymptotic expansion can be employed in order to solve it. However, the procedure is much more involved than the two previous cases, and those technicalities are not worth reproducing here. The final result for the large-scale velocity v¯ in the y-direction is:   3R ∂ 4 v¯ ∂¯ v ∂¯ v ∂ p¯ ∂ ∂¯ v ∂¯ v 3 2 +u ¯ + v¯ + = (R v¯ − R) − − λ¯ v. ∂t ∂x ∂y ∂y ∂x ∂x 2 ∂x4 Here, u¯ is the large-scale velocity in x-direction, p¯ the large-scale pressure, R is a Reynolds number, and λ is a Rayleigh friction. A key term to note is the first on the right-hand side, which represents an eddy diffusion term with the eddy diffusion coefficient νe defined by: νe = R3 v¯2 − R. It is important to note that the eddy diffusion is not always positive, but when the large-scale flow is weak enough (i.e., R2 v¯2 < 1), we obtain a negative diffusion. The important general message here is that an eddy diffusivity can sometimes be defined as a function of the background flow in a closed form in this manner.

5

A review of turbulence parameterizations

Turbulence is such a complex phenomena that it is still difficult to simulate fully numerically, with strong limitations created by the parameterization of smaller scales. For this reason, an overview of this research domain also provides a good perspective on parameterizations in atmospheric modelling. It may also be useful for some readers to become familiar with the terminology of turbulence studies, because some familiarity with the vocabulary is very helpful for interdisciplinary communication. Among the many possible topics, we take as an example a series of efforts for computing the kinetic energy spectrum of fully developed turbulence in a parameterized manner. The main goal is to recover Kolmogorov’s similarity-theory spectrum, but this type of model may produce more. This section heavily relies on Ch. VII of Lesieur (1987). An earlier extensive review on this subject is found in Tatsumi (1980). This is not an attempt at a full exposure of a particular domain of turbulence research, but rather a sketch of the development of a particular type of parameterization in turbulence research. For this reason, only key statements of the mathematical development are presented without derivation.

Sub-grid parameterization problem

49

By doing so, this will hopefully give a good perspective of what turbulence research has achieved. The goal is to compute the kinetic energy spectrum of turbulent flows. As a first step, we write down the Navier–Stokes equation in Fourier space:    ∂ 2 + νk u(k, t) = −i [P · u(p, t)][k · u(q, t)]dp. (5.1) ∂t p+q=k

Here, ν is the viscosity, u the Fourier-transformed velocity, k, p, and q are wavenumber vectors, k the scalar wavenumber, and P is a matrix with components ki kj Pij = δij − 2 . k The matrix operator P has the simple physical interpretation that it extracts the part of the flow that is divergence free in the Fourier space. 5.1

The moment expansion

In the following, there is an attempt to compute a statistically most-likely energy spectrum of a given system. In seeking a statistical answer, an ensemble average is designated by a bracket. An ensemble may be generated by running experiments either in a laboratory or with a numerical model with the same set-up many times. In the following, such an operation is performed only conceptually. The bracket also indicates an average over the wavenumber-vector angle, because the angular dependence will rapidly be dropped in the following, and only the absolute wavenumber k is considered (see Eq. 5.2). A formal procedure for solving this statistical problem is to develop a series of equations for the moments given by: < u(k)u(p) >, < u(k)u(p)u(q) >, < u(k)u(p)u(q)u(r) >, ··· Note that the moments have a tensor form, with an n-th moment forming an n-th order tensor. By setting k = p in the second moment equation, we obtain an equation for the kinetic energy spectrum, K(k, t):   ∂ + 2νk 2 K(k, t) = T (k, t). (5.2) ∂t

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Part I: Basic parameterization concepts and issues

Here, an average over a wavenumber angle has already been performed on the spectrum. The right-hand side, T (k, t), represents an energy transfer rate. In its complete form, T (k, t) contains terms containing fourth-order moments, and thus the above equation is not closed, but we must add the fourthorder moment equations. In this manner, we obtain a series of equations in moment expansion. At each order, we always obtain a term from a higher order, and thus the next equation is also needed. In order to make the problem tractable, it is necessary to truncate the expansion at a certain order. For this purpose, we need to close the system by introducing a certain closed expression for the required higher order moments. Such a condition imposed on the system is called “closure”. This is a common feature of parameterization problems: a problem is often not closed in a formal manner under a given formulation, but a certain closure condition must be introduced. The need for such a condition is relatively clear when a problem is solved under a certain expansion procedure. However, the situation is more general. For example, the mass-flux approach as used as a standard formulation for convection parameterization does not involve any expansion procedure. However, a closure problem persists. 5.2

Quasi-normal approximation

For now, we focus on the problem closed at the second order as given as by the kinetic energy equation of Eq. 5.2. In order to close the problem, it is necessary to write down the fourth-order moments in terms of the second-order moments. A typical approximation proposed as a closure is: < u(k)u(p)u(q)u(r) >= < u(k)u(p) >< u(q)u(r) > + < u(k)u(q) >< u(p)u(r) > + < u(k)u(r) >< u(p)u(q) > .

(5.3)

Here, the sum is designed to be consistent with a symmetry of the fourthorder moments. This condition would be satisfied if the distribution of the velocities were Gaussian (normal). For this reason, this approximation is called the quasi-normal (QN) approximation. This is a typical trick in developing a parameterization: one way or another, it is often convenient (or necessary) to assume a certain distribution

Sub-grid parameterization problem

51

of a subgrid-scale variable. These variables are, unfortunately, often not Gaussian (as is the case for many turbulent flows). Nevertheless, assuming a Gaussian distribution is often a good starting point for solving a problem. By substituting the QN approximation of Eq. 5.3 into the energy equation, Eq. 5.2, along with obvious simplifications by invoking homogeneity (invariance under arbitrary translation) and isotropy (invariance under rotation and reflection), we obtain a closed expression for the energy transfer term, which may be expressed in the form:  t  dpθkpq (t − τ )Tˆ(k, p, q, τ ), (5.4) dτ T (k, t) = 0

p+q=k

where θkpq (t) = exp[−ν(k 2 + p2 + q 2 )t].

(5.5)

This system under QN was first solved by Ogura (1963). However, as it turns out, this system leads to a negative energy spectrum. 5.3

Eddy-damping quasi-normal approximation

Orszag (1970) traced a principal problem with the QN approach to a tendency of the third-order moment defined by:

∂ + ν(k 2 + p2 + q 2 ) < u(k)u(p)u(q) >= Kkpq (5.6) ∂t with an unspecified forcing term on the right-hand side. The third-order moments tend to build up too much. An obvious procedure to apply where there is a tendency for overdevelopment of a quantity is to add an additional damping term μkpq to the system. Thus, Orszag proposes to modify Eq. 5.6 to:

∂ + ν(k 2 + p2 + q 2 + μkpq ) < u(k)u(p)u(q) >= Kkpq . ∂t For consistency, a similar eddy-damping term must also be added to the energy equation, Eq. 5.4, because the term ν(k 2 + p2 + q 2 ) in Eqs. 5.4 and 5.6 has the same origin. This is achieved by modifying the definition of θkpq given by Eq. 5.5 to: θkpq (t) = exp[−ν(k 2 + p2 + q 2 + μkpq )t].

(5.7)

This approximation is called the “Eddy-Damping Quasi-Normal” (EDQN) approximation.

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There are two lessons to learn here. First, the eddy damping is clearly an artificial device. However, in parameterizations, introducing such an artificial device is a common practice, although often without strong justification. (Introducing something more legitimate is often a challenge.) Second, it is important to take a great deal of care regarding self-consistency in parameterizations. Although the introduction of eddy damping itself may be artificial, a self-consistency argument introducing it to both skewness and energy-spectrum equations is key to the success of EDQN. 5.4

Markovization

Note that Eq. 5.4 takes the form of a convolution, keeping a memory of the previous state in the energy transfer rate Tˆ (k, p, q, τ ). A drastic approximation is possible, assuming that such a memory effect can be neglected, and we may set Tˆ(k, p, q, τ ) ≃ Tˆ(k, p, q, t) in Eq. 5.4. As a result, an integral over τ can be performed independent of the transfer rate Tˆ(k, p, q, τ ), and we obtain:  t  ˆ dpT (k, p, q, t) θkpq (t − τ )dτ. (5.8) T (k, t) = p+q=k

0

This approximation simplifies the computation procedure a great deal, and is called “Markovization”. When Markovization is introduced into a pure QN system, it is called the “Quasi-Normal Markovian” (QNM) approximation. When introduced into a system with an eddy-damping feature, it is called the “Eddy-Damping Quasi-Normal Markovian” (EDQNM) approximation. From the point of view of the discussions so far, Markovization may be considered another example of approaches neglecting the transient nature of small-scale processes. This is a useful approach, although whether it is a good approximation for fully developed turbulence is another question. 5.5

Stochastic approach

Adding stochasticity is another typical approach for solving a parameterization problem. However, we should not be confused between adding stochasticity to a problem and adding a random number numerically to a system. The example here demonstrates how stochasticity can be introduced to simplify a system. Importantly, the final answer for the formulation is not necessarily stochastic.

Sub-grid parameterization problem

53

The idea, originally introduced by Kraichnan (1961), is to replace the Navier–Stokes equation, Eq. 5.1, by a modified equation having basically the same structural properties as the Navier–Stokes equation, but somehow easier to solve than the original system. Specifically, a random noise is added to the Navier–Stokes equation of Eq. 5.1 so that it reads:    1  ∂ + νk 2 uα (k, t) = − 2 [P · uβ (p, t)][k · uγ (q, t)]Φαβγ dp. i ∂t N p+q=k β,γ

(5.9) Here, Φαβγ is a random variable under a Gaussian distribution. Indices α, β, and γ are introduced for indicating different realizations, and are not to be confused with the indices for vector components. For each index, N realizations are considered. Thus, P · uβ (p, t) and k · uγ (q, t) are obtained from realizations β and γ respectively as a vector and a scalar. These pieces of information are used for updating the state for a realization α by Eq. 5.9. An interesting result is obtained when this noise has no time correlation, but is “white”: < Φαβγ (t)Φαβγ (t′ ) >∝ δ(t − t′ ). It can be shown that the system simply reduces to EDQNM under this approximation. This is insightful, because an equivalent result is obtained by two qualitatively different paths: a closure assumption and a stochasticity. Note that the final formulation is deterministic, although stochasticity is introduced in deriving it. It is important to remember that the path of reaching a solution in parameterization is not unique. Even a different approach sometimes leads to the same conclusion. 5.6

Variations and lessons

Many more approaches can be developed as variations of the methods presented so far. For example, the time integral  t θkpq (t − τ )dτ 0

in Eq. 5.8 may further be replaced by a constant (Markovian Random Coupling Model, or MRCM). A Random Coupling Model (RCM) can also be obtained by assuming Φαβγ to be independent of wavenumber and time, though it remains random. Interactions between the modes may be limited to a particular category by taking a drastic truncation under an expansion

54

Part I: Basic parameterization concepts and issues

procedure. This leads to the idea of a Direct Interaction Approximation (DIA). Furthermore, a Lagrangian history may be introduced to the above (LHDIA). The list would continue, and a family tree of approaches may be drawn (cf., Fig. VII-1 of Leiseur, 1987. A more cynical cartoon may be found in Fig. 9.1 of Fritsch, 1995). The purpose here is not at all to reconstruct this “family tree”. Rather, this demonstrates a very general tendency for endless prolificity of the parameterization problem. The problem is not closed, and there is no obvious solution to achieve a closure. We often need to attempt various possibilities in order to obtain better results, and unfortunately without always having clear principles to follow. Parameterization in atmospheric modelling is no exception, including convection parameterization. This set aims at lucidity, and at logical consistency, by trying to learn lessons from the history of turbulence parameterization presented here as a short overview. In this respect, there appears to be a lot to learn from Orszag’s (1970) work on EDQN, in order to avoid making endless, fruitless effort in trying to improve a parameterization. Orszag proposed EDQN only after a careful analysis of the behaviour of the original QN model. This lesson tells us that a careful analysis of the behaviour of a given parameterization is the best way to improve it. It is important to remember that Orszag did not look at anything in terms of real turbulence measurements in order to solve this problem. This is an important lesson to remember, because this goes against the current trend for analysing more details of the processes associated with convection from full numerical simulations, hoping that such effort will ultimately contribute to improving convection parameterization. In order to improve a parameterization, one has to understand the behaviour of this parameterization well enough first, before attempting to add anything else.

6

Approaches for subgrid-scale parameterizations

What are the general approaches for subgrid-scale parameterizations? We may consider the three major possibilities: (1) moment-based approaches; (2) approaches based on distributions of subgrid-scale variables; and, (3) the mode decomposition. The moments can be considered a type of expansion series of a physical variable with the first three corresponding to the mean, variance, and

Sub-grid parameterization problem

55

skewness. The expansion by moments provide a systematic framework for parameterizations. Alternatively, the problem of parameterization can be reduced to that of defining a distribution of values of physical variables at the subgrid-scale. This distribution is often called the probability distribution, although this terminology is somewhat misleading. The third possibility, the mode decomposition, was originally expounded by Yano et al. (2005b). It contained the mass-flux convection parameterization as a special case. The next three subsections describe these three possibilities. 6.1

Moment-based approach

A description based on a moment expansion is a standard approach in turbulence research. For this reason, the subgrid-scale boundary-layer turbulence is also often formulated in this manner (cf., Stull, 1988; Garratt, 1992). Here, the n-th moment is defined by < ϕn > for any physical variable, ϕ for n = 1, 2, · · · , although in practice a deviation from the mean is often considered by using < ϕ′n > instead. Here, < > designates an average over a given grid box for the present purpose, equivalent to the overbar introduced in Sec. 2.2. Note that the averaging operator can be interpreted in various different ways. Section 2 discussed the interpretation of spatial averaging, and the issue will be further discussed in the next chapter. In the context of moment-based approaches, a major alternative is to interpret the brackets in terms of an ensemble average. Here, as in statistical mechanics, we imagine many realizations of turbulent-scale realizations (or of small scales in general) under a given macroscopic state (or largescale state). Such a set of realizations is called an “ensemble”. A subtle difference between the spatial and ensemble averages must be kept in mind, because they may lead to different conclusions. A prognostic equation for the n-th moment is in general form given by:   ∂ 1 ∂ + < u > ·∇ + w ¯ < ϕ′n > n ∂t ∂z   ∂ ′ ′n−1 ′ ′n−1 > ·∇+ < w ϕ > + ∇· < u′ ϕ′n > + n ρ ∂z

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Part I: Basic parameterization concepts and issues

1 ∂ ρ < w′ ϕ′ > =< ϕ′n−1 F ′ > . − < ϕ′n−1 > ∇· < u′ ϕ′ > + ρ ∂z

(6.1)

It transpires that this set of equations constitutes an expansion series. The higher-order moments under consideration include the correlation terms with the velocity < u′ ϕ′n >, < w′ ϕ′n >. The prognostic equations for these quantities are not explicitly stated here for the sake of economy of presentation. An obvious problem is that this series of equations is not closed, but higher-order moments are always required in order to compute the temporal tendency of a moment at a given order. In order to truncate this infinite expansion series at a finite order, a certain set of assumptions must be introduced, which is the closure. This approach has been extensively developed in turbulence research, which is also applied to many boundary-layer problems. However, the focus has usually been on conservative processes for the dry atmosphere. Much less is known when the system involves the condensation processes of water as well as the other cloud microphysics (cf., Mironov, 2009). The similarity theory discussed in Vol. 2, Ch. 24 is considered a variant of moment-based approaches. 6.2

Subgrid-scale distribution

An alternative perspective for approaching the problem of describing subgrid-scale physical processes is to consider the distribution of values for a given set of variables over a given grid box. Once this distribution is defined, the apparent source term (Eq. 2.3) can be evaluated readily. This is the basic idea of the distribution-based approach. Before we discuss the actual approach, a short remark on semantics is warranted, because this distribution is often misleadingly called the “probability density function” (PDF). However, there is no necessity for a probability notion behind this distribution-based description: here, no uncertainty (in a Bayesian sense: cf., Gregory, 2005; Jaynes, 2003) nor stochasticity (i.e., physical randomness: cf., Gardiner, 1985; Horsthemke and Lefver, 1984; Paul and Baschnagel, 1999) is considered. In other words, the description is completely deterministic. For this reason, we propose to call this approach the “subgrid-scale distribution”, and the function to be determined as the “distribution density function” (DDF) rather than the PDF, although the acronym PDF nevertheless appears in the following due to certain established terminologies.

Sub-grid parameterization problem

6.2.1

57

A formal approach for DDF: Liouville equation

Many studies exist which have adopted a subgrid-scale distribution as a basis for constructing a parameterization, especially for total water and related quasi-conservative quantities for cloud parameterizations (Bechtold et al., 1992, 1995; Bony and Emanuel, 2001; Bougeault, 1981; Golaz et al., 2002; Mellor, 1977; Richard and Royer, 1993; Sommeria and Deardorff, 1977; Tompkins, 2002; Treut and Li, 1991, cf., Vol. 2, Ch. 25). However, unfortunately, a clear exposition of a basic principle is missing in the literature. In order to present it, we re-write the basic physical equation 2.1 as:   1 ∂ ∂ ϕ = S ≡ − ∇ · uϕ + ρwϕ + F. (6.2) ∂t ρ ∂z A general formulation exists for describing the time evolution of a distribution p(ϕ) of this variable. When the total source term S does not involve any spatial derivatives (the advection effects may be neglected), its evolution is defined by the well-known Liouville equation (cf., Landau and Lifshitz, 1980): ∂ ∂ p(ϕ) = − [p(ϕ)S]. (6.3) ∂t ∂ϕ Note that this equation has a simple physical interpretation that in the phase space of ϕ the shape of the distribution is conserved by following its movement given by S. In other words, the equation is understood in analogy with the mass conservation law: here, the distribution is conserved in place of the mass. The Liouville equation is relatively straightforward to solve when the source term on the right-hand side is purely given in terms of a single point as is often the case with many microphysical processes (with the major exception of sedimentation). It appears that this fact is not widely appreciated. The effort required for computation of Eq. 6.3 is equivalent to bin calculations of the same microphysical process (cf., Caro et al., 2002; Khain et al., 2000, 2004) when only a single-variable distribution is considered. A bin approach for microphysics (cf., Vol. 2, Ch. 23) may even be considered a special case of the Liouville equation: a distribution of a variable in general is the key element in the former, whereas a size distribution of a particular hyrometeor type is more specifically of interest in the latter. Although the computation of joint distributions can be performed in the same manner, the computational cost increases rapidly as more variables

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are considered. For this reason, a compromise would be to focus on singlevariable distributions. However, once the total source S is allowed to contain a spatial derivative (e.g., advection), the formulation suddenly becomes intractable. An exact formulation for evaluating the time evolution of the distribution is still available, as will be fully discussed in Vol. 2, Ch. 25. However, performing this computation becomes impractically demanding, because a full joint distribution for the spatial points must somehow be evaluated. Thus, in summary, the Liouville equation of Eq. 6.3 is a powerful principle for evaluating the time evolution of the subgrid-scale distribution when the physical processes are described solely in terms of a single physical point. However, once a spatial dependence plays a role, for example by advection (transport), the computation of the distribution suddenly becomes intractable. 6.2.2

Alternative approach: Assumed PDF (DDF)

An alternative approach can be identified by noticing a close link between the distribution function p(ϕ) and the moments < ϕn >. First, all the moments can readily be evaluated once a distribution is known:  < ϕn >= p(ϕ)ϕn dϕ (6.4)

for n = 1, 2, · · · , where the integral is performed over the full range expected for a given physical variable ϕ. Conversely, we also intuitively expect that once all the moments are known, the distribution can be perfectly defined, at least in principle. However, a formal procedure for defining a distribution from given a series of moments (by analogy with the Taylor expansion) is not known. The so-called assumed-PDF (or assumed-DDF) approach, originally proposed by Golaz et al. (2002), takes advantage of this link. The basic idea is to introduce a certain generic distribution function with a finite number of free parameters. Then, a set of moments defined by Eq. 6.4 can be used for defining the free parameters for this distribution function, and henceforth the distribution function itself. Time evolution of the given set of moments may be evaluated using moment-based approaches, as discussed in Sec. 6.1. The estimated tendency for the moments, in turn, defines the tendency for the distribution. This idea has been applied by more precisely assuming a double Gaussian distribution for temperature, moisture, and the liquid water.

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59

Since the moment-based approach is best developed for describing eddy transport, the assumed-PDF approach also works best when evolution of a distribution due to eddy transport is the point of concern. However, when the process is non-conservative with the involvement of cloud microphysical processes in particular, as already remarked on in Sec. 6.1, the momentbased approach works less well, and the assumed-PDF approach also suffers from the same problem. Some previously proposed subgrid-scale distribution schemes may be reinterpreted from the point of view of the assumed-PDF approach. For example, the cloud-fraction scheme by Tompkins (2002) can essentially be reinterpreted as a type of assumed-PDF approach by taking the first three moments for the total water for determining an assumed distribution form. However, these three moments are treated in a less consistent manner in his approach. 6.2.3

Time-splitting method

As the last two sub-subsections show, there are two principal methods for evaluating the subgrid-scale distribution: the Liouville principle and the assumed-PDF. The Liouville principle works well when all the processes are locally described, whereas the assumed-PDF approach works well when only the transport processes are in consideration. Advantages of both methods can be taken by evaluating the temporal tendency for the parts for the local physics and for transport (advection) in a split manner:     ∂ ∂ ∂p = p p + . ∂t ∂t physics ∂t transport Here, the local physical tendency is evaluated based on the Liouville principle:   ∂p ∂ = − (pF ), ∂t physics ∂ϕ whereas the tendency due to the transport is evaluated from the tendency for the required moments (∂ < ϕn > /∂t)transport due to the same transport process. The sum of the two tendencies provides a total temporal tendency for the subgrid-scale distribution. A time-splitting approach proposed here for the subgrid-scale distribution evaluation is commonly used in many numerical problems (LeVeque, 2002, and cf., Vol. 2, Ch. 19), though the timesplitting for subgrid-scale distribution calculation has never been attempted in the literature. Such a time-splitting approach would be helpful for the

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moment-based approach in order to take account of the non-conservative processes, as suggested in Sec. 2.5 of Mironov (2009). Another closely related analogue would be the transformed spectrum approach developed for global atmospheric modelling (Bourke, 1972, 1974). Under this approach, all the physical tendencies are computed in physical space, whereas the transport (advection) is evaluated in spectral space. By the same token, the procedure here is to compute the physical tendency directly with the Liouville equation, whereas to evaluate the advection tendency in a moment phase space. The PDF(DDF)-based approaches are further discussed in Vol. 2, Ch. 25. 6.3 6.3.1

Mode decomposition approach Basic idea

Any physical variable ϕ can be decomposed in terms of a set of “mode” functions (e.g., χl ) as long as the latter are properly defined. This is the starting point of the mode decomposition approach. In order to see the procedure in a more straightforward manner, we take a one-dimensional system with a coordinate x defined over a range [0, L]. Then a physical variable ϕ(x) can be approximated by N mode functions χl (x) as: ϕ(x) ≃

N 

ϕ˜l χl (x)

(6.5)

l=1

with ϕ˜l the coefficient of the l-th mode. The basic idea of the mode decomposition approach is to choose a set of modes in a manner such that the physical variable ϕ(x) can be approximated using Eq. 6.5 with as few modes as possible. Such a procedure is called “compression” in the context of a wavelet-based approach (cf., Mallat, 1998). Image compression is the most popular application of this methodology, but a numerical model itself can also be “compressed” in this manner. When an original full physical system can be compressed to such a degree that it can run as fast as a conventional parameterization, such a compression can also be considered a parameterization itself. Lorenz’s (1963) strange attractor can be interpreted as such an example that parameterizes Rayleigh–Benard convection by truncating a full system into three modes in Fourier space.

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61

A major limitation of the mode-decomposition approach is that the compressed system remains prognostic, requiring time integration with short timesteps, as for Eq. 3.11b derived from a very simple full system in Sec. 3. A conventional parameterization is usually diagnostic, without involving any time integral, as given by Eq. 3.4 for the same system in Sec. 3. Thus, further reductions are required to make a mode decomposition-based parameterization diagnostic in general. Nevertheless, such a prognostic system, at least, serves as a prototype for developing a diagnostic scheme. The mass-flux formulation, a standard formulation for convection parameterization (to be extensively discussed in Part II) can be considered a special example of mode decomposition-based parameterization. Its basic formulation is derived in Ch. 7 from this perspective. In the reminder of this section, we outline how such a procedure is possible in a very general manner. 6.3.2

Mode decomposition procedure

Decomposition is facilitated when a set of modes χl (x) satisfies an orthogonality given by:  1 L χl (x)χm (x)dx = σl δl,m (6.6) L 0 with a normalization constant σl . Here, δl,m is Kronecker’s delta. In that case, the coefficient of each mode is estimated by a simple projection of the mode onto a variable by the integral:  L 1 ϕ(x)χl (x)dx. (6.7) ϕ˜l = σl L 0 Here, in order to perform the calculations for Eq. 6.7, the orthogonality of Eq. 6.6 is a sufficient condition. For example, although the mass-flux decomposition as adopted in mass-flux parameterization does not satisfy a completeness, a set can be chosen so that it satisfies an orthogonality. A generalization of the above description for a variable ϕ(x, y, z) in a three-dimensional space (x, y, z) is not unique. Consistent with the idea of mass-flux formulation, we here take a set of horizontal modes χl (x, y) so that N  ϕ˜l (z)χl (x, y). (6.8) ϕ(x, y, z) = l=1

As a result, ϕ˜l (z) defines the vertical profile of a mode. Here, we redefine Eqs. 6.6 and 6.7 by replacing the integrals with those over the twodimensional domain defined by the coordinates (x, y).

62

6.3.3

Part I: Basic parameterization concepts and issues

Prognostic equation for the subgrid-scale processes

Under an orthogonal mode expansion satisfying Eq. 6.6, the system of Eq. 2.1 is transformed into a description in terms of the mode coefficients ϕ˜l by multiplying Eq. 2.1 with the mode χl and integrating over the horizontal grid-box domain: ∂ 1 ∂ ϕ˜l = −(∇ ρ(wϕ)l + F˜l , H · ϕu)l − ∂t ρ ∂z

where the horizontal and vertical flux terms are given by:  (∇ (aj,k,l u˜j + bj,k,l v˜j )ϕ˜k H · ϕu)l =

(6.9)

(6.10a)

j,k

l= (wϕ)



cj,k,l w ˜j ϕ˜k

(6.10b)

j,k

with the coefficients aj,k,l , bj,k,l , and cj,k,l , defined by the orthogonality of the modes as:  Ly  Lx ∂(χj χk ) 1 χl dxdy (6.11a) aj,k,l = Lx Ly σl 0 ∂x 0  Ly  Lx ∂(χj χk ) 1 χl dxdy (6.11b) bj,k,l = Lx Ly σl 0 ∂y 0  Ly  Lx 1 cj,k,l = χj χk χl dxdy (6.11c) Lx Ly σl 0 0 for a horizontal grid-box domain Lx × Ly . The derived set of equations (the system of Eqs. 6.9–6.11), constitutes a full prognostic description of a full physical system in Eq. 2.1 over the gridbox domain under mode decomposition. When the number N of modes is kept small enough, such a system may serve as a subgrid-scale parameterization, or at least as its prototype. 6.3.4

Choice of modes

The basic set of modes must be chosen so that it fits well with the geometrical structures of the subgrid-scale processes in consideration. Fourier decomposition would be the best known example of mode decomposition. However, it would probably not be the best choice for most atmospheric subgrid-scale parameterization problems. A glance at any three-dimensional visualization of a typical atmospheric subgrid-scale process shows that it consists of an ensemble of spatially

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63

isolated coherent structures such as cumulus convective towers, convective downdraughts, stratiform cloud ascents, and mesoscale downdraughts. Wavelets (cf., Mallat, 1998) would be a natural choice to make for efficiently representing spatially–isolated coherent structures. Yano et al. (2004) demonstrate that wavelets can easily compress atmospheric convective systems down to a level of 1% without substantially losing their overall structure. However, a drawback of wavelet decomposition is the difficulty of evaluating the forcing term F in wavelet space. In spite of various mathematical drawbacks discussed in Yano et al. (2005b), the mass-flux decomposition turns out to be a more favourable approach for this reason, as will be discussed in Part II.

7

Overview of the convection parameterization problem

There are two major historical origins that motivated the development of convection parameterization for atmospheric global circulation models (GCMs). The first is the hot-tower hypothesis proposed by Riehl and Malkus (1958), to be reviewed in Ch. 6. The second is associated with the development of tropical cyclogenesis theories, which culminated in the so-called conditional instability of the second kind (CISK: Charney and Eliassen, 1964). This second aspect is recounted by Kasahara (2000) from a historical perspective. Not only for tropical cyclone development, but also for understanding tropical large-scale disturbances such as Madden–Julian oscillations, convection parameterization is a key ingredient, as already emphasized in the introduction, and as will be discussed further in Ch. 5. Part II begins by discussing Riehl and Malkus’s (1958) hot-tower hypothesis, because this hypothesis more or less directly leads to the concept of mass flux that plays a key role in developing the mass-flux convection parameterization. It would be fair to say that the mass-flux formulation remains a focal point of the remainder of Vol. 1. Such an emphasis is legitimate considering the fact that a significant majority of the current convection parameterizations adopted by numerical models both for operational forecasts and climate projections use this formulation. However, various different convection parameterizations have been developed from various different historical roots. A rudimentary reason for introducing convection parameterization is to prevent a state of explicit conditional instability in which the moist adiabatic slope becomes negative.

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Manabe et al.’s (1965) adjustment scheme, one of the first such attempts, was developed exactly for this very purpose, more or less as a stop-gap procedure. It may even be amusing to notice that this scheme developed for a GFDL (Geophysical Fluid Dynamics Laboratory) GCM was used as a default for over 30 years. The basic idea of adjustment approaches is to relax the atmospheric state to a reference state ϕref with a certain timescale τ so that a convective tendency is defined by:   ϕ − ϕref Q=− . τ Manabe et al.’s (1965) original scheme assumes an instantaneous adjustment taking a limit of τ → 0. (Or more precisely, we set τ to the numerical timestep Δt.) The idea of the adjustment scheme has often been revisited in various different ways; a scheme by Betts (1986) may be mentioned in particular. Betts’s scheme is worth some attention, because it takes special care in defining a reference profile for the adjustment based both on observations and theoretical considerations. In contrast, Neelin and Yu (1994) use a very simple relaxation scheme in the context of a two-layer configuration of a general circulation model. In contrast to the adjustment idea, the basic idea of mass-flux parameterization is to consider convective processes as producing vertical transport. Riehl and Malkus’s hot-tower hypothesis is key to understanding the basic idea. This hypothesis essentially proposes a special method for transporting a surface thermodynamic quantity directly to tropopause level without substantial mixing with the environment. Such a notion is sometimes called “non-local mixing” in order to contrast it against more common “local” mixing processes seen, for example, in the boundary layer. Thus, these two major convection-parameterization approaches (adjustment and transport) may be contrasted with a mixing perspective more often adopted in general parameterization problems, especially in turbulence contexts. In these situations, the subgrid-scale processes can be considered local mixing processes, which may be crudely represented as a kind of diffusion. Such a notion is usually referred to as “eddy diffusion”. A formal derivation of an eddy-diffusion formulation was given in Sec. 2.1. The moment-based approaches outlined in Sec. 6.1 are known to work well with such local mixing processes. However, the local mixing (diffusion) idea does not work for all subgridscale processes. Deep moist convection is considered a primary example, and that is the focus of this set.

Sub-grid parameterization problem

7.1

65

Mixed-layer model

The convective boundary layer is well mixed vertically. Once this basic nature is accepted as a premise, a consistent model for the convective boundary-layer parameterization can be developed, which is called the “mixed-layer model”. The case without water condensation is especially simple. We essentially assume that the dry convective boundary layer is always well mixed in terms of entropy (potential temperature) as well as various passive scalars including the moisture. Under this assumption, it can easily be shown that the boundary-layer depth increases with the square-root of time, given a constant surface flux. When the water vapour reaches saturation at a certain vertical level, a different approach is required above the condensation level: we need to assume instead that the moist entropy (represented by an equivalent potential temperature or a moist static energy) and the total water are well mixed vertically. For details of this extension, we refer to Lilly (1968) and Schubert et al. (1979). This description provides a basic understanding for the vertical structure of cloud-topped boundary layers (e.g., marine stratocumulus), and also partially contributes to understanding trade cumulus. The idea of a mixed-layer model may be, intuitively speaking, considered as a type of local-mixing model by extending the idea of eddy diffusion. However, this understanding would be overly naive for two reasons. First, a conventional eddy diffusion model never achieves a perfect vertical homogenization of a variable, because a finite gradient is always required in order to drive a mixing process under this principle. Second, the mixing is not necessarily local, but it may extend through the whole layer. As emphasized by schematics in Figs. 14–16 of Schubert et al. (1979), the mixing may be realized by a convective cell extending through the whole well-mixed layer. 8

Occam’s razor

A principle that is likely to be important in considering the subgrid-scale parameterization problem in general is the principle of Occam’s razor. This principle is often invoked in order to justify a simpler scientific explanation against a more complicated one. Judgement of whether this principle should be adopted as a general basic principle for scientific research is left for the readers. However, the principle of Occam’s razor certainly plays an important role in constructing a subgrid-scale parameterization.

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The purpose of a parameterization is not to reproduce every detail of the motions at subgrid scales. In fact, we do not require any subgrid-scale information (e.g., subgrid-scale variability) as an output of a parameterization. The sole purpose of a parameterization is to obtain grid box-averaged (large-scale) impacts (feedbacks) of a given subgrid-scale process. Thus, a parameterization must describe a subgrid-scale process just accurately enough in order to provide these outputs satisfactorily. For this purpose, the economy of the formulation becomes a central issue. In short, we do not wish to run a full physical model within a grid box (an idea of super-parameterization: Grabowski and Smolarkiewicz, 1999, Randall et al. 2003: cf., Sec. 2.3). Instead we seek something extremely simple, but good enough for large-scale model outputs. Thus, the principle of Occam’s razor clearly applies here. The issue may furthermore be translated as that of gain against a given investment. An investment (a complexity of parameterization) must be optimized in such a manner that further investment (increment of complexity of a parameterization) should be stopped when the gain (return) becomes saturated. This problem is formulated in a very formal manner under Bayesian statistics as detailed, for example, in Gregory (2005); Jaynes (2003).

9

The status of parameterization studies

To complete this general introduction to the subgrid-scale parameterization problem, we end with some rather lengthy quotations, because together these provide a general flavour of the current status of the field in a more objective manner. 9.1

Turbulence perspectives

In Sec. 5.6.2 of his book, Wyngaard (2010) evaluates the current status of parameterization studies as follows, focusing more specifically on the second-moment turbulence model (parameterizations that specify the covariance in turbulence flows, such as the Reynolds stress): The use of second-moment turbulence models grew rapidly in the early 1970s. In this period the modelling technique now called large-eddy simulation, or LES . . . also appeared (Deardorff, 1970). LES attracted great interest, but because it is

Sub-grid parameterization problem

vastly more demanding of computer resources it was not initially competitive with second-moment modelling. In a review on turbulence, Liepmann (1979) criticized much of this early second-moment modelling: Problems of technological importance are always approaches by approximate methods, and a large body of turbulence modelling has been established under prodding from industrial users. The Reynolds-averaged equations are almost always applied in such work, and the hierarchy of equations is closed by semi-empirical arguments which range from very simple guesses . . . to much more sophisticated hierarchies . . . I am convinced that much of this huge effort will be of passing interest only. Except for rare appraisals such as the 1968 Stanford contest for computation of turbulent boundary layers, much of this work is never subjected to any kind of critical or comparative judgement. The only encouraging project is that current progress in understanding turbulence will . . . guide these efforts to a more reliable discipline. These were some early, in-depth assessments of the performance of second-moment turbulence models, mainly in engineering but in geophysical applications as well. The need was more pressing in engineering, and the requisite data were much more accessible there. In time the salient features of these models became evident. One is their lack of universality – their tendency to unreliability in flows different from those used to develop them. In acknowledging this attribute, Lumley (1983) cautioned that one should not expect too much from these models, which he termed “calibrated surrogates for turbulence.” He felt they should ‘work satisfactorily in situations not too far removed geometrically, or in parameter values, from the benchmark situations used to calibrate them.’ He went on to write: Many of the initial successes of the models . . . have been in flows . . . where details of the models are irrelevant. Thus emboldened, the modellers have been overenthusiastic in promoting their models . . . often without considering in depth the difficult questions that arise. Consequently, there is some disillusionment with the models . . . This reaction is probably justified, but it would be a shame if it resulted in a cessation of efforts to put a little more physics and mathematics into the models.

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By the mid-1980s, LES was being widely used in research applications in both geophysical and engineering flows. It showed a universality lacking in second-moment turbulence models. A decade later, Bradshaw (1994) wrote: . . . even if one makes generous estimates of required engineering accuracy and requires predictions only of the Reynolds stresses, the likelihood is that a simplified model of turbulence will be significantly less accurate, or significantly less widely applicable, than the Navier–Stokes equations themselves; i.e., it will not be ‘universal.’ Irrespective of the use to which the Reynoldsstress model will be put, lack of universality may interfere with its calibration. For example, it is customary to fix one of the coefficients . . . so that the model reproduces the decay of grid turbulence accurately. This involves the assumption that the model is valid for grid turbulence as well as in the flows for which it is intended – presumably shear layers, which have a very different structure from grid turbulence . . . It is becoming more and more probable that really reliable turbulence models are likely to be so long in development that large-eddy simulations (from which, of course, all required statistics can be derived) will arrive at their maturity first. In a later meeting, Bradshaw (1999) summarized: Perhaps the most important defect of current engineering turbulence models . . . is that their nonuniversality (the boundaries of their range of acceptable engineering accuracy) cannot be estimated at all usefully a priori . . . very few codes output warning messages when the model is leaving its region of proven reliability. The 20 years spanned by these comments saw Liepmann’s pessimism about turbulence modelling, then Lumley’s appeal for broader understanding of this nature and more patience with the model development process, next Bradshaw’s tacit acceptance that before turbulence models become adequately reliable they may be replaced by LES, and finally Bradshaw’s doubts that we know enough about turbulence-model reliability. The turn to LES is evident in geophysical applications, and more recently to evaluate turbulence models and parameterizations (Ayotte et al., 1996). . . . because of their larger scales and smaller

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speeds geophysical flows tend to be much more strongly influenced by buoyancy. Because of the tendency of turbulence models to non-universality, those for geophysical applications need their own development and assessment process . . .

9.2

Climate perspectives

An overall tone emphasized by Wyngaard (2010) is an emphasis of universality. Although he does not make this point explicitly, universality must be supported by a general physical principle. This overall tone may be contrasted with that of the most recent IPCC report (Flato et al., 2013). Its tone is more technologically oriented, and the key focus is the extent to which the current physical parameterizations work well in climate models. We see little about universality. Section 9.1.3.1 of Flato et al. (2013) is exclusively devoted to parameterization issues. After a short paragraph introducing the notion of parameterization, it begins with a remark “Atmospheric models must parameterize a wide range of processes . . .” with a list of parameterized processes. Recent advances in representing cloud processes in general with new approaches in shallow cumulus convection and moist boundary-layer turbulence are especially emphasized. Section 9.2.1.2 of the same report, a section that emphasizes the importance of isolating processes, also emphasizes that parameterizations can be tested under this context. It lists recent isolation-based tests performed since the last IPCC report. This section ends with a remark “These studies are crucial to test the realism of the process formulation that underpin climate models.” It would be reasonable to note that “realism” is a key word here. Elsewhere in the report, Sec. 9.5.2.2 attributes recent improvements in simulating blocking to improvement of parameterizations, whereas Sec. 9.5.2.3 attributes model errors associated with the Madden–Julian oscillations to the fact that “convection parameterization do not provide sufficient build-up of moisture in the atmosphere . . .” In its summary (Sec. 9.5.4.5), the report emphasizes the fact that a major source of model uncertainties is due to “representation of processes (parameterization)”. It is followed by a list of some specific examples.

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Concluding reflections

The two distinct perspectives detailed above from two distinct contexts should provide a sense of where the subgrid-scale parameterization problem stands. In the atmospheric science community, subgrid-scale parameterization is often considered a highly technical problem of details, and a typical prejudice might say, mostly a problem of tuning. A recent review (Randall et al., 2003) even provocatively concludes that it is in “deadlock”. In concluding this chapter, let us consider a rather provocative question: do we wish parameterization to be a science or rather an alchemy? Alchemy is based on many mysterious concepts comprehensible only to the original inventors and their close followers, while modern science refuses such mystification: everything must be derived from simple, clear concepts that are understandable (after proper education) by everyone. Stating this as a matter of principle, the philosophy of science clearly points out that the distinction between alchemy and science is not so clearcut in practice. Even in modern science, many ambiguous concepts are inevitably introduced in the process of research. Nevertheless, the basic principle of modern science is to avoid such ambiguous concepts as much as possible. Details do matter in science, and we cannot make any progress without dwelling on details. However, it is important to recognize when the complexities of the details are in danger of reaching a level of alchemy where the basic physical principles are lost from sight. In other words, we should keep a general, basic principle in mind in pursuing any scientific problem. These general statements are important here, because a parameterization often begins to develop a flavour of alchemy. Another slightly different, but equally useful point of reference is the difference between Copernican and Ptolemaic astronomies. Modern science takes the Copernican astronomy, whereas in a previous age, understanding of the movements of the planets were given under the Ptolemaic astronomy. The central idea of the latter is continuously adding epicycles so that more accurate predictions of planetary movements can be achieved. In retrospect, these epicycles do not correspond directly to physical principles behind the actual planetary movements. The parameterization problem can also easily fall into the trap of continually adding epicycles without concerning itself with the physical principles. Importantly, if solely a precise prediction is the issue, there is nothing fundamentally wrong with this type of approach.

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The only problem is that since it lacks in universality, it begins to break down gradually as a longer-term prediction is pursued. An important question to be asked would be whether it is possible to maintain a healthy field of parameterization research by avoiding a risk of falling into a trap of alchemy or epicycles. By considering various examples for parameterizations, both idealized as well as from turbulence research, this chapter has attempted to provide some insights into this general question. An important guiding principle would be to maintain as far as possible a universality of a parameterization formulation throughout all the phases of the formulational development. Such an attitude also helps to maintain a global picture of the problem without being lost in details of an alchemical nature. However, a parameterization is not an abstract problem. We wish to implement a specific physical process into a model in a parametric manner. As soon as we plunge into a specific problem, details do matter. In such an endeavour, a theoretical formulation can never be developed without at least some approximations and hypotheses. Unfortunately, as a general rule, these approximations and hypotheses gradually develop a flavour of epicycles, and the whole effort gradually becomes more like an alchemy. That would be the moment when we need to return to a more general principle. The goal of this set is to provide enough background knowledge to enable readers to make more sound judgements on parameterization issues, especially for convection. The intention is to provide a good balance of both the big picture and the important details. 11

Bibliographical notes

Starr (1968) is an unorthodox introduction to the subgrid-scale parameterization problem focusing on the eddy momentum issues. Though some materials presented therein may be outdated, the whole argument of the book still remains fascinating. Stensrud (2007) is a textbook that covers various physical parameterizations adopted in current weather prediction models. Emanuel and Raymond (1993) review many existing operational convection parameterization schemes along with some useful overview reviews.

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Chapter 3

Scale separation

J.-I. Yano Editors’ introduction: The previous chapter introduced the notion of a scale separation as an aspect of developing a parameterization. This chapter explores that notion more comprehensively. Three methodologies are explained for splitting a flow into some part that is to be represented explicitly and directly by a numerical model and some part that is to be represented through a parameterization approach. The simplest example is a partition into grid-scale and subgrid-scale motions, and in this case the grid separates between scales. This can be generalized to a partition into filtered and residual motions. In this case there is not necessarily a simple separation of scales. This is both the great merit and the great drawback of the approach. Clearly there is nothing special in a physical sense about a chosen grid spacing (and an associated chosen numerical scheme) but a more general filtering procedure introduces additional terms into the equations for the filtered motions that cannot necessarily be ignored. Finally, a multi-scale asymptotic expansion asserts that there are distinct scales in the problem. In principle, the expansion is infinite but if such distinct scales do exist and are identifiable then in the limit that they are fully separated, the higher order terms will vanish. A general remark to be made here is that a specific methodology is not normally explicitly stated in the specification of numerical weather prediction and climate projection models. It is often not implicitly stated either, as language appropriate to each of the three methodologies is often intermingled in discussions. Rather the methodologies provide ways of thinking about parameterizations for such models. This situation is arguably not important in practice, but arguably it may create problems. For instance, it is difficult to consider carefully questions of consistency or double counting between different parameteri73

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zations within a model (cf., Vol. 2, Ch. 26) without first being completely clear about what exactly each parameterization is attempting to represent.

1

Introduction

The basic idea of subgrid-scale parameterization may be best understood in terms of a dichotomy between the grid (resolved, large-scale) and the subgrid (unresolved) scales. This dichotomy is usually called the “scale separation” and is the theme of this chapter. The seminal paper by Arakawa and Schubert (1974) begins with a statement: “The many individual cumulus clouds which occur in a large-scale atmospheric disturbance have time and space scales much smaller than the disturbance itself.” This statement can be taken as a tentative definition for the concept of the scale separation. Accordingly, the principle of scale separation may mathematically be stated as: ΔX ≫ Δx, (1.1) where ΔX and Δx are characteristic scales for large-scale and subgrid-scale processes, respectively. A subgrid-scale parameterization would be relatively easy to construct if a system consisted of two clearly separated scales. However, unfortunately, the observed atmospheric variability does not present such a clear scale gap justifying a naive two-scale interpretation. Nonetheless, the concept of subgrid-scale parameterization can be developed by assuming an idealized two-scale system. In a tropical context, such an idealization can relatively easily be envisioned by assuming that tropical circulations consist only of two scales: that for deep moist convection and that of the mean Hadley– Walker circulation. This idea will be developed in a more concrete manner in Ch. 6. This chapter contains general discussions towards this goal. Section 3 contains three derivations for a statement of the subgrid-scale problem presented in Ch. 2, Sec. 2. The first is a straightforward averaging of a system over grid boxes fixed in space. The second is based on the multiscale asymptotic analysis already introduced in Ch. 2. Finally, in the third approach, the concept of filtering is introduced, which is often considered the most formal procedure for defining a subgrid-scale parameterization problem. The remainder of the chapter is devoted to conceptual discussions based on these derivations. By paying close attention to its various aspects, the intention here is to clarify this often-misunderstood concept.

Scale separation

2

75

Scale gap

The concept of subgrid-scale parameterization would be easier to understand when there is a clear scale gap between the resolved (i.e., large-scale) and unresolved (subgrid). Often it is even argued that the existence of such a clear scale-gap is a prerequisite for justifying a subgrid-scale parameterization. Such an argument may be developed as follows. 2.1

Need for a scale gap

Atmospheric motions cover a wide range of scales, from the largest eddies of order tens of thousands of kilometres, on the size of the globe, through to the smallest eddies of order a few millimetres, contributing to molecular dissipation. However, it is not numerically feasible to resolve the complete range of scales, and so we face a need for parameterizing small-scale motions, as already argued in the last chapter. The question now is how to separate what is to be resolved and explicitly simulated from what is to be parameterized. The idea of filtering, as will be further discussed in Sec. 3.3, provides one way of considering this issue. From this perspective, the distinction between what is resolved and what is parameterized is established by applying a filter, or averaging operation, to the full set of physical equations. That part of the flow which remains after the filtering (averaging) is identified as that which must be explicitly represented within the numerical model, while motions that are removed by the filtering must be parameterized. In this manner, the concept of filtering provides a formal procedure for splitting the flow into resolved and parameterized components. In Sec. 3.3, we will show how the application of a filter leads to a formal expression for the set of equations to be solved. In the given set of filtered equations, we can furthermore formally identify which terms must be parameterized, and which need not. An alternative perspective may be constructed by considering the resolved variables to be those which can be measured by an observational network, given that in a forecasting context this will typically constitute the basis of the model initialization and verification. For example, the conventional rawinsonde network can only represent what is commly called the “synoptic scale” in meteorology, and has a horizontal scale in the hundreds to thousands of kilometres, which is also associated with the evolution of weather patterns over a few days. In effect, the focus on such a scale may

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be considered as a form of filtering. A local afternoon shower would not be well resolved under such an observational network. Indeed, any observational network filters out small scales, in a relative sense, based on those aspects of the meteorology that it is designed to study. Of course, there are interactions between the resolved scales and the subgrid scales (the scales to be parameterized). That is why we need a parameterization. Hence, as well as identifying the scales of the practical interest, intuitively we should also seek to identify a split which minimizes such interactions as far as practical. We might anticipate that a relatively weak interaction could help to make the parameterization problem simpler, or perhaps even less important. This is the basic motivation for seeking a scale gap. In other words, a scale gap in atmospheric motions is a very appealing concept because a clear and pronounced separation between the resolved and parameterized scales should make possible a reasonably simple form of parameterization. The idea of a two-scale system, for example, consisting only of convection and large-scale flows, may be considered a special case of the scale gap. The scale gap generalizes this idea by allowing that both the resolved and the unresolved processes may consist of many scales, not necessarily literally designated by the single symbols ΔX and Δx. But it insists that those sets of scales are well separated from each other. Such a separation would be likely to guarantee different physical processes driving the resolved and the parameterized scales. Distinguished physical processes separating those scales would furthermore facilitate us to develop a physical principle for a parameterization. 2.2

Mesoscale gap: A review

Where might we find such a scale gap in the atmosphere? As a positive answer to this question, the notion of the so-called mesoscale gap is often invoked, which may furthermore justify the scale-separation principle. This notion roughly says that atmospheric variability presents a local minimum in its power spectra over a mesoscale. If that should be the case, we can construct a parameterization based on a strict sense of scale separation as long as the model resolution remains at a scale just above that of the mesoscale gap. In support of this notion, Fig. 1 of Van der Hoven (1956) is often quoted, which ostensibly represents a spectral power minimum at a onehour timescale, which is interpreted as a sign of a mesoscale gap. Readers

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are strongly encouraged to consult this original article themselves, because a major problem has been found in citing this result without actually reading the paper. A careful reading of the original reference will reveal various data issues, which are not discussed here. The most important point to notice is the fact that what is plotted is not actually the frequency power spectrum itself, but a spectrum multiplied by the frequency. Thus, the hidden message of this spectrum is that the atmospheric variability roughly follows a power law proportional to an inverse of the frequency, what is often called “1/f noise”. Compared to this distinctively decaying tendency of the power over the four decades of the frequency considered, the minimum plotted is tiny. Notice that this spectrum multiplied by the frequency is plotted in a linear scale. A very good paper to demonstrate this simple point is Vinnichenko (1970), which literally replots Van der Hoven’s spectrum without multiplying by frequency as shown in his Fig. 3. It demonstrates that the mesoscale gap is nothing dramatic compared to the overall tendency of 1/f -noise spectrum in atmospheric variability. However, this does not say either that a mesoscale gap does not exist (see Heggem et al., 1998; Ishida et al., 1984; Saito and Asanuma, 2008). The mesoscale gap appears to be a recurring phenomena depending on a regime of the mesoscale variability, and thus it is clearly a fascinating topic in its own right. However, the main conclusion does not change: the mesoscale gap is so weak that even if it may exist, it hardly constitutes a legitimate justification of the scale-separation principle. 2.3

Power-law spectra

It is well known that atmospheric variability of many variables follows a power law without any distinguished spectral peaks, perhaps apart from those associated with diurnal and annual cycles. For example, the atmospheric kinetic energy follows a −5/3 power law for a wide range of horizontal scales from less than 1 km up to 103 km (Nastrom and Gage, 1985; Wikle and Cressie, 1999). A basic reasoning for the existence of a power-law spectrum behaviour is because the atmospheric motions are fully turbulent (cf., Charney, 1971), residing in a regime called the “inertial subrange” in which non-linear advection processes dominate (cf., Fritsch, 1995). In the internal subrange of a turbulent flow, a spectrum is continuous without any clear scale gap. Thus, from the scale-gap point of view presented in Sec. 2.1, the scale-separation principle is not applicable either.

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It may be important to emphasize that the nature of the ubiquitous power-law spectrum in atmospheric variability is still to be fully understood. The current theoretical investigations (for the free troposphere) are almost exclusively focused on dry atmospheric dynamics. The role of moist deep convection (on which this set is focused) in establishing powerlaw behaviour is still unknown. Nevertheless, the phenomenological fact of power-law spectra without a scale gap does not change due to a lack of good theoretical investigations. 2.4

Fractality and scaling

A more emphatic argument against the notion of a scale gap is fractality, which is identified by examining common geometrical structures associated with atmospheric phenomena. A good starting point for understanding this point of view is to consider a global cloud image as shown in Ch. 2, Fig. 2.1: zooming in on this image at any rate, we find similar geometrical structures regardless of the zooming rate. In other words, it is hard to tell the difference between scales by just looking at a pair of image segments zoomed in on at different rates. Such a simple demonstration is found, for example, in Fig. 2 of Yano and Takeuchi (1987). The simple exercise here suggests that clouds tend to repeat a similar structure from the smallest possible (likely down to the molecular dissipation scale) up to the largest possible scales (i.e., the size of the Earth). Such a notion may be called “self-similarity”. Mandelbrot (1982) proposed to call such geometrical objects “fractal”. The global fractal distribution of clouds has already been established for more than two decades (Lovejoy, 1982; Yano and Takeuchi, 1987). In recent years, the fractality of global cloud distribution has been established over a much wider spectral range (Lovejoy et al., 2008), and fractality is also established in many other physical variables including winds, pressure, and temperature (Lovejoy et al., 2009a,b). Overviews for these observational studies are provided by Lovejoy and Schertzer (2010) and Tuck (2008). Mathematical and theoretical bases for the fractal-based approaches are given by Lovejoy and Schertzer (2012). Extensive discussions of theoretical implications are provided by Tuck (2008). Fractality is also sometimes referred to as “scaling”, when the emphasis is placed more on its statistical nature such as a power law. From this point of view, a close link between the power-law spectra discussed in the last subsection and fractality must be emphasized. A fractal geometry is

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characterized by a fractal (non-integer) dimension, as extensively discussed in Mandelbrot (1982). It can be shown that the fractal dimension is linearly related to a power exponent of the spectrum for the same object. 2.5

Is fractality useful for constructing subgrid-scale parameterizations?

Power laws, fractality and scaling may be argued to point to an alternative paradigm for developing subgrid-scale parameterizations without relying on the scale-separation principle. Here, it may be emphasized that if the scale separation can indeed be justified only with the existence of a scale gap, as discussed, then scale separation cannot be applied to atmospheric processes. In the remainder of the present chapter, it will be argued that it is possible to introduce a scale-separation principle to a system without the existence of a scale gap. A multi-scale asymptotic-expansion approach turns out to be significant in justifying this principle without the existence of a scale gap. It is fair to say that a fractal paradigm may suggest an alternative framework for constructing subgrid-scale parameterizations. Thus, is fractality useful for constructing subgrid-scale parameterizations? In considering this possibility, we should first note that the notions of fractality, scaling, and even power laws are merely a geometrical characterization of atmospheric processes. These notions by themselves do not describe any physical principles, and lack the prognostic power needed for making a prediction. Crudely speaking, these geometrical characterizations can take care of only half of the overall subgrid-scale problem: i.e., downscaling. For a data set (either model output or observational data) with limited horizontal resolution, such geometrical information provides us a means for estimating distributions of a variable in the unresolved scale. Essentially, geometrical information enables us to perform an interpolation for realizing a downscaling. The problem of parameterization also requires to estimate the feedback from these subgrid-scale variabilities to the resolved scale. This question may, intuitively, be considered as that of determining a value for the starting point of an interpolation. This issue is considered that of closure (in the terminology introduced in the last chapter). Of course, there are some physical bases leading to the notion of frac-

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tality, scaling, and power laws. The most obvious one, as often invoked by Lovejoy and his colleagues, and as briefly remarked upon in Sec. 2.3, is to consider atmospheric processes as turbulence. From this perspective, some parameterization theories developed for turbulence may be useful. (See some reviews on this subject in the last chapter.) However, as also already remarked in Sec. 2.3, these turbulence-based perspectives developed in atmospheric investigations are mostly limited to the dry atmosphere. A potentially more powerful tool for dealing with fractal, self-similar systems is the renormalization group (RNG) approach. The basic idea of RNG is to apply a filtering procedure sequentially so that a self-similar atmospheric structure is averaged out step by step. Crudely speaking, fractality suggests that the atmosphere consists of multiple scales rather than two scales. Thus, the scale-separation principle must be applied sequentially so that each scale in the hierarchy is correctly averaged out. Unfortunately, the alternative possibilities outlined here are yet to be fully developed.

3

Subgrid-scale parameterization in two-scale systems

In this section, the subgrid-scale parameterization formulation presented in Ch. 2 will be derived assuming a two-scale system consisting of resolved and unresolved scales. This derivation is provided in three different ways. In the first derivation, an approach of grid-box averaging is taken, as already outlined in Ch. 2. The second is based on the multi-scale asymptotic expansion. This approach was adopted in Ch. 2 in order to elucidate the basic concepts behind the parameterization problem by taking a very simple mathematical model. The third is based on filtering, which is often considered the most formal procedure for defining a subgrid-scale parameterization. As in Ch. 2, as a starting point, we take a prognostic equation:

∂ 1 ∂ ϕ + ∇ · uϕ + ρwϕ = F ∂t ρ ∂z

(3.1)

for an unspecified variable ϕ. In the following, an anelastic approximation will implicitly be assumed, with the density ρ a function of height only.

Scale separation

3.1

81

Grid-box averaging

Grid boxes are introduced into the system with an index (i, j) designating each box. Its centre position is (xi , yj ), and it extends from [xi−1/2 , xi+1/2 ] and [yj−1/2 , yj+1/2 ], respectively, in the x and y directions. We assume square boxes of size Lx × Ly . An average over a grid box is defined by:  yj+1/2  xi+1/2 1 ϕdxdy. (3.2a) ϕ¯ij = Lx Ly yj−1/2 xi−1/2 Here, it is important to add superscripts i and j in order to indicate a grid-box number. An average of the horizontal divergence term is given by:  yj+1/2  xi+1/2 1 ∇ · uϕdxdy = Lx Ly yj−1/2 xi−1/2

 yj+1/2  xi+1/2 1 ∂uϕ ∂vϕ + dxdy. Lx Ly yj−1/2 xi−1/2 ∂x ∂y The first term in the integrand can be further rewritten as:  yj+1/2  xi+1/2  yj+1/2 ∂uϕ 1 1 xi+1/2 [uϕ]xi−1/2 dy = dxdy = Lx Ly yj−1/2 xi−1/2 ∂x Lx Ly yj−1/2

 xi+1/2 yj+1/2 uϕi+1/2,j − uϕi−1/2,j 1 1 xi+1/2 = = [uϕj ]xi−1/2 . (uϕ)dy Lx Ly yj−1/2 Lx Lx xi−1/2

(3.2b) Here, i±1/2,j

ϕ¯

1 = Ly



yj+1/2

yj−1/2

ϕ|x=xi±1/2 dy

is introduced. The second term can also be rewritten in a similar manner, with a corresponding notation. Then, we obtain:

 yj+1/2  xi+1/2 1 ∂uϕ ∂vϕ + dxdy = Lx Ly yj−1/2 xi−1/2 ∂x ∂y uϕi+1/2,j − uϕi−1/2,j vϕi,j+1/2 − vϕi,j−1/2 ij + ≡ ∇ · uϕ . Lx Ly

(3.3)

Furthermore, we separate each variable into an average along a grid-box edge and a deviation, such as: u=u ¯i±1/2,j + ui±1/2,j′ .

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Note that it is important to add the superscript i ± 1/2, j to the deviation term, because the definition of the deviation depends on the reference average. As a result, uϕ = (¯ ui±1/2,j + ui±1/2,j′ )(ϕ¯i±1/2,j + ϕi±1/2,j′ ) = u¯i±1/2,j ϕ¯i±1/2,j + ui±1/2,j′ ϕ¯i±1/2,j + u¯i±1/2,j ϕi±1/2,j′ + ui±1/2,j′ ϕi±1/2,j′ . By taking an average of the above, uϕi±1/2,j = u¯i±1/2,j ϕ¯i±1/2,j + ui±1/2,j′ ϕi±1/2,j′

i±1/2,j

i±1/2,j

by noting, for example, ui±1/2,j′ = 0. As a result, the horizontal convergence term can be separated into two contributions as: ij ij ¯ ·u ¯ ϕ) ∇ · uϕ = (∇ ¯ ij + ∇ · u′ ϕ′

(3.4)

where the grid-box mean and the subgrid-scale contributions are, respectively, defined by: u ¯i+1/2,j ϕ¯i+1/2,j − u ¯i−1/2,j ϕ¯i−1/2,j Lx i,j+1/2 i,j+1/2 ϕ¯ − v¯i,j−1/2 ϕ¯i,j−1/2 v¯ , + Ly

¯ ·u ¯ ϕ) (∇ ¯ ij =

∇·

ij u′ ϕ′

= +

ui+1/2,j′ ϕi+1/2,j′ v i,j+1/2′ ϕi,j+1/2′

i+1/2,j

i,j+1/2

(3.5a)

− ui−1/2,j′ ϕi−1/2,j′ Lx

− v i,j−1/2′ ϕi,j−1/2′ Ly

i−1/2,j

i,j−1/2

.

(3.5b)

The vertical flux term is similarly given by: ij

wϕij = w ¯ ij ϕ¯ij + wij′ ϕ¯ij′ .

(3.6)

By substituting all of the above into Eq. 3.1, a grid box-averaged equation is finally given by: 1 ∂ ∂ ij ¯ · uϕ) ϕ¯ + (∇ ρw ¯ ij ϕ¯ij = Qij , ¯ ij + ∂t ρ ∂z

(3.7)

where the apparent source term is defined by: ij

Qij = −∇ · u′ ϕ′ −

1 ∂ρwij′ ϕij′ ρ ∂z

ij

+ F¯ ij .

(3.8)

Under this derivation, we see that the subgrid-scale contribution is intermingled with issues of numerical discretization of the system. For example,

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in Eq. 3.3, the grid box-averaged horizontal divergence is defined in terms of a first-order finite difference. If a higher-order scheme is to be adopted for the horizontal divergence term, as a result, we need to redefine the gridbox average by a certain weighted average for consistency. That would further lead to a change in the final expression for the subgrid-scale terms in Eq. 3.5b. It is important to note that once this grid-box average is generalized into a certain weighted average, it is no longer a grid-box average that is under consideration in a literal sense. This fact should also make us realize that under most of the formulations for numerical algorithms, the grid box does not exist in a literal sense, but is only useful in a conceptual sense. The analysis of the present section suggests that the grid-box concept can be taken in a literal sense only if we adopt a first-order finite difference. Once we move to a higher-order scheme, this notion is no longer useful in a literal sense, because all the numerical manipulations associated with horizontal derivatives are performed by crossing a few grid points. In deriving the mass-flux formulation in Part II, it turns out be to more convenient to stay with a finite volume framework rather than finite differences. In this case, too, we find that the grid-box concept is useful only when a leading-order finite volume formulation is adopted. 3.2

Multi-scale asymptotic expansion

The concept of scale separation may be more intuitively understood by invoking an asymptotic expansion approach. A multiple of distinguished scales may be introduced under an asymptotic expansion. However, the simplest choice within this framework would be to take two scales, which correspond to the resolved and subgrid scales. Introducing a two-scale description both in time and space, the time derivative is rewritten: ∂ ∂ ∂ =ǫ + (3.9a) ∂t ∂T ∂t in terms of the slow T and the fast t times with ǫ designating a small parameter for scale separation. The horizontal coordinates are also rewritten in a similar manner, for example: ∂ ∂ ∂ =ǫ + . (3.9b) ∂x ∂X ∂x We denote the nabla operator with a bar and prime designating operations on the large-scale and subgrid-scale coordinates, respectively: ¯ + ∇′ . (3.10) ∇ = ǫ∇

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Here, we use the same small parameter ǫ both for time and space separations. This is a natural choice for equations taking the form of Eq. 3.1. However, this is not always the case, as already seen in Ch. 2, Sec. 4.1. Accordingly, all the physical variables are separated into two parts: those solely depending on the large scale, and those that also depend on the subgrid scales: ϕ = ϕ(X, ¯ Y, z, T ) + ϕ′ (x, y, z, t, X, Y, T )

(3.11a)

with a similar decomposition for the horizontal velocity u. Decomposition of the vertical velocity is slightly more complicated due to the fact that by mass continuity, the large-scale velocity becomes smaller by the factor ǫ: w = ǫw(X, ¯ Y, z, T ) + w′ (x, y, z, t, X, Y, T ).

(3.11b)

Note that we do not introduce the two scales in the vertical direction z by following the conventional wisdom of meteorology, i.e., the vertical scale is comparable both in large and subgrid scales. Stratification of the atmosphere is the main reason that prevents such a distinction from being made. A bar is added for the large-scale variable in order to suggest that it can be obtained by averaging the full variable ϕ over subgrid-scale (small-scale) coordinates, i.e.,  Y +L/2  X+L/2 1 ϕdxdy (3.12) ϕ(X, ¯ Y ) = lim 2 L→∞ L Y −L/2 X−L/2 (cf., Deardorff, 1970; Lilly, 1967). As a simple corollary, we can prove: ϕ′ = 0.

(3.13)

Here, it is assumed that short timescale fluctuations are also smoothed out by spatial averaging. For a more careful treatment explicitly considering this aspect, see Majda (2007). It transpires that the bar can be considered as an averaging operator:  Y +L/2  X+L/2 1 dxdy. (3.14) lim L→∞ L2 Y −L/2 X−L/2 This operation may be considered as what is typically called a “large-scale average” in slight contradiction with the fact that it is defined in terms of average over the subgrid-scale coordinates. Recall that as a consequence of the asymptotic scale separation as given by Eq. 1.1, the integral in Eq. 3.14 over an infinite domain over subgrid-scale coordinates is considered as a single large-scale point.

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After introducing these two-scale descriptions, the three terms in Eq. 3.1 are, respectively, rewritten as: ∂ϕ ¯ ∂ϕ′ ∂ϕ′ ∂ϕ =ǫ + +ǫ , ∂t ∂T ∂t ∂T ¯ · (ϕ¯ ¯ + ϕ′ u′ ) + ∇′ · (ϕu ¯ + ϕ′ u′ ), ∇ · ϕu = ǫ∇ ¯u + ϕu ¯ ′ + ϕ′ u ¯ ′ + ϕ′ u F = ǫF¯ + F ′ . Note that F¯ is defined by a formal application of Eq. 3.14 on F , but divided by ǫ. Substitution of these three expressions into Eq. 3.1 leads to the full equation under the two-scale description. To O(1) in the ǫ–expansion, Eq. 3.1 gives: ∂ ′ ∂ϕ′ ¯ + ϕ′ u′ ) + + ∇′ · (ϕu w (ϕ¯ + ϕ′ ) = F ′ . ¯ ′ + ϕ′ u ∂t ∂z This equation describes the evolution of the subgrid-scale processes at every large-scale point (X, Y ). As will be seen immediately below (cf., Eq. 3.16), the large-scale variable ϕ¯ changes only slowly with the time T . For consistency, the subgridscale processes should not have any feedback to the large scale to the leading order. Especially, the vertical eddy-transport term should not produce any large scale-averaged tendency to this order, and thus a constraint w′ ϕ′ = O(ǫ) must be satisfied, although it would be difficult to satisfy such a constraint in practice. (A formal method for achieving this goal would be to assume a certain linearity to the leading order so that these eddy terms do not appear. In this case, a higher-order in ǫ must be explicitly considered in order to recover a fully non-linear description of the system.) As a practical procedure, in order to ensure that this equation does not produce any large scale-averaged tendency, we subtract the large scale-averaged eddytransport contribution from the above equation so that the final equation is: 1 ∂ ∂ϕ′ + ∇′ · (¯ ρ[w′ (ϕ¯ + ϕ′ ) − w′ ϕ′ ] = F ′ . u + u′ )(ϕ¯ + ϕ′ ) + (3.15) ∂t ρ ∂z ¯ ϕ¯ = 0 has been invoked. In this final expression, a trivial relation that ∇′ · u Note that Eq. 3.15 is still not self-contained, because it is generally not possible to evaluate the perturbation forcing F ′ directly due to the inherently non-linear nature of the forcing. This term must rather be evaluated by taking a difference of the two terms: F ′ ≡ F − ǫF¯ . In other words, F ′ is not known without knowing F¯ .

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Note that by separating out the large-scale contributions in the equation as externally prescribed variables, the evolution of the subgrid-scale variables can be evaluated over any size of domain by Eq. 3.15 under any lateral boundary conditions. Especially, although it may seem intuitively incorrect, it is consistent to solve this equation over a periodic domain as is often the custom in simulating convection with cloud-resolving models (CRMs). To O(ǫ), Eq. 3.1 gives:   1 1 ∂ ∂ ′ ¯ ϕ¯ ¯ +ϕ′ u′ )+ (ϕ+ϕ ¯ ρ w( ¯ ϕ¯ + ϕ′ ) + w′ ϕ′ = F¯ . )+ ∇·( ¯u + ϕu ¯ ′ +ϕ′ u ∂T ρ ∂z ǫ Note that the vertical eddy-transport term taken out from the leading-order equation is put back here for consistency. It is important to keep in mind that this whole term is O(ǫ). In order to obtain an equation that is closed in terms of the large-scale ¯ , we apply an averaging operator, Eq. 3.14, to the above. variables ϕ¯ and u Note that this operator is interchangeable with the differential operators for the large-scale variables. This is obvious, once a multi-scale coordinate is accepted in a formal manner, and X and x are interpreted as two independent coordinates. We also recall the property of Eq. 3.13, thereby obtaining:   1 ′ ′ 1 ∂ ∂ ′ ′ ¯ ϕ¯ + ∇ · (ϕ¯ ¯u + ϕ u ) + ρ w ¯ϕ¯ + w ϕ = F¯ . (3.16) ∂T ρ ∂z ǫ Furthermore, we introduce the apparent source term Q in the above: 1 ∂ ∂ ¯ · ϕ¯ ϕ¯ + ∇ ¯u + ρw ¯ ϕ¯ = Q, ∂T ρ ∂z

(3.17)

where the apparent source is defined by: ¯ · ϕ′ u′ − Q = −∇

1 ∂ ρw′ ϕ′ + F¯ . ǫρ ∂z

(3.18)

This is the term to be parameterized. The advantage of the multi-scale asymptotic expansion is in providing a formal separation between the large (resolved, grid-box average) and the subgrid (unresolved) scales both for dependent variables, Eqs. 3.11a, b, and independent variables (coordinates: Eqs. 3.9a, b, 3.10) by explicitly introducing an expansion parameter ǫ. The formal separation, furthermore, allows us to obtain separate equations for the large and subgrid scales, respectively, as Eqs. 3.16 and 3.15. Note especially that the subgridscale equation, Eq. 3.15, is not a simple difference between the large-scale

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equation, Eq. 3.16, and the original full system, Eq. 3.1. The obtained subgrid-scale equation, Eq. 3.15, rather retains only the terms significantly contributing to the subgrid scales. A formal application of the asymptotic expansion enables such discriminations to be made. Using this procedure, we find that advection of the subgrid-scale variables by large-scale vertical advection does not contribute; the subgrid-scale variables must be horizontally advected by the total wind ¯ + u′ . In contrast, the filtering approach to be discussed in the next u subsection does not make such discriminations. It rather provides exact equations at all steps, but without suggesting the significance of each term. 3.3

Filtering approach

Filtering is another formal approach for describing subgrid-scale parameterization. Arguably, without introducing an asymptotic expansion, a filtering approach can define the parameterization problem in an exact manner without approximation. This is a consequence of not explicitly introducing a notion of scale separation. This advantage of the filtering approach can also be interpreted as a disadvantage: a small expansion parameter introduced in the asymptotic expansion may guide the formulation for a parameterization, as will be more explicitly demonstrated in Part II. The present section heavily relies on Ch. 13 of Pop (2000). Reynolds (1990) emphasizes that the filtering can be constructed independently of a numerical grid, which may only be introduced later, and thus distinction between the resolved and the subgrid scales is better replaced by that between the filtered and the residual scales. The filtering approach may be, to some extent, considered a generalization of the grid box-averaging procedure discussed above. A filtering procedure may be, generally, defined by:  h h G(r)ϕ(x − r, t)dr. ϕ¯ = −h

−h

We may recover the grid-box average, Eq. 3.2a, by setting G = 1/(Lx Ly ), h = Lx /2 = Ly /2. However, there is a major difference: the points considered for averaging are no longer discretized as in the grid box-averaging procedure, but continuous in space. Here, the filter function G must be normalized such that:  x+h/2  y+h/2 G(r)dr = 1. x−h/2

y−h/2

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Part I: Basic parameterization concepts and issues

The filter function may, generally, also depend on the spatial position x. However, this possibility is not considered in the following. A limit of h → ∞ may also be taken for mathematical convenience. The most convenient filter function forms would be a top hat or Gaussian. A half-sine shape is also often adopted. A deviation from a filtered state may be defined by: ¯ ϕ′ = ϕ − ϕ. However, unlike the multi-scale asymptotic method (cf., Eq. 3.13), usually ϕ′ = 0.

(3.19)

The filtering operation and horizontal derivatives are interchangeable, and, for example, ∂ ϕ¯ ∂ϕ = . ∂x ∂x This is proved backwards as follows:  h ∂ ϕ¯ ∂ = G(r)ϕ(x − r)dr ∂x ∂x −h  h ∂ϕ(x − r) dr = G(r) ∂x −h where ϕ is the only quantity for the derivative to act upon as long as G is independent of x, and h is a constant. The filtered version of Eq. 3.1 is: 1 ∂ ∂ ϕ¯ + ∇ · ϕ¯ ¯u + ρw ¯ ϕ¯ = Q. ∂t ρ ∂z This is effectively identical to Eq. 3.17 except that no multiple scale is introduced, and thus no averaging operator is introduced to the nabla operator, for example. The corresponding apparent source term is defined by: ¯u) − Q = −∇ · (ϕu − ϕ¯

1 ∂ ρ(wϕ − w ¯ ϕ) ¯ + F¯ . ρ ∂z

Here, it is important to note that ϕu − ϕ¯ ¯u = ϕ′ u′ and wϕ − w ¯ ϕ¯ = w′ ϕ′ .

Scale separation

89

Instead, the left-hand side takes much more complicated expressions. For example, Germano (1986) shows that the Reynolds stress term can be decomposed as: ui uj − u ¯i u¯j = Lij + Cij + Rij , where Lij = u ¯i u¯j − u¯i u¯j is called the Leonard stress, ¯j ¯i u′j + u′i u¯j − u¯i u′j − u′i u Cij = u is the cross stress, and Rij ≡ u′i u′j − u¯′i u¯′j is called the subgrid-scale (SGS) Reynold stress. Rather involved expressions resulting from acting with the filtering operator may be considered both an advantage and a disadvantage of the filtering approach. A major advantage of the filtering approach is that it contains the possibility of taking into account transitional scales between the resolved and unresolved because of the generality of Eq. 3.19. An obvious drawback as a result is that an actual subgrid-scale representation becomes rather involved in a complete form, and thus makes it difficult to treat in practice. 4

Concepts of scale separation

A general formulation for subgrid-scale parameterization was presented by three different approaches in the last section. Each approach elucidates the issue of the scale separation in a different manner. The grid box-averaging approach is perhaps the most straightforward at first sight. Explicit numerical-model values are defined by a grid-box average. By carefully writing a numerical discretization of a model, we have obtained a formal expression for the subgrid-scale processes as well. However, this formal expression does not provide useful implications for (or the philosophy behind) the scale separation. The parameterization problem is introduced purely as a part of the issue of developing a numerical algorithm: processes below the scale of the grid box must be treated in a special manner. In particular, under this approach, the issue of a scale gap does not play a role at all.

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Part I: Basic parameterization concepts and issues

Probably the most important message from the grid box-averaging approach is in suggesting a close link between the parameterization problem and the numerical formulation for a physical model. Specifically, the grid box-averaging procedure must be modified depending on the numerical algorithm adopted in the model. In other words, the notion of a grid-box average cannot be adopted in a literal sense in general. A specific expression for the apparent source term must also be modified accordingly. In this respect, this approach even suggests that the parameterization problem is part of the development of a numerical algorithm. In contrast, the last two approaches introduce the parameterization problem independently of any numerical algorithm. Thus, even the notion of subgrid becomes elusive. A numerical algorithm is introduced only a posterori. In order for the numerical algorithm to be consistent with the already-defined large-scale equation, the grid size LG must be sufficiently smaller than a typical large scale ΔX. Thus: ΔX ≫ LG .

(4.1)

On the other hand, the scale separation suggests ΔX ≫ Δx. However, nothing can be said about the mutual scales between the grid size and the subgrid scale. The grid size may even be taken smaller than a typical subgrid scale without any logical contradiction. In this manner, we realize again that the concept of a grid box itself can even be misleading in the context of subgrid-scale parameterizations. The argument developed here works well when a flow is laminar. In such a case, a characteristic scale ΔX of the flow can be identified more or less straightforwardly. In order to simulate such a flow, the grid size LG must be chosen smaller than the scale of interest Δx, i.e., ΔX > LG . The convergence of the solution can be verified by decreasing the grid-box size LG to the regime of Eq. 4.1. Unfortunately, once a flow becomes turbulent, it is no longer possible to describe a well-defined numerical convergence of a solution. Instead, we find more and more fine-scale structures realized as the resolution ∼ 1/LG is increased. In a turbulent situation, the notion of the numerical convergence is valid only in a certain asymptotic sense. This further leads to a question of reliability of fine-scale details close to the grid size. This last question, which seeks to know the effective resolution of a given numerical model, would probably be better distinguished from the notion of the typical resolved scale ΔX introduced here. The asymptotic expansion approach best elucidates the concept behind the scale separation, at least in a very formal manner. Introduction of

Scale separation

91

a small parameter ǫ explicitly takes care of the issues associated with a distinction between the large and the subgrid scales. This parameter marks a certain existence of a scale gap under the asymptotic approach. The notions behind the asymptotic expansion, and implications for the scale separation as well as the scale gap will be discussed in the next section. The filtering approach is perhaps the most technical of the three approaches considered. Arguably, a wide freedom of choice of filter functions makes this approach highly flexible. The absence of scale gap in a particular system can be much compensated for by an adaptation of a sophisticated filter function, as well as by careful consideration of additional terms arising from a formal application of filtering. However, those sophistications may come at the expense of clarity, and tend to obscure the key issue of scale separation. Overall, a major advantage of the asymptotic approach is the ability to present a subgrid-scale equation, Eq. 3.15, with an appropriate approximation. Although the grid-box average and filtering approaches can also provide a subgrid-scale equation as a difference between the original full equation and an averaged equation, the resulting equations in those cases retain various additional terms that can be neglected under an asymptotic expansion. The large-scale advection term is a notable example. 5 5.1

Asymptotic limit Subtleties of the notion of asymptotic limit

So far, it has been suggested that the concept of scale separation is probably best understood as a type of asymptotic limit. Thus, the proposal is to base the notion of the scale separation on the asymptotic limit to Δx/ΔX → 0. The notion of the asymptotic limit is rather subtle, and some good presentations can be found in basic textbooks such as Bender and Orszag (1978) and Olver (1974). Pedlosky (1987) wrote his textbook on geophysical fluid dynamics entirely from the perspective of asymptotic expansion. For interpretations of the standard mass-flux formulation from the perspective of asymptotic expansion, see Yano (1999) and Yano (2003a). Here, a few key points should be emphasized: most importantly, it is necessary to clearly distinguish the notions associated with the two words “asymptotic” and “analytic”. When we take an analytic limit to, say, ǫ → 0, it means that we literally take the parameter ǫ infinitesimally small. From the point of view of numerical calculations, we have to take the value of ǫ

92

Part I: Basic parameterization concepts and issues

as small as possible in order to obtain the answer correctly. However, the asymptotic limit is something else: the asymptotic limit to Δx/ΔX → 0 means that we take the ratio Δx/ΔX very small, but maintain it as finite. The best known example in fluid mechanics is when taking an asymptotic limit of vanishing viscosity. Under this asymptotic limit, we can ignore the effects of viscosity almost everywhere in the fluid, and take the fluid as if perfectly inviscid, except over a very thin (indeed, with an asymptotically vanishing thickness) boundary layer, where the fluid must be taken as viscous. In other words, the effects of viscosity do not entirely vanish when the asymptotic limit of vanishing viscosity is taken. By the same token, we may take an asymptotic limit to ΔX/Δx → ∞, but it does not mean that we analytically consider that the domain over a large-scale point (X, Y ) as infinitely large as measured by subgrid-scale coordinates. It is only asymptotically large, but not literally infinite. For example, Lumley (1965) even argues that a factor of 3 is enough in order to justify such an asymptotic limit. The idea of asymptotic expansion is based on that of writing down variables and equations ranked by the power of a parameter ǫ assumed to be asymptotically small, as developed in Sec. 3.2. The asymptotic expansion, however, does not say how small the parameter must be. The situation can be confusing, because often we reset this small parameter as ǫ = 1 in the end. It was said in the beginning that the limit Δx ≪ ΔX would be taken, but then after all the mathematical manipulations, it could be said that it is not disadvantageous in practice to take it back to Δx ∼ ΔX. To reiterate, the idea of asymptotic scale separation resides in separating a total variable ϕ into two components, that belonging to the large scale ϕ¯ and that belonging to the subgrid scale ϕ′ . Such a separation is rather formal, but this formality is useful as long as the derived asymptotic expansion is useful for describing the whole system in an efficient manner. Such an efficient description is the goal of subgrid-scale parameterization. As it turns out, asymptotic expansions are often useful and are valid over a much wider range than intuitively expected. In fact, an asymptotic expansion is considered to be powerful if the result still works well even when the expansion parameter is set to ǫ = 1. In many situations, we find that is the case. A particular example is found in Fig. 6 of Yano (1992). Here, an asymptotic expansion is essentially (without going into the details) made assuming a certain smallness of s. The plots show that the asymptotic-expansion solutions (solid lines) fit well to the full numerical solutions (dashed lines) for the whole range of s.

Scale separation

93

To some extent the asymptotic limit could even be considered as a kind of thought experiment: we just suppose that the large scale and subgrid scale are well separated, and begin all our mathematical manipulations following this hypothesis. The existence of a clear spectral gap between the large and subgrid scales would be a robust basis for this approach, but it is not an absolute prerequisite. It is rather a matter of convenience to separate between the large and subgrid scales. As a result, the validity of an asymptotic expansion cannot be automatically measured by any observational estimates of the expansion parameter, for example, ǫ ≡ Δx/ΔX. Rather, its validity can be tested only a posterori by numerical validations. It may fail terribly even with an apparently small ǫ, but we may obtain an excellent result even when ǫ is not small at all by any conventional measure. Consequently, under an asymptotic expansion, the notion of the scale separation can only be qualitative, or even merely nominal. The validity of the notion can be verified only a posterori from the operation of the model, not by looking for a spectral gap, for example. 5.2

Limits of the scale-separation principle

In order to see the limits of parameterizations based on scale separation more clearly, let us first address a common misunderstanding. Suppose that we want to parameterize propagating squall-line systems. Because each squall line moves with time, a concern arises: what should we do if a given squall line moves out from one grid box and into another? Here, it must be emphasized that under the asymptotic scale-separation principle, such a situation never arises. A propagating squall line would always remains within the same grid box, or more precisely never move away from the original large-scale grid point (X, Y ). By taking an asymptotic limit ΔX/Δx → ∞ the grid box is infinitely large, from the point of view of subgrid-scale phenomena, and thus regardless of how far the squall line propagates it is never able to move away from the domain. It is often argued that parameterization is fundamentally statistical (cf., Sec. 8). In order to justify a statistical description of the squall-line system, we should have many squall lines within a single grid box. The goal of parameterization is not to describe the contribution of any single squall line to the large scale, but is only concerned with the ensemble effect of many squall lines on a large scale. However, it to wise to recall the subtleties of the asymptotic notion just discussed: “many” need not be very many, and even just one squall

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line in a grid box could be enough. In all likelihood, the most important implication of the scale-separation principle is that we should also find another squall line in the next grid box so that the continuity of a squallline dominant environment is guaranteed. In this case, propagation of a train of squall lines would be successfully parameterized under the scaleseparation principle. This type of situation may be most vividly demonstrated by a simulation of mesoscale organization under super-parameterization by Grabowski (2006) (cf., Ch. 2, Sec. 2.3), that explicitly solves a system analogous to Eq. 3.15 without introducing a parameterization. See specifically his Fig. 3: here, propagation of mesoscale organization is simulated by using only eight large-scale grid boxes over a horizontally one-dimensional periodic domain. Each grid box represents convective-scale variability by using the method of super-parameterization and by assuming a horizontal periodicity for solving all the subgrid-scale equations analogous to Eq. 3.15. In other words, individual convective elements simulated by super-parameterization never move away from one large-scale grid box into another. Nevertheless, the propagation of mesoscale organization (which could be interpreted as a modulation of convective variability for the sake of the argument here) is successfully simulated, because the adopted super-parameterization formulation is consistent with the scale-separation principle, as formulated in Sec. 3.2. However, if one really wants to represent an isolated squall line in a grid box (such that there is no squall line in neighbouring grid boxes), then one is really looking for a subgrid-scale parameterization beyond scale separation. That issue will be addressed in Vol. 2, Ch. 19.

6

Timescales

Discussions so far have focused on the spatial-scale separation. The large scale has been defined in terms of a spatial averaging or filtering, simply assuming that only a long timescale remains after spatial averaging. However, the issues of timescales do warrant special attention, because we can define more timescales than a simple distinction between large and subgrid scales. The following distinctions should be considered important: i) A characteristic life-cycle for convection: this concept roughly corre-

Scale separation

95

sponds to what is called the eddy-turnover timescale in turbulence studies. Shortness of this timescale compared to the large timescale (as defined in iii) below) justifies the introduction of a steady-plume hypothesis, to be discussed in Part II. ii) A characteristic timescale over which convection modifies a large-scale thermodynamic state: this timescale is called the “convective timescale” by Arakawa and Schubert (1974) in proposing their convective quasiequilibrium hypothesis. However, this timescale should not be confused with the scale of a convective life-cycle defined above in i). This point is sometimes mistaken (see Adams and Renn´ o (2003), for example, and compare Yano (2003a)). For this reason, this timescale may be better called the “convective response” timescale. Importantly, this timescale is the same as that characterizing the response of large-scale forcing to convective activity, under the hypothesis of convective quasiequilibrium. This hypothesis is the topic of the next chapter. iii) Convective adjustment timescale: some convection parameterizations such as the Betts (1986) scheme assume relaxation towards a predefined equilibrium state with a fixed characteristic timescale. This timescale may be called the “convective adjustment timescale”. The concept is similar to the convective response timescale, but is defined in a more specific context. iv) A characteristic timescale on which the large-scale thermodynamic state evolves (i.e., the so-called “large-scale timescale”): it is important to realize that when convective quasi-equilibrium is satisfied, this last timescale is much longer than the characteristic timescale for largescale forcing. Here, three timescales (life-cycle, convection response, and adjustment) are associated with convection, and two (forcing and evolution) associated with the large scale are identified. The convective response timescale is equal to the large-scale forcing timescale when convective quasi-equilibrium is satisfied. Thus, we should not naively interpret that scale separation suggests that every scale associated with subgrid scales is much shorter (or smaller) than those associated with the large scales. Here is a clear exception. Under convective quasi-equilibrium, the actual timescale for large-scale evolution is much longer than that for the forcing and the convective response, and thus a separation of two timescales is established. Although there is no attempt to establish the order of the three convec-

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tive timescales identified above, it is possible that the convective life-cycle is much shorter than the convective response timescale. In other words, several convective life-cycles may be required in order to establish an equilibrium state by convection responding to large-scale forcing. More specifically, it should be realized that the convective life cycle has a timescale associated with a single convective event. On the other hand, the convective response timescale is that of the collective effect by an ensemble of convection under a given large-scale state, and that this collective response is measured over a large scale (or more specifically over a grid-box average). A simple corollary is that the convective response timescale becomes longer as the large-scale (or grid box size) in consideration becomes larger. Note that large-scale forcing usually becomes weaker for larger scales, and thus its associated timescale is longer. This last point has a very curious implication, because as the space-scale separation between convection and the large scale is better established, the timescale separation (i.e., quasiequilibrium state) is less satisfied. Thus, the space- and timescale separations must be carefully distinguished. 7

Fractional area occupied by convection

An important quantity in convection parameterization is the fractional area σc occupied by convection over a given large-scale ΔX or a grid box LG . This fractional convective area may be considered as another measure of a scale separation. If a characteristic scale of convection, which may also define a characteristic size of convection, is comparable to the grid-box size, then the area occupied by a single convective element over a given grid box would be comparable to the size of the grid box itself. Consequently, a small fractional area occupied by convection is a prerequisite for guaranteeing a space-scale separation between convection and the large scale. The mass-flux formulation, to be introduced in Part II, is developed under an asymptotic limit of σc → 0. This asymptotic limit can be considered a variant of the multiscale asymptotic argument developed in Sec. 3.2. Here, however, a subtle difference between the two asymptotic limits Δx/ΔX → 0 and σc → 0 must be kept in mind. If there are N convective elements of size Δx within a large–scale of size ΔX, the two quantities are related by:  2 Δx . σc ∼ N ΔX

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Thus, these two asymptotic limits are equivalent as long as the number N of convective elements remains of the order unity. On the other hand, when N is very many, a situation in which Δx/ΔX → 0 but σc ∼ 1 could be found. Thus, the two asymptotic limits may not necessarily agree. Such a special situation is considered by Yano (2012). Here, the continuity of the solution already developed in Sec. 5.2 can again be emphasized: it is important to realize that the fractional area σc for convection is a large-scale variable that characterizes subgrid-scale processes. This quantity should change smoothly from one grid box to the next in a manner that is consistent with a scale-separation principle.

8

Statistical nature of parameterization

Here is another confusion occasionally found in the literature and more often in informal settings. It is often said that parameterizations are fundamentally statistical, and the scale separation is invoked as a basis for justifying such a statistical description of parameterization. Such an idea is already suggested in Sec. 2.1. The idea is stated in a more concise manner in Kuo (1974): The grid area A is taken as much larger than the area occupied by a single small-scale system such as a cumulus cloud and its surrounding descending region such that a large number of cloud cells are included in A. The exact location and time of occurrence of the individual cells are considered as unknown from the large-scale point of view, and a random distribution in space A and time δt [“large-scale” time step as designated by Kuo] can be assumed such that their average transport of sensible heat and release of latent heat in A and during the time interval δt are taken as determinable from the large-scale flow variables. The last point mentioned above is a necessary requirement for the feasibility of a parametric representation of the net influence of small-scale cumulus convection and other subgrid-scale processes on the large-scale system.

After presenting a basic formulation akin to what will be presented in Ch. 7, Sec. 2, he summarizes: According to the equations given above, our parameterization problem for large-scale flow is simply to strive to find expression for the statistical contribution.

This claim has several serious implications. First, by pushing a certain statistical description for subgrid-scale processes, this claim also pushes

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the basic notion of scale separation further. The scale separation discussed so far is only concerned with a difference of characteristic scales between subgrid and large scale. There may be only one cumulus element within a grid box and this would be consistent with a scale separation as long as this element is much smaller than the grid-box size. On the other hand, in order to justify a statistical description of convection, the number of cumulus convection elements within a grid box must be many, because otherwise a statistics can not be developed. Clearly this is a stronger statement than scale separation (cf., Ch. 4, Sec. 5.2). General statements for subgrid-scale parameterization were derived above in three different ways, but without introducing any notion of statistics in any of those derivations: a subgrid-scale parameterization can be formulated without introducing any statistical notion (such as ensemble average; see immediately below). This point would be better revealed following a specific formulation for convection parameterization based on mass flux in Part II. Finally, the notion of “statistics” is often not well defined in this context. The notion often remains philosophical and vague without being invoked in any concrete manner in the actual development of the formulation. For example, after the above discussions, Kuo (1974) introduces the notion of the average half-life of convection in his formulation, although it would be more reasonable to expect that such an elementary timescale would be eliminated by a statistical procedure (cf., Sec. 6). The only elaboration found in the literature along these lines is to reinterpret the averaging procedure (either as grid-box average, multi-scale asymptotic expansion, or filtering) as an ensemble average. As long as the number of cumuli within a grid box is large, the averaging may be reinterpreted as that for an ensemble. However, in the literature, the advantage of considering an ensemble is often not clear in convection parameterizations. This may be contrasted with turbulence parameterizations, where the notion of an ensemble average is more explicitly invoked. The argument so far does not exclude the possibility that a robust formulation for parameterization may be provided by a certain statistical description. However, such a statistical theory for convection is still to be fully developed. It remains to be seen how a convection parameterization can be developed based on such a statistical theory.

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Concluding remarks

This chapter has attempted to review the notions behind scale separation. This concept is often considered a basis for developing a subgrid-scale parameterization. This chapter has attempted to tone down this common notion by not discussing or even suggesting how a parameterization can be developed based on this concept. A very general statement of the subgrid-scale parameterization problem is derived in three different ways. By going through these exercises, the intention was to show what the notion of scale separation behind each derivation is. This author prefers to interpret the notion of scale separation in the light of an asymptotic expansion. In particular, the mass-flux formulation to be introduced in Part II could be interpreted as an example developed under an asymptotic expansion. However, the two alternative perspectives are also equally valid, and a filtering-based interpretation is widely accepted in the meteorological community. For this reason, Vol. 2, Chs. 20, 25 and 28 will adopt a filtering approach for discussing stochasticity, the subgrid-scale distribution density, and applications of Lie algebra to parameterizations, respectively. 10

Bibliographical notes

Overviews of filtering methods are provided by Leonard (1974), Lilly (1967), and Reynolds (1990). Ch. 13 of Pop (2000) also provides a good introduction on this subject. Mandelbrot (1982) remains the best introduction for fractals, though for further references, see Sec. 2.5. A good introduction for numerical computations of geophysical flows is found in Durran (1999).

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Chapter 4

Quasi-equilibrium

R.S. Plant and J.-I. Yano Editors’ introduction: Chapter 2 introduced the notion of quasi-equilibrium as a key concept used to develop parameterizations in current use. This chapter explores that notion more comprehensively. Although quasiequilibrium is very frequently invoked, there are many different interpretations and extensions of the idea that appear in the literature and these may be relevant for many different purposes. Unfortunately, few authors at the present time are taking care to explain exactly what they mean by quasi-equilibrium in their articles. This has led to a degree of confusion, misinterpretation, and even misuse of a key concept. This chapter aims to examine systematically the various interpretations on an equal footing. It explains what the different interpretations are, what assumptions are required for each interpretation to be valid, and why and when those distinctions are important.

1

Introduction

As discussed in Ch. 2, quasi-equilibrium is considered an important basis for constructing subgrid-scale parameterizations along with the concept of scale separation. Scale separation was examined in Ch. 3, and this chapter will now examine quasi-equilibrium. In short, in order to justify many forms of parameterization, the process in consideration must be of much smaller scale than those explicitly resolved in a model (i.e., scale separation), and at the same time, the process must be approximately in equilibrium with the resolved large-scale processes (i.e., quasi-equilibrium). In particular, in terms of convective quasi-equilibrium (CQE), as originally proposed by 101

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Arakawa and Schubert (1974), it is assumed that the convection can be treated as almost in equilibrium with the large-scale processes. In examining the scale-separation concept in the previous chapter, a rather cautious attitude was taken towards the validity of the concept. The same cautious attitude is also taken here for the concept of quasiequilibrium. The subtleties behind the concept of scale separation were described in the previous chapter. In this chapter, it will be emphasized that the concept of quasi-equilibrium is not only somewhat subtle, but often also somewhat ambiguous in various contexts for interpreting atmospheric processes. Thus, the focus of this chapter is to elucidate the meaning of convective quasi-equilibrium for various atmospheric contexts. In doing so, the basic notions behind Arakawa and Schubert’s (1974) formulation of CQE will also be examined. A full definition of CQE for use as a mass-flux parameterization closure hypothesis will be presented in Ch. 11. The starting point is to consider the meaning of “equilibrium” as a word, including its etymology. The subsequent sections discuss the various interpretations of quasi-equilibrium that have been applied in parameterization developments. 2 2.1

What is equilibrium? Etymology

The etymology for “equilibrium” is found in the Latin aequilibrium, which consists of two stems, aequi and libera, meaning weight and balance, respectively. Thus, etymologically speaking, equilibrium suggests weighted balance. In this respect, the concept of “equilibrium” is very close to “balance”, but implies something more. However, it is curious to note that French, for example, has only one word ´equilibre for referring to both “equilibrium” and “balance”. The concept of equilibrium is fundamental in thermodynamics. It seems fair to say that the concept of thermodynamic equilibrium, in turn, was developed from a tradition of French authors such as Carnot and Clausius. As noted, the French word ´equilibre essentially corresponds to both “equilibrium” and “balance” in English. It may even be something of a historical accident that ´equilibre thermodynamique was translated into English as “thermodynamic equilibrium”. According to the Oxford English dictionary, “equilibrium” dates back to 1608 in English, and its earliest textual references from around the 1660s onwards are clearly scientific. On

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the other hand, the word “balance” has been known in English since the 13th century. The word equilibrium has a more mystifying power in English than in French, and so perhaps does the notion of “convective quasi-equilibrium” (CQE). For this reason, we extensively examine the connotations behind the concept of thermodynamic equilibrium in the next section. This will be the first step in gauging how these connotations have influenced thoughts on the concept of CQE. In current English scientific language, the word “balance” is used for dynamical rather than thermodynamical concepts. For this reason, it will also be pertinent to review dynamical balance as a counterpart to thermodynamic equilibrium (Sec. 4). The word “balance” is associated with fewer additional connotations than “equilibrium”. Thus, if CQE had originally been simply coined as “convective quasi-balance” instead, its interpretation may have been less controversial. 2.2

What is Arakawa and Schubert’s (1974) convective quasi-equilibrium?

Arakawa and Schubert’s (1974) convective quasi-equilibrium is introduced in the context of the problem called “closure”, which will be discussed in depth in Ch. 11. In short, closure refers to a condition for defining the total strength (amplitude) of convection within a parameterization. Leaving aside the technical formulation for Ch. 11, Arakawa and Schubert’s (1974) convective quasi-equilibrium essentially states that the parameterized convection strength can be obtained by assuming the convection to be close to a state of equilibrium (or quasi-equilibrium) with the large scale. The idea of such an equilibrium is defined by a balance between a largescale forcing FL and a convective response Dc (for any physical variable); i.e., FL − Dc = 0.

(2.1)

The given FL can then be used for determining Dc , which itself depends on the convection strength. Thus, the latter is defined by this principle from Dc , thereby closing the parameterization. In this manner, Arakawa and Schubert’s (1974) convective quasiequilibrium can be interpreted in terms of both equilibrium and balance. The goal of this chapter is to asses the merits of taking either interpretation by discussing the various implications that may follow.

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Note that the above condition can be introduced for a budget of any physical variable, and it essentially amounts to dropping a time tendency term in the given budget equation. Such a condition is often more conveniently introduced in terms of a vertically integrated quantity, as discussed in Ch. 11. Again, leaving aside the details for the present, Arakawa and Schubert (1974) more specifically adopt the cloud-work function A as such a quantity, which may be considered a generalization of CAPE (convective available potential energy; cf., Ch. 1). 3 3.1

Thermodynamics What is thermodynamics?

Thermodynamics is inherently interested in equilibrium. For this reason, in order to understand what equilibrium is, we must first understand something about thermodynamics. Arguably the most successful application of classical thermodynamics is to the heat engine problem. In this problem a heat engine cycle is defined as progressing through a sequence of equilibrium states, with each transition being necessitated as a response to an external condition that is imposed on the engine. For example, the heat engine may be subjected to an externally specified temperature which is changed sequentially in discrete steps. On each occasion that the external condition is changed, a new equilibrium state of the system is calculated, the state of interest being that which would be reached after waiting for an indefinitely long time. The sequencing of external conditions is continued in such a way as to progress through a closed cycle of such states. Then the obtained useful energy (called “work”) is evaluated, this quantity being the main interest of the heat engine problem. In performing a heat engine calculation, the question of how long it takes to complete the cycle is not considered. How fast one can turn the engine is clearly a question of enormous practical importance, but a classical heat engine problem (such as the Carnot cycle) does not pose the question. In other words, in classical thermodynamics the concept of time remains implicit.

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Equilibrium as a thermodynamic concept

As an example of a process leading to thermodynamic equilibrium, take two boxes (with the same heat capacity, for simplicity) with two different temperatures, T1 and T2 . Bring these two boxes into contact, but with no interactions with the outside world: i.e., assuming that interactions happen only between the boxes. Heat is exchanged between the boxes in affecting a transition towards an equilibrium. More specifically, heat is transferred from the box with higher temperature to the box with lower temperature and the process continues until the temperatures of the two boxes are equal. In this case, the final temperature is the mean (T1 + T2 )/2 of the original temperatures and the final state is called an equilibrium. The example just given, albeit an extremely simple one, is a typical problem considered in thermodynamics: define an equilibrium thermodynamic state of a system in contact with an external system, under given constraints (e.g., the volume remains constant, and the initial temperatures are given). In the example, one of the boxes is the system of interest, and the other box is an external system. In many cases the external system is considered to be much larger than the system of interest (such as in the heat engine problem). It is sometimes called a “thermal bath” when the temperature is fixed, and in that case the final equilibrium temperature of the system would be the same as the external temperature. Carnot’s Reflections (1824) are widely recognized as a key foundation of thermodynamics, although they had to be more carefully formulated and rescued from obscurity by Clapeyron (1834). Interestingly, Clapeyron’s article (1834) does not speak of equilibrium, although Carnot (1824) uses the concept frequently without ever defining quite what he means by it. The closest Clapeyron comes is in the description: “an equilibrium considered as destroyed by any cause whatever, by chemical action such as combustion, or by any other.” This, and the broader sense of his article, is certainly consistent with the interpretation given above. Importantly, the thermodynamic equilibrium is always a motionless state, except for an obvious situation such as a uniform translation. From a thermodynamic point of view, dissipation plays a key role when dealing with motion. For example, if a coffee cup is stirred then a rotating flow arises, and persists until dissipated away by friction so that the system recovers a quiescent state of equilibrium. This is distinct from mechanics, which prefers to take a dissipationless limit for describing many motions

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(this will be expanded upon in Sec. 4.2). We may even perpetualize motion, for example by imposing a temperature gradient on a given system, either horizontally or vertically. However, from a thermodynamic point of view, this simply forces the system perpetually away from thermodynamic equilibrium. As the discussion above suggests, the application of a thermodynamic equilibrium to any real physical system is necessarily approximate. Therefore, while time invariance of macroscopic properties may be a useful practical description of a system in a state of statistical equilibrium it does not provide an appropriate definition of that state, since the macroscopic properties will undergo fluctuations for any system of finite extent. Rather, in order to define the statistical equilibrium state for a finite system, one may consider that “a system is in a condition of equilibrium when the information one has about it has reached a time-independent minimum” (Andrews, 1963, p. 24). In such a condition one can derive probability distributions from a variational principle (cf., Sec. 3.3) such that the extent of the fluctuations becomes a known aspect of the equilibrium state. In this way, a time-independence remains inherent to the concept of equilibrium. It is often desirable to try to apply the equilibrium concept without necessarily waiting a very long time for a complete adjustment to a state of time-independence. As discussed by Landau and Lifshitz (1980) for instance, complete adjustment can be a particularly awkward restriction of the concept in practice because the timescale for such adjustment will increase with the system size. It is therefore often necessary to invoke a partial equilibrium in which the probability distributions of observables take functional forms derived from the limit of a complete statistical equilibrium but containing parameters that are considered to change slowly in space and time. The notion of partial equilibrium can also be applied in a local sense, considering that thermodynamic equilibrium is established at any individual macroscopic point, although the system as a whole (such as the climate) never reaches a thermodynamic equilibrium because it is perpetually perturbed by external forces. The need for such a local equilibrium concept to derive the standard equation set for geophysical flow is made explicit (for example) in Ch. 1 of Salmon (1998). 3.3

Variational principle

The main goal of thermodynamics is to describe equilibrium states in as general a manner as possible. For this purpose, a variational principle

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is often invoked, in which the thermodynamic equilibrium is defined by a minimization (or maximization) of a certain thermodynamic quantity (potential). For the example of joining two boxes in the previous subsection, the equilibrium state can be deduced by maximizing the entropy. The textbook by Kondepudi and Prigogine (1998) emphasizes the variational principle. A more concise description of the principle can be found, for example, in Sec. 7.4 of Adkins (1983), and in Ch. 2 of Chandler (1987). The strength of the variational principle is that it enables one to determine a thermodynamic equilibrium state in many situations with a few general rules. Due to its generality, this principle can also be applied to many other physical problems. In other words, the concept of equilibrium can be generalized by invoking the variational principle. Indeed, it was a consideration of equilibrium in its broadest sense that led Maupertius, Euler, Lagrange, and others to introduce the variational principle into classical mechanics (Hepburn, 2007), contemperaneously with Johann Bernoulli’s characterization of static equilibrium through an optimality principle (Hildebrandt and Tromba, 1996). Ch. 5 of Salmon (1998) reviews applications of the variational principle to geophysical flows. However, a major limitation of the variational principle in thermodynamics is that it restricts attention to the most likely state of a given system. In other words, the problem is formulated in such a way that a state of maximum probability is sought, often equated with a state of maximum entropy. In many cases, applications of a variational principle may require an assumption that a large number of ensemble members can be associated with the system of interest. Clearly the maximum entropy principle makes this assumption: only if the ensemble size is large enough can we assume that the most likely state will correspond to actual realizations (cf., Sec. 5.2). This discussion leads to a statistical notion of equilibrium that will be discussed further in Sec. 5. 3.4

Why equilibrium is a useful concept

The great strength of classical thermodynamics resides in excluding the need for consideration of the time evolution of a system by focusing on the final, equilibrium state of a given system under given constraints. Within this framework, even a full initial condition of the system is not important as long as the constraints are well specified. In the above example with the two boxes, it suffices merely to specify the initial temperatures of the boxes; there is no need to ask further questions, such as the initial positions

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of the boxes and their volumes. This approach greatly facilitates the computations by focusing only on the final state, and removing the need to consider a complicated initial value problem for the complete evolution. In many practical applications, it is indeed only the final state that is of interest, especially if the given system approaches an equilibrium rapidly. The heat engine problem is a good example for making this point: in many practical engineering applications, a system can be considered to reach equilibrium almost instantaneously whenever an operation is performed. The transient adjustment process is not of interest. The geostrophic adjustment problem originally developed by Rossby (1938) can be considered a similar approach based on this concept of equilibrium. The problem asks for the final state of a flow under a geostrophic constraint, starting from an initial non-geostrophic state (assumed to be a state of rest in Rossby’s original problem) with a height anomaly (recall that Rossby’s original problem considers a shallow-water system). By invoking a principle of potential-vorticity conservation, the final state under geostrophy can be determined without solving a complicated initial value problem. Conceptually speaking, a synoptic-scale weather system can be considered to evolve under a continuous sequence of geostrophic adjustments. The process of geostrophic adjustment is fast (e.g., a few hours) relative to a typical timescale (e.g., a day and beyond) of interest in traditional synoptic weather forecasts. Hence, there may be no need to consider explicitly the details of the adjustment process. This reasoning essentially leads to the idea of quasi-geostrophy. To some extent, the concept of convective quasi-equilibrium can be interpreted in a similar manner, as discussed in Sec. 4.3. Nevertheless, a limitation of the geostrophic adjustment concept should also be emphasized: it is not necessarily consistent with the cascade point of view based on geostrophic turbulence (Ambaum, personal communication, 2010). Certain issues beyond this metaphor of reality are discussed in Sec. 4.4. 3.5

Homeostasis

Homeostasis can be considered a biological counterpart to quasiequilibrium, generalizing the original thermodynamic concept. It refers to self-regulating processes in biological systems that maintain the constancy

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of properties such as acidity, salinity, and other compositional aspects of the blood, as well as body temperature, against changing external conditions. The concept especially refers to the ability of biological systems to maintain their stability against external perturbations such as abnormal food intake or a change of external temperature. Etymologically, “homeostasis” consists of two stems: the prefix homeo means “similar” or “like” in Latin, whereas stasis comes from the Greek meaning “standstill”. Thus, as a whole, the word can be translated as “quasi-equilibrium”. The concept was originally introduced by a physiologist, Cannon (1929, 1932). In his words: The constant conditions which are maintained in the body might be termed equilibria. That word, however, has come to have fairly exact meaning as applied to relatively simple physico-chemical states, in closed systems, where known forces are balanced. The coordinated physiological processes which maintain most of the steady states in the organism are so complex and so peculiar to living beings – involving, as they may, the brain and nerves, the heart, lungs, kidneys and spleen, all working cooperatively – that I have suggested a special designation for these states, homeostasis.

By the same token, although atmospheric convection may not be as complex as biological systems, it is far more complex than “relatively simple physico-chemical states”. Moreover, atmospheric convection is an open system, as are biological systems. In this respect, it may be more relevant to call it “convective homeostasis” rather than “convective quasi-equilibrium”. At a very conceptual level, probably the thermodynamic analogy interpretation of CQE is the best aligned with the concept of homeostasis, in the sense that it suggests stability of the system. This concept furthermore makes a very good counterpoint to the concept of self-organized criticality (SOC), which will be discussed in Sec. 7.2. However, a qualitative difference between convective homeostasis and biological homeostasis must be emphasized. In biological systems, homeostasis maintains an internal state (e.g., constant body temperature) regardless of the external conditions (e.g., how cold or warm the outside environment is). On the other hand, convective homeostasis must be environment dependent: the term may be a useful one in referring to a state that is uniquely defined by its environment and in which the stability of that state is maintained by its own self-regulation.

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Thermodynamic analogy

In order to treat a convective system by analogy to a thermodynamical system, we need to suppose that the convective system constitutes only a small part of the whole atmospheric system, with the latter working like a thermal bath. More precisely, we suppose that the given large-scale atmospheric state can be considered as a fixed external environment to the convective system similar to a thermal bath for a thermodynamical system. In this manner, the convective system is then slaved to the given large-scale environment. As a result, if the large-scale state changes (through some suitably slow process) by a forcing FL , then the convective state is considered to adjust almost immediately (as a fast process) in order to produce a response Dc which satisfies Eq. 2.1. This is the same concept as a heat engine adjusting itself to a new equilibrium as an external condition (its environment) is modified (cf., Sec. 2.1). It is important to note that cause and effect are clearly distinguished under this thermodynamic analogy: the large scale is regarded as the cause, and the convective system responds to any changes in the large scale as an effect. This may not necessarily be true for atmospheric convection as will be discussed later in Sec. 3.10. 3.7

Radiative-convective equilibrium

The notion of radiative-convective equilibrium originates from the context of a radiative transfer calculation within a static, one-dimensional atmosphere. Under such conditions, a stationary solution for the vertical profile of temperature T (z) can be sought, such that QR = 0,

(3.1)

where QR is the radiative heating rate of the atmosphere. The obtained radiative equilibrium state is typically unrealistic in the sense that the lapse rate in the troposphere is too steep and even convectively unstable, contradicting observations. For example, such a superadiabatic state was found in one of the original full radiative equilibrium calculations for the atmosphere by M¨oller and Manabe (1961). Phenomenologically, convection would arise in such a situation, and act to adjust the vertical profile to a stable configuration. Hence, for the problem of radiative equilibrium to be of practical interest it must be at least minimally modified into the problem of radiative-convective equilibrium, by adding

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the effects of moist convection to the above equation: QR + Qc = 0.

(3.2)

Here, Qc refers not only to diabatic heating (due to latent heating, both by condensation and freezing of water) but also to vertical transfers of heat associated with convective motions. Manabe and Strickler (1964) introduced a critical lapse rate into their calculations, adjusting the computed lapse rate to the critical value wherever the former exceeds the latter. They proposed to call this procedure “convective adjustment”, and to call the profile so-obtained the “radiativeconvective equilibrium” (RCE). Note that under this procedure, the convective heating Qc is completely implicit, and the term represents all dynamical and thermodynamical processes that were neglected in the radiative transfer calculations. Today, the concept of RCE has been rather taken away from this original context, and often refers to a final state obtained by integrating a CRM for an extensive period, without imposing any large-scale vertical motion (e.g., Cohen and Craig, 2006; Grabowski, 2003; Parodi and Emanuel, 2009; Stephens et al., 2008). In such applications, the radiative heating rate may be computed with a complex radiation code that is fully responsive to the simulated cloud field, but alternatively, it may sometimes be treated in extremely simplified manner: for example, by imposing a prescribed, fixed tropospheric cooling rate. In the latter case, the cooling rate is not necessarily typical of the rates in the tropical atmosphere. Indeed, the purpose of an investigation may be to assess the scaling of convective properties with the strength of the cooling (e.g., Parodi and Emanuel, 2009; Shutts and Gray, 1999). Finally, the problem can be further generalized to the situation in which the given atmospheric column is no longer static, but is subject to vertical motion. Vertical heat transport from the column-averaged vertical motion w must then be added to the above equation: ∂θ + QR + Qc = 0. (3.3) ∂z Here, θ is the potential temperature, and then the terms QR and Qc must be reinterpreted in terms of the potential temperature. Such a generalization may include simulations where w is imposed from the outset (e.g., Sui et al., 1994). Alternatively, w can be diagnosed based on the assumption that Eq. 3.3 is satisfied at every time step (e.g., Sessions et al., 2010, also see Sec. 4.5). −w

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Note that there is no ambiguity about the above generalizations of the concept of RCE as long as the system evolves to a configuration that remains perfectly steady with time, even with a non-vanishing vertical velocity w = 0. However, as soon as the system becomes time evolving, the concept of RCE suddenly becomes subtle, even elusive. The degree to which RCE is satisfied under time-evolving situations is not an easy question to answer. However, the basic assertion of CQE is, essentially, that Eq. 3.3 remains a good approximation even when the system is evolving with time. This issue will be discussed further in Sec. 4.5. 3.8

Convective neutrality

A more direct application of the thermodynamic analogy to convective quasi-equilibrium is to suppose that the atmosphere is always maintained at a state of convective neutrality by the large-scale processes. Conceptually, this could also be considered a result obtained by an application of a variational principle discussed in Sec. 3.3. For example, Emanuel (1987) adopted such a formulation in developing his linear theory for the MJO (Madden–Julian oscillation: cf., Ch. 5). More precisely, by adopting a two-level model for his thermodynamic descriptions, he sets the boundarylayer equivalent potential temperature θeb to be equal to a saturated version ∗ at a middle troposphere level. Thus: of its counterpart θem ∗ θeb = θem .

This gives a very simple condition for convective neutrality for lifting parcels. 3.9

Equilibrium control

Mapes (1997a) proposed an interpretation of CQE that is based on a liftingparcel argument. The interpretation is introduced by means of a contrast with an alternative principle known as “activation control”, which will be discussed in Sec. 7.1. Consider an air parcel which is being lifted upwards within an atmosphere that is conditionally unstable. As described in Ch. 1, Sec. 3.2, the parcel first experiences negative buoyancy due to the adiabatic cooling associated with lifting, until it reaches saturation. Latent heating effects may then be sufficient to overcome adiabatic cooling, such that the parcel begins to feel positive buoyancy, and is accelerated upwards.

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The initial energy barrier can be measured by the convective inhibition (CIN). Further parcel ascent converts potential energy into kinetic energy by following a downslope of the potential energy. A relevant question to ask is whether the generation rate of convective kinetic energy within the atmosphere is controlled primarily by “changes in amount of the downhill plunge” (equilibrium control) or by “the rate at which parcels are lifted over the activation energy barrier (i.e., CIN) by intense small scale lifting processes” (activation control). Mapes (1997a) points out that equilibrium control lies behind many conceptions for deep convection. His Sec. 4.1 lists six historical or observational points that he believes have led to the general adoption of the equilibrium control assumption, and then argues that the evidence for equilibrium control is susceptible to other interpretations. One such interpretation is discussed next. 3.10

Cause and effect in CQE

If the thermodynamic analogy discussed above applies literally to CQE, convection acts in direct response to an (assumed known) large-scale forcing in order to maintain a state of quasi-equilibrium (cf., Sec. 3.6). Although it must be accepted that convective activity does feed back on the forcing (for example, through cloud-radiative interactions), the forcing is essentially treated as an external constraint imposed on the convective system. However, the observed smallness of the tendency of CAPE, or the cloudwork function A in comparison with the strength of the large-scale forcing dA/dt ≪ FL does not in itself say anything about the causality. Mapes (1997a) points out that the same result would be entirely consistent with a complete reversal of the assumed causality. That is, instead of convection responding to the large scale, it could rather be that the large scale is responding to convective processes. The alternative point of view may be called “heating response control”. In order to illustrate the apparent plausibility of this alternative interpretation, Mapes (1997a) presented a Gedankenexperiment using a linear shallow-water model with a localized white noise forcing. The shallow-water system has a long history as an analogue model for the tropical atmosphere (e.g., Gill, 1980), while the white noise is designed to mimic random convective heating. Under this analogue model, CAPE is measured by fluctuations in the height of the shallow water. It was shown that the fluctuation of this analogue of CAPE becomes much smaller than the imposed forc-

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ing strength after times equivalent to a few hours. This happens because thermal anomalies generated by convective heating are rapidly smoothed out by gravity waves in the tropical atmosphere. As a result, a smooth temperature field is left behind once the gravity waves have readjusted the atmospheric thermodynamic structure in response to convective heating. Although this alternative interpretation is fully consistent with dA/dt ≪ FL , of course it does not follow that the equilibrium control picture is necessarily wrong. The actual causality may be dominated by equilibrium control, or heating response control, or more plausibly, through a genuinely two-way interaction between convection and its environment (cf., Sec. 5.4). Indeed, as Mapes (1997a) discussed, a suitable picture may depend on both the scale and the phenomena of interest: the equilibrium control picture has proved fruitful in seeking to understand aspects of global-scale behaviour, while at the other extreme, both observational and model-based case studies of individual convective storms are interested in exactly where and when a storm occurs and naturally adopt a heating response control picture. Perhaps neither picture is truly satisfactory on its own to address other questions and other phenomena. These arguments seem to imply that a more general framework is needed: one which reduces to heating response or equilibrium control within appropriate limits. A good first step would be to look for data in which the outcome of very many individual heating response-controlled events can be shown to produce an equilibrium control situation on a much broader scale. It could be fruitful to explore some recent large-domain, cloud system-resolving model data (e.g., Holloway et al., 2012; Liu et al., 2009; Shutts and Palmer, 2007) from this perspective. An important issue with the experiment of Mapes (1997a, 1998) is that the interpretation of the tropical atmosphere as being driven by white noise convective forcing is not consistent with observations. A simple application of asymptotic analysis in Yano et al. (2000) shows that in that case the CAPE power spectrum must be proportional to the square of the frequency. That is not what is observed. Yano et al. (2000) examined various alternatives. Among those, they showed that when the large-scale forcing FL is prescribed as in typical CRM simulations, convection actively responds in order to maintain the system close to the equilibrium as defined by Eq. 2.1. Slight deviations from this equilibrium behave as white noise, and as a result CAPE evolves as a Brownian motion. However, this is not what is observed either. Instead, the frequency spectrum of CAPE has the form of 1/f noise. This leads to

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the notion of self-organized criticality, to be discussed in Sec. 7.2. More importantly, we should recognize that Mapes’s (1997a) criticisms raise legitimate concerns about the validity (certainly about the range of validity) of the thermodynamic analogy-based interpretation of CQE, but they do not discredit the whole idea of CQE, especially if it is interpreted as a balance condition (cf., Sec. 4.3). The concept of heating response control, i.e., the large-scale dynamics responding to convective forcing, will be revisited in the context of the free ride principle in Sec. 4.5.

4 4.1

Mechanics Thermodynamics and mechanics

Before the advent of quantum mechanics and relativity, classical physics could be divided into two distinct disciplines: thermodynamics and mechanics (dynamics). The basic concepts of classical thermodynamics were reviewed in the last section. In contrast to thermodynamics, the main interest of classical mechanics is the time evolution of a given system. Perhaps the greatest success of classical mechanics is in explaining the movements of the planets and moons in the solar system. The degree of success in such an application is judged by the predictability (for example, precise timings of solar and lunar eclipses). Arguably, the concept of thermodynamic equilibrium as discussed in the last section is inherently foreign to mechanics, because motion is inherent to mechanics. Notions of equilibrium, however, began to play an important role in the development of mechanics in the 18th century (Hepburn, 2007) through a French school, perhaps most notably by d’Alembert, which led to the discovery of a variational principle by Lagrange (cf., Sec. 3.3). It seems fair to say that modern meteorology, originating from the Bergen school led by Vihelm Bjerkness in the late 19th century, developed as an outgrowth of classical mechanics with its main interest being the prediction of weather. For this reason, one might even argue that the notion of equilibrium is inherently foreign to modern meteorology. 4.2

Balance as a dynamical concept

The closest analogy to thermodynamic equilibrium in classical mechanics may be the concept of stationary or steady solutions. A steady solution

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occurs if an inertial frame exists in which all of the particles of a given system remain at rest. The notion can, furthermore, be generalized to a case in which a mode of movement of the system does not change with time. For example, a planet rotating around a star along a fixed orbit with a fixed period, can also be considered to be in a steady state. Steady solutions can be said to be subject to a balance condition. Under a perfect steady state with no motion, all the forces acting on a given particle sum to zero:  Fj = 0, (4.1) j

where Fj designates an individual force acting on the particle. In the example of the orbiting planet with steady motions, all of the forces acting on the planet remain perpendicular to the direction of movement and so are balanced along a given curved coordinate. Note that both Arakawa and Schubert’s (1974) convective quasiequilibrium (Eq. 2.1) and radiative-convective equilibrium (Eq. 3.3) take this form. Thus, the concept of quasi-equilibrium can easily be translated into that of dynamical balance. These balanced states are clearly similar to the thermodynamic concept of equilibrium in the sense that both can be characterized by a certain time invariance. However, the two concepts carry quite different implications, as illustrated in Fig. 4.1. Balance does not necessarily imply stability, although equilibrium implies stability. Balance is simply the identification of a state which may be of interest, with no consideration made as to whether and how that state may actually arise. On the other hand, equilibrium implies that the given system will arrive at the state of interest (assuming that the external conditions so allow) after certain transient adjustments. A good example to illustrate the distinction is a standing egg: we can make an egg stand upright after careful adjustments of its position. However, this state is hardly stable, and an egg would never reach that position spontaneously. Thus, the state is balanced, but not in equilibrium. Although convective quasi-equilibrium is most often considered in the context of a thermodynamic or statistical mechanics interpretation of equilibrium (cf., Sec. 6), some discussions based on the concept of dynamical balance can also found in the literature. An example is Emanuel’s (2000) discussion on the entropy budget in his Sec. II. However, in the context of convection, we should emphasize the role of dissipation as a major difference between the balance concept of classical mechanics and the equilibrium

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Fig. 4.1 Schematics to illustrate the difference between (a) thermodynamic equilibrium and (b) dynamic balance. Richer implications of thermodynamic equilibrium are seen (a relationship with the environment, between macroscopy and microscopy, and the role of the law of large numbers).

concept of thermodynamics (Sec. 3.2). Clearly, convection is subject to dissipation, although its importance may depend on the situation. The concept of dynamical balance can be generalized under the Hamiltonian formulation into various other conservative systems. However, the distinction between conservative and non-conservative systems, or alternatively between non-dissipative and dissipative systems is important in this framework as well: the Hamiltonian formulation is essentially developed for non-dissipative systems. Dissipative behaviour requires a more general geometric description such as the metriplectic approach reviewed by Guha (2007). Here, recall that a Hamiltonian system can be described in terms of the Poisson bracket (cf., Goldstein et al., 2002). The basic idea of the metriplectic approach

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is to include both a Poisson bracket and a symmetric (metric) bracket. Whether such a mathematical structure can also be used to study CQE is an intriguing possibility. An example application relevant to atmospheric science can be found in Bihlo (2010). 4.3

Balance condition and slow manifold

Thus, CQE may be alternatively interpreted as a dynamical balance condition. It is the authors’ belief that the concept of the “slow manifold” best elaborates this point of view. The slow manifold may be considered a generalization of quasi-geostrophic flows. However, care should be taken to note the a subtle difference between quasi-geostrophy and the slow manifold. Quasi-geostrophy implies an approximate solution to an exact system (i.e., geostrophy), whereas the slow manifold refers to where in phase space an exact solution actually resides, albeit after some filtering may have been applied. The concept of the slow manifold was originally proposed by Leith (1980) and revisited by Lorenz (1986) (see also Lorenz, 1992). An analogy between the ideas of the slow manifold and quasi-equilibrium was made by Schubert (2000). In his defence, Schubert does not use the terminology “slow manifold”, but it is easy to infer this concept behind his essay. Schubert (2000), more specifically, draws attention to the analogy between Arakawa’s original ideas of quasi-equilibrium and quasi-geostrophic theory: the CQE condition is considered to filter out the transient adjustment of a convective cloud ensemble in the same sense that quasigeostrophic theory filters out transient inertia-gravity waves. By extending this analogy, the state of CQE may be considered as analogous to the slow manifold. To illustrate his point, Schubert (2000) first considers a linear onedimensional shallow-water system with an exponentially decaying mass source of the form: α2 t exp(−αt).

(4.2)

The cases of α ≪ 1 and α ≫ 1 correspond respectively to slow and fast forcing, and the time-integrated mass source is normalized to unity by definition. Schubert (2000) shows that provided the forcing timescale is slow, the inertia-gravity wave mode appears only at an initial stage of the evolution starting from a stationary initial condition. At later times, only the geostrophic mode remains. One way to filter out the transient inertia-

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gravity waves a priori is to introduce a balance condition so that the wave modes never arise. By the same token, we might consider a balance condition being applied to a convection parameterization in such a way that fast convective adjustments are filtered out from the model evolution. Schubert (2000) argues that this is the original idea behind the CQE hypothesis. In their linear stability analysis of convectively coupled waves, Neelin and Yu (1994) and Yu and Neelin (1994) also emphasize the importance of the distinction between fast and slow modes. In their formulation with convective adjustment, the fast modes always damp with the fast convective timescale, thereby ensuring the maintenance of the equilibrium defined as a basic state of the model. On the other hand, the slow modes may be interpreted as explaining aspects of observed convectively coupled equatorial waves. These perspectives lead to further implications by analogy with the issues encountered in the original slow manifold problem as developed by Lorenz and others. 4.4

Slow manifold and Lighthill’s theorem

The main issue encountered with the concept of the slow manifold is whether it is actually possible to construct a system consisting solely of the slow timescale processes. If that is the case, then we can develop a fully self-consistent description of geophysical flow on a slow manifold. The same question can be asked of the concept of CQE, because it implicitly assumes that a self-contained description of large-scale flows is possible while leaving implicit the fast convective-scale processes. A more specific way of addressing this question is to consider a full system initialized only with slow modes (i.e., the system initially resides on the slow manifold, or alternatively is in a state of convective quasi-equilibrium), and to ask whether the system evolves in such a way as to remain on the slow manifold (or alternatively to ask whether the CQE condition remains satisfied). As discussed in the previous subsection using an example from Schubert (2000), the case with a linear system is relatively obvious: as long as the slow and the fast modes are orthogonal in phase space (as is the case for geostrophic and inertia-gravity modes, cf., Greenspan, 1968), the system remains on the slow manifold. However, once the system becomes non-linear, the question is far from trivial. Lighthill’s theorem casts light on this question.

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In general, Lighthill’s (1952; 1954) theorem as reinterpreted by the McIntyre school (Ford, 1994; Ford et al., 2000, and summarized in McIntyre, 2001) says that a system initialized under geostrophic balance (or another balanced condition) will spontaneously generate gravity waves. The theorem suggests that a slow manifold can exist only in a limited sense, and that it is not possible to construct a system purely consisting of slow modes (i.e., geostrophic modes, or Rossby modes). Recent work by Ring (2009) illustrates this issue more concretely. Consider an expansion for some variable A: A = A0 + ǫA1 + ǫ2 A2 + · · ·

(4.3)

in terms of the expansion parameter ǫ, which is a Rossby number. Thus, the expansion describes systematic departures from geostrophy. Note that Neelin and Yu (1994) perform a similar expansion in their linear stability analysis, taking the convective adjustment timescale to be their expansion parameter. In the specific case studied by Ring (2009), the expansion is performed for a shallow-water system on an f -plane. It is shown that even when a (fast) inertia-gravity mode is absent from the initial state (i.e., A2 (t = 0) = 0) A2 nonetheless grows so quickly that geostrophy breaks down due to the non-linearity of the system. In the final stage, the contribution of the inertia-gravity waves reaches 10% of the total non-zonal energy of the system. This level of contribution from the inertia-gravity ˇ waves is also comparable with an estimate from global data analysis (Zagar et al., 2009). A simple extrapolation of the above result into the context of CQE suggests that even a system initialized without explicit convective modes may rapidly and continually develop fast convective modes. In the parameterization context, this implication might seem rather pessimistic because it suggests that it is fundamentally not possible to keep the fast convective processes implicit and completely exclude them from the parameterization. It appears that the absence of the slow manifold in the strict sense is already well established in the literature. For this reason, a proposal has been made to replace the original concept of the slow manifold by the “slow quasi-manifold”, a system consisting primarily of the slow modes, but allowing for finite departures within a stochastic layer (Ford et al., 2000). The notion of the quasi-manifold is based on an anticipation that the departure is small enough that the system remains within a “fuzzy” zone close to the original manifold. A complementary view of this situation is that there are infinitely many, slightly different slow manifolds that could

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be constructed, and that it is their non-uniqueness that leads to the notion of a quasi-manifold (e.g., Cox and Roberts, 1994). Applying the analogy to convection suggests that it may be important to take into account a “fuzzy” zone arising from convective-scale fluctuations that interact more directly with the large-scale processes in order to formulate CQE in a more robust manner. In this respect, Neelin and Zeng’s (2000) quasi-equilibrium tropical circulation models (QTCM), mentioned in Sec. 4.3, may arguably be considered as a slow manifold formulation for tropical large-scale circulations, at least in a conceptual sense. In their case, there is no specially designed filtering procedure or initialization performed in order to maintain the system on a slow manifold, but rather the fast modes are effectively eliminated by the damping provided by an adjustment form of the convection parameterization. This construction is almost guaranteed to avoid a problem with Lighthill’s theorem, although gradual leaking from fast modes into slow modes by non-linear interactions could still be an issue. For a more general examination of the issue, we suggest that the convective energy cycle system discussed in Ch. 11, Sec. 10 might fruitfully be considered under a coupling with large-scale dynamics. If all linear perturbations around a CQE solution are damping, as is the case with QTCM, the system is almost guaranteed to stay on the slow manifold. However, an important question to be addressed is whether, if some perturbations turn out to be exponentially growing, the slow manifold is no longer a welldefined concept, but at the very least a “fuzzy” zone away from the strict slow manifold must be considered. The concept of self-organized criticality (SOC), to be discussed in Sec. 7.2, would appear to provide a mechanism for establishing a “fuzzy” zone that is relatively narrow. 4.5

A link to the notion of “free ride”

Despite the caution expressed towards the end of Sec. 3.3, it turns out that radiative-convective equilibrium (RCE) as defined by Eq. 3.3 is generally a good approximation for large-scale tropical atmospheric processes. Almost any tropical sounding (but especially those over the oceans) can demonstrate this point, as shown in Fig. 4.2 for example. This summarizes the relationship for very many soundings from the TOGA-COARE (Tropical Ocean Global Atmosphere-Coupled Ocean-Atmosphere Response Experiment) campaign. Charney’s (1963) adiabatic scale analysis also demonstrates this point (cf., Delayen and Yano, 2009; Yano et al., 2009).

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Fig. 4.2 An observational demonstration of the “free ride” principle: for (a) the thermodynamic and (b) the moisture equations. The horizontal axis is the large-scale forcing, and the vertical axis is the convective forcing. Values at 500 hPa are shown from the c American Meteorological Society (AMS) TOGA-COARE IFA (Intensive Flux Array).  2001, from Fig. 1 of Yano (2001). Reproduced by permission of the AMS.

It is therefore tempting to apply the constraint as a dynamical balance condition for studying large-scale tropical circulations by analogy with geostrophic balance for the mid-latitudes (cf., Yano and Bonazzola, 2009). The central importance of this balance is emphasized by Neelin and Held (1987). Fraedrich and McBride (1989) propose to refer to the balance as a “free ride”, whilst Sobel et al. (2001) in a different context introduce the name “weak temperature gradient” (WTG) approximation. For further discussions see Ch. 11, Sec. 14.1 . 5

Equilibrium as a statistical concept

In Sec. 3.3, the statistical nature of the quasi-equilibrium concept was proposed. This section focuses on this aspect. 5.1

Macroscopic and microscopic views

The macroscopic view of thermodynamics may be contrasted with the microscopic view from statistical mechanics. A goal of statistical mechanics is,

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presumably, that of deriving and even of proving the principles of thermodynamics, originally derived in a completely empirical, phenomenological manner from more fundamental principles of atomic theory. Note that the qualifier “presumably” has been added in order to remain cautious about this point of view. In contemporary, standard physics education, atomic theory is often considered to be more fundamental than macroscopic phenomenologies. Convective quasi-equilibrium can also be interpreted from a perspective of statistical mechanics. For this reason, a review by Emanuel (2000) emphasizes the importance of “statistical equilibrium thinking” in order to properly understand convective quasi-equilibrium. However, this perspective has not always been accepted as common wisdom in the history of science: recall the philosophical debates between Ernest Mach, Stephan Boltzmann and others a little more than hundred years ago. Mach strongly argued that macroscopic phenomenology is robust enough for establishing the physics without invoking what was then a speculative atomic theory. Mach’s position almost carried the debate, greatly distressing Boltzmann, and possibly contributing to his suicide. The positions taken by Planck are nicely illustrative of the debates. In the preface to his Treatise on Thermodynamics, published in 1897, he described “the most fruitful” treatment of the subject as that which “starts direct from a few very general empirical facts” and contrasted it with “essential difficulties ... in the mechanical interpretation of the fundamental principles of Thermodynamics” (Planck, 1922, p. viii). By the time of the second edition in 1905, following the success of his 1901 quantization postulate for black-body radiation, he had somewhat reluctantly adopted a more statistical perspective, arguing that “the full content of the second law can only be understood if we look for its foundations in the known laws of the theory of probability” (Planck, 1922, p. x). This historical anecdote suggests that, against contemporary urgings, it may not necessarily be a good idea to turn to more elementary theories in order to re-establish something already phenomenologically established. A major strength of classical thermodynamics is in providing a robust theoretical framework without having to rely on more elementary atomic theory. In the context of atmospheric convection, the use of cloud-resolving models (CRM) is nowadays extremely popular for studying convective systems. However, it is always worth exercising some caution with this major trend, reminding ourselves that while CRM studies might be elementary, they are not necessarily fundamental. CRMs themselves contain many un-

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certainties, especially in cloud physics. In this respect, CRM studies are not necessarily more robust than a macroscopic phenomenological approach. 5.2

Law of large numbers

As discussed in the last subsection, the advent of statistical mechanics from the 1870s led to a link between microscopic dynamics and thermodynamics, and this gave rise to other conceptions of equilibrium. For example, in a standard undergraduate textbook on thermodynamics (Adkins, 1983) we find the statement (p. 7) “equilibrium is itself a macroscopic concept. We may only apply the idea of equilibrium to large bodies, to systems of many particles.” The view of equilibrium as a large-scale statistical concept implies that strictly it holds only in the limit of infinite system size. This argument can be grossly summarized by invoking the “law of large numbers”: statistical expectation values (mean, variance, etc.) become increasingly more reliable as the sample size increases. More precisely, the so-called law of large numbers in probability theory (e.g., Ch. 10 of Feller, 1968) guarantees convergence of the mean for a sequence of mutually independent events to the probabilistic expectation. Furthermore, the central limit theorem guarantees that the fluctuation of this mean value around the expectation value decreases as ∼ N −1/2 as the sample number N increases. These two mathematical theorems provide an explanation of why macroscopic thermodynamic quantities, such temperature and pressure, can be measured in a stable and reliable manner. Sometimes the same argument is invoked in order to justify convective quasi-equilibrium. A review by Emanuel (2000), for example, stresses the separation of convective and large scales. Convective processes are of much smaller scales (∼ 1 km) than a typical synoptic scale (∼ 103 km). As a result, the number of convective elements (or convective towers, more precisely) contained within a typical synoptic-scale disturbance may be substantial, if the elements are not too widely spaced. Thus, it is tempting to appeal to the law of large numbers and the central limit theorem. The limitations of this argument must be kept in mind. The number of gas molecules contained in a 1 mm3 volume is more than 1016 for standard atmospheric parameters (103 hPa and 300 K). On the other hand, for the sake of an estimate, let us assume that a convective tower is found every 10 km in two horizontal directions. Even in this relatively dense situation, we find only 100 convective towers within an area of size 104 km2 . The convection within such an area may be in equilibrium in a certain sense,

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but clearly with far more statistical uncertainties than the case of gas in a 1 mm3 volume. It is especially important to keep in mind that the central limit theorem guarantees only a relatively slow convergence towards the statistical expectation with increasing sample size N . A good example for making this point is the population dynamics of a two-species predator-prey system. Phenomenologically, many such systems have an oscillatory evolution, whereas a conventional description based on the number density of each species (the Lotka–Volterra equation) gives only a steady state after an initial transient dependency. This apparent dilemma can be resolved by explicitly calculating the evolution of the number of each species as a particular realization of a stochastic process: oscillatory behaviour is then revealed (McKane and Newman, 2005). The reason for the discrepancy is traced to a discrepancy between an expectation value (as calculated by the Lotka–Volterra system) and a single actual realization of a stochastic process. The two solutions can be qualitatively different, even if an apparently large system of ∼ 103 individuals is considered, because purely internal noise can induce resonance effects. This study with a biological system emphasizes the need for caution and careful interpretation when applying statistical ideas of equilibrium that are based on large samples to real systems, such as a system of convective clouds, which have relatively small populations. Sol´e and Bascompte (2006) is a good starting point for biological descriptions of population dynamics. In particular, their book includes extensive discussions on self-organization, an issue to be discussed in Sec. 7.2. Van Kampen (2007) provides more general discussions on probabilistic descriptions of physical (as well as biological) systems of finite extent. Some possibilities for and perspectives on applying explicitly stochastic descriptions to convective systems can be found in Vol. 2, Ch. 20. 5.3

Scale-separation principle

Partly for the reason discussed in the previous subsection, the notion of scale separation is often invoked in order to justify convective quasi-equilibrium. As discussed in Gombosi (1994) or Ch. 1 of Salmon (1998), for example, the vast difference in scales between those of interest in typical fluid-mechanics problems and the mean free path between molecular collisions is important in order to justify the statistical thinking that leads to the Navier–Stokes equations. Emanuel (2000) draws the analogy with convective scales that

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are much smaller than those scales of interest in synoptic meteorology, and thus a similar statistical thinking might be justified, although obviously such an approximation must be very much more tentative in the convective case. It may be worth restating in the present context that although scale separations are implied both in space and time, the relationship to quasiequilibrium in the sense described in Sec. 3.6 emphasizes the fundamental importance of a separation in the timescales. Timescale separation is not necessarily equivalent to space-scale separation, as discussed in Ch. 3, Sec. 6. Here, the convective timescale is expected to be much shorter than the timescales of interest for the evolution of the large-scale atmosphere. Davies et al. (2009) explicitly demonstrate the importance of timescale separation in order to establish CQE. They consider a dynamical system consisting of a fixed periodic large-scale forcing and a convective relaxation timescale. Convection is no longer slaved to the large-scale forcing when the forcing period is reduced such that it approaches the relaxation timescale. The same point has also been demonstrated in analogous CRM experiments (Davies et al., 2013). The notion of a timescale separation, while recognizing a non-zero convective timescale, leads to the idea of considering convective processes as adjustments towards an equilibrium state. Due to the timescale difference, the adjustment is accomplished relatively rapidly. Thus, CQE can be reinterpreted as a fast adjustment process, as proposed by Neelin and Yu (1994). This concept is further discussed in the next subsection. Overall, one may argue, the system remains on a slow manifold due to such rapid adjustment processes, in the same sense in which geostrophic adjustment maintains mid-latitude dynamics on a slow manifold. Emanuel (2000) concludes his essay by invoking this notion (cf., Sec. 4.3). Thus, from the point of view of considering quasi-equilibrium as a consequence of timescale separation, we are led to two alternative views: moist convective adjustment and the slow manifold. More generally, there is an extensive literature in applied mathematics for systems consisting of two distinguishable and well-separated timescales. Ch. 3 discussed some of those methods, while other possibilities can be found in the review by van Kampen (1985).

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Quasi-equilibrium as a relaxation process

As suggested in the last subsection, quasi-equilibrium may be considered a fast adjustment process towards an equilibrium so that the system always remains close to the equilibrium (cf., Ch. 2, Sec. 3). In this manner, more specifically, CQE may be reinterpreted as a “moist adjustment” process, the point of view adopted by Neelin and Yu (1994) and Yu and Neelin (1994), for example, in their treatment of tropical convective dynamics interactions (see also Mapes, 1997b, as a review). Several convection parameterizations have been developed based on the idea of moist convective adjustment. Manabe et al. (1965) used a hard adjustment in which any convectively unstable atmospheric profile is instantaneously reset to a moist adiabat, in the same spirit as convective adjustment in Manabe and Strickler’s (1964) RCE. The Kuo (1974) scheme can be regarded as imposing a soft adjustment over a finite timescale (Arakawa, 2004), but perhaps the most familiar such parameterization to the modern reader is that of Betts (1986). Based on this scheme, full global atmosphere models of intermediate complexity have been developed, known as “quasi-equilibrium tropical circulation models” (Neelin and Zeng, 2000). However, differences between the convective adjustment interpretation of CQE and Arakawa and Schubert’s (1974) original definition should be noted. Arakawa and Schubert’s (1974) original formulation attempts to define an equilibrium directly without explicitly considering a transitional phase. On the other hand, in the reinterpreted version, the assumed short, but nonetheless non-zero timescale for adjustment is explicitly recognized. In this manner, convection is no longer slaved to the large-scale state, but two-way interactions between convection and the large-scale state are established. More precisely, in the formulation of Neelin and Yu (1994) and Yu and Neelin (1994), the large-scale state is adjusted towards a reference profile by convection, with the reference profile itself being dependent on the large-scale state. Note also that the reference moisture profile is defined in terms of the background temperature profile. Convection naturally modifies the background thermodynamic state. Such a modulation effect is considered by Kuang (2011) by coupling a cloud-resolving model with selfcontained dynamics to large-scale gravity waves, as introduced in Sec. 5 of Yano et al. (1998). Here, a slight modification is to replace the temporal tendency of the large-scale vertical velocity with a Reynolds-like damping tendency. Such a two-way interaction approach allows the specification of a ver-

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tical profile for tropical flows, as formulated in Neelin and Yu (1994) and numerically presented by Fig. 2 of Yu and Neelin (1994). The main assumption is that a perturbation thermodynamic profile satisfies a moist adiabat, while the vertical profile for the basic state can remain unspecified. Holloway and Neelin (2007) present supporting evidence for this assumption above the boundary layer, almost up to the top of convection. Thus, a reference perturbation temperature can be determined, which is then used in Neelin and Yu’s (1994) parameterization. However, we must recognize that the “adjustment” interpretation of CQE is different from the interpretation of CQE through a thermodynamic analogy as discussed in Sec. 3.6. Although this type of Newtonian relaxation description is very convenient for many purposes, it is clearly a qualitative description with no obvious correspondence to any approximation of a more physically based formulation. Indeed, the Introduction of Betts (1986) is careful to make no such claim, stressing instead the operational utility (which is undeniable, even today) and that the relaxation “sidesteps all the details of how the subgrid-scale cloud and mesoscale processes maintain the quasi-equilibrium structure we observe.” An indication that the simplest Newtonian relaxation is not in fact an accurate description of the adjustment process is provided by Raymond and Herman (2011). These authors demonstrate that the relaxation timescale may be height dependent, and moreover that there may not be a welldefined or unique target profile. Another challenge arises from observations of power-law behaviour (cf., Ch. 3, Sec. 2.3), which suggests that a single convective relaxation timescale cannot be well defined as the rate of decay to equilibrium will depend on the initial departure from the equilibrium state. In the Betts–Miller scheme (Betts, 1986), the relaxation is performed towards a target profile of temperature and moisture. This leads to a further reinterpretation of CQE as a means of maintaining the atmosphere close to a reference profile. Over the tropics, intuitively the most likely reference profile would be a moist adiabat, as emphasized by Emanuel (2007). This then leads further to an anticipation that CQE maintains the tropical atmosphere close to a state of convective neutrality (cf., Sec. 3.8) whenever convection is a dominant process. An early application of the reference state idea can be found in Lord et al. (1982) (their Eqs. 8 and 9) where convective forcing is evaluated in terms of departures from a set of timeaveraged reference values for the cloud-work function. Xu and Emanuel (1989) argue that the convectively neutral reference state has zero CAPE,

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for a definition of CAPE based on reversible ascent (i.e., assuming that all condensed water is retained by the lifting air parcel). The analysis of sounding data by Roff and Yano (2002) shows that such a state of zero reversible CAPE is indeed realized as a time mean over the tropical Western Pacific, but also emphasizes that deviations from the mean state are substantial over synoptic timescales. In summary, the moist adjustment reinterpretation of CQE leads to a different perspective from the original definition by Arakawa and Schubert (1974). Whilst Arakawa and Schubert’s (1974) original formulation (Eq. 2.1) defines CQE as a response to forcing, the adjustment reinterpretation considers CQE as a function of a large-scale state (typically a thermodynamic vertical profile). The latter obviously has the advantage of intuitive appeal, especially as further developed into a focus on the concept of transition to strong convection (cf., Sec. 7.5). 5.5

Absence of scale separation?

Although one should be mindful that power-law behaviour has undoubtedly been claimed too readily and too strongly in the scientific literature (Clauset et al., 2009), nonetheless the ubiquitous presence of power laws and scaling behaviour in the atmosphere does raise a major challenge to interpretations of CQE based on the scale-separation principle. However, we must distinguish between the elusiveness of the scale-separation principle in the face of observed scale-free behaviour for various aspects of the atmosphere, and the very clear usefulness of the scale-separation principle, as already emphasized in Ch. 3. Understanding the behaviour of an idealized system in a suitable limit may of course provide valuable insights into the much more complicated behaviour of a real system. In order to illustrate these points, quasi-geostrophic theory again provides a good example. Arguably this theory is also based on a scaleseparation principle, by singling out the scale of the Rossby deformation radius as a characteristic scale for large-scale flow. However, observations do not single out this scale as having any particular importance in the face of atmospheric scaling behaviour. Nonetheless most dynamicists would consider that the absence of clear, simple observational support in this sense hardly diminishes the usefulness of quasi-geostrophic theory. Singling out a particular scale is clearly useful in deriving the quasi-geostrophic system. Quasi-geostrophic theory is even capable of explaining the observed scaling behaviour of kinetic energy (cf., Charney, 1971).

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By the same token, despite the lack of observational evidence for a single, simple characteristic convective timescale, that does not exclude the usefulness of the concept for deriving a theoretical principle. The usefulness of the principle must then be judged a posterori from its applications, such as the performance of parameterizations. Again, it should be emphasized that the interpretation of CQE as a balance condition stands without invoking a scale-separation principle. The existence of mesoscale organization provides a similar objection to the scale-separation principle from phenomenology. A more basic issue with such organized structures is less the use of a quasi-equilibrium hypothesis, but rather the absence of an explicit representation of the structures in convection parameterizations. To the best of the authors’ knowledge, the only mass-flux parameterization to have included a mesoscale downdraught component is that of Donner (1993). Moncrieff’s (1981; 1992) archetype model could be a more formal answer to this challenge (cf., Ch. 14, Sec. 8.1). However, no operational centres seem to have taken this proposal seriously to this date.

6

Statistical cumulus dynamics

A rigorous justification of the analogy of convective quasi-equilibrium with thermodynamic equilibrium would be “an eventual goal of statistical cumulus dynamics” as stated in Arakawa and Schubert (1974). As foreseen by Arakawa and Schubert (1974) themselves, improvements in cloud-resolving modelling are expected to lead to the development of statistical theories describing ensembles of cumulus clouds. Such theories should reduce to quasi-equilibrium within suitable limits. However, in spite of rapid modelling improvements in recent decades, rigorous theories of statistical cumulus dynamics remain in their infancy (cf., Cohen and Craig, 2006; Craig and Cohen, 2006; Davoudi et al., 2010; Khouider et al., 2010; Plant, 2009, 2012). Nonetheless, some important aspects of a statistical theory that would support the concept of CQE can easily be identified. In particular, we should be able to identify a large enough number of convective elements (e.g., convective plumes) within a single large-scale domain in order to ensure that the ensemble statistics which might be predicted by the statistical theory would be representative of the modelled domain-mean statistics. In other words, a large-scale, macroscopic state must itself be well defined and

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be actually realized in practice, at least to a good approximation. In much of the parameterization literature, the large-scale domain is equated with a grid box of the parent model, but in practice the size of the large-scale domain must be much larger than a single grid box in a typical GCM in order to guarantee a smooth description of large-scale processes (cf., Ch. 3, Sec. 3.1). The law of large numbers discussed in Sec. 5.1 can then support a statistical cumulus dynamics approach. A simple corollary is the need for a separation of scales between convective and large-scale processes.

6.1

Statistical mechanics

As a first step for constructing the statistical cumulus dynamics, let us review the basics of statistical mechanics specifically following a presentation by Jaynes (1978). The general goal of statistical mechanics is to define a distribution of (microscopic) states under a given set of physical constraints (on the macroscopic state). These physical constraints can be fairly general as long as they follow a general form given by Eq. 6.3 below. The basic principle invoked in order to define a distribution is to assume that it gives the maximum number of combinations (partitions) amongst the possible distributions. So long as the system contains a fairly large number of states, this would be a valid assumption, as proved by many successes of statistical mechanics so far. The obtained distribution can furthermore be used in order to infer some macroscopic relationships, as shown below. To be more specific, let us assume that a given system has n discrete states. The goal of the statistical mechanics is to obtain a set of frequencies {pi } for finding these states with i = 1 . . . n the index for a state. We suppose that the system consists of N “particles”. Here, the word particles may be translated to a different word depending on the application (e.g., “events”). When we apply this idea to the convective system, the particles and the states may be translated into the convective elements and convective types. When N is fairly large, we expect that the distribution of the system follows the maximum possible combination of the n states for those N particles. The possible number of combinations for having Ni particles in the state i is given by: W =

N! N1 !N2 ! · · · Nn !

(6.1)

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or by taking the logarithm of the above, S ≡ ln W = −N

n 

pi ln pi ,

(6.2)

i=1

where pi = Ni /N is the frequency with which the i-th state is occupied by particles. In deriving Eq. 6.2, we invoke the Stirling approximation: x! ≃ x log x. The quantity S is often called the “information entropy”, or “entropy” for short. We also suppose that the given system is under m constraints given in the form: n 

pi fk (xi ) = Fk

(6.3)

i=1

for k = 1, . . . , m (≤ n). Here, fk is a function of xi , a physical variable attached to the state i, and Fk is a constant. The most likely distribution is obtained by maximizing the entropy given by Eq. 6.2 under the constraints of Eq. 6.3. In order to make this calculation, we first add all of the constraints to Eq. 6.2 multiplied by constants λk (k = 1, . . . , m) called “Lagrange multipliers”:   n m   I = ln W + pi fk (xi ) − Fk . λk i=1

k=1

The problem then reduces to that of maximizing I under a variational principle. The variation of I can be written as: δI = −N

n 

(ln pi − 1)δpi +

i=1

m 

λk

n 

fk (xi )δpi = 0.

i=1

k=1

This leads to a final answer:

 m   1 exp − λk fk (xi ) , pi = Z(λ1 , . . . , λm )

(6.4)

k=1

where Z(λ1 , . . . , λm ) =

n  i=1

is known as the partition function.



exp −

m 

k=1



λk fk (xi )

(6.5)

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It is also straightforward to obtain: ∂ ln Z Fk = − ∂λk m  S = ln Z + λk Fk

(6.6) (6.7)

k=1

λk =

∂S ∂Fk

(6.8)

∂pi = (−fk + Fk )pi . ∂λk Various macroscopic relationships can also be derived from the partition function. For example, consider: n  pi q(xi ), q(x) = i=1

the angled brackets denoting an expectation value. Then: n ∂q(x)  ∂pi q(xi ) = ∂λk ∂λ k i=1

= −fk q + fk q. (6.9) In particular, if we take q(x) = fj (x), then: ∂fj  = −fj fk  + fj fk . (6.10) ∂λk Note that, in general, fk may depend on some external parameters, {αr }, such that fk = fk (x; α1 , · · · , αs ). Consider now an arbitrary small change {δFk , δαr }. From Eqs. 6.5 and 6.7, we have that m m n    λk < δfk > pi δfk (xi ) = − λk δZ/Z = − k=1

i=1

k=1

and

δS = δZ/Z +

m 

λk δFk

k=1

respectively. Hence: δS = or

m 

λk (δFk − δfk ) ,

k=1

δS =

m 

λk δQk

(6.11)

k=1

where

δQk = δFk − δfk  = δfk  − δfk .

(6.12)

134

6.2

Part I: Basic parameterization concepts and issues

Example: The Boltzmann distribution

As an example, consider a system constrained by the known total energy E as well as the total particle number N . These constraints are stated as: n 

Ni Ei = E

(6.13)

i=1

n 

Ni = N.

(6.14)

N −βEi e Z(β)

(6.15)

i=1

Under these constraints, we obtain: Ni = with Z(β) =

n 

e−βEi .

(6.16)

i=1

6.3

Application to convection

Properties of the convective equilibrium state may also be determined using a similar procedure. Although a different approach was taken by Craig and Cohen (2006), the distribution of individual cloud mass fluxes analogous to Eq. 6.15 can also be derived from a maximum entropy condition. It is necessary only to replace the total energy by the total mass flux in the above. The goal of statistical cumulus dynamics would be to construct a selfcontained theory in a systematic manner under the methodology outlined here. It seems certain that we could learn much along these lines about the statistical behaviour of convective systems. But could it provide a definitive answer to the question of convective quasi-equilibrium, as Arakawa and Schubert (1974) predicted, so that a robust solution to the convective closure problem (cf., Ch. 11) can be obtained? As outlined above, in order to develop a theory for statistical mechanics, we first need to specify a set of macroscopic constraints, characterizing the large-scale atmospheric state. On the other hand, the question of the convective quasi-equilibrium state as well as convective closure is precisely intended to establish the large-scale states. Thus, the idea of exploiting statistical mechanics in order to find the large-scale convective state faces a chicken-and-egg dilemma.

Quasi-equilibrium

7 7.1

135

Beyond convective quasi-equilibrium Activation control

The activation control principle proposed by Mapes (1997a) was introduced in Sec. 3.9 as an alternative idea for the control of large-scale variations of deep convection. This principle emphasizes, in contrast to CQE (equilibrium control), the importance of overcoming an energy barrier of CIN. The viability of this alternative principle will now be discussed. Note first of all that the activation control idea focuses on shorter timescale processes than those considered by Arakawa and Schubert (1974). As will be fully discussed in Ch. 7, their mass-flux parameterization is formulated by assuming steady-state plumes from the outset. Under the steady-plume hypothesis, the vertical structure of the plume is assumed to be in equilibrium. By contrast, activation control focuses on an initial transient stage of convective growth, as a boundary-layer eddy breaks through the local energy barrier. Whether that short timescale process has a key importance that needs to be recognized in the large-scale evolution of convective systems is an open question. The implications from Lighthill’s theorem discussed in Sec. 4.4 remind us that the possibility cannot easily be discounted. It is also interesting to note that the concept of activation control implicitly adopts the perspective of atmospheric convection as consisting of a series of ascending “bubbles”, rather than as an ensemble of steady plumes as assumed by Arakawa and Schubert (1974). Bubble theory is further discussed in Ch. 12, and also reviewed in Yano (2014a). It was once seen as a strong alternative theory for describing atmospheric convection, but largely abandoned during the 1970s. However, strong echoes persist to this day, most notably in the ongoing debates on entrainment (cf., Ch. 10, Yano, 2014a), and in the usual textbook and academic course introductions to CAPE. Mapes’s (1997a) arguments on activation control urge us to seriously reconsider the alternative possibility of bubble theory. Unfortunately, the distinction between two timescales, that associated with individual convective plumes and that associated with the ensemble of convective plumes, is not always clear in Mapes’s (1997a) article. Recall that the latter aspect can be prognostically described by the convective energy cycle even under the standard mass-flux formulation (cf., Ch. 11, Sec. 10). Issues raised by Mapes (1997a) in his Sec. 6 are not associated with lifting parcels, and could be equally well interpreted in terms of the

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evolution of an ensemble of convective plumes as described under the energy cycle. In this description, CIN enters the problem only as a part of the vertical integral defining A, and only if the vertical integral includes the boundary layer (cf., Ch. 11, Eq. 8.2). Recall that the lower limit zB of the integral is usually taken at the cloud base, which is close to the top of the boundary layer (cf., Romps and Kuang, 2011). The issue raised in Mapes’s (1997a) Sec. 6 is that variations in deep convection occur due to processes that simultaneously increase CAPE and reduce CIN so that it is ambiguous which controls the variation. An answer is suggested by the convectiveensemble energy cycle discussed in Ch. 11, Sec. 10: the combination that matters is dictated by a weighting function provided by the vertical plume profile. Finally, a limit to thinking solely in terms of the CIN barrier should be emphasized. The barrier is typically expressed in terms of a neutrally buoyant air parcel artificially lifted from the middle of a well-mixed boundary layer. The ascending air within a well-mixed boundary layer is likely to be positively buoyant. Thus, if the buoyancy variable to be integrated over is defined as a weighted average using the vertical velocity of actual local air parcels, then we no longer see any negatively buoyant barrier zone, except perhaps for an inversion layer at the top of the well-mixed layer. An example of this analysis from CRM data is shown in Fig. 11.6 of Ch. 11, Sec. 11.5. The idea of activation control is also discussed in Sec. 11.2 of Emanuel (1994) as a concept of triggered convection. Some dimensional analyses are also presented there. See Ch. 11, Sec. 11.5 for further discussions. Most importantly, if activation control were to be accepted as a guiding principle, then a rather drastic modification of the formulation of convection parameterization would be required (cf., Ch. 7, Sec. 7). Mapes (1997a, 1998) does not address such formulation issues. Unfortunately, subsequent attempts to implement the ideas (Fletcher and Bretherton, 2010; Hohenegger and Bretherton, 2011; Kuang and Bretherton, 2006; Mapes, 2000) have all been made within a traditional framework that assumes CQE (cf., Ch. 11, Sec. 11.5). Thus, they lack self-consistency, an issue that is further discussed in Yano (2011). The concept of a trigger function, as described by Kain and Fritsch (1992), for example, can also be understood as arising from an activation control perspective (cf., Sec. 7.4: see also Ch. 11, Sec. 13.3).

Quasi-equilibrium

7.2

137

Self-organized criticality (SOC)

The concept of self-organized criticality (SOC) was originally proposed by Bak et al. (1987) to explain 1/f noise behaviour in the sand-pile system. The concept refers to a state of a macroscopic system which is analogous to the critical state of a thermodynamic system at a phase transition (cf., Stanley, 1972; Yeomans, 1992). However, the major difference from the thermodynamic phase transition is that the system remains at, or close to, the critical state due to its self-maintaining tendency. An SOC system remains in a type of equilibrium state called “criticality” but this is due more to the system’s own critical behaviour rather than externally imposed conditions. In other words, it is not the environment that defines the equilibrium, as the conventional thermodynamic analogy suggests, but rather the internal dynamics. Bak et al. (1987) demonstrate their idea by introducing a simple, multi-variable dynamical system that perpetually remains in a marginally unstable state. Common features of the textbook systems exhibiting SOC (e.g., Jensen, 1998) are that the system is slowly driven by some external forcing, with threshold behaviour of the individual degrees of freedom, and that there are interactions between those degrees of freedom. Internal interactions drive the system towards criticality, developing large variability and structures on many scales without any need for external tuning. Clearly SOC has a very different emphasis on interactions than the conventional interpretation of CQE, in which the convective plumes do not interact in any direct sense, only via their influence on the environment. The possible treatment of interactions in a convective system is discussed in Vol. 2, Ch. 27, alongside further perspectives on SOC. An important ingredient of SOC from a dynamical systems point of view is its linear instability around the critical state. From the perspective of the slow manifold and Lighthill’s theorem discussed in Sec. 4.4, the convective system remains within a “fuzzy” interface zone due to the combination of its own instability and self-maintaining tendency. In both interpretations, an important contribution from fast convective processes is suggested unlike the conventional interpretation of CQE. The state of SOC is often associated with 1/f noise behaviour of the frequency spectrum (i.e., the power spectrum is a power law with an exponent of −1), and thus findings of 1/f noise behaviour (Yano et al., 2000; Yano et al., 2001, 2004) in various tropical time series, including CAPE, are suggestive that tropical convection is also at SOC.

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Part I: Basic parameterization concepts and issues

Stronger evidence has been found by Peters and Neelin (2006) (see also Neelin et al., 2008; Peters et al., 2002). The behaviour of a system close to a state of criticality can be characterized by a power-law relationship: P ∼ (I − Ic )α ,

(7.1)

where I is a control variable, Ic the critical point defined in terms of this control variable, P a variable that represents self-organized behaviour, and α a positive critical exponent less than one (Stanley, 1972, Ch. 11). By identifying the column-integrated water vapour (CWV) and precipitation rate from satellite retrievals with I and P , respectively, it is found that tropical convection exhibits such behaviour with α = 0.215. However, some subsequent observational analyses and modelling simulations have experienced some difficulties in recovering the same results. Raymond et al. (2007, 2009) have pointed to qualitatively different behaviour characterized by a relation: P ∼ (Ic − I)−1 .

(7.2)

Other analyses include Holloway and Neelin (2009) and Holloway and Neelin (2010), who investigated ARM (Atmospheric Radiation Measurement) data but concluded that it was “impossible to test the power-law relationship at high total column water”. A large-domain CRM experiment by Posselt et al. (2012) produced a scatter plot for precipitation that splits into two directions suggestive of the two possibilities in Eqs. 7.1 and 7.2. Global model data analysis by Bechtold (2009) does show a flattening tendency of the precipitation rate as a function of column-integrated water for the largest values of column water (his Fig. 17), as would be expected from Eq. 7.1. However, at lower values, this flattening shape is preceded by a curve that is well fit by an exponential for precipitation rates varying over two orders of magnitude. Note that the singular relationship of Eq. 7.2 indicates a tendency for stabilization of the system by a negative feedback: as the CWV approaches Ic , the precipitation rate dramatically increases. Such a tendency not only prevents the system from reaching the critical point, but it tends to stabilize the system by rejecting a highly moist state. This could be considered a good example of the convective homeostasis discussed in Sec. 3.5 maintaining the stability of the system by self-regulation. On the other hand, the relationship of Eq. 7.1 indicates a critical behaviour in the system, with a slower increase of the precipitation rate with increasing column-integrated water above Ic . As a result, above the critical

Quasi-equilibrium

139

point, the system will tend to accumulate more and more moisture into a given atmospheric column under sufficiently strong forcing. The accumulated column water is lost only gradually by precipitation. Such behaviour is a reflection of the inherent instability of a system under SOC. It may furthermore be remarked that SOC is potentially important for understanding convective organization (cf., Peters et al., 2009; Yano et al., 2012a). As discussed, an SOC system is inherently unstable at the critical point and will therefore tend to evolve further away from that point, relative to the departures from the equilibrium point that might be expected for a thermodynamic system. In the latter case, one would expect to find a system that stays at its equilibrium point stably for substantial periods of time. To distinguish the two cases, it is important to study the frequency of occurrence of CWV. Such an analysis (Lintner et al., 2011; Neelin et al., 2008, 2009) reveals a Gaussian core but with tails that are much longer than would be expected from a Gaussian, suggesting some occasional, substantial deviations away from the critical point. Thus, the identified critical point cannot be straightforwardly interpreted only in terms of a standard thermodynamic equilibrium state. 7.3

Activation and SOC: Complementary or contradictory with CQE?

Conceptually both activation control and SOC propose very different principles in comparison with CQE as interpreted through a thermodynamic analogy in Sec. 3.6. Both of these alternative principles emphasize that convective processes are not passively defined as an equilibrium dictated by the given large-scale environment, but rather that they represent their own autonomous actions. The activation control principle emphasizes the importance of the local threshold: i.e., the individual air parcel or boundary-layer eddy that triggers a convective element. However, it emphasizes less how an individual convective element modifies its environment, how such modifications may then affect subsequent triggering, and consequently also how a system comprised of multiple convective elements behaves collectively. Mapes (1997a) states that “clearly this situation is hopeless in detail” and so advocates an empirical approach. On the other hand, SOC does emphasize the collective behaviour of many convective elements. Individual convective elements are considered more as fluctuations. Details of their triggering and of their local environ-

140

Part I: Basic parameterization concepts and issues

mental modifications are not considered to be important but rather the focus is on the collective behaviour that emerges from the general character of their interactions. An ensemble average of the fluctuations provides crucial feedbacks to large-scale behaviour due to non-linear interactions between the convective elements. This is a qualitative difference of a critical phenomenon (cf., Wilson, 1983) from a normal thermodynamic equilibrium. In the latter case, microscopic (convection) fluctuations may be simply averaged out at the macroscopic scale (large-scale). Clearly, activation control and SOC can be compatible: the former focusing on initiation of individual elements while neglecting details of how collective behaviour arises, while the latter ignores details of the initiation and focuses on collective behaviour. However, even if activation control and SOC are relevant, then CQE considered as a balance condition, as discussed in Secs. 4.3 and 4.4, may nonetheless remain valid. In both of the alternative paradigms, however, the most serious implication is that fast convective processes have crucial impacts on the evolution of large-scale processes. Indeed, SOC suggests that the equilibrium solution would be unstable under linear perturbations. To what extent do we need to consider explicitly the fast, fluctuating processes? At the time of writing, it is not immediately clear whether the quasi-equilibrium description can still be retained, or whether it is necessary to move to a more prognostic or stochastic formulation. Neelin et al. (2008) interprets that SOC can be regarded as an extension of an adjustment interpretation of CQE. In this respect, an important ingredient to be added to CQE in order to accommodate SOC is the transition to strong precipitating convection (Neelin et al., 2009) above a critical threshold. This perspective is further discussed in Sec. 7.5. 7.4

Phenomenological limitations

Arguably the concept of CQE has been developed with the tropical atmosphere in mind, and mainly for maritime situations. In such situations, both temperature and moisture are horizontally homogeneous, relatively speaking, leading to the “free ride” principle and an equivalent balance for the moisture (cf., Sec. 4.5). These situations are consistent with CQE in observational diagnoses (e.g., Betts, 1986; Donner and Phillips, 2003; Holloway and Neelin, 2007; Zhang, 2009). However, the situations over land as well as in mid-latitudes are likely to be very different. Both temperature and moisture are much more horizontally heterogeneous and it is less obvi-

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141

ous how and when CQE is supportable from observational diagnoses (e.g., Zhang, 2003; Zimmer et al., 2011). Consider the situation over land in summer in the mid-latitudes. The North American Great Plains is perhaps the most studied example of this situation. It is phenomenologically known that (assuming fine weather) the surface heats up strongly during the day, so that CAPE becomes large around noon indicating a conditionally highly unstable atmosphere. Thus, a simple application of CQE would predict the onset of convection well before noon (cf., Guichard et al., 2004). However, in practice convection is typically triggered in the late afternoon: thus, an external triggering (by either a synoptic or a boundary-layer process) seems to be required to realize the conditional instability. See Ch. 11, Sec. 13.2 for further discussion of this triggering. A strict CQE hypothesis does not work in this type of situation, and many operational schemes introduce a trigger condition for just this reason. Kain and Fritsch (1992) demonstrate the sensitivity of mesoscale simulations to the formulation of triggering in some circumstances. Sud et al. (1991) investigate the use in a GCM of critical onset values of the cloudwork function in an adapted form of the Arakawa and Schubert (1974) parameterization. Alternative possibilities for taking into account these phenomenological limitations include adopting a prognostic description, as discussed in Ch. 11, Sec. 10, or else to distinguish between boundary-layer and free-tropospheric processes within the closure formulation, as discussed in Vol. 2, Ch. 15. The concept of transition to strong convection, to be discussed in the next subsection, may also be considered as an alternative possibility for overcoming phenomenological limitations of CQE. 7.5

Transition to strong convection

From a purely phenomenological point of view, probably the most important aspect revealed by a series of observational analyses initiated by Peters et al. (2002), which further led to a SOC interpretation as already discussed in Sec. 7.2, is the fact that there is a well-defined onset of convection at I = Ic , beyond which the major proportion of tropical precipitation occurs. The concept of such an onset is something missing, or at least implicit, in the original sense of CQE. Neelin et al. (2008) use the phrase “transition to strong convection” to describe the onset, and further analyses examining its characteristics

142

Part I: Basic parameterization concepts and issues

are presented by Holloway and Neelin (2009), Holloway and Neelin (2010), Neelin et al. (2009), and Sahany et al. (2012). By focusing on identifying the onset and its dependencies, these studies attempt to develop a phenomenological theory relatively independent of implications in terms of SOC. Note that the (I, P ) space description characterizes convection as a function of a state (i.e., column-integrated water vapour, or CWV) rather than of forcing. Neelin et al. (2009) note that the critical behaviour represented by Eq. 7.1 for I > Ic can be interpreted as a non-linear extension of a linear relaxation convection scheme originally developed by Betts (1986). The scheme by Betts (1986) can be recovered as a special case of Eq. 7.1 with α = 1, and so the transition from strong convection back to the onset state can be considered as being a natural extension of the adjustment reinterpretation discussed in Sec. 5.4. Neelin et al. (2009) and Sahany et al. (2012) furthermore show that variations in the onset value Ic can be defined to a good approximation as a function of the column-integrated tropospheric temperature, denoted T. Thus, the onset is characterized as a critical thermodynamic state in terms of both the CWV and temperature. Intriguingly, their analyses suggest that the onset is independent of the sea surface temperature (SST), which instead appears to be manifest as a stronger drive towards onset from below resulting in a frequency distribution of CWV that is shifted towards the onset boundary. Holloway and Neelin (2009) examine the evolution of the vertical structure of the atmosphere associated with the transition to strong convection by constructing various composites, emphasizing the importance of water vapour in the lower free troposphere. Holloway and Neelin (2010) focus more on lag-lead relationships between CWV and precipitation and so argue that high values of CWV occur primarily as a result of external forcing mechanisms rather than as a response to strong convection. Holloway and Neelin (2009) and Sahany et al. (2012) furthermore demonstrate that that the onset boundary in (CWV, T) space can be approximately reproduced by some relatively simple bulk plume models, suggesting a link to conditional instability. The main requirement for the plume model is that it should have sufficiently strong entrainment in the lower free troposphere, so that the environmental water vapour then plays a sufficiently important role in the calculated plume buoyancy. The emphasis on the onset of convection by these authors is, to some extent, reminiscent of points emphasized by the activation control princi-

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143

ple (Mapes, 1997a) discussed in Sec. 7.1. On the other hand, there is an important difference in these more recent analyses. Neelin et al. (2008) emphasize that the concept of transition to strong convection need not be distinct from CQE, but in fact is closely related to it (cf., Sec. 5.4). 8

Conclusions

The concept of convective quasi-equilibrium (CQE) originally proposed by Arakawa and Schubert (1974) has multiplied in its interpretations over the years. The purpose of this chapter has been to present a coherent picture of the various, sometimes competing interpretations. With the given literature, there are two major ways to interpret Arakawa’s original philosophical argument for justifying the CQE hypothesis. First, an unbiased reading of Sec. 7 of Arakawa and Schubert (1974) suggests that they have a thermodynamic analogy with the convective system in mind, an argument developed here in Sec. 3.6. On the other hand, Schubert (2000) retells the history of the development of Arakawa’s CQE concept by analogy with quasi-geostrophic theory. We have expanded this argument by invoking the concept of slow manifold in Sec. 4.3. A third possibility is a statistical one as advocated by Emanuel (2000) and discussed in Sec. 5.1. CQE is a key concept in order to understand the role of deep moist convection in the atmosphere. The concept serves a wide range of purposes. It has been used as a guiding principle to develop almost all convective parameterizations that are used in weather forecasting and climate modelling. More fundamentally, it also provides a basic theoretical framework in order to understand the role of convection in large-scale tropical dynamics. Although the concept is frequently invoked, there are different interpretations that may be relevant for different purposes. Unfortunately, few authors take care to explain their own interpretation before applying the concept, and as a result there now appears to be some confusion in some quarters. The intention here was to consider the various interpretations of CQE as systematically as possible, as summarized in Fig. 4.3. However, because so much has already been said about CQE, it is fair to acknowledge that only selective materials have been examined. The authors’ focus has been to examine the existing interpretations under the two basic interpretations identified above: as a thermodynamic analogy, and as a dynamical balance. It has also been remarked that a biological counterpart to quasi-equilibrium, homeostasis (cf., Sec. 3.5), may help our understanding of CQE.

144

Part I: Basic parameterization concepts and issues

Statistical Cumulus Dynamics

Radiative Convective Equilbrium

Scale Separation Thermodynamic Analogy

Activation Control

Equilibrium Control

SelfOrganized Criticality

Homeostasis

Convective Adjustment (Relaxation)

? =

Dynamic Balance

Free Ride

Slow Manifold

Lighthill’s Theorem ? Fig. 4.3 A flowchart for summarizing the links between various concepts discussed in this chapter. The arrows indicate directions of evolution of the concepts. Where linking arrows are shown in both directions, it suggests that the two concepts are almost equivalent. On the other hand, the double arrow suggests two conflicting concepts. The concepts of convective adjustment and the slow manifold are linked by an equal sign with a question mark, because they are closely related but clearly not equivalent.

Probably the main issue for CQE interpretations based on the thermodynamic analogy has been best expressed by Mapes’s (1997a; 1998) criticisms of the assumed causality (cf., Sec. 3.10). However, if CQE is interpreted as a dynamical balance condition, then no particular form of the causality has then to be assumed, and such criticisms immediately become irrelevant. Indeed, this review as a whole suggests that it would be more fruitful to consider CQE as being primarily a balance condition, with the thermodynamic analogy being a more specific view that can be useful in particular, more limited situations. The concept of a slow manifold provides a robust, but not yet fully exploited theoretical basis for developing CQE theories from a dynamical balance interpretation. Neelin and Zeng’s (2000) QTCM may be considered as taking such a first step.

Quasi-equilibrium

145

However, various obstacles for further developing CQE theories have also been identified. The most basic of these is the relationship between CQE and the free ride principle that has been used to constrain large-scale tropical dynamics (Sec. 4.5). By referring to essentially the same balance as CQE, simultaneous use of these two principles may lead to a tautological situation in which neither the large-scale circulation nor the convective heating is predictable (cf., Ch. 11, Sec. 14.1). A potentially serious issue for CQE is the possibility that the atmospheric convective system is at self-organized criticality (SOC: Vol. 2, Ch. 27). In contrast with the thermodynamic analogy for CQE, this would suggest the need for explicit consideration of contributions from fast convective processes to the large-scale processes. A similar issue may be anticipated by analogy with the slow manifold, as a consequence of Lighthill’s theorem (Sec. 4.4). The latter theorem might help us to tame the issues arising from the possibility of convective SOC. There still remain many investigations to be performed at a theoretical level. For example, it may be revealing to formulate and study possible statements of a variational principle (Sec. 3.3) for the thermodynamic analogy to CQE. Moreover, although a reinterpretation of CQE as a slow manifold suggests the applicability of rich resources from dynamical systems thinking, as well as Hamiltonian dynamics, this simple point is yet to receive full attention. Studies of atmospheric convection over many years have no doubt greatly enriched both our understanding and interpretation of convective quasi-equilibrium. Rich satellite data that has become available over the last decade for convection studies may especially be highlighted (Vol. 2, Ch. 16). Not least, this has led to the series of papers discussed in Sec. 7.5 through which the concept of the “transition to strong convection” has emerged as a promising direction. However, our conceptual understanding of CQE is hardly converged. The authors’ point of view is that Arakawa and Schubert’s (1974) equations defining the CQE hypothesis provide the basic, and far from exhausted, statement of this fundamental issue. It is furthermore proposed to make it customary to define the meaning of convective quasi-equilibrium in a given context whenever the phrase is invoked, because its meaning has so multiplied over the years that it can be hard to judge otherwise exactly what some references in the literature mean by CQE.

146

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Part I: Basic parameterization concepts and issues

Bibliographical note

c American This chapter is based on Yano and Plant (2012a), which is  Geophysical Union 2012.

Chapter 5

Tropical dynamics: Large-scale convectively coupled waves

ˇ Fuchs Z. Editors’ introduction: Although convection plays an important role in the weather and climate of the mid-latitudes, it is an even more central aspect of the weather and climate of the tropics. Variability in the tropics is predominantly controlled by large-scale convectively coupled waves within which dynamics, convection, radiation, and boundary-layer turbulence interact in complicated ways. Capturing those interactions is a difficult problem for numerical models generally, and in particular it is a severe test of the ability of convective parameterizations to respond to and influence other features of the model. Before proceeding to analyse the structure of mass-flux parameterization in Part II, this chapter considers the convectively coupled Kelvin wave in some detail, in order to reveal key aspects of the required behaviour of parameterizations. Volume 2, Ch. 15 deals further with this issue, considering how current parameterizations actually do perform in reproducing such waves in an operational global forecast model.

1

Introduction

Understanding the dynamics of the tropical atmosphere to this day remains a challenge, and there are still many fundamental questions for which we do not have answers. What makes it so complex is that the primary mechanisms of the mid-latitudes do not act in the same way in the tropics. Perhaps most important of these are geostrophic balance and baroclinic instability: they do exist, but they are not the primary mechanisms. The Coriolis parameter becomes smaller as we approach the equator, making it harder to establish a geostrophic balance. The surface temperature is 147

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Part I: Basic parameterization concepts and issues

close to uniform and on large scales the meridional temperature gradients needed for baroclinic instability are not strong enough. The most important mechanism in the tropics is deep convection, which is generally associated with high surface temperature (the higher the temperature, the stronger the deep convection), in particular high sea surface temperatures (SSTs) as the ocean dominates the tropical area. The ocean is a major source of the water vapour that gets lifted into the atmosphere; when it reaches condensation level it creates substantial cloud cover and that cloud cover disturbs the basic balance between the incoming solar radiation and outgoing longwave radiation. When the clouds are stratiform and dense the outgoing radiation is trapped, similar to the greenhouse effect. If we could claim to understand deep convection, we would be able to predict when and where it occurs. However, that is not the case and therefore it is hard to obtain a reasonably accurate weather forecast for the tropical areas, especially when one tries to forecast local convection. Even the intertropical convergence zone (ITCZ), the largest area of deep convection, related to the Hadley cell and the trade winds, not only shifts around the equator depending on the season, but is also known sometimes to vanish on any particular day at a particular place. Field projects in the tropics are extremely important due to the vast areas of ocean where we lack the data needed for research as well as for forecasts. In 2001 as a graduate student I was a part of Eastern Pacific Investigation of Climate Processes in the Coupled Ocean-Atmosphere System (EPIC) field project. For almost two months we flew from a small town, Huatulco, Mexico, 16◦ N, to the equator and back (Fig. 5.1). On our way to the equator we also performed case studies following the same zig-zag flight pattern every mission. We dropped sondes for the atmosphere as well as the ocean (Fig. 5.2). We had radar data as well. But, perhaps most importantly for me, it was a spectacular eyes-on experience. The ITCZ that we flew through would show itself in the full glory one day, while it would completely disappear the next day even though the SST remained the same and there was no obvious reason for it to vanish. As we flew, we would see an impressive development of the cumulus towers on one side of the plane (Fig. 5.3), while on the other side there could have been a perfectly calm sunny day. If all the local conditions were the same why did convection develop exactly at that spot. To this day we do not have many answers when it comes to tropical meteorology, but if I learned one thing while participating in EPIC, it is that it is truly fascinating.

Tropical dynamics: Large-scale convectively coupled waves

Fig. 5.1

(a) Fig. 5.2

EPIC2001 mission.

(b) Deployment of sondes, the EPIC campaign.

149

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Part I: Basic parameterization concepts and issues

Fig. 5.3

Somewhere over the ocean on the way to the equator.

Tropical cyclones, easterly waves, hurricanes, monsoons, the Madden– Julian oscillation (MJO: Madden and Julian, 1971) and convectively coupled equatorial waves (CCEW) are some of the disturbances that we know exist in the tropics. From satellite observations and field projects we know that CCEW that are found in the tropics are Kelvin waves, equatorial Rossby waves, inertio-gravity waves, and mixed Rossby-gravity waves. However, the greatest variability observed in the tropics is due to the MJO (Fig. 5.5). The MJO and CCEW are generally proceeded by low-level convergence, moistening, and upward motion in the lower troposphere, while upper tropospheric descent, cooling, and drying still prevail. Low-level moistening and warming due to shortwave radiation and surface fluxes result in destabilization of the boundary layer and the development of shallow convection. This is then gradually followed by the onset of deep convection associated with a rapid lifting of the moisture into the middle troposphere by congestus clouds. After a passage of heavy rainfall, stratiform precipitation follows. Initiation of shallow convection with a bottom-heavy vertical mass-flux profile, which develops into deep convection, and decays with

Tropical dynamics: Large-scale convectively coupled waves

Fig. 5.4

151

P3 flight through the hurricane.

stratiform convection characterized by a top-heavy vertical mass-flux profile is a signature of coupling between the free waves and local convection. That leads to a notion of convectively coupled waves. Historically, the analytical study of equatorial waves starts with work by Matsuno (1966). Matsuno analysed a shallow water system of equations in an adiabatic three-dimensional atmosphere and modelled the free equatorial waves. Those waves are called “free” as they do not couple with convection. However, they give us a starting point by showing how they may look in a moisture-free equatorial atmosphere. Those waves are also called “equatorially trapped waves” as they are confined to the area around the equator (their signal decreases towards the higher latitudes). Equatorial waves are long: their wavelengths are of the order of thousands of kilometres and in that sense they are very different from the mid-latitude gravity waves, for all that we may compare them with Rossby waves. When equatorially trapped waves pass through the moist (diabatic) atmosphere, they couple with convection, becoming convectively coupled equatorial waves. All free Matsuno-like waves couple with convection, but there is an additional

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Part I: Basic parameterization concepts and issues

Fig. 5.5 Space-time spectrum of OLR symmetric about the equator for equatorial waves: Kelvin waves, Rossby waves (ER), inertio-gravity waves (IG) (Wheeler and Kiladis, 1999) and MJO – dark contours at wavenumbers 1 to 4.

wave that emerges due to the moisture: the moisture mode. A moisture mode arises when two conditions are satisfied: precipitation increases with tropospheric humidity, and convection itself, possibly working with convectively coupled surface flux and radiation anomalies, tends to increase the humidity (Sugiyama, 2009a,b). There is increasing evidence that the MJO is in essence a moisture mode (Maloney and Esbensen, 2005; Raymond and Fuchs, 2009; Sobel and Maloney, 2013). Convectively coupled equatorial waves in the real three-dimensional moist atmosphere are not easy to model. Even the best general circulation numerical models (GCMs) that incorporate non-linearity as well as the extensive physics that we know about the atmosphere sometimes fail.

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153

In the cases when GCMs do a fairly good job in modelling the CCEWs, it is still hard to understand why, due to their complexity. Cloud-resolving models (CRMs) offer another approach for simulating CCEWs; they are simpler and can offer more physical insight (Fuchs et al., 2014). Yet another way is looking at analytical linear models; they are simple enough, linear, and most importantly give a straight answer as to which physical mechanism is the most important for the onset of the coupling of the largescale disturbance and convection. Historically, we can see three broad categories of the simplified models that try to characterize the role that moist convection, surface fluxes, radiation and cloud-radiation effects play in the large-scale dynamics of the tropical atmosphere. These are convergence-driven models, quasi-equilibrium models, and multiple vertical heating profile models. A theory for coupling between the large-scale tropical dynamics and convection called the “conditional instability of the second kind” (CISK) was originally proposed by Charney and Eliassen (1964). This theory became so popular that many researchers even take this acronym as a synonym for any mechanisms coupling between the large-scale dynamics and convection (see Ooyama, 1982, as a critical review). More specifically, Charney and Eliassen’s CISK assumes that convection is coupled to the Ekman pumping at the top of the planetary boundary layer. Hayashi (1970, 1971), and Lindzen (1974) propose that a similar coupling mechanism is at work between the equatorial waves and convection, but replacing the boundarylayer Ekman pumping by a low-level mass or moisture convergence. This mechanism has been named “wave-CISK” by Lindzen. The convergencedriven models or wave-CISK models ignore surface energy fluxes and variations in tropospheric humidity and tend to produce modes with the largest growth rates at the smallest scales, which is the opposite of what is expected in the tropics. Quasi-equilibrium models, applying the concept discussed in Ch. 4, Sec. 3.8, start with Emanuel’s (1987) linearized model for the MJO, in which he introduced a highly simplified form of the convective quasi-equilibrium hypothesis (cf., Ch. 4). Deep convection is assumed to be near equilibrium with mechanisms which create convective available potential energy (CAPE), while the importance of the role of surface fluxes for moisture or entropy is emphasized. More specifically, control of the surface flux rate by the surface wind is emphasized. For this reason, Yano and Emanuel (1991) propose to call this coupling mechanism the “wind-induced surface heat exchange” (WISHE). This mechanism is further pursued by Emanuel

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Part I: Basic parameterization concepts and issues

(1993b), Emanuel et al. (1994), and Neelin and Yu (1994). All the coupling mechanisms reviewed so far essentially consider only deep convection and its interaction with the first baroclinic mode. Mapes (2000), in turn, points out a potential importance of the interaction between the second baroclinic mode with convection. Here, a key process is the interplay of stratiform clouds, shallow convection, and downdraughts, whose heating tendencies give a direct link to the second baroclinic mode. This mechanism is called the “stratiform cloud instability”. Many twomode models were developed afterwards by Majda and Shefter (2001b,a), Majda et al. (2004), Khouider and Majda (2006, 2008) and Kuang (2008). To avoid the quasi-equilibrium and imposed two-vertical-mode assumptions from the above mentioned models, Fuchs and Raymond (2007) and Raymond and Fuchs (2007) developed a vertically resolved model that produces the observed complex vertical structure of convectively coupled Kelvin waves assuming a simple, sinusoidal vertical heating profile. The Raymond and Fuchs (RF07) model incorporates three crucial factors in its convective closure: the importance of convective inhibition, the control of precipitation by the saturation fraction (a column-averaged relative humidity) of the troposphere, and the effects of surface moist entropy fluxes. Convectively coupled Kelvin waves (CCKW) are large-scale disturbances that account for a substantial fraction of tropical rainfall variability; the second largest after the MJO (cf., Kiladis et al., 2009). CCKWs modelled by RF07 have a vertical structure and propagation speed that match the observations (Straub and Kiladis, 2002). A CCKW is unstable (i.e., growing in time), and there were no a priori assumptions made about its vertical structure, unlike in multiple vertical heating profile models. This chapter, by introducing the RF07 model, explains the mechanisms behind the coupling of free Kelvin waves with convection. It will also discuss the moisture mode, an additional mode that is unstable in the RF07 model, and compare it to the CCKW.

2

Model for convectively coupled Kelvin waves

The convectively coupled Kelvin waves are three-dimensional. However, they have a longitudinal dynamics identical to the convectively coupled gravity waves in a two-dimensional, non-rotating domain. In other words, the gravity mode from two-dimensional models maps onto the equatorial Kelvin wave in the real rotating Earth’s atmosphere. When developing a

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155

vertically resolved analytical model, the fact that looking at the mode in the two-dimensional (x, z) plane tells us how it looks in three dimensions becomes very beneficial. Here, x is taken in an eastward direction. Note that under this two-dimensional configuration with the Coriolis term turned off, a mode corresponding to the Kelvin wave propagates both eastward and westward. The mode that propagates in the eastward direction has a physical meaning in three dimensions. 2.1

Governing equations

The governing system of equations that the derivation starts from are primitive equations that include the momentum equations in the x, y, and z directions: → → → − − → 1 1− d− v = − ∇p − 2 Ω × − v +→ g + F tr , (2.1) dt ρ ρ the continuity equation: dρ → + ρ∇ · − v = 0, (2.2) dt and the three thermodynamic equations for buoyancy b, mixing ratio q, and moist entropy e: db = SB dt

(2.3)

dq = SQ dt

(2.4)

de = SE , (2.5) dt are buoyancy, moisture, and moist entropy source

where SB , SQ , and SE terms. The governing equations must then be simplified while keeping in mind that the ultimate goal is to be able to solve them analytically. Thus, assume → → − a non-rotating atmosphere 2 Ω × − v = 0 in hydrostatic equilibrium. The → − friction F tr is neglected, and only the (x, z) plane is considered. Further→ more, it is assumed that the fluid is incompressible, retaining only ∇· − v =0 as the continuity equation. Finally, the equations are linearized around the state of rest using the perturbation method, expressing every variable as a superposition of the equilibrium state and its perturbation. The basic state of every variable satisfies the governing system of equations, and the

156

Part I: Basic parameterization concepts and issues

perturbation values are small enough for their products to be neglected. The Boussinesq approximation is also applied. The linearized, slab symmetric governing equations that include the horizontal momentum equation, hydrostatic equation, mass continuity, and thermodynamic equations for buoyancy b, mixing ratio q, and moist entropy e in a non-rotating atmosphere with Boussinesq approximation are then: ∂u ∂Π + =0 ∂t ∂x

(2.6)

∂Π −b=0 ∂z

(2.7)

∂u ∂w + =0 ∂x ∂z

(2.8)

∂b + ΓB w = SB ∂t

(2.9)

∂q + Γ Q w = SQ ∂t

(2.10)

∂e + ΓE w = SE , (2.11) ∂t where all the variables are perturbations. Π is the scaled Exner function perturbation, obtained by factorizing the pressure gradient term. The ′ scaled potential temperature perturbation (or buoyancy) is b = gθ /TR , ′ where θ is the perturbation potential temperature and TR = 300 K is a constant reference temperature. The moist entropy perturbation is scaled by g/Cp , where g is the acceleration of gravity and Cp is the specific heat al¨ a of air at constant pressure. ΓB is the constant square of the Brunt–V¨ais¨ frequency. The quantity ΓE = de0 /dz, where e0 (z) is the scaled ambient profile of moist entropy. The scaled moist entropy source term is SE . The scaled mixing ratio anomaly is given by q = e − b and has the scaling factor gLv /(Cp TR ), where Lv is the latent heat of condensation. All the variables are assumed to have the x and t dependence of the wave form exp[i(kx − ωt)] with k and ω being the zonal wavenumber and frequency. Equations 2.6 to 2.11 lead to an equation for the vertical velocity w: k2 d2 w(z) 2 + m w(z) = SB (z) dz 2 ω2

(2.12)

Tropical dynamics: Large-scale convectively coupled waves

157

and polarization relations for the buoyancy and the scaled moist entropy perturbation: b = (i/ω)(SB − ΓB w)

(2.13)

e = (i/ω)(SE − ΓE w),

(2.14)

1/2 kΓB /ω.

where the vertical wavenumber is m = The heating profile is assumed to be:

SB = (m0 B/2) sin(m0 z)

z < h,

(2.15)

where B is independent of z and m0 = π/h, h being the height of the tropopause. The shape of the heating profile is here not allowed to vary with the phase of the wave. Fuchs et al. (2012) generalized this assumption and varied the heating profile assuming it to take the form: SB = Bm0 X exp(m0 νz) sin(m0 z),

(2.16)

2

where X = (1 + ν )/[1 + exp(πν)]. Varying the nondimensional parameter ν from negative to positive values allows us to change the vertical heating profile from bottom-heavy to top-heavy. ν = 0 brings us back to the neutral or unperturbed heating profile. Recall the importance of the heating profile, as suggested in Sec. 1. In the following, a detailed derivation will be presented for the case with the heating profile of Eq. 2.15 for simplicity, but the results will be presented for both cases. Solution of Eq. 2.12 with an upper radiation boundary condition yields:  

iπ m0 B z) + Φ exp − sin(m sin(mz) (2.17) w(z) = 0 2ΓB (1 − Φ2 ) Φ and substitution of this into Eq. 2.13 results in:  

iπ im0 B z) + exp − b(z) = − Φ sin(m sin(mz) . (2.18) 0 2ακ(1 − Φ2 ) Φ Here, 1/2

κ = hΓB k/(πα)

(2.19)

is the dimensionless wavenumber and Φ = ω/(ακ) = m0 /m

(2.20)

is the dimensionless phase speed. In the following, a dispersion relation will be presented in terms of this dimensionless phase speed Φ. The real part of Φ gives the phase speed of the waves, while the imaginary part gives a wave-growth rate, a measure of the instability of the waves.

158

2.2

Part I: Basic parameterization concepts and issues

Thermal assumptions

To a good approximation, the vertically integrated dry entropy perturbation source term SB depends on the precipitation rate minus the radiative cooling rate, the integrated moisture source term SQ depends on the evaporation rate minus the precipitation rate, and the integrated moist entropy source term SE depends on the evaporation rate minus the radiative cooling rate:  h SB (z)dz = P − R (2.21) B= 0

Q=



h

SQ (z)dz = E − P

(2.22)

SE (z)dz = E − R.

(2.23)

0

Ξ=



h

0

We assume an already-linearized evaporation rate by following Miller et al. (1992) to be: CΔqU us , (2.24) E= (U 2 + W 2 )1/2 where C is the transfer coefficient, Δq is the scaled difference between the saturation mixing ratio at the sea surface temperature and the subcloud mixing ratio, the parameter us is the perturbation surface zonal wind obtained from w via mass continuity, W ≈ 3 m s−1 is a constant needed to account for gustiness, and U is a constant ambient zonal wind. The radiative cooling rate takes the form:  h q(z)dz (2.25) R = αε 0

or

R = εP1 .

(2.26)

Here, α is a moisture adjustment rate and P1 is to be defined immediately below. The variable ε incorporates the effect of cloud-radiation interactions, which are assumed to cause a radiative heating anomaly in phase with precipitation. The precipitation P is the term where the most interesting closure insight appears. Here, it is written as a sum of two contributions: P = P1 +P2 . P1 = α



h

q(z)dz 0

(2.27)

Tropical dynamics: Large-scale convectively coupled waves

159

represents the moisture feedback and P2 = μCIN [λs E − λt b(D)]

(2.28)

represents the CIN feedback. P2 is further subdivided into a part P2s related to surface flux variations: P2s = μCIN λs E,

(2.29)

and a part P2t related to variations in the buoyancy above the boundary layer: P2t = μCIN λt b(D).

(2.30)

λs is a constant representing the sensitivity of convective inhibition (CIN) to the surface evaporation rate, μCIN is a parameter that governs the sensitivity of the precipitation rate to this inhibition, and λt is a constant representing the sensitivity of precipitation to variations in b at a nondimensional level D. D is expressed as a fraction of the tropopause height h and Dh is taken to be around 2 km. The term in Eq. 2.30 basically expresses that there will be less precipitation if there is a stable layer at elevation D, as one would intuitively expect. To summarize, the vertically integrated heating B from Eq. 2.21 can be written as the sum of three contributions, B = P1 + P2 − R:  h  h q(z)dz + μCIN [λs E − λt b(D)] . SB dz = B = P − R = α(1 + ε) 0

0

(2.31) Using the definition for the scaled mixing ratio anomaly given by q = e − b and the polarization relations of Eqs. 2.13 and 2.14 for buoyancy and entropy perturbations, as well as a definition for the vertical integral of the moist entropy source term SE from Eq. 2.23 we get:

  h 1+ε iκΦ + ε E + (1 − ΓM ) μCIN [λs E − λt b(D)] + ΓB wdz . B=− 1 − iκΦ 1 − iκΦ 0 (2.32) ΓM is a version of the gross moist stability of Neelin and Held (1987) and is defined here as:   h h ΓM = ΓE wdz ΓB wdz . (2.33) 0

0

Finally, combining Eqs. 2.17, 2.18, 2.24, 2.32, and 2.33 results in the dispersion relation κΦ3 + iΦ2 − κΦ − i + i(1 + ε)(1 − ΓM )F (Φ) − ΛG(Φ)/κ + (ε + iκΦ)[χt L(D, Φ) + χs ΛG(φ)/(1 + ε)]/κ = 0.

(2.34)

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Part I: Basic parameterization concepts and issues

Here, the dimensionless WISHE parameter Λ is defined as: Λ=

(1 + ε)m0 CΔqU 1/2

2αΓB (U 2 + W 2 )1/2

,

(2.35)

and it measures the degree of WISHE (wind-induced surface heat exchange) working on the system. The dimensionless parameters χt = λt μCIN m0 /(2α) and χs = λs μCIN represent the sensitivity of precipitation to buoyancy anomalies above the boundary layer and to surface entropy flux variations respectively. Fuchs and Raymond (2007) assumed that the scaled ambient moist entropy e0 (z) takes a piecewise linear form in the troposphere, which results in a gross moist stability ΓM equal to Δe[2H − 1 + cos(πH)] Δe[2H − 1 + cos(πH) + K(H, Φ)] ≈ , 2H(1 − H)F (Φ) 2H(1 − H) (2.36) where the last approximation corresponds to a limit of |Φ|2 ≪ 1, which is valid for all of the interesting modes studied here. Here, H is the fractional height relative to the tropopause of the minimum in the ambient moist entropy profile, and Δe = Δe0 /hΓB is the scaled difference between the surface and tropopause values of moist entropy (assumed to be the same) and the minimum value at the height Hh. Note that the approximate form for ΓM above is real and independent of Φ so that it can be treated as a constant external parameter. Furthermore, the auxiliary functions F (Φ), G(Φ), K(H, Φ), and L(D, Φ) are defined as:  π  π  Φ2 exp −i 1 − cos , (2.37) F (Φ) = 1 + 2 Φ Φ ΓM =

G(Φ) = 1 + exp(−iπ/Φ),  

 π  π πH K(H, Φ) = Φ2 exp −i + cos H − 1 − H cos , Φ Φ Φ L(D, Φ) = exp(−iπ/Φ) sin(πD/Φ) + Φ sin(πD). 3

(2.38) (2.39) (2.40)

Results

The goal is to solve the dispersion equation (Eq. 2.34) for non-dimensional phase speed Φ where its real part will give the phase speed of the modelled waves, while the imaginary part will show whether the waves are growing

Tropical dynamics: Large-scale convectively coupled waves

161

in time. The next step is then to convert the phase speed and the growth rate into physical space. The parameters used to calculate the dispersion curves in Fig. 5.6 were discussed at length in RF07 and their values are given in Table 5.1. These parameters are cloud-radiation interaction (CRI) parameter ε, scaled height of moist entropy minimum H, scaled magnitude of entropy minimum Δe, scaled height of CIN threshold layer D, sensitivity to convective inhibition χt , gross moist stability GMS ΓM , and sensitivity to surface entropy flux variations χs . The calculation of the roots of the dispersion relation in Eq. 2.34 was performed numerically.

Re(ω)/k (m/s)

50 25 0 -25

A - phase speed

-50

Im(ω) (1/day)

0.3

0

B - growth rate -0.3 0

5

10 l

15

20

Fig. 5.6 Real part of phase speed (upper panel) and imaginary part of frequency (lower panel) for the modelled modes for the unmodified heating profile. The solid lines show convectively coupled Kelvin waves, dotted lines show free Kelvin waves, and dashed lines show the moisture mode.

The upper panel of Fig. 5.6 shows the phase speed Re(ω/k) in ms−1 while the bottom panel shows the growth rate Im(ω) in units of day−1 . The phase speed and the growth rate are shown as a function of planetary wavenumber l defined as the circumference of the Earth divided by the zonal wavelength. There are three types of modes: one that corresponds to a free Kelvin wave, eastward and westward propagating with the phase speed

162

Part I: Basic parameterization concepts and issues Table 5.1 Non-dimensional free parameters used in calculation of the dispersion relation of Eq. 2.34. Parameter ε H ∆e ΓM Λ D χs χt

Value 0.2 0.5 0.26 ≈0 0, −0.28 0.17 7 12

Comment cloud-radiation interaction scaled height of moist entropy minimum scaled magnitude of entropy minimum approximate gross moist stability WISHE parameter scaled height of CIN threshold layer sensitivity to surface entropy flux sensitivity to stable layers

of 48 ms−1 and decaying; a convectively coupled Kelvin wave (CCKW) propagating eastward and westward with a phase speed of about 18 ms−1 and growing in time; and, the moisture mode, that is also growing in time. The CCKW has a maximum growth rate for the planetary wavenumber l = 7, which, together with the phase speed of 18 ms−1 , agrees well with the observations (Straub and Kiladis, 2002; Wheeler and Kiladis, 1999). The moisture mode is stationary when there are no mean easterlies present. It propagates slowly eastward under a presence of surface mean easterlies when χs = 0 so that WISHE is active (not shown). A more careful analysis (RF07) can show that the CCKW is unstable due to variations in CIN correlated with convection (see also Fig. 5.8 below), while the moisture mode is unstable due to cloud radiation interactions and negative gross moist stability (i.e., when the effective gross moist stability is negative). 3.1

Variations in vertical heating and moisture profile

It will now be shown how changes in the heating and moisture profiles affect the unstable modes in Fig. 5.6 (i.e., the convectively coupled Kelvin wave and the moisture mode). The heating profile from a dispersion relation from Fuchs et al. (2012) (not shown here) is varied from top-heavy to bottomheavy by varying the non-dimensional parameter ν in Eq. 2.16. Positive ν corresponds to a top-heavy heating profile, negative to bottom-heavy, while ν = 0 corresponds to the unmodified heating profile of RF07. The moisture profile is varied to yield different values of the gross moist stability for a fixed value of ν. Figure 5.7 shows the CCKW dispersion curves for different heating pro-

Tropical dynamics: Large-scale convectively coupled waves

163

files: the growth rate is highly dependent on the heating profile. The Kelvin mode is strongly unstable for the top-heavy heating profile while decaying for the bottom-heavy profile. The wavelength of the maximum growth rate shifts as well; for the top-heavy heating profile the maximum growth rate occurs at the larger planetary wavenumber l = 12, while in the control case it occurs at l = 7. The phase speed increases from 17 ms−1 for the bottom-heavy to 21 ms−1 for the top-heavy heating profile. As the physical mechanism responsible for destabilizing the convectively coupled Kelvin wave is variations in CIN, when varying χt it is found that the mode is unstable for a range of χt values, specifically for χt ≥ 1. The shift of the maximum growth rate to larger planetary wavenumbers is notable as χt values become larger (see Fig. 5.8). Varying the moisture profile via different values of GMS shows that the unstable CCKWs are not sensitive to GMS itself. However, they are more likely to occur when GMS is positive since positive GMS is generally correlated with the top-heavy heating profiles (not shown). 50

Re(ω)/k (m/s)

A - phase speed 25 0

top heating unmodified heating bottom heating

-25 -50 1

Im(ω) (1/day)

B - growth rate 0.5 0 -0.5 -1 0

5

10 l

15

20

Fig. 5.7 Convectively coupled Kelvin wave for top, unmodified, and bottom-heavy heating profiles.

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Part I: Basic parameterization concepts and issues

50

Re(ω)/k (m/s)

A - phase speed 25 χt = 12 χt = 6 χt = 3 χt = 1 χt = 0

0 -25 -50 1 B - growth rate

Im(ω) (1/day)

0.5 0 -0.5 -1 -1.5 0

5

10 l

15

20

Fig. 5.8 Convectively coupled Kelvin wave for different χt values for the top-heavy vertical heating profile.

The moisture mode, on the other hand, does not care about the CIN closure and if GMS is kept the same, it develops regardless of the shape of the heating profile. However, more negative GMS values result in more unstable modes as also occurs in the real atmosphere; unstable moisture modes are expected when effective   GMS is more negative. This can be understood if we take the limit of Φ2  ≪ 1, which is true for the moisture modes, from Eq. 2.34. The dispersion relation of Eq. 2.34 then approximately reduces to Φ = i (ε − ΓM ) /κ and there is no direct dependence on  CIN param the eter. Also (not shown here) if one takes the same limit Φ2  ≪ 1 from the more complex dispersion relation obtained with different heating profiles, one obtains the same relation Φ = i (ε − ΓM ) /κ from which it is possible to see that there is also no dependence on the ν parameter, i.e., on the vertical heating profile.

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3.2

Vertical structure

Figure 5.9 shows the vertical structure of the eastward-moving convectively coupled Kelvin wave in the x−z plane. Panels (a) and (b) show respectively the buoyancy anomalies at the planetary wavenumber of maximum growth rate for the unmodified heating profile and the top-heavy vertical heating profile. For both vertical heating profiles the characteristic boomerang structure from observations (Straub and Kiladis, 2002; Wheeler et al., 2000) and from cloud-resolving numerical simulations (Peters and Bretherton, 2006; Tulich et al., 2007) is seen. The westward tilting contours of buoyancy or temperature anomaly reach higher in the case of top-heavy heating profiles. 1

z/h

z/h

1

0.5

west

0.5

east

west

0

east

0 0

0.2

0.4

0.6 x/λ

(a)

0.8

1

0

0.2

0.4

0.6

0.8

1

x/λ

(b)

Fig. 5.9 Heating anomaly and buoyancy perturbation for the convectively coupled Kelvin wave. Shading represents the heating anomaly, light shading positive and dark shading negative. Contours represent the buoyancy anomaly, solid lines positive, and dashed lines negative. (a) the unmodified heating profile and (b) the top-heavy vertical heating profile.

Figure 5.10 shows the vertical structure of the moisture mode for planetary wavenumber l = 2 in the x − z plane. It shows the buoyancy anomalies and the convective heating for the unmodified and bottom-heavy heating profiles. The buoyancy anomalies are in phase with heating as the mode is stationary and there is no tilted structure. The vertical structure of the moisture mode is very different from that of the convectively coupled Kelvin wave. The buoyancy or potential temperature anomaly contours are arbitrary in Fig. 5.9 and Fig. 5.10, but their

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1

z/h

z/h

1

0.5

west

0.5

east

west

0

east

0 0

0.2

0.4

0.6 x/λ

(a)

0.8

1

0

0.2

0.4

0.6

0.8

1

x/λ

(b)

Fig. 5.10 As in Fig. 5.9, heating anomaly and buoyancy perturbation, only for the moisture mode. (a) the unmodified heating profile and (b) the bottom-heavy vertical heating profile.

ratio to the heating is fixed. We can use that ratio to evaluate the relative magnitude of the temperature anomaly between the two unstable modes. It turns out that the ratio between the potential temperature anomaly and heating for the convectively coupled Kelvin waves is an order of magnitude larger than for the moisture mode, confirming that in essence the moisture mode is a weak temperature gradient (free ride) mode (Sobel et al., 2001, Ch. 11, Sec. 14.1).

4

Conclusions

Out of many disturbances present in the tropical atmosphere, this chapter focused on the large-scale equatorial wave, or convectively coupled Kelvin wave (CCKW). The CCKW forms similarly to most large-scale disturbances in the tropics; it starts with shallow convection, develops into deep convection, and decays into stratiform convection. The linearized, slab symmetric, vertically resolved model was presented in this chapter. It included the CIN closure, the moisture closure, WISHE, and cloud-radiative interactions. As the gravity mode from two-dimensional models maps onto the equatorial Kelvin wave in the real rotating Earth’s atmosphere, the twodimensional model that was presented is sufficient to model the CCKW. Two types of unstable large-scale tropical modes, CCKWs and moisture

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167

modes, arise from two different convective forcing mechanisms, namely, coherent variations in CIN correlated with convection and a moisture feedback mechanism associated with small or negative values of gross moist stability. The results that were presented included a heating profile that can range from top-heavy to bottom-heavy (maximum heating in the upper and lower troposphere respectively) while the explicit mathematical analysis was presented for the neutral heating profile. It is demonstrated that top-heavy heating profiles favour convectively coupled Kelvin waves whereas bottomheavy profiles favour moisture modes. Starting from the primitive equations and applying some educated assumptions leads to a simplified system of equations for the diabatic atmosphere, a system that can be linearized and analytically solved. A particular virtue of this model is that it has a full vertical dependence and yet can still be fully analytically solved. This is often not the case. We are able to include many different convective closures or feedbacks and directly see in the results which one is responsible for the onset of, for instance, convectively coupled Kelvin waves. The downsides are that it is necessary to make the assumptions, however educated they may be, and that in order to solve the system analytically, it has to be linearized. These limitations depart from the situation that one finds in the real atmosphere. There are many tropical features that are likely to be under substantial non-linearity: see the scale analysis by Yano and Bonazzola (2009) and the observational analysis by Yano et al. (2009). According to the latter study, as well as a linear wave analysis by Delayen and Yano (2009), the MJO is likely to be strongly non-linear (see also Wedi and Smolarkiewicz, 2010). Observations for CCKWs, although limited, provide some insight into the mechanisms responsible for the Kelvin wave vertical structure. An example of a particularly clean CCKW (Straub and Kiladis, 2002) was observed during the Tropical East Pacific Process Study project (TEPPS; Yuter and Houze, 2000). During this project, the research vessel Ronald H. Brown was stationed near 125◦ W, 8◦ N for approximately two weeks in August 1997. The ship was equipped with a C-band scanning Doppler radar, and local weather observations were made, along with six radiosonde launches per day. RF07 analysed the radiosonde observations from this project to obtain the time series of deep convective inhibition (DCIN) and saturation fraction (precipitable water divided by saturated precipitable water). They showed that the deep convection and resulting precipitation were related to the moistening of the atmosphere, but the onset of precipi-

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tation was delayed approximately one day by the existence of a stable layer. In that case, DCIN played a significant role in the timing of the precipitation. The presented analytical model here shows the very same thing: the CIN variations that are responsible for the coupling of the Kelvin waves and convection.

5

Linear stability analysis and convection parameterization

This chapter has been devoted to a linear stability analysis of large-scale tropical atmospheric flows with a simple representation of convection. The approach here can be generalized in various different directions. In particular, a more sophisticated (operational) convection parameterization could be adopted. An analysis with more realistic parameterizations would help to more directly interpret the results of full GCM runs. However, once the complexity of the convection representation (parameterization) adopted in a model exceeds a certain degree, it is no longer practical to perform a linear analysis in an analytical manner as presented in this chapter. A numerical approach must be adopted. Presumably for this very reason, not many studies have been reported on the linear analysis of atmospheric models with more operationally oriented convection parameterizations. Brown and Bretherton (1995) is a major exception, presenting an investigation of a linear stability of the tropical atmosphere in combination with the Emanuel (1991) scheme by a numerical means. The goal of this chapter has been to understand the convectively coupled tropical waves. However, the linear analysis method can equally be exploited in order to understand the basic behaviour of a given convection parameterization. Even assuming that a given parameterization is constructed in a physically sound manner, it may still behave in an ill-defined fashion. The linear stability analysis can be used to detect such a problem of a parameterization. The wave-CISK instability discussed in Sec. 1, for example, provides such an insight: this instability presents a tendency for the fastest growth rate to occur for the smallest scales. The implication from this analysis is that any convection parameterization controlled by low-level convergence would tend to behave in a similar manner, leading to grid-point scale numerical instabilities.

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Although we call such behaviours “grid-point scale numerical instabilities”, whether a given instability is numerical or physical is always a difficult issue to judge. Individual convective elements would certainly grow at the finest scale of a large-scale model numerically. To that extent, the tendency for grid-point storms is, in fact, physically consistent with the expected dynamical behaviour of atmospheric convection. However, this is not quite a parameterization: the goal of parameterization is to keep the contributions of these individual convective elements to an implicit side of the model so that only their smooth ensemble effect can be seen, as emphasized by Ooyama (1982) as well as in Ch. 3 of this set. Yano et al. (1998) discuss the usefulness of linear stability analysis for understanding the basic behaviours of different convection parameterizations, along with some other methodologies, within the framework of a shallow-water analogue model.

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PART II

Mass-flux parameterization

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Introduction to Part II

Currently, the vast majority of operational forecast and climate-projection models adopt the mass-flux formulation for deep-convection parameterization. Part II systematically examines this formulation. Historically, the mass-flux formulation emerged from Riehl and Malkus’s hot-tower hypothesis proposed for the tropical large-scale circulation. This hot-tower hypothesis significantly contributed to the development of deepconvection parameterizations, firstly by identifying a clear need for deepconvection parameterization in global modelling, and secondly by suggesting a specific way in which deep-convection parameterization could be formulated. For these reasons, Part II begins in Ch. 6 by reviewing Riehl and Malkus’s hot-tower hypothesis. This was postulated in order to resolve an issue with the mean heat budget in the tropical atmosphere. The budget cannot be closed if heat transports within the tropical atmosphere occur through large-scale motion only, but some additional transport mechanism is required. This is the hot tower, and is closely associated with modern notions of tropical deep convection. The concept of a “mass flux” then arises as a measure of the rate of vertical transport by means of the hot towers. Chapter 7 introduces the traditional basic mass-flux formulation based on this picture. An important issue for the mass-flux formulation is the physical link between the hot towers and the mass flux. Each hot tower may be considered to be associated with a single mass flux value (as a function of height and time). A self-consistent formulation for the hot-tower mass flux can be systematically derived simply by imposing a geometrical constraint of a single hot tower embedded within a horizontally homogeneous environment. Such a formal derivation is accomplished by introducing a concept of the segmentally constant approximation (SCA), and this formulation is 173

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discussed carefully in Ch. 7. Chapter 8 supplements this presentation by more explicitly laying down the formulations for describing the heat and the moisture budgets under mass-flux parameterization. A key question to ask next is how many distinct mass-flux types need to be introduced into a parameterization. Here, the most stark choice is between a single (bulk) mass flux, and a range of multiple (spectral) mass fluxes. In many respects the two formulations are almost equivalent, but in other respects equivalency requires linearity assumptions, particularly in relation to microphysics. These issues are explained and discussed in Ch. 9. In traditional approaches, the issue of calculating the mass flux is separated into two distinct parts: one is to calculate its vertical profile, and the other is to determine an overall amplitude. In Ch. 10 the notion of entrainment and detrainment is introduced in order to formulate the problem of the vertical profile of mass flux. The amplitude part of the calculation is traditionally called the closure problem, and is the subject of Ch. 11. The last three chapters of Part II are devoted to further basic issues for completing the mass-flux convection parameterization formulation. In many of the actual, operational parameterizations, representations are required not only for the updraught mass flux of the convective towers, but also representations may be sought for the updraught vertical velocity (Ch. 12), for downdraughts (Ch. 13), and for momentum transfers (Ch. 14).

Chapter 6

Hot-tower hypothesis and mass-flux formulation

J.-I. Yano Editors’ introduction: The necessity of a convective parameterization for realistic simulation of the atmosphere on regional and global scales was stressed throughout Part I. Here, a further argument is presented which also suggests a possible form for such a parameterization. Specifically, in order to close the entropy budget of the tropical atmosphere, there must be some mechanism other than mean vertical advection which transports nearsurface values of moist entropy up towards the tropopause. The postulated mechanism, and the simplest way this might be achieved, is the so-called “hot tower”, a narrow region of fast vertical motion which interacts only weakly, if at all, with its immediate surroundings. Modest generalizations then suggest the consideration of a set of entraining–detraining plumes within which the transport is well described by a variable known as the “mass flux”. Mass-flux parameterization will be discussed extensively and elaborated upon greatly during Part II, but the basic idea is very simple.

1

Introduction

Presently, the majority of operational-forecast as well as climate-projection models adopt the mass-flux formulation for deep-convection parameterization. The purpose of this chapter is to introduce the basic concept of mass flux used in this formulation. The concept of mass flux is probably best understood by tracing it back to its historical root: the hot-tower hypothesis of Riehl and Malkus (1958). In other words, the mass-flux formulation has historically emerged from Riehl and Malkus’s hot-tower hypothesis, which was proposed in order to 175

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understand the tropical heat budget associated with the tropical large-scale mean circulation. In essence, the hypothesis rests on an extremely simple self-consistency argument for the mean heat budget in the tropical atmosphere: we cannot close the tropical mean heat budget unless we assume the existence of isolated, tall, convective vertical heat transport that happens without substantial interaction with the environment. Riehl and Malkus named this hypothetical entity the “hot tower”, a prototype for the contemporary notion of tropical deep convection. In order to represent such an entity as a subgrid-scale process, an “isolated” vertical transport process is required, which may be described by introducing a “mass flux”, a measure of vertical transport rate. This is the basic idea behind mass-flux convection parameterization. Once a mass flux is defined, it is relatively straightforward to define the vertical transport of both heat (entropy) and moisture. Thus, the key issue of mass-flux parameterization becomes that of defining the mass flux associated with convection. For this reason, the second half of this chapter is devoted to introducing the traditional, basic formulation for the mass flux.

2

Tropical large-scale mean circulation and heat budget

Both the need for convection parameterization and the concept of mass flux are best understood in terms of the hot-tower hypothesis of Riehl and Malkus (1958). Their basic idea can be understood in the context of a general picture for the tropical large-scale mean circulation. It consists of the two major branches: a meridional (latitudinal) circulation called the “Hadley circulation”, and zonal (longitudinal) circulations called Walker “circulations”. The tropical meridional (Hadley) circulation is characterized by mean ascent at the equator, and descent in the subtropics. The so-called Walker circulation consists of mean ascent and descent over the Western and the Eastern Pacific, respectively. Tropical, global, zonal circulations consist of two additional ascending branches over north Africa and South America, with descending branches in between. It has long been recognized that the tropical mean ascending branches are associated with active moist convection with enhanced precipitation, whereas the mean descending branches are stable, non-convective, and dry.

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Such a situation may be schematically represented by drawing a large, single convective cloud over an ascending branch of a circulation (Fig. 6.1a).

(a)

(b)

Fig. 6.1 Schematics for tropical large-scale mean circulations. (a) A naive classical view, which only consists of large-scale circulations, and (b) the modern view proposed by Riehl and Malkus (1958), in which the convective towers (hot towers) are embedded into the mean ascending branch.

A very simple point made by Riehl and Malkus is that we cannot naively look at the tropical large-scale mean circulation in this manner. Instead, we have to explicitly assume that deep moist convection consists of many localized entities, which may be called “hot towers” or convective towers. The upwards motions within these hot towers must be so intense that the environment outside of these localized entities is actually descending, even within a mean-ascending branch of the mean circulations (Fig. 6.1b). The basic argument by Riehl and Malkus is amazing simple: it solely hinges on examining the mean vertical profile of the atmospheric entropy. The observed mean thermodynamic profile forbids the construction of a self-consistent closed heat budget solely in terms of the large-scale mean ascent. Here, note that Riehl and Malkus considered only the meridional

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circulations in their study. However, the same argument equally applies to the zonal circulations. 3

Hot-tower hypothesis by Riehl and Malkus (1958)

A typical vertical profile of the total atmospheric entropy1 in the tropical atmosphere is schematically shown in Fig. 6.2 by a solid line. Note that the entropy is a conserved quantity under an adiabatic process, and thus it is a convenient variable for considering a heat (thermodynamic) budget. Here, it is important to remember the fact that latent heating associated with the condensation of water does not change the total atmospheric entropy. A major loss of atmospheric entropy occurs through radiative cooling, and the atmosphere gains entropy from the surface both as sensible heat (i.e., supplied to an internal energy) and water vapour. Furthermore, entropy is transported at the lower and the upper levels of the troposphere by mean convergence and divergence associated with the Hadley–Walker circulation. An additional minor loss is associated with precipitation as well as fusion of water. However, both contributions are expected to be minor and are not considered in the following. Here, it may also be noted that “heat budget” is very loose terminology albeit typically used in the meteorological literature. In thermodynamics, “heat” is merely a flux and not a thermodynamic quantity to be conserved. “Heat budget” actually means a thermodynamic budget associated with heat exchanges, as defined in the sense of the first law of the thermodynamics (cf., Ch. 1, Sec. 3.1). Atmospheric total entropy Sm may be related to the equivalent potential temperature θe by: Sm = Cp ln θe , where Cp is the heat capacity of moist air at constant pressure, and the equivalent potential temperature is given by: θe = (p0 /p)R/Cp T exp(Lv qv /Cp T ) to a good approximation with p the pressure, p0 = 1000 hPa the reference pressure, R the gas constant for air, T the temperature, Lv the latent heat of condensation, and qv the mixing ratio of the water vapour (cf., Ch. 1, Sec. 3.2). 1 Atmospheric entropy is usually called “moist entropy”, although for misleading reasons. The quantity is simply referred to as “entropy” throughout this chapter.

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Height

Tropopause

Smc Sm

Moist Entropy, Sm Fig. 6.2 A schematic of a typical vertical profile S¯m (solid line) of the atmospheric entropy (moist entropy) in the tropical atmosphere: it decreases with height up to the middle troposphere, and then turns to increase with height. Also shown is a possible profile Smc (dashed line) for a hot tower. The entraining and the detraining levels of the hot tower are indicated by a closed and an open circle, respectively.

The entropy, furthermore, approximately reduces to moist static energy when a hydrostatic balance is satisfied. In fact, the original budget analysis by Riehl and Malkus (1958) is based on the moist static energy. However, a more basic conservative quantity is invoked here for the analysis. Riehl and Malkus’s main argument simply hinges on examining the typical vertical profile of entropy in the tropical atmosphere, which is typically highest at the surface, decreases with height towards the middle troposphere, then begin to increase with height, as schematically shown by a solid line in Fig. 6.2. The atmospheric entropy decreases with height from the surface because moisture exponentially decreases with height, and moisture contributes an exponential factor in the definition of the equivalent potential temperature. However, above a certain height, a purely dry contribution begins to dominate since moisture begins to deplete. Since the atmosphere is typically stably stratified in a dry sense, entropy increases with height, as is the case in the upper half of the troposphere. Under this given vertical profile, Riehl and Malkus (1958) ask the question of how the heat (thermodynamic) budget can be closed over an as-

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cending branch of the Hadley–Walker circulation. Entropy is supplied from the surface by the sensible heat flux and evaporation, and then lost to space by radiation, predominantly at the tropopause level. Additionally, the strongest import and export of the entropy happens at a lower and an upper part of the troposphere, respectively, associated with convergent and divergent flows in the Hadley–Walker circulations. (Recall that the entropy is conserved even under condensation processes.) Thus, in order to close the thermodynamic budget, the entropy must be transported upwards from the surface level to the tropopause level. The main question is: how is the entropy transported upwards all the way through the troposphere so that the budget can be closed? A simple, physical consideration immediately tells us that the entropy budget cannot be closed by a large-scale upward movement w ¯ in the ascending branch of the Hadley–Walker circulation. Here, by large scale, we refer to the type of circulation that can readily be defined by a conventional radiosonde network with, say, a 1,000 km resolution. More precisely, Riehl and Malkus (1958) considered the climatology over the ascending region of the Hadley–Walker circulation (equatorial trough zone). Probably, this original interpretation of w ¯ (i.e., climatological average) makes it easier to understand the following argument, although a generalization of the notion of “large-scale” is relatively straightforward, as will be fully addressed later. Such a climatological (large-scale) ascent modifies the entropy S¯m with a rate: d (3.1) −w ¯ S¯m . dz This formula shows that such a transport of atmospheric entropy is always in the direction of a down slope of its vertical profile. Hence, under an ascending motion, the entropy is transported upwards in the lower half of the troposphere and downwards in the upper half of the troposphere by following the given vertical profile of the entropy schematically shown in Fig. 6.2. The obtained vertical advection rate is schematically shown by a solid line in Fig. 6.3. Importantly, the argument here is valid as long as the air is ascending all the way through the troposphere and does not depend on details of the vertical-velocity profile. Consequently, under the given large-scale circulation, the atmospheric entropy cannot be transported upwards all the way through the troposphere, but it accumulates in the middle troposphere, and thus the entropy budget is not closed. In other words, in order to accomplish an upwards

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Height

Tropopause

+ Moist entropy transport rate Fig. 6.3 Schematic curves for vertical transport rates of the atmospheric entropy as functions of height with two given vertical profiles in Fig. 6.2, under a vertical ascent. The climatological profile S¯m (solid line in Fig. 6.2) gives upwards (positive) and downwards (negative) transports in the lower half and upper half of the troposphere respectively (solid line). Meanwhile, a vertically homogeneous profile Smc (dashed line in Fig. 6.2) produces transport rates showing a sharp sink and source at the surface and the tropopause levels respectively (dashed line) as shown by Eq. 4.10 below. Those sinks and sources balance with the surface flux and radiative cooling, respectively, and thus the entropy budget can be closed.

transport of entropy all the way through the troposphere, a downward mean motion must be introduced in the upper half of the troposphere, which is not consistent with the observed structure of the Hadley–Walker circulation. Hence, something else must be going on. Riehl and Malkus (1958) proposed that a way to avoid this dilemma is to assume the existence of tube-like entities penetrating through the whole troposphere. Within these tubes, the surface entropy is more or less directly transported to the tropopause level without mixing with the air outside the tubes. In order for such a process to be realized, the vertical transport within the tubes must be fast enough so that surface air can reach the tropopause without substantial mixing with the surrounding air, as indicated by a dashed line in Fig. 6.2. The resulting vertical advection rate is schematically shown by a dashed line in Fig. 6.3: it provides a sharp sink

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and source at the surface and the tropopause levels respectively (dashed line) as shown by Eq. 4.10 below. Those sink and source balance with the surface flux and radiative cooling, as well as convergence and divergence, and thus the entropy budget is closed. The proposed tubes must be tall (rather than flat) so that their relative isolation is well ensured. These tubes are subsequently named the “hot towers”, phenomenologically corresponding to the deep convective towers of several-kilometre scales in the contemporary terminology. The proposed deep-convective transport process is fast and efficient, but only locally. The fractional area occupied by these hot towers within the whole ascending branch of the Hadley–Walker cell is relatively small, and thus the integrated effect of the transport by convective towers averaged over the whole Hadley–Walker cell is not at all faster than the one realized by the large-scale mean (climatological) transport in its order of magnitude.2 The main effect of introducing hot towers into the description of the Hadley–Walker circulation is to create a transported quantity wellpreserved against lateral mixing with the environment, as schematically represented by a dashed line in Fig. 6.2. As a result, the direction of the vertical entropy flux is changed, as summarized by Fig. 6.3. 4

Basis of the mass-flux formulation within the framework of Riehl and Malkus

Now to introduce a basic notion of the mass flux-based description for the convective vertical transport by elaborating the idea of convective tubes. The basic idea of Riehl and Malkus (1958) is to divide the mean vertical motion w ¯ averaged over an ascending branch of the Hadley–Walker circulation into two components: the convective updraught Mc and the environmental descent Me . Consequently, the climatological-mean ascent w ¯ within a Hadley– Walker circulation may be written: w ¯=

1 (Mc + Me ). ρ

(4.1)

Here, M∗ with subscript c or e is called the “mass flux”. Note that the mass flux M∗ is divided by the density ρ in order to recover the velocity. 2 In

standard scaling, the convective vertical velocity is assumed to be wc ∼ O(σc−1 ), thus we obtain Mc ∼ O(1) ∼ Me . See Sec. 4 for the notations.

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It can be interpreted as multiplication of a typical vertical velocity, say, w∗ associated with a corresponding component, and the fractional area σ∗ occupied by the corresponding component. Thus: 1 M∗ = σ∗ w∗ . ρ

(4.2)

Hence, the horizontal area over the whole ascending branch of the Hadley–Walker circulation is divided into two subdomains: hot towers (convective updraughts) and the environment, with fractional areas σc and σe respectively. Thus: σc + σe = 1.

(4.3)

Within each component, we assume that the air moves vertically with a horizontally homogeneous velocity wc or we . The transport rate within each component is given by the product of area and velocity, as given by Eq. 4.2. The idea is schematically represented by Fig. 6.4. The areas for convective updraughts are not necessarily confined to a single encircled subdomain, but it is more likely that they are distributed all over the whole ascending region. The total fractional area obtained summing all these individual elements is designated by σc . As already emphasized, each updraught element is well isolated. The remaining area is considered the environment, which is typically descending. Here, we assume that values of physical variables are horizontally homogeneous for each subdomain, although each of them is actually separated into a number of separate elements.3 As a result, the vertical transport rate of entropy Sm within each subdomain is given, in flux form4 , by: ∂ M∗ Sm∗ . (4.4) ∂z By taking a sum of the contributions from all of the subdomains, we obtain the total rate of vertical transport over the whole ascending branch of the Hadley–Walker circulation as: −

∂ ∂ ∂ Mc Smc + Me Sme , M Sm = ∂z ∂z ∂z

(4.5)

3 Alternatively, all of the physical variables could be considered as ensemble averages. In this case, however, additional assumptions would be required for deriving the following results. 4 The flux form is adopted in place of an advective form, as in the last section, for ease of deriving a final expression, seen in Eq. 4.10.

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Fig. 6.4 A schematic representing a distribution of convective updraughts (dark circles) within a square horizontal box (top view). The remaining area (shown in white) within the box corresponds to the environment.

where, as in Eq. 4.1, the overbar indicates climatological mean. At this point, it should be emphasized that the main reason for dividing the whole ascending branch of the Hadley–Walker circulation into two subdomains is not due to, for example, a convective vertical velocity wc much larger in magnitude than an environmental value we . Rather, it can primarily be attributed to the qualitatively different vertical profiles Smc for the transported variable within convective elements as compared to Sme in the environment. Such distinctively different vertical profiles for the environment and the convective updraughts are schematically shown in Fig. 6.2. In order to make this point in a totally trivial manner, let us consider an extreme situation in which the two vertical profiles are identical, i.e.: (4.6) Smc = Sme = S¯m . Substitution of Eq. 4.6 into Eq. 4.5, and with the help of Eq. 4.1, leads to: ∂ ∂ ∂ ∂ ¯ M c Sm + M e Sm = w ¯ Sm M Sm = (4.7) ∂z ∂z ∂z ∂z i.e., no convective transport effect is seen in the large-scale (climatological) average, although the convective vertical velocity wc may locally be substantially larger in magnitude than the environmental value we . Within the framework of Riehl and Malkus (1958), the entropy is assumed to be transported to a tropopause level zt from a surface level zs (not necessarily right at the surface, but at the top of the boundary layer, for example), without mixing. This implies a vertical entropy profile constant with height: (4.8) Smc = S¯m (zs ).

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185

We assume a constant convective mass flux throughout the updraught zone (i.e., zs < z < zt ): ⎧ ⎪ z ≤ zs ⎪ ⎨0 Mc = Mc,0 = const. zs < z < zt , (4.9) ⎪ ⎪ ⎩0 z≥z t

where zt is the top of the convective updraught. As a result, the convective vertical transport is, by introducing Dirac’s delta δ(z): ∂ ∂ − Mc Smc = −Smc Mc ∂z ∂z = −μSmc [δ(z − zs ) − δ(z − zt )], (4.10) where μ ≡ Mc,0 is the rate at which entropy is entrained into the hot tower at z = zs and that it is detrained at z = zt . It must be stressed again that the character of the transport given by Eq. 4.10 is qualitatively different from the one for the environment as given by Eq. 3.1 and as schematically indicted by a dashed line in Fig. 6.3. In this manner, the hot tower provides a distinct sink and source of entropy, respectively, at surface and tropopause levels, that balances with the large-scale tendency for the increase and decrease of entropy at the respective levels. Here, note that a description based on Dirac’s delta is, of course, an idealization of a more realistic continuous distribution of the convective sink and source. Here, it is also important to note how mass flux plays a role in defining the convective vertical transport. The mass flux is a key quantity that characterizes the feedback of convective processes to the large scale, as will become gradually clearer as subsequent discussions unfold.

5

Basis of the mass-flux formulation: Implementation into a grid box

In the last section, the notion of mass flux within the framework of Riehl and Malkus (1958) was introduced into a description of the whole ascending branch of the Hadley–Walker circulation. In implementing a mass-flux formulation as a convective parameterization, the idea must be applied to individual grid boxes. In other words, the square area marked by solid lines in Fig. 6.4 must be reinterpreted as representing a single grid-box domain. In order to justify all of the previous reasoning under this generalization, two conditions must be satisfied.

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Part II: Mass-flux parameterization

Scale separation

First of all, individual convective towers must be well isolated within a given grid box. In other words, the size of the individual convective towers must be much smaller than the size of the grid box. This principle is often referred to as “scale separation” (cf., Ch. 3). More precisely, this condition requires that a typical fractional area σc /N occupied by a convective tower must be much less than unity, i.e., σc /N ≪ 1. Here, N is the total number of convective towers (hot towers) in a grid box. Under the traditional formulation, the condition is further generalized into: σc ≪ 1

(5.1)

assuming that N remains finite when an asymptotic limit to σc /N → 0 is taken (cf., Ch. 3, Sec. 7). Recall that, by definition, the grid box-averaged entropy S¯m is given by a weighted average of the contributions from convective areas and the environment: S¯m = σc Smc + σe Sme .

(5.2)

The scale separation stated by Eq. 5.1 suggests that, because the weighting from the convection becomes negligibly small with this asymptotic limit, the grid-box mean can be approximated by the environmental value: S¯m ≃ Sme .

(5.3)

This approximation substantially simplifies the mass-flux convection parameterization formulation, as will be shown below. However, a major exception to the rule in Eq. 5.3 is the vertical velocity, because a finite contribution of the convective mass flux Mc to the transport process is essential in mass-flux convection parameterization. For this reason, it cannot be approximated as 1 w ¯ ≃ Me ρ which would simply negate the whole point of this parameterization. In order for the convective mass flux to remain finite under an asymptotic limit to σc → 0, the convective vertical velocity must asymptote as wc → ∞. In other words, the convective vertical velocity is a large quantity in this problem as already suggested by the basic notion of the hot tower. However, there is a further implication from the scale-separation principle: all of the quantities associated with the mass-flux formulation, i.e.,

Hot-tower hypothesis and mass-flux formulation

187

M∗ and Sm∗ with ∗ = c, e, must be smooth functions of model large-scale variables. Especially, convective variables such as Mc and Smc should not change drastically from one grid box to the next in order to ensure a smooth evolution of a global model under mass-flux parameterization. An important corollary is that, as a result, the solution of the problem should be relatively insensitive to the grid-box size of the model. It further suggests that the concept of the grid box used in the argument so far could be better replaced by a characteristic scale of the large-scale (climatological) circulation. There is a general discussion of this issue in Ch. 3, Sec. 5.2. 5.2

Convective quasi-equilibrium

The discussion of the last section was based on the climatological mean state, and no time dependence was considered. However, in a global modelling context, the system evolves with time, and thus decomposed variables such as M∗ and Sm∗ must also evolve with time. Nevertheless, in standard implementations of mass-flux convective parameterization, analogous procedures as outlined by Eqs. 4.8–4.10 are employed at every timestep for diagnosing M∗ and Sm∗ by assuming an instantaneous adjustment of convection to the environment (grid-box state). This hypothesis is called the “convective quasi-equilibrium” (cf., Ch. 4). Intuitively, if the scale separation is satisfied, and the convective scale is much smaller than that of the large-scale circulation in consideration, then such an assumption is expected to be satisfied. The exact manner of formulating a quasi-equilibrium is, however, far from trivial. This procedure is called the “closure problem”, and is an important issue in the actual implementation of a parameterization to be discussed further in Ch. 11. These are the two principles that must be introduced in order to apply the mass-flux formulation in a global model by generalizing the framework of Riehl and Malkus. It may also be emphasized that a much more complex vertical structure than Eq. 4.10 is adopted in operational applications. Nevertheless, the basic argument of Riehl and Malkus (1958) should remain valid, and a distinct profile of the entropy for the convective updraught should also remain vertically coherent, although some erosion inevitably happens through a limited interaction with the environment. In other words, in spite of the simplifications behind their theory, Riehl and Malkus (1958) provide an important pedagogical basis for understanding the basic concept of the mass flux.

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6 6.1

Part II: Mass-flux parameterization

Formulation: Heuristic derivation Tendency equation for environment

How is the large-scale evolution of a system described under the hot-tower hypothesis? With the discussion given so far, the formulation can now be presented in a heuristic manner. A more systematic derivation will be presented in the next chapter. The basic equation for atmospheric entropy Sm may be written as: 1 ∂ ∂ Sm + ∇ · uSm + ρwSm = QR + FS . (6.1) ∂t ρ ∂z Here, the second term on the left-hand side represents the horizontal advection effect, associated with the horizontal wind u, which describes the inflow and the outflow of the entropy at the surface and the tropopause level. The third term is the vertical advection. On the right-hand side, the source terms for entropy are presented: radiative heating QR and surface flux FS . Note that the contribution of the surface flux FS is always limited vertically to a surface layer, by definition. As already discussed, the total large-scale tendency associated with the Hadley–Walker circulation is, by neglecting the vertical transport, conceptually represented by: −∇ · uSm + QR + FS = Q0 [δ(z − zs ) − δ(z − zt )], where Q0 represents both the gain and the loss of entropy at the surface and the tropopause levels, respectively. Of course, this is an idealization, because in reality the tendency is not literally restricted to a narrow zone at the surface and the tropopause, as approximated by Dirac’s delta here. This total large-scale tendency is to be balanced by a tendency from the vertical convective transport given by Eq. 4.10. The two tendencies balance when Q0 = μS¯m (zs ). A closed expression of the large-scale equation with a mass-flux convection parameterization is obtained by more explicitly paying attention to the fact that the grid-box average is approximated by an environmental value under the scale separation (Eq. 5.3). Thus, we write down an equation for the environment as an approximation of that of the total grid-box average. Such an equation is obtained by averaging the entropy equation, Eq. 6.1, over the environment part of the grid box (the white area in Fig. 6.4). With a more careful derivation along these lines left for the next chapter, it is given by:   ∂Sme 1 ∂ ∂ Sme + ∇ · uSme + Me Sme = QR + FS + . (6.2) ∂t ρ ∂z ∂t c

Hot-tower hypothesis and mass-flux formulation

189

Here, the vertical advection term is replaced by the environmental descent −Me , and an extra tendency due to convection is added as (∂Sme /∂t)c to the right-hand side. Averaging over the environmental segment in Fig. 6.4 carefully, this term arises as a contribution due to the exchange of mass across the boundaries between the environment and the hot towers (convective towers). As the discussions so far suggest, the term is proportional to Eq. 4.10, and thus:   Smc ∂ ∂Sme Mc . (6.3) =− ∂t ρ ∂z c 6.2

Entrainment and detrainment

Here, for later purposes, the expression above is further generalized. Under the hot-tower hypothesis considered so far, the convective mass flux Mc has an abrupt shape given by Eq. 4.9. As a result, ∂Mc /∂z has an abrupt change, too: the only source (∂Mc /∂z > 0) and sink (∂Mc /∂z < 0) for the convective variable Smc are found only at the surface and the tropopause, respectively, as represented by Dirac’s delta in Eq. 4.10. However, we can generalize the vertical profile of Mc to any form (to be specified later). The source and sink then take continuous functions. In general, when ∂Mc /∂z > 0, then environmental air entrains into convection, and when ∂Mc /∂z < 0, then convective air detrains out into the environment. The sense of the two terms here, to “entrain” and to “detrain”, must be clear within the context. We define rates of entrainment and detrainment E and D by: E=

∂Mc , ∂z

(6.4a)

when ∂Mc /∂z > 0, and D=−

∂Mc , ∂z

(6.4b)

when ∂Mc /∂z < 0, respectively. From the perspective of the budget for the environment, when entraining, the environmental value is lost into convection, whereas when detraining, the convective value is passed into the environment. Thus, Eq. 6.3 must, in general, be written as:   1 ∂Sme (6.5) = [DSmc − ESme ]. ∂t ρ c

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Note that both entrainment and detrainment are often defined in terms of their fractional rates ǫ and δ: 1 ∂Mc , (6.6a) ǫ= Mc ∂z when ∂Mc /∂z > 0, and 1 ∂Mc , (6.6b) δ=− Mc ∂z when ∂Mc /∂z < 0, respectively. A full discussion on the concept of entrainment and detrainment is given in Ch. 10. 6.3

Large-scale tendency equation

A closed expression for the large-scale tendency equation is finally obtained by substituting Eq. 6.5 into Eq. 6.2, and recalling the approximation of Eq. 5.3: 1 ∂ 1 ∂ ¯ Sm + ∇ · uS¯m + Me S¯m = QR + FS + [DSmc − E S¯m ]. ∂t ρ ∂z ρ This equation can be rewritten in several different forms. First, by recalling Eq. 4.1, we may rewrite it as: 1 ∂ 1 1 ∂ ∂ ¯ Sm + ∇ · uS¯m + ρw ¯ S¯m = QR + FS + Mc S¯m + [DSmc − E S¯m ]. ∂t ρ ∂z ρ ∂z ρ (6.7) Now, the left-hand side represents the total Lagrangian tendency, i.e., the tendency of entropy moving with a large-scale wind. The right-hand side gathers the virtual source terms, or the apparent source in the terminology of Yanai et al. (1973). The last three terms are the terms due to convection. An alternative expression may be obtained by recalling the mass continuity for the grid-box average and for the convective component, respectively, given by: 1 ∂ ρw ¯ = 0, ∇·u+ ρ ∂z 1 ∂ Mc = E − D. ρ ∂z Note that the second equation is a direct consequence of the definition of the entrainment and the detrainment rates (Eqs. 6.4a,b). These constraints lead to an expression in advection form: ∂ ∂ ¯ Sm + u · ∇S¯m + w ¯ S¯m = QR + FS + Qc (6.8) ∂t ∂z

Hot-tower hypothesis and mass-flux formulation

191

with the convective tendency given by: D Mc ∂ ¯ Sm + (Smc − S¯m ). (6.9) Qc = ρ ∂z ρ Recall that under the hot-tower hypothesis, Smc is defined by an environmental value at an entrainment level (z = zs in the discussion so far). To allow entrainment to take place at other levels, a more sophisticated formulation is required, as will be discussed in the next chapter.

7

Convective mass flux

Note that the final expression for the convective source term given by Eq. 6.9 is fairly general regardless of the variable in consideration. There are only two unknown variables in this equation: the convective mass flux Mc and the convective variable Smc . As will be shown in next chapter, the latter can, in principle, be derived when the former is known. Thus, the key problem of mass-flux convection parameterization boils down to that of determining the convective mass flux Mc . Under the hottower hypothesis considered in this chapter, Mc is prescribed by a particular form, given by Eq. 4.9. Such a profile of convective mass flux can balance only with a particular vertical profile for the large-scale tendency. Thus, the formulation must be generalized from the case with the hot-tower hypothesis discussed so far. All of the associated issues will be discussed in the following chapters. This section provides a short outline. 7.1

Separation of variables

The issue of determining the convective mass flux can be separated into two major parts: its vertical profile η(z) and amplitude MB (t). This is enabled by a separation of variables: Mc = η(z)MB (t).

(7.1)

Note that such a separation of variables would not always be possible, but it becomes possible when convective quasi-equilibrium is assumed. Under this hypothesis, convection is at equilibrium with a large-scale state. As a result, the structure (i.e., vertical profile) of convection is steady with time. The strength (i.e., amplitude) of convection is, in turn, defined by a large-scale timescale. Thus, the amplitude evolves with time by following the evolution of the large-scale state.

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However, note that the argument for the separation of variables here is purely conceptual, and the actual time dependence of the problem is slightly more complicated. Although the vertical structure of the convective element is defined purely in a diagnostic manner, strictly speaking it is modified with time as the large-scale state evolves on which the diagnostic equation depends. This aspect becomes explicitly a problem under certain closure assumptions. 7.2

Cloud model

The problem of defining the vertical structure of convection is, in general, called the “cloud model”. A standard procedure is to define it through prescribed fractional entrainment and detrainment rates. By combining Eqs. 6.6a and b and invoking the separation of the variables from Eq. 7.1, we obtain: 1 dη = ǫ − δ. η dz Note that the use of fractional entrainment–detrainment rates is essential here. Were the full entrainment–detrainment rate E − D to be used instead, then such a separation of variables becomes no longer possible. 7.3

Closure problem

Once a vertical profile η(z) of convection is prescribed by a cloud model, the remaining problem becomes that of defining the amplitude (strength) MB (t) of convection. Traditionally, the amplitude is defined at the convection base, which is often equated with the cloud base. For this reason, a subscript B is used. This problem is called “closure”. Under the principle of convective quasi-equilibrium, the convective amplitude is defined solely in terms of the large-scale state or the large-scale forcing. This is a basic strategy of many of the closure hypotheses, which will be discussed in Ch. 11.

8

Bibliographical note

c the author, 2009, CC This chapter is based on Yano (2009), which is  Attribution 3.0 License.

Hot-tower hypothesis and mass-flux formulation

9

193

Historical notes

Riehl (1954) is a classic book on tropical meteorology, which is still revealing in many aspects even today. Riehl and Malkus (1958) must definitely be considered a classic. However, this original article devotes most of its discussions to the details of the issues associated with observational data in performing their heat budget analysis. The core idea of hot towers is only succinctly presented in p. 524–526 of their text. It may also be worthwhile to note that their budget analysis is limited to a two-layer configuration: they considered the bulk budget by dividing the whole troposphere into two layers. A continuous formulation would have made the budget analysis more involved without much gain, except that a limitation of the hot-tower hypothesis might have been revealed more explicitly, as suggested in the discussions in the main text. Finally, the phrase “hot tower” does not actually appear in the original paper, although it is used in their revisit, Riehl and Malkus (1979).

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Chapter 7

Formulation of the mass-flux convective parameterization

J.-I. Yano Editors’ introduction: This chapter presents the formulation of a conventional mass-flux parameterization as a sequence of steps starting from a full atmospheric equation set. This makes clear the various assumptions required, and the extent of the simplifications. It also makes it clear that intermediate levels of description for convection are possible by stopping the sequence at some earlier step. Such intermediate models with clear links to mass-flux parameterization have perhaps not received sufficient attention to date. As suggested in the previous chapter, the starting point is to impose a geometrical constraint by decomposing the vertical transports using top-hat functions. The space is divided into plumes within which variables are treated as constant in the horizontal but able to vary in the vertical. Key assumptions in the sequence are: to neglect the horizontal configuration of the various plumes by introducing entrainment and detrainment in such a manner as to satisfy continuity requirements; to introduce a scale separation; to neglect the rapid evolution of each plume; to separate the vertical structure calculations from all explicitly prognostic calculations; and simplifying the estimate of within-plume forcing terms. Each of these matters is discussed in turn.

1

Introduction

The basic idea of mass-flux convection parameterization was introduced in the last chapter by tracing this idea to its historical root in Riehl and Malkus’s hot-tower hypothesis. An outline of mass-flux convection parameterization was also presented in a rather heuristic manner. The purpose 195

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of the present chapter is to derive its full formulation. Generalizations of the mass flux idea are also attempted in this chapter. Riehl and Malkus’s hot-tower hypothesis says that the ascending branches of the tropical large-scale circulations essentially consist of two major components: hot towers (convective towers), and their surrounding environment. The goal of convection parameterization is to represent the small-scale hot towers as a subgrid-scale component of a model, as schematically represented in Ch. 6, Fig. 6.4. This chapter begins by explaining how such a configuration, dividing a grid box into components of hot towers and environment, can be considered as a geometrical constraint posed to a model that fully describes the processes within a grid box. Such a geometrical constraint will be called the “segmentally constant approximation” (SCA). Once this geometrical constraint is introduced, the formulation for the mass-flux convection parameterization can be derived from there by stepby-step simplifications of this system.

2

Basic ideas

First, let us recall various basic concepts already discussed in Part I. In deriving the formulation for mass-flux convection parameterization, we begin with a grid box as schematically represented by Ch. 6, Fig. 6.4. Of course, such a grid box does not exist in a literal sense in most of the numerical model formulations, as already emphasized in Ch. 3. Nevertheless, the concept is useful, and here it is affected that such a box literally exists. From this point of view, the grid box itself can be reinterpreted as its own numerical modelling domain with various processes going on. The multiple-scale asymptotic expansion introduced in Ch. 3 would be the most convenient way to visualize such a picture. These processes are called the “subgrid-scale processes” in respect of the large-scale (global) modelling. Models are available that can explicitly simulate the processes over a gridbox domain of typical global models. These are called “cloud-resolving models” (CRMs). In fact, attempts are under way to actually run a CRM for each grid box of a global model, as originally proposed by Grabowski and Smolarkiewicz (1999). These attempts are called “super parameterization” (cf., Ch. 2, Sec. 2.3). However, such an approach is numerically expensive, especially

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197

if a CRM is to take a full grid-box domain. Note that in actual implementations of super-parameterization approaches, the CRM domain size is typically much smaller than the actual grid-box domain, and it often only spans one horizontal direction. That is a short reason as to why a parameterization is required. A schematic provided by Riehl and Malkus’s hot-tower hypothesis provides a starting point for constructing a parameterization by adopting a geometrical constraint suggested in the hypothesis. The basic strategy here is to construct a subgrid-scale parameterization in a stepwise manner starting from a CRM. The starting point is to introduce a geometrical constraint SCA to a CRM.

3

CRM formulation: Non-hydrostatic anelastic model (NAM)

Before introducing SCA to a CRM, it is necessary to review the basic formulation of CRMs. CRMs typically take a mathematical formulation called the “non-hydrostatic anelastic approximation” (NAM). This is essentially a natural extension of the Boussinesq approximation developed for describing thermal convection of laboratory scales, to atmospheric convection associated with stratification of both density and entropy. Recall that the Boussinesq approximation amounts to the same concept as a standard incompressible description for fluid flows with the exception that it retains an effect of changes of density associated with change of temperature in the form of a buoyancy force contributing to the vertical momentum equation. The anelastic approximation furthermore adds to the formulation background states for density and entropy as functions of height ρ = ρ(z) and ¯ θ¯ = θ(z). Here, the bar on θ, and on other variables in the following, designates a background state, and entropy is represented by a potential temperature θ, which may furthermore be replaced by an equivalent potential temperature θe , as required. Note that the bar is not added to the density, because no perturbation density is to be considered in the following. The basic set of equations for NAM consists of the prognostic equations for the horizontal u and vertical w velocities, and the potential temperature θ: 1 1 ∂ ∂ u + ∇H · uu + ρuw = − ∇H p ∂t ρ ∂z ρ

(3.1a)

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Part II: Mass-flux parameterization

θ′ 1 ∂ 1 ∂p ∂ w + ∇H · wu + ρw2 = − +g ¯ ∂t ρ ∂z ρ ∂z θ

(3.1b)

1 ∂ ∂ θ + ∇H · θu + ρwθ = Q (3.1c) ∂t ρ ∂z and the mass continuity: 1 ∂ ∇H · u + ρw = 0, (3.2) ρ ∂z with g the acceleration due to gravity. Here, p is a pressure perturbation defined as a deviation from hydrostatic balance, and Q represents diabatic heating (including surface flux). Note that θ′ is a perturbation potential ¯ temperature defined by θ′ = θ − θ. Additionally to those equations, we may add equations for the moisture mixing ratio qv as well as for mixing ratios of various hydrometeors (e.g., cloud water, precipitating water, snow) designated by qµ , with the subscript μ designating the hydrometeor type: 1 ∂ ∂ qv + ∇H · qv u + ρwqv = Fv (3.3a) ∂t ρ ∂z ∂ 1 ∂ qµ + ∇H · qµ u + ρwqµ = Fµ , ∂t ρ ∂z

(3.3b)

where F with the corresponding subscript represents a source term (conversion rates to and from the other hydrometeors and water types, etc.) Note that all the prognostic equations (Eqs. 3.1a, b, c, 3.3a, b) can be summarized into a form: 1 ∂ ∂ ϕ + ∇ · uϕ + ρwϕ = F (3.4) ∂t ρ ∂z for an arbitrary prognostic variable ϕ. For a compact and general derivation of a formulation, in the following Eq. 3.4 is taken as a starting point of the deduction, along with the continuity equation, Eq. 3.2. 3.1

Poisson problem

The above set of equations, Eqs. 3.1 and 3.2, is technically closed once appropriate thermodynamic variables required for defining the buoyancy are included as a part of the set, through Eq. 3.1c. However, a problem with solving this system in practice is that there is no obvious way for evaluating the perturbation pressure p directly.

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199

Physically speaking, the perturbation pressure p must be defined in such a way that the mass continuity, Eq. 3.2, remains satisfied at the next timestep by integrating the momentum equation defined by Eqs. 3.1a and 3.1b. A required constraint is obtained by applying the divergence operator to the momentum equation multiplied by density ρ, which is constrained by mass continuity, Eq. 3.2. As a result, we obtain a Poisson equation for the perturbation pressure p: ∂ ρb, ∂z where v ≡ u + zˆw is the three-dimensional velocity. ∇2 p = −∇ · ∇ · ρvv +

4 4.1

(3.5)

Segmentally constant approximation (SCA) Basic concept

A key concept behind the hot-tower hypothesis is to divide a grid-box domain into the two subcomponents convection and environment, so that the grid-box mean is recovered by taking a weighted average of these two subcomponents: ϕ¯ = σc ϕc + σe ϕe

(4.1)

(cf., Ch. 6, Eq. 5.2). Here, the variables within convection and the environment are represented by single values ϕc and ϕe at each vertical level assuming that the distribution of each physical variable is horizontally homogeneous over a given subcomponent. In other words, each subcomponent can be considered a constant segment. This leads to the notion of the “segmentally constant approximation” (SCA). In the discussion of the last chapter, it was assumed that only one type of hot tower exists within a grid box. However, there is no reason to restrict the hot towers to only one type, and more generally, the convection would consist of a spectrum of hot towers (convective towers, or plumes). Note that a mass-flux convection parameterization is called “bulk” when it considers only one type of convection, and is called “spectral” when it considers multiple types. Most current convective parameterizations use the bulk formulation. Studies after Riehl and Malkus (1958) have shown that a downdraught Md , both in convective scales and mesoscales, contributes significantly to vertical transports (Zipser, 1969, 1977), and thus we may also add this effect into a decomposition of the vertical motion. Adding downdraughts

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is a relatively straightforward extension of the hot-tower hypothesis, with the schematic of Ch. 6, Fig. 6.4 being modified into Fig. 7.1.

Fig. 7.1 A schematic representing a distribution of convective updraughts (dark circles) and downdraught (grey circles) within a square horizontal box (top view). The remaining area (shown in white) within the square box corresponds to the environment.

Along this line of argument, it is even straightforward to consider N types of convective subcomponents embedded into a homogeneous environment. By adding subscript i as a subcomponent label, Eq. 4.1 can be generalized to: ϕ¯ =

N 

σi ϕi + σe ϕe .

(4.2)

i=1

The area occupied by the i-th subcomponent is designated by Si , and its boundary by ∂Si . Furthermore, a distribution of the variable ϕ over a grid box is defined as: N  ϕ= Ii (x, y, z)ϕi + Ie (x, y, z)ϕe (4.3) i=1

where Ii (x, y, z) is an indicator for the i-th segment, which is defined by:  1, if (x, y, z) ∈ Si Ii (x, y, z) = (4.4) 0, if (x, y, z) ∈ / Si . A similar definition also applies to the environment with the index i replaced by e. 4.2

Prognostic equation

A prognostic equation for each subcomponent (segment) i is obtained by averaging (integrating) the full prognostic equation, Eq. 3.4, over the area

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Si . An average of a variable ϕ over the i-th segment is designated by a subscript i, and defined by:  1 ϕdxdy, (4.5) ϕi = Si S i  where Si dxdy indicates a horizontal integral over the i-th segment, and Si is the area for the i-th subcomponent (segment). Using Leibniz’s theorem (Eq. A.1, as described in Sec. 10) we obtain:  1 1 ∂ ∂ σi ϕi + ρσi (wϕ)i = σi Fi . (4.6) ϕ(u∗ − r˙ b ) · dr + ∂t S ∂Si ρ ∂z n Here, S = i=1 Si is the total area of the model domain (which corresponds to the grid-box domain of the host model), and σi = Si /S is the fractional area occupied by the i-th subcomponent (segment). The second term on the left-hand side represents an effective horizontal divergence, including that due to the displacement r˙ b of the subcomponent (segment) boundary ∂Si . Here, an effective horizontal velocity u∗ that takes into account an inclination ∂ri,b /∂z of the boundary is defined by: ∂ri,b . (4.7) ∂z The direction of the integral in the second term is defined as a left direction with respect to the normal vector outward from the boundary. u∗ = u − w

4.3

Mass continuity

Mass continuity under SCA is obtained by directly averaging the mass continuity equation, Eq. 3.2, over a segment Si . By applying Leibniz’s theorem, Eq. A.1, we obtain:  1 1 ∂ ρσi wi = 0. (4.8) u∗ · dr + S ∂Si ρ ∂z

Alternatively, mass continuity can be obtained by setting ϕi = 1, Fi = 0 in Eq. 4.6:  1 1 ∂ ∂ σi + ρσi wi = 0. (4.9) (u∗ − r˙ b ) · dr + ∂t S ∂Si ρ ∂z

Furthermore, a difference between Eqs. 4.8 and 4.9 leads to:  1 ∂ r˙ b · dr. σi = ∂t S ∂Si

(4.10)

Eq. 4.10 provides a direct link between the displacement r˙ b of the segment boundary and the rate of change of fractional area σi occupied by the

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segment. Keep in mind that only two of the three equations (Eqs. 4.8, 4.9, and 4.10) are independent. Eqs. 4.6, 4.8, 4.9, and 4.10 constitute a basic set of equations describing the evolution of the SCA system. Under SCA, the problem of evaluating continuous physical fields is replaced by that of evaluating segmentally constant values ϕi . Yet, in order to close the system, some issues must still be resolved; these are discussed in the next three subsections. 4.4

Vertical flux: Standard mass-flux approximation

As described in the last chapter, the basic idea of the mass-flux formulation is to describe convective vertical transport in terms of convective mass fluxes: Mi = ρσi wi

(4.11)

(i = 1, . . . , N ). In other words, a physical value ϕi within each subcomponent segment is transported at the rate given by Mi . This amounts to a simplification of the vertical transport rate to: (wϕ)i = wi ϕi .

(4.12)

Note that the vertical flux term may more generally be written as: (wϕ)i = wi ϕi + (wi′′ ϕ′′i )i ,

(4.13)

where the second term on the right-hand side represents a contribution of deviations from the SCA. Here, the deviations are defined by wi′′ = w − wi and ϕ′′i = ϕ − ϕi . In the standard mass-flux formulation, this second term is neglected, leading to Eq. 4.12. 4.4.1

Eddy diffusion and mass-flux approach

Although in what follows the second term in Eq. 4.13 will be neglected, it may be relatively straightforward to take account of this term by following the standard procedures of the moment expansion method for boundarylayer turbulence (cf., Mellor and Yamada, 1974). The simplest choice would be just to take an eddy-diffusion formulation, and thus: ∂ϕi , ∂z where Ki is the eddy-diffusion coefficient for the i-th segment. A contribution of such effects for the environment component is taken into account by Soares et al. (2004), for example. (wi′′ ϕ′′i )i = −Ki

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203

Although it might be hoped that this term would disappear when the fractional area σi , occupied by the i-th segment, is small enough, that is not necessarily the case. The vertical transport rate for the i-th segment would be controlled by local vertical gradients on the scale Δz, which would be smoothed out by SCA to the resolved vertical scale ΔZ. In order to account implicitly for those unresolved vertical slopes, the eddy-diffusion coefficient should scale with (ΔZ/Δz)2 , which could compensate for a small fractional area. The LES (large-eddy simulation) analysis by Siebesma and Cuijpers (1995) indeed suggests that this term can be significant. However, it appears that no convection scheme has included such convective-scale fluctuations to date. 4.5

Segment boundaries

Values at segment boundaries (interfaces) are also to be specified. Specifically, segment-boundary values for the physical variables ϕ are needed in order to calculate fluxes crossing the segment boundaries. The simplest choice would be to take an upstream (upwind) value by following previous studies (e.g., Arakawa and Schubert, 1974; Asai and Kasahara, 1967). This choice also produces a stable scheme under a finite volume approach (cf., LeVeque, 2002). The issues are, however, more involved when the segment boundary coincides with the grid-box boundary. In this case, communication between neighbouring grid boxes must be introduced in order to enable proper horizontal flux calculations. This issue is addressed in the next section. A more serious issue is in determining the displacement rate r˙ b of the boundaries. There is no obvious principle to define this quantity because SCA is an imposed approximation to a full physical system. 4.6

Winds and pressure

It is important to keep in mind that in solving the SCA system of equations, information on the positions of the subcomponent segment boundaries ∂Si is required. Such geometrical information is explicitly required when solving for the winds and the pressure. Indeed, the winds and the pressure fields turn out to be the most difficult variables to evaluate under SCA. It is not possible to apply SCA to all three components of the wind. In the two-dimensional implementation by Yano

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Part II: Mass-flux parameterization

et al. (2010a), SCA is applied to the vertical velocity only, and then the vertical momentum equation (i.e., Eq. 3.1b here) is solved by casting it into the form of Eq. 4.6 above. The divergent-wind field is evaluated from the mass continuity equation, Eq. 3.2, with the vertical divergence term given under SCA. This leads to a segmentally linear distribution of divergent winds. The rotational component can be evaluated by applying SCA to the vorticity field. Note that the resulting rotational winds are also segmentally linear. The pressure problem is the hardest to handle. The pressure must be diagnosed by a Poisson equation, Eq. 3.5, that is obtained by taking the divergence of the three-dimensional momentum equation. Here, the horizontal distribution of the associated source term is far from being segmentally constant, and so is the inverted pressure field. The procedure introduced in Yano et al. (2010a) is to invert the Poisson problem in standard grid space. All of the physical variables under SCA are remapped into a continuous field, using the relationship of Eq. 4.3 for this purpose. A more discretized version of the Poisson solver is still be to developed at the time of writing. 4.7

Discussions

The model derived under SCA from NAM is coined NAM-SCA by Yano et al. (2010a). This paper also presents a self-contained formulation for NAM-SCA under a two-dimensional configuration. Furthermore, a threedimensional SCA system has been derived from the primitive equation system by Yano (2012). A full formulation for a three-dimensional NAM-SCA is rather involved, and only its sketch will be given in the remainder of this chapter. It is interesting to approach the NAM-SCA model from the point of view of numerical algorithms, since it is closely related to the finite-volume approach (cf., LeVeque, 2002). This correspondence becomes more evident if a relatively large number of segments is introduced. However, even in this case, the number of elements is still radically reduced in comparison with a CRM, in an analogous manner to how compression is performed in wavelet space (cf., Mallat, 1998): i.e., retaining a high local resolution where a high variability is found, and removing non-active elements as much as possible. As a result, NAM-SCA may be considered as a compressed-CRM (NAM), or in other words a numerically highly efficient CRM (NAM). Furthermore, the introduction of a NAM-SCA in place of a standard CRM (NAM) into a GCM may be considered as a compressed super-parameterization.

Formulation of the mass-flux convective parameterization

205

The number and distribution of the segments is prescribed first as an initial condition in time-integrating the NAM-SCA. However, as in the other finite-volume approaches, time-dependent adaptive mesh-refinement may be introduced, as in Yano et al. (2010a). Thus, some segments may be removed when they become inactive and new segments are added by following the evolution of disturbances with time. On the other hand, when very low numbers of segments are taken, a philosophical departure from the traditional finite-volume approach becomes evident. Under the strongest truncation, only a single constant segment may be placed within a grid box in order to represent the evolution of a single convective plume, as assumed by Yano and Baizig (2012). This configuration is geometrically identical to that of the bulk mass-flux parameterization. Thus, under this limit, NAM-SCA becomes a prototype for the bulk mass-flux parameterization. More generally, when N convective elements are considered, NAM-SCA constitutes a prototype for spectral type mass-flux convection parameterization by introducing only its purely geometrical aspect to a CRM (NAM).

5

5.1

Lateral exchange between convection and environment: Entrainment–detrainment hypothesis Lateral exchange between convection and environment

Eq. 4.6 may be rewritten for the environmental segment by replacing i by e:  1 1 ∂ ∂ σe ϕe + ρσe we ϕe = σe Fe . (5.1) ϕ(u∗ − r˙ b ) · dr + ∂t S ∂Se ρ ∂z It is seen that the environment is primarily affected by convection through the lateral exchange process, expressed by the second term on the left-hand side of the equation. Thus, this becomes a key term in order to understand the interactions between convection and the environment. This section examines the lateral exchange term in detail. For this purpose, the first step is to categorize the environment boundary ∂Se which consists of the grid-box boundary ∂SG and the boundary with the segments for the other subgrid-scale subcomponents ∂Si . Here, by following the schematic in Fig. 7.1, and also for simplicity, it is assumed that the whole grid-box boundary is only adjacent to the environment. This is a standard assumption adopted in current mass-flux parameterizations.

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Part II: Mass-flux parameterization

Thus: ∂Se = ∂SG −

N 

∂Si .

(5.2)

i=1

Note that a minus symbol is added to the boundaries with the subgrid-scale subcomponent segments, because the direction of various counter integrals to be considered must be reversed. Substitution of Eq. 5.2 into the second term of Eq. 5.1 leads to:    N  1 1 1 ∗ ∗ ϕ(u − r˙ b ) · dr = ϕ(u − r˙ b ) · dr − ϕ(u∗ − r˙ b ) · dr. S ∂Se S ∂SG S ∂Si i=1 (5.3) In the first term on the right-hand side, u∗ − r˙ b can be replaced by the actual horizontal velocity u assuming that the grid-box boundary is perpendicular and fixed with time. Then, it is seen that this term is equivalent to the horizontal divergence evaluated within the large-scale model, thus:   1 1 ¯ · ϕe ue . ϕ(u∗ − r˙ b ) · dr = ϕu · dr = ∇ (5.4) S ∂SG S ∂SG ¯ in order to indicate emphatically that the Here, the bar is added to ∇ divergence operation is for the large-scale variables. Note that ue is the large-scale horizontal velocity for the environment, defined in the same sense as ϕe .

5.2

Entrainment–detrainment formulation

Let us now focus on the second term on the right-hand side of Eq. 5.3. In order to calculate this term, it is necessary to specify a value of ϕi,b at the segment boundary ∂Si . This choice is not obvious, because ϕ is discontinuous over the segment boundary. Here, based on a physical intuition, it is assumed that a value from the upwind side is transported on crossing the boundary. This approximation is called either “upwind” or “upstream” in numerical computations. In the present context, the value of ϕi,b is chosen depending on whether the given subcomponent segment is diverging or converging, and thus:   ϕi , if ∂Si (u∗ − r˙ b,i ) · dr > 0 (5.5) ϕb,i =  ϕe , if ∂Si (u∗ − r˙ b,i ) · dr < 0.

Note that the above formulae are correct only if the physical variable is conservative when crossing a subgrid-scale component-segment interface.

Formulation of the mass-flux convective parameterization

207

If it is non-conservative, then a change associated with entrainment and detrainment must also be added to the above definitions, as seen in Ch. 8, Eqs. 4.5a and 4.5b. The definition of the segment boundary value by Eq. 5.5 permits ϕ to be taken out of the integrand, leading to:   ϕb,i 1 ϕ(u∗ − r˙ b ) · dr = (u∗ − r˙ b ) · dr. S ∂Si S ∂Si

Next, note that the remaining integral corresponds (apart from a density factor) to the detrainment Di and entrainment Ei already introduced in the previous chapter. Thus:    Di , if ∂Si (u∗ − r˙ b,i ) · dr > 0 ρ ∗ (5.6) (u − r˙ b ) · dr =  S ∂Si −Ei , if ∂Si (u∗ − r˙ b,i ) · dr < 0.

It is furthermore assumed that

Di = 0 when



(u∗ − r˙ b,i ) · dr ≤ 0,

∂Si

and

Ei = 0 when



(u∗ − r˙ b,i ) · dr ≥ 0,

∂Si

respectively, then:  1 1 ϕ(u∗ − r˙ b ) · dr = [Di ϕi − Ei ϕe ]. S ∂Si ρ Substitution of Eqs. 5.4 and 5.7 into Eq. 5.3 leads to:  N  1 ¯ · ϕe ue − 1 ϕ(u∗ − r˙ b ) · dr = ∇ [Di ϕi − Ei ϕe ], S ∂Se ρ i=1

(5.7)

(5.8)

and substitution of Eq. 5.8 into Eq. 5.1 leads to a final expression of the equation for the environment: N

 1 ∂ ∂ ¯ · ϕe ue − 1 σe ϕe + ∇ ρσe we ϕe = σe Fe . (5.9) [Di ϕi − Ei ϕe ] + ∂t ρ i=1 ρ ∂z

This equation may be compared with Ch. 6, Eq. 6.2. The influence of convection on the environment is expressed in terms of the entrainment and detrainment processes in Eq. 5.9.

208

5.3

Part II: Mass-flux parameterization

Prognostic equation for the i-th subcomponent segment

The corresponding prognostic equation for the i-th subcomponent segment is obtained by substituting Eq. 5.7 into Eq. 4.6: 1 1 ∂ ∂ σi ϕi + [Di ϕi − Ei ϕe ] + ρσi (wϕ)i = σi Fi . (5.10) ∂t ρ ρ ∂z Note that in deriving Eq. 5.10, it has been assumed that the i-th subcomponent does not cross the grid-box boundary. Otherwise, an additional term is required as shown in Sec. 7.4. 5.4

Prognostic equation for the grid-box mean

By taking a sum of Eqs. 5.9 and 5.10, we obtain a prognostic equation for the grid-box mean:    N   ∂ ϕ¯ ∂ ϕ¯ 1 ∂ ∂ ¯ ϕ¯ + ∇ · uϕ¯ + ρw ¯ ϕ¯ = + , (5.11) ∂t ρ ∂z ∂t e i=1 ∂t i where the terms on the side have been defined by:  right-hand  ∂ ϕ¯ 1 ∂ ρσi wi ϕ′i (5.12a) = σi Fi − ∂t i ρ ∂z   ∂ ϕ¯ ¯ · uϕ′ − 1 ∂ ρ(1 − σc )we ϕ′ , (5.12b) = (1 − σc )Fe − ∇ e e ∂t e ρ ∂z and ϕ′i = ϕi − ϕ¯ is the deviation from the grid-box mean. 5.5

Mass continuity

Under the entrainment–detrainment formulation of Eq. 5.6, the mass continuity given by Eq. 4.9 reduces to: ∂ ∂ ρσi wi = Ei − Di (5.13) ρ σi + ∂t ∂z for a subcomponent segment, and: ∂ ¯ · ue + ∂ ρσe we = D − E (5.14) ρ σe + ρ∇ ∂t ∂z for the environment. Recall that the large-scale convergence contribution to the mass continuity of the environment stems from Eq. 5.4. Here, total entrainment E and detrainment D rates are introduced by: N  Ei E= i=1

D=

N  i=1

Di .

Formulation of the mass-flux convective parameterization

5.6

209

Entrainment–detrainment hypothesis

In rewriting Eqs. 4.6 and 5.1 into Eqs. 5.9 and 5.10, entrainment and detrainment E and D are nothing other than short-handed expressions for the total convergence and divergence associated with a given subcomponent segment, as defined by Eq. 5.6. In order to get the entrainment and detrainment rates right, it is still necessary to establish an exact distribution of u∗ . The entrainment–detrainment hypothesis more emphatically claims that it is possible to replace the horizontal-velocity information u∗ by more compact information expressed in terms of entrainment and detrainment rates E and D. The advantage of introducing entrainment and detrainment in this way stems from the complex procedure for determining the horizontal wind field under the NAM-SCA formulation as discussed in the last section. At a conceptual level, the issue may be best understood by further examining the mass continuity equation, Eq. 4.8, which may be rewritten as:  1 ∂ 1 ρσi wi . (5.15) u∗ · dr = − S ∂Si ρ ∂z This equation states a relation between the horizontal divergence (left-hand side) and the vertical divergence (right-hand side). Hence, if either side is known, the other is immediately diagnosed as a result. However, the exact use of this equation depends on the whole model formulation. Under the NAM-SCA formulation, the mass continuity equation, Eq. 5.15, is used to define the divergent-wind field (left-hand side) from the vertical velocity (right-hand side), which is prognostically calculated. The basic idea of the entrainment–detrainment formulation is to reverse this procedure: the horizontal-wind divergence (left-hand side) is prescribed in terms of the parameters entrainment and detrainment, and then the vertical velocity (right-hand side) can be diagnosed from the given horizontalwind divergence. This simplifies the computations a great deal. First of all, there is no longer the need to evaluate the divergent-wind field through a complex procedure, as outlined in the last section. It is also no longer necessary to integrate the vertical-momentum equation in time in order to evaluate the vertical velocity. Within this entrainment–detrainment framework, the vertical velocity is simply diagnosed by mass continuity. A further major advantage is that, as a result, there is no longer any need to specify the positions of each subcomponent segment. As emphasized in Sec. 4.6, this geometrical information is required mainly for evaluating the

210

Part II: Mass-flux parameterization

pressure and the horizontal velocity. By removing these needs, it is also no longer necessary to specify the positions of segments. A fractional area σi occupied by a given segment becomes the sole quantity necessary for geometrically characterizing a subcomponent segment. More precisely, it is the mass flux, Eq. 4.11, that is diagnosed from the mass continuity of Eq. 5.15 within the entrainment–detrainment framework. As already emphasized in the last chapter, this quantity also characterizes the convective vertical transport of physical variables. This is the main reason that to pursue a mass-flux form of parameterization. As a result, the role of the vertical velocity is replaced overall by the mass flux, but with some caveats (see Sec. 6.9).

6

Asymptotic limit of vanishing fractional area for convection

6.1

Scale-separation principle: σi → 0

The standard mass-flux formulation, as originally derived by Arakawa and Schubert (1974), is built upon an asymptotic limit that the fractional area σi covered by each convective element is much smaller than σe for the environment, thus: σi ≪ σe . It may furthermore be assumed that the total fraction: σc =

N 

σi ,

i=1

occupied by convection is much smaller than unity: σc ≪ 1. Thus: σe ≡ 1 − σc ≃ 1. Under a standard notation for asymptotic expansion, we may write: σi → 0,

σc → 0, and σe → 1.

See Ch. 3 for some basic concepts behind asymptotic limits.

(6.1)

Formulation of the mass-flux convective parameterization

6.2

211

Consequences of the asymptotic limit σi → 0

Within this asymptotic limit, a scaling can be taken that the magnitude of the convective variables are of the same order of magnitude as those of the environment, i.e., O(ϕi ) ∼ O(ϕe ), except for the vertical velocity and forcing. For these, it is assumed that O(σi wi ) ∼ O(we ) and O(σi Fi ) ∼ O(Fe ). As a result of these scalings, the grid-box mean may be approximated by the environmental values: ϕ¯ ≃ (1 − σc )ϕe ≃ ϕe

(6.2)

by recalling the relation: ϕ¯ = σe ϕe +

N 

σi ϕi .

i=1

However, keep also in mind ¯− σe we ≃ we ≃ w

N 

σi wi ,

N 

σi Fi ,

i=1

σe Fe ≃ Fe ≃ F¯ −

i=1

and we = w ¯ and Fe = F¯ . These two inequalities cannot be neglected even approximately. The approximation of Eq. 6.2 implies that: ∂ ∂ ϕ¯ ≃ (1 − σc )ϕe , ∂t ∂t

(6.3)

and thus the tendency for the grid-box mean can be approximated by that of the environmental component. Also, note that the tendency given in Eq. 5.12b is approximated by:   ∂ ϕ¯ ≃ (1 − σc )Fe ≃ Fe , (6.4) ∂t e because ϕ′e → 0 under the asymptotic limit.

212

6.3

Part II: Mass-flux parameterization

Convective-scale collective balance

A simple corollary from Eq. 6.3 is that the total convective tendency is negligible compared to that for the grid-box mean, i.e., N

∂  σi ϕi ≃ 0. ∂t i=1

(6.5)

It is proposed that this constraint be called the “convective-scale collective balance”. Taking the sum over components of Eq. 5.10 and imposing the condition of Eq. 6.5, we obtain: N N N  1 ∂  1 σi Fi , (Di ϕi − E ϕ) ¯ =− Mi ϕi + ρ i=1 ρ ∂z i=1 i=1

(6.6)

where the approximation of Eq. 6.2 has also been applied. This relation constrains the total entrainment–detrainment rate by the total vertical flux divergence and the physical forcing. 6.4

Steady-plume hypothesis

However, keep in mind that the approximation ∂ σi ϕi ≃ 0 (6.7) ∂t for the individual convective component i is not necessarily justified under the formulation presented so far. The asymptotic limit σi → 0 merely implies that the temporal tendency associated with ϕi follows the scaling ∂/∂t ∼ O(σi−1 ) → ∞ so that the left-hand side of Eq. 6.7 remains of the order unity as a whole, i.e., (∂/∂t)σi ϕi → O(1). This scaling states a simple fact that when the scale of convection is much smaller than that of the large scale, convection evolves proportionally faster. Here, it is recognized that the existence of such a fast timescale in subgrid-scale components does not induce fast processes in the large-scale components (a consequence that must be avoided) as long as the condition of convective-scale collective balance, Eq. 6.6, is satisfied. However, such a fast process is not of great interest in parameterization, and thus better avoided in the description from a practical point of view. For this reason, traditionally, the condition of Eq. 6.7 is introduced as a further simplification of the treatment. The assumption of Eq. 6.7 is called the “steady-plume hypothesis”, because each convective element is also

Formulation of the mass-flux convective parameterization

213

traditionally modelled by a plume, as will be discussed in detail in Ch. 10. By introducing this additional condition, it is still necessary to address the sub-ensemble of the subgrid components designated by each index i, but not the evolution of an individual subgrid-scale component. In other words, if the evolution of individual convective elements is to be considered explicitly (e.g., for triggering), then the condition of Eq. 6.7 must be removed. It is important to realize that the steady-plume hypothesis is not an essential part of the mass-flux formulation. The hypothesis could be removed without changing anything else that is central to the formulation, because the whole formulation remains self-consistent regardless of whether the steady-plume hypothesis is maintained, provided that the convectivescale collective balance of Eq. 6.6 is satisfied. Although it is sometimes argued that such a short timescale convective evolution should explicitly be considered in a parameterization, it appears not to be widely recognized in the literature that the introduction of such a fast timescale process is, in principle, straightforward for the reason explained above. We require only a method to describe the fast process. On the other hand, if a fast process, such as a trigger (cf., Ch. 11, Sec. 13.2) is introduced into a parameterization, then the above steadyplume hypothesis must clearly be removed for consistency. Unfortunately, the existing literature is careless also in this respect. However, the merits for introducing a transient behaviour of convection into a parameterization are not well established. Some of the literature tends to simply argue for its importance without carefully explaining why or how. Notice that the neglect of transience by introducing the steadyplume hypothesis of Eq. 6.7, does greatly simplify the parameterization formulation. Also, the discussions in Ch. 2, Sec. 2, based on a simplified mathematical model, do suggest that such transient behaviour can usually be neglected for parameterization purposes. Here, note also a link between the steady-plume hypothesis and the entrainment–detrainment formulation. If the spatial location of individual elements is to be considered as important to the formulation, then it would not be advisable to make a steady-plume hypothesis that would fix the steady plumes at particular locations for all times. If the locations of all the elements matter, then triggering and plume life-cycles would matter, too. On the other hand, using an entrainment–detrainment formulation itself does not force the decision as to whether to introduce steady plumes or not.

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Part II: Mass-flux parameterization

6.5

Equations for the i-th subgrid component

The prognostic equation for i-th subgrid component within the limit of σi → 0 does not change from the general case given by Eq. 5.10, except for the replacement of ϕe by ϕ. ¯ Introduction of the steady-plume hypothesis of Eq. 6.7, however, does simplify the matter, and the prognostic equation reduces to a diagnostic form: 1 ∂ 1 [Di ϕi − Ei ϕ] ρσi (wϕ)i = σi Fi ¯ + ρ ρ ∂z or ∂ Mi ϕi = Ei ϕ¯ − Di ϕi + ρσi Fi . (6.8) ∂z By taking the limit σi → 0 in Eq. 5.13, the mass continuity for the i-th convective element becomes: ∂ Mi = Ei − Di . (6.9) ∂z Thus, the asymptotic limit of vanishing fractional areas for convection also leads to a diagnostic description of mass continuity for the convective elements. With the help of Eq. 6.9, Eq. 6.8 can furthermore be rewritten as:

Mi ∂ ϕi σi Fi = −ǫi (ϕ¯ − ϕi ) + ρ ∂z or ∂ ϕi = ǫi (ϕ¯ − ϕi ) + ρσi Fi /Mi , (6.10) ∂z where ǫi = Ei /Mi is the fractional entrainment rate. This equation states that the subgrid-scale (convective) state is nudged towards an environmental state according to the fractional entrainment rate as it moves upwards. For example, moist convective updraught air becomes less moist as it moves upwards by entraining the drier surrounding air. 6.6

Equations for the grid-box mean

The equation for the grid-box mean is obtained by substituting Eqs. 5.12a and 6.4 into Eq. 5.11: N 1 ∂ 1 ∂  ∂ ¯ ϕ¯ + ∇u · ϕ¯ + ρw ¯ ϕ¯ = − Mi ϕ′i + F¯ , ∂t ρ ∂z ρ ∂z i=1

(6.11)

Formulation of the mass-flux convective parameterization

215

where F¯ =

N 

σi Fi + Fe .

i=1

An alternative expression for the grid-box mean equation is more directly obtained from that for the environment given by Eq. 5.9. By applying the approximations of Eqs. 6.1, 6.2, and 6.3 etc., we obtain: N

1 1 ∂ ∂ ¯ · ϕ¯ [Di ϕi − Ei ϕ] ¯ + ϕ¯ + ∇ ¯u − ρwe ϕ¯ = Fe . ∂t ρ i=1 ρ ∂z

(6.12)

A further alternative expression is obtained with the help of mass-flux continuity as expressed by Eq. 6.9:

where

1 ∂ ∂ ¯ · ϕ¯ ϕ¯ + ∇ ¯u + ρw ¯ ϕ¯ = F˜c + Fe , ∂t ρ ∂z

(6.13)



N  1 ∂ ϕ ¯ F˜c = + Di (ϕi − ϕ) ¯ Mc ρ ∂z i=1

(6.14)

and Mc =

N 

Mi

i=1

is the total convective mass flux. An important implication of Eq. 6.12 is that no information on the forcing Fc at the convective scale is directly required in order to evaluate the tendencies of the grid-box mean itself. A major influence of convection on the environment is through the detrainment of convective air as given by the third term on the left-hand side. The mass continuity for the environment reduces from Eq. 5.14 to: ¯ ·u+ ρ∇ 6.7

∂ ρwe = D − E. ∂z

(6.15)

Conservative subgrid-scale processes

As a full formulation for the standard mass-flux parameterization takes shape, it should be becoming clear how the mass flux Mi plays a key role in this formulation. The point is best seen for conservative subgrid-scale processes.

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Part II: Mass-flux parameterization

For a variable ϕi that is conserved under the subgrid-scale processes associated with the i-th subcomponent, then Fi = 0 and so Eq. 6.8 reduces to: ∂ Mi ϕi = Ei ϕ¯ − Di ϕi , (6.16) ∂z and Eq. 6.10 reduces to: ∂ ϕi = ǫi (ϕ¯ − ϕi ). (6.17) ∂z Here, the variable in consideration need not be conservative in a strict sense. As long as the timescale for non-conservative processes associated with the variable is much longer than that describing subgrid-scale processes (e.g., convection), then the variable may be considered to be conservative for a parameterization purpose. For example, although moist entropy may be lost by radiative cooling, since that is a slow process, the moist entropy may be considered a conserved variable for the convection parameterization. Within this set-up, the full advantage of the entrainment–detrainment hypothesis introduced in Sec. 5.6 can now be appreciated. Once the fractional entrainment rate ǫi is known, the determination of ϕi is a simple matter of vertically integrating Eq. 6.17. Furthermore, the mass flux Mi can also be evaluated simply by vertically integrating Eq. 6.9 with the given entrainment and detrainment rates. Note, however, that a bottom boundary condition must be specified. This is the closure condition that was briefly discussed in the previous chapter (Ch. 6, Sec. 7.3). The vertical profile for the subcomponent state ϕi also requires a bottom boundary condition. A common custom is to take its value as being equal to the environmental value. Some convection schemes, however, also add perturbations to the environmental value (e.g., Gregory and Rowntree, 1990; Kain and Fritsch, 1990, 1992: see also Ch. 11, Sec. 13.3). In summary, the mass-flux convection parameterization is so constructed that once the convective mass flux Mi is determined then all the other subgrid-scale variables can also be determined in a more or less straightforward manner. The only remaining problem is to define the mass flux at the convective base. 6.8

Separation of variables

An important aspect of the procedure for the subgrid components presented above is that all are evaluated diagnostically rather than prognostically. This is an important consequence of the steady-plume hypothesis, which

Formulation of the mass-flux convective parameterization

217

suggests that these subgrid-scale processes are in equilibrium against a given large-scale state, and thus no fast evolution of the subgrid-scale must be explicitly evaluated prognostically. This hypothesis, furthermore, separates the procedure of determining the vertical structure of the subgrid components and that of determining their total magnitudes. The former is left as an issue of subgrid-scale processes as more specifically given by Eqs. 6.8 and 6.9. Meanwhile, the latter is left as a closure problem, which is to be determined based on a large-scale state. This separation can be conceptually understood in analogy with the principle of a separation of the variables, albeit with some subtleties. As discussed in Ch. 6, under a typical formulation it is the fractional entrainment and detrainment rates ǫi and δi rather than the entrainment and detrainment rates themselves that are prescribed. These are defined by: ǫi =

1 ∂Mi , Mi ∂z

δi = 0, when ∂Mi /∂z > 0, and: ǫi = 0, δi = −

1 ∂Mi Mi ∂z

otherwise. As a result, Eq. 6.9 reduces to: 1 ∂ M i = ǫ i − δi . Mi ∂z

(6.18)

When the mass flux equation is rewritten in this manner, a separation of variables becomes possible, and we may write it as Mi = ηi (z)MB,i (t). This is a standard assumption that is adopted in many mass-flux convection parameterizations. Here, MB,i (t) is the mass-flux value at the convection base (or the base of any subgrid component). However, note that the separation of variables obtained above is a consequence of specifying fractional entrainment and detrainment rates. If the entrainment and detrainment rates Ei and Di were to be specified instead, for example, then the separation of variables would no longer be possible, but instead we would obtain an expression Mi = MB,i (t) + M0,i (z), where M0,i (z) is a homogeneous solution with M0,i (zB ) = 0 at the convection base z = zB .

218

Part II: Mass-flux parameterization

Moorthi and Suarez (1992) take an alternative approach by considering 1 ∂Mi , ǫi = MB,i ∂z which is assumed to be constant under an entraining plume formulation. As a result, the mass flux becomes a linear function of height: ηi (z) = 1+ǫi z. This modification helps to avoid various numerical difficulties arising from assuming an exponential growth of the plume mass flux with height (as implied by a constant fractional entrainment), but it still maintains a separation of variables. In either case, the problems of determining the vertical structure and defining the total intensity are separated into two distinct issues under the steady-plume hypothesis. Note again that such a separation is possible only when the intrinsic processes (e.g., trigger) associated with a subgrid component can be neglected. However, a subtle aspect of the arguments above must also be mentioned. Under the standard procedure, the vertical profile ηi of a plume is strictly speaking defined by a fixed rule for the entrainment–detrainment rate, with the latter depending on the large-scale (environmental) state ϕ ¯ which is denoted as a vector consisting of a set of variables. Thus, the vertical profile ηi itself also evolves with time through the large-scale (envi¯ i.e., ηi = ηi (z, ϕ). ¯ Nevertheless, the point remains that ronmental) state ϕ, the time evolution of convective intensity is mostly described by MB,i (t). 6.9

Non-conservative subgrid-scale processes

When the subgrid-scale processes in consideration are non-conservative, and Fi = 0, however, the procedure suddenly becomes highly involved. Here, the mass flux can still be computed in the same way as before. On the other hand, in order to evaluate the subcomponent (subgrid-component) state ϕi now, Eq. 6.8 must be integrated vertically in place of Eq. 6.16. In doing so, we face two difficulties. First, the source term Fi must somehow be evaluated. The issues are far from trivial and will be discussed in Vol. 2, Ch. 18 in the context of cloud microphysics. However, in terms of the formulational structure of mass-flux parameterization, this does not pose any fundamental difficulty. Fi can be evaluated, in principle, when the subgrid-scale state (even from a previous time step) is known. The second difficulty is more fundamental, because in order to account for a non-conservative tendency to a given subgrid-scale variable, the fractional area σi occupied by the subgrid component in consideration must

Formulation of the mass-flux convective parameterization

219

also be known. This factor is required because the subgrid-scale local tendency influences the grid-box mean only weighted by this factor. However, we do not have any explicit equation for defining σi . Worryingly, we are assuming an asymptotic limit to σi → 0, thus it could be problematic trying to define this quantity as a finite value. There are two major approaches to overcoming this difficulty: 1) to introduce a formulation for the source term in terms of σi Fi instead of Fi so that the need to specify σi explicitly can be avoided; or alternatively, 2) to evaluate σi in some manner. Many of the earlier mass-flux based parameterizations adopt the first approach. In the second approach, the strategy most often taken is to try to evaluate the vertical velocity wi instead of σi itself, because once the mass flux Mi is known, σi can be evaluated from Mi and wi . However, the evaluation of wi , the convective vertical velocity, is also a problem, as will be discussed in Ch. 12.

6.10

Summary

A step-by-step derivation of the mass-flux convection parameterization as adopted by many operational models today has been presented here. There are two major issues to be resolved: the entrainment–detrainment rates must be specified, and a closure problem remains. These will be discussed in Chs. 10 and 11, respectively.

7

Generalizations: Removal of the environment and the inclusion of other subgrid-scale processes

The derivation so far can be generalized in various respects. First of all, although the focus has been on convective elements (updraughts, downdraughts), the presentation has been given in such a manner that the description can be applied to any subgrid-scale processes. However, in order to realize this generality in practice, various additional considerations are required. Most importantly, the whole derivation has been performed under the assumption that all of the subgrid-scale subcomponent segments are surrounded by the environment. However, this assumption, which may be called the “environment hypothesis”, is not quite necessary until the entrainment–detrainment hypothesis is introduced in Sec. 5. Especially,

220

Part II: Mass-flux parameterization

the SCA-based prognostic equation, Eq. 4.6, is general in the sense that no environment is assumed. As shown schematically in Fig. 7.2, the convective updraught does not solely interact with the environment, but also with other subgrid-scale processes. The present section introduces the necessary modifications to the standard formulation discussed so far when the environmental hypothesis is removed, and all of the subgrid-scale components are allowed to interact directly with each other. stratiform cloud

downdraught

convective updraught cold pool

Fig. 7.2 A schematic representing interactions between different subgrid-scale components: convective updraught, downdraught, cold pool, and stratiform cloud.

In order to see the necessary modifications directly, in this section an explicit environment component is totally removed from the formulation. As a result, all of the subgrid-scale components are treated equally. Though we may decide to call one of them “environment” in the end, this component no longer plays any special role such as the traditional concept of environment has played.

7.1

Lateral exchange between subgrid-scale components

The major complexity introduced by removing the environment hypothesis is the treatment of lateral exchange between subgrid-scale components. Such a lateral exchange is no longer exclusively between the environment and each subgrid-scale component, but more combinations become possible. Thus, each subgrid-scale component may be adjacent both with the gridbox boundary and, potentially, with all the other subgrid-scale components. Thus, in analogy with Eq. 5.2, any subgrid-scale component boundary can

Formulation of the mass-flux convective parameterization

221

be separated as: ∂Si = ∂SG,i +

N 

∂Si,j .

(7.1)

j=1,j=i

Here, note a change of sign in the second term in respect of Eq. 5.2. A convention here is that when air detrains from a certain component (ith), the same air would entrain into another component (j-th), and thus ∂Si,j = −∂Sj,i . The lateral exchange term for each subgrid-scale component is given, by analogy with Eq. 5.3, as: 1 S



ϕ(u∗ −r˙ b )·dr =

∂Si

1 S



∂SG,i

ϕ(u∗ −r˙ b )·dr+

N 

j=1,j=i

1 S



ϕ(u∗ −r˙ b )·dr.

∂Si,j

(7.2) The first term on the right-hand side of Eq. 7.2 can, furthermore, be rewritten in an analogous manner as for Eq. 5.4, noting that the given subgridscale component does not share the whole grid-box boundary, but only a part of it. A simple statistical consideration leads us to realize that the most likely fraction occupied over the grid-box boundary by a given component is equal to the fractional area σi for the given component. Thus:   1 1 ¯ · σi ϕi ui . ϕ(u∗ − r˙ b ) · dr = σi ϕi ui · dr = ∇ (7.3) S ∂SG,i S ∂SG Note that σi is defined along the line integral along the grid-box boundary, and thus it must enter the derivative operator. 7.2

Entrainment–detrainment formulation

The segment boundary value must also be modified accordingly, and it leads to:   ϕi , if ∂Si,j (u∗ − r˙ b,i ) · dr > 0 (7.4) ϕb,i =  ϕj , if ∂Si,j (u∗ − r˙ b,i ) · dr < 0. With all possible interactions between the subcomponents considered, the entrainment–detrainment takes a matrix formulation as a result. Thus:    Di,j , if ∂Si,j (u∗ − r˙ b,i ) · dr > 0 ρ ∗ (u − r˙ b ) · dr = (7.5)  S ∂Si,j −Ei,j , if ∂Si,j (u∗ − r˙ b,i ) · dr < 0.

222

Part II: Mass-flux parameterization

By substituting these definitions into the second term on the right-hand side of Eq. 7.2, we obtain a final expression:  N  1 ¯ · σi ϕi ui + 1 ϕ(u∗ − r˙ b ) · dr = ∇ [Di,j ϕi − Ei,j ϕj ]. (7.6) S ∂Si ρ i=1,i=j

As a result, Eq. 5.10 is generalized as: N ∂ 1  ¯ · σi ϕi ui + 1 ∂ ρσi (wϕ)i = σi Fi . σi ϕi + [Di,j ϕi − Ei,j ϕj ] + ∇ ∂t ρ ρ ∂z i=1,i=j

(7.7) Note that Eq. 7.7 recovers both Eqs. 5.9 and 5.10 as special cases. Conversely, some assumptions behind the derivations for both Eqs. 5.9 and 5.10 are revealed by comparing this pair with Eq. 7.7. 7.3

Scale separation

In the absence of a segment with a status equivalent to the environment within the standard formulation, it is not immediately clear how to introduce a scale-separation principle under the generalization. One approach would be to assume that one set of subcomponents has fractional areas much smaller than unity, and another set has areas much closer to unity. In this manner, for example, the environment may be considered as consisting of clear-sky and stratiform-cloud components. For the small-scale subgrid components, again a steady hypothesis may be introduced so that the fast evolution of these components need not be evaluated explicitly. However, note that in this case, it is necessary to retain two full prognostic equations, for the clear-sky and the stratiform-cloud components, in order to determine the large-scale evolution. There is no longer a single stand-alone equation for the grid-box mean in this case. Yano (2012) points out the possibility of considering an intermediate situation for the scale separation such that all of the individual subgrid– elements occupy only small fractions of a grid box, but we still define each subgrid-scale category in such a manner that each occupies a finite fractional area. The first assumption means that the scale separation is satisfied individually for each subgrid-scale element (e.g., an individual convective tower). On the other hand, the second assumption means that so far as each subgrid-scale category as a whole (e.g., all convective towers within the grid box) is concerned, the scale separation is no longer satisfied and σi ∼ 1. Here, recall that the index i is for the whole i-th category, rather than an individual element or segment.

Formulation of the mass-flux convective parameterization

7.4

223

Further generalizations

The mass-flux formulation can be generalized for various subgrid-scale processes as already discussed. This generalization becomes possible by interpreting that the central concept of mass-flux parameterization resides in the geometrical constraint of the segmentally constant approximation (SCA). Thus, the SCA-prognostic equation, Eq. 4.6, is a starting point for all types of mass flux-based parameterization. The standard mass-flux parameterization is developed by introducing various additional hypotheses and approximations. Thus, the question of generalization concerns those assumptions and hypotheses that it is desirable to retain or to remove. The discussions in this section so far have been limited to the removal of the environmental hypothesis. In the last subsection, the possibility of further removing the scale-separation hypothesis was also discussed. By the same token, it is also possible to ask the question of whether the entrainment– detrainment hypothesis should be retained or not. Note that the generalization of the mass-flux formulation becomes a critical issue with increasing horizontal resolutions of numerical models. This issue is discussed further in Vol. 2, Ch. 19. 7.5

Summary

This section has outlined a generalization of the mass-flux subgrid-scale parameterization to a case with the environment hypothesis removed. A full derivation for this case is given by Yano (2012), but starting from the primitive equation system instead of NAM. A full derivation from NAM is given by Yano et al. (2010a) when a two-dimensional geometry is defined for NAM. Though currently only preliminary results are available towards this generalization, considering the importance of generalizations of the massflux formulation, this is an area where much investigation is expected in the near future.

8

Bibliographical notes

c the author, The present chapter is based on Yano (2014b), which is  2014, and published by Elsevier B.V. It is an open access article under the CC BY-NC-ND license. For further background information on the present chapter, see Yano et al. (2005b, 2010a); Yano (2012). For further discussions

224

Part II: Mass-flux parameterization

on the approximation of Eq. 4.11, see Yano et al. (2004). For a simple demonstration of NAM-SCA for a single convective-plume configuration, see Yano and Baizig (2012). The anelastic approximation, adopted as a starting point for deriving the standard mass-flux formulation in this chapter, warrants its own attention. A systematic derivation was first attempted by Ogura and Phillips (1962). See also Lipps and Hemler (1982) and Durran (1989) for subsequent efforts. The procedure for defining the horizontal winds discussed in Sec. 4.6 is, to some extent, analogous to the one adopted under the contour dynamics. For the latter, we refer to Dritschel (1989) and Dritschel and Ambaum (1997).

9

Historical notes

An upstream approximation for the subgrid-component segment boundaries is first introduced by Asai and Kasahara (1967). Original historical references for the mass-flux convection parameterization are: Arakawa and Schubert (1974); Fraedrich (1973, 1974); Ooyama (1971, 1972). Ooyama (1971) is considered a first historical attempt for constructing a mass-flux convection parameterization, and it provides its philosophical basis. Although this paper uses the term “bubble” throughout, a careful reading shows us that this can easily be translated into “steady plume” without changing the mathematical formulation. Ooyama (1972) provides insights on the fundamental issues of convection parameterization. Many of the remarks in this lecture note are amazingly relevant even today. Yanai et al. (1973) introduce a bulk formulation from a point of view of the observational data analysis.

10

Appendix A: Leibniz’s theorem for differentiation of an integral

For any function f (x, y, t) depending on two spatial coordinates x, y, and the time t, a time derivative of its integral over a spatial area defined by S is divided into two parts:    ∂f (x, y, t) d dxdy + f (x, y, t)˙rb · dr, (A.1) f (x, y, t)dxdy = dt S ∂t ∂S S

Formulation of the mass-flux convective parameterization

225

where the second term on the right-hand side refers to a line integral over the boundary ∂S of the area S, and rb is a position vector of the boundary. 11

Appendix B: Bulk formulation and total eddy transport

A bulk mass-flux formulation can be obtained by setting N = 1 (without counting the environment) in the general discussions in the main text. As a result, convection is represented by a single component, to which a subscript c is added. An alternative approach, in which the convective components are averaged over to obtain a single, effective component is discussed in detail in Ch. 9. In the simple approach of setting N = 1, a domain mean value for any physical variable is given by: ϕ¯ = (1 − σc )ϕe + σc ϕc . Under the bulk formulation, especially in the context of the shallowconvection parameterizations, the problem is often formulated in terms of ¯ rather than in the total eddy transport expressed as ϕ′ w′ = ϕw − ϕ¯w terms of the convective transport ϕc wc . The latter is a key quantity that is focused on in the main text. The former is given under the bulk formulation ¯ c − w) ¯ (cf., Eq. 20, Yanai et al., 1973). As by ϕ′ w′ = σc (1 − σc )(ϕc − ϕ)(w a result, it is useful to introduce an eddy mass flux Mc′ by: Mc′ = ρσc (1 − σc )(wc − w) ¯

(B.1)

so that the total eddy transport is concisely given by: ϕ′ w′ =

1 ′ M (ϕc − ϕ). ¯ ρ c

Note that Arakawa and Schubert (1974) and Ooyama (1971) along with many other studies, adopt the mass flux as defined by Eq. 4.11 in the main text. On the other hand, Betts (1973, 1975) and Tiedtke (1989) adopt the mass-flux definition given by Eq. B.1.

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Chapter 8

Thermodynamic effects of convection under the mass-flux formulation

J.-I. Yano Editors’ introduction: The chapters immediately preceeding this have focused on the identification and mathematical description of convective elements within a grid box. Attention now returns to the primary objective of any parameterization, which is to provide appropriate feedbacks to the resolved-scale flow. In order to discuss more concretely how the thermodynamic feedbacks are influenced by different choices within a mass-flux framework, it is necessary to specify some aspects of the entrainment– detrainment formulation and the microphysics, and some simple options are illustrated.

1

Introduction

This chapter, to some extent, repeats elements of Ch. 7. However, it derives more precisely how convection modifies the large-scale thermodynamic state as described under the mass-flux parameterization. The general form was presented in the last chapter, but here more specifically the heat and moisture budgets are considered. A more precise goal of this chapter is to reproduce the major results of Arakawa and Schubert (1974) based on the general results of Ch. 7. Since the focus of Arakawa and Schubert (1974) was on the thermodynamic effects of convection in the large-scale dynamics, thus the title of this chapter follows. By following Arakawa and Schubert (1974), it is considered that only the convective processes are parameterized, although as suggested in the last chapter, a generalization to the other subgrid-scale processes is relatively straightforward. 227

228

2

Part II: Mass-flux parameterization

Thermodynamic equations

Following Arakawa and Schubert (1974), the budgets for the dry static energy s and the water vapour mixing ratio q are considered. Here, the dry static energy is defined by: (2.1) s = Cp T + gz with Cp the specific heat at constant pressure for dry air, T the temperature, g the acceleration due to gravity, and z the geometrical height (from the surface). The next step is to adopt the budget equations: 1 ∂ρsw ∂s + ∇H · su + = QR + Lv (c − e) (2.2a) ∂t ρ ∂z ∂q 1 ∂ρqw + ∇H · qu + = e − c, (2.2b) ∂t ρ ∂z where QR is the radiative heating rate, c the condensation rate of water vapour, e the evaporation rate of condensed water, and Lv the latent heat (for condensation). Note that the freezing effect is neglected for now. For a completeness, let us also write down a budget equation for the cloud water qc : 1 ∂ρqc w ∂qc + ∇H · qc u + =c−r (2.2c) ∂t ρ ∂z with r the loss of cloud water by precipitation. Furthermore, a budget equation for the total water, qT = q + qc , is also obtained by taking a sum of Eqs. 2.2b and 2.2c: ∂qT 1 ∂ρqT w + ∇H · qT u + = e − r. (2.2d) ∂t ρ ∂z Note that the evaporation term e includes that from the precipitating water. Additionally, the moist static energy h will reveal itself to be a useful quantity to consider, and it is defined by: h = s + Lv q = Cp T + gz + Lv q. (2.3) Its budget equation is obtained by taking a sum of Eqs. 2.2a and 2.2b: 1 ∂ρhw ∂h + ∇H · hu + = QR . (2.2e) ∂t ρ ∂z Thus, the moist static energy is a conserved quantity over water condensation processes. Only the external entropy source contributes as a source for this variable, not only the radiative heating, but also including the surface fluxes, which would appear as a boundary condition for solving the budget equation. Eqs. 2.2a–e may be considered a special case of a generic prognostic equation introduced by Ch. 7, Eq. 3.4.

Thermodynamic effects of convection under the mass-flux formulation

3

229

Equations for the grid-box mean

A generic prognostic equation for the grid-box mean is given by Ch. 7, Eq. 6.13, which may be rewritten as:     ∂ ϕ¯ ∂ ϕ¯ ∂ ϕ¯ + , (3.1) = ∂t ∂t conv ∂t LS where the convective and the large-scale tendencies are defined by:

   N ∂ ϕ¯ ∂ ϕ¯  1 D Di (ϕi − ϕ) ¯ (3.2a) Mc + = ∂t conv ρ ∂z i=1 and



∂ ϕ¯ ∂t



LS

1 ∂ ¯ · ϕ¯ ρw ¯ ϕ¯ + Fe . = −∇ ¯u − ρ ∂z

(3.2b)

Here, a superscript D is added to a detrained value to indicate that it may be different from the original convective value (cf., Ch. 7, Sec. 5.2). As special cases of the above, the prognostic equations for the dry static energy and the water vapour mixing ratio are given by:     ∂¯ s ∂¯ s ∂¯ s = + (3.3a) ∂t ∂t conv ∂t LS ∂ q¯ = ∂t or



N

∂¯ s  = ∂t i=1 N

and 

∂ q¯  = ∂t i=1 ∂¯ s ∂t



LS



∂ q¯ ∂t



+

conv

∂ q¯ ∂t 

∂¯ s ∂t





∂ q¯ ∂t





∂¯ s ∂t



+



∂ q¯ ∂t



+

conv,i

conv,i





(3.3b)

LS

(3.4a)

LS

(3.4b)

LS

1 ∂ ¯ · s¯u ρw¯ ¯ s + QR,e + Lv (ce − ee ) + = −∇ ¯− ρ ∂z

∂ q¯ ∂t



LS

1 ∂ ¯ · q¯u ρw ¯ q¯ + ee − ce + = −∇ ¯− ρ ∂z 

∂¯ s ∂t



conv,i



∂ q¯ ∂t





∂¯ s 1 D + Di (si − s¯) = Mi ρ ∂z

∂¯ s ∂t



(3.5a)

BL

(3.5b)

BL

(3.6a)

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Part II: Mass-flux parameterization



∂ q¯ ∂t



conv,i

=

∂ q¯ 1 + Di (qiD − q¯) . Mi ρ ∂z

(3.6b)

Here, the subscript e in the forcing terms in Eqs. 3.5a and 3.5b indicate those from the environment. With a slight confusion of terminology, the large-scale forcings in Eqs. 3.5a and b include a contribution from the boundary-layer subgrid-scale processes as given in their last terms. Here, “large scale” should more strictly mean “non-convective”, and this rather inconvenient terminology will be adhered to here, because it is a standard terminology, and the other subgrid-scale processes can be neglected for the most part. 3.1

Apparent sources and basic constraints

As discussed in Ch. 2, Sec. 2.2, in observational diagnoses, these thermodynamic convective tendencies are called “apparent sources”, and often designated as:  

1 ∂¯ s ¯R +Q (3.8a) Q1 ≡ Cp ∂t conv   Lv ∂ q¯ Q2 ≡ (3.8b) Cp ∂t conv

expressing both terms in units of [K/day]. Note that, following the conventional definition, the radiative heating is included as a part of the apparent heat source in Eq. 3.8a. As already emphasized in Ch. 2, Sec. 2.2, these source terms are relatively straightforward to diagnose from a given observational network, because by referring to the relations of Eqs. 3.3a and b:

  1 ∂¯ s ∂¯ s ¯ (3.9a) − + QR Q1 = Cp ∂t ∂t  LS

∂ q¯ Lv ∂ q¯ Q2 = − , (3.9b) Cp ∂t LS ∂t the terms on the right-hand side can be diagnosed from observational data obtained from such a network. It is usual to assume that ce = ee = 0 in these diagnoses. Some basic constraints for the apparent sources can be derived (cf., Yanai et al., 1973) in the following manner. First, notice that the sum of Eqs. 3.3a and b leads to a budget equation for the moist static energy: ¯ R = − 1 ∂ ρw′ h′ . (3.10) Cp (Q1 − Q2 ) − Q ρ ∂z

231

Thermodynamic effects of convection under the mass-flux formulation

Here, the left-hand side is expressed in a general form, without assuming a mass-flux parameterization, by referring back to Ch. 2, Eq. 2.3, but neglecting horizontal eddy terms as a standard approximation. Furthermore, a vertical integral of the source terms can be expressed to a good approximation by: ¯ R > +Lv P + FS Cp < Q1 > =< Q Cp < Q1 > = P − FL , Lv

(3.11a) (3.11b)

where P is the precipitation rate, FS is the surface sensible heat flux, FL is the surface evaporation rate, and angled brackets < > designate the vertical integral (from the surface to the top of the atmosphere). Note the boundary-layer subgrid-scale terms in Eqs. 3.5a and b. Unfortunately, these constraints are not strong enough to guide the construction of a subgrid-scale parameterization in any obvious manner. However, any given parameterization must be consistent with those constraints.

3.2

Observational diagnoses of mass-flux parameterization parameters

The formulation outlined in the last subsection provides more than just observational estimates of the thermodynamical apparent sources, as originally identified by Yanai et al. (1973). Once non-convective processes (namely, radiative heating in the presentation of the last subsection) are removed from the above diagnoses (e.g., by running a radiative transfer code independently for a given large-scale state), an observational estimate of convective tendencies is known. Once a convective tendency is known, it is possible to invert Eq. 3.2a for a given left-hand side as well as a given large-scale state ϕ. ¯ Such a diagnosis was first performed by Yanai et al. (1973) for a single plume mode (a bulk approximation, i.e., N = 1) for estimating the vertical profiles for mass flux M (z), entrainment E(z), and detrainment, D(z). The diagnosis was extended to a spectral formulation (N > 1) assuming an ensemble of entraining plumes by Ogura and Cho (1973), and Nitta (1975). The observational diagnoses were further extended to include downdraughts by Johnson (1976), Nitta (1977) and Cheng (1989: see Ch. 13 for this generalization). The vorticity budget was analysed by Sui and Yanai (1986). Houze Jr. et al. (1980) pointed out the usefulness of combining sounding-network observations (as adopted in

232

Part II: Mass-flux parameterization

the above studies) with precipitation estimates by radar. See Yanai and Johnson (1993) for a review. There are some technicalities for performing these diagnoses, which are beyond the scope of the work presented here. However, for example, the inversion of the mass-flux spectrum from Eq. 3.2a is, in principle, a straightforward matrix inversion problem, provided positiveness of the mass flux is ensured. Some additional assumptions are required in order to include downdraughts. However, these assumptions are the same as those required in operational schemes, and thus the diagnoses can be used as a verification of a choice of those assumptions (as well as a choice of free parameters) in a given scheme. Furthermore, a simple extension of this diagnostic approach would be able to verify various closure assumptions (cf., Ch. 11) by observations. In short, those observational networks can extensively be used for verification of convection parameterization formulations in a very direct manner. Although the analysis of LES and CRM data is currently more popular, they actually provide far more information than is required for verification of convection parameterization. A researcher can easily be sidetracked towards an issue not actually addressed within an operational scheme that one is expected to improve.

4

Detraining convective air

A full condition for the detraining air from convection can be determined only after a cloud model (the set of entrainment–detrainment assumptions) is specified. Nevertheless, some preliminary considerations are possible without specifying it. That is the purpose of this section. The only assumption introduced here is that the detraining air from convection is at saturation. Then the mixing ratio of the water vapour within convection is given by: qi = q ∗ (Ti , p¯), assuming in-cloud pressure is equal to that in the environment. Here, q ∗ designates the saturated water vapour mixing ratio. The above may be approximated by a linear expression: γ (si − s¯), (4.1) qi ≃ q¯∗ + Lv with   Lv ∂ q¯∗ γ≡ , (4.2) Cp ∂ T¯ p¯

Thermodynamic effects of convection under the mass-flux formulation

233

and q¯∗ is the saturated mixing ratio for a given large-scale state (T¯, p¯). Some useful expressions are obtained in terms of the moist static energy introduced by Eq. 2.3. We may combine Eq. 4.2 and this definition in order to obtain hi = si + Lv q¯∗ + γ(si − s¯), or by introducing the saturated moist static energy by h∗ = s + Lv q ∗ , then hi − ¯h∗ ≃ (1 + γ)(si − s¯). By inverting the relation, 1 ¯ ∗ ). (hi − h (4.3) si − s¯ ≃ 1+γ Furthermore, with the help of Eq. 4.1, γ 1 ¯ ∗ ). (hi − h (4.4) qi − q¯∗ ≃ 1 + γ Lv Eqs. 4.3 and 4.4 will be used heavily in the following reductions. Here, both the dry static energy s and the water vapour q are not conserved by the detrainment process, as already noted generally in Ch. 7, Sec. 5.2, when the environment is cloud free (i.e., q¯c = 0). In that case, it may be assumed that all of the condensed water (i.e., cloud water) reevaporates into water vapour after detrainment. Also, considering the conservation of enthalpy, we obtain: sD i

qiD = qi + qci ,

(4.5a)

= si − Lv qci .

(4.5b)

This is a standard assumption adopted in this chapter as well as in Ch. 11. On the other hand, when q¯c = 0, then qiD = qi and sD i = si may be set, assuming that all the detrained air could enter an environmental cloud. 5

Convective buoyancy and convection top condition

In order to define a mass flux from a prescribed entrainment–detrainment profile, the bottom zB and top zT,i of the convection must be specified. By following Arakawa and Schubert (1974), it is assumed that the bottom of the convection is placed at the top of the planetary boundary layer. Thus, the convection bottom is common to all the convection types. The top height, however, is likely to be different for each convection type. Since convection is driven by buoyancy, the most natural and convenient assumption is to take a level of neutral buoyancy as the top of convection. It should immediately be realized that this assumption is rather naive but the limitations of this assumption are not discussed. A good, short discussion on this subject is found in Sec. 3 of Ooyama (1971).

234

Part II: Mass-flux parameterization

Convective buoyancy bi is defined in terms of the difference between the virtual temperature Tv,i for the given convection type and that of the environment T¯v . Thus: Tv,i − T¯v . bi = g T¯v The level of neutral buoyancy is defined such that bi = 0, or equivalently that the virtual temperature Tv,i for the given convection type becomes equal to that of the environment T¯v . Thus Tv,i = T¯v at z = zT,i . Here, the virtual temperature is defined by: 1 + q/ˆ ǫ ˆ − qc )T, Tv = T ≃ (1 + δq 1 + qT with the last approximation following that of Arakawa and Schubert (1974). Here, ǫˆ = Rd /Rv ≃ 0.611 and δˆ = Rv /Rd − 1 are defined terms of the gas constants for dry air Rd and water vapour Rv ; qT = q + qc is the total water mixing ratio, and qc is the cloud-water (liquid-water) mixing ratio. The possible existence of ice is neglected for now. Furthermore, also by following Arakawa and Schubert (1974), we can examine the level of neutral buoyancy in terms of the virtual static energy sv , defined by: ˆ − qc ). (5.1) sv = Cp Tv + gz ≃ s + Cp T (δq Thus, the convective buoyancy is measured in terms of the difference between the convective and the environment virtual static energies. The convective buoyancy can then be further approximated as: g (sv,i − s¯v ), (5.2a) bi = Cp T¯ and the level of neutral buoyancy is correspondingly obtained from sv,i = s¯v

(5.2b)

at z = zT,i . Substitution of Eq. 5.1 into the right-hand side of Eq. 5.2a leads to: ˆ i − q¯) − (qc,i − q¯c )], (5.3a) sv,i − s¯v = si − s¯ + Cp T¯ [δ(q and substitution of Eq. 5.1 into Eq. 5.2b leads to: ˆ i − q¯) − (qc,i − q¯c )] = 0 si − s¯ + Cp T¯ [δ(q

(5.3b)

at z = zT,i . Furthermore, substitution of Eqs. 4.3 and 4.4 into the righthand side of Eq. 5.3a leads to: ∗ ¯∗ ˆ q ∗ − q¯) − (qc,i − q¯c )]. ˆ hi − h + Cp T¯ [δ(¯ sv,i − s¯v = (1 + γ ε˜δ) (5.4) 1+γ

235

Thermodynamic effects of convection under the mass-flux formulation

Thus, the condition of neutral buoyancy at z = zT,i is given by: ∗ ¯∗ ˆ q ∗ − q¯) − (qc,i − q¯c )] = 0 ˆ hi − h + Cp T¯ [δ(¯ (1 + γ ε˜δ) 1+γ

(5.5)

with ε˜ =

Cp T¯ . Lv

(5.6)

Thus, Eq. 5.3b can be rewritten as: h∗ − h∗i = ¯

(1 + γ)Lv ε˜ ˆ ∗ [δ(¯ q − q¯) − (qc,i − q¯c )] 1 + γ ε˜δˆ

(5.7)

at z = zT,i . The further substitution of Eq. 5.7 into Eqs. 4.3 and 4.4 leads to: si = s¯ −

qi = q¯∗ −

Lv ε˜ 1 + γ ε˜δˆ

ˆ q ∗ − q¯) − (qc,i − q¯c )] [δ(¯

(5.8a)

ˆ q ∗ − q¯) − (qc,i − q¯c )] [δ(¯

(5.8b)

γ ε˜ 1 + γ ε˜δˆ

at z = zT,i . Thus, these are the values of dry static energy and water vapour at the detrainment level. Note that we may assume q¯c = 0 as in Arakawa and Schubert (1974) for now. Keep in mind that the expression in Eq. 5.4 is valid only above the saturation level. In the following, Eqs. 5.8a and b are used for evaluating the convective values at the detrainment level of entraining plumes. However, when half-sinusoidal plumes are considered in Sec. 10, the convective values at the detrainment level are rather based on Eqs. 4.3 and 4.4, because the detraining air is no longer neutrally buoyant in this case. 6

Vertical profiles of convective variables

A vertical profile of a convective-scale variable can be determined by Ch. 7, Eq. 6.10 derived in the last chapter: ∂ ϕi = ǫi (ϕ¯ − ϕi ) + ρσi Fi /Mi . (6.1) ∂z By referring to Eqs. 2.2d and e, the equations for moist static energy and total water within convection are found to be: ∂ ¯ − hi ) (6.2a) hi = ǫi (h ∂z

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Part II: Mass-flux parameterization

∂ ˜i qT,i = ǫi (¯ qT − qT,i ) − R ∂z ˜ i defined by: with the precipitation term R ˜ i = ρσi ri /Mi . R

(6.2b)

(6.3)

Here, both the radiation effect on convective-scale moist static energy, and the evaporation of precipitating water in the convective-scale total-water budget are neglected. For the purpose of presenting an analytical solution, Eq. 6.1 can be interpreted in two different ways. The first is to consider the forcing Fi as externally prescribed. The second is to consider it as a function of the convective-scale variable ϕi . These two interpretations allow the above equation to be written in two different ways:   ∂ + ǫi ϕi = ǫi ϕ˜¯ (6.4a) ∂z

and



where

 ∂ ¯ + ǫ˜i ϕi = ǫi ϕ, ∂z ϕ˜¯ = ϕ¯ +

(6.4b)

ρσi Fi ǫi Mi

(6.5a)

and ρσi Fi . (6.5b) Mi ϕi The left-hand side of Eq. 6.4a can be rewritten as:   z  z   

 ∂ ∂ + ǫi ϕi = exp − ǫi dz ′ ǫi dz ′ . ϕi exp ∂z ∂z zB zB z ′ After multiplying by exp( zB ǫi dz ) on both sides, we obtain:  z  z 

 ∂ ǫi dz ′ ǫi dz ′ . = ǫi ϕ˜¯ exp ϕi exp ∂z zB zB ǫ˜i = ǫi −

This equation can be integrated easily, and we obtain:    ′

   z z ′′ dz ′ exp − ¯ exp ǫi ϕ˜ ϕi = ϕi,B + ǫi dz zB

zB

z

zB

 ǫi dz ′ .

Note that when we start from Eq. 6.4b instead, we obtain:  ′

    z   z z ′ ′ ′′ dz exp − ǫi ϕ¯ exp ǫ˜i dz . ǫ˜i dz ϕi = ϕi,B + zB

zB

zB

(6.6a)

(6.6b)

Thermodynamic effects of convection under the mass-flux formulation

237

Note that as long as the vertical integrals are performed numerically, both Eqs. 6.6a and b lead to the same answer within the accuracy of the numerics. However, the two solutions are based on different physical interpretations of the forcing Fi . Under the first solution, Eq. 6.6a, the forcing is externalized, and treated as a given, whereas the second solution, Eq. 6.6b, treats the forcing as a dependent function of the convective variable ϕi . Physically speaking, in many problems, the solution in Eq. 6.6a is wrong, and the solution in Eq. 6.6b is right: the forcing Fi often depends on ϕi , and so it cannot be externalized. However, there are merits to externalizing the forcing in solving this problem. Most importantly, and more generally, the forcing Fi may be an intricate function of many convective variables (say, ϕi ) and thus it becomes difficult to solve in the form of Eq. 6.6b, even numerically. A practical approach could be to take the solution in Eq. 6.6a with the forcing Fi estimated by values from the previous timestep. An iteration would give a final solution. Furthermore, as will be discussed in the chapter on closure (Ch. 11), the externalization of a forcing leads to an interesting interpretation of some closure problems. In seeking the level of neutral buoyancy in order to define a convection top, in the following, Eqs. 6.2a and b are solved first by adopting the same strategy for solving Eq. 6.1 as outlined above. It is also necessary to make use of the fact that above the saturation level, the water vapour always remains saturated within convection. Thus, qi is readily obtained from the given Ti and qT,i . With given Ti , qi , and qc,i , the virtual temperature Tv,i for convective air is then readily evaluated. However, in order to solve Eq. 6.2b for the total water, the precipitation term ρσi ri /Mi must first be known. That is the subject of the next section.

7

Precipitation formulation

˜ i ≡ ρσi ri /Mi in Eq. 6.2b, two possiIn specifying the precipitation term R ˜ i = c0 qc,i by following Arakawa bilities are considered. The first is to set R and Schubert (1974). Lord and Arakawa (1982) take c0 = 2 × 10−3 m−1 as a default value but also investigate sensitivity with c0 = 0.5×10−3 m−1 and 3 × 10−3 m−1 . Meanwhile, Hack et al. (1984) emphasize the importance of making the precipitation efficiency c0 a function of the convection height. Phenomenologically speaking, shallower convective clouds tend to be nonprecipitating, and deep convective clouds tend to rain more intensively, and

238

Part II: Mass-flux parameterization

this aspect should somehow be taken into account. For this reason, they propose to take a dependence presented by their Fig. 3. Unfortunately, in general it is not clear how to relate the convection height to the fractional entrainment rate λ (i.e., ǫi from Ch. 7, Eq. 6.10) without knowing the water loading, for example. This is a critical aspect missing from the original article. For this reason, in the following, it is instead assumed that the precipitation efficiency c0 is more directly related to the fractional entrainment rate, λ as shown in Fig. 8.1.

Fig. 8.1 The assumed precipitation efficiency c0 as a function of the fractional entrainment rate λ.

An alternative approach is to define the retained cloud water qc,i relative to the cloud water q˜c,i expected without precipitation. The latter may be defined in terms of the total water q˜T,i : ∂ q˜T,i = ǫi (¯ qT − q˜T,i ). ∂z

(7.1)

q˜c,i = q˜T,i − qi

(7.2)

Then:

is the total cloud water expected when there is no precipitation. Recall that qi = q ∗ (Ti , p) by definition above the condensation level. Whenever q˜c,i > 0, we may assume that a fixed fraction ǫp of the reference cloud water q˜c,i is lost by precipitation. Thus, the retained cloud water is: qc,i . qc,i = (1 − ǫp )˜

(7.3)

Note that by design, the cloud water turns into a diagnostic variable. The actual precipitation rate can be estimated backwards by substituting the

Thermodynamic effects of convection under the mass-flux formulation

239

obtained qT,i = qi + qc,i into Eq. 6.2b: ∂ ˜ i = ǫi (¯ R qT,i . qT − qT,i ) − (7.4) ∂z As with the first formulation, the precipitation efficiency should increase for deeper clouds. For the entraining plume considered in the next section, a linear dependence will be assumed:   λ (7.5) + ǫp,min , ǫp = (ǫp,max − ǫp,min ) 1 − λmax with ǫp,min = 0.05, ǫp,max = 0.75, and λmax = 1 × 10−4 m−1 under an entraining-plume model to be introduced in the next section. When halfsinusoidal profiles are considered in Sec. 10, the precipitation efficiency is defined by: ǫp = (ǫp,max − ǫp,min)

zT,i − zT,min + ǫp,min , zT,max − zT,min

(7.6)

where zT,min = 2 km and zT,max = 13 km. 8

Entraining-plume model

Some general considerations have been given so far on the thermodynamic effects of convection on the large-scale state under the mass-flux formulation. However, in order to proceed further, it is necessary to specify a cloud model in order to specify the vertical profiles ηi of convection. In the following, two possibilities are considered. The first is the entraining-plume hypothesis as originally adopted by Arakawa and Schubert (1974). As an alternative possibility, a set of simple vertical profiles for convective plumes are taken, specifically, a half-sinusoidal form. Under this alternative approach, the entrainment and the detrainment rates are diagnosed backwards from a specified vertical mass-flux profile. The entraining-plume hypothesis, as introduced by Arakawa and Schubert (1974) assumes a constant fractional entrainment rate ǫi for each convective type. Following this hypothesis, a convective plume grows in size with height by simply entraining air from the surroundings (environment) all the way to the top. At the top, it is assumed that all of the convective air is detrained into the environment. Thus: ǫi = λi

(8.1a)

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Part II: Mass-flux parameterization

is set for zB ≤ z < zT,i , where λi is a constant fractional entrainment rate, and zT,i is the level of neutral buoyancy. As for the fractional detrainment rate, δi = Mi (z = zT,i )δ(z − zT,i )

(8.1b)

is set with δ(z) Dirac’s delta. Here, the complete detrainment within an infinitesimally thin layer is clearly an idealization. In the following graphical presentations, this infinitesimally thin layer is replaced by a single vertical layer of the numerical discretization. Note that the vertical resolution of the data adopted in the following demonstration is 50 hPa. Recall that the convective mass-flux can be evaluated by Ch. 7, Eq. 6.9 or ∂ Mi = Ei − Di . ∂z Substitution of Eqs. 8.1a and b leads to:

for zB obtain:

∂ Mi = λi Mi (8.2) ∂z ≤ z < zT,i . Eq. 8.2 can be readily integrated vertically, and we Mi = MB,i ηi ,

(8.3)

where MB,i is the mass flux value at the convection base, and a normalized mass-flux vertical profile is given by: ηi = eλ(z−zB ) .

(8.4)

Substitution of Eq. 8.3 into Eq. 8.1b, in turn, gives: δi = MB,i eλ(zT ,i −zB ) δ(z − zT,i ). 9

(8.5)

Demonstration: Entraining plume case

For a demonstration, we take a climatological tropical sounding produced by Jordan (1958) by analysing a Caribbean (West Indies) sounding data set. Figure 8.2 shows the vertical profiles for s, h, and h∗ obtained from this climatological sounding. Note that all of these static energies are translated into units of temperature [K] by dividing them by Cp . The long-dash line for the moist static energy h may be compared to the schematics in Ch. 6, Fig. 6.2.

Thermodynamic effects of convection under the mass-flux formulation

241

Fig. 8.2 Vertical profiles for dry (solid line), moist (long-dash line), and saturated (short-dash line) static energies obtained from Jordan’s (1958) topical climatology data.

The normalized mass-flux profiles ηi for selected choices of the fractional entrainment rate λi and the two choices of the precipitation formulation described in Sec. 7 are shown in Fig. 8.3. Note that the Jordan sounding data is provided with vertical interval of 50 hPa with 1050 hPa at the surface. These two plots show the mass-flux profiles with this same resolution, with the convection base assumed to be at the 950 hPa level. In both cases, it can be seen that when the entrainment rate is larger, the mass-flux increases more rapidly with height, but results in shallower convection due to dilution of the buoyant air by entrainment. In other words, deeper convection tends to maintain a more nearly constant massflux with height. Note that in the limit to λi → 0, the hot tower discussed in Ch. 6 is recovered. The convective responses (∂¯ s/∂t)conv,i and (∂ q¯/∂t)conv,i , defined by Eqs. 3.6a and b are shown for the same fractional entrainment rates λi in Figs. 8.4 and 8.5 for the two choices of the precipitation formulation described in Sec. 7. The convection-base mass flux is taken to be MB = 10−2 kgm−2 s−1 based on a typical tropical vertical velocity w ∼ 10−2 ms−1 and the air density ρ = 1 kg m−3 , and assuming σc wc ∼ w. Alternatively, it may be supposed that σc ∼ 10−2 and wc ∼ 1 ms−1 . As shown by Eqs. 3.6a and b, the convective response consists of two parts: heating and drying (the first term) associated with the compensating descent −Mi , and cooling and moistening (the second term) associated with re-evaporation of detrained cloudy air. The compensating descent −Mi is

242

Part II: Mass-flux parameterization

Fig. 8.3 Vertical profiles for the normalized mass-flux ηi for the fractional entrainment rates λi = 2 × 10−5 m−1 (solid lines), 4 × 10−5 m−1 (long-dash lines), 6 × 10−5 m−1 (short-dash lines), and 8 × 10−5 m−1 (chain-dash lines), assuming that (a) the precipitation rate is locally proportional to the cloud-water mixing ratio, as assumed by Arakawa and Schubert (1974); (b) the precipitation rate is proportional to the non-precipitating cloud water amount with a proportionality constant ǫp , defined by Eq. 7.5.

added to the environmental vertical motion for mass continuity. Typical convective heating and drying rates are of the order of 3 Kd−1 . The compensating heating and drying are also of this order, as a simple scale argument can show independently. On the other hand, cooling and moistening due to re-evaporation of detrained air can exceed values of 10 Kd−1 , which is unrealistically large. Note that the re-evaporation is stronger for shallower convection, in which precipitation is weaker. When the entrainment rate is large enough (i.e., λi ≥ 4 × 10−5 m−1 ), the cooling and moistening reach absolute values of 20 Kd−1 . A simple scale analysis given in Sec. 12 supports the notion that these numerically obtained values are physically reasonable. The effect decreases with increasing convection heights (decreasing entrainment rates), and it almost vanishes when λi = 2×10−5 m−1 under Arakawa and Schubert’s microphysical formulation (solid lines in Fig. 8.4), because the condensed water almost all precipitates out in this limit.

Thermodynamic effects of convection under the mass-flux formulation

243

Fig. 8.4 Convective responses according to Eqs. 3.6a and b are plotted in panels (a) and (b) respectively, when the Arakawa and Schubert (1974) microphysics formulation is adopted. Shown in the units of [K/day] by multiplying Eq. 3.6a by 1/Cp and Eq. 3.6b by Lv /Cp . The fractional entrainment rates considered are λ = 2 × 10−5 m−1 (solid line), 4 × 10−5 m−1 (long-dash line), 6 × 10−5 m−1 (short-dash line), and 8 × 10−5 m−1 (chain-dash line). A typical convective-base value MB = 10−2 kgm−2 s−1 is used.

Clearly, such strong cooling and moistening is not consistent with the known budget analysis for the apparent sources from observations (e.g., Lin and Johnson, 1996; Nitta, 1977; Thompson Jr. et al., 1979). Observations typically show that the apparent source both for heat (entropy) and moisture is more like a half-sinusoidal shape than those shown in Figs. 8.4 and 8.5. If the cooling and moistening spikes associated with detrainment could be taken out, the convective response would look more realistic, so arguably this is an unrealistic aspect of the entraining-plume model adopted here. An alternative perspective is that with such a simple entraining-plume model it is necessary to have a good range of convective types with different λ in order to produce a total profile that is realistic.

244

Part II: Mass-flux parameterization

Fig. 8.5 As for Fig. 8.4 but when the microphysics formulation with the precipitation efficiency ǫp is adopted, as defined by Eq. 7.5.

10

Demonstration with half-sinusoidal plumes

The assumption of the entraining-plume model that all of the air detrains at the convection top appears to be unrealistic, or at least an oversimplification. In order to see the sensitivity of the convective response to the choice of a detrainment hypothesis, in this section a set of half-sinusoidal shaped mass fluxes is taken instead:   z − zB ηi = sin π zT,i − zB for zB ≤ z ≤ zT,i , otherwise ηi = 0. The mass flux is still defined by Mi = MB ηi with MB ∼ 10−2 kgm−2 s−1 . Noting that   π dηi z − zB cos π = , dz zT,i − zB zT,i − zB the fractional entrainment–detrainment rate for these plumes is then:   π 1 dηi z − zB = , cot π ǫ i − δi ≡ ηi dz zT,i − zB zT,i − zB

Thermodynamic effects of convection under the mass-flux formulation

245

with both ǫi and δi taking positive definite values. It will be assumed that only either entrainment or detrainment exists at a given vertical level, and the above formula then provides a unique answer for entrainment and detrainment rates. Importantly, using this formulation, the convection top is no longer neutrally buoyant, but convection is forced to terminate at the prescribed height zT,i , regardless of the buoyancy of the convective air. The corresponding results to Figs. 8.4 and 8.5 with the half-sinusoidal profiles are shown in Figs. 8.6 and 8.7 respectively.

Fig. 8.6 As in Fig. 8.4 but when half-sinusoidal plume profiles are used. Arakawa and Schubert’s microphysics formulation is adopted. Here, zT,i = 4 km (solid lines), zT,i = 7 km (long-dash lines), zT,i = 10 km (short-dash lines), and zT,i = 13 km (chaindash lines). For the purpose of defining the precipitation coefficient c0 corresponding values of λ = 8 × 10−5 m−1 (solid lines), 6 × 10−5 m−1 (long-dash lines), 4 × 10−5 m−1 (short-dash lines), and 2 × 10−5 m−1 (chain-dash lines) are assumed.

The results are qualitatively similar to the case with the entraining plumes, except that the detrainments spread to a deeper layer, and thus the cooling and moistening associated with evaporative cooling of detrained air are much more modest at each vertical level. The convective heating and moistening profiles for the shallowest case (solid line: zT,i = 4 km) may

246

Part II: Mass-flux parameterization

Fig. 8.7 As for Fig. 8.6 but when the microphysics formulation with the precipitation efficiency ǫp is adopted, as defined by Eq. 7.6.

be compared to an observational diagnosis by Nitta and Esbensen (1974) for trade cumuli (see their Fig. 8). On the other hand, the strong cooling again found at the top of deeper convection is clearly unrealistic. Thus, although a sinusoidal profile partially alleviates the problem of strong cloud-top cooling and moistening, it does not do so in a fundamental manner. It is speculated that the more fundamental problem is the assumption of immediate evaporation of detrained cloudy air. In more realistic situations, the cloudy air does not evaporate so rapidly, but it forms a part of stratiform and cirrus clouds adjacent to deep convection. Although the associated stratiform and cirrus clouds dissipate eventually, it happens on a much longer timescale. Thus, strong cooling is not observed at any instant at the convection top. A simple corollary from this discussion is the importance of coupling between deep convection and stratiform clouds. Unfortunately, the subject is beyond the scope of this set.

Thermodynamic effects of convection under the mass-flux formulation

11

247

Bibliographical notes

This chapter overall follows the formulation of Arakawa and Schubert (1974). Haman (1969) is a classic work, predating even Ooyama (1971), investigating the thermodynamic effects of convection on the large scale in a systematic manner. 12

Appendix A: Simple scale analysis

The purpose of this appendix is to estimate the convective response values shown in Figs. 8.4 and 8.5 by a simple scale analysis. Let us begin from the basic fact that latent heating by condensation of a unit of water vapour q = 1 gkg−1 leads to an increase of temperature by qLv /Cp = 2.5 K with the latent heating Lv = 2.5 × 106 Jkg−1 and the heat capacity of air at constant pressure Cp = 103 Jkg−1 K−1 . Here, note that 1 gkg−1 of water vapour is a relatively small amount. However, the associated temperature increase is comparable to a typical buoyancy anomaly within tropical convection. A typical mass-flux value under convective quasi-equilibrium is M ∼ 10−2 kgm−2 s−1 , or M/ρ ∼ 10−2 ms−1 in units of vertical velocity. As a result, adiabatic warming associated with compensating environmental descent is estimated as: M dθ ∼ 3 × 10−5 Ks−1 ∼ 3 Kday−1 , ρ dz where dθ/dz ∼ 3×10−3 Km−1 . This is consistent with the graphical results in Figs. 8.4 and 8.5 for compensating descent. At the convective cloud top, all of the mass flux detrains under the entraining-plume hypothesis. The associated negative heat flux is estimated as: (M/ρ)(qc Lv /Cp ) ∼ 2.5×10−2 ×qc Kms−1 ∼ 2.5×103 ×qc Kmday−1 , (A.1) assuming that the cloud-water mixing ratio qc is given in units of [gkg−1 ]. If the detrainment happens over a 1 km layer, it amounts to a cooling rate of 2.5 Kday−1 with qc = 1 gkg−1 . The last piece of the estimate is the amount of cloud water qc expected at the convective cloud top at the height of, say, H ∼ 10 km. In order to obtain this estimate, note that within a convective updraught condensative heating is well balanced by adiabatic cooling (a local realization of the free

248

Part II: Mass-flux parameterization

ride principle: cf., Ch. 11, Sec. 14.1). Thus: dθ Lv dq ∗ + ≃ 0. Cp dz dz This leads to an estimate for a rate of decrease of saturated water-vapour with height given by: dq ∗ Cp dθ ∼− ∼ −10−6 m−1 . dz Lv dz Thus, the accumulation of condensed water from lifting over a height H ∼ 10 km is estimated as:   dq q∗ = − H ∼ 10−6 m−1 × 104 m ∼ 10−2 kgkg−1 ∼ 10 gkg−1 . (A.2) dz Substitution of Eq. A.2 into Eq. A.1 leads to an estimate of the cooling rate of 25 Kd−1 for a 1 km-deep detrainment layer.

Chapter 9

Spectral and bulk mass-flux representations

R.S. Plant and O. Mart´ınez-Alvarado Editors’ introduction: Previous chapters in Part II have introduced the ideas of convective plumes as an ingredient in mass-flux parameterization, how the properties of such plumes may be calculated, and how the existence of such plumes feeds back onto the mean state. Most mass-flux parameterizations in operational use make calculations for a single plume only, despite the fact that moist convection is far from being uniform. Even within the relatively small area visible to the human eye, a field of convective clouds can exhibit a rich variety of forms. Thus, the single plume appearing in such parameterizations cannot be a direct attempt to model the vertical structure of all of the convective elements that may be present within an area, but must instead be considered as an effective plume produced from a suitable average over the full cloud field. The purpose of this chapter is to explain the previous sentence in detail, and scrutinize its consequences. Specifically, it formulates a suitable average, and proceeds to ask whether such an average over a set of plumes with distinct properties does indeed yield a single effective plume.

1

Introduction

The purpose of this chapter is to compare bulk and spectral mass-flux parameterizations. The model for an individual convective element or plume (labelled i) can be formulated through the budget equations which were derived in Ch. 8. Bulk and spectral parameterizations are distinguished through the way in which the collective effects of an ensemble of plumes are treated. In a spectral model, plumes are grouped together into types 249

250

Part II: Mass-flux parameterization

characterized by some parameter (e.g., λ) so that the mass flux due to plumes of each type can be represented as:  Mi . (1.1) M(λ)dλ = i∈(λ,λ+dλ)

In Arakawa and Schubert (1974), it is assumed that each plume type is characterized by a given, constant entrainment rate, but other categorizations are possible in general. Indeed, a generalization to multiple such parameters is trivial, although not common. For example, one exception is Nober and Graf (2005) in which plume types are defined through both the radius and the vertical velocity at the cloud base. In a bulk parameterization there is no consideration of plume types and the collective effects are treated through an implicit summation over all plumes to produce a single, effective bulk plume. In both types of parameterization, it is assumed that there are sufficient plumes to be treated statistically, such as might be found within a region of space-time “large enough to contain an ensemble of cumulus clouds but small enough to cover only a fraction of a large-scale disturbance” (Arakawa and Schubert, 1974, p. 675). The existence of any such well-defined region in practice is an important question discussed in Ch. 3. A spectral parameterization certainly requires more computations, with multiple plume types to be explicitly considered. Historically, this was an important consideration, and (at least in part) has motivated the development of various bulk parameterizations for operational models. In recent times, it has been less clear that the run time of a convective parameterization should be quite such a strong consideration in its formulation. Certainly if we compare the computational resources that are being devoted to model the climate with convection being represented explicitly rather than parameterized, the overheads of a spectral parameterization appear modest indeed. Nevertheless, at the time of writing, the majority of both operational and climate models still adopt bulk formulations. In parameterizations for mesoscale models another argument has sometimes been advanced for single-plume as opposed to spectral formulations: since the grid elements are relatively small, “it is assumed that all convective clouds in an element are alike” (Fritsch and Chappell, 1980, p. 1724). This argument (see also Frank and Cohen, 1987; Fritsch and Kain, 1993) fails to recognize that although there may be only a small number of clouds present in a relatively small grid box, the properties of those clouds may not be knowable a priori but rather are randomly drawn from those of the

Spectral and bulk mass-flux representations

251

statistical ensemble. As discussed in Vol. 2, Ch. 20, the consideration of smaller grid boxes leads not to a single-plume formulation but rather to stochastic sub-sampling of a spectral formulation. The objective in this chapter is to ask whether a bulk parameterization is a valid simplification of a spectral parameterization in principle. In particular, there will be an explanation and discussion of the approximations and ansatzes required to construct the bulk analogue of a spectral parameterization. Important historical references are the spectral formulation of Arakawa and Schubert (1974) and the bulk formulation of Yanai et al. (1973). The latter is not a parameterization as such, but rather a system for diagnostic analyses. However, the rationale for bulk parameterizations in the literature does seem to be by appeal to the Yanai et al. (1973) system. This is stated explicitly by Gregory and Rowntree (1990) and Gregory (1997) for example. The remainder of this chapter is structured as follows. There is a brief review of the equations for an individual plume in Sec. 2 before a formalizing of the averaging required to develop a bulk parameterization in Sec. 3. A majorly problematic aspect of constructing a bulk parameterization is the treatment of detrainment, and this is considered separately for detrainment below the plume top in Sec. 4 and at the top in Sec. 5. The bulk budget is discussed again in Sec. 6, with implementation of possible assumptions for detrainment and with further explanation of the implications of a bulk system, for instance in terms of modelling the microphysics of convective cloud. Some hybrid approaches are discussed in Sec. 7, and more general discussions are presented in Sec. 8.

2

Plume equations

Let us start the analysis by recalling the budget equation corresponding to mass continuity for a single entraining–detraining plume, as given by Ch. 7, Eq. 6.9: ∂Mi = Ei − Di . ∂z

(2.1)

The equation describing some mass-specific thermodynamic or moisture variable ϕ within the plume is given by Ch. 7, Eq. 6.8: ∂Mi ϕi = Ei ϕ − Di ϕi + ρσi Fi . ∂z

(2.2)

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Part II: Mass-flux parameterization

In particular, for the plume variables the dry static energy s=cp T +gz, the water vapour q, and the liquid water l will be taken, leading to: ∂Mi si = Ei s − Di si + Lv ρci + ρQRi ∂z

(2.3a)

∂Mi qi = Ei q − Di qi − ρci ∂z

(2.3b)

∂Mi li = −Di li + ρci − Ri . (2.3c) ∂z The notation introduced to describe the various forcing terms is such that QRi is the radiative heating rate within the plume, Ri is the rate of conversion of liquid water to precipitation, and ci is the rate of condensation. These forcing terms could in principle be functions of any of the large-scale variables (denoted with an overbar) or any of the variables describing the relevant plume (subscript i). The above equations do not describe mesoscale circulations (e.g., Yanai and Johnson, 1993), downdraughts (e.g., Johnson, 1976, and see also Ch. 13) or phase changes involving ice (e.g., Johnson and Young, 1983). These are, of course, considerable limitations for practical applications, but attempting to include them here would only serve to distract from the main points of the analysis. For the same reason cloud-radiation interactions will also be neglected in the following by dropping the QR forcing term. It must also be remarked that the steady-plume hypothesis (cf., Ch. 7, Sec. 6.4) has been assumed so that no time derivative terms appear in Eqs. 2.1 to 2.3c. Assuming the in-plume air to be saturated at and above the cloud base leads to the relations (cf., Ch. 8, Eqs. 4.3 and 4.4):  1  ∗ (2.4) hi − h si − s ≈ 1+γ  γ  ∗ (2.5) hi − h , Lv (qi − q ∗ ) ≈ 1+γ where h=s+Lv q is the moist static energy, the star denotes saturation and  Lv ∂q ∗  γ= . (2.6) cp ∂T  p

There are many plume models which can be described using these equations, with various formulations for the entrainment and detrainment rates, and various microphysical assumptions for ci and Ri . Some simple choices

Spectral and bulk mass-flux representations

253

were introduced in Ch. 8 and more comprehensive discussions of current practice can be found in Ch. 10 for entrainment and detrainment and Vol. 2, Ch. 18 for the microphysics. As far as possible, no particular specifications will be made in this chapter, assuming that some suitable plume model is available which can be used to integrate the plume equations (Eqs. 2.1 to 2.3) upwards from the cloud base up to a terminating level at which the plume buoyancy vanishes. 3

The bulk plume: Basic idea

The basic idea of the bulk representation is to account for all plumes in terms of a single, effective plume. This results in effective in-plume values that are obtained from a mass-flux weighting operation. Using the subscript B to denote a bulk value, the operation is:  Mi ϕi , (3.1) ϕB = i i Mi

where the summation extends over all plumes within the convective ensemble. Essentially, the hope is that an ensemble of convective plumes provide an almost-linear system such that averaging over the plume spectrum will deliver a closely analogous equation set for the bulk plume. The effect of the convective ensemble on the large-scale thermodynamic state of the atmosphere was presented in Ch. 8 as Eq. 3.2a:   ∂ϕ  ∂ϕ  + (3.2) Di (ϕi − ϕ). Mi = ρ  ∂t conv ∂z i i From the perspective of a bulk formulation, this may be considered as:  ∂ϕ  ∂ϕ  =M + ρ (3.3) Di ϕi − Dϕ  ∂t conv ∂z i where

M=

 i

Mi

;

D=



Di .

(3.4)

i

The first and third terms on the right-hand side of Eq. 3.3 are immediately in a convenient form. However, the second term requires attention and cannot easily be simplified: it seems to demand knowledge of all of the in-plume values of ϕ across the spectrum. As shall become apparent in the remainder of this chapter, it is the detrainment contributions that prove most problematic in bulk formulations.

254

4

Part II: Mass-flux parameterization

Detrainment below plume top

At the level of neutral buoyancy for each plume, all of the mass flux is lost by detrainment within a relatively thin layer. It will be convenient for these discussions to distinguish between the detrainment in this layer and that which may be taking place at lower levels. In order to do so let us follow here the terminology of Gregory and Rowntree (1990) by referring to the detrainment at plume top as the “terminal” detrainment (with subscript term), and the detrainment at lower levels as the “mixing” detrainment (with subscript mix). Thus, denoting the top of each plume by zi , we define: For z < zi

For z = zi

then Dmix,i = Di

then

and Dterm,i = 0

(4.1)

Dmix,i = 0 and Dterm,i = Di .

(4.2)

We can also divide the total detrainment D appearing in the bulk formulation into mixing and terminal contributions D = Dmix + Dterm , where   Di ∀i such that z < zi (4.3) Dmix,i = Dmix = i

i

Dterm =



Dterm,i =

 i

i

Di

∀i such that z = zi .

(4.4)

Let us now consider the mixing detrainment contributions to the term i Di ϕi from Eq. 3.3. The terminal detrainment part is discussed in the next subsection.  One option, which would simplify i Di ϕi dramatically, would be to neglect the mixing detrainment for all plumes. This reduces each entraining– detraining plume to a so-called “entraining plume” which undergoes detrainment only in a thin layer at the top of the plume. Such plumes are the basis of the Arakawa and Schubert (1974) parameterization. An alternative way of obtaining a bulk structure to the equations would be simply to redefine the mixing detrainment rate for the bulk plume in such a way as to impose the desired structure. Thus we could set:  Dmix,i ϕi = Dmix,ϕ ϕB , (4.5) 

i

thereby defining Dmix,ϕ to be:

 i Dmix,i ϕi Dmix,ϕ = M  . i Mi ϕi

(4.6)

Spectral and bulk mass-flux representations

255

The bulk detrainment rate is then different in general for each plume variable, and is different from the detrainment rate required for mass continuity of the bulk plume (obtained by setting ϕi = 1). Bulk detrainment would only become independent of the variable in consideration if the mixing detrainment for individual plumes is proportional to mass flux and if the proportionality constant is the same for all plumes. 5

Terminal detrainment assumptions

 Let us now consider the contributions to i Di ϕi that arise from plumes reaching their level of neutral buoyancy. This level defines the top of each individual plume, and the usual assumption (Arakawa and Schubert, 1974) will be followed that all plumes terminating at a given level have the same zi ). Such an assumption is very value of in-plume liquid water there,  l≡li ( convenient, particularly for observational analyses, in that it allows the cloud top to be regarded as a parameter defining the plume types (e.g., Ogura and Cho, 1973). 5.1

Terminal detrainment in the spectral formulation

The neutral-buoyancy condition is the equality of the environmental and the in-plume virtual dry static energy. This condition is, as considered earlier (cf., Ch. 8, Sec. 4): ∗

zi ) ≡  h∗ = h + hvc , hi (

where the virtual contribution is given by (cf., Ch. 8, Eq. 5.7):  (1 + γ)Lv ε˜  ˆ ∗ δ(q − q) −  l hvc = − 1 + γ ε˜δˆ using the quantity ε˜ defined by Ch. 8, Eq. 5.6: ε˜ =

cp T Lv

(5.1)

(5.2)

(5.3)

ˆ and with δ=(R v /Rd ) − 1 = 0.608. The terminal detrainment relations can also be written for other variables, specifically: si = s = s + svc

(5.4)

li =  l

(5.6)

qi = q∗ = q ∗ + qvc

(5.5)

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Part II: Mass-flux parameterization

with virtual contributions of (cf., Ch. 8, Eqs. 5.8a and b):   γ ε˜  ˆ ∗ Lv ε˜  ˆ ∗ l ; qvc = − l . svc = − δ(q − q) −  δ(q − q) −  1 + γ ε˜δˆ 1 + γ ε˜δˆ (5.7)  The contributions of terminal detrainment to i Di ϕi therefore take the forms:  Dterm,i si = Dterm s (5.8) i

 i

Dterm,i qi = Dterm q∗

 i

l. Dterm,i li = Dterm

(5.9)

(5.10)

Thus, unlike the mixing detrainment, the terminal detrainment contributions can be obtained directly from the sum over plumes and there is no dependence in the bulk formulation on the variable in question. Inspection of Eqs. 5.4 to 5.7 shows that all of the information required to calculate the detraining values of the variables is known to a bulk formulation, with the exception of  l. The liquid-water content associated with terminal detrainment at some level z can only be calculated by integrating the equations for the individual plume that has its level of neutral buoyancy at zi = z.  It is worth remarking also that the simplifications to i Di ϕi for terminal detrainment arise only because the values of in-plume variables that contribute to the buoyancy are constrained by the neutral buoyancy condition. For other variables that may be of interest in a convective parameterization, if attempting to compute chemical tracer or momentum transports for example, no such simplification occurs. 5.2

Terminal detrainment in the bulk formulation

At the heart of a bulk formulation is an ansatz that the liquid water on terminal detrainment from each individual plume is given by the bulk value: zi ) =  l = lB . (5.11) li (

This relation was introduced by Yanai et al. (1973, p. 615) and is described there as being a “gross assumption” but “needed to close the set of equations”. Its practical importance is not clear from the literature. Note that Yanai et al. (1976) compared results from bulk and spectral diagnostic models using data from the Marshall Islands. At least in terms of the profiles of

Spectral and bulk mass-flux representations

257

total mass flux and terminal detrainment flux, the differences were modest. However, the comparison is complicated by “data corrections” made for the spectral but not for the bulk analysis (Yanai et al., 1976, Sec. 3), and the remarks of Tiedtke (1989, p. 1781) on this matter still hold today: it is difficult to know how well such comparisons might hold more generally. Two approaches are presented here in order to explore the role of the ansatz in the construction of a bulk parameterization. First, a somewhat arbitrary spectrum of individual entraining plumes will be computed within a typical tropical sounding and then summed to create a bulk plume. Second, the output is taken from a bulk parameterization operating in a numerical weather prediction model for the mid-latitudes and the bulk plume decomposed into a spectrum of entraining plumes. Figure 9.1 illustrates the behaviour of individual entraining plumes launched into Jordan’s (1958) mean hurricane season sounding. Examples are plotted of 200 separate plumes, with each being assigned the same arbitrary mass flux at the updraught base, with no detrainment below the plume tops and with a range of entrainment rates that results in levels of neutral buoyancy between 850 and 150 hPa. The figure shows mass flux Mi (z) and liquid water li (z) profiles for such plumes. Also shown is a profile for the liquid water  l(z) that detrains at the top of each individual plume, reaching neutral buoyancy at height z, as well as the profile of lB (z) computed from averaging over the full set of plumes according to Eq. 3.1. Two effects produced by the bulk ansatz are immediately apparent. First, by taking lB (z) to be the liquid water detrained at plume top, a bulk formulation will produce terminal detrainment of liquid water at levels in between the lifting condensation level and the top of the most strongly entraining plume in the spectrum: here between 950 and 850 hPa. (Such detrainment is not present in the full system of individual plumes.) Second, a bulk formulation will systematically overestimate liquid-water terminal detrainment at all other levels occupied by the plume spectrum. In this particular example, the overestimate is by ∼ 20%. An overestimate will occur because the terminating plumes have lower liquid-water content than the plumes which remain buoyant. Thus, any procedure for averaging over plumes must produce a bulk value larger than the actual liquid water detraining within plumes terminating at that level. The second example involves decomposing a bulk mass-flux profile into a spectrum of constituent entraining plumes. This approach was first considered by Lawrence and Rasch (2005) for a study of the vertical transport produced by deep convection of chemical species such as ozone and other

258

Part II: Mass-flux parameterization b) 100

200

200

300

300

400

400

Pressure (hPa)

Pressure (hPa)

a) 100

500

600

500

600

700

700

800

800

900

900

1000 0.000

0.005

0.010

0.015

0.020

Mass flux (Pa/s)

(a)

0.025

0.030

0.035

1000 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Liquid water (g/kg)

(b)

Fig. 9.1 (a) Vertical profiles of mass flux for entraining plumes launched into the Jordan (1958) sounding. Each plume has an arbitrary updraught-base mass flux of 0.01 Pas−1 and a range of entrainment rates are used to produce a range of detrainment levels, these being indicated by the diamond symbols. (b) The corresponding profiles of plume liquid water (solid lines). Also shown are the profiles for the terminally detraining liquid water (dotted line) and the bulk liquid water for the plume spectrum (dashed line).

short-lived tracers. The difference between bulk and spectral representations of the transport can be important for such species. The example sounding used is taken from a numerical weather prediction model, specifically for 0600 UTC 30 September 2011 at T+48 in a simulation with Met Office Unified Model (MetUM, Davies et al., 2005). The sounding is taken through the warm conveyor belt of a mid-latitude cyclone which produced significant convective precipitation embedded within a cold frontal structure that was also producing heavy precipitation from layer cloud. This is a situation in which the convection and large-scale cloud parameterizations will interact strongly, and so it is interesting to consider the condensate detrainment from convection, which will (either implicitly or explicitly) produce a source term for the cloud scheme. Full details of the simulation and a description of the weather system can be found in Mart´ınez-Alvarado and Plant (2014). Figure 9.2a shows the sounding, which is close to saturation in the lower and middle troposphere, and which supports elevated convection from a base at around 750 hPa.

259

Spectral and bulk mass-flux representations 100

í4 0

í8 0

200

300

í2 0

í6 0

100

0

Pressure (hPa)

400

500

600

700

800

40

í4 0

400

500

600

900

60

í2 0

í6 0

í8 0

300

20

200

700 1000 0.00

80

0

800 900 1000

0.01

0.02

0.03

0.04

0.05

Mass flux (Pa/s)

(a)

(b)

Fig. 9.2 (a) Vertical sounding through a warm conveyor belt as simulated by an NWP model. The solid and dashed lines represent the temperature and dew point profiles, respectively. (b) Vertical profiles of mass flux for entraining plumes launched into this sounding. Each plume has an arbitrary updraught-base mass flux and its own rate of entrainment.

Similarly to Fig. 9.1a, individual entraining plume ascents for this sounding are shown in Fig. 9.2b. The MetUM uses a bulk parameterization based on the scheme of Gregory and Rowntree (1990), albeit substantially modified since (Derbyshire et al., 2011). The mass-flux profile produced by the model is shown by the dashed line in Fig. 9.3a. That profile is decomposed by finding the set of positive coefficients {ci } that minimizes the difference between the bulk profile, denoted M (z α ), and the c-scaled spectrum of plumes according to:

2   α α M (z ) − ci Mi (z ) , (5.12) min α

i

where the superscript α labels the vertical model levels. The resulting spectrum of entraining plumes is shown by the solid lines in Fig. 9.3 and their summation is shown by the dotted line. Although the plume model for the individual plumes is extremely simple, it is nonetheless able to provide a good approximation to the bulk profile. The liquid-water profiles, and the implied liquid-water detrainment  l

Part II: Mass-flux parameterization

100

100

200

200

300

300

400

400

Pressure (hPa)

Pressure (hPa)

260

500

600

500

600

700

700

800

800

900

900

1000

0

1

2

3

4

Mass flux (Pa/s)

(a)

5

6

1000

0.0

0.2

0.4

0.6

Liquid water (g/kg)

0.8

1.0

(b)

Fig. 9.3 (a) Vertical profile of the bulk mass flux produced by an NWP model (dashed line) and the spectral decomposition (solid lines) that generates the closest profile (in a least-squares sense) according to Eq. 5.12. The dotted line represents the approximated bulk mass flux. (b) The corresponding profiles of plume liquid water (solid lines). Also shown are profiles of terminally detraining liquid water from the plume spectrum (dotted line) and the bulk liquid water for the plume spectrum (dashed line).

from the spectral decomposition are shown in Fig. 9.3b. As for the case of the Jordan (1958) sounding, a bulk formulation in which terminal liquidl will produce water detrainment is made proportional to lB rather than  spurious detrainment at lower levels. For this warm conveyor belt sounding, the effect is rather striking since none of the constituent entraining plumes in the spectrum reach their level of neutral buoyancy below about 500 hPa. Indeed, the constituent plumes detrain only at a few different levels and so the continuous profile of lB will overestimate the overall amount of condensate detrained. On the other hand, there is a relatively weak spread in liquid-water content li amongst the plumes as compared to the earlier l at those levels example, and so lB does provide a reasonable estimate for  for which terminal detrainment actually occurs.

Spectral and bulk mass-flux representations

5.3

261

Neutral buoyancy condition for bulk plume

Another issue arising in a bulk formulation is how one should define the level of neutral buoyancy for the bulk plume. Note that in the bulk formulation of Yanai et al. (1973), the bulk plume is assumed to terminate at the level z for which ∗

z) = h , hB (

(5.13)

from which it follows that sB ( z) = s

;

TB ( z) = T

;

qB ( z) = q∗ .

(5.14)

A small historical curiosity is that Yanai et al. (1973) claim that the same assumption is used in a version of Arakawa and Schubert (1974) (referenced by Yanai et al. (1973) as 1973 and with a status of “to be published”). It can only be presumed that there were changes in producing the final version of Arakawa and Schubert (1974), as the neutral-buoyancy condition there includes the virtual contributions. According to Yanai et al. (1976, p. 978), “the virtual temperature correction has been neglected for simplicity”. While these authors were commendably straightforward about the arbitrary but convenient nature of the bulk ansatz in Eq. 5.11, this appeal to simplicity is perhaps slightly misleading. Inspection of Eqs. 5.2 and 5.7 shows that the virtual correction simply cannot be accounted for fully within a bulk model since these terms involve the in-plume liquid water  l appropriate for the terminal detrainment level of each plume. As discussed above, this quantity can only be computed by integrating the individual plume equations for a plume that has its top at the level in question. This is not known within a bulk formulation. The virtual corrections and their potential importance (or otherwise) ∗ are not often discussed. Johnson (1976, p. 1894) noted that hvc /h ≪1 and so argued that the virtual correction could indeed be reasonably neglected in Yanai et al. (1973). However, Nordeng’s (1994, p. 25) experiments with a bulk parameterization found some significant effects from changing the treatment of virtual terms in the terminal detrainment condition used. It is straightforward to evaluate the specific-humidity component of ∗ hvc /h explicitly, and for the Jordan (1958) sounding the ratio has a maximum amplitude of around 0.005 in the lower troposphere. Moreover, svc /s and qvc /q∗ peak at 0.002 and 0.04 respectively, so that it would indeed appear very reasonable to neglect virtual effects as being small corrections to in-plume variables at the terminal detrainment layer. However, it must be stressed that the purpose of Eq. 5.1 (or of Eq. 5.13) is to determine

262

Part II: Mass-flux parameterization

the terminal detrainment level zi (or z). Figure 9.4a illustrates that the environmental gradient of saturated moist static energy can be small in the tropical upper troposphere. Hence, there is not necessarily any basic disagreement between Johnson (1976) and Nordeng (1994) because even small errors in the specification of the neutral-buoyancy condition have the potential to result in considerable errors in a calculation of cloud top. b) 200

300

300

400

400

Pressure (mb)

Pressure (mb)

a) 200

500

600

500

600

_ h* 700

700

^ h* 800

800

900

900 336

338

340

342

344

346

348

350

Moist static energy (kJ/kg)

(a)

0

20

40

60

80

100

^ correction (m) Estimated z

120

140

(b) ∗

h∗ (dashed line; Eq. 5.1) for the Fig. 9.4 (a) Vertical profiles of h (solid line) and  Jordan (1958) sounding. The difference between them represents a virtual contribution to the determination of the level of neutral buoyancy. (b) Error in the calculation of z, as discussed in the main text and estimated from Eq. 5.15. ∗

The errors are difficult to estimate reliably when ∂h /∂z is small, particularly if there is any noise in the sounding data. For this reason, in Fig. 9.4b we plot for the Jordan (1958) sounding the quantity     1  svc   qvc  , (5.15) + Δ z= 2  ∂s/∂z   ∂q ∗ /∂z 

which provides a simple estimate of the effect of virtual contributions on the evaluation of cloud top, and which should at least be reliable for plumes terminating in the lower troposphere. The corrections are ∼ 150 m. Of course, one could adopt a bulk formulation with an estimated virtual correction by taking Eqs. 5.1, 5.4, and 5.5 to define the top of the bulk plume, but with  l→lB (cf., Nordeng, 1994). This should be an improveˆ ∗ −q)− l| δturb

but

momentum convergence –

Organized detrainment is, in general, formulated as a massive lateral outflow of mass around the neutral buoyancy level although the precise details differ among the cited parameterizations. The above-cited parameterizations typically use Eq. 2.1 assuming a fixed radius of R ≃ 500 m for shallow clouds and R ≃ 2000 m for deep convection. Another class of entrainment–detrainment parameterizations that does not explicitly distinguish between dynamical and turbulent mixing is based on the buoyancy-sorting concept introduced by Raymond and Blyth (1986). This buoyancy-sorting concept is transformed into an operational parameterization by Kain and Fritsch (1990) (cf., Sec. 4.1). In their parameterization, an ensemble of mixtures of cloudy and environmental air is considered where each ensemble member has a different concentration of environmental air. If resulting mixtures are positively buoyant, they remain in the updraught and are part of the entrainment process while negatively buoyant mixtures are rejected from the updraught and are part of the detrainment process. A number of recently proposed shallow cumulus convection schemes are based on or make use of this buoyancy-sorting concept (cf., Secs. 4.1 and 4.3). It is important to note that all of these studies are performed within a framework of the environment hypothesis (Ch. 7). Although LES studies (e.g., Heus and Jonker, 2008) point to a limit of the environment hypothesis and the importance of taking into account an immediate environment, such a formulation is yet to be fully considered in a parameterization context. As the discussions of this chapter will show, when the full suite of interactions between the subgrid-scale components are taken into account, the entrainment–detrainment formulation takes a matrix form. Although it would be straightforward to generalize the existing methodologies (de Rooy and Siebesma, 2010; Siebesma et al., 2003; Siebesma and Cuijpers, 1995) for obtaining entrainment–detrainment estimates from LES and CRM data for

Entrainment and detrainment formulations for mass-flux parameterization

279

this purpose, a theoretical basis for developing such a matrix formulation is clearly missing. 2.5

Lateral versus vertical mixing

Entrainment of environmental air into clouds tends to dilute cloud properties and degrade the buoyancy characteristics of cloudy air, both of which affect the vertical transport by clouds. Knowing the characteristics of the air entering the cloud, which is strongly related to knowing the source height of the entrained air, is therefore naturally regarded as a crucial issue. In this respect it is most surprising that two radically opposing views, referred to here as “lateral entrainment” and “cloud-top entrainment”, have been able to coexist for such a long time in the community. The origins of these views go back at least as far as Stommel (1947) (lateral entrainment) and Squires (1958) (cloud-top entrainment). In the former view, cloudy air, carrying the properties of the cloud base, is continually diluted during its ascent by mixing air entrained into the cloud via the lateral cloud edges. It is this view that has served as the basis for parameterizations of moist convective transport in operational models. Conversely, in the cloud-top entrainment view, environmental air is predominantly entrained at or near the top of the cloud, after which it descends within the cloud via penetrative downdraughts, finally diluting the rising cloudy air by turbulent mixing. A conceptual picture of how this could look is given in Fig. 10.2. Clearly the two views outlined above differ enormously in respect of the source height of entrained air, and therefore they differ also crucially in respect of the properties of air that dilute the cloud. It should be noted that in principle both views are not mutually exclusive since both mechanisms could be active at the same time, but the question really is which of the two mechanisms dominates. A rather contrived argument would be needed to anticipate that both mechanisms are equally effective. Ample evidence for the importance of lateral entrainment can be found in the literature (e.g., Lin and Arakawa, 1997; Raga et al., 1990). Also, though perhaps more indirectly, the lateral entrainment view derives justification from the appreciable predictive quality of the moist convective parameterizations that are based on it (e.g., Kain and Fritsch, 1993; Siebesma and Cuijpers, 1995). Compelling observational evidence for the cloud-top entrainment view came from the elegant analysis of Paluch (1979), who plotted in-cloud values of conservative variables (total specific humidity and equivalent poten-

280

Part II: Mass-flux parameterization

Fig. 10.2 A conceptual picture of cloud-top entrainment, developed by following Fig. 13.8 of Stull (1988).

tial temperature) in a diagram which now carries her name. Rather than displaying significant scatter, the in-cloud values observed during a cloud transect at a particular level were found to collapse onto a distinct line. Such a “mixing line” is commonly taken as strong evidence for a two-point mixing scenario: if one mixes air from two (but not more) different sources, any mixture must show up on a line in such a diagram due to the nature of conservative variables. By extrapolating the line, Paluch identified the two source levels as cloud-base and cloud-top (or a level significantly higher than the level of the cloud transect). It is important to note that, at face value, the analysis of Paluch seems to leave no room for significant lateral entrainment since mixing with more than two sources would yield significant scatter away from the mixing line. Later studies (e.g., Betts, 1982, 1985; Boatman and Auer, 1983; Jensen et al., 1985; Lamontagne and Telford, 1983) confirmed the findings of Paluch, thus providing further support for the importance of cloud-top entrainment. The historical shift from a lateral entrainment-oriented view towards a cloud-top entrainment-oriented view can very clearly be noted in the overview paper by Reuter (1986), for example. Criticism and warning comments, not so much directed at the location of data points in the Paluch diagrams itself, but rather at the interpretation drawn from them, were given by B¨oing et al. (2014), Siebesma (1998) and Taylor and Baker (1991). Siebesma (1998) pointed to the strong selfcorrelation that exists in the conserved variables chosen for the Paluch diagrams, which makes it hard for the data not to line-up (see for example Fig. 1 of Heus et al., 2008). Taylor and Baker (1991) drew attention to the

Entrainment and detrainment formulations for mass-flux parameterization

281

phenomenon of buoyancy sorting, which suggests that essentially only a biased selection of data points can show up in Paluch diagrams. Simply put, a bias is introduced because it is less likely to observe buoyant air parcels coming from above, as well as it is unlikely to observe negatively buoyant parcels coming from below. Consequently, most observed data points are related to buoyant parcels coming from below and negatively buoyant parcels from above. As explained in detail in Taylor and Baker’s paper, this effect puts serious limits on the possible values one can observe at a particular cloud level: essentially the data points are confined to a triangle that very much resembles a line. They conclude that: The graphical analysis of non-precipitating cloud composition shows that the apparent mixing line structure of single level in-cloud points on a conserved tracer diagram can result from a continuous series of entrainment events occurring throughout the cloud depth if buoyancy sorting is dominant throughout the flow.

In Sec. 3.7 recent LES studies are discussed that seem to provide a final answer to the controversy between cloud-top and lateral mixing, which may furthermore be described as entrainment processes. 2.6

The use of LES in studying entrainment and detrainment

Obtaining accurate observations of entrainment and detrainment is notoriously difficult. Nevertheless, some inventive studies by, for example, Raga et al. (1990) and Yanai et al. (1973) linked observations to entrainment rates. LES models, in turn, have matured since the 1990s, initiating a strong revival of entrainment–detrainment studies. The resolution of these models is high enough to capture the largest eddies, which are responsible for the majority of the convective transport. Comparison with various different field observations, such as ARM (Atmospheric Radiation Measurement Brown et al., 2002), BOMEX (Barbados Oceanographic and Meteorological EXperiment Siebesma et al., 2003), and ATEX (Atlantic Trade wind EXperiment Stevens et al., 2001) has shown that modern LES is capable of accurately simulating cumulus cloud dynamics and resolving the intricacies of entrainment processes, even down to non-trivial geometrical properties of the cloud edge (Siebesma and Jonker, 2000). Before lateral mixing can be studied in LES, one first has to define the cloud and the environment (called the sampling method). Often applied

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Part II: Mass-flux parameterization

is the cloud-core sampling where all LES grid points that contain liquid water (ql > 0) and are positively buoyant (θv > θv ) are considered to be part of the cloudy updraught. Here, ql is the liquid-water content, θv is the virtual potential temperature (being a measure of density), and θv is the horizontally averaged virtual potential temperature. In the 1990s, computer resources were too limited to perform LES of deep convection. However, early LES results provided important insight into shallow convection including lateral mixing. For example, Siebesma and Cuijpers (1995) showed in a careful analysis of LES results that the turbulent transport can be accurately described with a bulk mass-flux approach, especially when the cloud-core sampling method is applied. Diagnosing ǫ and δ from LES can give a strong indication of the ǫ and δ that should be used in an NWP or climate model bulk mass-flux scheme. Therefore, LES provide a powerful tool to study the qualitative and quantitative behaviour of the lateral mixing coefficients. For example, Siebesma and Cuijpers (1995) established that the typical entrainment and detrainment values for shallow convection usually applied at that time were an order of magnitude too small. Several LES studies of shallow convection have also revealed that the detrainment varies much more than the entrainment and therefore has a much larger impact on variations in the mass-flux profile (de Rooy and Siebesma, 2008). A more theoretical basis for this result is given by de Rooy and Siebesma (2010) who showed that a significant part of the variations in δ is associated with the cloud layer depth. This LES finding is applied in the convection parameterization schemes of de Rooy and Siebesma (2008) and Neggers et al. (2009) (cf., Sec. 4.3). LES studies have further revealed the influence of environmental humidity conditions (e.g., Derbyshire et al., 2004) and properties of the updraught itself (e.g., de Rooy and Siebesma, 2008) on the mass-flux profiles. Large-eddy simulation studies have also been used to investigate lateral mixing in a more fundamental way. For example Heus and Jonker (2008) and Jonker et al. (2008) described the influence of a subsiding shell on lateral mixing (cf., Sec. 3.8). Zhao and Austin (2005b), among others, investigated the mixing between clouds and their environment during the life cycle of single clouds. Finally, two recent LES studies derived local entrainment and detrainment coefficients (Dawe and Austin, 2011b; Romps, 2010, and see also Sec. 3.6). Whereas Dawe and Austin (2011b) accomplished this by carefully determining the net velocity through the cloud-environment interface, Romps (2010) used an inventive definition of entrainment and detrainment. Compared to LES results diagnosed within the bulk mass-

Entrainment and detrainment formulations for mass-flux parameterization

283

flux framework, both of these studies diagnosed significantly larger lateral mixing coefficients. This discrepancy can be explained by the fact that the lateral transport in Romps (2010) and Dawe and Austin (2011b) involves less distinct properties between cloud and environment. With increased computer resources, LES models are now capable of simulating deep convection (e.g., B¨ oing et al., 2012; Khairoutdinov and Randall, 2006; Khairoutdinov et al., 2009; Kuang and Bretherton, 2006). The first of these studies will be discussed in Sec. 4.3 because it systematically explores the sensitivity of lateral mixing in deep convection to the relative humidity and stability of the free atmosphere. Although LES studies of deep convection have been very insightful, the important role of the microphysics on cloud dynamics is a complicating factor as these processes also need to be parameterized. 3 3.1

LES diagnosis of entrainment and detrainment processes Introduction

The purpose of this section is to introduce a precise notion of the mixing processes and to explore their behaviour in a more fundamental way. First, basic definitions will be provided and applied to a rising dry plume which is governed purely by an entrainment process. Subsequently, entrainment and detrainment will be reviewed in the context of the steady-state cloud model of Asai and Kasahara (1967), which will make the notion of organized versus turbulent entrainment and detrainment more precise. There will follow a review of the various ways of determining the exchange rates from LES. Finally, LES results will be used to discuss the long-lasting controversy between the relative importance of lateral and vertical mixing in cumulus convection, and the importance of the vicinity of the cloud in the lateral mixing process. 3.2

General definitions

A convenient starting point is the conservation law of a scalar variable ϕ: ∂ϕ + ∇ · vϕ = F (3.1) ∂t (cf., Ch. 7, Eq. 3.4), where v denotes the three-dimensional velocity vector and where all possible sources and sinks of ϕ are collected in F . For the sake of simplicity a Boussinesq flow is assumed, implying that the density

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Part II: Mass-flux parameterization

in Eq. 3.1 is constant and has been divided out of Eq. 3.1. A domain with a horizontal area A is considered, with an interest in the lateral mixing between a cloudy area Ac and a complementary environmental area Ae at a given height z, as sketched schematically in Fig. 10.3. Note that the areas are denoted by A in this chapter rather than S as in Ch. 7. Also, a single convective component will be considered throughout this chapter, designated by the subscript c.

Fig. 10.3 Schematic diagram showing an ensemble of clouds at a certain height. A, Ac , and Ae represent respectively the total horizontal domain area (A = Ac + Ae ), the cloudy area (Ac , white), and the environmental area (Ae , gray). The interface between the cloudy area and the environment is plotted as a dashed line and has a total length c Royal Meteorological Lb . (Adapted from de Rooy and Siebesma (2010), which is  Society, 2010.)

At this point it is not necessary to be more specific about the precise definition of the cloudy area but it should be noted that it may consist of many different “blobs” (or clouds) that can change in shape and size as a function of time and height. By integrating Eq. 3.1 horizontally over the cloudy area Ac (z, t) and applying the Leibniz integral rule and the Gauss divergence theorem, a conservation equation of the cloudy area for ϕ can be deduced (Siebesma, 1998):  1 ∂σc wϕc ∂σc ϕc + = σc Fc n ˆ · (u − ui )ϕdl + (3.2) ∂t A ∂Ac ∂z ˆ is (cf., Ch. 7, Eq. 4.6), where σc = Ac /A is the fractional cloud cover, n an outward pointing unit vector perpendicular to the interface, u is the horizontal velocity vector at the interface, and ui is the velocity of the

Entrainment and detrainment formulations for mass-flux parameterization

285

interface. Overbars and variables with subscript c denote averages over the cloudy part. In the special case ϕ = 1 and Fc = 0 we recover the continuity equation:  1 ∂σc ∂σc wc + =0 (3.3) n ˆ · (u − ui )dl + ∂t A ∂Ac ∂z (cf., Ch. 7, Eq. 4.9). Equation 3.3 has a simple geometrical interpretation. The net change of the cloud fraction is a result of the net lateral inflow of mass across the cloudy interface on the one hand and the vertical mass-flux divergence on the other hand. It must be emphasized that it is the mass velocity u relative to the interface velocity ui that enters in the interface term. This way it is guaranteed that there is no net inflow if a cloud is simply advected by the mean wind. Since entrainment is usually associated with the inflow of mass into the cloudy area, whereas detrainment is associated with the complementary outflow, it seems natural to define these processes as:  1 n ˆ · (u − ui )dl, (3.4a) E=− A ∂Ac−  1 n ˆ · (u − ui )dl. (3.4b) D= A ∂Ac+ This is a generalization of the definition already introduced in Ch. 7, in which each subgrid-scale component is assumed either purely entraining or detraining as a whole (Ch. 7, Eq. 5.6). Here, this definition is generalized by further subdividing the subcomponent boundary into the entraining (∂Ac− : n ˆ · (u − ui ) < 0) and the detraining (∂Ac+ : n ˆ · (u − ui ) > 0) subsegments. ˆdl = dr. Here, note some modifications of notations from Ch. 7: ui = r˙ i , n Recall that ui had a different meaning in Ch. 7. By setting E ≡ ǫM and D ≡ δM , Eq. 3.3 reduces under steady state conditions to 1 ∂M = ǫ − δ. (3.5) M ∂z Although it is relatively straightforward to determine E − D as a residual from Eq. 3.3, it is by no means trivial to determine entrainment and detrainment rates separately, neither in laboratory experiments nor in numerical simulations. This point will be discussed further in Sec. 3.6. 3.3

Dry plumes

While entrainment and detrainment are easily defined mathematically, the physical processes involved are not always fully understood and in fact can

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Part II: Mass-flux parameterization

depend on how one defines the interface across which the mixing processes are defined. Buoyant dry plumes, which rise in a non-turbulent environment, provide a relatively simple example. They entrain environmental air and show virtually no detrainment. Such plumes rise and grow almost indefinitely, until they are diluted to the extent that they are absorbed in the chaos of molecular motions. If the length of the perimeter of the plume is defined by Lb , and ub is defined as the net mean velocity at the boundary of the plume:  1 b n ˆ · (u − ui )dl, (3.6) u ≡ ub ≡ Lb ∂Ac and assuming steady state and a circular geometry (i.e., Ac = πR2 and Lb = 2πR) it is then straightforward to rewrite Eq. 3.3 as: Lb ub 1 2ub 1 ∂M = ≃ , M ∂z Ac wc R wc

(3.7)

which provides a justification for the famous entrainment relationship for plumes (Morton et al., 1956) and a physical interpretation of the proportionality constant in Eq. 2.1. 3.4

Steady-state single cloud

Atmospheric clouds differ from dry plumes. Entrainment of unsaturated environmental air leads to the evaporation of cloud liquid water. Some cloud parcels will lose their buoyancy and ultimately their liquid water and are then by definition detrained. This naturally demands explicit inclusion of the detrainment process. It is not possible to make more precise statements on the entrainment and detrainment processes without being more specific as to the physics that plays a role in these processes. A popular model has been proposed by Asai and Kasahara (1967), in which a steady-state cloud is assumed to be cylinder-shaped with a radius R. Further they presume a scale separation between turbulent exchange across the cloud interface and a larger scale inflow or outflow resulting from the buoyancy-driven mass-flux convergence or divergence inside the cloud. This is achieved by applying a Reynolds decomposition of the flux across cloud boundary for thermodynamic conserved variables ϕ: uϕb ≡ ub ϕb + u′′ ϕ′′

b

(3.8)

in which by convention ub is positive if it is pointing outwards from the cloud, and the mean property of ϕ along the cloud boundaries ϕb may

Entrainment and detrainment formulations for mass-flux parameterization

formally be defined as: ϕb ≡ ϕb ≡

1 Lb



ϕdl.

287

(3.9)

∂Ac

This scale separation allows the introduction of turbulent entrainment and detrainment on the one hand and organized entrainment driven by convergence and organized detrainment driven by divergence on the other. More specifically, following Asai and Kasahara (1967), we approximate the turbulent flux by an eddy diffusivity approach and make an upwind approximation of the organized inflow and outflow (i.e., ϕb = ϕc if ub > 0, and ϕb = ϕe if ub < 0). One can then derive for the various terms on the right-hand side of Eq. 2.3 that: 2ξ R

(3.10a)

1 ∂wc wc ∂z

(3.10b)

1 ∂wc , wc ∂z

(3.10c)

ǫturb = δturb = ǫdyn = H(−ub ) δdyn = −H(ub )

where H denotes the Heaviside function, wc is the average vertical velocity in the cloud, and ξ is a dimensionless constant analogous to the constant of proportionality between horizontal (here radial) and vertical velocity fluctuations in the mixing length theory, which is of the order O(1). If these results are coupled with updraught equations for temperature, moisture, and vertical velocity, and fed with appropriate boundary conditions at cloud base, one typically finds net condensational heating in the lower part of the cloud that feeds the buoyancy, leading to an acceleration of the updraught. This acceleration has a negative feedback since it will induce an inflow due to the organized entrainment that will eventually slow down the updraught, leading to divergence and an organized detrainment in the upper part of the cloud. A few remarks should be made here. First, the fact that the turbulent mixing is assumed to be symmetric in terms of an equal entrainment and detrainment has been criticized by Randall and Huffman (1982). In their model, the interface is defined as the boundary of the mass of turbulent air associated with the cloud. They therefore model the turbulent mixing solemnly as an entrainment process and not as a turbulent mixing process as in Asai and Kasahara (1967). Second, the form of the organized entrainment and detrainment is a direct result of the strong assumption that the

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Part II: Mass-flux parameterization

cloud has a constant radius R. With wc predicted by an updraught equation, ǫdyn and δdyn are determined by Eqs. 3.10b and 3.10c. Therefore, the constant R assumption can be seen as the organized entrainment and detrainment closure of the Asai and Kasahara (1967) model. If the interface is defined as the buoyant part of the cloud, a thermodynamic constraint should determine how R varies with height. The buoyancy-sorting principle presented by Kain and Fritsch (1990) is a step in that direction. In their model (cf., Sec. 4.1), equal masses of environmental and cloudy air are assumed to form various mixtures. It is then assumed that negatively buoyant mixtures are detrained whereas positively buoyant mixtures are entrained. However, in that case the closure problem is shifted to the choice of how much mass is available for supplying such mixtures and which probability distribution function to choose for the occurrence of the various mixtures. Another interesting idea was proposed by Neggers et al. (2009). In their approach, a probability function of temperature and moisture within the cloud is reconstructed from different updraughts. Such a joint PDF allows the determination of the area of the cloud that is positively buoyant and hence the variation of its radius as a function of height. 3.5

Determination of entrainment and detrainment from large-eddy simulations: Bulk estimates

Large-eddy simulations have proven to be an extremely useful tool in determining entrainment and detrainment rates in cumulus clouds, initially for shallow cumulus (e.g., Siebesma and Cuijpers, 1995; Siebesma et al., 2003; Swann, 2001) but more recently also for deep convection (e.g., B¨oing et al., 2012; Khairoutdinov et al., 2009; Kuang and Bretherton, 2006). These studies have provided useful guidance for parameterizations of detrainment and entrainment in large-scale models (e.g., de Rooy and Siebesma, 2010; Gregory, 2001; Siebesma and Holtslag, 1996). The traditional way to diagnose E and D is not through the direct use of Eqs. 3.4a and 3.4b, but rather through an effective bulk entrainment and detrainment rate defined as:  1 n ˆ · (u − ui )ϕdl, (3.11a) Eϕ ≡ − Aϕe ∂Ac−  1 n ˆ · (u − ui )ϕdl, (3.11b) Dϕ ≡ Aϕc ∂Ac+

289

Entrainment and detrainment formulations for mass-flux parameterization

where the exchange rates Eϕ and Dϕ have been indexed to indicate that there might be a ϕ dependence. Substituting these definitions in Eq. 3.2 then directly gives: ∂σc ϕc ∂σc wϕc = M (ǫϕ ϕe − δϕ ϕc ) − + σc Fc . ∂t ∂z

(3.12)

By combining Eq. 3.12 and Eq. 3.5, the bulk fractional entrainment and detrainment rates can be diagnosed from LES output. For this diagnosis, the sub-plume term in wϕc is usually ignored, steady state is assumed and if a conserved variable is considered then the source term Fc is zero, so that the entrainment can be diagnosed according to Betts (1975): ∂ϕc = −ǫ(ϕc − ϕe ), ∂z

(3.13)

where Eq. 3.5 has been used to eliminate δ from the expression. (See Sec. 8 for more technical details.) Most importantly, the exchange rates deduced in this way are used in a similar way in parameterizations. Indeed, virtually all parameterizations use Eq. 3.12 as a starting point and therefore need to be fed by the same effective bulk entrainment rates that are diagnosed in this way by LES. The price to be paid is that the bulk exchange rates ǫϕ and δϕ are now not necessarily a property of only the turbulent flow, but can be dependent also on the physical variable ϕ (cf., Yano et al., 2004, and Ch. 9). 3.6

Determination of entrainment and detrainment from large-eddy simulations: Direct estimates

The use of the “true” exchange rates as defined by Eqs. 3.4a and 3.4b is far from trivial from a numerical point of view, mainly because it is difficult and until recently it was unclear how to diagnose the local velocity ui of the interface. While in reality u and ui are of the same order of magnitude, the cloudy surface in an LES model may shift one grid box in one timestep, giving rise to very high, unrealistic ui values that are dictated by the timestep value rather than anything physical. However, two recent independent studies, Dawe and Austin (2011b) and Romps (2010), have been able to tackle this problem and derive E and D directly based on Eqs. 3.4a and 3.4b. Dawe and Austin (2011b) follow a straightforward method by applying a subgrid interpolation to determine the position of the cloud surface more accurately. Romps (2010) follows a different approach: instead of a bulk

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Part II: Mass-flux parameterization

cloud sampling, he defines a local activity operator A such that:  1 for ql (x, t) ≥ qthresh and w(x, t) ≥ wthresh A(x, t) = 0 for ql (x, t) < qthresh or w(x, t) < wthresh .

(3.14)

A point in space (or a grid box) entrains when it switches from A = 0 to A = 1 and detrains when it switches from A = 1 to A = 0. Romps (2010) then diagnoses E and D as follows:

∂ (3.15a) E = max 0, (ρA) + ∇ · (ρuA) ∂t

∂ D = max 0, − (ρA) − ∇ · (ρuA) , (3.15b) ∂t

where ∂(ρA)/∂t + ∇ · (ρuA) is referred to as the activity source, and is built up from the local tendency of active air (first term) and the resolved advection of active air (second term). The activity source is positive if there is a source of active air (entrainment) and negative if there is a sink of active air (detrainment). E and D are diagnosed by summing the activity source over the complete period for which the grid cell is adjacent to the cloud surface. This can be seen as an implicit subgrid interpolation of the cloud surface and it also ensures that a pure advective cloud has E = D = 0. Note that active air is not a prognostic field being advected. Instead, the activity operator is applied at the end of every timestep to determine which grid cells are active. As Dawe and Austin (2011b) and Romps (2010) evaluate E and D locally, there are some important differences from the bulk diagnostic approach, which will be more fully discussed later. For example, bulk estimates of ǫ and δ are tracer dependent whereas direct measurements of the lateral mixing coefficients are only related to the local properties of the flow (Romps, 2010). However, the most striking result was that the bulk plume approach underestimates entrainment and detrainment by roughly a factor of two (Romps, 2010). This is elucidated in Fig. 10.4a, which shows the conceptual picture following the bulk concept. The air that detrains from the cloud (with properties ϕD ) is supposed to have the same property as air averaged over all clouds (ϕD = ϕc ≡ ϕc ). Fig. 10.4b illustrates the situation in the Romps (2010) local approach. Some grid boxes are diagnosed as detraining according to the direct measurement technique, represented by grey squares in the figure. In general the relatively less buoyant cloudy grid boxes will detrain (it is possible that a grid box that has recently entrained may detrain again). Consequently, the potential temperature

Entrainment and detrainment formulations for mass-flux parameterization

291

of the detraining grid boxes will on average be lower than the potential temperature averaged over the complete cloudy area. Similarly, it will normally not be the most humid grid boxes that detrain. Thus, detraining air will on average have properties in between the average cloudy and average environmental air. The same arguments hold for entrainment. Because the difference between detraining and environmental air, or entraining and cloudy air, is larger in the bulk approach than in the direct Romps (2010) framework, the corresponding ǫ and δ values should be smaller in the bulk approach in order to obtain the correct lateral fluxes. Recently, this discrepancy between bulk and directly measured ǫ and δ values has been further investigated and quantitatively explained by Dawe and Austin (2011a).

Fig. 10.4 Schematic diagram showing the situation relevant for detrainment in the bulk mass flux picture (left panel), and for directly measured detrainment (Romps, 2010) (right panel). In the bulk picture, detraining cloudy air always has the average properties of the cloudy area ϕc . Within the direct measurement method the detraining grid boxes are represented by the grey squares and, as indicated by the grey scale, the detraining air has properties in between the average cloudy and environmental air.

A potentially important result of Dawe and Austin (2011b) and Romps (2010) is the change of the cloud properties due to detrainment because in their approach detraining air does not have the average cloud properties (cf., Fig. 10.4). This is in contrast with the entraining-plume model of Betts (1975) (Eq. 3.13) as used in the bulk mass-flux concept, where only ǫ determines the dilution. On the other hand, if ǫ is diagnosed in LES within the bulk framework, it will describe the correct cloud dilution as long as it is applied in a bulk scheme. One might say that this diagnosed bulk ǫ implicitly takes into account the negative dilution due to detrainment. Direct entrainment and detrainment calculations are very useful to understand the underlying processes. At the same time it should be realized that ultimately the different approaches lead to the same, correct dilution

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of the cloud properties and turbulent transport, as long as the mixing coefficients are diagnosed and applied consistently in the same framework. Therefore, bulk diagnosed entrainment and detrainment values are more appropriate to use in a bulk mass-flux parameterization. 3.7

On the origin of entrained air in cumulus clouds

The controversy between lateral and vertical mixing, discussed in Sec. 2.5, was recently studied in more detail within the framework of large-eddy simulations by Heus et al. (2008). Interestingly this LES study does reproduce the Paluch mixing lines, but refutes the conclusion that the diagram implies. The study followed a large number of Lagrangian particles which could be traced back in time so as to reveal the true origin of the entrained air. The technique made it possible to obtain a wealth of statistical data by analysing all clouds in the ensemble and additionally averaging in time, and revealed the precise fraction of particles that entered near the top of the cloud, the fraction of particles that entered the lateral edge of the cloud, and the fraction that entered via the cloud base. Figure 10.5, reproduced from the original paper, shows a density plot of the relative entry level (height of first entry of a particle in the cloud, normalized by cloud depth) and the relative level of observation in the cloud. The horizontal dark grey band is indicative of particles that entered via the cloud base, the dark diagonal band is indicative of particles entering via the lateral edge near the observation level, while the light grey lower right triangle shows the fraction of particles that were laterally entrained at levels below observation level. But, in the context of the present discussion, the most significant feature of Fig. 10.5 is the white upper-left triangle, representing in a statistical sense the striking absence of particles entrained from above. If cloud-top entrainment, followed by penetrative downdraughts, were to be a significant mechanism, then the upper-left triangle in Fig. 10.5 should have been filled to a reasonable extent. But clearly it is not. The question is still open as to why cloud data points tend to line up in a Paluch diagram in the way that they do, thus providing compelling but flawed support for the importance of cloud-top entrainment for cumulus dynamics. Heus et al. (2008) found some evidence for the buoyancy-sorting mechanism suggested by Taylor and Baker (1991), but actually not enough to serve as a full explanation. B¨ oing et al. (2014) argue that a mixing line will always arise when an ensemble of updraughts which experience lateral mixing with different entrainment rates is considered. In order to support

Entrainment and detrainment formulations for mass-flux parameterization

293

this view, they used an analytical model, as well as an atmospheric parcel model where the entrainment rate varies between parcels. Cloud Size>300 m

Relative Entry level (í)

1

0.8

0.6

0.4

0.2

0 0

0.2 0.4 0.6 0.8 Relative Observation level (í)

1

Fig. 10.5 The relative height at which particles entered the cloud as a function of relative observation level, for all clouds in the ensemble with a vertical size larger than c American Meteorological Society. 300 m. (Figure reproduced from Heus et al. (2008).  Used with permission.)

3.8

Relevance of the near vicinity of clouds

From the premise that lateral entrainment is important, it follows that the immediate vicinity of clouds must be important as well because it defines the properties of the air to be entrained into the cloud. Observational studies by, for example, Jonas (1990), Rodts et al. (2003), and more recently by Heus et al. (2009b) and Wang et al. (2009) reveal how the (thermo)dynamic and microphysical properties of the near-cloud environment differ significantly from the environmental properties further away from the cloud. Studying individual cloud transects, Jonas (1990) noted the presence of a thin shell of subsiding air surrounding clouds and pointed to the importance of it for understanding the droplet spectrum in clouds, taking into account that the shell in principle could contain air with cloud-top properties. As to the cause of the descending shell, Jonas (1990) identified two mechanisms: mechanical forcing and evaporative cooling resulting from mixing saturated and unsaturated air at the cloud edge. As a follow-up, Rodts et al. (2003) studied a large number of horizontal cloud transects

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Part II: Mass-flux parameterization

measured during the SCMS (Small Cumulus Microphyics Study) campaign (Knight and Miller, 1998), and created a statistical average by normalizing each transect by the corresponding cloud width; they confirmed the persistent occurrence of the descending shell of air surrounding the cloud and found their data to be more consistent with the mechanism of evaporative cooling. Detailed large-eddy simulations by Heus and Jonker (2008) of the SCMS case as well as BOMEX, reproduced the existence of the descending shell in the simulations and, based on a budget analysis of vertical momentum in the model, negative buoyancy resulting from evaporative cooling was identified as the main driver for the downward motion. Since the shell surrounds clouds along their entire perimeter (which can be substantial due to its geometrical properties (Lovejoy, 1982; Siebesma and Jonker, 2000)), Jonker et al. (2008) used LES to precisely quantify the total downward mass flux through the shell. To this end, they analysed the data conditioned on the distance to the (nearest) cloud edge. Rather surprisingly, the total downward mass flux in the cloud shells was found to compensate virtually all the total upward mass flux of the cloud field, not so much due to the negative velocity in the shell (which is rather modest compared to the upward velocity in the cloud core), but rather due to the large area that is associated with the cloud-edge region. This view was confirmed in an observational study by Heus et al. (2009b), who analysed RICO (Rain in Cumulus over the Ocean) data adopting the strategy of conditionally averaging quantities in respect of the distance to the cloud edge. One implication of the large downward mass flux in the vicinity of clouds is that the total downward mass flux in the environment distant from the clouds must be quite small. As shown by Verzijlbergh et al. (2009), this in turn has a tremendous effect on the efficacy of vertical transport of species in the regions away from clouds, because the upward and downward transport in the cumulus layer takes place only where clouds are located (which is usually only a small fraction of the total space). Apart from the dynamical aspects, the microphysical structure of the cloud-edge region is also of fundamental interest (for a recent observational study see Wang et al. (2009)).

Entrainment and detrainment formulations for mass-flux parameterization

4

295

Entrainment and detrainment in mass-flux parameterizations

For almost all NWP and climate models, convection is a sub-grid process which has to be parameterized. One of the key questions is how the parameterization should account for the influence of environmental conditions (e.g., relative humidity). As will become clear in this section, a wide variety of approaches exist. In convection parameterizations, a distinction is usually made between shallow and deep convection. Most developments described here, however, are relevant for shallow as well as deep convection. Section 4.4 deals with differences between shallow and deep convection. 4.1

Kain–Fritsch-type buoyancy-sorting schemes and updates

Methods for the parameterization of ǫ and δ show a wide range of complexity. At one end of the scale are the simple bulk mass-flux schemes with constant ǫ and δ values which are loosely based on Eq. 2.1. However, it has been shown that such simple fixed values for the mixing coefficients are too limited since their values appear to be dependent on cloud layer depth (de Rooy and Siebesma, 2008) and on the environmental conditions (e.g., Derbyshire et al., 2004; Kain and Fritsch, 1990). To take the environmental conditions into account Kain and Fritsch (1990) and Raymond and Blyth (1986) introduced the buoyancy-sorting concept. This widely applied type of scheme together with some recent updates are described here. Although not specifically designed for shallow convection, the parameterization of Kain and Fritsch (1990) is widely applied as such. In their approach, different mixtures of in-cloud and environmental air are made. Negatively buoyant mixtures are assumed to detrain, whereas positively buoyant mixtures entrain. Due to evaporative cooling, θv of the mixture can drop below that of the environment, leading to detrainment. This process is illustrated in Fig. 10.6, which shows the θv of a mixture of cloudy air with a fraction χ of environmental air. For example, purely cloudy air has χ = 0 and obviously θv (χ = 0) = θv,c . The critical mixing fraction χcrit is defined as the fraction of environmental air needed to make the mixture just neutrally buoyant. In the original Kain–Fritsch (KF) scheme, mixtures with χ < χcrit are assumed to entrain while mixtures with χ > χcrit are assumed to detrain. To derive the fractional entrainment and detrainment coefficients within

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θv

θv,c

positively buoyant mixtures

χcrit

0

negatively buoyant mixtures

fraction environmental air (χ)

θv,e

1

Fig. 10.6 The virtual potential temperature of a mixture of cloudy air and environmental air is shown as a function of the fraction χ of environmental air. The virtual potential temperatures of the cloudy and environmental air are θvc and θve respectively. χcrit is the fraction of environmental air necessary to make the mixture just neutrally buoyant.

the KF concept, the amount of mass used for mixing (not discussed), as well as the probability density function (PDF) for the occurrence of the various mixtures must be determined. As there is no a priori knowledge on which PDF should be chosen, it is natural to assume that all mixtures have an equal probability of occurrence, which leads to (Bretherton et al., 2004): ǫKF = ǫ0 χ2crit

(4.1)

δKF = ǫ0 (1 − χcrit )2 ,

(4.2)

where ǫ0 is the fractional mixing rate, i.e., the fractional mass available for mixing, which in the original KF concept is kept constant. LES results from a shallow cumulus case based on observations made during BOMEX (Holland, 1972) have been used in order to evaluate ǫ based on Eq. 4.1 and the results compared with LES-diagnosed values based on Eq. 3.13. Even taking a best estimate of ǫ0 , Fig. 10.7 shows a low correlation. Better results can easily be obtained if ǫ is estimated with a simple decreasing function with height (cf., Fig. 10.8). When the original KF concept was used in practice, several deficiencies were reported, many of them related to the corresponding lateral mixing coefficients. These deficiencies, including some modifications to address

Entrainment and detrainment formulations for mass-flux parameterization

297

 



H.)





 



H/(6





Fig. 10.7 Fractional entrainment rates diagnosed (using the core sampling) from LES, ǫLES , plotted along with the corresponding estimates according to Eq. 4.1 with ǫ0 = 0.02 (optimal value), ǫKF . These results are for the BOMEX case (Siebesma et al., 2003).







H ]]ERW







 



H/(6





Fig. 10.8 As in Fig. 10.7 but with estimates according to ǫ = (z − zb + 500)−1 , where z is the height in m and zb is cloud base height.

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Part II: Mass-flux parameterization

them, are well summarized by Kain (2004), who pointed out that, according to Eq. 4.1, dry conditions (corresponding to small χcrit ) will result in small ǫ values and consequently little dilution of the updraught. Hence, the original KF concept can lead to the counterintuitive result of deeper cloud layers in combination with drier (more hostile) environmental conditions. This behaviour of the KF model was also confirmed by Jonker (2005). In contrast, LES results show considerably shallower cloud depths for drier environmental conditions (Derbyshire et al., 2004). To address the abovementioned deficiency, some of the newer versions of the KF scheme prescribe a lower limit to the entrainment of 50% of the environmental air being involved in the mixing process and/or try to use state-dependent values for ǫ0 . Another deficiency, this time related to Eq. 4.2 and discussed by Bretherton et al. (2004), is the excessive detrainment if all negatively buoyant mixtures are rejected from the updraught. Bretherton et al. (2004) dealt with this problem by introducing a minimum length scale: negatively buoyant parcels can continue to move upwards and do not detrain until that length scale is reached. 4.2

Stochastic mixing model

An important variation to the entraining–detraining plume model is the stochastic-mixing model formulated by Raymond and Blyth (1986) and implemented by Emanuel (1991) into a parameterization. Here, note that despite its name, the model does not contain any stochastic component in its formulation, unlike those stochastic schemes discussed later (Sec. 4.6). This model is based on similar ideas to those found in KF and others but more explicitly considers the movements of sub-plumes or secondary buoyancy updraughts within individual plumes. Although the Emanuel (1991) scheme is formulated in terms of a bulk model (cf., Ch. 9, Sec. 7), a generalization to a spectral model is clearly feasible. Under this stochastic-mixing formulation, multiple sub-plumes are launched from the bottom level of convection. Each sub-plume subsequently multiplies, so that it is necessary to consider two major categories of sub-plumes: those originally launched from the convection base, and those generated as a consequence of the further multiplications of the original sub-plumes. First considered is a set of plumes with mass fluxes Mj,k (j = 1, . . . , N ) at a vertical level k being initiated from the convection base. The number

Entrainment and detrainment formulations for mass-flux parameterization

299

of sub-plumes N is set equal to the number of vertical levels for the convectively unstable layer, with the highest level corresponding to the level of neutral buoyancy (kLNB ). These sub-plumes are partitioned such that each has as an exit level j = k. At each exit level, the sub-plume mass is further divided into N equal sub-masses. Each sub-mass is mixed with the environment with a different mixing ratio. As a result, N new sets of sub-plumes containing particular mixtures with the environment are generated at each vertical level. These ′ with l = 1, . . . , N . secondary sub-plumes are designated as Mj,k,l These secondary sub-plumes also move vertically, seeking their own neutrality level l where they exit and mix into the environment. In principle, the secondary sub-plumes could be allowed to multiply further and continue the process to further generations of sub-plume. Thus, the stochastic mixing process could continue indefinitely, but in the actual implementation, the sequence is terminated with the secondary sub-plumes. Note that in defining the mass flux for the secondary sub-plumes, when the sub-plume moves downward (i.e., l < k), a downward positive mass flux is assumed. As a result, the net mass flux at a level k is given by: Mk =

k LNB

k′ =k

Mj,k′ +

k−1 LNB  k

k′ =1 l=k

′ Mj,k ′ ,l −

k LNB k−1 

′ Mj,k ′ ,l

k′ =k l=1

(cf., Eq. 25, Emanuel, 1991), where the second term is the sum of the secondary sub-plumes ascending from lower levels, whereas the third term is the sum of the secondary sub-plumes descending from upper levels. In defining the mixing fraction for the secondary plumes, the liquid potential temperature for each secondary sub-plume is considered. However, the treatment of each plume does not differ otherwise from the standard mass flux approach. In the end, after rather elaborate computations based on the stochastic mixing, Emanuel (1991) only uses the bulk updraught mass flux Mk for each vertical level k in the final parameterization implementation (cf., Ch. 9, Sec. 7). Note however that the downdraught is evaluated differently (cf., Ch. 13.1). 4.3

Parameterizing the shape of the shallow mass-flux profile

From the discussion in Sec. 4.1 it can be concluded that, although the KF concept contains interesting and important ideas, there are some fundamental problems. In practice, these problems are solved by rather drastic

300

Part II: Mass-flux parameterization

modifications and tuning parameters which more or less undermine the physical attractiveness of the concept. In this subsection, two recent, alternative shallow-convection parameterizations (de Rooy and Siebesma, 2008; Neggers et al., 2009) are described that make use of the parameter χcrit introduced by KF. More importantly, these parameterizations distinguish themselves from other mass-flux schemes because the entrainment and the shape of the mass-flux profile are treated separately. In such a framework, ǫ can be dedicated to an adequate description of the change of the cloud properties with height (via Eq. 3.13) and therewith the cloudy updraught termination height, without being used for the mass-flux profile (via Eq. 3.5). In this way, several problems with conventional convection schemes can be circumvented. As will be discussed shortly, the argument for a separate treatment of ǫ and the mass-flux profile is based on the much larger variation of δ in comparison with ǫ and therewith its much larger impact on variations in the mass-flux profile. This is incompatible with the KF concept because if we use ǫKF and δKF (Eqs. 4.1 and 4.2), these coefficients vary in a similar but opposite way with χ2crit , and therefore have a similar impact on variations in the mass-flux profile related to the variations in χ2crit . However, as first identified by de Rooy and Siebesma (2008), the variations from case to case and hour to hour observed in mass-flux profiles can be almost exclusively related to the fractional detrainment. Numerous LES studies support this by revealing order-of-magnitude larger variations from case to case and hour to hour in δ than in ǫ (see Fig. 10.11 and also B¨oing et al., 2012; Derbyshire et al., 2011; Jonker et al., 2006). Nevertheless, its implication for parameterizing convection is almost never used or even discussed. Apart from this empirical evidence, de Rooy and Siebesma (2010) also provide a sound theoretical basis for the observed large variation in δ and the strong coupling to variations in the mass-flux profile. Based on a general total water-specific humidity budget equation and within the usually applied bulk mass-flux framework, they derived a general picture for a shallow convection cloud ensemble as shown in Fig. 10.9. As in Asai and Kasahara (1967), a distinction is made between small-scale diffusive turbulent lateral mixing, expressed by ǫturb and δturb and larger-scale advective transport across the lateral boundaries, described by ǫdyn and δdyn (see also Sec. 2). Figure 10.9 is in line with Arakawa and Schubert (1974) but now also includes a turbulent detrainment term which counteracts ǫturb , leading to a constant mass flux of individual clouds until the detrainment at their top. As a result of different cloud sizes in the ensemble, the cloud-top de-

Entrainment and detrainment formulations for mass-flux parameterization

301

trainment of the various clouds shows up as a dynamical detrainment term throughout the cloud layer. Figure 10.9 also reveals that it is mainly the dynamical detrainment that regulates the shape of the cloud-layer massflux profile. Indeed, de Rooy and Siebesma (2010) show that the vertical structure of the mass flux is largely determined by the detrainment while the detrainment in turn is determined by the vertical structure of the cloud fraction, all in agreement with the conceptual picture sketched in Fig. 10.9. Accepting that the shape of the mass-flux profile is determined by the detrainment, and assuming a monotonic decrease of the mass flux from the cloud-base height zb to cloud-top height ztop gives an zeroth order estimate of the detrainment of: 1 ∂M 1 δ∼ ∼ . (4.3) M ∂z ztop − zb This equation also illustrates the cloud-layer depth dependence of δ first mentioned by de Rooy and Siebesma (2008) and clearly recognizable in several studies (see Fig. 10.11 and also Gentine et al., 2013; Jonker et al., 2006). The empirical and theoretical arguments above support the approach of de Rooy and Siebesma (2008) in describing variations in the nondimensionalized mass-flux profile with a detrainment coefficient. Their flexible δ ensures a certain mass-flux profile. So in principle, the mass-flux profile itself is parameterized in their approach and thus the concept of δ becomes obsolete. Note that the cloud-layer height dependence is taken care of by evaluating and prescribing the mass flux with a non-dimensionalized height and mass flux. In de Rooy and Siebesma (2008), it is also shown that variations in δ, and thus the mass-flux profile, depend not only on cloud depth (cf., Eq. 4.3) but also on environmental conditions such as the vertical stability and the relative humidity. These dependencies are well described by χcrit which can be written approximately (cf., de Rooy and Siebesma, 2008) as: χcrit =

δθv cp π , Lv qs (β − α)(1 − RH) − αql,u

(4.4)

where cp denotes the specific heat, Lv the latent heat, π the Exner function, δθv the difference in virtual potential temperature between the cloudy updraught and the environment, RH the relative humidity of the environment, ql,u the liquid water in the cloudy updraught, and α and β are constants. Note that χcrit increases both with the buoyancy of the cloudy updraught and with the environmental relative humidity. LES results show

302

Part II: Mass-flux parameterization

δdyn

ztop

ε turb

δdyn

δturb

δdyn

δdyn

zbot

ε dyn

ε dyn

ε dyn

ε dyn

Fig. 10.9 Schematic diagram of a shallow-convection cloud ensemble with dynamical entrainment ǫdyn at the cloud base (zb ) and dynamical detrainment δdyn at the top of individual clouds. From cloud base to the top of the individual clouds, turbulent lateral mixing takes place represented by ǫturb and δturb . For individual clouds, the mass flux is constant with height. The deepest cloud reaches height ztop , the top of the cloud layer. This picture is valid for divergent conditions (i.e., ∂M/∂z < 0) which is usually the case for shallow convection.

that detrainment rates decrease with increasing values of χcrit . This relation can be easily understood from physical considerations. Small χcrit values correspond to marginally buoyant, often relatively small clouds, rising in a dry, hostile environment (cf., Sec. 4.1). It is likely, and confirmed by LES, that under such conditions the mass flux will decrease rapidly (large detrainment). The opposite is true for large, buoyant clouds rising in a friendly, humid environment, corresponding to large χcrit values (small detrainment). Via this χcrit dependence, δ and the mass-flux profile vary not only with the relative humidity of the environment but also with the properties (buoyancy) of the updraught itself. For further details see de Rooy and Siebesma (2008). Instead of the critical mixing fraction, Neggers et al. (2009) used a moist zero-buoyancy deficit qtx − qt , where qtx is the total water-specific humidity at the mixing fraction χcrit . Note that the moist zero-buoyancy deficit is

Entrainment and detrainment formulations for mass-flux parameterization

303

proportional to χcrit . By using this deficit to estimate the cloud fraction and thus also the cloudy updraught velocity (via the PDF of wc ), Neggers et al. (2009) established a link between the mass flux and statistical cloud schemes (cf., Vol. 2, Ch. 25, Sec. 3.4). A systematic LES study exploring the influence of the entrainment and detrainment rates in deep convection by B¨ oing et al. (2012) confirmed the importance of the critical mixing fraction for mass-flux parameterization. In this study, ninety large-eddy simulations were used to explore the sensitivity of deep convection to the relative humidity and stability of the free troposphere. Figures 10.10a and 10.10b show respectively how the entrainment and detrainment rate at 2,500 m above ground level vary with these parameters (see B¨ oing et al., 2012, for details). Although both entrainment and detrainment decrease in an unstable and humid environment, the absolute variations in entrainment are much smaller. This confirms the notion that variations in the mass-flux profile mainly depend on detrainment. The variations in entrainment only become visible when plotted on a more sensitive colour scale (Fig. 10.10c). The changes in detrainment coincide with changes in other measures of convective instability such as cloud-top height and precipitation rate (Fig. 10.10d and Fig. 10.10e), and with the critical mixing fraction (Fig. 10.10f). B¨ oing et al. (2012) showed that a dependence of detrainment on χcrit is robust at other levels as well. This implies that in deep convection, the same mechanisms determine the mass-flux profile as in shallow convection. Even though the mass flux is directly determined largely by detrainment rather than entrainment, the dependence of detrainment on χcrit still allows entrainment an indirect role in deep convection. The critical mixing fraction depends largely on the buoyancy of the plume, and therefore on the entrainment rate. In line with the arguments of this subsection, an adaptive detrainment parameterization was proposed by Derbyshire et al. (2011). LES results in their study (Figs. 5 and 6) confirm that δ varies much more strongly with environmental RH than ǫ. Finally, note that for the mass-flux parameterizations discussed in this section as well as for certain entrainment formulations, the cloud-top and/or cloud-base height are needed. These heights can be determined by releasing a parcel with a certain temperature and humidity excess at the surface. Subsequently, the cloud top is diagnosed as the height where the vertical velocity of the parcel becomes zero. For more details on an appropriate vertical velocity equation, see Ch. 12. The fractional entrainment is necessary to calculate the dilution and is also applied in the vertical velocity

304

Part II: Mass-flux parameterization

b)

Init. stability [K m-1 ]

×10-3 4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.870

-1 ε[m ]

75

80

85

Init. H [%]

90

-3 ×10 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.870

75

80

85

Init. H [%]

×10-4

90

8.0 7.2 6.4 5.6 4.8 4.0 3.2 2.4 1.6 0.8 0.0

80

85

Init. H [%]

90

e)

×10-3 Ensemble cloud top [m]14000 13000 4.5 12000 4.4 11000 4.3 10000 9000 4.2 8000 4.1 7000 4.0 6000 3.9 5000 3.870 75 80 85 90 4000

Init. H [%]

f)

Precipitation [W m-2 ]

×10-3 4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.870

75

80

85

Init. H [%]

90

400 360 320 280 240 200 160 120 80 40 0

×10-3 4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.870

Init. stability [K m-1 ]

Init. stability [K m-1 ]

75

-3 ×10 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

d)

-1 ε[m ]

×10-3

Init. stability [K m-1 ]

[m-1 ]

δ

Init. stability [K m-1 ]

c)

×10-3 4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.870

Init. stability [K m-1 ]

a)

χcr

75

80

85

Init. H [%]

90

0.30 0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12 0.10

Fig. 10.10 Dependence of a) ǫLES , b) δLES , c) ǫLES (using a smaller colour scale), d) cloud top, e) precipitation rate, and f) χcrit on the initial relative humidity and stability. For details of the simulations, see B¨ oing et al. (2012). a)–c) and f) are plotted for data c American Geophysical Union 2012, from Fig. 2 of at 2,500 m above ground level.  B¨ oing et al. (2012).

Entrainment and detrainment formulations for mass-flux parameterization

305

equation. If the entrainment itself is a function of cloud-base height, for example, some kind of iteration is necessary. By first releasing a test parcel with a different kind of entrainment formulation (e.g., ǫ = βz −1 ) an estimate of cloud-base and cloud-top height can be made. These heights would then be applied in the cloud-base height-dependent entrainment formulation in the next parcel release. Usually this process results in fast convergence. 4.4

Differences between deep and shallow convection

Besides varying with the environmental conditions, entrainment and detrainment rates may vary considerably between shallow and deep convection. Figure 10.11 shows as an example profiles of ǫ and δ derived from LES. Here, ǫ and δ are computed from Eqs. 3.5 and 3.13 with ϕ = s, where s is the frozen moist static energy (see Bretherton et al., 2005): s = cp T + gz + Lv rv − Lf ri ,

(4.5)

where rv is the water vapour mixing ratio, Lf is the latent heat of freezing, and ri is the mixing ratio of ice. The frozen moist static energy here may be considered a middle point between MSEd and LIMSE introduced later in Vol. 2, Ch. 22, where various definitions of moist static energies are discussed. Panels (a) and (b) of Fig. 10.11 are based on LES simulations conducted for the Kwajalein Experiment (KWAJEX) over the west Pacific warm pool for the period 23 July to 4 September 1999. The entrainment and detrainment rates are stratified and averaged as a function of cloud depth. Panels (c) and (d) contain the results for mid-latitude continental convection, from large-eddy simulations driven by measurements made at the Atmospheric Radiation Measurement (ARM) Southern Great Plain station between 18 June and 3 July 1997. Due to the large variability in synoptic conditions and consequently in entrainment and detrainment rates, only specific times are shown in (c) and (d). The time spans from 0900 to 1700 local time (LT) on 27 June 1997 and encompasses a typical diurnal cycle of surface-forced convection, from shallow (0900 LT) to deep convection with maximum precipitation at 1700 LT. Figure 10.11 clearly indicates that the transition from shallow to deep convection is accompanied by a reduction in entrainment and detrainment rates. This is true for both the tropical oceanic (Fig. 10.11a,b) and midlatitude continental (Fig. 10.11c,d) convection. Similar reductions in ǫ have

306

Part II: Mass-flux parameterization

Fig. 10.11 Profiles of fractional (a,c) entrainment and (b,d) detrainment rates based on large-eddy simulations of the (a,b) KWAJEX and (c,d) ARM cases. Plots (a) and (b) show entrainment and detrainment rates averaged for clouds of various depths (less than 2 km, between 2–3 km, 3–4 km, 4–5 km, and 5–6 km). Plots (c) and (d) show entrainment and detrainment rates at two hourly time intervals with to corresponding to 0900 LT on 27 June 1997.

been noticed by Genio and Wu (2010) for TWP-ICE (Tropical Warm Pool International Cloud Experiment), Kuang and Bretherton (2006) for an idealized oceanic transition case, and Khairoutdinov and Randall (2006) for the LBA (Large-Scale Biosphere-Atmosphere) experiment over Amazonia. The variations in δ appear also much larger than the ǫ variations in Fig. 10.11. de Rooy and Siebesma (2008) identified the importance of the detrainment rate and its cloud-depth dependence (clearly visible in Fig. 10.11b and d) in controlling the mass-flux profile (cf., Sec. 4.3). Other than this, other effects are thought to be responsible for reducing ǫ and δ. Firstly, detrainment from previous clouds moistens the environment; the entrained air becomes moister, the evaporative cooling due to the mixing of cloudy and environmental air is reduced, and consequently the detrainment decreases (see de Rooy and Siebesma, 2008). The effect on ǫ, however, seems less clear. As described in Sec. 4.5, Bechtold et al. (2008) apply an entrainment rate which decreases with increasing RH of

Entrainment and detrainment formulations for mass-flux parameterization

307

the environment. This explicit dependency on RH has a highly beneficial effect in the ECMWF model. On the other hand, in the widely applied scheme of Kain and Fritsch (1990), ǫ increases with increasing RH of the environment (Eq. 4.1). Thus, the influence of environmental RH on ǫ is yet far from established. Secondly, several studies (e.g., Khairoutdinov et al., 2009; Khairoutdinov and Randall, 2006; Kuang and Bretherton, 2006) have suggested that the formation of cold pools plays a key role in the transition from shallow to deep convection. In particular, the formation of cold pools induces larger clouds which entrain less and thus are more suitable to reach greater depth. Recent attempts to unify shallow and deep convective parameterizations have therefore added explicit relations to tighten their entrainment and detrainment rates to precipitation or its evaporation, as measures of cold pool activity (cf., Hohenegger and Bretherton, 2011; Mapes and Neale, 2011). See Sec. 9 for further discussions. 4.5

Relative-humidity-dependent entrainment

A possible approach to account for the influence of environmental conditions on convection is to use an entrainment explicitly depending on relative humidity. Here, an example is presented that proved successful in the context of numerical weather and seasonal prediction (cf., Vol. 2, Ch. 15). However, some alternative entrainment formulations are also considered. The sensitivity of moist convection in respect of environmental moisture gained considerable attention since the work by Derbyshire et al. (2004) who set up a series of single-column and cloud-resolving simulations (CRMs) where the atmosphere was relaxed to different values of the ambient RH. The convection schemes employed in their single-column models were not able to match the sensitivity of the mass-flux profiles in respect of environmental RH as represented by the CRMs. Motivated by their study and observations that mid-tropospheric humidity modulates tropical convection (Redelsperger et al., 2002), Bechtold et al. (2008) revised the European Centre for Medium range Weather Forecast (ECMWF) convection scheme including an entrainment formulation that explicitly accounts for RH dependency. It was shown in Hirons et al. (2013a,b) and Jung et al. (2010) that the revised convection scheme also greatly improves mid-latitude and tropical variability in the ECMWF model on various scales, including the representation of the Madden–Julian Oscillation (MJO) (Madden and Julian, 1971). Several recent CRM studies

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Part II: Mass-flux parameterization

and model developments (e.g., Chikira and Sugiyama, 2010; Genio and Wu, 2010; Kim and Kang, 2012) focused on state-dependent entrainment rates, their sensitivity to environmental profiles, and their impact on the large-scale circulation and variability. A tropical wave analysis, and insight into how convective mixing impacts the large-scale tropical variability, is presented in Vol. 2, Ch. 15. On the basis of the Derbyshire et al. (2004) model set-up, some necessary ingredients for a bulk entrainment model in order to represent the sensitivity of the mass-flux profiles in respect of environmental RH are now discussed. The model set-up simply consists of running the ECMWF singlecolumn model (SCM) for a 24 hour period while relaxing with a timescale of 2 h the free atmosphere (above 2 km) humidity fields to specified values of 25, 50, 70, and 90% RH. The background boundary-layer moisture profile is identical for all runs. Different updraught-entrainment formulations are evaluated. The base version is a formulation that has evolved from Bechtold et al. (2008) and constitutes the ECMWF operational formulation since 2010: ǫ = ǫ0 (1.3 − RH(z))fscale 3  ∗ q (z) ǫ0 = 1.8 × 10−3 m−1 ; fscale = , (4.6) q ∗ (zb ) where ǫ depends at each height z on the RH and a scaling function fscale , which is a cubic function of the ratio of the saturation-specific humidities q ∗ at level z and at convective cloud base zb . The scaling function aims to mimic the effect of an ensemble of clouds. Furthermore, entrainment is only applied when the buoyancy of the updraught is positive and the distinction between deep and shallow convection is made by multiplying the entrainment rates in Eq. 4.6 by a factor of two if the cloud thickness of a test parcel is smaller than 200 hPa. Figure 10.12 shows the entrainment profiles with Eq. 4.6 for the different RH regimes in Derbyshire et al. (2004). LES results presented by Derbyshire et al. (2011) (their Fig. 5) for the same RH regimes also reveal increasing ǫ values with decreasing RH in the lower part of the cloud layer. Moreover, the order of magnitude of the entrainment rates is confirmed by Genio and Wu (2010), using CRM data. Fractional detrainment rates are parameterized differently, namely as the sum of a vertically constant turbulent detrainment and a term that is proportional to the decrease in updraught kinetic energy when the buoyancy is negative (cf., Bechtold et al., 2008). The turbulent detrainment

Entrainment and detrainment formulations for mass-flux parameterization

309

5 25 50 75 90

4

z (km)

3

2

1

0 0

0.5

1

ε (kmí1)

1.5

2 í3

x 10

Fig. 10.12 Entrainment profiles using Eq. 4.6 for background RH of 25% (dash-dotted line), 50% (dashed line), 70% (dotted line), and 90% (solid line).

rate for deep convection is set to δturb = 0.75 × 10−4 m−1 , whereas for shallow convection it is equal to the entrainment rates (in accordance with de Rooy and Siebesma (2010), Fig. 10.9). This simple detrainment formulation together with the entrainment rate that strongly decreases with height assures a convective mass-flux profile that from a certain level on can decrease with height even for positively buoyant convection throughout. A comparison between the operational ECMWF SCM version using Eq. 4.6 and CRM data is shown in Fig. 10.13a and g. The CRM data is taken from the Met Office model which was one of the two participating CRMs in Derbyshire et al. (2004) (the CRM data has been scaled by a factor of 0.6 as the mass-flux scaling between the participating SCM and CRMs varied by a factor 0.5–1.5) . Comparing also to the other SCM and CRM results in Derbyshire et al. (2004), the current results suggest that the model is broadly able to represent the increase of the cloud height and amplitude of the mass flux with increasing RH, and in particular to represent reasonably well the shallow convection regime for the 25% RH profile. The cloud-top heights for the deep regimes are lower in the ECMWF model than in the CRM, but global model evaluation against satellite observations (not shown) suggests realistic cloud-top heights in the ECMWF model (Ahlgrimm et al., 2009; Ahlgrimm and Forbes, 2012).

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Part II: Mass-flux parameterization

(b) base 0.7

5 0

0.05

10 5 0

0.1

0

5 0

0.05

5 0

0.1

0

0

0.05

0 0.05 0.1 í2 í1 Mass flux (kg m s )

0.05

0.1

10 5 0

0.1

(h) cst z (km)

z (km)

5

0

(f) noRH 0.5

10

(g) CRM 10

5 0

0.1

z (km)

10

0

10

(e) noRH 0.7 z (km)

z (km)

(d) noRH 0.9

0.05

10 5 0

0

0.05

0.1

(i) Greg z (km)

0

(c) base 0.5 z (km)

10

z (km)

z (km)

(a) base 1

0 0.05 0.1 í2 í1 Mass flux (kg m s )

10 5 0

0 0.05 0.1 í2 í1 Mass flux (kg m s )

Fig. 10.13 Time and/or domain-averaged convective mass-flux profiles for background RH of 25% (dash-dotted line), 50% (dashed grey line), 70% (dotted line), and 90% (solid line) as obtained from ECMWF SCM simulations with different entrainment formulations and from CRM simulations: (a)–(c) entrainment according to Eq. 4.6 with scale factors of 1, 0.7, and 0.5; (d)–(f) entrainment according to Eq. 4.6 without relative humidity dependence and with scale factors of 0.9, 0.7, and 0.5; (g) Met Office CRM data scaled by a factor of 0.6; (h) constant entrainment; and, (i) the entrainment formulation of Gregory (2001).

Next, sensitivity studies to the entrainment profile are performed including a series of variations to Eq. 4.6. In the first series, the entrainment in Eq. 4.6 is multiplied by a factor of 0.7 or 0.5. The corresponding results in Fig. 10.13b and c respectively show that as a consequence the cloud-top heights rise and the shallow convection regime transitions to a congestus regime compared to the base version (Fig. 10.13a), but still a distinct sensitivity to the RH remains. In the second series of sensitivity experiments, Eq. 4.6 is simplified by dropping the RH-dependent term so that ǫnoRH = βǫ0 fscale ,

(4.7)

Entrainment and detrainment formulations for mass-flux parameterization

311

and by applying different scaling factors β = 0.9, 0.7, and 0.5. The corresponding results in Fig. 10.13d-f appear rather similar to the base version (Fig. 10.13a-c) but the mass-flux amplitudes are smaller and there is less sensitivity of the cloud-top heights between the different scaling factors. Simplifying even further, the vertical scaling function is also dropped to obtain a vertically constant entrainment rate ǫcst = βǫ0 . Choosing β = 0.05 in Fig. 10.13h, one notices immediately the lack of RH sensitivity. This is due to the low entrainment rate, but this low value is necessary in order to produce reasonable cloud-top heights for the deep convection regimes. Finally, the entrainment formulation suggested by Gregory (2001) and which has been advocated in Genio and Wu (2010) and Chikira and Sugiyama (2010) is applied: ΔTv b , (4.8) ǫgreg = β 2 ; b = g wc Tv where b is the buoyancy, wc the updraught vertical velocity, g is gravity, and Tv and ΔTv are the virtual temperature and the cloudy updraught virtual temperature excess, respectively. A method to determine the updraught vertical velocity and virtual temperature excess is presented in Vol. 2, Ch. 15. The corresponding results in Fig. 10.13i were obtained using β values close to those in the literature: i.e., β = 0.03 for deep convection and β = 0.06 for shallow convection. Note that, based on LES diagnoses, de Rooy and Siebesma (2010) found a much larger optimal value, β = 0.12 for Eq. 4.8, which will increase the sensitivity to RH but also leads to lower cloud tops. Fig. 10.13i suggests that Eq. 4.8 does not provide sufficient sensitivity in terms of cloud-top height, and also in terms of the shape of the mass-flux profiles. To remedy the characteristic top-heavy massflux profiles inherent to this formulation a detrainment formulation might be required which is different from the present ECMWF one. Finally, an entrainment formulation of the type ǫz = βz −1 was also tested but the results (not shown) for the present case were similar to the Gregory (2001) formulation. The conclusions drawn from this SCM study are that a bulk massflux formulation can broadly reproduce the sensitivity of convection to the mid-tropospheric moisture field. However, this requires strong entrainment rates of O(10−3 ) in the lower troposphere. The results also suggest that the entrainment rates should strongly decrease with height, in order to control the mass-flux profile, and also to allow for strong values near the cloud base, and that the inclusion of a further explicit dependence of the entrainment rate on RH leads to further more realistic sensitivities to the

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Part II: Mass-flux parameterization

large-scale moisture fields. As suggested in Sec. 4.3, an alternative approach is to parameterize the entrainment and the mass-flux profile independently, increasing the flexibility to simultaneously obtain correct cloud-top heights, sensitivity to the environment, and mass-flux profiles. 4.6

Stochastic entrainment

A key issue in modern research in cumulus convection pertains to the variability among convecting elements. It is not yet clear if most of this variability can be attributed to variability in the thermodynamic properties close to cloud base, or if variability in the entrainment process above cloud base also plays a fundamental role. Recent LES studies (Romps and Kuang, 2010a) and SCM simulations (Su˘selj et al., 2013) have highlighted the important role of stochastic entrainment in producing realistic thermodynamic structures for a variety of shallow convection case studies. The parameterization of Neggers et al. (2002) partially tackles the issue of producing a realistic amount of variability between updraughts. In this formulation, the entrainment rate is inversely proportional to the product of the updraught vertical velocity and a (constant) timescale. In this way, the positive feedback between the updraught vertical velocity and the entrainment rate effectively increases the variability between updraughts. One of the key problems with this type of parameterization is that the results are quite sensitive to the value of the timescale. An alternative perspective of lateral entrainment is provided by studies such as Raymond and Blyth (1986) and Romps and Kuang (2010a,b), who suggest that entrainment should be represented as a stochastic process. In Su˘selj et al. (2013) an entrainment parameterization similar to the one suggested by the LES studies of Romps and Kuang (2010a) is implemented in a single column model. It is essentially assumed that most entrainment occurs as discrete events. In practice, the simplest form for an entrainment event is assumed: when the updraught grows a distance dz, the probability of entrainment is dz/L0 , where L0 represents the mean distance the updraught needs to grow to entrain once. The entrainment coefficient can be determined stochastically from a Bernoulli distribution and along the finite length z the probability of an entrainment event follows a Poisson distribution. With this entrainment parameterization, two stochastic processes drive the moist updraught properties: the stochastic initialization of the moist updraughts at cloud base and the stochastic entrainment rate. The results

Entrainment and detrainment formulations for mass-flux parameterization

313

of Su˘selj et al. (2013) suggest that using only these two important and independent stochastic mechanisms, realistic thermodynamic structures can be simulated for a variety of shallow cumulus events.

5

Conclusions and discussions

Large-eddy simulations have become an extremely useful tool for diagnosing entrainment and detrainment. This has resulted in new insights as well as improved parameterizations for entrainment and detrainment in operational NWP and climate models. A number of these new insights have been discussed here. For example, on the origin of entrained air, different theories have coexisted for several decades. However, detailed particle tracking studies in large-eddy simulations have shown unambiguously (cf., Sec. 3.7) that cloud-top entrainment plays no significant role in the mixing process compared to lateral mixing (B¨ oing et al., 2014; Heus et al., 2008). New insight is also provided by diagnosing local entrainment and detrainment rates from the basic definitions of Eqs. 3.4a and 3.4b rather than using the approximate bulk method of Eqs. 3.11a and 3.11b (Dawe and Austin, 2011b; Romps, 2010, cf., Sec. 3.6). The local exchange rates are larger by a factor of two than the ones diagnosed using the bulk approach. However, parameterizations within a bulk framework should apply entrainment and detrainment rates as diagnosed according to the bulk approach as they will provide the optimal turbulent fluxes (the prime objective of any convection parameterization). When ǫ and δ are diagnosed in an LES or are applied in a bulk parameterization framework, the boundary between cloud and environment is more or less artificial and will, for example, change with the chosen sampling method. Moreover, as shown by Heus and Jonker (2008) and Jonker et al. (2008) in reality a buffer zone exists between the plume core and the environment indicating that there is no well-defined boundary between cloud and environment. Nevertheless, well-known sampling methods and the parameterization schemes based upon them, have been shown to be capable of accurately describing the turbulent fluxes. From the many LES studies on entrainment and detrainment over the last decade, mainly for shallow cumulus convection, a physically consistent picture of the cloud dynamics and mixing is emerging (de Rooy and Siebesma, 2010). Cumulus convection is constituted by an ensemble of

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Part II: Mass-flux parameterization

cumulus clouds, many small and shallow ones and fewer larger and deeper ones. They all share the same cloud base height but have different cloud-top heights. The main inflow occurs at the cloud base and can be interpreted as an organized entrainment. All clouds are diluted through equal turbulent entrainment and detrainment, but the smaller clouds are exposed to larger rates than the larger clouds, simply due to their dimensional surface to volume ratio. Organized detrainment takes place at the cloud top and therefore it is the cloud-size distribution, or more precisely the closely related cloud-top height distribution, that determines the shape of the massflux profile of the whole cloud ensemble. This can be affected by external factors such as the atmospheric stability or free tropospheric relative humidity. Higher relative humidity and/or decreasing atmospheric stability supports relatively deeper clouds, shifting the organized detrainment to higher altitudes, and leading to a mass-flux profile with a weaker gradient with height. What are the consequences for shallow cumulus parameterizations? For bulk parameterizations one should use entrainment parameterizations that decrease with height. This reflects the fact that near the cloud base the bulk entrainment is dominated by the many small clouds while higher in the cloud layer the entrainment is smaller since it is dominated by the larger clouds. The organized detrainment can vary significantly from case to case and is therefore the key process that determines the shape of the mass-flux profile (de Rooy and Siebesma, 2008, 2010). Fortunately, the detrainment appears to be well correlated to χcrit (cf., Sec. 4.3), which can be determined by using a simple entraining-plume model. Alternatively, one can use χcrit directly to parameterize the mass flux (de Rooy and Siebesma, 2008) or the cloud-core fraction (Neggers et al., 2009). In the latter case, the mass-flux profile can be constructed by combining it with the vertical velocity equation that is routinely used in the entraining-plume model. Instead of a bulk model one can also employ a multi-plume model such as that pioneered by Arakawa and Schubert (1974). In that case, closure assumptions on the shape of the cloud-size distribution are required. Such a parameterization is computationally more expensive as it requires several plume updraughts. On the other hand, one might argue that it is potentially conceptually simpler as some of the assumptions that need to enter into a bulk parameterization are not necessary any more as they are sorted out by the various plumes explicitly. (See Ch. 9 for further discussions.) As shown by B¨oing et al. (2012) the general conclusions presented here for shallow-convection parameterization also hold for deep convection. For

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315

example, the detrainment as well as the mass-flux profile appear to be well correlated with χcrit . Nevertheless, deep convection results in additional complications that need to be parameterized. Most importantly, precipitation induces downdraughts (cf., Ch. 13) that promote the formation of cold pools. The mesoscale organization associated with the cold pools supports deep convection and accelerates the transition from shallow to deep convection (B¨ oing et al., 2012). Attempts have been made to include this process by adding explicit relations to tighten ǫ and δ to precipitation or its evaporation (cf., Sec. 4.4, Hohenegger and Bretherton, 2011; Mapes and Neale, 2011). Based on these increased physical insights on entrainment and detrainment one might expect to observe some convergence of cumulus parameterizations in operational weather and climate models. However, an extensive variety of parameterizations has been developed for ǫ and δ without a sign of any convergence towards certain approaches. Moreover, there is no consensus about the necessary dependencies included in the formulations of the mixing coefficients. A remarkable example is the dependency of the entrainment on relative humidity. In two recent and operationally applied approaches by Bechtold et al. (2008) and Kain and Fritsch (1990) (cf., Sec. 4.1 and 4.5), this dependency is simply opposite. What are the possible causes of the non-converging developments? Most importantly, many parameterizations are still developed without a direct comparison of the formulations against LES-diagnosed ǫ and δ (see e.g., Kain and Fritsch, 1990, Fig. 10.7). Yet, LES experiments offer an excellent opportunity to validate potential expressions within the corresponding model framework. One obvious example that demonstrates how LES results are simply ignored by developers is the large observed variation from case to case and hour to hour in δ, even by orders of magnitude. Apart from the strong empirical evidence, these large variations are recently also explained from theoretical considerations (de Rooy and Siebesma, 2010). Variations in δ can be related to different environmental conditions, like relative humidity and stability. However, the largest variations in δ are often related to variations in the cloud-layer depth (de Rooy and Siebesma, 2008), as can be observed, for example, during daytime convection over land with a deepening cloud layer. To the best of the authors’ knowledge, only two shallow-convection parameterization schemes capture these large variations in δ, namely those by de Rooy and Siebesma (2008) and Neggers et al. (2009) (cf., Sec. 4.3). Another reason for the lack of convergence is that the development and

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Part II: Mass-flux parameterization

the actual implementation of parameterizations is a slow process. This is partly due to the relatively low number of scientists that are working directly on the development and the implementation of convection parameterizations (Jakob, 2010). At the same time it is becoming increasingly more difficult to implement new parameterization schemes that demonstrate an immediate increase of skill, since operational models are highly optimized and contain compensating biased errors. Therefore reducing one bias through implementing an improved parameterization often results in a deteriorating model skill and retuning or removing a compensating bias becomes necessary. This requires thorough knowledge of many of the relevant parameterization processes and obviously slows down the process of model improvement through parameterization development. Because of this increasing complexity, keeping parameterizations as simple as possible is strongly recommended. That is, one should try to capture only the most substantial processes responsible for LES-diagnosed and/or observed variations in lateral mixing. 6

Bibliographical notes

c Royal MeteoroThis chapter is based on de Rooy et al. (2013), which is  logical Society, 2013. 7

Appendix A: Operational implementations

The purpose of this appendix is to provide some technical details of operational implementations of the entrainment–detrainment formulations by expanding some descriptions of the main text, especially those found in Sec. 2.4. By taking a historical perspective, it is shown how the operational implementations have evolved from Morton et al.’s (1956) original idea. Many of the basic historical ideas are often maintained even today in the operational implementations, but under various elaborations, as outlined here. 7.1

Entraining-plume models

Many operational schemes, at least as their basis, adopt the entrainingplume model originally formulated by Morton et al. (1956), and subsequently adopted by Arakawa and Schubert (1974). This component of

Entrainment and detrainment formulations for mass-flux parameterization

317

entrainment may be termed “turbulent” in contrast to an additional component added in some schemes. This term could be confusing, as Telford (1966) proposes that an additional “turbulent” entrainment must be added on top of the entrainment originally formulated by Morton et al. (1956). Nevertheless, readers should judge the validity of this proposal by examining the exchange between Morton (1968) and Telford (1968). However, even in talking about the entraining-plume model itself, there is already a minor complication due to a difference between Morton et al. (1956) and Arakawa and Schubert (1974). In Morton et al.’s (1956) original formulation, the fractional entrainment rate is assumed to be inversely proportional to the plume radius. As a result, the fractional entrainment decreases with height as the plume radius increases with height due to entrainment. However, Arakawa and Schubert (1974) simply assume that the fractional entrainment is a constant for a particular plume type. Morton et al.’s (1956) original entraining-plume model was adopted by Donner (1993) in his spectral formulation. Gregory and Rowntree (1990) crudely translated Morton et al.’s (1956) entrainment formulation of Eq. 2.1 into a pressure-coordinate system by setting ǫ = ǫ0

p , ps

where ǫ0 is a constant and ps is the surface pressure. The majority of operational schemes including Nordeng (1994), Tiedtke (1989), and the ECMWF model, adopt Arakawa and Schubert’s (1974) entraining-plume formulation as the basis for their entrainment. Even in that case, it is still conceptually useful to invoke a relationship between the plume radius and the fractional entrainment rate, as introduced by Morton et al. (1956), in order to define the latter. The plume radius simply becomes a fixed parameter characterizing a given plume. Bechtold et al. (2001) and Kain and Fritsch (1990), for example, defined the value of the fractional entrainment rate in this manner. An unfavourable feature of entraining-plume formulations with a fixed fractional entrainment is the fact that the mass flux increases exponentially with height. In order to tame the rate of increase, Moorthi and Suarez (1992) instead assume that the mass flux increases only linearly with height. A corresponding entrainment formula may then be inferred (Ch. 7, Eq. 6.8). Most operational schemes, however, avoid the difficulty by adding extra detrainment (cf., Sec. 7.4).

318

7.2

Part II: Mass-flux parameterization

Dynamical entrainment by organized flows

An additional contribution to entrainment is often added, supplementing the traditional entrainment formulation just discussed. This is considered to arise from organized flows, as opposed to turbulent mixing at the cloud edges. The idea of such an entrainment process was proposed by Houghton and Cramer (1951), who named it the “dynamical entrainment” (cf., Sec. 2.2) Their idea of dynamical entrainment, attributed by them to Austin (1948), is extremely simple: there must be an inflow to a convective plume (or tower) proportional to a large-scale convergence. In order to demonstrate this idea, they constructed a simple theoretical model, in which they assumed that the horizontal section of a convective element (a plume or a convective tower) does not change with height. In other words, all of the converging airflow simply merges into the convective air, without shrinking the convective horizontal section, by a simple geometrical reconfiguration. It is clear that this is an extreme limiting case to take, which maximizes the effect of dynamical entrainment. Much of their theoretical development is devoted to developing a closed expression for the dynamical entrainment rate in terms of the thermodynamic state of the atmosphere. An additional key assumption in the derivation is that the convective air is purely accelerated by buoyancy without feeling a pressure effect. The same idea is termed by Tiedtke (1989) the “entrainment by organized flow”. Tiedtke (1989) more specifically proposes (cf., Ch. 11, Sec. 14.1) to take this contribution to be equal to the convergence rate weighted by moisture: ρ v · ∇¯ q ), E = − (¯ q¯ where v is the three-dimensional wind vector. Nordeng (1994) modified the above formulation by considering a convective vertical velocity equation in the form: ∂wi (A.1) = bi − ǫi wi2 . wi ∂z He invoked an approximate relationship between the fractional entrainment rate and the fractional vertical divergence: 1 ∂ρ 1 ∂wi + , (A.2) ǫi = wi ∂z ρ ∂z assuming that the fractional area σi for convection is constant with height. If we furthermore assume that the density stratification (1/ρ)(∂ρ/∂z) is

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319

negligible, this relation reduces to: 1 ∂wi . (A.3) wi ∂z Substitution of Eq. A.3 into Eq. A.1 leads to: bi ∂wi = . ∂z 2wi Substitution of this relation back to Eq. A.2 gives a final expression: bi 1 ∂ρ . ǫi = + 2wi2 ρ ∂z ǫi ≃

This formula was used by Nordeng (1994) as a definition of the organizedflow entrainment rate. 7.3

Modulation factor for entrainment rate

Another way to control the problems of a simple entraining-plume hypothesis is to apply a modulation factor fscale to the original entrainment rate formulation. Thus: ǫ = ǫbase fscale ,

(A.4)

where ǫbase is a base fractional entrainment defined from an entrainingplume hypothesis. Both Bechtold et al. (2001) and Kain and Fritsch (1990) introduce their buoyancy-sorting formulations for entrainment and detrainment in this form. The ECMWF model also introduces a dependence on the relative humidity as a part of this modulation factor (cf., Sec. 4.5). 7.4

Turbulent and mixing detrainments

One awkward aspect of the entraining-plume model is the fact that all of the convective air suddenly detrains out at the convection top, which can lead to various singular behaviours. Some of these are demonstrated in Ch. 8. Such a final detrainment of convective air may be called “forced detrainment”. A remedy to avoid this behaviour is to add detrainment at all of the vertical levels along with the entrainment. This detrainment process may be termed “mixing” or “turbulent”, as in the formulations of Gregory and Rowntree (1990) and Tiedtke (1989), respectively. Tiedtke (1989) simply sets the turbulent detrainment equal to the turbulent entrainment δturb = ǫturb . As a result, the mass flux becomes constant with height, although the convective variables are still diluted by entrainment.

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Part II: Mass-flux parameterization

On the other hand, Gregory and Rowntree (1990) assume more generally that the mixing detrainment is of a fixed fraction of the turbulent entrainment at every level. By default, they set δturb /ǫturb = 1/3. The ECMWF model sets the above ratio to about 0.04 with a different modulation factor fscale,d being applied to the turbulent detrainment rate as opposed to the entrainment modulation, including a different dependence on the relative humidity. 7.5

Forced or organized detrainment

Supplementing the turbulent or mixing detrainment described in the last subsection, the original detrainment at the convection top in the entrainingplume model is called “forced detrainment” by Gregory and Rowntree (1990). This process detrains all of the convective air by default at the level of neutral buoyancy. The behaviour of this forced detrainment is still found to be brutal in many operational models; for example, when the mixing detrainment is assumed to be weaker than the turbulent entrainment as in Gregory and Rowntree (1990). Thus, a further effort is required to make the forced detrainment smoother in the vertical. In Gregory and Rowntree (1990), the forced detrainment is activated when the convective buoyancy is less than a threshold bc , which is set to 0.2 K by default. Under this formulation, the convective plume terminates at a level where the mass flux is less than a prescribed minimum value. The ECMWF model includes a similar process but uses a different name, “organized detrainment” (Nordeng, 1994), in which the mass flux is designed to tail off as the convective kinetic energy tails off when convection penetrates into a stable layer. Here, the convective kinetic energy is estimated from the convective vertical velocity, which is itself evaluated from an analogous equation to that of Levine (1959), and as discussed in Ch. 12.

8

Appendix B: LES diagnosis of entrainment and detrainment

There are two major approaches for diagnosing entrainment and detrainment from LES: Siebesma and Cuijpers (1995) and Swann (2001). The major difference is that Siebesma and Cuijpers (1995) take out the eddytransport term separately in the diagnosis budget whereas Swann (2001)

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321

attempts to recover the total vertical transport by a mass-flux formulation. In order to proceed, it is first necessary to write down the equation corresponding to Eq. 3.2 but for the environment:  1 ∂σe (wϕ)e ∂σe ϕe + = σe Fe . n ˆ · (u − ui )ϕdl + (B.1) ∂t A ∂Ae ∂z

In this section, it is assumed that the domain is doubly periodic, being consistent with standard LES experiments, and thus w ¯ = 0 and ∇ · ϕu = 0, and by continuity:   1 1 n ˆ · (u − ui )ϕdl + n ˆ · (u − ui )ϕdl = 0. (B.2) A ∂Ac A ∂Ae

Note that this constraint is not at all fundamental, and the derivation without it is just a bit more involved. 8.1

Siebesma and Cuijpers’s (1995) method

In this method, Siebesma and Cuijpers (1995) first separate out a part of the vertical transport that can be explained by a mass-flux formulation from a remaining part: ρσc wϕc = Mc ϕc + ρσc w′′ ϕ′′

c

(B.3)

in Eq. 3.2, and simlarly for Eq. B.1, noting the relation in Eq. B.2. The entrainment–detrainment hypothesis assumes that:  1 Eϕe − Dϕc = n ˆ · (u − ui )ϕdl. (B.4) A ∂Ac

As suggested by Eqs. 3.4a and 3.4b, the above line integral can be performed separately over the entraining and the detraining segments of the interface so that the entrainment and detrainment rates may be obtained directly from the above. However, Siebesma and Cuijpers (1995) avoid this procedure with a caution against numerical procedures for dividing the line integral into such segments. Instead, they invoke the mass continuity: ∂Mc ∂σc + . (B.5) E−D =ρ ∂t ∂z By subtracting Eq. B.5, multiplied by ϕc or by ϕe , from Eq. B.4, we obtain, respectively:

 ∂Mc 1 ∂σc + n ˆ · (u − ui )ϕdl − ϕc ρ (ϕe − ϕc )E = (B.6a) A ∂Ac ∂t ∂z

 ∂Mc 1 ∂σc + n ˆ · (u − ui )ϕdl − ϕe ρ . (B.6b) (ϕe − ϕc )D = A ∂Ac ∂t ∂z

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Part II: Mass-flux parameterization

These are the basic formulae that Siebesma and Cuijpers (1995) adopt for diagnosing entrainment and detrainment from LES. In order to obtain better-closed expressions, the first term on the right-hand sides of Eqs. B.6a and B.6b is estimated by removing this term with the help of Eqs. 3.2 and B.1 respectively. In using Eq. B.1, we also invoke the relation in Eq. B.2. 8.2

Swann’s (2001) method

Instead of Eq. B.3, Swann (2001) sets: σc wϕc = Mc ϕ∗c ,

(B.7)

where ϕ∗c is an effective value for ϕ over a convective area, defined through the above equation so that the mass-flux formula perfectly recovers the total vertical transport. Under this formulation, Eq. B.4 is also replaced by:  1 ∗ n ˆ · (u − ui )ϕdl. (B.8) Eϕe − Dϕc = A ∂Ac Thus, Eqs. B.5, B.7, and B.8 are combined together in order to diagnose E, D, and ϕ∗c . As a result, his budget equations become: Mc ∂ϕ∗c E ∂σc ∂ϕc + (ϕc − ϕ∗c ) =− − (ϕ∗c − ϕe ) + σc Fc ∂t ∂t ρ ∂z ρ ∂ϕe Mc ∂ϕe D ∗ 1 ∂ e = + (ϕc − ϕe ) + σe Fe − ρσe w′′ ϕ′′ . σe ∂t ρ ∂z ρ ρ ∂z σc

(B.9a) (B.9b)

Note that the environmental eddy-transport term (the last term in Eq. B.9b) still appears in his budget. His convective-scale budget is consistent with the standard formulation as long as ∂σc /∂t is negligible as expected under the standard asymptotic limit discussed in Ch. 7. His diagnosis method is adopted, for example, by Stirling and Stratton (2012). 9

Appendix C: Tokioka parameter and “organization”

When a spectral representation of convection is adopted for a parameterization with a full range of possible convective types, a common problem is that the deepest convection with a zero entrainment rate tends to dominate. Observations suggest that convection may gradually evolve from

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shallow to deep modes taking several days, for example, in association with the Madden–Julian oscillation (cf., Ch. 5). In order to overcome this shortcoming, Tokioka et al. (1988) propose to introduce a minimum fractional entrainment rate according to the formula: α ǫmin = , hb where hb is the depth of the planetary boundary layer (PBL), and α is a constant, commonly referred to as the Tokioka parameter in the literature. As a consequence of this constraint, deep convection tends to be suppressed when the PBL is shallow, and thus the system has to wait until the PBL deepens sufficiently to support the onset of deep convection. The test experiments of Tokioka et al. (1988) show that in order to produce a tropical 30–60 day oscillation mode, which is missing under the default formulation (i.e., α = 0), they need to set α = 0.1 in an aqua-planet configuration. They suggest α = 0.05 is optimal in order to reproduce realistic climatology. A similar problem is encountered within bulk formulations, and in a sense this is more serious, because an appropriate entrainment parameter must be chosen dependent on the environmental state. Mapes and Neale (2011) call this issue the “entrainment dilemma”: too large an entrainment rate tends to suppress convection too much, leading to an excess accumulation of instability, whereas too small an entrainment rate tends to induce convection too easily. A compromise between these two extremes must be chosen under a given environment. In order to overcome this problem, they propose to introduce a new variable called “organization” so that the interactions between convection and the environment can be better described. In their actual implementation, this organization is controlled by the rain evaporation rate, which in turn controls the entrainment rate. As a result, stronger rain evaporation tends to induce more deep convection with a lower entrainment rate. Note that the same idea is introduced by Hohenegger and Bretherton (2011) without introducing a new variable. It is also important to note that the introduced sensitivities of entrainment on the environmental humidity, as discussed in Sec. 4.5, are part of the general efforts to overcome this entrainment dilemma.

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Chapter 11

Closure

J.-I. Yano and R.S. Plant Editors’ introduction: This chapter discusses the methods of closure for mass-flux parameterizations, or in other words, the calculation of the massflux amplitude at some suitable reference level (traditionally the convection base). The closure is developed in a very general way in terms of the budget of some vertically integrated quantity that is sensitive to convective strength. Common closures used in the literature are the moistureconvergence closure, the stationarity of the cloud-work function, and various closures using CAPE, or some CAPE-like variant. All of these closures can be understood in terms of the general formalism presented and so their various hypotheses or approximations can be compared on an equal footing. 1

Introduction

As outlined in Ch. 7, the core of mass-flux convection parameterization is in defining a spectrum of the mass flux Mi , with i an index for a convection type. Alternatively, a single bulk mass flux M may be considered (Ch. 9). Once the mass flux is defined, all of the convective-scale variables are defined by: ∂ ϕi = ǫi (ϕ¯ − ϕi ) + ρσi Fi /Mi ∂z

(1.1)

in terms of the mass flux and the fractional entrainment rate ǫi = Ei /Mi , with Ei the entrainment rate. Once both Mi and ϕi are known, the convective feedback to the large-scale variable (grid-box mean) ϕ¯ can be evaluated by Ch. 7, Eq. 5.14 (see also Eq. 4.1). Thus, the main problem in mass-flux convection parameterization re325

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duces to that of defining the mass flux Mi . It is usually decomposed under a separation of variables into a vertical dependence and temporal dependence: (1.2) Mi = ηi (z)MB,i (t), where ηi (z) is a normalized vertical profile and MB,i (t) is the timedependent amplitude of convection. The problems of determining these are, respectively, called the “cloud model” and the “closure”. The cloud model is usually formulated by taking prescribed fractional rates ǫi and δi for entrainment and detrainment so that: 1 ∂ ηi = ǫi − δi . (1.3) ηi ∂z The issues of prescribing the entrainment and detrainment were the topic of the previous chapter, while the closure problem is the topic of this chapter. Here, note that MB,i (t) is usually defined as the mass-flux value at the convection base, but mostly for historical reasons. This convention is also adopted in this chapter and as a result ηiB = 1 by definition, where a subscript B indicates a convection-base value. The closure may be considered a major constraint to be introduced into a given parameterization. In respect of thermodynamic variables, the purpose of a closure is to make general constraints such as those already presented in Ch. 8 by Eqs. 3.10, 3.11a and b more specific. At least, a closure must be consistent with these physical constraints. However, further constraints are required in order to define a closure. Some dicta useful for defining such constraints are discussed in Sec. 2.4. In the main part of this chapter, beginning from Sec. 2, the closure problem is considered as that of a constraint on a vertically integrated quantity. This framework covers a wide range of existing closure hypotheses, but not all of them. Some of the major alternative possibilities are discussed in Sec. 12. A background theme of this chapter is, by following up on the discussions from Ch. 9, the choice between the bulk and the spectral formulations. Specifically, the conditions for reducing a spectrum-based closure to a bulk closure formulation are examined in Sec. 6.

2 2.1

A general formulation for the closure Basic idea

A common approach for defining the mass-flux convection parameterization closure is to assume that a certain vertical integral I is quasi-stationary

Closure

327

under interactions between convection and the large-scale dynamics. Thus: ∂ I=0 ∂t

(2.1)

to a good approximation. The basic idea behind this closure formulation may be better seen by explicitly writing down a budget equation in the form: ∂ I = FI,L − DI,c . ∂t

(2.2)

Here, FI,L is a rate that the quantity I is generated by large-scale processes, and DI,c is a rate that I is consumed by convection. The closure of Eq. 2.1 implies that as soon as I is generated by a largescale process, it is consumed by convection almost immediately so that a balance: FI,L − DI,c = 0

(2.3)

is maintained. This is a slightly more careful derivation of the quasiequilibrium condition (Ch. 4, Eq. 2.1) already discussed in Ch. 4. The idea was originally proposed by Arakawa and Schubert (1974) and the specific formulation by Arakawa and Schubert (1974) is described in detail in Sec. 8. Note that in general, the two terms FI,L and DI,c on the right-hand side of Eq. 2.2 are not necessarily positive definite. Thus, some details of the physical interpretation of the balance condition of Eq. 2.3 may be modified. Nevertheless, it is reasonable to expect that the principle of Eq. 2.1 or Eq. 2.3 remains a useful guiding principle for the convectionparameterization closure. 2.2

A perspective of transformations

In practical convective parameterizations, it is usual to start from some first guess of the cloud-base mass flux and integrate the cloud model of Eq. 1.3 vertically to the level of neutral buoyancy. The effects of the massflux profile on the large-scale state can then be determined, as discussed in Ch. 8. Finally, the first guess is rescaled on the basis of the closure condition in order to determine the actual mass flux and hence the overall modification of the large-scale state. In a spectral parameterization it may be necessary to rescale the mass flux separately for each cloud type. A general normalization transformation

328

Part II: Mass-flux parameterization

T is defined as being a positively valued rescaling of the cloud-base mass flux for each cloud type: MB,i → MB,i Ti

(2.4)

with the full transformation being the set {Ti }. In these terms, the parameterization closure can be thought of as the method adopted in order to select a particular normalization transformation. A normalization transformation for which T = {Ti } is independent of i may be called a “global transformation”. A global transformation amounts  to a simple multiplying factor for MB = i MB,i and is sufficient in order to close a bulk parameterization scheme, or indeed to close a spectral parameterization in which the spectral distribution of convective types is prescribed (cf., Ch. 9, Sec. 7). The closure formulation is presented in this chapter in terms of an unspecified number of convection types with mass-flux profiles given by {ηi }. A bulk parameterization is adopted by the majority of operational schemes, and for the most part this will be considered as being equivalent to the analysis of a single convection type only. As discussed in detail in Ch. 9, such an equivalence may not be strictly valid, as a weighted sum over convection types does not quite yield a single effective convective type, at least not without the introduction of further assumptions. The implications for parameterization closure are described in detail in Sec. 6. However, some general remarks about bulk and spectral closures may be useful to set the scene here. Under a bulk formulation, a single mass flux Mc (z) must be determined by properly choosing both its amplitude MB and the profile ηc . These are taken care of by the closure and the entrainment–detrainment formulations, respectively, and both are important issues. On the other hand, when a spectral formulation is taken, we have additional freedom in choosing a set of convection types, whose vertical profiles are customarily determined by a relatively simple cloud model, or a certain entrainment–detrainment formulation (e.g., entraining-plume hypothesis, cf., Ch. 8). Alternatively, we may more directly choose a set of convective vertical profiles (e.g., half-sinusoidal shapes). Under a spectral formulation, when a large enough number of convective vertical profiles are adopted, we would expect to have enough freedom for determining the vertical structure of convective feedback to the large scale without carefully adjusting a cloud model for determining the entrainment–detrainment rate for each convection type. In this case, the

Closure

329

key issue reduces to that of a closure that defines a spectrum of convection. Regardless of the details of the choice, the issues of entrainment and detrainment become secondary. The notion of a large enough number of profiles for this context may be understood in analogy with the concept of “completeness” defining the capacity of a given set of modes (e.g., a Fourier expansion) for representing a general functional form. In order to represent any plausible convective response, a suitable set of ηi should be available. However, no formal condition is yet known for this purpose. Although the presentation will be made in terms of a discrete set of spectral profiles, it is straightforward to generalize it to a continuous spectrum. The simplest example would be an entraining-plume model with continuous values for the entrainment rate. Similarly, a potentially important possibility is that of taking more than one parameter for the convection-type representation. Again, this is entirely straightforward and is not included in the presentation only in order to avoid an unnecessary complication of the notation. An interesting example could be to include the precipitation efficiency, introduced in Ch. 8, Sec. 7, as an additional parameter. This perspective is important, because if a closure under a precipitation-efficiency spectrum works, it suggests that a more sophisticated representation of microphysical processes may be feasible along these lines without necessarily introducing a fully prognostic microphysical scheme (cf., Vol. 2, Ch. 18). 2.3

Large-scale and convective processes

The distinction between large-scale and convective processes for separating the right-hand side of Eq. 2.2 into two tendencies may not be immediately obvious for all purposes. Under the mass-flux formulation framework, however, the distinction is relatively straightforward, and follows from the properties of the various terms in ∂I/∂t under a normalization transformation. The quantities of interest to us (denoted V for some arbitrary variable) transform in one of the following ways: (1) Normalization-invariant variables are unaffected by the transformation, so that V→V for all possible T . Such variables might be directly dependent on plume dynamics, but only through intensive properties of each plume type. A good example, as can be seen from Eq. 1.1, would be a conservative convective variable such as the moist static energy hi .

330

Part II: Mass-flux parameterization

Such variables must be independent of the overall amount of convective transport MB , and also of its distribution across the plume spectrum {MB,i }. They evolve only in response to changes in the large-scale state, since this would modify ϕ ¯ appearing in Eq. 1.1. Their evolution can therefore be characterized by a large-scale timescale τLS . (2) Globally invariant variables are unaffected by a global transformation, so that V→V if and only if Ti is independent of i. Such variables are independent of the overall amount of convective transport but are sensitive to its distribution across the plume spectrum. A good example would be the bulk moist static energy, obtained as a weighted sum of the hi as explained in Ch. 9. The evolution of such variables can be characterized by a timescale τspec . (3) Normalization-rescaled variables transform as V→VT i for all possible T . Such variables transform alongside one type within the convective spectrum only, depending extensively on the given type i, but being independent of any rescaling of the rest of the spectrum. The trivial example is MB,i , but less trivial examples are Mi (z = zB ), Di , etc. These variables evolve in response to changes in the strength of the particular type, which can be characterized by a timescale denoted τi . Note that this timescale should not be confused with the corresponding cloud lifespan because this is not considered within the mass-flux parameterization formulation that has been discussed throughout Part II which makes the steady-plume hypothesis (Ch. 7, Sec. 6.4), filtering out terms such as ∂ρσi /∂t within the cloud model. (4) Globally rescaled variables transform as V→VT if and only if Ti is independent of i. Such variables depend extensively on the overall amount of convective transport and may also be sensitive to its distribution across the spectrum. Their evolution is governed by the timescale τadj originally introduced by Arakawa and Schubert (1974): if all forcing for convection were to be removed then the overall convective transport would decay on this timescale. The two groups of terms in Eq. 2.2 can be separated simply from their transformation properties. The large-scale forcing terms FI,L are globally invariant whereas the convective terms DI,c are globally rescaled. Nevertheless, some subtleties will be remarked upon during the following discussions. In particular, it is important to realize from these definitions that the so-called “large-scale” processes, in fact, include some “non-convective” subgrid-scale processes, especially those coming from the boundary layer.

Closure

331

This distinction becomes important later in discussing CAPE-based closures. If a spectral formulation is adopted then the integral quantity I to be considered as stationary acquires an index i labelling each convective type (Sec. 3). The corresponding large-scale forcing terms and convective terms similarly become FI,i,L and DI,i , respectively, also acquiring a convectiontype index. These terms can be specified as being normalization invariant for the forcing terms, and globally rescaled for the convective terms. Note that the convective terms are globally rescaled rather than normalization rescaled because the action of other convective types j = i may affect the integral Ii , characterizing the extent to which the environment supports the i-th type of convection. This point will be demonstrated explicitly in Sec. 5; note, for example, the structure of Eq. 5.7. It may be helpful to clarify the meaning of some of the timescales by considering a step-change in the large-scale forcing, such as a sudden increase in the tropospheric cooling. (An example of such an experiment can be found in the CRM study of Cohen and Craig (2004).) The mass fluxes for each convection type will respond to that change, each type with its own specific timescale τi . The overall convective transport, as measured by MB , will approach a new, steady value on the timescale τadj that is presumably intermediate between the τi of the more quickly and more slowly responding types. A more complete adjustment of the system occurs on the timescale τspec , over which the relative contributions of the various convective types MBi /MB approach their new values. Ultimately, a new large-scale thermodynamic state is also established on the timescale τLS . Unfortunately, there is no useful information from the literature that would provide good estimates of all of these separate timescales and their possible dependencies or relationships. In reality, the large-scale forcing often varies slowly, so the different rates of approach to adjusted, steady values of various variables may not be clear. However, it would not appear overly difficult to devise idealized CRM simulations and/or to examine various lag correlations with a view to identifying these timescales (cf., Ch. 3, Sec. 6). The physical interpretations and implications of a quasi-equilibrium formulation for closure, as stated in Sec. 2.1, are discussed in full detail in Ch. 4 (see also Ch. 3). It is not necessary to repeat such discussions here but simply to recall the concept of a timescale separation underpinning the quasi-equilibrium view, which amounts to an assumption of τLS ≫ τadj for a bulk quantity I or to an assumption of τspec ≫ τadj for a spectral quantity Ii .

332

2.4

Part II: Mass-flux parameterization

Choice of I: Physical considerations

The general formulation for convective closure to be presented does not specify which variable is the most physical to take for the vertical integral I. Nevertheless, the generality of the formulation allows us to examine all existing closure hypotheses defined in terms of a column vertical integral in a unified manner. The advantages and disadvantages of existing closure hypotheses can then be discussed on an equal basis. A merit of presenting a general formulation may be understood by accepting the following dicta: Dictum 1: Any physically based diagnostic convective closure must have a prognostic counterpart. This statement is rather trivial if a closure condition is derived based on a stationary condition for a budget equation of a vertically integrated quantity, as proposed here. However, in the literature, there are various different types of closure hypotheses proposed, which are not necessarily derivable based on this procedure. This dictum essentially says that a closure should be derived from a budget equation. If not, a given closure hypothesis must be deemed to be unphysical. A stronger version of the dictum can be further developed, which may be stated as: Dictum 2: Any physically based diagnostic closure condition must have a prognostic counterpart that can be integrated in time in a self-contained manner. This dictum may greatly narrow the possibilities as discussed further in Sec. 9.3. Under the mass-flux formulation, the goal of the closure is to define MBi . The first point to be emphasized is that as long as any variable controlled by convection is chosen for the vertical integral (and assumed to be steady) a given closure condition can define the mass-flux magnitude (provided certain mathematical conditions, such as invertibility of a matrix are satisfied). However, it is natural to expect that in a prognostic treatment, such a self-contained description should produce a prognostic equation for the mass flux or an equivalent quantity.

Closure

3 3.1

333

Formulations for the vertical integral I A closure depending only on the large-scale variables

The simplest choice for the vertical integral I is:  zT f (ϕ)dz, ¯ I=

(3.1)

zB

where f is an unspecified function of an unspecified physical variable ϕ, ¯ which is defined as a grid-box average (i.e., what is usually called a “largescale” variable). Here, the integral may be performed from the convection base zB to the convection top zT . The choice of these two levels is left open in most of the analysis, as it was in Ch. 8. Nevertheless, a few words are warranted here. The convection base zB would be most conveniently defined at the top of the planetary boundary layer (well-mixed convective boundary layer), as assumed by Arakawa and Schubert (1974). In many CRM/LES diagnostic analyses, the convection base is simply defined as a cloud base, or a lifting condensation level. The ECMWF model, for example, also takes this latter definition. Alternatively, we may take the convection base simply at the surface. The convection top zT is relatively straightforward to define, following a basic idea of convective plumes, as reviewed in Ch. 10: it would be equated with the level of neutral buoyancy. However, this is not the only option, and an alternative was discussed in Ch. 8, Sec. 10. The function f may, in general, depend on multiple variables. In that case, ϕ¯ must be replaced by a vector representing those multiple variables. A possibility for this generalization is kept implicit in much of the following formulation, because it only introduces a minor modification of a notation in so far as a general derivation is concerned. The important point is that the function depends solely on large-scale variables. A moisture closure by Kuo (1974), for example, takes such an assumption (cf., Sec. 7). However, this limited allowed dependency may be generalized in several ways, as discussed next. 3.2

Dependence on the convective variables

The vertical integral might also depend on convective-scale variables ϕi as well. This generalization is crucial when a spectral formulation is taken so that MB,i is defined for each component separately by a vertical integral Ii defined for each convection type. In other words, a closure solely in terms

334

Part II: Mass-flux parameterization

of the large-scale variables works only for a bulk formulation. Here, the same notation is taken for the convective-scale variable as for the large-scale variable solely for the sake of the simplicity. In general, the two could be different variables. In this generalization, the vertical integral may be defined by:  zT i f (ϕi , ϕ)dz, ¯ (3.2) Ii = zB

and the quasi-stationarity condition becomes: ∂ Ii = 0. (3.3) ∂t The index i indicates that this is a closure condition for the i-th type of convection. It is also expected that the convection top zT i depends on the convection type i, whereas the convection base zB is assumed to be common to all of the types. 3.3

Dependence on the mass-flux profile

The vertical profile of the mass flux ηi may play an important role in constraining the intensity of convection for a given component, i. For this reason, ηi may also be added as an additional dependence in the function f . Thus:  zT i f (ηi , ϕi , ϕ)dz. ¯ (3.4) Ii = zB

3.4

Integral in two parts

The vertical integral may, furthermore, be divided into two parts (or more) for various reasons. Among those, the most likely possibility is a change in the function to be integrated when crossing the water condensation level (the level at which water vapour reaches saturation) zci . Thus:  zT i  zci Ii = f2 (ηi , ϕi , ϕ)dz. ¯ (3.5) f1 (ηi , ϕi , ϕ)dz, ¯ + zB

zci

It is assumed that the two functions are continuous over the interface, z = zci . Thus: f1 (z = zci ) = f2 (z = zci ).

(3.6)

The convective quasi-equilibrium hypothesis of Arakawa and Schubert (1974) can be considered a special case of a vertical integral defined by Eq. 3.5.

Closure

335

In the following, the closure condition is derived under the constraint of Eq. 3.3 with the vertical integral Ii defined by Eq. 3.5. In order to derive an explicit expression for Eq. 3.3, prognostic equations both for the large-scale (grid-box average) and the convective-scale variables must first be known. These are reviewed in the next section. A general closure formulation is developed in Sec. 5 and the Kuo (1974) and Arakawa and Schubert (1974) closures are derived as special cases of this general formulation in Secs. 7 and 8 respectively.

4

Prognostic equations for the physical variables

The prognostic equations for the physical variables under the mass-flux formulation were derived in Chs. 7 and 8. Here, for convenience, the main results needed to derive a general closure condition are summarized and collected. 4.1

Large-scale variables

The prognostic equations for the large-scale variables ϕ¯ are given by Ch. 7, Eq. 6.12. For the purpose of the present chapter, it is written in the following form:

   ∂ ϕ¯ ∂ ϕ¯ 1  ∂ ϕ¯ D Di (ϕi − ϕ) ¯ + Mc + = , (4.1) ∂t ρ i ∂z ∂t L where Mc is the total mass flux defined by:  Mi , Mc =

(4.2)

i

and 

∂ ϕ¯ ∂t



L

1 ∂ ¯ · ϕ¯ ρw ¯ ϕ¯ + Fe = −∇ ¯u − ρ ∂z

(4.3)

is the tendency due to the large-scale processes. The terms on the righthand side are defined in Ch. 7. A superscript D is added to ϕi in Eq. 4.1 in order to indicate a detrainment value which may not be equal to ϕi if the variable is not conservative. In the following, the large-scale tendency is dealt with as a single term without regard to the contributions of the individual terms.

336

4.2

Part II: Mass-flux parameterization

Convective-scale variables: Diagnostic solution

The convective-scale variables are dealt with diagnostically under the standard mass-flux formulation as carefully discussed in Ch. 7, and summarized by Ch. 7, Eq. 6.10. Explicit solutions for convective-scale variables were further presented in Ch. 8, Sec. 6. The present subsection slightly generalizes the presentation made there. A diagnostic equation for a convective-scale variable ϕi may be, in the most general form, given by:   ∂ + ǫ˜i ϕi = ǫi ϕ˜¯i , (4.4) ∂z where ϕ˜¯i = ϕ¯ +

ρσi ˜ Fi ǫi Mi

(4.5a)

ǫ˜i = ǫi −

ρσi ˆ Fi . Mi ϕi

(4.5b)

and

The final terms in both of the expressions in Eqs. 4.5a and b are obtained by dividing the forcing term Fi into two arbitrary contributions: Fi = F˜i + Fˆi .

(4.6)

The division can be made in any manner desired so as to obtain a more convenient analytic expression for the particular variable ϕi . The ideal division would be to make the parameter ǫ˜i a function of height only (with a possible extension to the case with additional dependence on ϕi ), and for ϕ˜ ¯i to have a simple closed expression (ideally independent of the convectivescale variables: see immediately below). ¯ In general, ϕ˜ ¯i may depend on other physical variables such as χi and χ so that Eq. 4.5a takes the form: ¯ ϕ˜¯i = ϕ¯ + F˜i (χi , χ),

(4.7a)

ρσi ˜ F˜i = Fi . ǫi Mi

(4.7b)

where

This possibility will be considered later in Sec. 5.3. For now, however, ϕ¯˜i is assumed to be a function of height only with no additional functional dependence.

337

Closure

As discussed in Ch. 8, Sec. 6, Eq. 4.4 above can be readily solved, and the solution is:

 z 1 ′ ˜ (4.8) ϕi = ǫi η˜i ϕ¯i dz , ϕiB + η˜i zB where

η˜i = exp



z

ǫ˜i dz zB

′



.

(4.9)

Note that η˜i = ηi even when ǫ˜i = ǫi , unless a purely entraining plume is assumed. Keep in mind that this chapter is pursuing a general formulation without that assumption. It is also convenient to introduce  z  ηˆi = exp (4.10) ǫi dz ′ zB

in preparation for later use. 4.3

Convective-scale variables: Prognostic equations

A prognostic equation for a convective variable ϕi can be obtained by taking a time derivative of Eq. 4.8. This procedure is consistent with the spirit of the bounded-derivative method (Browning et al., 1980; Kreiss, 1980): i.e., when a balance condition (diagnostic relation) is assumed for a given variable, its prognostic equation can be obtained by taking a time derivative of the given balance condition. In order to proceed towards this direction, it is necessary to note that: ∂ η˜i = −z˜˙ B ǫ˜iB η˜i , (4.11) ∂t where  z ∂˜ǫi ′ 1 dz . z˜˙ B = z˙B − (4.12) ǫ˜iB zB ∂t Before taking the time derivative of Eq. 4.8, we first rewrite it as:  z ǫi η˜i ϕ˜¯i dz ′ . η˜i ϕi = ϕiB + zB

The time derivative of the left-hand side is: ∂ϕi ∂ η˜i ϕi = η˜i − z˜˙ B ǫ˜iB η˜i ϕi . ∂t ∂t The time derivative of the integral on the right-hand side gives:    z  z ∂ ∂ ǫi η˜i − z˜˙ B ǫ˜iB ϕ˜¯i dz ′ − z˙B ǫiB ϕ˜¯iB . ¯i dz ′ = ǫi η˜i ϕ˜ ∂t zB ∂t zB

338

Part II: Mass-flux parameterization

Putting these two expressions together, and simplifying the result with the aid of Eqs. 4.8 and 4.12 we obtain:  

 z ∂ ρσi ˆ ∂˜ǫi ′ ∂ϕi Fi = −z˙B ǫiB Δϕ˜iB + − z˙B dz ϕiB − η˜i ∂t ∂t Mi ϕi zB ∂t B +



z

ǫi η˜i

zB

∂ ϕ˜¯i ′ dz , ∂t

(4.13)

where Δϕ˜i,B = ϕ˜¯iB − ϕiB . The following prognostic equation will also be needed:

   ∂ ϕ˜¯i 1  ∂ ϕ˜ ∂ ϕ¯ ¯i D = Di (ϕi − ϕ) ¯ + Mc + , ∂t ρ i ∂z ∂t L where



∂ ϕ˜ ¯i ∂t



L

=



∂ ϕ¯ ∂t



∂ + ∂t L



 ρσi ˜ Fi , ǫi Mi

(4.14)

(4.15)

(4.16)

which follows immediately from Eqs. 4.1 and 4.5a. In general, the second term on the right-hand side of Eq. 4.16 depends on convective-scale variables, but it will initially be assumed to depend only on the large-scale variables, with modifications to be considered later. 5

General formulation

In deriving the general formulation for closure, the case with Eq. 3.4 is considered first in the following subsection, because of its relative simplicity. The modifications arising from separating the integral into the two parts (Eq. 3.5) are discussed separately in Sec. 5.2. Finally, further modifications when the forcing term also depends on convective-scale variables are discussed in Sec. 5.3. 5.1

A single vertical integral

Taking Eq. 3.4 to define Ii , we can rewrite the definition as:  zT i ∂Ii ∂f = (ηi , ϕi , ϕ)dz ¯ + z˙T i f (z = zT i ) − z˙B f (z = zB ) ∂t ∂t zB

(5.1)

Closure

339

simply by invoking Leibniz’s theorem. The time derivative of the integrand may be expanded as: ∂f ∂ϕi ∂f ∂ ϕ¯ ∂f ∂f ∂ηi (ηi , ϕi , ϕ) + + . ¯ = ∂t ∂ηi ∂t ∂ϕi ∂t ∂ ϕ¯ ∂t

(5.2)

Strictly speaking, ∂f /∂ηi , ∂f /∂ϕi , and ∂f /∂ ϕ ¯ above are functional differentials rather than partial differentials, but different symbols are not introduced here, since there should be no ambiguity. The tendency ∂ηi /∂t may be derived in analogous manner as for Eq. 4.11, and ∂ηi = −z˙B (ǫiB − δiB )ηi . ∂t Thus, the first integral can be rewritten as:  zT i  zT i ∂f ∂ηi ∂f dz = −z˙B (ǫiB − δiB ) ηi dz. ∂η ∂t ∂η i i zB zB

(5.3)

(5.4)

The tendencies ∂ϕi /∂t and ∂ ϕ/∂t ¯ are given by Eqs. 4.13 and 4.1, respectively. By substituting them into the last two integrals, we obtain:

 zT i ∂f ∂ ϕ¯ ∂f ∂ϕi + dz = ∂ϕi ∂t ∂ ϕ¯ ∂t zB 

zT i

zB



z



1 ∂f η˜i ∂ϕi

 

  z ∂ ρσi ˆ ∂˜ǫi ′ Fi − z˙B dz ϕiB − −z˙B ǫiB Δϕ˜iB + ∂t Mi ϕi zB ∂t B

∂ ϕ˜ ¯i ′ dz ǫi η˜i + ∂t zB



∂f + ∂ ϕ¯

     ∂ ϕ¯ 1  ∂ ϕ ¯ + Di (ϕD ¯ + Mc dz. i − ϕ) ρ i ∂z ∂t L (5.5)

A further reduction of Eq. 5.5 is possible by substituting the explicit tendency equation for ϕ˜ ¯i , as given in Eq. 4.16. When doing so, it is convenient to exchange the order of integration so that:  zT i  zT i  z  zT i ∂ ϕ˜¯ 1 ∂f ′ ∂ ϕ˜ ¯ 1 ∂f ǫi η˜i i dz dz. (5.6) ǫi η˜i i dz ′ dz = η ˜ ∂ϕ ∂t ∂t η ˜ i i i ∂ϕi zB zB z zB It is also necessary to decompose the total mass flux based on Eq. 4.2, recalling that by definition Di = δi Mi . After some lengthy manipulations, we find that:  ∂Ii = Kij MjB + z˙T i GT i + z˙B GBi + FI,L,i . (5.7) ∂t j

340

Part II: Mass-flux parameterization

Here, the matrix 

 zT i  1 ∂ ϕ¯ ∂f D Kij = ¯ + aϕi η˜i + ǫi ˜ ηj δj (ϕj − ϕ) dz ρ ∂ ϕ¯ ∂z zB

(5.8a)

defines the fractional rate at which the action of the j-th convective type increases the vertically integrated quantity Ii associated with the i-th convection type. GT i = f (z = zT i )

and GBi = − (ǫiB − δiB )



zT i

zB

− f (z = zB )

(5.8b)



 ρσi ˆ ∂f Fi ηi ϕiB dz − ˜aϕi B ǫiB Δϕ˜iB + ∂ηi Mi ϕi B (5.8c)

are changes associated respectively with the top and the bottom boundaries moving with time. Finally, the last term 

  

 zT i ∂ ϕ¯˜i ∂f ∂ ϕ¯ ∂ ˜ ˜ ˜ ǫi aϕi η˜i + dz FI,L,i = aϕi B − cϕi ϕiB + ∂t ∂t L ∂ ϕ¯ ∂t L zB

represents the large-scale forcing. This may be furthermore rewritten by decomposing ϕ˜ ¯i into its large-scale and forcing-dependent parts to produce:

   zT i ∂f ∂ ϕ¯ ∂ ǫi ˜aϕi η˜i + dz aϕi B − ˜cϕi ϕiB + FI,L,i = ˜ ∂t ∂ ϕ¯ ∂t L zB    zT i ρσi ˜ ∂ ˜ Fi dz. ǫi aϕi η˜i (5.8d) + ∂t ǫi Mi zB

In order to shorten the expressions presented above, the following coefficients have been introduced:  zT i 1 ∂f ′ ˜ dz (5.9a) a ϕi = η˜i ∂ϕi z  zT i ∂˜ǫ ˜aϕi i dz. ˜cϕi = (5.9b) ∂t zB

The subscript ϕi has added to the definitions of ˜aϕi and ˜cϕi to indicate that they depend on the variable ϕi . Although the tendency for the vertically integrated quantity Ii is mostly defined by the changes taking place within the integral range [zB , zT i ], there are also terms indicating an influence from underneath in the form of bottom-boundary contributions. For example, the first term in the largescale forcing given in Eq. 5.8d accounts for such a boundary-layer contribution.

341

Closure

5.2

Two-part vertical integral

When the vertical integral is separated into two parts, as in Eq. 3.5, the derivation proceeds in a similar manner. The starting point is:

 zci ∂f1 ∂ηi ∂f1 ∂ϕi ∂f1 ∂ ϕ¯ ∂Ii = + + dz ∂t ∂ηi ∂t ∂ϕi ∂t ∂ ϕ¯ ∂t zB

 zT i ∂f2 ∂ϕi ∂f2 ∂ ϕ¯ ∂f2 ∂ηi + + + dz ∂ηi ∂t ∂ϕi ∂t ∂ ϕ¯ ∂t zci + z˙T i f2 (z = zT i ) − z˙B f1 (z = zB ).

(5.10)

Here, by the assumption of the continuity (as expressed by Eq. 3.6) of the two functions over z = zci , a contribution of the values at the integral boundary z = zci cancels out. However, some care is required in changing the order of the integrals. In place of Eq. 5.6 before, it is necessary to use:  zci  zci   1 ∂f1 z ∂ ϕ˜¯i zci 1 ∂f1 ′ ∂ ϕ˜ ¯i ′ dz dz = ǫi η˜i dz dz, (5.11a) ǫi η˜i ˜i ∂ϕi zB ∂t ∂t z η˜i ∂ϕi zB η zB 

zT i

1 ∂f2 η˜i ∂ϕi



z

∂ ϕ˜ ¯i ′ dz dz = ∂t



zT i

∂ ϕ˜¯i ∂t



zT i

1 ∂f2 ′ dz dz. η ˜ zB max(z,zci ) i ∂ϕi zB zci (5.11b) Following a similar reduction, we arrive at the same general form as Eq. 5.7, although with some different definitions for the terms: 

 zci  1 ∂ ϕ¯ ∂f1 − ϕ) ¯ + aϕi η˜i + Kij = ǫi ˜ ηj δj (ϕD dz j ∂ ϕ¯ ∂z zB ρ

 zT i ˜ ∂ ϕ¯ ǫi bϕi η˜i D ηj δj (ϕj − ϕ) ¯ + + dz ρ ∂z zB

 zT i 1 ∂f2 ∂ ϕ¯ ηj δj (ϕD ¯ + + dz, (5.12a) j − ϕ) ¯ ∂z zci ρ ∂ ϕ ǫi η˜i

ǫi η˜i

GT i = f2 (z = zT i ), GBi

(5.12b)

 zT i ∂f2 ∂f1 ηi dz + dz = − (ǫiB − δiB ) ηi ∂ηi ∂ηi z zB  ci

 ρσi ˆ ˜ − (˜ aϕi B + bϕi B ) ǫiB Δϕ˜iB + Fi ϕiB − f1 (z = zB ), Mi ϕi B (5.12c) 

zci

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Part II: Mass-flux parameterization

∂ FI,L,i = (˜ a ϕi B + ˜ bϕi B ) − (˜cϕi + ˜dϕi ) ϕiB ∂t    

 zci ∂ ϕ˜¯i ∂f1 ∂ ϕ¯ + dz aϕi η˜i ǫi ˜ + ∂t L ∂ ϕ¯ ∂t L zB      zT i  zT i ∂ ϕ˜¯i ∂f2 ∂ ϕ¯ ˜ ǫi bϕi η˜i dz + dz. + ∂t L ∂ ϕ¯ ∂t L zB zci

(5.12d)

The definitions of the coefficients are also modified as follows, and some new coefficients introduced:  zci 1 ∂f1 ′ ˜ dz , (5.13a) a ϕi = η ˜ i ∂ϕi z  zT i 1 ∂f2 ′ ˜ ϕi = b dz , (5.13b) ˜i ∂ϕi max(z,zci ) η ˜cϕi =



zci

˜aϕi

zB

˜ dϕi =



zT i zB

∂˜ǫi dz ∂t

˜ϕ ∂˜ǫi dz. b i ∂t

(5.13c)

(5.13d)

It is easy to check that the expressions in this subsection reduce to those of the previous one when zci = zT i or zci = zB . 5.3

When forcing depends on convective-scale variables

In the derivation so far, it has been assumed that the quasi large-scale tendency (∂ ϕ˜ ¯i /∂t)L can be treated as a part of the large-scale forcing. In general, this is not the case, as indicated by Eq. 4.7. The present subsection considers further modifications of the closure formulation in order to accommodate this generalization. More generally, the tendency of the quasi-fractional entrainment rate ǫ˜i may also depend on the convective-scale variables, as indicated by Eq. 4.5b. This further generalization is in fact straightforward, and only makes the final results more involved. Furthermore, only the generalized treatment of (∂ ϕ˜ ¯i /∂t)L turns out to be necessary for rederiving Arakawa and Schubert’s (1974) convective quasi-equilibrium closure in Sec. 8 and so such a generalized treatment of ∂˜ ǫi /∂t is not explicitly considered here. Under this generalization, (∂ ϕ˜¯i /∂t)L does not solely represent a largescale tendency, but also contains some convective contributions that stem from the second term on the right-hand side of Eq. 4.16. The term should

343

Closure

be separated into contributions associated with the large scale and with convection:     ˜   ∂ F˜i ∂ ∂ Fi ρσi ˜ + (5.14) Fi = ∂t ǫi Mi ∂t ∂t L

c

by referring to the definition of Eq. 4.7b. As a result, Eqs. 4.15 and 4.16 then read:

     ∂ ϕ˜¯i ∂ F˜i ∂ ϕ¯ 1  ∂ ϕ˜ ¯i D + (5.15) = Di (ϕj − ϕ) ¯ + Mc + ∂t ρ i ∂z ∂t ∂t L c

and 

∂ ϕ˜ ¯i ∂t



L

=



∂ ϕ¯ ∂t



L

+



∂ F˜i ∂t



.

(5.16)

L

A new type of term (the second term on the right-hand side) appears in Eq. 5.15 and leads to a corresponding new type of term ΔFIi on the right-hand side of Eq. 5.10, which may be written as:      zci  zT i ∂ F˜i ∂ F˜i ˜ aϕi η˜i ǫi ˜ ǫi bϕi η˜i dz + dz. (5.17) ΔFIi = ∂t ∂t zB zB c

c

In order to write this contribution in the form of the other convective terms in Eq. 5.10, we begin by writing:       ∂ F˜i ∂χi ¯ ∂ F˜i ∂ χ ∂ F˜i = , + ∂t ∂χi ∂t ∂χ ¯ ∂t c c

and the tendencies (∂χi /∂t) and (∂ χ/∂t) ¯ c can then be expressed using equivalent equations to Eqs. 4.13 and 4.1 for the convective-scale and large¯ respectively. Note that here, the total tendency is scale variables χi and χ, considered for χi , whereas only the convective tendency is considered for χ¯ so that the necessary new contributions are properly acounted for. In the following, it is assumed that χ is a conserved variable so that no further forcing terms applying to χ must be added. This assumption serves only to simplify the final expression, but a further generalization or the inclusion of a further dependence of F˜i on the mass-flux profile ηi could also be possible, if required. With these assumptions the convective-scale contribution reads:

 z ∂χ ¯ 1 ∂χiB ∂χi ǫi ηˆi dz ′ . = + (5.18) −z˙B ǫiB ΔχiB + ∂t ηˆi ∂t ∂t zB

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Part II: Mass-flux parameterization

Recall that ηˆi is defined by Eq. 4.10. The tendency may furthermore be divided (somewhat arbitrarily) into the term containing the convective tendency for χ ¯ and the remaining terms, adding the subscript c and L, respectively:      1 z ∂χ ¯ ∂χi ǫi ηˆi = dz ′ (5.19a) ∂t c ηˆi zB ∂t c 

    z ∂χi ∂χ ¯ ∂χiB 1 + ǫi ηˆi = −z˙B ǫiB ΔχiB + dz ′ . (5.19b) ∂t L ηˆi ∂t ∂t L zB Accordingly, the tendency (∂ F˜i /∂t)c may also be divided into the two major contributions:       ∂ F˜i ∂ F˜i ∂ F˜i = + , ∂t ∂t ∂t c

c,c

c,L

where 

     ∂ F˜i ∂χi ∂ F˜i ¯ ∂ F˜i ∂ χ = + , ∂t ∂χi ∂t c ∂χ ¯ ∂t c c,c     ∂ F˜i ∂χi ∂ F˜i = . ∂t ∂χi ∂t L

(5.20a)

(5.20b)

c,L

Substituting for (∂χi /∂t)c and (∂ χ/∂t) ¯ c in Eq. 5.20a and reversing the order of integration for the double integrals, the required correction due to (∂ F˜i /∂t)c,c can be reduced to have the same form as the other convective terms in Eq. 5.7:  ΔKij MjB . ΔFIi = j

with the correction to the interaction matrix being given by: 

 zci  zci ∂ F˜i η˜i ∂ F˜i ′ ǫi ∂χ ¯ − χ)+ ¯ η˜i aϕi dz ηj δj (χD ǫi a ϕi +ηˆi ΔKij = dz j ∂χ ¯ ηˆi ∂χi ∂z z zB ρ 

 zB  zT i ǫi ∂χ ¯ ∂ F˜i η˜i ∂ F˜i ′ D η˜i bϕi +ηˆi ¯ dz ηj δj (χj − χ)+ ǫi bϕi + dz ρ ∂χ ¯ ηˆi ∂χi ∂z zB z (5.21a)

This correction must be added to Kij , defined by Eq. 5.12a.

Closure

345

The term (∂ F˜i /∂t)c,L , on the other hand, contributes as an additional term for the large-scale forcing: 

  zT i zci η˜i ∂ F˜i η˜i ∂ F˜i ˜ ǫi bϕi dz + dz ǫi ˜aϕi ΔFI,L,i = − z˙B ǫiB ΔχiB ηˆi ∂χi ηˆi ∂χi zB zB  

 zci  z ∂χ ¯ η˜i ∂ F˜i ∂χiB + + ǫi ηˆi a ϕi ǫi ˜ dz ′ dz ηˆi ∂χi ∂t ∂t L zB zB

   z  zT i ˜ ∂χ ¯ η˜i ∂ Fi ∂χiB ′ ˜ + dz dz. (5.21b) ǫi ηˆi ǫi bϕi + ηˆi ∂χi ∂t ∂t L zB zB Here, this term includes contributions from changes of the convection base. For completeness, let us now summarize the expressions for Eqs. 5.12a–d under this modification: 

 zci  1 ∂ ϕ¯ ∂f1 D ˜ ¯ + Kij = ǫi aϕi η˜i + ηj δj (ϕj − ϕ) dz ∂ ϕ¯ ∂z zB ρ

 zT i ˜ ǫi bϕi η˜i ∂ ϕ¯ ηj δj (ϕD − ϕ) ¯ + + dz j ρ ∂z zB

 zT i 1 ∂f2 ∂ ϕ¯ ηj δj (ϕD − ϕ) ¯ + dz + j ¯ ∂z zci ρ ∂ ϕ 

 zci  zci ǫi ∂ F˜i ∂χ ¯ η˜i ∂ F˜i ′ η˜i aϕi +ηˆi + − χ)+ ¯ dz ηj δj (χD ǫi a ϕi dz j ∂χ ¯ ηˆi ∂χi ∂z zB ρ z 

 zB  zT i ∂ F˜i η˜i ∂ F˜i ′ ǫi ∂χ ¯ D η˜i bϕi +ηˆi ¯ ǫi bϕi dz ηj δj (χj − χ)+ + dz, ρ ∂χ ¯ ηˆi ∂χi ∂z z zB (5.22a)

GT i = f2 (z = zT i ) 

(5.22b)

 zT i ∂f1 ∂f2 ηi dz + dz ∂ηi ∂ηi zB z  ci

 ρσi ˆ Fi ϕiB − (˜ a ϕi B + ˜ bϕi B ) ǫiB Δϕ˜iB + Mi ϕi B

   zT i zci ˜ ˜i η˜i ∂ Fi η ˜ ∂ F i ˜ ϕi − ǫiB ΔχiB a ϕi ǫi ˜ ǫi b dz+ dz −f1 (z = zB ), ηˆi ∂χi ηˆi ∂χi zB zB

GBi = − (ǫiB − δiB )

zci

ηi

(5.22c)

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Part II: Mass-flux parameterization

˜ϕi B ) ∂ − (˜cϕi + ˜dϕi ) ϕiB FI,L,i = (˜ a ϕi B + b ∂t    

 zci ∂ ϕ˜¯i ∂f1 ∂ ϕ¯ + dz aϕi η˜i ǫi ˜ + ∂t L ∂ ϕ¯ ∂t L zB      zT i  zT i ∂ ϕ˜¯i ∂f2 ∂ ϕ¯ ˜ ǫi bϕi η˜i dz + dz + ∂t L ∂ ϕ¯ ∂t L zB zci

   zci  z ∂χ ¯ η˜i ∂ F˜i ∂χiB + dz ′ dz + ǫi ηˆi a ϕi ǫi ˜ ηˆi ∂χi ∂t ∂t L zB zB  

 zT i  z ˜ ∂χ ¯ η˜i ∂ Fi ∂χiB ′ ˜ + + ǫi bϕi ǫi ηˆi dz dz. ηˆi ∂χi ∂t ∂t L zB zB

(5.22d)

In this section, a general formulation for convection-parameterization closure has been considered. For this purpose, the formulation is gradually generalized, and the need for such a stepwise generalization may by itself important to emphasize. At first sight, it may appear that a closure based on a vertical integral as in Eq. 3.2 is already fairly general. However, in order to recover Arakawa and Schubert’s (1974) convective quasi-equilibrium closure (cf., Sec. 8), the formulation must be further generalized, which includes a subtle role for a forcing term in the convective-scale variable equation, as seen in the last subsection. 6

Reduction of spectral closure to bulk closure

Some possible advantages of taking a spectral formulation have been discussed in Ch. 9 as well as Sec. 2 of this chapter. The general closure formulation presented above has taken a spectral formulation partially for these reasons. At the same time, the discussions have also recognized that the parameterization would become much more compact if a spectral formulation can be reduced into a bulk form. This section asks the same questions of reduction as in Ch. 9, but here in the context of the closure problem. For the closure of a spectral parameterization the stationarity of a vertical integral Ii needs to be specified for each convective type i. As discussed in Sec. 2.2, this can be formulated for practical calculations as the construction of a normalization transformation (i.e., a set of rescaling coefficients for each cloud type) to be applied to some first-guess distribution. For the closure of a bulk parameterization, a global transformation (single-valued rescaling) is sufficient to provide the rescaling of a first-guess cloud-base

Closure

347

mass flux. An underlying spectral distribution of convection types is implicit in the choice of the entrainment and detrainment rates. We consider here whether closure conditions can be formulated in such a way that the closure of bulk and spectral formulations can be made equivalent. More specifically, suppose that the closure of a spectral formulation is given. Is it then possible to develop a consistent closure of a corresponding bulk formulation, in the sense that it would respect the following, apparently simple conditions: (1) given a method to generate a suitable normalization transformation for the spectral model, it must be possible to obtain from that a method that yields a global transformation that can be used to close the bulk model (i.e., it should be possible to make a direct link between the two closures). (2) the method used to generate the global transformation that closes the bulk model should respect all of the same physical constraints used to formulate the closure of the spectral model.

6.1

Closure based on Ibulk

Consider the quasi-equilibrium of the integral quantity Ii . The integral Ii itself is normalization invariant, and, as seen in Sec. 2.2, its time derivative has contributions which are normalization invariant (the forcing terms) and which are globally rescaled (the convective terms). Thus, timescales τLS and τadj are appropriate for the forcing and convective terms, respectively, and the physical constraint imposed is the separation of those timescales τLS ≫τadj . The closure transformation (Sec. 9.1, Eq. 9.3) can be constructed from this constraint. The budget equation for the integral quantity Ii has the general form: ∂Ii = FIi ,L − DIi , ∂t

(6.1)

as stated in Sec. 2, with FIi ,L the forcing and DIi the convective terms. As demonstrated in Sec. 5, the budget can be expressed more explictly in the form of Eq. 5.7. This equation contains contributions from variations in the convective base and convective top that can be incorporated into the definition of the forcing terms, in line with the above prescription for the distinction of the two sets of terms. Thus, we are led to consider a budget

348

Part II: Mass-flux parameterization

for Ii in the form:  ∂Ii = Kij MjB + FIi ,L . ∂t j

(6.2)

Recalling the mass-flux weighting operation of the bulk approach (see Ch. 9, particularly Ch. 9, Eq. 3.1) a bulk form of the integral quantity can be introduced as:  MiB Ii , (6.3) Ibulk ≡ i MB  where MB = i MiB . It is immediately clear from the definition that Ibulk is a globally invariant quantity. Its time derivative is easily obtained as:   ∂MiB ∂Ii 1  1  MiB ∂MB ∂Ibulk = + − MiB Ii . (6.4) ∂t MB i ∂t MB i ∂t MB ∂t

The important point to be made is that this time derivative cannot be decomposed into normalization-invariant and globally rescaled terms. As a consequence, an assumption of its stationarity does not correspond to quite the same separation of timescales as implied by the stationarity of each of the integral quantities Ii . For example, consider the first term on the right-hand side of Eq. 6.4. Substituting from Eq. 6.2, this reads: ⎞ ⎛  ∂Ibulk 1  1  ∂Ii Kij MjB + FIi ,L ⎠+· · · = +· · · = MiB MiB ⎝ ∂t MB i ∂t MB i j

(6.5)  K M is a globally rescaled variable and the The combination ij jB j weighted sum of this combination over convective types i likewise produces a globally rescaled contribution to ∂Ibulk /∂t, associated with the timescale τadj . However, the normalization-invariant variable FIi ,L does not produce normalization-invariant contributions to ∂Ibulk /∂t after the weighted averaging. ˜ϕ B from the To illustrate this, the contribution to ∂Ibulk /∂t involving b i first term on the first line of Eq. 5.22d can now be stated explicitly. This is:  ∂ϕiB zT i 1 ∂f2 1  ∂Ibulk = MiB dz + · · · (6.6) ∂t MB i ∂t zci η˜i ∂ϕi The integral over z has an integrand that is normalization invariant, and limits that depend on the convection type. In particular, the upper integration limit may be markedly different for the different convection types.

Closure

349

The overall contribution to ∂Ibulk /∂t is therefore globally invariant, and cannot be evaluated without knowledge of the full plume spectrum. The time derivative of Ibulk is composed of globally invariant and globally rescaled terms, such that its stationarity could be used to close a bulk parameterization on the assumption of the corresponding timescale separation τspec ≫τadj . Such a closure would satisfy condition (1) above, as it is obtained from the closure of the spectral formulation. However, it does not satisfy condition (2), as it is not based on the same timescale separation as for the spectral formulation, which relies on τLS ≫τadj . It must be stressed that the quasi-equilibrium of Ibulk is not necessarily problematic: it could be more or less physically plausible that the quasiequilibrium of the set of Ii , or it may be equally plausible with τspec ≈ τLS . The point is simply that quasi-equilibrium of Ibulk is not a fully equivalent closure constraint for a bulk parameterization to the corresponding closure constraint for the spectral parameterization. The closure has to be reconsidered and reassessed for the two formulations. 6.2

Closure based on I0

It has been shown that a bulk value of the integrated closure quantity Ii may not be used to close a bulk parameterization in a manner that is equivalent to the quasi-equilibrium closure of the corresponding spectral parameterization. However, there could be other ways in which to close a bulk parameterization which may be physically consistent with the spectral parameterization closure. Let us consider the use of some particular integral quantity Ii for the bulk parameterization closure. In other words, we select one particular convective type as having special status, labelling that special type with i = 0, and then use I0 for the bulk closure. This is an interesting possibility to consider partly because the standard CAPE is a special case of the cloudwork function for the particular case of a non-entraining plume. The decomposition of ∂I0 /∂t into large-scale and convective contributions applies just as for any other Ii , and so trivially a quasi-equilibrium closure based on ∂I0 /∂t≈0 would be physically based on τLS ≫τadj , thereby satisfying condition (2) for an equivalent closure of a bulk parameterization. However, we also need to consider condition (1): whether the I0 closure can allow us to compute the global transformation that allows determination of the rescaled MB . Let us examine first FI0 ,L , the normalization-invariant/large-scale forc-

350

Part II: Mass-flux parameterization

ing part of ∂I0 /∂t. One of the contributions to this is analogous to the term shown explicitly in Eq. 6.6 and is specifically:  ∂ϕ0B zT 0 1 ∂f2 ∂I0 = dz + · · · (6.7) ∂t ∂t ˜0 ∂ϕ0 zc0 η The explicit form of terms in FI0 ,L would not normally be used in a parameterization. However, in order for a closure based on I0 to satisfy condition (1), then it must be possible in principle to evaluate all such terms directly using a bulk model. For a general integral quantity I, that is clearly not the case, as the above example shows. However, the examination of all of the forcing terms arising from an ensemble of entraining plumes reveals that this is the case if Ii is taken as the cloud-work function Ai and I0 is taken as the CAPE (Plant, 2010). Let us therefore consider the special case of the stationarity of CAPE in a little more detail. Turning to the globally rescaled/convective terms in ∂CAPE/∂t, a careful inspection (as in Sec. 8 below, remembering to set ǫi = 0 and ηi = 1 to obtain the appropriate results for CAPE) shows that many convective terms can indeed be evaluated based on knowledge that would be available in a bulk treatment of an underlying spectral formulation: i.e., the environmental sounding and the total mass flux Mc (z). However, there are also convective terms that are associated with detrainment. Although the total detrainment profile D(z) is known in a bulk treatment, the terms also depend upon the values of convective variables on detrainment. As discussed in full detail in Ch. 9, these values are problematic, and particularly the cloud liquid water on detrainment which can only be computed by integrating the budget equations for a single plume that reaches its level of neutral buoyancy at the level in question (Ch. 9, Sec. 5.1). Thus, strictly speaking, the stationarity of CAPE does not satisfy condition (1) for a valid equivalent closure of a bulk parameterization. This issue can be avoided by invoking the liquid-water detrainment ansatz of Yanai et al. (1973) in which the detraining liquid water is arbitrarily set equal to the bulk-plume liquid water (Ch. 9, Eq. 5.11). This ansatz is required not only to compute the vertical profile of the equivalent bulk-plume but also in order to permit an equivalent CAPE closure for the bulk treatment of a spectrum of plumes. The practical impact of the ansatz on closure calculations is difficult to discern: certainly the authors are unaware of any attempt in the literature to assess the impact. As will be discussed in Sec. 11, various authors (e.g., Kain, 2004; Kain et al., 2003; Zhang, 2009) have investigated the use of dilute CAPE for the

Closure

351

closure of bulk parameterization. That does not alter the main arguments presented in this subsection. 6.3

Summary remarks on bulk and spectral parameterizations

The discussion in this section has focused on the formal validity (or otherwise) of the closure formulations that might be obtained to derive global transformations for bulk parameterizations. Closures based on the stationarity of some quantity such as CAPE, or a cloud-work function, assume a timescale separation (cf., Ch. 4) between the slow mechanisms of atmospheric destablization and the relatively fast mechanisms of the convective response. If the stationary quantity is defined for each convective type then the natural analogue for a bulk closure condition is the stationarity of Ibulk . However, it has been shown that the timescales associated with the evolution of Ibulk are not necessarily the same as those associated with Ii . This timescale issue can be avoided in the particular case of a bulk parameterization based on CAPE. However, a point of difficulty for any bulk parameterization is the detrainment of condensate (Ch. 9), and this enters into considerations of the rate of change for any environmental instability measure when the vertically integrated measure encompasses the detrainment layer of any plume within the (presumed) spectrum of convective types. CAPE is clearly no exception to this statement and so a condensate detrainment ansatz such as Yanai et al. (1973) becomes a necessary ingredient which is buried within the CAPE closure of a bulk parameterization. Thus, a bulk treatment of an underlying spectral convective parameterization cannot be closed using either a bulk integration quantity Ibulk or a specific integration quantity I0 in a way that is completely consistent with the corresponding direct closure of the spectral parameterization. It is important, however, that these caveats and cautions about bulk parameterization closure should not leave the reader with the impression that closure of a spectral parameterization is somehow a simple matter. A physical constraint that ultimately provides a single number to rescale Mc (zB ) is sufficient to close a bulk parameterization, but would provide none of the necessary information required by a spectral parameterization about the spectral distribution of the mass flux. Some spectral parameterizations apply a set of physical constraints (e.g., use of the stationarity of a set of integral quantities {Ii }) to generate explicitly a suitable normalization transformation that rescales the mass flux for each convective type

352

Part II: Mass-flux parameterization

(Arakawa and Schubert, 1974; Nober and Graf, 2005). Other parameterizations combine instead a global transformation for an overall rescaling with some additional constraints to determine the relative contributions of the different types (cf., Ch. 9, Sec. 7). This might be by appeal to observations (e.g., Donner et al., 1993), or by making theoretical arguments (e.g., Plant and Craig, 2008), or even “mainly for the sake of simplicity” (Zhang and McFarlane, 1995, p. 412). Regardless of the approach taken, however, setting the spectral distribution is not trivial. 7

Kuo’s (1974) moisture-based closure

The moisture-based closure proposed by Kuo (1974) can be considered an example of a closure solely based on large-scale variables as introduced in Sec. 3.1. It is a popular idea that the column-integrated water vapour  zT ρ¯ q dz (7.1) Iq = zB

controls convection, and Kuo’s (1974) closure is essentially recovered by assuming stationarity of this quantity. The range of the vertical integral is usually taken from the surface to the top of the atmosphere, although other interpretations could be taken in the following. By referring to Eqs. 5.7 and 5.8a–d in Sec. 5.1, the tendency equation for Iq is given by:   ∂Iq ∂Iq = KMB + + z˙T ρT qT − z˙B ρB qB , (7.2) ∂t ∂t L where K=



zT

ηc

zB



∂Iq ∂t



L



=

∂ q¯ δc (qc − q¯) + dz ∂z 

zT

zB

ρ



∂ q¯ ∂t



dz.

(7.2a)

(7.2b)

L

The index i for a convection type is dropped for now, or else it is replaced by a subscript c whenever helpful for clarity. In the following, it will also simply be assumed that z˙T = z˙B = 0. The large-scale tendency of moisture may be written as:   ∂ q¯ 1 ∂ 1 ∂ ρw ¯ q¯ − ρw′ q ′ , = −∇ · u ¯ q¯ − ∂t L ρ ∂z ρ ∂z

Closure

353

the third term being the eddy moisture flux. It substitution into Eq. 7.2b leads to:

   zT ∂Iq ∂ ρw ¯ q¯ dz + E, (7.3) =− ∇ · ρ¯ uq¯ + ∂t L ∂z zB where the vertical integral of the third term has been replaced by E, which is simply the surface evaporation rate if zB and zT are taken to be the Earth’s surface and the top of the atmosphere respectively. By assuming the steadiness of Iq , we obtain a moisture-based closure:   1 ∂Iq MB = − . K ∂t L Under this closure, the tendency for the column–integrated water vapour becomes totally stationary, which is unrealistic. For this reason, in his original formulation Kuo (1974) introduces a major provision in making the column-integrated water vapour tendency slightly non-stationary by setting:   ∂Iq ∂Iq =b . (7.4) ∂t ∂t L Here, b is a small positive parameter that controls this weak unsteadiness (0 < b ≪ 1). After this modification, the closure changes to:   1 − b ∂Iq . (7.5) MB = − K ∂t L The major interest of Kuo (1974) was to obtain the convective moisture tendency (∂ q¯/∂t)c , although this is misleadingly called the “convective heating rate” after multiplying it by L/Cp . By referring to Ch. 8, Eq. 3.6b, it is given by:

  ∂ q¯ ∂ q¯ 1 D + Dc (qc − q¯) . = Mc ∂t c ρ ∂z This may be expressed as:     ∂ q¯ 1 − b ∂Iq = −f˜(z) , ∂t c K ∂t L where

∂ q¯ 1 D ˜ f (z) = ηc + δc (qc − q¯) . ρ ∂z

(7.6a)

(7.6b)

Eq. 7.6a can be considered the major conclusion of Kuo (1974). In Kuo’s original formulation, the vertical profile f˜(z) is determined in a rather

354

Part II: Mass-flux parameterization

arbitrary manner. It is also often criticized that the small parameter b remains arbitrary. However, these weaknesses of the original formulation should not discredit the overall soundness of Kuo’s (1974) moisture-based closure. Furthermore, once Kuo’s (1974) closure is recast into the mass-flux framework as presented here, these weaknesses turn out be to be less serious than they might at first appear: the issue of the vertical profile f˜(z) simply reduces to that of determining a vertical profile for the mass flux ηc , as can be seen from Eq. 7.6b. The introduction of the parameter b may be judged more positively as a simple attempt to overcome the limit of a strict diagnostic closure condition, as stated by Eqs. 2.1 or 3.3. An alternative approach towards this goal is discussed in Sec. 11 Kuo’s formulation has been further pursued by, for example, Anthes (1977), Krishnamurti et al. (1976) and Molinari (1985). 7.1

Bougeault (1985)

Bougeault (1985) may be considered as an implementation of Kuo’s (1974) idea into the mass-flux formulation. His first attempt was to modify Kuo’s (1974) closure so that the moistening by detrained air Dc (qcD − q¯) did not contribute as a part of the closure balance as in Eq. 7.6b above, but simply acted to increase the large-scale moisture. Thus: ∂ q¯ = Dc (qcD − q¯). ∂t As a result, the closure of Eq. 7.2 with b = 0, reduces to:    zT ∂Iq ∂ q¯ η dz = − MB . ∂z ∂t L zB However, Bougeault’s (1985) formulation differs more in that he also decides to define the height-independent detrainment Dc with an additional closure condition. Specifically, it is assumed that the convective tendency for the moist static energy h is held stationary with time:

 zT  zT  ¯  ∂ ¯h ∂h D ¯ + Dc (hc − h) dz = 0. Mc dz = (7.7) ρ ∂t c ∂z zB zB This condition is used to determine the detrainment rate Dc , given the mass flux Mc . Note that this additional closure condition in Eq. 7.7 is an example of a diagnostic closure without a prognostic counterpart, and so does not satisfy Dictum 1 of Sec. 2.4.

Closure

8

355

Arakawa and Schubert’s (1974) convective quasi-equilibrium closure

The convective quasi-equilibrium closure proposed by Arakawa and Schubert (1974) can also be considered as a special case of the general closure formulation presented in Sec. 5. The choice of Ii is based on analysis of the convective kinetic energy budget, a strategy that implicitly follows Dictum 2 of Sec. 2.4. As will be derived in Sec. 10, the convective kinetic energy budget is given by: ∂Ki = MB,i Ai − Di , (8.1) ∂t where Ki is the convective kinetic energy for the i-the convection type, Di is a corresponding convective dissipation rate, and Ai is a quantity called the “cloud-work function” by Arakawa and Schubert (1974). It is seen that the quantity Ai controls the growth of convective kinetic energy. The cloud-work function is defined by:  zT i ηi bi dz (8.2) Ai = zB

in terms of the convective buoyancy for the i-th convection type bi = ρα(sv,i − s¯v )

(8.3)

with α=

g . ρCp T¯v

(8.4)

Note that Arakawa and Schubert (1974) apply the approximation T¯v ∼ T¯ in the definition above. In Eq. 8.3, sv is the virtual static energy sv = Cp Tv + gz,

(8.5)

where Cp is the specific heat at constant pressure, g is the acceleration due to gravity, and Tv is the virtual temperature defined by: ˆ − qc )T, Tv = (1 + δq

(8.6)

with δˆ = Rv /Rd − 1 defined in terms of the gas constants for dry air Rd , and water vapour Rv . Thus, Arakawa and Schubert’s (1974) closure condition can be rederived by taking Ai (Eq. 8.2) for Ii in Sec. 5. In doing so, it turns out that the virtual static energy sv is not a convenient variable to work with above the

356

Part II: Mass-flux parameterization

condensation level zci , because it is no longer conserved. For this reason, above the condensation level, the convective buoyancy bi is rewritten, by invoking the relation of Ch. 8, Eq. 5.4: ¯ ∗ ) + ρα˜ ˆ q ∗ − q¯) − qc,i ], bi = ρβ(hi − h εLv [δ(¯ where β=α



Lv γ= Cp

1 + γ ε˜δˆ 1+γ 

∂ q¯∗ ∂ T¯





,

(8.7)

(8.8)

,

p¯

and ε˜ =

Cp T¯ . Lv

Thus the cloud-work function defined by Eq. 8.2 reduces to a vertical integral of the form of Eq. 3.5 with the two functions being most conveniently defined by: f1 = ραηi (sv,i − s¯v )

(8.9a)

¯ ∗ ) + ραηi ε˜Lv [δ(¯ ˆ q ∗ − q¯) − qc,i ]. f2 = ρβηi (hi − h

(8.9b)

Here, note ρα˜ εLv = g/(1 + δˆq¯). In the following, the variables α and β, associated with the large-scale virtual temperature profile T¯v , are treated as constant with time. Such an approximation is consistent with the free ride principle proposed by Fraedrich and McBride (1989) or the weak temperature gradient approximation of Sobel et al. (2001). Another way to look at this approximation is to transform the integrals appearing in the closure equations into pressure coordinates by invoking hydrostatic balance: dp = −ρgdz. With help from the ideal gas law p = ρRd T¯v , we find that dz = −Rd d ln p. T¯v

Closure

357

Thus, the major contribution from changes in α could be absorbed into a part of the integration variable, and as a result the change of α with time would only appear in the top and the bottom boundary conditions. The derivation of the closure condition is essentially the same as in Sec. 5. However, as a major difference, it is necessary to take care of the multi-variable dependence of the functions f1 and f2 . Thus, the time derivative of the cloud-work function reads:

 zci ∂f1 ∂svi sv ∂f1 ∂¯ ∂f1 ∂ηi ∂Ai = + + dz ∂t ∂svi ∂t ∂¯ sv ∂t ∂ηi ∂t zB  +

zT i

zci



∂f2 ∂hi ∂f2 ∂qci ∂f2 ∂ ¯h∗ ∂f2 ∂ q¯∗ ∂f2 ∂ q¯ ∂f2 ∂ηi + + ¯∗ + ∗ + + dz ∂hi ∂t ∂qci ∂t ∂ q¯ ∂t ∂ q¯ ∂t ∂ηi ∂t ∂ h ∂t −z˙B f1 (z = zB ) + z˙T i f2 (z = zT i ).

(8.10)

The forms of the functional derivatives are straightforward to derive from the definitions of f1 and f2 . Here, by noting that an analogous relationship to Ch. 8, Eq. 5.4 also holds for the time derivative, we find: ∂f2 ∂ q¯∗ ∂f2 ∂ q¯ ∂f1 ∂¯ ∂f2 ∂ ¯ sv h∗ + + = . ∂ q¯∗ ∂t ∂ q¯ ∂t ∂¯ sv ∂t ∂¯ h∗ ∂t The expression for Eq. 8.10 is then greatly simplified, and we obtain:

 zci  zT i  zci ∂Ai ∂f2 ∂hi ∂f2 ∂qci ∂f1 ∂svi ∂f1 ∂ηi = dz + + dz dz + ∂t ∂s ∂t ∂h ∂t ∂q ∂t vi i ci zB zci zB ∂ηi ∂t 

zT i



zT i

∂f1 ∂¯ sv dz − z˙B f1 (z = zB ) + z˙T i f2 (z = zT i ). ∂¯ sv ∂t zB zc i (8.11) Following the procedure of Sec. 5, the next step is to derive a closure expression, assuming at least for the present that all terms of the form F˜i can be considered purely as large-scale forcings. As a result, we obtain a relation: ∂Ai  † = Kij MjB + z˙T i GT i + z˙B GBi + FIi . (8.12) ∂t j +

∂f2 ∂ηi dz + ∂ηi ∂t

† Here, Kij is a provisionally obtained interaction matrix, and the forcing term FIi is actually found to contain two distinctive contributions: those coming from the large-scale and those from the convective processes. Thus:

FIi = FI,L,i + FI,c,i .

(8.13)

358

Part II: Mass-flux parameterization

The final step then required is to rewrite the convective contributions to the forcing term into a matrix form:  ΔKij MjB , (8.14) FI,c,i = j

by following the procedure of Sec. 5.3. It is straightforward to obtain each term in Eq. 8.12 by following the procedure already given by Eq. 5.12, while taking account of the multivariable nature of the problem. The results are:

 zci 1 ∂¯ sv † ǫi ˜asvi η˜svi ηj δj (sD = − s ¯ ) + Kij dz v vj ∂z zB ρ  zT i  ˜ ¯

ǫ i b hi ¯ + ∂h η˜hi ηj δj (hD − h) + j ρ ∂z zB 

bqci ∂ q¯c ǫi ˜ D η˜qci ηj δj (qcj − q¯c ) + + dz ρ ∂z

 zT i 1 ∂f1 ∂¯ sv D ηj δj (svj − s¯v ) + dz, (8.15a) + ρ ∂¯ sv ∂z zB (8.15b) GT i = f2 (z = zT i ), 

 ρσi ˆ Fs sviB + sviB GBi = − (ǫiB − δiB )Ai − ˜asvi B ǫiB Δ˜ Mi svi vi B 

 ˜ iB + ρσi Fˆh hiB −˜ bhi B ǫiB Δh M i hi i B

  ρσi ˆ ˜ Fq qciB − f1 (z = zB ), (8.15c) qciB + − bqci B ǫiB Δ˜ Mi qci ci B     ∂ ˜h B ∂ − ˜dh hiB asvi B − ˜csvi sviB + b FIi = ˜ i i ∂t ∂t      zci ∂ s˜¯vi ∂ dqci qciB + ǫi ˜asvi η˜svi + ˜ bqci B − ˜ dz ∂t ∂t L zB      zT i  zT i ∂ q˜¯ci ∂ ˜¯hi ˜ ˜ ǫi η˜qci bqci dz + + ǫi η˜hi bhi dz ∂t ∂t L zB zB L    zT i sv ∂f1 ∂¯ + dz. (8.15d) ∂¯ sv ∂t L zB

Let us now consider the treatment of forcing terms for the convectivescale variables. The most static energy hi is the simplest since this is conserved, and so the associated entrainment ǫ˜hi = ǫi , the hat-forcing Fˆhi = 0,

Closure

359

and ˜ dhi = 0. For the virtual static energy svi there may be a forcing due to evaporation and this is partitioned as being an effective modification of svi rather than as an effective entrainment. Thus, the associated entrainment ǫ˜svi = ǫi , the hat-forcing Fˆsvi = 0, and ˜csv i = 0, while the profile of svi itself is determined by:   ∂ + ǫi svi = ǫi s˜¯vi , (8.16a) ∂z

where

ˆ ρσi ei . s˜ ¯vi = s¯v − Lv (1 − ε˜δ) (8.16b) ǫi Mi Here, ei is the evaporation rate from the i-th convective element, and it shoud be noted that Arakawa and Schubert (1974) assume ei = 0. Furthermore, from the definition of f1 , note that: ∂f1 = −ραηi . ∂¯ sv A major additional hidden contribution from the convective-scale stems from the term involving (∂ q˜¯ci /∂t)L in Eq. 8.15d. A closer look at the convective-scale cloud-water budget is required in order to obtain an explicit form for this term. This is facilitated by examining the convective totalwater budget. The only sink term for convective total-water qti is the precipitation ri . Thus:   ∂ ρσi + ǫi qti = ǫi q¯t − ri , (8.17a) ∂z ǫi Mi where qti = qi + qci and q¯t = q¯ + q¯c . A division into two contributions is considered for the precipitation rate ri of the i-th convective type by setting: ρσi ρσi ri = c˜0 qci + r˜i . (8.17b) Mi Mi Thus, the precipitation is treated as being potentially an effective entrainment, potentially as an external forcing modifying qti , and potentially as some combination of the two. Arakawa and Schubert (1974) took the first of these options and set c˜0 to be a constant as well as r˜i = 0. The additional term r˜i = 0 is introduced here to provide possible further freedom for the convective precipitation rate formulation. Rewriting Eq. 8.17a as an equation for the convective cloud water qci , it takes the form:   ∂ + ǫ˜i qci = ǫi q˜¯ci , (8.18) ∂z

360

Part II: Mass-flux parameterization

where ǫ˜i = ǫi + c˜0 and q˜ ¯ci = q¯ + q¯c −



 1 ∂ ρσi r˜i . + 1 qi − ǫi ∂z ǫi Mi

(8.19a)

(8.19b)

Partitioning the forcing in this way means that, according to Eq. 4.9:  z

′ η˜i = exp (8.20a) ǫi dz = ηˆi zB

in Eq. 8.15 for all of the variables except for qci for which we have:  z

′ η˜qci = exp ˜ǫi dz . (8.20b) zB

According to Eq. 8.19b above, q¯˜ci depends on q¯ and qi . Thus, the tendencies in these variables must be taken into account in computing the tendency for q˜ ¯ci . According to the workings of Sec. 5.3, (∂ F˜qci /∂t)c must be evaluated, which is given by:   

 ∂ 1 ∂ ∂ F˜qci = + 1 qi q¯ − ∂t ∂t ǫi ∂z c c

by referring to Eq. 8.19b. As it turns out, working directly on those tendencies is not entirely convenient, because the moisture is not a conserved quantity. Rather, following the approach of Arakawa and Schubert (1974), these tendencies are rewritten in terms of those for dry and moist static energies: 

   ∂ ¯h 1 ∂¯ 1 ∂ γ ∂hi s 1 1 ∂ ∂ + 1 qi = − + − . q¯ − ∂t ǫi ∂z ǫi Lv ∂z 1 + γ ∂t Lv (1 + γ) ∂t Lv ∂t (8.21) The derivation is given in Sec. 16. As a result, we obtain:         ¯ s 1 ∂ γ ∂hi ∂h 1 1 ∂¯ ∂ F˜qci =− + − , ∂t ǫi Lv ∂z 1 + γ ∂t c Lv (1+γ) ∂t c Lv ∂t c c,c

and corresponding to Eq. 5.17 in Sec. 5.3, we obtain:      ¯   zT i  zT i ∂h ∂¯ s ηi di ΔFIi = ǫi ηˆi ci − ǫi di ηi ρdz + ρdz 1 + γ ηˆi ∂t c ∂t c zB zB (8.22)

361

Closure

for a correction to the forcing term. Note that in order to obtain this final result, (∂hi /∂t)c is expressed in an analogous manner to Eq. 5.19a. Here, some coefficients are introduced by: 

 z′  zT i g 1 exp −˜ c0 (z ′ − z) − δi dz ′′ dz ′ (8.23a) di = ρLv max(z,zci ) 1 + δ q¯ z 1 ci = ρ



z

zT i

ηi ∂ ρdi ηˆi ∂z



γ 1+γ



dz ′ .

(8.23b)

These two definitions may be considered as generalizations respectively of Eqs. B20 and B19 of Arakawa and Schubert (1974). Here, note that bqci η˜qci = −ρLv ηi di ., where bqci is a further coefficient to be defined shortly below. Further, recall that:

 ¯ ∂ ¯h 1 ∂h D ¯ MjB ηj δj (hj − h) + = ∂t c ρ j ∂z 

∂¯ s ∂t



=

c



∂¯ s 1 − s ¯ ) + MjB ηj δj (sD j ρ j ∂z

according to Ch. 8, Eqs. 3.6a and b. ΔFI,i,L can also be evaluated by following the method shown in Eqs. 5.19b and 5.21b in Sec. 5.3. After putting all these calculations together, the final result is:  

 zT i ηˆi ∂¯ sv D ηi ηj −α + ǫi ai δj (svj − s¯v ) + dz Kij = ηi ∂z zB  

 zT i ηi ∂ ¯h di D ¯ dz δj (hj − h) + ǫi ηˆi ηj bi + ci − + 1 + γ ηˆi ∂z zB

 zT i ∂¯ s ¯) + ǫi di ηi ηj δj (sD + dz j −s ∂z zB

 zT i ∂ q¯c D − q¯c ) + ǫi Lv di ηi ηj δj (qcj − dz, (8.24a) ∂z zB FL = z˙T i GT i + z˙B GBi + FI,L,i ,

(8.24b)

GT i = −ραηi (¯ sv − svi )|z=zT i

(8.25a)

where

GBi = −(ǫiB − δiB )Ai

362

Part II: Mass-flux parameterization

−ρB [ǫiB {aiB Δ˜ sviB + (biB + ciB )ΔhiB − Lv diB Δ˜ qciB } − αB ηiB ΔsviB +˜ c0 Lv diB qciB ]

FI,L,i

(8.25b)

∂ ∂ ∂ =ρB aiB sviB + (biB + ciB ) hiB − Lv diB qciB ∂t ∂t ∂t   zT i 

ηi ρLv di + c˜˙0 dz qciB η˜qci zB    

 zci ∂ s˜ ¯vi ∂¯ sv + ǫi ai ηˆi − αηi ρdz ∂t L ∂t L zB    ¯   zT i ∂h ηi di ρdz ǫi ηˆi bi + ci − + 1 + γ η ˆ ∂t L i zB      zT i  zT i ∂¯ s ∂ q¯c + ǫi di ηi ǫi Lv di ηi ρdz − ρdz ∂t L ∂t L zB zB    zT i  zT i  zT i ρσi ∂ ǫi Lv di ηi r˜i ρdz + c˜˙0 dz ′ ρdz ǫi Lv di ηi + ∂t ǫi Mi z zB zB (8.25c)

in which further coefficients have been introduced by:  1 zci ηi ρα dz ′ , ai = ρ min(z,zci ) ηˆi  1 zT i ηi bi = ρβ dz ′ . ρ max(z,zci ) ηˆi

(8.26a) (8.26b)

Recall that ci and di have been defined by Eqs. 8.23b and a. For completeness, there follows a list of the various values of convective variables on detrainment (cf., Ch. 8, Secs. 4 and 5):  si − Lv qci if q¯c = 0 (8.27a) sD i = si if q¯c = 0  qi + qci if q¯c = 0 D (8.27b) qi = if q¯c = 0 qi ⎧ ˆ v qi + Lv (˜ ⎪ si + ε˜δL εδˆ − 1)qci if q¯c = 0 ⎪ ⎪ ⎪ ⎨s¯ + L [˜ ˆ if q¯c = 0, and with neutral buoyancy v v ε(1 + δ) − 1]qci sD vi = ˆ i − qci ) ⎪ if q¯c = 0 ⎪si + ε˜Lv (δq ⎪ ⎪ ⎩ s¯v if q¯c = 0, and with neutral buoyancy (8.27c)

Closure

363

D D and clearly, hD i = si + Lv qi . The expressions obtained above are generalizations of Eqs. B17 and B18 of Arakawa and Schubert (1974). Notably, by assuming an entrainingplume in Arakawa and Schubert (1974), η˜i = ηˆi = ηi . Also note that GT i = 0 when the convection top is defined by the level of neutral buoyancy as in Arakawa and Schubert (1974). Moreover, according to Arakawa and Schubert (1974), c˜0 is a fixed constant, and thus c˜˙0 = 0, although again this term is retained for generality above.

8.1

Precipitation forcing

By assuming a generality of the precipitation formula in Eq. 8.17b, we find an additional term due to r˜i = 0 in the forcing term: the temporal tendency ∂˜ ri /∂t of the precipitation rate becomes a part of large-scale forcing (i.e., precipitation forcing). As it turns out, the order of magnitude of precipitation forcing is comparable to that of other aspects of the standard large-scale forcing. As shown in Eq. 8.25c, precipitation forcing is defined by:    zT i ∂ r˜i ǫi di ηi ρdz. (8.28) Lv ∂t ǫi wi zB Recall that r˜i measures a convective-scale precipitation formation rate as defined by Eq. 7.19, and wi is the convective vertical velocity. However, r˜i /wi is a rather non-trivial variable to interpret, with a unit of m−1 or gkg−1 m−1 , depending on the unit taken for the water mixing ratio. This is essentially a vertical gradient of the precipitating water generation rate. Only after multiplying by wi does the quantity reduces to a rate at which precipitating water is being generated at a given vertical level per unit time (with the unit of s−1 or gkg−1 s−1 ). The corresponding total convective precipitation is given by:  zT  1 r˜i dz, (8.29) ρa P = ρw z B i where ρa and ρw are the air and liquid water densities. The extent of the vertical integral in Eqs. 8.28 and 8.29 is defined by the vertical extent of the convection. A typical tropical precipitation rate is P ∼ 10 mmh−1 ∼ 3 × 10−6 ms−1 . The precipitation rate due to the i-th convective type may be, to an order of magnitude, estimated from Eq. 8.29 as: ρa r˜i Hi , Pi ∼ ρw

364

Part II: Mass-flux parameterization

with Hi providing a vertical scale for the convection. A typical value for r˜i /wi is then estimated as: Pi 3 × 10−6 ms−1 r˜i ∼ 3 × 10−7 m−1 . ∼ ∼ −3 wi (ρa /ρw )wi Hi 10 × 1 ms−1 × 104 m Here, it has been assumed that wi ∼ 1 ms−1 and Hi ∼ 104 m. An assumption behind this estimate is that the order of magnitude of the i-th convective precipitation is of the same order as the total. Next, note that the precipitation forcing given by Eq. 8.28 is controlled by a temporal change ∂(˜ ri /wi )/∂t of the precipitation formation measure. In order to estimate the latter, a characteristic timescale τ is introduced for convective precipitation formation. The two possible values τ ∼ 1 h∼ 3 × 103 s and τ ∼ 1 day∼ 105 s are considered. These lead to the estimates: ∂ r˜i r˜i /wi 3 × 10−7 m−1 ∼ ∼ 10−10 m−1 s−1 ∼ ∂t wi τ 3 × 103 s and r˜i /wi 3 × 10−7 m−1 ∂ r˜i ∼ ∼ 3 × 10−12 m−1 s−1 . ∼ ∂t wi τ 105 s Additionally, an order of magnitude estimate is needed for Lv di defined by Eq. 8.23a, which is given by: Lv di ∼

g 10 ms−2 × 104 m Hi ∼ ∼ 105 m5 s−2 kg−1 , ρ 1 kgm−3

assuming c˜0 = 0 and δi ∼ 0 (assuming an entraining plume, this term contributes only a factor of unity to the integrand). Finally, we obtain the order of magnitude estimate for precipitation forcing as:      r˜i r˜i ∂ ∂ dz ∼ ρLv di Hi Lv ρηi di ∂t wi ∂t wi ∼ 1 kgm−3 × 105 m5 s−2 kg−1 × 10−10 m−1 s−1 × 104 m ∼ 10−1 Jkg−1 s−1 ∼ 104 Jkg−1 day−1 with τ ∼ 1 h, and Lv



ρηi di

∂ ∂t



r˜i wi



dz ∼ ρLv di

∂ ∂t



r˜i wi



Hi

∼ 1 kgm−3 × 105 m5 s−2 kg−1 × 3 × 10−12 m−1 s−1 × 104 m

Closure

365

∼ 3 × 10−3 Jkg−1 s−1 ∼ 300 Jkg−1 day−1 with τ ∼ 1 day. These estimates are comparable to an order of magnitude estimate for large-scale forcing F ∼ 103 Jkg−1 day−1 (cf., Yano and Plant, 2012b). The very last estimate can also be obtained by recalling a typical value for CAPE (convective available potential energy) ∼ 103 Jkg−1 for the tropics as well as assuming a characteristic timescale of 1 day. The convective precipitation formulation based on the precipitation efficiency introduced in Ch. 8, Sec. 7 may be best incorporated as a part of a precipitating forcing, because once a fractional entrainment rate is specified, as under an entraining-plume hypothesis, the precipitation measure r˜i /wi is also specified. In general, when a sophisticated convective precipitation formulation is adopted, it becomes increasingly difficult to incorporate this process as a part of the convective response within the interaction matrix Ki,j . It may be more straightforward to treat it as a part of the large-scale forcing from the point of view of studying the closure relation. Generally speaking, such a precipitation forcing cannot be fully determined until the full convective response is known, and thus the procedure for solving the closure problem and evaluating precipitation forcing becomes an iterative procedure. 9 9.1

Discussions Short summary

The major finding of the analysis so far is that regardless of the details of the vertical-integral constraint, a closure condition defined by a stationarity of a vertically integrated quantity, as given by Eq. 2.1 or Eq. 3.3, always reduces to a form given by Eq. 5.7:  Kij MjB + z˙T i GT i + z˙B GBi + FI,L,i = 0. j

The formulation is not quite closed yet, because we still need to establish expressions for z˙T i and z˙B . In the case of Arakawa and Schubert (1974), they assume that: z˙T i = 0 and z˙B = −

1 McB + (z˙B )L , ρB

366

Part II: Mass-flux parameterization

where (z˙B )L is a large-scale tendency for zB . Note that the latter formula is derived based on their own boundary-layer formulation, and indeed it transpires that the formula depends on the boundary-layer formulation adopted in a given system (cf., Sec. 11.4). Nevertheless, their example serves the purpose of suggesting that, in general, these two terms proportional to z˙T i and z˙B can be partitioned into convective and large-scale terms as long as a certain linearity is satisfied. As a result, the general closure condition reduces to:  Kij MjB + FI,L,i = 0 (9.1) j

by redefining these two terms accordingly. Comparing Eq. 9.1 to Eq. 2.3, we see that the convective consumption term is defined by:  DI,c,i = Kij MjB . (9.2) j

The closure condition may equally be presented in a vector matrix form as: KMB + FI,L = 0. In principle, the closure condition can be solved by inverting the matrix K to obtain: MB = −K−1 FI,L .

(9.3)

The quasi-equilibrium closure is schematically summarized in Fig. 11.1. 9.2

Operational implementation

Although the solution of Eq. 9.3 may appear straightforward, we have to take into account technical aspects such as the positiveness of the mass flux: MB,i ≥ 0. In order to overcome this difficulty, a rather involved procedure for solving Eq. 9.1 is proposed by Lord and Arakawa (1982) and Lord et al. (1982), as further discussed in Sec. 13.1. As an alternative approach, Moorthi and Suarez (1992) proposed to consider only the diagonal terms of K in order to simplify the procedure, and thus the solution in Eq. 9.3 is replaced by: FI,L,i . (9.4) Mi,B = − Ki,i They called this procedure the relaxed Arakawa and Schubert (RAS).

367

Closure

K λ3

^ K λ3M3

K λ2

Total ^ K λ2M2 convective = damping

K λ1 ^ M 1

^ M 2

^ M 3

......

Largescale forcing

^ K λ1M 1 Cloud-type λ

On cloud-type λ

Fig. 11.1 A schematic to illustrate the original concept of convective quasi-equilibrium (CQE) by Arakawa and Schubert (1974). Each convective cloud is characterized by ˆ 2, M ˆ 3 , . . . (left) and damps the cloud-work function for ˆ 1, M its cloud-base mass flux, M ˆ 3 , . . . CQE is the assumption ˆ 2 , Kλ3 M ˆ 1 , Kλ2 M cloud-type λ (centre) with a rate Kλ1 M that the sum (i.e., the total damping rate) balances with the large-scale forcing for each c American Geophysical Union 2012, from Fig. 2 of Yano and Plant cloud type (right).  (2012a).

9.3

Physical considerations for a vertically integrated quantity

Two dicta were proposed in Sec. 2.4. The first dictum asserts that any physically based diagnostic convective closure must have a prognostic counterpart. The second (stronger) version further asserts that a prognostic counterpart to a physically based closure must be able to be integrated in time in a self-contained manner. Arakawa and Schubert (1974) chose the cloud-work function as the vertically integrated quantity Ii and based their argument implicitly on the stronger version of the dictum, although the dictum itself was not stated by them. They did not remark on the possibility of integrating this energycycle system in time self-consistently. Such a possibility was first considered by Pan and Randall (1998) and Randall and Pan (1993). More recently, Yano and Plant (2012b,c) proposed a different version, which is considered in the next section. A self-consistent closure framework can be developed simply by writing down a prognostic equation for the convection-base mass flux, which can

368

Part II: Mass-flux parameterization

essentially be derived by vertically integrating a prognostic mass-flux equation (i.e., physically a convective vertical velocity equation). It is straightforward to show (Sec. 10.7) that in this case, the evolution of the vertically integrated mass flux is controlled by the vertically integrated convective buoyancy. By then constructing a prognostic equation for the vertical integral of convective buoyancy, we obtain a self-contained prognostic system for describing the evolution of mass-flux amplitude. From this perspective, the stationarity of the vertically integrated convective buoyancy can be seen as a logical choice for an equilibrium convective closure under the mass-flux formulation. The consistency of Kuo’s (1974) moisture closure with the second (stronger) dictum is not obvious. It is widely believed that atmospheric moist convection is controlled by moisture, but there is no known selfcontained prognostic description under a coupling with the moisture closure. Finally, some potential issues with quasi-equilibrium closures may be remarked upon here. (1) Does the system actually evolve slowly? Although slowness of the evolution of Ii as observationally known suggests the validity of Eq. 9.3 for estimating MB , the inverse is not necessarily true. Note that by reducing the prognostic equation for the evolution of Ii to a diagnostic relation, we lose the ability to predict Ii directly. Instead, its evolution must be diagnosed, based on the predicted evolution of the thermodynamic fields. The situation is analogous to the use of hydrostatic balance. Recall that hydrostatic balance is obtained by setting Dw/Dt = 0 in the vertical momentum equation. The vertical velocity w can then be calculated diagnostically from mass continuity. This does not automatically mean that the evolution of the vertical velocity so-computed is actually slow. Indeed, the resulting evolution of vertical velocity could be much more rapid than an evolution that would support the hydrostatic balance assumption. (2) Does the system evolve at all under the steady constraint of Eq. 9.3? In contrast to the previous question, the issue is now whether the imposition of Eq. 9.3 might over-constrain the evolution of temperature and moisture. Consider the case that the large-scale forcing F is slowly varying in time. The quasi-equilibrium constraint implies that Ii is stationary with time, and the convective strength MB is obtained by assuming stationarity of a set of Ii . The vertical integral (or cloud-work

Closure

369

function, more specifically) for each cloud type is defined by a vertical integral, as in Eq. 3.5 or Eq. 8.2, of a function of temperature and moisture, the function being different for each type. The stationarity of {Ii } would therefore suggest that both temperature and moisture are stationary with time, if enough convective types with enough functional forms of the integrand are considered to obey the constraint, and if those functional forms are a complete basis set. Thus, it is not clear whether it is still possible to see temporal evolution of the thermodynamic variables. This is a rather serious issue, because it suggests that the convective quasi-equilibrium may destroy the predictability of thermodynamic fields (cf., Sec. 14.1 for a related issue). It should be appreciated that the question is not trivial, again through analogy with the hydrostatic balance. In that case, although the steadiness of the vertical velocity field is implied by the balance assumption, the vertical velocity field nonetheless evolves with time as estimated by mass continuity. Unfortunately, it is not immediately clear whether an analogous situation occurs for the convective quasi-equilibrium hypothesis or not. (3) Is the inverted solution stable against a perturbation? The obtained balanced solution may turn out to be unstable against any linear perturbation. In that case, in practice, the system would never stay at convective quasi-equilibrium. Note that this issue has a link to the concept of self-organized criticality (SOC), which is discussed in Ch. 4 and Vol. 2, Ch. 27.

10

Convective energy cycle

The convective energy cycle may be considered an example that fulfils the stronger dictum (Sec. 2.4) for justifying a closure condition based on the existence of a self-contained system under its prognostic extension. This section considers the convective energy cycle, first deriving the formulation and then discussing some simple applications.

10.1

Convective kinetic energy equation

The convective energy cycle can be derived in two ways. The first is a systematic approach of performing an energy integral on the full momentum equations (e.g., for a non-hydrostatic anelastic system). The procedure

370

Part II: Mass-flux parameterization

is rather tedious and less illuminating, though with the advantage of determining the dissipation term explicitly. The second and more intuitive approach is to consider the process of generation of kinetic energy as a thermodynamic work. Here, this second approach is taken. By definition, convection is a motion driven by buoyancy b. Thus, the process of generating convective kinetic energy is a work dW performed by buoyancy: dW = bdz,

(10.1)

when an air parcel is lifted by a distance dz. However, one should realize that this description is Lagrangian following the trajectory of an air parcel, whereas the intention here is to describe the evolution from a Eulerian point view. In order to convert to a Eulerian perspective, the displacement dz is related to a time increment dt through the vertical velocity w by: dz = wdt.

(10.2)

Substitution of Eq. 10.2 into Eq. 10.1 leads to: dW = bwdt or dW = bw, dt which is the local generation rate of kinetic energy by buoyancy forcing per unit mass. This is further translated into an energy generation rate per unit volume by multiplying by density ρ: dW = ρbw. dt The generation rate of kinetic energy for the i-th convective component per grid box is: ρ

dWi = Mi bi dt by further multiplying by the fractional area σi occupied by the i-th convection type. Note that this is an energy generation rate at each vertical level. The vertically integrated energy generation rate Gi is thus:  zT ,i  zT ,i dWi σi ρ Mi bi dz. dz = Gi = dt zB zB σi ρ

Closure

371

The final expression Gi = MB,i Ai

(10.3)

is obtained by invoking the separation of variables as in Eq. 1.2 and recalling that the cloud-work function Ai is defined by Eq. 8.2. The budget for the vertically integrated convective kinetic energy Ki may be defined in terms of the competition between the energy generation rate Gi and the dissipation rate Di . Thus, we obtain Eq. 8.1. In this way, we also see that the generation rate of the convective kinetic energy is controlled by the cloud-work function Ai .

10.2

Convective energy cycle formulation

In order to complete the convective energy cycle, let us consider the budget for the cloud-work function, which is given by:  ∂Ai Kij MjB + FI,L,i , = ∂t j

(10.4)

by referring to Eq. 9.1. Eqs. 8.1 and 10.4 contain three dependent variables, Ki , Ai , and MB,i for each convection type. Thus, once a relationship is introduced to relate two of these three variables, and an expression for the dissipation is formulated, then a closed system for the convective energy cycle can be obtained. This would constitute a prognostic extension of Arakawa and Schubert’s (1974) convective quasi-equilibrium closure. It is important to recall that this system is derived under the steadyplume hypothesis. Thus, it does not describe the evolution of individual convective towers, but only that of the sub-ensemble for a given convection type. However, note that in deriving this system, nothing particularly statistical is invoked, nor any explicit ensemble averaging procedure. All these remain implicit without requiring any additional assumption on top of the already assumed steady-plume hypothesis. Reflecting this fact, a particular feature of the system is that convection never grows when it is initialized with zero kinetic energy. In other words, we always require a pre-existing small seed in order for convection to grow.

372

10.3

Part II: Mass-flux parameterization

The choice of p

Amongst Ki , Ai , and MB,i , it is natural to expect the most direct link to occur between Ki and MB,i , and thus we set: p Ki ∝ MB,i ,

(10.5)

with p a constant. The case of p = 2 is considered by Pan and Randall (1998) and Randall and Pan (1993) whereas Yano and Plant (2012b,c) take p = 1. Various evidence suggests that the choice p = 1 is likely to be more realistic. The most direct evidence is found in Figs. 1–3 of Derbyshire et al. (2011). These show that changes of the bulk convective mass flux Mc produced by changing the background mean moisture, are associated with changes to the bulk fractional area σc for convection, but with a universal convective vertical-velocity vertical profile wc that is independent of the background moisture. A simple analysis (cf., Yano and Plant, 2012b) shows that if Mc variations are dominated by σc rather than wc variations, then p = 1 is suggested. Fig. 8 of Parodi and Emanuel (2009) also shows that the convective updraught velocity is invariant under a change of large-scale forcing, although it does change substantially with a change of the cloud microphysical state. Table 1 of Shutts and Gray (1999), obtained from their CRM experiments, also shows that the convective vertical velocity is approximately invariant with large-scale forcing. More indirect evidence is found by comparing the expected equilibrium solution under constant forcing with p = 1 and p = 2 against properties of states produced by idealized CRM experiments under constant forcing. Fig. 2 of Emanuel and Bister (1996) shows that CAPE is approximately invariant with the large-scale forcing, being consistent with the equilibrium solution expected with p = 1. Notice that the cloud-work function reduces to CAPE when a non-entraining convective mass-flux profile is assumed. 10.4

Structure of the interaction matrix Kij

The evolution of the cloud-work function Ai is controlled by the interaction matrix Kij through the first term on the right-hand side of Eq. 10.4. By referring to Eq. 8.24a, we can divide the interaction matrix into two contributions: those terms proportional to the fractional detrainment rate, δj , and those independent of δj . We may designate these contributions as

Closure

373

Kd,ij and Kv,ij , respectively: Kij = Kd,ij + Kv,ij . As an example, the interaction matrix is shown in Fig. 11.2 for Jordan’s (1958) sounding already adopted in Ch. 8, along with the two separate contributions. The plot is presented in terms of a constant fractional entrainment rate ǫi = λ, characterizing the convection type (cf., Ch. 8, Sec. 8), and representing the above matrix in a continuous form: Kλ,λ′ = Kd,λ,λ′ + Kv,λ,λ′ . The precipitation formulation of Arakawa and Schubert (1974) is also adopted, as discussed in Ch. 8, Sec. 7. Kd,λ,λ′ describes the change of the large-scale environment directly due to the detrainment Dλ′ of convective air, whereas Kv,λ,λ′ is the change due to the compensating descent −Mλ′ . Note that the detrainment effect Kd,λ,λ′ , being positive, always leads to the destabilization of the large-scale atmosphere (i.e., an increase of the cloud-work function) by re-evaporative cooling. Since this effect is only realized at the convective top detrainment level (or possibly at and below the convective top if the plumes are considered both entraining and detraining), only the convection types that extend higher than the convection type in consideration (i.e., only for λ < λ′ ) are affected. Thus, the lower-right portion of the Kd,λ,λ′ matrix remains zero. On the other hand, the compensating descent always leads to warming and stabilization so that Kv,λ,λ′ is negative. For this contribution, all of the convective types are affected by all of the other convective types. By putting those two opposing tendencies together, we find that shallow convection (upper part) tends to destabilize all convective types, albeit weakly, whereas deeper convection (lower part) tends to stabilize all convective types. Furthermore, the destabilization tendency by shallow convection is more pronounced on shallow convection (upper-right part), whereas the stabilization tendency by deep convection is more pronounced on deep convection (lower-left part). As a whole, convection, as defined by the interaction matrix, tends to stabilize the large-scale state against a destabilizing tendency from the large-scale forcing FL,i , as expected, and also being consistent with Arakawa and Schubert’s (1974) idea of convective quasi-equilibrium. However, a substantial part of the interaction matrix can be weakly negative, suggesting that convection can destabilize by itself, especially shallow nonprecipitating convection. This particular characteristic of shallow convection is considered further in Sec. 10.6.

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Part II: Mass-flux parameterization

(a)

(b)

(c) Fig. 11.2 Contours of the interaction matrix Kij : (a) the part due to detrainment Kd,ij , (b) the part due to ascent Kv,ij , and (c) the total. The contour interval is 0.1 Jm2 kg−2 except for the positive contours in the total, for which the interval is 0.01 Jm2 kg−2 . Negative values are indicated by dashed lines.

10.5

Bulk case with a single deep-convection mode

A basic behaviour of the convective energy cycle system can be understood by taking a single convective mode. The system given by Eqs. 8.1 and 10.4 reduces to: ∂K = MB A − D (10.6a) ∂t ∂A = −γMB + FI,L (10.6b) ∂t with the subscript i removed for convenience. The interaction matrix Kij is also replaced by a constant, −γ. A negative sign is introduced here in order

Closure

375

that γ is usually positive, as described above. Thus, this term constitutes a damping for deep convection: the effect of deep convection is to stabilize the environment by reducing the potential energy (i.e., cloud-work function), as suggested by Emanuel et al. (1994) with the term “convective damping”. Let us consider a simple case with a constant large-scale forcing FI,L . Imagine that we start the system from a state with a small cloud-work function value and a weak convective seed. The non-linearity in the first term on the right-hand side of Eq. 10.6a prevents any noticeable growth of convection initially. Meanwhile, the cloud-work function A increases linearly with time through the large-scale forcing FI,L . This process (or “recharge”) continues until the cloud-work function A becomes large enough for the energy generation term MB A to trigger convection by a non-linear feedback chain. The induced kinetic energy consumes the cloud-work function, and the reduced cloud-work function eventually terminates the convection event. In this manner, this simple system can explain an ensemble convective life cycle in a very concise manner. An example of such a solution is shown in Fig. 11.3. Here, D = K/τ is set with τ = 103 s. (See Yano and Plant (2012b) for discussion of this choice and of the other parameters.)

Fig. 11.3 An example of the solution in the phase space of kinetic energy (horizontal axis) and cloud-work function (vertical axis) for the one-mode convective energy cycle system (Eqs. 10.6a and b), representing an idealized life cycle of a convective ensemble with recharge (an upward cycle on the left-hand side) and discharge (a downward cycle on the right-hand side). The arrow indicates the direction of the path taken within phase space as time increases.

A similar exercise can also be applied to shallow, non-precipitating convection. As discussed in the last subsection, in this case γ is negative, and

376

Part II: Mass-flux parameterization

thus Eq. 10.6a leads to an explosive self-destabilization cycle associated with a continuous destabilization of the environment by evaporative cooling of the detrained air from shallow convection. Such an artificial behaviour arises by truncating the system only to a shallow-convection mode. In reality, shallow convection would interact with the other convective modes. 10.6

A two-mode description: Shallow-deep convective interactions

A natural extension of the analysis in the previous subsection is to consider the coupling of shallow and deep convective modes under this formulation. In order to distinguish the shallow and deep modes, subscripts s and d are added respectively. The kinetic energy equation remains the same as before, albeit with the addition of the subscripts: ∂Ks = Ms,B As − Ds (10.7a) ∂t ∂Kd = Md,B Ad − Dd . (10.7b) ∂t Interactions between the modes are realized through the equations for the cloud-work functions. Based on the analysis in Sec. 10.4, shallow convection is expected to destabilize both shallow and deep convection, and deep convection is expected to stabilize both shallow and deep convection. Thus: ∂As = γs Ms,B − βd Md,B + FI,s,L (10.8a) ∂t ∂Ad = −γd Md,B + βs Ms,B + FI,d,L , (10.8b) ∂t with constants βs , βd , γs , and γd chosen to be positive definite. As for the case with a single mode, when the system is initialized with low kinetic energies and cloud-work functions, it goes through an initial recharge phase before convection is triggered. However, once convection is triggered, different behaviours can arise. Shallow convection is a self-destabilizing process, and thus it begins to grow faster than deep convection. Shallow convection increases the cloudwork function for deep convection as well, and thus the former begins to promote the latter. As deep convection begins to grow, the whole system is gradually stabilized, and eventually both shallow and deep convection are suppressed. Then a new cycle begins.

Closure

377

Indeed, an external large-scale forcing is not necessary in order to sustain the two-mode system due to the self-destabilization tendency of the system induced by shallow convection. Even in the absence of large-scale forcing, a self-sustaining periodic cycle can be realized, as illustrated by Fig. 11.4. Here the parameters are as adopted in Yano and Plant (2012c).

Fig. 11.4 An example of the time evolution of the convective energy system under a coupling between shallow and deep convection. Here, Ks = αs Ms,B and Kd = αd Md,B have been set and the lines are Ks (long-dash line), A′s (short-dash line), Kd (solid line) and A′d (chain-dash line). Note that only the anomalous, non-equilibrium part of the cloud-work functions are shown, with A′s = As − (αs /τs ) and A′d = Ad − (αd /τd ). Note also that the kinetic energies Ks and Kd are presented in the units of Jkg−1 by dividing them by 103 kgm−2 and 104 kgm−2 , respectively. The rescalings roughly correspond to the air mass over depths of 1 km and 10 km, respectively.

More generally, a balance between the self-stabilizing and selfdestabilizing tendencies of the system is required for a perpetual closed cycle, and stabilization or destabilization may dominate for other parameter settings. An analysis of the different regimes of the two-mode system, including a derivation of the balance conditions, is provided by Plant and Yano (2013). An important message from the analysis here is that coupling between shallow and deep convection may be key for successfully simulating the transformation from shallow to deep convection. The specific coupling pro-

378

Part II: Mass-flux parameterization

posed here would be easy to implement into mass flux-based convection parameterizations. 10.7

A variation to the energy cycle formulation

The convective energy cycle considered so far appears to provide the most self-consistent description available for the time evolution of parameterized convection. However, a major drawback is the need to specify the exponent p in Eq. 10.5 (or to formulate some other plausible relationship) in order to close the system. As already suggested in Sec. 9.3, this obstacle may be overcome by more directly considering the evolution of vertically integrated mass flux instead of kinetic energy. Let us now consider this possibility. A prognostic equation for mass flux will be derived in Ch. 12. According to Ch. 12, Eq. 7.2 there, it is given by:

∂ ∂p′i ∂ Mi + wi Mi + Di wi − Ei we = σi − + ρgbi . (10.9) ρ ∂t ρ ∂z ∂z Here, the horizontal convergence term has been replaced by entrainment and detrainment terms. The vertical integration of Eq. 10.9 leads to, after various rearrangements:

 zT i ∂MBi + (wi,T − z˙T i )Mi,T − (wi,B − z˙B )Mi,B ηi dz ∂t zB  zT i  zT i ∂p′ (10.10) σi i dz + MBi Bi , (Di wi − Ei we )dz = − + ∂z zB zB where Bi is defined by:  zT i ηi bi dz. (10.11) Bi = w i zB Thus, the cloud-work function Ai is replaced by a new vertical integral Bi when the mass-flux budget is considered directly rather than the kinetic energy budget. Nevertheless, it is important to notice that the term directly responsible for generating convective mass flux via buoyancy in Eq. 10.10 takes a very similar form as the corresponding term in the kinetic energy budget in Eq. 8.1, with a non-linear tendency for the energy-generation rate (mass-flux generation rate) retained. Furthermore, if the fractional area σi can be assumed to be approximately invariant with height, then ηi /wi ∝ ρ and Bi essentially reduces to a density-weighted vertical integral of the convective buoyancy bi :  zT i ρbi dz. Bi ∝ zB

Closure

The above may be further rewritten as:  pB Bi ∝ bi dp.

379

(10.12)

pT i

with the help of the hydrostatic balance dp = −ρgdz, and defining pB = p(zB ) and pT i = p(zT i ). Bechtold et al. (2014) call the quantity on the righthand side of Eq. 10.12 the PCAPE (pressure-integrated convective available potential energy). Although this expression is close to the standard CAPE, when using PCAPE the parcel-lifted buoyancy is replaced by the actual buoyancy felt by a given convective element. It will be shown in Vol. 2, Ch. 15 that the introduction of PCAPE defined by Eq. 10.12 into a closure is key for successfully simulating the diurnal cycle of convection. 11

CAPE-based closure

Most current operational models adopt CAPE (convective available potential energy) as the basis for their closure. CAPE may be considered a special case of the cloud-work function introduced by Eq. 8.2, and it is obtained by setting the mass-flux vertical profile to unity, i.e., ηi = 1, in its definition. This would be appropriate for ascending air that does not interact with its surroundings through entrainment and detrainment processes. Thus:  zT bdz. (11.1) CAPE = zB

The buoyancy b may be expressed in terms of the virtual temperature Tv by: Tvc − T¯v b=g . (11.2) T¯v

Note that CAPE is a special case of a vertical integral defined by Eq. 3.2 without the mass-flux profile factor, and a closure based on CAPE may be considered a special case of the closure introduced in Sec. 3.2. The basic principle of the CAPE closure is to set: ∂CAPE = 0. (11.3) ∂t In the standard procedure, the convective virtual temperature Tvc is evaluated by a simple lifting-parcel argument already introduced in Ch. 1: a convective parcel is lifted from the bottom of convection (or any predefined lifting level) in a pseudo-adiabatic manner without mixing with the environment.

380

Part II: Mass-flux parameterization

The convective virtual temperature Tvc may alternatively be evaluated by following the standard formulation for convective-scale variables presented in Sec. 4.2. In this case, the air parcel mixes with the environment through entrainment, leading to less buoyancy than a simple parcel lifting would predict. The latter definition is sometimes referred to as “dilute CAPE” for this reason. Note that, in this latter case, the dropped subscript i for the convective type must be recovered. Hence, an entraining plume ηi = 1 is used for the evaluation of the convective-scale virtual temperature but the factor ηi is nonetheless removed from the definition of the integrand in the cloud-work function in order to obtain the dilute CAPE. Care should be taken because the literature is sometimes confused and fails to distinguish between the dilute CAPE and the cloud-work function. Note that when dilute CAPE is considered, Tvc is best treated as a convective-scale variable. On the other hand, when the more standard parcel lifting-based CAPE is considered, Tvc is better treated as a large-scale variable. Treating it as a convective-scale variable is confusing, because a convective-scale variable would experience entrainment by Eq. 4.4, whereas Tvc does not. In either case, the total CAPE tendency can be divided into convective and large-scale contributions:     ∂CAPE ∂CAPE ∂CAPE = + . (11.4) ∂t ∂t ∂t c L

Here, it is assumed that z˙T = z˙B = 0 for simplicity. The convective tendency can furthermore be given generally in the form:   ∂CAPE = KMB , (11.5) ∂t c

assuming a bulk formulation. The CAPE closure of Eq. 11.3, thus, leads to:   1 ∂CAPE . (11.6) MB = − K ∂t L

The precise expressions for the two factors on the right-hand side of Eq. 11.6 depend on whether we take a dilute parcel or a simple lifting parcel. When a dilute, non-entraining (i.e., lifted) parcel is considered, we find:  ¯ 

 zT ∂ Tv g D ¯ K = −Cp + αηc δc (Tvc − Tv ) + dz (11.7a) ∂z Cp zB   zT

    zT ∂¯ sv ∂CAPE ∂svc,B ρα ρα =− dz + dz (11.7b) ∂t ∂t η ˜ ∂t c zB zB L L

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381

by following the same procedure as in Sec. 7. Recall that α is defined by z Eq. 7.4, and η˜c = exp( zB ǫc dz ′ ) (cf., Eq. 4.9). These expressions may also be written as:  zT ∂ αηc (svc − s¯v )dz (11.8a) K=− ∂z zB 

  zT ∂ ∂CAPE (svc − s¯v ) dz. αηc (11.8b) =− ∂t ∂t zB L L We may assume:   zT  ¯ g ∂ Tv + dz (11.9) α K ≃ − Cp ∂z Cp zB as a first approximation more generally, as long as the entrainment rate is small enough. In many current operational implementations of the CAPE closure, however, the large-scale tendency (∂CAPE/∂t)L is usually not directly calculated but replaced by a term: CAPE , (11.10) τ where τ is known as the closure timescale. This replacement is necessary, because otherwise a parameterization “underestimates convective activity in situations where the large-scale forcing is weak, and where convective heating preceeds the dynamic adjustment” (Bechtold et al., 2014). Various other examples and further discussions of such an implementation can be found in Bechtold et al. (2001); Emanuel (1993a); Fritsch and Chappell (1980); Gregory (1997); Kain (2004); Willett and Milton (2006); Zhang and McFarlane (1995). By putting all those approximations and assumptions together, a final expression for the closure is given by:  zT  ¯  −1 g ∂ Tv CAPE + dz Cp α (11.11) MB = τ ∂z Cp zB (cf., Gregory et al., 2000; Zhang and McFarlane, 1995). 11.1

Parcel environment-based closure

The CAPE tendency may, alternatively, be divided into two contributions: one coming from the parcel (convective) virtual temperature Tvc , and the other from the environmental state T¯v . These are given by:    zT ∂CAPE g ∂Tvc dz (11.12a) = ¯ ∂t T v ∂t zB BL

382

and

Part II: Mass-flux parameterization



∂CAPE ∂t



env

=−



zT

zB

g ∂ T¯v dz, T¯v ∂t

(11.12b)

respectively. Here, contributions from the changes to the top and the bottom of the integral range are neglected as before. A change of T¯v in the denominator of the integrand has also been neglected, because this term may be absorbed into a part of the integration variable when the integral is performed in terms of pressure as discussed at the beginning of Sec. 8. The contribution from the parcel (convective) virtual temperature Tvc is considered as arising due to boundary-layer (BL) processes, because the parcel originates from the boundary layer and does not interact with the environmental air aloft. Meanwhile, the contribution from the large-scale virtual temperature T¯v is labelled as environmental. As a result, the total CAPE tendency is given by:     ∂CAPE ∂CAPE ∂CAPE = + . (11.13) ∂t ∂t ∂t BL env

Physical intuition would suggest that most of the CAPE variability originates from the boundary layer, so that:   ∂CAPE ∂CAPE ≃ . (11.14) ∂t ∂t BL

Emanuel (1995) and Raymond (1995) thus argued that the CAPE closure can be well approximated by considering only its boundary-layer contribution:   ∂CAPE ≃ 0. (11.15) ∂t BL

This idea is called “boundary-layer quasi-equilibrium”, and is discussed further in Sec. 11.3. However, observational data analysis by Donner and Phillips (2003) and Zhang (2002, 2003) led to rather unexpected conclusions. They considered data both from the tropics and the mid-latitudes. In order to present their results, both the boundary-layer and the environmental tendencies are divided here into contributions from convection and large-scale (i.e., non-convective) processes (with subscript c and L, respectively):       ∂CAPE ∂CAPE ∂CAPE = + (11.16a) ∂t ∂t ∂t BL BL,c BL,L       ∂CAPE ∂CAPE ∂CAPE = + . (11.16b) ∂t ∂t ∂t env env,c env,L

383

Closure

These studies confirm that CAPE variability is dominated by the boundarylayer terms, and thus that Eq. 11.14 is indeed a good approximation. However, they also find that such variability is not regulated by a close balance between the boundary-layer processes and feedback by convection, as the boundary-layer quasi-equilibrium hypothesis envisions. This can be seen from a scatter plot between (∂CAPE/∂t)BL,L (horizontal axis) and (∂CAPE/∂t)BL (vertical axis), as shown in Fig. 11.5. If the boundarylayer quasi-equilibrium of Eq. 11.15 is to be satisfied, it can be expected that values on the vertical axis tend to be much smaller than those on the horizontal axis, as seen in Arakawa and Schubert’s (1974) original demonstration of convective quasi-equilibrium (see their Fig. 13). However, the scatter plot demonstrates that this is not at all the case, and the large-scale tendency for (∂CAPE/∂t)BL is comparable to the total.

300

(dAp/dt) (J/kg/hr)

200

100

0

-100 -100

0

100 200 (dAp/dt)ls (J/kg/hr)

300

Fig. 11.5 Scatter plot between (∂CAPE/∂t)BL,L (horizontal axis) and (∂CAPE/∂t)BL (vertical axis). Data is from an observational site under an ARM (Atmospheric Radiation Measurement) programme over the North American Great Plains. (See Zhang (2002, 2003) for data details.) Here, convective and non-convective periods are distinguished by diamonds and stars respectively. [Courtesy of Guang Zhang]

The boundary-layer processes are rather noisy, and convective systems dominated by mesoscale organization cannot follow the fast changes. Note that the data analysis is performed over a typical synoptic-scale observational array, and thus a balance between individual convective elements and immediate local boundary-layer processes is not the concern here, and nor

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Part II: Mass-flux parameterization

is it for convection parameterization, as already emphasized in Ch. 3. Those authors propose that the CAPE closure should be replaced by that for the parcel environment, i.e.:   ∂CAPE ≃0 ∂t env or



∂CAPE ∂t



+

env,c



∂CAPE ∂t



≃ 0.

(11.17)

env,L

This leads to the parcel environment-based closure:   1 ∂CAPE MB = − . K ∂t env,L

(11.18)

An implementation of this parcel environment-based closure was attempted by Donner et al. (2001). 11.2

Operational implementation of parcel environmentbased closure

The idea of a parcel environment-based closure has recently been operationally implemented at ECMWF (Bechtold et al., 2014) under the following procedure. The key step is to replace both the convective and the large-scale tendencies by those coming from the environment:     ∂CAPE ∂CAPE ⇒ ∂t ∂t c env,c 

∂CAPE ∂t



L

⇒



∂CAPE ∂t



.

env,L

Under the first replacement above, Eq. 11.5 above is replaced by:   ∂CAPE = KMB , (11.19) ∂t env,c in which the convective efficiency K is still approximated by the full expression of Eq. 11.9. In order to replace (∂CAPE/∂t)L by (∂CAPE/∂t)env,L , it is necessary to start from       ∂CAPE ∂CAPE ∂CAPE = − (11.20) ∂t ∂t ∂t env,L L BL,L

Closure

and adopt the formulation (cf., Eq. 11.10):   CAPE ∂CAPE . ≃ ∂t τ L

385

(11.21)

The notation (∂CAPE/∂t)BL,L features a slightly delicate set of subscripts: the first subscript BL indicates a component of CAPE variability originating from the boundary layer, or more precisely from Tvc ; the second subscript L stands for “large scale”, more precisely referring to non-convective process. Strictly, L indicates that large-scale advection processes are to be included. However, in the implementation by Bechtold et al. (2014), the diagnosis of this term is limited to the boundary-layer processes modelled by the boundary-layer parameterization. The term can thus be written as:      zT g ∂Tvc ∂CAPE = dz, (11.22) ¯ ∂t ∂t BL zB T v BL,L where (∂Tvc /∂t)BL refers to the tendency arising from the boundary-layer parameterization. The lifting parcel originates from the boundary layer, and its lifting-level value may be estimated from an average over a layer [zs , zB ], where zs is the surface level. It is convenient to apply such an estimate for the change in Tvc to Eq. 11.22, rather than attempting to compute a tendency through the full convective depth [zB , zT ]. The estimation implies changing the range of the vertical integral and the introduction of a factor of (zT − zB )/(zB − zs ). Furthermore, the variability of Tvc is enhanced by latent heating associated with the condensation of water, which is not taken into account when the Tvc variability is estimated from values below the condensation level. Thus, an extra factor μ is introduced to the above. Noting that 1 K of temperature increase leads to an increase of saturated water vapour by ∼ 1 gkg−1 and that this increase leads to a latent heating of 2.5 K, we estimate that μ ≃ 3.5. As a result:    zB  ∂Tvc g ∂CAPE ≃ ∗ dz, (11.23) ∂t T zs ∂t BL BL,L

in which the adjustment factors have been absorbed into the inverse of an effective temperature: zT − zB 1 = μ < T¯v−1 > , (11.24) ∗ T zB − zs with the angled brackets denoting an average over [zB , zT ]. This effective temperature is estimated as: 300K 500m < T¯v > zB − zs ∼ 4K. (11.25) ∼ T∗ = μ zT − zB 3.5 104 m

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Part II: Mass-flux parameterization

Note that in the actual implementation, Bechtold et al. (2014) use a smaller value (1 K). Tests show that the scheme is relatively insensitive to the choice of this parameter for the range T ∗ = 1–6 K. Also note that CAPE is replaced by PCAPE (cf., Sec. 10.7) in their operational implementation. Substitution of Eqs. 11.21 and 11.23 into Eq. 11.20 leads to:     zB  ∂Tvc g CAPE ∂CAPE − αBL ∗ ≃ dz. (11.26) ∂t τ T zs ∂t BL env,L A further substitution of this expression into Eq. 11.18 gives a final expression for MB . Here, Bechtold et al. (2014) introduce an extra parameter αBL purely for convenience, such that when αBL = 1 the parcel-environment closure is adopted, and when αBL = 0 the standard CAPE closure is recovered. As discussed in detail in Vol. 2, Ch. 15, this implementation significantly improves the model performance for the convective diurnal cycle. However, operational experience with global models suggests that boundary layerbased closure is rather unreliable due to high variability in the boundary layer. As a result, simulations with a boundary layer-based closure tend to be too noisy. However, this does not mean that a boundary layer-based closure will never become operationally relevant. For example, isolated, scattered, deep convection over intense surface heating, such as over land, may be more directly influenced by boundary-layer processes. There are also indications that boundary layer-based closures work rather well for shallow convection (cf., Vol. 2, Ch. 24). 11.3

Boundary-layer quasi-equilibrium

Boundary-layer quasi-equilibrium, proposed by Emanuel (1995) and Raymond (1995), is now examined. In introducing the vertical integral I in Sec. 3, the range of integration was expected to cover the whole convective layer by default. However, under the boundary-layer quasi-equilibrium, the integral ranges over the planetary boundary layer (PBL), a layer just below convection base. Thus, intuitively, the idea may look a little odd, because this argument suggests that the intensity of convection is primarily defined by the atmospheric state outside of the convective clouds. However, there is merit to this thinking: as already noted, the CAPE variability is dominated by variations within the boundary layer, and thus it is reasonable to suppose that processes within the boundary layer may control convection, if indeed CAPE controls convection as a whole. Since

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the convective boundary layer is well mixed, hereinafter there is no need to take a vertical average explicitly. Instead, a budget equation for the boundary layer is considered directly. For the budget variable, let us take the boundary-layer equivalent potential temperature θeb , following Emanuel (1995) and Raymond (1995). Its budget may be written as:     ∂ ∂ ∂ θeb = θeb θeb , + (11.27) ∂t ∂t ∂t L c

with the meanings of the subscripts the same as before. The boundary-layer quasi-equilibrium closure is given by: ∂ θeb = 0 ∂t or     ∂ ∂ θeb θeb = 0. + (11.28) ∂t ∂t L c The large-scale tendency consists of horizontal advection and the surface flux, so that:   ∂ 1 θeb = −hb ub · ∇θeb + F0 (11.29a) hb ∂t ρ B L

∗ with the surface flux given by F0 = ρB CD u0 (θeb −θeb ) and with hb denoting the depth of the convective boundary layer. Here, CD is a drag coefficient, ∗ is a saturated value appropriate for u0 a near-surface wind speed, and θeb an underlying sea surface. The convective tendency consists of cooling and drying tendencies due to both convective downdraughts and environmental descent:     Md ∂ θeb = (11.29b) − we Δθeb , hb ∂t ρB c

where Md is the convective-downdraught mass flux (taken to be downwards positive), we is the vertical velocity of environmental descent (taken to be upwards positive), and Δθeb is the difference between the boundary-layer equivalent potential temperature and that in the troposphere. Raymond (1995) more precisely proposes to take it as a jump over the top of the wellmixed layer, whereas Emanuel (1995) takes the tropospheric value from a middle level. After substitution of Eqs. 11.29a and 11.29b into Eq. 11.28, we obtain a closed expression for the total convective mass flux:

1 ¯+ (−hb ub · ∇θeb + F0 ) , (11.30) MB + Md = −ρB w Δθeb

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recalling that we = w ¯ − MB /ρB . The convective updraught mass flux may be diagnosed from the downdraught by relating them through a parameter ǫp called the “precipitation efficiency” as introduced by Emanuel (1995): Md = (1 − ǫp )MB . (11.31) See Ch. 13 for more general discussions on downdraught formulations within mass-flux convection parameterization. 11.4

How is the boundary layer affected by convective updraught and downdraught mass fluxes?

In the boundary-layer quasi-equilibrium closure presented in the last subsection, the updraught and downdraught are seen to alter the properties of the boundary layer. However, following the formulation originally introduced by Arakawa and Schubert (1974), the convective updraught does not modify the boundary layer. To understand this apparent contradiction, we must understand a little more about the structure of the convective boundary layer, and so this subsection examines the budget equation for the boundary layer more carefully. A complementary perspective on the boundary-layer structure below moist convection can be found in Vol. 2, Ch. 24. In order to rederive the formulation by Arakawa and Schubert (1974), let us start from the prognostic equation for the equivalent potential temperature θeb averaged over the well-mixed boundary layer: 1 ∂θeb = −(ρu)b · ∇θeb + (F0 − FB ) + (QR )b . (11.32) ρ ∂t hb Here, F denotes the vertical eddy flux, with the subscripts 0 and B designating the surface and the top of the boundary layer (the bottom of the inversion layer) z = hb , QR is the radiative heating rate, and the subscript b is used to indicate mean boundary-layer values. The top of the well-mixed boundary layer is usually overlaid by a narrow layer called the inversion which can be characterized by a sudden jump of physical variables: (11.33) Δθeb = θ¯e − θeb , ¯ where θe is the free-tropospheric value immediately above the inversion layer. A budget equation over the inversion layer is obtained by vertically integrating the full budget equation  over this narrow layer:  1 dhb −w ¯B,e Δθeb = − FB . (11.34) dt ρB

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Note that (dhb /dt−w ¯ B,e ) is the displacement rate of the top of the boundary layer relative to the Lagrangian displacement rate of an air parcel in the ¯B − MB /ρB . Substitution of environment at the inversion level w ¯B,e = w Eq. 11.34 into Eq. 11.32 leads to:

  dhb ∂θeb Δθeb 1 = −(ρu)b · ∇θeb + F0 + −w ¯B + MB + (QR )b . ρ ρB ∂t hb hb dt (11.35) Eqs. 11.34 and 11.35 show how a given convective-base updraught mass flux MB modifies the boundary layer. A generalization to include the downdraught is straightforward, and it is only necessary to replace MB above by MB +Md , so that both the convective updraught and downdraught cool and dry the boundary layer. However, the equations are not closed yet, because the rate of change of the boundary-layer top must be specified. In order to resolve this issue, Arakawa and Schubert (1974) invoked a well-known fact (see e.g., Garratt, 1992) that the downward eddy fluxes from the bottom of the inversion layer into the well-mixed layer are well approximated as a fixed fraction k ≃ 0.2 of the surface values: FB = −kF0 .

(11.36)

Substitution of Eq. 11.36 into Eq. 11.32 leads to: ρ

1+k ∂θeb = −(ρu)b · ∇θeb + F0 + (QR )b . ∂t hb

(11.37)

In this formulation, convection no longer explicitly modifies the boundary layer. An alternative, simple assumption could be to set dhb /dt = 0 in Eq. 11.35. Although this is not realistic, it is a choice worthy of consideration, because most global models do not resolve the inversion layer well, and thus jumps such as Eq. 11.33 are not well-defined in the models. Hence, it could be more convenient to consider the height hb = zB as fixed. An associated required change to Eq. 11.35, when such a continuous formulation is adopted, would be to replace Δθeb with a value at the top of the boundary layer (convection bottom) θeB , as can be shown by simply repeating the derivation under these assumptions. As a final important step for implementing the idea of boundary-layer quasi-equilibrium operationally, it is necessary to recognize that it is difficult to identify the well-mixed boundary layer unambiguously numerically. Hence, the description of Eq. 11.35 should be replaced by a vertical integral over the boundary layer, with possible vertical gradients included. A final formula for such an

390

Part II: Mass-flux parameterization

implementation of boundary-layer quasi-equilibrium is:

 zB ∂ ∂θe + ρw′ θe′ − QR dz (11.38) ¯ (MB + Md )θeB = ρub · ∇θe + ρw ∂z ∂z zs (cf., Eq. 7 of Neggers et al. 2004; Eq. 19 of Bechtold et al. 2014). 11.5

Activation control

Mapes (1998) put forward the idea of boundary-layer control in a more intuitive manner by following a parcel-lifting argument, as used in evaluating CAPE. He called this argument “activation control” in order to contrast it with the more conventional quasi-equilibrium point of view. However, the idea itself has been applied within formulations that also make assumptions of quasi-equilibrium, leading to some confusion (cf., Ch. 7, Sec. 6.4; Yano, 2011). In order to understand Mapes’s (1998) argument, let us simply consider an air parcel lifted from the boundary layer. In a typical computation of CAPE, the lifting parcel quantities are defined as an average over the lowest part of the layer (typically 50 hPa or 500 m). The lifting parcel, computed using standard sounding data, would initially experience negative buoyancy due to dry adiabatic cooling. At a certain level (lifting condensation level, or LCL), the parcel becomes saturated, and above this level, the parcel experiences condensative heating. In a typical tropical situation, the parcel reaches a level of neutral buoyancy (LNB) at a certain point, and has a positive buoyancy above this. Such a state is called “conditional instability” (cf., Ch. 1) because the ascent is conditioned by a finite displacement of the air parcel in the vertical. Work must be performed on the parcel in order to lift it from the surface level to the LNB. This required work is usually called the “convection inhibition” (CIN). Mapes (1998) emphasizes the simple fact that this CIN barrier must be overcome before convection can be activated, and postulates that the convection may be controlled by the rate at which parcels are able to overcome the barrier. The idea is called the “activation principle”. Unfortunately, it is often not realized that such a notion is an inherently separate description from the basic formulation of mass-flux convection parameterization, which only deals with an ensemble (or sub-ensemble) of convection, but not a single convective event. In other words, mass-flux convection parameterization does not describe the process of the growth of convection from the boundary layer upwards to the free troposphere. More

Closure

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specifically, such a description is forbidden by adopting a steady-plume hypothesis (cf., Ch. 7, Sec. 6.4). For these reasons, activation control is rather a misleading concept to be applied to a standard mass-flux convection parameterization. A necessary step would be to remove the steady-plume hypothesis whilst maintaining the collective balance of the plumes (cf., Ch. 7, Sec. 6.3). It must also be acknowledged that the notion of CIN arises only based on a simple parcel argument. In a real convective well-mixed boundary layer, air parcels are usually driven by buoyancy, with ascending parcels from near the surface usually experiencing positive buoyancy. Thus, we return to the consideration of a normalized vertical integral of the buoyancy flux bw the importance of which was discussed in Sec. 10.1. To obtain an effective (typical) buoyancy b∗ , the buoyancy flux may be normalized as: b∗ ≡

ρwb , (ρw)∗

(11.39)

where the overbar designates a horizontal average and (ρw)∗ is a root mean square value over the atmospheric column volume: i.e., (ρw)∗ ≡ 1/2

< (ρw)2 > . Yano et al. (2005a) propose to call the vertical integral of b∗ the “potential energy convertibility” (PEC). Figure 11.6 (reproduced from Fig. 2 of Yano, 2003b) compares vertical profiles of the standard parcel-lifting buoyancy and the effective buoyancy b∗ , based on the definition of Eq. 11.39. The data was taken from three CRM runs from the TOGA-COARE period for different convective regimes. The convective inhibition is clearly visible under the standard liftingparcel calculations for the lowest 1 km in all of the convective regimes. Conversely, the effective buoyancy b∗ shows no negative buoyancy except for the suppressed regime after a dry intrusion event (dotted line), in which case an inversion at the top of the boundary layer produces a very narrow zone of negative buoyancy. As a result, in the actual convective-scale dynamics, the role of inhibition control may not be as strong as it at first appears from a simple parcel analysis. This also suggests that the activation control process is not as strong as Mapes (1998) argued, although this does not at all discredit his entire argument. However, it does pose a question as to the physical basis for adopting CIN as a closure variable. Nonetheless, efforts have been made to implement this idea into massflux convection parameterization. The basic concept for the implementation is to consider the activation control as a kind of stochastic process in which

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(a)

lifted parcel buoyancy

(b) PECíbased buoyancy (b*)

15

15

E W 10

10 height (km)

height (km)

D

5

5

0 0

2 (K)

4

0 í0.05

0.00

0.05 (K)

0.10

0.15

Fig. 11.6 Mean vertical profile for (a) lifted-parcel (CAPE) buoyancy, and (b) convection-driving (PEC) buoyancy b∗ . Three cases from the TOGA-COARE period are considered: a dry intrusion event (14–18 November 1992, marked by D with dotted lines), an easterly wind regime (10–14 December 1992, marked by E with long-dashed lines), and a westerly wind burst (20–24 December 1992, marked by W with dash-dotted lines). (Reproduced from Fig. 2 of Yano (2003b).)

the chance for a boundary-layer air parcel to overcome the CIN barrier decreases with increasing CIN. The chance for a parcel to escape should also be related to the strength of turbulent motions in the boundary layer, which can be measured by the turbulent kinetic energy (TKE). (This quantity is produced as an output by some boundary-layer parameterization schemes.) Thus it could be postulated that the probability of an air parcel overcoming the CIN barrier is proportional to exp(−CIN/TKE). Along this line of argument, but with a slight leap of logic, Mapes (2000) further proposed that the actual convection-base mass flux is proportional to exp(−CIN/TKE). This idea, which may be called the PBL-based closure, has been attempted by a number of authors including Bretherton et al. (2004); Hohenegger and Bretherton (2011).

Closure

12

393

Alternative approaches

This chapter proposes the budget of a vertically integrated quantity as a basis for developing a closure formulation for mass-flux convection parameterization. However, alternative perspectives exist. One possibility is to interpret the closure problem as that of a boundary condition at the convection base. A closure can then be regarded as a matching condition between the parametric descriptions of the sub-cloud layer and the convective layer. Along this line of argument, Grant (2001) ∗ as a constraint so that proposed to take a turbulent velocity measure wB ∗ MB ∼ wB . (For a full description of the approach see Vol. 2, Ch. 24.) It is worth stressing here, however, that Grant (2001) did not follow a standard mass-flux parameterization approach. In the standard approach, as presented in Part II, closure requires a general measure of convective strength after a mass-flux vertical profile has been normalized. To some extent it is merely a convenient custom to normalize the profile at the convection base, but there is no overwhelming reason to do so, especially if the mass-flux profile increases substantially from the convection base. Gregory and Rowntree (1990) proposed a rather simple closure which is also interesting in that it stresses the conditions around the convection base. Specifically, they assumed that the mass-flux amplitude is proportional to the convective parcel buoyancy, bB+ at the second convective model level. Here, the first convective level is defined as being that at which a parcel has a buoyancy of at least 0.2 K. Thus, MB ∝ bB+ . This closure is no longer used operationally. A dimensional analysis, as introduced in Ch. 12, Sec. 2, may also be a useful basis for defining a closure. For example, Shutts and Gray (1999) found, based on a dimensional analysis, that: (12.1) MB ∝ Fh /CAPE at equilibrium, where Fh is the surface moist static energy flux. A similar relationship is also derived in Vol. 2, Ch. 24, Eq. 3.1. Unfortunately, the particular relation of Eq. 12.1 is not suitable for direct use in predicting the mass flux as a practical closure condition. Consider a small perturbation that leads to a slight excess of the mass flux relative to the value required for equilibrium. The excess mass flux reduces the CAPE and so gives rise to a positive feedback that further increases the closure mass flux, leading to a runaway process of convection. Thus, the example demonstrates the importance of analysing the stability of a scheme as emphasized in Ch. 5, Sec. 5.

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13 13.1

Part II: Mass-flux parameterization

Related technical issues Switch conditions

Convection does not always exist at a given grid column, and thus we have to turn off the convection parameterization from time to time. Such a condition may be called the “switch condition”. Formally speaking, the switch condition may simply be stated as MB,i ≥ 0, and so if a closure leads to a solution MB,i < 0, convection must be turned off by resetting MB,i = 0. The implementation of such a switch condition under a bulk formulation is straightforward. Under a moisture-based closure or a stationarity-ofCAPE closure, this could be determined by whether or not the columnintegrated water vapour or CAPE has an increasing tendency by large-scale processes, respectively. However, when a spectral formulation is adopted, this introduces an extra complexity to the closure problem. In this case, the closure condition is defined as the inversion of a matrix, as explained in Sec. 9.1. Implementation of a switch condition implies that we should repeat a matrix-inversion problem after resetting any non-active convective modes with MB,i = 0. This could become a rather complex iteration procedure, because it may turn out that more convective modes must be turned off when the inversion is repeated. An implementation of the procedure for Arakawa and Schubert’s (1974) convective quasi-equilibrium closure is carefully discussed in Lord and Arakawa (1982); Lord et al. (1982). An alternative interpretation of a closure computation that leads to negative convective mass-flux modes might be to consider such modes as representing downdraughts. Although this interpretation has to date never been considered in the literature, it may be worthy of investigation. However, in this case, another auxiliary condition for turning off convection would nonetheless be required as a separate constraint, because a matrix inversion solution may prove unphysical: for instance, all of the mass fluxes might turn out to be negative. In operational implementations, a condition for convective instability is often tested first before any further convection parameterization calculations are performed. Following this procedure, the scheme tries to identify the bottom and top of the convective layer, typically by examining a liftedparcel buoyancy. If the closure itself is based on the vertical integral of the same buoyancy, this gives a fully self-consistent result.

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However, parcel testing is rather a complex procedure in itself, because the sequence of testing must carefully defined, as well as the definition of the lifting air. A typical operational document pays substantial attention to this issue. It is a purely practical but also non-trivial question because the whole motivation for applying a switch condition as a pre-condition is to avoid making full computations of the in-plume buoyancies through the cloud model in situations where that is unnecessary. Discussions on the practical importance of switch conditions, and sensitivities to assumptions made when formulating them, can be found in Jakob and Siebesma (2003), for example. 13.2

Trigger and testing

In the literature, a statement is often made that convection may or may not be triggered by testing a lifted air parcel. However, such statements may be misleading in the sense that they do not relate to the actual physical trigger of convective events. The convective energy cycle analysis in Sec. 10 suggests that an ensemble of convection never completely dies out (in other words, there would always be some very small possibility of producing a convective cloud, even within a hostile environment). It may, however, remain with very weak activity with a numerically negligible but non-zero mass-flux value. (Preliminary tests for more complex cases suggest that non-precipitating periods can easily be identified without ambiguity, and a new convective event can be induced without adding any additional trigger under this energy-cycle formulation.) These convective ensembles may never need to be triggered. However, such a treatment could be numerically inconvenient, and for practical reasons, it would be easier sometimes to turn off convection, and later turn on convection as required. However, such a technical switch condition should not be confused with a physical trigger. 13.3

Trigger function

Kain and Fritsch (1992) introduced the concept of a “trigger function”, which in fact refers to a bottom boundary condition for convective variables. A simple choice for this purpose could be simply to set the variables equal to values at the top of the well-mixed PBL, as originally suggested by Arakawa and Schubert (1974). However, many operational schemes add an extra perturbation before a convective instability test is performed by

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parcel lifting. For example, Gregory and Rowntree (1990) add a temperature perturbation of 0.2 K while the ECMWF model also adds a moisture perturbation of 0.1 gkg−1 . This is the basic idea of the trigger function. The trigger function introduced by Kain and Fritsch (1992) more specifically refers to a temperature perturbation ΔT added to a lifting parcel virtual temperature at the lifting condensation level in the context of the Fritsch–Chappell (1980) scheme. In the Fritsch–Chappell (1980) scheme, the convective instability of a lifted parcel is tested only at the LCL, and if the parcel ascent is stable then the test is simply repeated for another parcel beginning its ascent from the next vertical model level up. This structure is also inherited by the schemes of Kain and Fritsch (1990) as well as Bechtold et al. (2001). See also Kain (2004). Fritsch and Chappell (1980) defined the perturbation/trigger function as: 1/3

ΔT = c1 wG , where wG is the mesoscale-model vertical velocity at the LCL, and c1 is a constant chosen in order to realize ΔT = 1 K for wG = 1 cms−1 . Bechtold ¯ 1/3 , with Δx the et al. (2001) more specifically set wG = (Δx/Δxref )|w| model grid spacing, Δxref a reference grid length, and c1 = 6 Km−1/3 s1/3 . Kain (2004) modified this to: ΔT = c1 (wG − w∗ (z))1/3 with w∗ (z) =

 w0 zLCL /2000 zLCL ≤ 2000 m w0

zLCL > 2000 m

and w0 = 2 cms−1 , and zLCL the height of the LCL above the surface. Although this is the specific context where the concept of the trigger function is introduced, Kain and Fritsch (1992) proposed it as a more general (and “collective” in their own terminology) concept that constitutes “the complete set of criteria used to determine when and where deep convection occurs in a numerical model”. In essence, that concept is simply the “switch” from earlier in this section. 13.4

What is the “trigger”?

The notion of a convection trigger is very intuitive, and commonly used in the meteorological literature. Helpful reading in order to get a feeling for

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its use includes Shapiro et al. (1985) and Wilson and Schreiber (1986). The second paper more precisely talks about “initiation” rather than “trigger”. Defining the difference between these two concepts would be a good exercise for readers to try. Living in mid-latitudes, it is easily observed that an afternoon thunderstorm suddenly appears following a kind of trigger. Convection is a process of fluid motion in which everything occurs in a continuous manner without any sudden discontinuous change. Convection might appear to be triggered just as a bomb suddenly explodes. In order to explode a bomb in a controlled fashion, it is necessary to activate a mechanism called the “trigger”. But there is no definable mechanism that can be well specified as a trigger with convection. We may still say that “convection is triggered”, but this merely has an allegorical meaning, and to go beyond that it would be necessary to specify which process we are referring to by calling it the “trigger”. When referring to some specific process, it would be better to name that explicitly, rather than simply calling it a “trigger” in a vague manner. Note that an important earlier paper on convective storms by Newton (1950), for example, uses the term “trigger”, but always with quotation marks. Such a precaution has, unfortunately, been lost at some point in modern convection studies.

14

Observational investigations of the closure hypotheses

From the point of view of observational data analysis, the issue of convection-parameterization closure may be reduced to a question of what causes and controls convection. As reviewed in Yano et al. (2013a), in many of the observational data analyses, it is almost always forgotten that a closure assumption is best defined in terms of a budget equation such as Eq. 2.2: a large-scale forcing both causes and controls convection. Instead, simple correlations between a convection measure (such as precipitation rate, or outgoing longwave radiation) and another physical variable (such as CAPE, or column-integrated water vapour) are taken. The tendency terms have rarely been examined except for some obvious variables such as convergence. For this reason, unfortunately, observational investigations on the closure problem, directly or indirectly, have not been very successful so far. A more systematic data analysis from a budget point of view is still to be undertaken.

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Part II: Mass-flux parameterization

Free ride principle

There is a rather interesting observationally known constraint that is useful for further considering the issues of convective closure (cf., Ch. 4, Sec. 4.5). A naive application of this constraint even trivializes the convection parameterization problem. It is observationally well known that over the tropics, the vertical entropy advection is well balanced with the total diabatic heating. Writing down the heat equation (entropy budget) in terms of the potential temperature, and recognizing that diabatic heating Q can be roughly divided into that due to convection and radiation (with subscripts c and R, respectively), the above statement may be written as: ∂θ ≃ QR + Qc . (13.1) ∂z Fraedrich and McBride (1989) propose to call this balance the “free ride”, whereas Sobel et al. (2001) in a different context later name it the “weak temperature gradient” (WTG) approximation. Considering an intuitive appeal of the naming as well as its historical priority, it is referred to as “free ride” in the following. Although it is not emphasized in these two references, a similar balance is also found for the moisture budget, i.e.: w

Lv ∂q ≃ −Q2 , w Cp ∂z

(13.2)

to a lesser extent, but still as a relatively good approximation. Here, −Q2 is the condensative rate, and also has a contribution from vertical eddy transport (Ch. 8). This approximate form of the budget may also be considered as a part of the free ride balance. Observational evidence for the free ride balances is found in Fig. 1 of Yano (2001) and Figs. 2 and 3 of Yano et al. (2012b): cf., Fig. 4.2 in Ch. 4. These balance conditions have implications for the large-scale tropical dynamics discussed in Ch. 5. Leaving these discussions to the references therein, the purpose here is to make a more obvious point: convection parameterization can be closed simply by adopting the free ride balances, i.e.: ∂θ − QR (13.3a) Qc = w ∂z Q2 = −

Lv ∂q w . Cp ∂z

(13.3b)

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Furthermore, if the detrainment term is neglected in Eq. 4.1, these two formulae would suggest that:  −1 ∂θ (13.4a) Mc = ρw − ρQR ∂z and Mc = ρw

(13.4b)

respectively. Moreover, if the radiative forcing in Eq. 13.3a is negligible, Eq. 13.4a agrees with Eq. 13.4b. In fact, this rather simple formulation works surprisingly well in comparison with many current operational parameterizations, as discussed by Yano (2001). However, an obvious drawback of such a formulation is that it completely loses the predictability of either temperature (entropy) or moisture. A closure based on a vertical integral, as discussed throughout this chapter, can be recovered simply by vertically integrating Eq. 13.3a or 13.3b. These vertical integral-based closures would retain some predictability but would produce a free ride balance that is less well satisfied, by inferring a convective vertical structure separately by, for example, entrainment and detrainment. A closure formulation proposed by Geleyn et al. (1982) and Lindzen (1981, 1988) may, to some extent, be considered as a variation on the idea of a free ride closure. However, instead of applying a free ride formula directly, they propose to assume that the convective mass-flux divergence is equal to a moisture-weighted convergence rate, specifically: ρ ∂Mc = − (¯ v · ∇¯ q ), ∂z q¯ with v the three-dimensional velocity. Remarkably, Tiedtke (1989) took this as a definition of his “organized entrainment” in addition to a standard turbulent entrainment: Ch. 10, Sec. 7.2 presents this definition in the context of entrainment formulations more generally. Here, note that the distinction between the closure and the entrainment–detrainment formulation begins to blur (cf., Sec. 7.1). 15

Bibliographical notes

A complementary review on the subject of closure is given by Yano et al. (2013a) from a phenomenological point of view. A historical review by

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Arakawa and Chen (1987) may also be of interest. Details of the analysis in Sec. 10.5 are given in Yano and Plant (2012b), while details of that in Sec. 10.6 are given in Plant and Yano (2013); Yano and Plant (2012c). 16

Appendix A: Derivation of Eq. 8.21

For this derivation, we invoke Ch. 8, Eq. 4.4: qi − q¯∗ =

γ 1 ¯ ∗ ). (hi − h 1 + γ Lv

(A.1)

¯ ∗ = s¯ + Starting from the definition of the saturated moist static energy h Lv q¯∗ we immediately obtain: ∂ ¯h∗ ∂¯ s ∂ q¯∗ = + Lv . (A.2) ∂t ∂t ∂t By definition of the factor γ in Ch. 8, Eq. 4.2 we also have that: ∂¯ s ∂ q¯∗ = Lv ∂t ∂t and substitution into Eq. A.2 leads to: γ

γ 1 ∂ ¯h∗ ∂ q¯∗ = . ∂t 1 + γ Lv ∂t

(A.3)

From Eqs. A.1 and A.3 it follows that: γ 1 ∂hi ∂qi = . ∂t 1 + γ Lv ∂t We now take a vertical derivative of Eq. A.4, which produces:   ∂ ∂qi γ ∂hi 1 ∂ ǫi γ ∂ ¯ = + (h − hi ), ∂z ∂t Lv ∂z 1 + γ ∂t Lv 1 + γ ∂t

(A.4)

(A.5)

where, in deriving the second term on the right-hand side, we have invoked the relation: ∂hi ¯ − hi ) = ǫi (h (A.6) ∂z as derived in Ch. 7, Eq. 6.17, recalling that the moist static energy is conserved. Substitution of Eqs. A.4 and A.5 into the left-hand side of Eq. 8.21 leads to:

  ∂ γ ∂ ¯h ∂ q¯ ∂ ∂qi γ ∂hi + ǫi (qi − q¯) = + ǫi − ǫi Lv . (A.7) Lv ∂t ∂z ∂z 1 + γ ∂t 1 + γ ∂t ∂t

Closure

401

The aim is to obtain a closed expression in terms of the dry and moist static energies. For this goal, we note that, by definition: q¯ = q¯∗ −

1 ¯∗ ¯ (h − h). Lv

By taking the time derivative of the above: ∂ q¯ ∂ q¯∗ 1 ∂ ¯∗ ¯ = − (h − h). ∂t ∂t Lv ∂t The first term on the right-hand side can be rewritten using Eq. A.3, and after some rearrangement we obtain: 1 ∂ ¯h∗ ∂ ¯h ∂ q¯ =− + . (A.8) Lv ∂t 1 + γ ∂t ∂t Substitution of Eq. A.8 into Eq. A.7 leads to the result:

  γ ∂hi ǫi ∂ ¯ ∗ ¯ ∂ ∂qi ∂ Lv + ǫi (qi − q¯) = + (h − h). ∂t ∂z ∂z 1 + γ ∂t 1 + γ ∂t The final expression is obtained by inserting the relation of Eq. 8.21 to produce: ∂¯ s ∂ ¯h∗ = (1 + γ) . ∂t ∂t The relation itself follows directly from Eq. A.2.

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Chapter 12

Convective vertical velocity

J.-I. Yano Editors’ introduction: A major simplification of mass-flux convective parameterization is, as the name suggests, the ability to formulate the parameterization in terms of the mass flux alone, without the need to consider separately the fractional area and convective vertical velocity. Such a major simplification does have some significant drawbacks, however. Convective microphysics, for example, depends strongly on the vertical velocity, and knowledge of the mass flux is insufficient for anything beyond a relatively ad hoc, semi-empirical microphysical treatment. For this reason, various operational parameterizations also attempt to make use of an estimate of the vertical velocity. One approach would be to develop a means to estimate the fractional area of convective updraughts and then to diagnose the vertical velocity from the mass flux. However, defensible, physically based methods to estimate the area have proved difficult to construct, and so a more direct route is most often taken. Accordingly, this chapter presents and discusses methods that have been used to compute a convective vertical velocity.

1

Introduction

It was shown in Ch. 7 that the mass-flux convection parameterization formulation is closed once the mass flux is defined, as long as a convective variable in consideration is conservative. We may even say that this is the reason why the formulation is tagged as “mass flux”. However, once a nonconservative variable becomes a concern, this is no longer the case. Instead, it becomes necessary to calculate explicitly the convective vertical velocity, 403

404

Part II: Mass-flux parameterization

as well as the mass flux. The importance of the convective vertical velocity in order to properly treat microphysics problems in the mass-flux parameterization is especially emphasized by Donner (1993). Full discussions will be given in Vol. 2, Ch. 18. The goal of the present chapter is to review formulations for the convective vertical velocity within a mass-flux formulation. Most of the current mass-flux parameterizations which actually make an estimate of the convective vertical velocity either adopt the formulation by Levine (1959) or one of its variants. For this reason, the derivation of the original formulation by Levine (1959) is carefully examined here. This already reveals some historical ironies. Recall that the mass-flux convection parameterization assumes a plume as a basic element of convection. The hot tower, a concept introduced in Ch. 6, can be considered a special type of convective plume. The original formulation for convective vertical velocity by Levine (1959) is based instead on a bubble model for convection, rather than a plume. It was Simpson et al. (1965) and Simpson and Wiggert (1969) who took over this formulation and implemented it into the context of plume dynamics: a major historical twist. For this reason, this chapter begins with some reviews on the dynamics of the plumes and the bubbles.

2

Dimensional analysis (Batchelor, 1954)

Before moving on to review the formulation of the bubble dynamics by Levine (1959), let us first review a standard formulation for plume dynamics developed before this work. In order to provide a reference point for understanding Levine (1959) there follows a review of a dimensional analysis of plume dynamics by Batchelor (1954), which essentially yields a solution later provided by Morton et al. (1956) in a more concise manner. A form of a solution of a given system can often be determined solely by examining the dimensionality of the variables and the parameters in consideration. The basic principle of dimensional analysis is very simple: in order to express a solution of any variable, a mathematical expression must provide a consistent physical dimension. Sometimes, this condition is enough to define a full solution, or, if not a full solution, a universal function defined in terms of non-dimensionalized variables. The approach is best summarized by Batchelor (1954) himself:

Convective vertical velocity

405

Dimensional analysis proves to be an exceedingly powerful weapon, although I should add the warning, based on bitter experience of my own as well as a critical study of other people’s work, that it is also a blind, undiscriminating weapon and that it is fatally easy to prove too much and to get more out of a problem than was put in to it.

A dimensional analysis works well in the present problem when looking for a steady solution. A standard experimental plume is produced by a steady buoyancy force supplied at the bottom (or occasionally at the top) of an experimental apparatus, and a steady plume solution is sought experimentally. The heat flux H (Wm−2 ) is a key variable for controlling the whole process of the plume dynamics. Batchelor (1954) transformed its dimension by introducing a new variable F by H = (Cp ρ0 T /g)F, where Cp is the heat capacity at constant pressure (Jkg−1 K−1 ), ρ0 the reference fluid density (kgm−3 for the background state), T the temperature (K), and g the acceleration due to gravity (ms−2 ). The dimension for F is [m4 s−3 ]. It immediately follows that a unique way to express the heat flux in terms of a velocity w and a distance z is F ∼ w3 z. Here, these dimensional variables must be, more precisely, the plume-mean vertical velocity and the vertical height, respectively, in the present context of plume dynamics. The above expression can immediately be rearranged for the vertical velocity to obtain w ∼ (F/z)1/3 . Another key quantity is the buoyancy b, which may be defined by: T − T0 . b=g T0 Here, T0 is the temperature of the background environment. The dimension for the buoyancy is [ms−2 ]. Thus, a unique answer is b ∼ w2 /z ∼ F 2/3 z −5/3 . Furthermore, the radius of the plume, R, is R ∼ z, and the mass flux is M ≡ ρ0 πR2 w ∼ ρ0 F 1/3 z 5/3 . An interesting conclusion from this dimensional analysis is that the inflow velocity from the environment into the plume is proportional to the vertical velocity as shown by: 1 d M ∼ z −1/3 ∼ w. 2πR dz

3

Entraining plume (Morton et al., 1956)

Morton et al. (1956) solved the problem of the steady plume dynamics in a complete manner by more explicitly invoking three conservation principles:

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Part II: Mass-flux parameterization

conservation of (i) the plume volume; (ii) the momentum; and (iii) the mass (buoyancy) anomaly. These conservation principles are written down as: d (πR2 w) = 2πRα ˆw dz d (πρR2 w2 ) = πgR2 (ρ0 − ρ) dz d [πgR2 w(ρ0 − ρ)] = 0. dz Here, the inflow velocity into the plume is assumed to be α ˆ w, based on Batchelor’s dimensional analysis. In considering the momentum conservation, it is assumed that the plume is purely controlled by the buoyancy (right-hand side), and a contribution from the non-hydrostatic pressure is neglected. This would be true only for a plume strongly forced by buoyancy. Finally, the third conservation constraint states that the buoyancy flux is conserved when there is no density stratification. (Although an effect of density stratification is considered by Morton et al. (1956), it is not considered here.) These three equations may be rewritten in a slightly simplified manner by dropping common factors on both sides: d (R2 w) = 2Rα ˆw dz ρ0 − ρ d (R2 w2 ) = gR2 dz ρ0 

 ρ0 − ρ d = 0. gR2 w dz ρ0

(3.1a) (3.1b) (3.1c)

Eq. 3.1c can be immediately integrated vertically, and it gives:   ρ0 − ρ 2 =Q gR w ρ0

with Q a constant. As a result, Eq. 3.1b reduces to:

Q d (Rw) = . (3.2) dz 2Rw2 Eqs. 3.1a and 3.2 can be solved by assuming a solution of the form Rw ∼ z α and R2 w ∼ z β . In solving them, note that R ∼ z β−α and w ∼ z 2α−β , thus Rw2 ∼ z 3α−β . As a result, these equations lead to the two algebraic constraints: α − 1 = −3α + β α = β − 1,

Convective vertical velocity

407

and by solving them we obtain α = 2/3 and β = 5/3. By taking care of the constant factors, final answers for these dependent variables are: 1/3  9 α ˆQ z 2/3 Rw = 10  1/3 9 6 2 ˆ αQ ˆ R w= α z 5/3 , 5 10 which furthermore lead to solutions for more basic physical variables: 6 αz ˆ 5 1/3  5 9 w= α ˆQ z −1/3 6α ˆ 10 −1/3  5Q 9 ρ0 − ρ α ˆQ = g z −5/3 . ρ0 6α ˆ 10 R=

Finally, as an important reference, let us evaluate the fractional entrainment rate of this steady plume. Starting from Eq. 3.1a and defining the mass flux by M = ρπR2 w, we obtain: 2α ˆ 1 dM = . (3.3) M dz R This result is quoted in Ch. 10. The experimental results suggest the values α ˆ = 0.083 to 0.125. Note also that because R ∼ z in this plume dynamics, so also ǫ ∼ z −1 (cf., Ch. 10). ǫ=

4

Hill’s spherical vortex

Before presenting details of Levine’s formulation for convective vertical velocity, it is necessary to review Hill’s spherical vortex (Lamb, 1932). Levine (1959) used this model for defining the internal structure of his bubble, and therefore some understanding of Hill’s spherical vortex is required first. Bubbles are known to be associated with a vortex ring: a doughnutshaped vorticity concentration. Woodward (1959) is one of the first experiments which demonstrated this structure by detailed velocity measurements. Hill’s spherical vortex provides a convenient analytic formulation for describing such a structure. For this purpose, cylindrical coordinates (s, θ, z) are introduced, and it is assumed that the system is axisymmetric with no θ–dependence in the

408

Part II: Mass-flux parameterization

following. Recall that the vorticity equation is given by: ∂ ζ = ∇ × (v × ζ). ∂t Thus, a steady solution is given by ∇ × (v × ζ) = 0, where ζ = ∇ × v is the vorticity. Under the cylindrical coordinates, the equation to be satisfied can be written as: ∂ ∂ (vs ζ) + (vz ζ) = 0, (4.1) ∂s ∂z with ζ the vorticity in the azimuthal (θ) direction. In order to define a solution, first it is necessary to introduce a streamfunction ψ that defines the velocity by: 1 ∂ψ s ∂z 1 ∂ψ , vz = − s ∂s vs =

and the vorticity is given by:   1 ∂2 ∂2 1 ∂ ζ= + 2− ψ ≡ △ψ. s ∂z 2 ∂s s ∂s The aim is to find a vortex ring solution embedded inside a sphere of a radius R. Hill’s spherical vortex solution is given by: ψ=

1 2 2 As (R − r2 ), 2

(4.2)

with r2 = s2 + z 2 . As a result, the velocity components and the vorticity are given by: vs = −Asz vz = −A(R2 − r2 − s2 ) ζ = −5As. Direct substitution of these into Eq. 4.1 proves that Eq. 4.2 indeed constitutes a steady vortex-ring solution. This vortex solution constructed inside the sphere (r ≤ R) is connected to an irrotational flow outside the sphere with a vanishing flow at the surface. It is assumed that this solution is also asymptotically connected to a homogeneous vertical flow W at infinity. Such a solution is given by:   R3 1 ψ = W s2 1 − 3 , 2 r

Convective vertical velocity

409

with the velocity components given by: 3 R3 W sz 2 r5  

3 R 3 s2 vz = W 1− −1 . r3 2 r2 vs =

Deriving this solution from a general solution for the Laplace equation under cylindrical coordinates is rather lengthy. However, it can be easily verified that this solution is indeed irrotational (i.e., △ψ = 0) and satisfies the boundary condition vr = 0 at r = R. Finally, by continuity of the tangential velocity at the spherical surface, we find 3W . 2 R2 The drag force felt by this spherical vortex is an important aspect of defining a bubble vertical velocity equation in the next section. In order to know the drag force, we need to know the aerodynamic pressure p at the spherical surface. This can be defined by a Bernoulli function: A=−

p+

1 2 v , 2ρ0

which is conserved along a flow. At the spherical surface, by a direct calculation, we obtain: 9 W2 2 s . 4 R2 Substitution of the above into the Bernoulli function leads to: 9 p = ρ0 W 2 cos2 φ, 8 v2 =

(4.3)

with cos φ = z/R. Here, an arbitrary constant is defined in such a manner that the pressure averaged over the sphere vanishes. 5

Levine (1959)

Levine (1959) constructed his bubble solution by embedding Hill’s spherical vortex, which was reviewed in the last section. Levine’s bubble is spherical with a radius R. However, it is not a perfect sphere, but rather is left open below a level z = z0 , as schematically shown in Fig. 12.1. Here, the centre of the bubble is at z = 0, and an angle φ0 is defined by cos φ0 = z0 /R.

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Part II: Mass-flux parameterization

z

φ0

s z0

Fig. 12.1

Schematic representation of the geometrical structure of Levine’s bubble.

Levine (1959) presented an equation for the vertical velocity W at the centre position of the bubble after a relatively involved derivation. However, his final equation is straightforward to interpret:   ρ0 − ρ dM W = Mg (5.1) − M Fz . dt ρ0 Here, M is the total mass of the bubble, g is the acceleration due to gravity, and ρ0 and ρ are densities for the environment and the bubble respectively. The first term on the right-hand side is the buoyancy acting on the bubble, and the second, Fz , is the drag force per unit mass. Importantly, the pressure term does not appear in this problem by assuming an infinite space, and thus there is no bottom or top wall that induces back pressure upon the bubble. In other words, the bubble is sufficiently isolated from the other parts of the flow that the pressure term can be neglected. Another important assumption in the above formulation is that the environment is quiescent with no vertical movement. The drag force is still to be defined explicitly. For this purpose, it is assumed that there is no drag force acting on the bottom of the bubble (i.e., z ≤ z0 ) where it is open. Following this assumption, the total drag force acting on the bubble is:  φ0 p cos φR2 sin φdφ. Fz = 2π 0

Substituting Eq. 4.3, and performing an integral, we obtain: 9 Fz = πρ0 W 2 R2 (cos4 φ0 − 1). 16

(5.2)

Convective vertical velocity

411

A point to note is if Levine’s bubble is perfectly spherical and φ0 = π, then there is no drag force acting on the bubble, as for a frictionless solid sphere (the drag is felt only if there is a viscosity). This basic point is often forgotten in the literature. The picture of a partly open bubble is based on various earlier laboratory experiments, but notably on the results of Davies and Taylor (1950). However, this assumption could be misleading, because it amounts to assuming that the upper portion of the bubble behaves like a rigid body. A bubble should be, in fact, far from a rigid body. A good discussion on the issues of the drag force in retrospect is found in Malkus (1952). A standard formulation for representing the drag force is to set π Fz = ρ0 R2 W 2 CD , 2 with the drag coefficient defined in the present case by: 9 CD = (1 − cos4 φ0 ). (5.3) 8 In addition to the drag arising from the local pressure felt at the surface of the bubble, Levine further proposed to consider turbulent mass exchange which adds the terms:      dM dM W + dt + dt − to the total drag force M Fz . Here, (dM/dt)+ is the mass flux into the bubble (entrainment), and (dM/dt)− is the mass flux out (detrainment). These entrainment and detrainment terms are more precisely specified in terms of fractional entrainment and detrainment rates (3K2 /8 and 3K1 /8, respectively) so that:   3K2 dM (πρ0 R2 )W (5.4a) = dt + 8   dM 3K1 (πρR2 )W. (5.4b) = dt − 8

Here, the factor 3/8 is as given by Levine (1959). Thus, the total drag force is given by:

     dM dM π M Fz = (5.5) W + ρ0 R2 W 2 CD . + dt + dt − 2

The final goal is to write down a vertical velocity equation, which can be derived from the momentum equation of Eq. 5.1 as:     ρ0 − ρ dM 1 dW =g − M Fz + W . (5.6) dt ρ0 M dt

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Part II: Mass-flux parameterization

The rate of change of mass of the bubble is defined by:     dM dM dM = − . dt dt + dt −

(5.7)

Substitution of Eqs. 5.5 and 5.7 into the last two terms of Eq. 5.6 gives:   dM π dM =2 W + ρ0 R2 W 2 CD . M Fz + W dt dt + 2 By further substituting from Eq. 5.4a, we obtain: 3K2 π dM = (πρ0 R2 )W 2 + ρ0 R2 W 2 CD . dt 4 2 Finally, noting that the bubble mass is M = (4π/3)ρ0 R3 , we obtain:  2  W 3 3K2 W dM = + CD . (5.8) Fz + M dt 8 2 R M Fz + W

Substitution of Eq. 5.8 into Eq. 5.6 leads to the final answer:   2   ρ0 − ρ W dW 3 3K2 =g + CD . − dt ρ0 8 2 R

The above equation may be reinterpreted for a convective vertical velocity wc in general by rewriting the above:  2  wc 3 3K2 dwc = gb − + CD , (5.9) dt 8 2 R

where b = (ρ0 − ρ)/ρ0 is a non-dimensional buoyancy. Note that the buoyancy may be defined by various forms, as discussed in the Appendix of Yano et al. (2005a). An important modification is required when there is a vertical movement we in the background environment. Then:   dwc (wc − we )2 3 3K2 = gb − + CD . (5.10) dt 8 2 R

This modification is unfortunately forgotten most of the time in the literature. Here, d/dt = ∂/∂t + wc (∂/∂z). 6

Subsequent history

The convective vertical velocity equation derived by Levine (1959) was subsequently adopted by Simpson et al. (1965) and Simpson and Wiggert (1969) into their steady-plume model. Note that the steady version of the formulation by Levine (1959) is obtained by setting d/dt = wc (∂/∂z) in

Convective vertical velocity

413

Eq. 5.10. The purpose was to verify weather modification experiments, with the goal of assessing the capacity of cloud seeding to destroy harmful convective storms. Silver iodide was taken as a seed for ice condensation nuclei. The results never emphatically proved a human capacity for destroying convective clouds. Nevertheless, the analyses were required to be careful enough to verify their experimental results. For these verifications, a theoretical model was required. Their choice was the steady entrainingplume model originally proposed by Morton et al. (1956: cf., Sec. 3), in combination with Levine’s (1959) bubble model (cf., Sec. 5) in order to evaluate the vertical velocity. More precisely, Simpson et al. (1965) adopted exactly the same equation as Levine (1959), specifically Eq. 5.9 above, and they set K2 = 0.55 and CD = 0.65. Note that substitution of this value for CD back into Eq. 5.3 gives an angle φ0 = 143.7◦. However, Simpson and Wiggert (1969) later realized that the buoyancy acceleration under this model is too strong to be realistic. In order to compensate for this, they introduced a parameter γ, called the “virtual mass coefficient”. This is essentially a factor through which the effective mass of a bubble (or an air parcel) is enhanced. As a result, the convective vertical velocity equation is modified into:  2  wc gb 3 3K2 dwc = − + CD . (6.1) dt 1+γ 8 2 R

Here, Simpson and Wiggert (1969) take γ = 0.5 along with K2 = 0.506 and turn off the drag force by setting CD = 0. Recall that no drag force occurs for a bubble when a perfect spherical shape is assumed. Unfortunately, in spite of the rather dubious basis of Eq. 6.1, it is still widely used in convective parameterizations (e.g., Bechtold et al., 2001; Bretherton et al., 2004; Sud and Walker, 1999; Zhang et al., 2005). Sud and Walker (1999) took Eq. 6.1 with γ = 1.5, while Bechtold et al. (2001) ∗ = (3K2 /2) + CD = 0.2. took it with γ = 0.5 and CD Emanuel (1991) as well as Fritsch and Chappell (1980) further simplified ∗ Eq. 6.1 by setting γ = 0, CD = 0, and integrating it vertically:  z 2 bdz + w02 , wc = g zB

−1

where w0 = 1 ms in Fritsch and Chappell (1980) and w0 = 0 in Emanuel (1991). See, for example, Fig. 10 of Siebesma et al. (2003) for a verification of this equation from large-eddy simulations. Some modifications of this equation are attempted in the literature, but mostly in a non-fundamental manner.

414

7

Part II: Mass-flux parameterization

Convective vertical velocity under SCA

In Ch. 7, it was emphasized that the mass-flux convection parameterization can be constructed under a geometrical constraint of segmentally constant approximation (SCA). A general prognostic equation for the vertical momentum is similar to Ch. 7, Eq. 3.1b: 1 ∂ρw2 1 ∂p′ ∂ w + ∇ · (uw) + =− + gb. (7.1) ∂t ρ ∂z ρ ∂z Here, w is the vertical velocity, u the horizontal velocity, ρ the density (assumed a function of height only), p′ the dynamic pressure (deviation from the hydrostatic pressure), and b the non-dimensional buoyancy as before. Note that in the above equation, no drag force term is included. The above equation is averaged over a horizontal convective segment Sc in order to obtain the convective vertical velocity equation. An associated closed boundary is designated by ∂Sc . The result is:

 ∂ 1 ∂ 1 1 ∂p′c σc wc + ρσc wc2 + + gbc . wc,b (u∗c,b − r˙ c,b ) · dr = σc − ∂t ρ ∂z S ∂Sc ρ ∂z (7.2) Here, all the variables for the convective element are designated by the subscript c. Furthermore, σc is the fractional area occupied by convection, S is the grid box size, r˙ c,b is a local displacement rate of the convectiveelement boundary, and u∗c,b is the normal velocity to a constant segment boundary defined by: ∂rc,b , (7.3) u∗c,b = uc,b − wc,b ∂z with rc,b designating the position of the segment boundary. The subscript b designates the values at the convective-element boundary. The standard mass-flux formulation adopts the upstream condition for defining these boundary values. Note that an eddy contribution to the vertical flux is neglected, being consistent with the standard mass-flux formulation (Yano et al., 2004). Recall that the corresponding mass continuity is given by:  1 ∂ 1 ∂ (u∗ − r˙ c,b ) · dr = 0. (7.4) σc + ρσc wc + ∂t ρ ∂z S ∂Sc c,b Note that Eq. 7.2 is given in flux form. A corresponding advective version is obtained by subtracting Eq. 7.4 multiplied by wc from Eq. 7.2. The result is:

 ∂ ∂ 1 1 ∂p′c + gbc . (wc,b −wc )(u∗c,b −r˙ c,b )·dr = σc − σc wc +σc wc wc + ∂t ∂z S ∂Sc ρ ∂z (7.5)

Convective vertical velocity

415

Furthermore, by introducing the entrainment–detrainment hypothesis (cf., Ch. 7, Sec. 5) in the last term of the left-hand side, Eq. 7.5 may be rewritten as: ∂ 1 ∂p′c ∂ wc + wc wc = −ǫ(wc − we )2 − + gbc . (7.6) ∂t ∂z ρ ∂z Here, ǫ is the fractional entrainment rate. Comparing Levine’s vertical velocity equation of Eq. 5.10 with the derivation based on SCA (Eq. 7.6), there are several stark differences. The most noticeable is the presence of a perturbation pressure term in Eq. 7.6, which is entirely missing in Eq. 5.10. This issue is further discussed in the next section. The second is a contrast between the drag force term found in Eq. 5.10 and the entrainment term found in Eq. 7.6. An equivalence of the two terms can be formally established by setting   3K2 3 + CD . ǫ= 8R 2 Recall that Simpson et al. (1965) as a whole adopt Morton et al.’s (1956) entraining-plume hypothesis with the fractional entrainment rate defined by Eq. 3.3. Thus, by the consistency, we obtain:   3 3K2 + CD = 2α ˆ. 8 2 The values adopted by Simpson et al. (1965), and Simpson and Wiggert (1969) for the left-hand side are, respectively, 0.553 and 0.285, whereas the value for the right-hand side is about 0.2 according to Morton et al. (1956). Thus, Simpson et al. (1965) adopt a much stronger entrainment rate than that expected from a pure entraining plume. Such a treatment is arguably inconsistent. 8

The issue of perturbation pressure

The role of perturbation pressure in a convection problem (or any fluid dynamics problem) is probably best seen by examining the mass continuity in the problem. The latter is given in continuous form by: 1 ∂ ρw = 0. (8.1) ∇·u+ ρ ∂z However, in standard fluid dynamics computations, the mass continuity is not directly invoked, but the three-dimensional fluid-velocity equation is

416

Part II: Mass-flux parameterization

updated numerically at every time step under a given perturbation pressure. The role of the perturbation pressure is to ensure the consistency of the velocity evolution with mass continuity. As already discussed in Ch. 7, Sec. 3.1, such a constraint is given by: ∇2 p′ = −∇ · ∇ · ρvv + ∇ · ρgb,

(8.2)

where v ≡ vH + zˆw is the three-dimensional velocity, and b is the nondimensional buoyancy. Eq. 8.2 can be solved for p′ by inverting the Laplacian ∇2 once the right-hand side source term is known. The source term can be evaluated, in turn, once v and the non-dimensional buoyancy are known. As discussed in Ch. 7, Sec. 4.6, it is possible to solve this Poisson problem under the mass-flux formulation or SCA under a formal procedure. See Yano et al. (2010a) for more details. However, in the current operational implementations, this perturbation pressure problem is not properly addressed by examining the Poisson equation in Eq. 8.2. Instead, various local approximations are attempted. The classic example is the virtual mass coefficient γ, introduced by Simpson and Wiggert (1969), as already discussed in Sec. 6. However, it should be realized that such a local approach could be fundamentally misleading, and arguably even simply wrong, considering the nonlocal nature of the Poisson problem: even when the source term (right-hand side) is well localized in space, its Laplacian inversion has a long-distance influence. The same problem is identified in the process of inverting potential vorticity into a streamfunction. An illuminating example from the latter context can be found in Hoskins et al. (1985). For this reason, a more serious effort is required in order to solve the Poisson problem of Eq. 8.2 properly in order to define the convective perturbation pressure, albeit in a very approximate manner.

9

Convection and buoyancy

Convection is driven by buoyancy. However, the atmospheric convection literature tends to further simplify the picture by imagining that a convective air mass is locally driven by buoyancy. The chapter on cumulus convection in the book by Houze Jr. (1993), for example, promotes such a picture. This statement is indeed true for a localized thermal bubble considered by Levine (1959), as reviewed in Sec. 5. However, it is no longer true for a thermal plume, in which dynamic pressure plays a crucial role,

Convective vertical velocity

417

as discussed in the last two sections. This is a compelling reason for developing a more elaborate description of convective vertical velocity within parameterizations. A series of field measurements has been taken by penetrating into deep convective cores with aircraft in order to more directly examine the relationship between convective vertical velocity and local buoyancy. Here, “core” refers to a convective centre where the vertical velocity is strongest in magnitude (updraughts or downdraughts). In these measurements, the local buoyancy is defined as a deviation of the local virtual temperature from a background state (defined as an area average). Such measurements show that, against intuitive anticipation, the convective updraught cores are not predominantly positively buoyant, and nor are the convective downdraught cores predominantly negatively buoyant. Histograms for the local buoyancy are almost comparable for both updraught and downdraught cores, with the buoyancy being widely spread, roughly equally on the positive and negative sides. The middle point (as well as the mean value) is found almost at the point of neutrality, although slightly positive within the updraughts (e.g., Fig. 7 of Lucas et al. (1994) without water loading; Figs. 5 and 10 of Wei et al. (1998) with water loading). A scatter plot between updraught vertical velocity and local buoyancy (e.g., Fig. 8(a) of Jorgensen and LeMone (1989); Fig. 6 of Wei et al. (1998)) does not suggest much correlation, while the corresponding plot between downdraught velocity and local buoyancy (Fig. 8(b) of Jorgensen and LeMone, 1989) suggests that stronger downdraughts are more likely to be associated with more positive buoyancies. As emphasized in Ch. 11, in order for convection to be generated, buoyancy and vertical velocity must be positively correlated. However, that is not necessarily the case everywhere, and the above field measurements suggest indeed it is often not the case, because the pressure force tends to redistribute momentum spatially. The parcel theory first introduced in Ch. 1 and exploited in many places so far is a very useful tool for inferring and understanding atmospheric convective instability in a simple way. However, it is important to remember that real convection does not behave like a simple ensemble of air parcels, as the parcel theory envisions. Rather they interact with each other through the pressure force, and that simple point makes the evaluation of convective vertical velocity rather involved.

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Chapter 13

Downdraughts

J.-I. Yano Editors’ introduction: A downdraught is a commonly observed feature of the circulation pattern of many convective systems. An individual cumulonimbus may be associated with an adjacent area of rapidly descending air, which may in turn be associated with strong surface winds and the formation of a pool of cold, low-lying air. Similarly, organized convective systems are associated with coherent mesoscale streams of descending air. Downdraughts have contrasting effects on the life cycle and organization of convective systems. On the one hand, the descending air modifies the structure of the underlying boundary layer, which may dampen or inhibit convection. On the other, particularly when embedded within a sheared environment, the associated near-surface outflow can produce near-surface convergence and uplift that induces new convective elements. This chapter discusses how such descending air streams can be incorporated into the mass-flux parameterization framework, and explains the additional assumptions that are needed in order to describe them. Most operational mass-flux parameterizations do include downdraughts within their formulation, but the chapter concludes with a provocative question: is their explicit treatment necessary for parameterization purposes?

1

Introduction

Global Atmospheric Research Programme’s (GARP) Atlantic Tropical Experiment (GATE) was organized over the tropical Atlantic during 1974 (Houze Jr. and Betts, 1981). It would be fair to say that GATE remains the best field campaign ever organized in tropical convection studies for the 419

420

Part II: Mass-flux parameterization

level of preparation and organization, as well as in terms of the outcomes. An almost-forgotten historical anecdote may be relevant here. The campaign was organized in the middle of a period of reduced tensions during the Cold War. Partially in a symbolic acknowledgement of the political environment, the involvement of the Soviet Union was called for. The campaign was also originally planned over the western Pacific, where tropical convection is known to be most vigorous. At the last moment, however, a US authority realized that it would not be wise to organize such a campaign over a strategically rather sensitive region. Thus, the campaign was diverted to the tropical Atlantic instead. The original plan of the GATE project over the western Pacific was somehow revisited by TOGA-COARE (Webster and Lukas, 1992). However, it did not provide as much new learning as GATE. The timing of the GATE field campaign was almost perfect: the year 1974 was also the year of publication of two milestone papers on convection parameterization: Arakawa and Schubert (1974) and Kuo (1974). Many observational diagnoses of mass-flux parameterization naturally followed (e.g., Yanai and Johnson, 1993), and some of those reviewed in this chapter pointed to the importance of downdraughts in atmospheric convection. The most fundamental objective of GATE was a better understanding of the relationship between tropical convection and large-scale flow patterns (Houze Jr. et al., 1980). This information was expected to further improve convection parameterization formulations. As it turned out, the campaign revealed a much more complex structure of tropical convection than that envisaged under Riehl and Malkus’s (1958) hot-tower hypothesis (cf., Ch. 6). Arakawa and Schubert’s (1974) mass-flux convection parameterization constructed following that vision is a considerable oversimplification of realistic tropical convection. The downdraught, both in convective and mesoscales, was identified as the most important additional ingredient to be added to the convection parameterization. Another important finding was the widespread existence of mesoscale organization within tropical convection, which was not widely appreciated before the GATE campaign. Thus, this chapter also touches on the issue of convective organization in association with the mesoscale downdraught. Houze Jr. (1977) summarized a perspective on convection parameterization obtained by the GATE project as follows: Generally, the overall squall-line airflow pattern proposed and later refined by Zipser (1969, 1977) . . .. These mean air motions are the result of a complex interaction of convective and mesoscale components of the

Downdraughts

421

squall-line system, some of which have time and space scales smaller than those of the system as a whole. If the convection associated with tropical squall lines and its interaction with the large-scale environment are to be accurately modelled and parameterized, precise and detailed descriptions on all scales, extending down to the scale of individual cumulonimbus cells, are needed.

Johnson (1980) furthermore says: It is apparent that theories for the parameterization of moist convection in large-scale numerical prediction models must in some way include the effects of mesoscale as well as convective-scale downdraughts, if accurate predictions of boundary-layer moisture are required.

2

Discovery of downdraughts

In the mid-latitude context, it was probably the Thunderstorm Project (Byers and Braham, 1949) that revealed, along with many other features, that convective storms not only consist of concentrated upward motions (updraughts) associated with convective towers, but also of concentrated downward motions, which may be called “downdraughts”, or even “downbursts” when they are especially strong. Over the tropics, a naive picture of hot towers originally proposed by Riehl and Malkus (1958) persisted initially. However, observational studies (Zipser, 1969, 1977), notably during GATE, gradually revealed the existence of downdraughts in tropical convection as well. A major difference between downdraughts and updraughts is that the former are usually associated with unsaturated air, whereas the latter are associated with cloudy, saturated air. Here, the terminology of the literature is slightly confused. Some literature suggests a distinction between saturated and unsaturated downdraughts. However, saturated downdraughts are rare. Zipser (1977) pointed out that downdraughts over a tropical heavy precipitation area are almost saturated. However, such near-saturated air is so cold that it is still drier than the boundary-layer environment air. The vertical extent of downdraughts varies from case to case, although the minimum height is likely to be at the cloud base. Downdraughts often appear to originate from a mid-tropospheric level (400 to 600 hPa) where the minimum equivalent potential temperature is found over the tropics. This conclusion can be drawn simply by assuming that the equivalent potential temperature is well conserved along a Lagrangian trajectory of the air (cf., Zipser, 1969, 1977). A more careful diagnosis based on a mass-flux

422

Part II: Mass-flux parameterization

formulation by Johnson (1976) also supports this inference. Note that rain evaporation within a downdraught may cool the atmosphere, but it does not reduce the equivalent potential temperature. Downdraughts also often extend to mesoscale sructures. For example, a downdraught identified by Zipser (1969) over the eastern Pacific extended over 500 km as shown by a sketch in his Fig. 6. This further suggests its importance in mesoscale convective dynamics. 3 3.1

Basic considerations Microphysical difference from the updraught

The downdraught may be, crudely speaking, considered as an upside-down version of the updraught. However, the problem is not quite symmetric due to the different actions of the microphysical processes. Roughly speaking, the updraught, within clouds, is driven by condensation of water vapour into cloud droplets, whereas the downdraught, within rain, is driven by evaporation of precipitating water into water vapour. A drastic size difference between cloud droplets and precipitating water droplets leads to a qualitative difference between the updraught and the downdraught. Both cloud-droplet growth and the evaporation of precipitating water are controlled by the process of water vapour deposition. A careful derivation of a formulation for describing this process is given separately in Sec. 10. By referring to the details there, the rate of change of mass of a water droplet with a radius r is given by: dm = 4πrDe (ρv − ρ∗v ) dt (cf., Sec. 10, Eq. A.7), where ρv is the water vapour density and ∗ indicates a saturated value; De is an effective thermal diffusion rate with a precise definition given by Eq. A.16: ˜ v, De = f˜dD where f˜ is a ventilation factor, d˜ is a thermodynamic factor defined by Eq. A.15, and Dv is the molecular thermal diffusion rate. Alternatively, we may write: dm = 4πrDe ρ∗v s, dt where s = RH − 1 is the super saturation.

(3.1)

Downdraughts

423

When the number concentration of the droplets is N , the rate of change of the liquid-water content is given by: dm = 4πN rDe ρ∗v s. (3.2) dt Here, when many droplets are considered, r is reinterpreted as a mean radius. The water vapour deposition or re-evaporation rate is balanced by a supply or a withdrawal of the water vapour by vertical advection, thus: N

dq ∗ = 4πN rDe ρ∗v s. (3.3) dz For simplicity for now, it is assumed that the environment is approximately at saturation. Otherwise, dq ∗ /dz must be replaced by dq/dz as in the next subsection. From now on, we may consider the vertical velocity w to be a positive definite quantity, but oriented upwards and downwards for updraughts and downdraughts, respectively. Also, as a result, for the downdraught, s is reinterpreted as a sub-saturation rate. Super-saturation for updraughts or sub-saturation for downdraughts is estimated from Eq. 3.3 by: ρw

s=

dq ∗ w . 4πN rDe q ∗ dz

(3.4)

Next, the vertical gradient of the saturated water vapour may be estimated by separating out the temperature gradient factor as: dq ∗ dT dq ∗ = . dz dz dT Typically, dT /dz = 6 × 10−3 Km−1 , and Lv 1 dq ∗ ≃ ≃ 6 × 10−2 K−1 ∗ q dT Rv T 2 by neglecting a contribution from a pressure perturbation in an analysis given by Eq. A.9 in Sec. 10. Thus: 1 dq ∗ ∼ 4 × 10−4 m−1 . q ∗ dz Another factor relatively independent of whether updraughts (clouds) or downdraughts (rain) are under consideration is a part of the denominator, which is:   De 4π ∼ 10−4 m2 s−1 f˜

424

Part II: Mass-flux parameterization

by setting De /f˜ ∼ 10−5 m2 s−1 . Here, the ventilation factor f˜ is factorized out because the value may be substantially different between clouds and rain. Then the total fixed factor becomes: f˜ dq ∗ ∼ 4 sm−3 , 4πDe q ∗ dz and its substitution into Eq. 3.4 leads to: s ∼ 4 [sm−3 ] ×

w . ˜ fNr

(3.5)

Note also that a typical value of liquid-water content (LWC) is: LWC ∼ ρ∗v ∼ 10−2 kgm−3 , which may also be estimated from the number density N and the droplet radius r as: 4 LWC = πr3 ρl N, 3 where ρl ≃ 103 kgm−3 is the liquid-water density. When LWC is considered as another approximate constant constraint, Nr =

LWC ∼ 3 × 10−6 r−2 . (4π/3)r2 ρl

A further substitution of the above into Eq. 3.5 leads to: s ∼ 106 [sm−3 ] × 3.1.1

wr2 . f˜

(3.6)

Updraught

For the updraught within clouds, we may set w ∼ 1 ms−1 , r ∼ 10 μm∼ 10−5 m, and f˜ ∼ 1, which leads to: s ∼ 106 [sm−3 ] × 10−10 [m3 s−1 ] ∼ 10−4 ∼ 0.01%. Thus, the super-saturation level is extremely low within a cloud due to the small cloud droplet size, which also leads to a high concentration of particle number density, which may be estimated as: N=

LWC ∼ 3 × 10−6 r−3 ∼ 3 × 109 m−3 . (4π/3)r2 ρl

This estimate supports the assumption made in earlier chapters that the updraught air is at saturation above the lifting condensation level.

Downdraughts

3.1.2

425

Downdraught

In contrast to the updraught, the downdraught is dominated by rain water, whose typical size is r ∼ 1 mm∼ 10−3 m. Substitution of this value leads to: s ∼ 106 [sm−3 ] × 10−6 [m3 s−1 ] ∼ 1, which suggests the downdraught is almost at zero relative humidity. Note that as a result of the large particle size, the number density of rain is typically as small as N ∼ 3 × 103 m−3 by taking the same formula as for the cloud droplets above. This extreme estimate may be somewhat amended by invoking the turbulent nature of the downdraught, which may lead to a relatively large ventilation factor, say, f˜ ∼ 10. Then, we obtain s ∼ 0.1 instead. Thus, unlike the updraught, the downdraught is fundamentally unsaturated. Although the literature often makes a distinction between saturated and unsaturated downdraughts, downdraughts are never as close to saturation as the updraughts. This point is emphasized by Emanuel (1991) in introducing his parameterization, although his introduction of an unsaturated downdraught is not an originality (cf., Sec. 5.1). 3.2

Lagrangian trajectory analysis

How does the thermodynamic state of the downdraught air evolve as it descends? A simple Lagrangian trajectory analysis is performed in order to answer this question in this subsection. For this purpose, Eq. 3.3 is cast in the form dq = 4πrDf˜Δq, w dz where the dryness of the downdraught air is measured by: Δq = qW − q

(3.7)

with the saturated water vapour mixing ratio at the surface of rain water being assumed to be that of the wet-bulb temperature TW : qW = q ∗ (TW ). This equation may be alternatively written as: Δq dq =− , (3.8) dz ΠE with w . (3.9) ΠE = − 4πrDf˜

426

Part II: Mass-flux parameterization

Note that ΠE is defined to be positive as w < 0 for the downdraught. Here, also note that the vertical velocity w is taken to be negative downwards in this subsection in a change from the last subsection. Eq. 3.8 may be further rewritten as:   dqW Δq dΔq = − (3.10) dz ΠE dz θe by noticing that the equivalent potential temperature θe is approximately conserved along a downdraught air trajectory. This equation describes the change of the dryness Δq of the downdraught air along its Lagrangian descent. 3.2.1

Simple solution

In order to infer a general behaviour of the solution for Eq. 3.10, it is assumed that both ΠE and (dqW /dz)θe are constant in this subsection. Under these assumptions, Eq. 3.10 can readily be solved analytically, and we obtain:    

  zLFS − z dqW zLFS − z 1 − exp − ΠE , Δq = ΔqLFS exp ΠE dz θe ΠE (3.11) where the subscript LFS denotes the originating level of the downdraught, or the “level of free sinking” (LFS). If we let Δz = zLCL − z be the difference of the height z in respect of the lifted condensation level (LCL), then:   dqW (3.12a) Δq = −Δz dz θe   dθW Δθ = Δz . (3.12b) dz θe Here, Δz, Δq, and Δθ are defined such that they are usually all positive when the air parcel is below LCL. Substitution of the first expression above into Eq. 3.11 leads to:

      dqW zLFS − z dqW , + (ΔzLFS − ΠE ) exp Δq = − ΠE dz θe dz θe ΠE (3.13) where ΔzLFS = zLCL − zLFS . This equation shows how the downdraught air dryness evolves as it descends over time.

Downdraughts

427

An alternative approach for inferring the dryness of the downdraught is to apply a Taylor expansion to the definition of qW , which leads to: qW = q ∗ − γΔq, where γ=

Lv dq ∗ , Cp dT

as already introduced in Ch. 8. Also note that by definition itself, qW = q + Δq. By combining these two expressions for qW , we obtain: q ∗ − q = (1 + γ)Δq. By further substituting Eq. 3.12a, we obtain:   dqW Δz , 1 − RH = − ∗ (1 + γ) q dz θe

(3.14)

recalling that the relative humidity is RH = q/q ∗ by definition. This expression shows how the dryness of the downdraught increases as the downdraught air descends away from the cloud base (LCL). 3.2.2

General solution

In order to obtain an analytical expression for a more general case, without assuming the constants as above, here it is instead assumed that the downdraught temperature path parallels the environmental lapse rate. Thus: dθ dθ¯ ¯ = = Γ. dz dz Note also that along a moist adiabat, the equivalent potential temperature is conserved, thus: Cp T dθ dq =− . (3.15) dz Lv θ dz By substituting this relation into Eq. 3.8, we further obtain: Δq Cp T ¯ Γ. (3.16) = ΠE Lv θ Meanwhile, from a direct vertical integral, we obtain:   dqW dq − Δq = ΔqLFS + (z − zLFS ) dz dz or     dqW dq dqW Δq = − ΔzLFS + (z − zLFS ) dz dz dz

428

Part II: Mass-flux parameterization

by applying Eq. 3.12a to the first term on the right-hand side. Furthermore, by noting Cp T dqW =− ΓW , dz Lv θ we obtain: Cp T ¯ [ΓW ΔzLFS + (ΓW − Γ)(z − zLFS )]. (3.17) Δq = Lv θ By comparing it with Eq. 3.16, we find ¯ = ΓW ΔzLFS + (ΓW − Γ)(z ¯ ΠE Γ − zLFS ).

(3.18)

The dryness of the downdraught air may also be related to its coldness ΔT by: Cp ΔT. (3.19) Δq = Lv Substitution of Eqs. 3.15 and 3.19 into Eq. 3.8 leads to: ¯ = Δθ. ΠE Γ

(3.20)

¯ − zLFS ). Δθ = ΓW ΔzLFS + (ΓW − Γ)(z

(3.21)

Thus:

By further noting Δθ = ΓW Δz, we obtain: ¯ ΓW − Γ Δz = ΔzLFS + (z − zLFS ). (3.22) ΓW Eqs. 3.17, 3.18, 3.21, and 3.22 show that Δq, ΠE , Δθ, and Δz increase ¯ > 0, which is the as the downdraught air moves downwards when Γw − Γ usual situation in the lower troposphere. In this section, it has been shown how the dryness of the downdraught air can be analysed along its Lagrangian trajectory. As long as the environmental lapse rate remains as a moist adiabat, the downdraught air becomes drier as it descends downwards. 4

Importance of downdraughts

There are various theoretical studies that have led to an expectation that the inclusion of downdraughts is important for understanding the interactions between convection and large-scale dynamics (see Ch. 5 for a general review). More specifically, as seen from Ch. 11, Eq. 11.3 and Ch. 11, Eq. 11.4, the downdraught modifies the effective stratification of the system

Downdraughts

429

through the precipitation efficiency ǫp , with the former being proportional to 1 − ǫp . Emanuel (1993b) and Yano and Emanuel (1991) argued that this point accounts for the small phase speed of the MJO (Madden–Julian oscillation), consistent with observations. Emanuel (1989), in turn, argued that the existence of unsaturated downdraughts as well as shallow convection is crucial for the generation of tropical cyclones. With these considerations, it would thus be desirable to include downdraughts into operational mass-flux convection parameterizations. That is the subject of the next section.

5

Mass-flux formulation for downdraught

At a formal level, it is rather straightforward to include downdraughts in the standard mass-flux convection parameterization formulation described in Ch. 7, Sec. 7. The total convective mass flux is decomposed into updraughts and downdraughts, and thus the grid box-averaged mean vertical velocity is decomposed as: 1 w ¯ = [Mc − Md + σe we ], ρ where Mc and Md are the total mass fluxes for updraughts and downdraughts, respectively, being defined as a sum of an ensemble of elements: Mc =

Nu 

Mi,u

i=1

Md =

Nd 

Mi,d ,

i=1

in which subscripts u and d are added in order to distinguish between the updraughts and downdraughts. Note that the downdraught is defined to be positive downwards. The approximation σe ≃ 1 may be taken under the standard formulation. Recall that under the standard formulation, each updraught mass flux profile is described by a plume model: ∂ Mi,u = (ǫi,u − δi,u )Mi,u ∂z in terms of fractional entrainment and detrainment rates ǫi,u and δi,u . Each downdraught mass flux may be described in a similar manner, but by taking an upside-down plume, or “inverse plume”: ∂ (5.1) Mi,d = −(ǫi,d − δi,d )Mi,d ∂z

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Part II: Mass-flux parameterization

in terms of fractional entrainment and detrainment rates ǫi,d and δi,d . Here, a negative sign is added to the right-hand side, because the downdraught expands and contracts by entrainment and detrainment, respectively, as it moves downwards. As for the case of the updraughts, an ensemble of entraining plumes may be taken by setting ǫi,d = ǫi constant and δi,d = 0. The total tendency of a generic physical variable ϕ can be decomposed into contributions from convection and large scales:     ∂ ϕ¯ ∂ ϕ¯ ∂ ϕ¯ = + . ∂t ∂t conv ∂t LS When the downdraughts are also considered, the convective tendency may be further divided into parts coming from the updraughts and the downdraughts:       ∂ ϕ¯ ∂ ϕ¯ ∂ ϕ¯ = + , ∂t conv ∂t u ∂t d where 

∂ ϕ¯ ∂t



u

 Nu 1 ∂ φ¯  D Mc + Di,u (ϕi,u − ϕ) ¯ = ρ ∂z i=1

(cf., Ch. 7, Eq. 6.14). The tendency for the downdraughts can also be defined in a similar manner, but taking into account the fact that it is taken positive downwards:

   Nd ∂ φ¯  1 ∂ ϕ¯ D Md + Di,d (ϕi,d − ϕ) ¯ . = ∂t d ρ ∂z i=1

In order to evaluate the large-scale tendency due to the downdraught, it is necessary to know the detrained value ϕD i,d from the downdraught. As in the case with the updraught, the physical variables within the downdraught can be evaluated by vertically integrating a diagnostic equation: ∂ Mi,d ϕi,d = Ei,d ϕ¯ − Di,d ϕi,d + ρσi,d Fi,d ∂z

(5.2)

or ∂ ϕi,d = ǫi,d (ϕ¯ − ϕi,d ) + ρσi,d Fi,d /Mi,d ∂z in analogy with Ch. 7, Eq. 6.8 and Ch. 7, Eq. 6.10. As a major difference from the updraught, the above vertical integral must be performed downwards. The following subsections discuss specific issues that must be resolved to complete the downdraught formulation.

Downdraughts

5.1

431

Rain evaporation rate within downdraught

Within the downdraught, the rain evaporation rate must be evaluated, and so in principle, an appropriate mircrophysical formulation must be introduced. A key variable to consider within the downdraught is the water vapour mixing ratio, as already emphasized in Sec. 3. By referring to the general formula in Eq. 5.2, the essence of the formulation presented therein must be translated in terms of: ∂ Mi,d qi,d = Ei,d q¯ − Di,d qi,d + ρσi,d evpi,d , (5.3) ∂z where evpi,d is the evaporation rate within the downdraught. Note that ˜d within the downdraught is given by vertically the total evaporation E integrating ρσi,d evpi,d over the whole downdraught layer, and also summing over all the downdraught types:   zLFS ˜ ρσi,d evpi,d dz. Ed = i

zB

Emanuel (1991) explicitly evaluated the evaporation rate based on a bulk microphysical formulation for his bulk unsaturated downdraught formulation. However, the majority of schemes avoid such explicit microphysical calculations in one way or another. For example, Fritsch and Chappell (1980) circumvented this issue by assuming that within the downdraught, the air remains at a constant relative humidity RH= 80% by default. The downdraught air entrains environmental moisture as it descends. By maintaining a constant relative humidity, the rain evaporation rate can be diagnosed backwards by examining the difference between the constant RH air and the air modified by entrainment. Tiedtke (1989) and Zhang and McFarlane (1995) adopted the same idea but simply maintained the downdraught at saturation. A benefit of these assumptions is that the evaporation rate can then be diagnosed backwards from Eq. 5.3 without invoking any microphysical formulation directly. Donner (1993) more simply assumed a constant fractional evaporation rate relative to the condensation rate by following Leary and Houze’s (1980) observational diagnosis. ˜d may be related Alternatively, the downdraught re-evaporation rate E to the surface precipitation rate PCP via the precipitation efficiency ǫp introduced in Ch. 11, Sec. 11.3: 1 − ǫp E˜d = PCP. (5.4) ǫp

432

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Part II: Mass-flux parameterization

Entrainment and detrainment

Entrainment and detrainment rates for the downdraught are usually defined in a similar manner as for the updraught. Thus, Cheng (1989) and Johnson (1976) assumed a reverse entraining plume, while Tiedtke (1989) assumes an equal rate for both entrainment and detrainment. However, in many cases, some simplifications are attempted relative to a fairly complex entrainment–detrainment formulation used for the updraughts. In particular, a forced detrainment aspect is usually not considered for the downdraughts.

5.3

Closure

The formulation presnted so far for the downdraught closely parallels that for the updraught. Further specifications of the downdraught are also still paralleled, though some special aspects must be considered. The vertical mass-flux profile may be obtained by vertically integrating Eq. 5.1 with given entrainment and detrainment rates, as in the case of the updraught. However, unlike the case of the updraught, this vertical integral is performed downwards starting from the top. The top of the downdraught is referred to as the “downdraught originating level”, or “level of free sinking” (LFS) in the literature. This level must first be specified, and that is where the first major uncertainty is encountered. For the updraughts, the starting point is relatively obvious: usually at the top of the planetary boundary layer or the cloud base, but not far above. On the other hand, downdraughts could start at any vertical level. Johnson (1976), for example, proposed to take the starting point to be a fractional height (in pressure coordinates p) relative to the top of the corresponding updraught plume, i.e., pi,LFS = pB − βLFS (pB − pi,T ), where βLFS is a fractional constant. When an entraining plume is assumed, the corresponding updraught plume means the one with the same fractional entrainment rate ǫi . Alternatively, Fritsch and Chappell (1980), who introduced the term LFS originally, proposed to take the origin level as the highest level where the temperature of a saturated mixture of equal amounts of updraught and environmental air is less than the environmental temperature, so that a negative buoyancy could be induced. This definition is also followed by Bechtold et al. (2001) and Tiedtke (1989) as well in the Met Office Unified Model. Here, an equal mixing ratio is the simplest assumption to take, and

Downdraughts

433

clearly further elaborations would be possible, based on an LES analysis for example. Zhang and McFarlane (1995), on the other hand, simply took the LFS as equal to the lowest updraught detrainment level, or to the level of minimum moist static energy if the latter is lower. Next, the magnitude of the downdraught must be specified at the originating level (LFS): an equivalent issue to the closure problem for the updraught. It is commonly expressed as a fractional value βd of the corresponding updraught mode: Mi,d,LFS = βd Mi,B . By observational diagnosis, Johnson (1976) suggests a range of values, βd = 0.2 to 0.4. Tiedtke (1989) assumes βd = 0.2, the ECMWF model takes βd = 0.35, and the Unified Model takes βd = 0.05. Alternatively, βd may be related to the precipitation efficiency, as already suggested in Ch. 11, Sec. 11.3, and as also assumed by Bechtold et al. (2001). Zhang and McFarlane (1995) proposed the formula:   PCP , βd = μ ˜d /βd PCP + E with a constant assumed to be μ = 0.2. Here, PCP is the total precip˜d /βd is the unscaled total evapitation rate within the updraught, and E oration within the downdraught. The unscaled value for this evaporation term must be adopted here, because the actual total downdraught evaporation is not known until the closure is established. The basic idea of this formulation is to maintain the magnitude of the rain evaporation within the downdraught at an appropriate level. Thus, the closure factor is scaled down more strongly when the unscaled evaporation rate is larger. An alternative closure proposed by Ducrocq and Bougeault (1995), following a bulk approach, is to assume that:  zLFS ∂q ˜d , Md dz = E ∂z zB ˜d is the total re-evaporation of precipitating water within the downwhere E ˜d is then related to the surface precipitation rate PCP draught. This rate E via the precipitation efficiency ǫp , as in Eq. 5.4. The values of the initial convective downdraught variables at the originating level (LFS) must also be specified. Two major choices are: (i) to take these equal to the saturated environmental value at the LFS: ϕi,d (zLFS ) = ϕ¯∗ (zLFS ),

(5.5)

434

Part II: Mass-flux parameterization

or (ii) to consider an equal mixture of the saturated environment and the convective updraught values: 1 (5.6) ϕi,d (zLFS ) = [ϕ¯∗ (zLFS ) + ϕi,u (zLFS )]. 2 Johnson (1976) used the condition of Eq. 5.5 for the moist static energy in this diagnostic study. The Unified Model uses Eq. 5.6 for the potential temperature and the water vapour, but uses Eq. 5.5 for the liquid water, and applies these relations without the saturation condition. Bechtold et al. (2001) further generalized the idea of Eq. 5.6 by setting ¯ LFS ) + (1 − χ)ϕi,u (zLFS ) ϕi,d (zLFS ) = χϕ(z with a mixing fraction χ. More precisely, this condition is used for the total water in the formulation with the fraction being defined by: θe,c − θ¯e∗ . χ= θe,c − θ¯e Some schemes also introduce triggering perturbations. For example, Fritsch and Chappell (1980) assumed an initial downdraught vertical velocity of 1 ms−1 with a downward positive definition. 5.4

Interactions of updraughts and downdraughts

The downdraughts are driven by the evaporation of precipitating water, as discussed in Sec. 5.1. In order to describe this process properly, however, the precipitating water formed within the updraughts must first be transported into the downdraughts. Cheng (1989) specifically introduced a tilt to the updraught in developing his downdraught scheme for this purpose. The full formulation is rather elaborate, but essentially a rather ad hoc relationship is introduced between the convective-scale horizontal wind and the convective tilt, and then a diagnostic equation for the convective-scale horizontal wind is introduced. The details are not essential here, and an equivalent process can also be described by simply introducing a detrainment rate from the convective updraught to the downdraught without the need to explicitly determine a tilt of the former. For example, the Unified Model simply assumes that a fraction of the precipitating water formed within the updraught is immediately transported to the downdraught at every vertical level. Emanuel (1991) more carefully evaluated the transport of precipitating water from the updraught to the downdraught by assuming that all the detrained precipitating water is entrained into the downdraught (cf., his Eq. 9).

435

Downdraughts

5.5

Modifications required for the updraught

Some modifications are required for the updraught formulation in order to introduce the downdraught properly. Importantly, the downdraught is driven by evaporative cooling and by the drag force associated with precipitating water. In order to introduce such a description, the precipitating water within the updraught must be retained instead of assuming an instantaneous fall out. In short, an adequate cloud microphysics is required for the updraught. This point was probably first clearly emphasized by Donner (1993). Donner (1993) follows the bulk microphysics of Anthes (1977: see also Kessler 1965, 1969; Kuo and Raymond, 1980). 5.6

Problems with the standard formulation

So far, a straightforward method has been presented for the inclusion of the convective downdraught into the standard mass-flux formulation of Ch. 7, Sec. 6. Mathematically speaking, all comes out naturally. However, in trying to draw a schematic in order to better understand the scheme, physical problems behind it begin to emerge, as shown in Fig. 13.1.

zT zm = βLFS zT down draught

up draught PBL

Fig. 13.1 Schematic of the convective downdraught, as introduced into a standard massflux formulation. PBL denotes the planetary boundary layer.

Under the standard mass-flux formulation, the convective elements are assumed to interact directly only with the environment. Thus, under this formulation, an updraught and a downdraught are considered as two separate entities within the grid box. The only interaction between updraught and downdraught is seen when the downdraught is initialized at the originating level (LFS), where it may be assumed that part of the convective updraught air may be transported into the downdraught.

436

Part II: Mass-flux parameterization

However, this treatment is also ambiguous: although the downdraught may be initialized at the originating level (LFS) by making use of convective updraught values, the convective updraught budget is not itself affected by this assumption. To some extent, this is an artificial assumption. A more serious aspect is that the convective updraught and downdraught would more directly interact with each other in reality. However, the possibility for such a mutual direct interaction is clearly excluded under the standard formulation. A procedure for generalizing the standard formulation by relaxing all these standard assumptions has already been discussed in Ch. 7, Sec. 7. It would be relatively straightforward to apply this principle to the convective downdraught formulation. However, caution must be exercised with the question of whether such a generalization is really necessary in practice or not. As already emphasized elsewhere, parameterization is merely a schematic representation of subgridscale physical processes. It is not necessarily a precise description of those, and only a crude caricature of reality may sometimes suffice. A similar issue is further discussed in Sec. 8. Having said that, it should also be emphasized that more care is often taken than suggested by the simple schematic of Fig. 13.1. For example, as discussed in Sec. 5.4, precipitating water is transported from updraught to downdraught at every vertical level in many formulations. It should also be realized that it becomes increasingly difficult to pursue the realism of convective processes (if necessary) under the environmental hypothesis introduced in Ch. 7, Sec. 7. For example, the precipitating water may pass through the updraught, the downdraught, and the environment, rather than just between the first two as often assumed in parameterizations. In order to partially recognize this problem, Emanuel (1991) introduced an additional adjustable parameter called the “fraction of precipitation that falls through the environment”.

6

Mesoscale downdraught and organization

A bulk mass-flux formulation for mesoscale downdraughts and updraughts was proposed by Leary and Houze Jr. (1980), and is reviewed in this section. Leary and Houze Jr. (1980) performed diagnostic analysis of GATE observations based on this formulation, and it is subsequently implemented into the convection parameterization of Donner (1993) with some minor

Downdraughts

437

modifications. Leary and Houze Jr.’s (1980) formulation also naturally follows from a general mass-flux formulation under SCA (segmentally constant approximation), as presented in Ch. 7. 6.1

Mass-flux decomposition

Following Leary and Houze Jr. (1980), the whole grid box is divided into ¯ is first several subcomponents. The grid-box domain-averaged mass flux M divided into the environment (e) and the remaining part: ¯ = Me + M, M and the second term is further divided into the convective updraught (c), the convective downdraught (d), and the mesoscale component (m): M = Mc + Md + Mm in which the mesoscale component consists of the updraught (mu) and the downdraught (md): Mm = Mmu + Mmd . As indicated by Fig. 13.2, the top of the convective system is denoted by zT . Within the mesoscale component, it is assumed that the updraught and the downdraught are found above and below the level zm , respectively, and thus:  Mmu z ≥ zm Mm = Mmd z < zm being consistent with a typical mesoscale convective structure (cf., Ch. 14, Sec. 8.1). Note that this constraint on the mesoscale structure can easily be removed for a generalization. Leary and Houze Jr. (1980), rather arbitrarily, set zT = 14 km, zm = 4.5 km, and zLFS = 5 km, whereas Donner (1993) took zm to be the level at which the least penetrative cumulus clouds in his spectral representation began to detrain water vapour to the environment. Note that these rather arbitrary assumptions can easily modified and generalized. In their observational diagnosis, Leary and Houze Jr. (1980) assumed the total domain area for analysis to be, again rather arbitrarily but consistent with their radar observations, A = 2 × 105 km2 . The convective and the mesoscale subdomains are assumed to occupy areas of Ac = 5×103 km2 and Am = 2.5 × 104 km2 , respectively, so that the ratio of the mesoscale to

438

Part II: Mass-flux parameterization

zT

Mc

zLFS

Md

Mmu zm

Mmd

Fig. 13.2 Schematic of the mesoscale downdraught and updraught formulation proposed by Leary and Houze Jr. (1980).

the convective subdomain is Am /Ac = 5. Donner (1993) simply used the same ratio in order to close his problem. Both convective updraught and downdraught were assumed by Leary and Houze Jr. (1980) to be entraining plumes with the entrainment rate λ = 0.1 km−1 . Both Leary and Houze Jr. (1980) and Donner (1993) assumed a parabolic shape for the mesoscale updraught profile and a piecewise constant profile for the mesoscale downdraught. (See Sec. 8 for further discussions.) 6.2

Water budget

As in the standard mass-flux convection parameterization formulation, the magnitude of the mesoscale mass fluxes must be closed by a certain condition. Leary and Houze Jr. (1980) used the observed water budget for this purpose. The water budget introduced by Leary and Houze Jr. (1980) is schematically shown in Fig. 13.3. Here, the water vapour condenses at rates of Cu and Cmu , respectively, within the convective and mesoscale updraughts. The condensed water vapour is transported from convection to the mesoscale component with the rate CA . The precipitation rates from convection and the mesoscale component are defined as Rc and Rm , respectively. As well as these processes, cloud water is detrained into the environment both from the convective and the mesoscale components and ˜ me , respectively. Precipitating water is ˜ ce and D the rates are defined as D re-evaporated before reaching the surface from convective and mesoscale

439

Downdraughts

˜cd and E ˜md , respectively. downdraughts with the rates E

✡

✡

✆

✟

✟

✝

✂

✠

✠

✁

✂

✁

✡

✡

☎ ✄

✄

✂

☎

✝

✞

✞

✂

✝

000000000000000000000000000000000000000000000000000000000 111111111111111111111111111111111111111111111111111111111 111111111111111111111111111111111111111111111111111111111 000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000 111111111111111111111111111111111111111111111111111111111 000000000000000000000000000000000000000000000000000000000 111111111111111111111111111111111111111111111111111111111 000000000000000000000000000000000000000000000000000000000 111111111111111111111111111111111111111111111111111111111 000000000000000000000000000000000000000000000000000000000 111111111111111111111111111111111111111111111111111111111 000000000000000000000000000000000000000000000000000000000 111111111111111111111111111111111111111111111111111111111 000000000000000000000000000000000000000000000000000000000 111111111111111111111111111111111111111111111111111111111 000000000000000000000000000000000000000000000000000000000 111111111111111111111111111111111111111111111111111111111 000000000000000000000000000000000000000000000000000000000 111111111111111111111111111111111111111111111111111111111 000000000000000000000000000000000000000000000000000000000 111111111111111111111111111111111111111111111111111111111

Fig. 13.3 Water budget in the mesoscale downdraught and updraught formulation proposed by Leary and Houze Jr. (1980). See text for details.

According to Houze Jr. (1977), the fractional mesoscale precipitation rate is Rm /(Rc + Rm ) = 0.4 for the GATE period. Houze Jr. and Leary (1976) diagnosed the fractional evaporative rate from the mesoscale downdraught relative to the total cloud-water source for the mesoscale com˜md /(Cmu + CA ) = 0.13. Leary and Houze Jr. (1980) ponent to be E assumed these ratios for their diagnosis. Finally, Leary and Houze Jr. (1980) rather arbitrarily assumed that the fractional detrainment evaporation from the convection and the mesoscale into the environment relative to the condensative supply to each component to be E˜ce /Cu = 0.07 and E˜me /(Cmu + CA ) = 0.1, respectively. These last two ratios were also adopted by Donner (1993). 6.3

Closure

Under the steady-plume hypothesis (cf., Ch. 7, Sec. 6.4), the total water supply to the convective updraught must be balanced by the total convective condensation rate: Cu = MB I1 , where I1 =



zT zB

∂qc dz, ηu λ(qe − qc ) − ∂z

as suggested by Ch. 7, Eq. 6.10.

440

Part II: Mass-flux parameterization

Similarly, the magnitude of the convective downdraught may be assumed to be a fraction of that for the convective updraught: MLFS = βd MB . By invoking the water budget, it can be shown that the above condition is equivalent to: βd = (E˜cd /Cu )I1 /I2 , where I2 =



zm

ηd

zB



∂qd λ(qe − qd ) + dz. ∂z

In a similar manner, the magnitude of the mesoscale downdraught is constrained by the re-evaporation of rain within, thus: ˜md = Mmd (zm )I3 , E where I3 =



0

zm

ηmd

dqm dz, dz

and ηmd is the normalized vertical profile for the mesoscale downdraught. These relations close the mesoscale-downdraught formulation. 6.4

Diagnoses

Leary and Houze Jr. (1980) examined the thermodynamic budget for the GATE convective system under the above formulation for three cases: ˜md = 0 (i) no mesoscale component, i.e., Cmu = E ˜md = 0 (ii) no mesoscale condensation, i.e., Cmu = 0 but E (iii) a full analysis assuming Cmu = CA . The discussion of their results can be found in Sec. 8. 7

Is the downdraught driven by evaporative cooling of rain?

The formulation presented so far has been based on the premise that the downdraughts are primarily driven by evaporative cooling of precipitating water. Zipser (1969, 1977) originally inferred this mechanism from observation, and an earlier numerical modelling study by Brown (1979) also supports this idea.

Downdraughts

441

However, the observational analysis of buoyancy inside downdraughts does not quite support this idea. For example, Jorgensen and LeMone (1989) found that the downdraught is associated with positive pseudoadiabatic buoyancy (cf., Ch. 12, Sec. 9). The analysis suggests that evaporative cooling alone is not strong enough to make the downdraught air negatively buoyant. According to their analysis, it is water loading, rather than negative pseudo-adiabatic buoyancy caused by an air-density anomaly, that is responsible for maintaining the downdraught. If this is the case, the basic parameterization description of the downdraught should be reformulated accordingly. Similar analyses by Igau et al. (1999) and Wei et al. (1998) including water loading also arrived at the same conclusion. The implication is serious, because it suggests that the downdraught is not locally driven by negative buoyancy.

8

Do we really need parameterized downdraughts?

The observational evidence is clear that downdraughts both at convective and mesoscales play a critical role in convective dynamics. However, is it actually crucial to include an explicit downdraught in a convection parameterization? Here, it is necessary to clearly distinguish two issues: (1) the importance of a given process for understanding a given system, for example, in terms of the thermodynamic budget; and (2) the importance of including a process in a parameterization. Thus, although the downdraught is important in the convective thermodynamic budget, it does not automatically follow that it must be included as a part of parameterization. The question of whether we need a certain process to be included in parameterization is itself somewhat odd. The goal of a parameterization is not to put all of the important processes together, but rather to represent a whole entity (here, convection) in a parametric manner. Imagine that we try to represent convective vertical transport using an eddy-diffusion approach. In that case, it would just be necessary to find an eddy-diffusion coefficient that gives us the right amount of vertical transport. Whether this vertical transport ultimately derives from updraught or downdraught motions, for example, is totally irrelevant. In the description of a mass-flux parameterization we retain a certain reality in the language of the formulation by referring to the terms updraught, downdraught, etc. This tends to make us think that we are modelling convection directly, rather than parameterizing it, and it may further tempt us

442

Part II: Mass-flux parameterization

to make it more realistic by adding to or elaborating the processes. Frank (1983) may be considered as a review from such a perspective. We also need to think carefully about which details really matter in convection parameterization. In this respect, it would be helpful to realize that in many places in the mass-flux formulation, only the sum of the updraught and downdraught mass fluxes matters. For example, the boundary layer is modified in this manner as shown by Ch. 11, Eqs. 11.3 and 11.4. Various observational diagnoses may also be better interpreted from this perspective. For example, Johnson’s (1976) observational diagnosis including downdraughts has the effect of reducing the diagnosed contribution from shallow convection. He concluded that this demonstrated the importance of downdraughts, which had been previously wrongly represented by shallow convection. However, from a parameterization point of view, the result might also be interprted as suggesting a possibility that the downdraught can simply be substituted by shallow convection. However, Houze Jr.’s (1977) diagnosis urges some caution with this view: according to him, modification of the boundary layer by downdraughts is more clearly seen than that by updraughts, suggesting an asymmetry in their role, although the two mass fluxes enter in a symmetric manner in the boundary-layer equation. Moreover, it would be easy to find a literature showing strong sensitivities of parameterization to the inclusion of downdraughts, and it is easy to argue for the importance of including downdraughts based on such results. However, a strong sensitivity does not necessarily prove the importance of a particular process in consideration. Even a completely artificial parameter within a parameterization may present a very strong sensitivity to its value. In the latter case, that would suggest an ill-posed formulation of a parameterization leading to such an artificially exaggerated sensitivity, rather than any physical significance of the result. Although the downdraught is designed to be something physically meaningful, this latter possibility cannot be easily excluded: an overall defect of a parameterization formulation may lead to a rather artificially strong sensitivity to the downdraught, for example. The same could be said about the role of mesoscale organization. For example, for the three cases of Sec. 6.4 that were considered by Leary and Houze Jr. (1980), the results for the partition of the mass fluxes as well as their total based on the water budget are strongly sensitive to the mesoscale treatment. The total mass flux changes by a factor of two depending on

Downdraughts

443

whether a mesoscale is totally included or excluded. However, compared to this sensitivity, the obtained eddy transport of moist static energy is relatively insensitive to whether a mesoscale is included, the maximum difference being only 50%. Tests of the Donner (1993) scheme within a GCM with or without mesoscale downdraughts also lead to a similar conclusion (see also Donner et al., 2001). 9

Bibliographical notes

Discussions in Sec. 3 are based on Betts and Dias (1979) and Kamburova and Ludlam (1966). Note that Betts and Dias (1979) used the pressure as a vertical coordinate, but the analysis here is presented with a geometrical coordinate. Sec. 5 closely follows Cheng (1989). 10

Appendix A: Water vapour deposition growth of droplet

In a super-saturated state, a water droplet grows by deposition of water vapour on to it. The mass growth rate of a droplet with a radius r may be described by:   ∂ρv dm 2 = 4πr Dv . (A.1) dt ∂R R=r Here, the diffusive moisture flux given by Dv ∂ρv /∂R drives the droplet growth, the factor 4πr2 is the surface area of the droplet, Dv is the molecular diffusion coefficient for water vapour, and R is the distance from the centre of the droplet. In order to describe the droplet growth by water vapour deposition, it is necessary to know the water vapour distribution around this droplet. Since the diffusive timescale is much shorter (∼ 10−4 s) than any convective timescale of concern, it suffices to solve a steady diffusion problem: ∇2 ρv = 0,

(A.2)

where ρv is the water vapour density, and ∇2 = 1/R(∂/∂R)R2∂/∂R. The solution to Eq. A.2 is: r ∞ (ρ − ρrv ), (A.3) ρv = ρ∞ v − R v r ∞ ∞ in which ρv → ρ∞ v , as R → ∞ and ρv = ρv at R = r. ρv = ρv (T ) r ∗ r and ρv = ρv (T ) may also be set by introducing the same notations for the

444

Part II: Mass-flux parameterization

temperature T , and assuming that the water vapour is at the saturation point at the droplet surface. The solution of Eq. A.3 immediately gives the water vapour flux at the droplet surface:   ∂ρv Dv [ρv (T ∞ ) − ρ∗v (T r )]. (A.4) Dv = ∂R R=r r From now on, the superscript ∞ indicating the value at the infinity (environment) may be dropped without loss of clarity, because these are values obtained by standard instrumental measurements. Substitution of Eq. A.4 into Eq. A.1 almost leads to a final answer: dm = 4πrDv [ρv (T ) − ρ∗v (T r )]. (A.5) dt This equation simply states that a water droplet grows by an excess of water vapour available under supersaturation. However, the equation is not as simple as it first appears: the excess of water vapour concerns a value far from the water droplet (i.e., an environmental value) but also the saturated value ρ∗v (T r ) right at the droplet surface. The temperature T r at the droplet surface must be slightly higher than that of the environment due to the latent heating of condensation. The molecular diffusion evacuates the latent heat from the droplet to produce a steady state, and thus: dm = 4πrK(T r − T ) (A.6) dt by following an analogous derivation as for the water vapour diffusion process. Here, K is the heat diffusion coefficient. For practical purposes, it would be tempting to replace the saturated water vapour density at the droplet surface with that for the environment. However, the approximation ρ∗v (T r ) ≃ ρ∗v (T ) is not a good one. Nevertheless, ρ∗v (T ) is more readily measurable than ρ∗v (T r ). Therefore, when such a replacement is introduced into Eq. A.5, compensation is achieved by replacing the diffusion coefficient Dv with an effective value, say, De , so that: dm = 4πrDe [ρv (T ) − ρ∗v (T )], (A.7) dt ˜ v with the correction factor d, ˜ where the effective coefficient is De = dD L

defined by: ρv (T ) − ρ∗v (T r ) . d˜ = ρv (T ) − ρ∗v (T )

(A.8)

445

Downdraughts

In order to know this effective value explicitly, it is necessary to evaluate the difference ρ∗v (T ) − ρ∗v (T r ) first. For this purpose, it is noted that the saturated specific humidity is given by: q∗ =

Rd e∗ Rd e∗ , ≃ Rv p − e∗ Rv p

where e∗ is the saturated water vapour pressure. Thus, the change of q ∗ arises from changes of both the saturated water vapour and the pressure: 1 dq ∗ 1 de∗ 1 dp = ∗ − . ∗ q dT e dT p dT The first term is given by: Lv 1 de∗ = e∗ dT Rv T 2 based on the Clausius–Clapeyron law. The second term is given by: 1 dp 1 = . p dT T As a result, 1 1 dq ∗ = ∗ q dT T



 Lv −1 . Rv T

(A.9)

An approximate integration of the above over a small interval of [T, T r ] leads to:   Lv T − Tr q ∗ (T ) − q ∗ (T r ) ≃ − 1 . (A.10) q ∗ (T ) T Rv T By referring to Eq. A.6, Lv dm . 4πrK dt Substitution of Eq. A.11 into Eq. A.10 leads to:   dm Lv q ∗ (T )Lv ∗ ∗ r −1 q (T ) − q (T ) = − 4πrKT Rv T dt or   dm Lv ρ∗ (T )Lv −1 . ρ∗v (T ) − ρ∗v (T r ) = − v 4πrKT Rv T dt Tr − T =

(A.11)

(A.12)

On the other hand, Eq. A.5 can be written as: ρv (T ) − ρ∗v (T r ) =

1 dm . 4πrDv dt

(A.13)

446

Fig. 13.4

Part II: Mass-flux parameterization

Fractional diffusion coefficient De /Dv as a function of the temperature.

Thus, by taking the difference between Eqs. A.13 and A.11, we obtain: 

 dm 1 Lv ρ∗ (T )Lv + v −1 . (A.14) ρv (T ) − ρ∗v (T ) = 4πrDv 4πrKT Rv T dt

Substituting Eqs. A.11 and A.14 into the definition A.8, we finally obtain:  ∗ 

−1 ρv (T )Lv Dv Lv −1 . (A.15) d˜ = 1 + Rv T KT

The correction factor d˜ is plotted as a function of the temperature in Fig. 13.4. Here, the following values are used: the latent heat Lv = 2.5 × 106 Jkg−1 , the gas constant for water vapour Rv = 4.62 × 102 Jkg−1 K−1 , and the diffusion coefficient for water vapour Dv = D0 (T /T00 )1.94 , with D0 = 2.11 × 10−5 m2 s−1 and T00 = 273.16 K, and for heat K = 2.4 × 10−2 Jm−1 s−1 K−1 . The saturated water vapour pressure is evaluated by directly integrating the Clausius–Clapeyron equation, leading to:  

L 1 1 ∗ − e = e00 exp , Rv T00 T where e00 = 611 Pa. It is seen that the factor d˜ is a decreasing function of the temperature, equal to about half around the freezing point, asymptotic to unity towards low temperatures, and to zero towards the high temperatures. As the temperature increases, the saturated water vapour density

Downdraughts

447

increases exponentially, and thus a smaller temperature difference between the droplet surface and the environment is required in order to supply an excess water vapour to the droplet surface by diffusion. An important last effect to take into account is the contribution arising from turbulent flow around a droplet which enhances the diffusion rate. This contribution is typically taken into account through an additional factor f˜, so that: ˜ v. De = f˜dD

(A.16)

By definition, f˜ ≥ 1, and it increases with the increasing turbulent nature of the flow, as typically measured by a non-dimensional Reynolds number. Finally, the above formulation has been given for water vapour deposition growth. However, this same formulation is available when a water droplet is placed in a sub-saturated situation. In this case, the droplet would decrease its size with time by evaporation. The former is a key process within an updraught, whereas the latter is key within a downdraught.

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Chapter 14

Momentum transfer

J.-I. Yano Editors’ introduction: Part II has developed the mass-flux formulation for the convective transport of an arbitrary physical variable ϕ. However, special attention has been paid to thermodynamic variables, partly because of the great importance of their convective tendencies within the context of the operation of the parent climate or weather forecast model, and partly because of their contributions to the buoyancy, which itself drives the convection. This presentation of mass-flux parameterization is completed with a chapter devoted to another variable that has a somewhat special status, specifically the momentum. Some aspects of the momentum transport follow very naturally within a mass-flux plume model along lines that are by now well established. A special consideration, however, is necessitated by the effects of momentum transports on the pressure field. Recall that an appropriate treatment of pressure was perhaps the most awkward aspect of the calculations required when a segmentally constant approximation is introduced to the primitive equations. Recalling also that this is a key geometrical constraint underlying mass-flux parameterization, it should be no surprise that convective transports of momentum proves to be a non-trivial issue. They are considered here both in the context of plume dynamics and for organized mesoscale structures.

1

Introduction

Atmospheric convection not only transfers heat and moisture vertically, but also a dynamic quantity: the momentum. This chapter examines the issues of momentum transfer by convection. A separate treatment of this item is 449

450

Part II: Mass-flux parameterization

required because there are some special features arising. The notations in this chapter follow those already introduced in earlier chapters, and thus the definitions are not always restated. The general formulation presented in Ch. 7 is always applicable here, even though the specific case of the momentum equation is considered. 2

Vertical-momentum transfer

The literature focuses on the convective transfer of the horizontal momentum, but let us begin by considering the vertical momentum. Of course, it is important to understand why this term is usually not considered. The vertical momentum equation for convection has already been discussed in Ch. 12. Referring to Ch. 12, Eq. 7.2:

1 ∂ 1 ∂p′i ∂ ¯ σi wi + Mi wi + ∇H · σi wi ui = Ei we − Di wi + σi − + gbi . ∂t ρ ∂z ρ ∂z (2.1) Here, unlike in Ch. 12, we take a spectral notation with i indicating the i-th convection type. The lateral exchange effect with the environment is presented in terms of the entrainment Ei and the detrainment Di . An additional contribution from the large-scale divergence, omitted in Ch. 12, is included here. As discussed in Ch. 7, this term can be dropped under the standard asymptotic limit σi → 0, except for the convective vertical velocity, which is expected to be scaled as wi ∼ O(σi−1 ) under this limit. ¯ even as an approximation. Also, recall that we = w The convective transfer of the vertical momentum ultimately influences the environmental vertical velocity, whose prognostic equation may be written as:

N 1 ∂ 1 ∂pe 1 ∂ ¯ σe we + ∇H · we ue − Me we = σe − −g [Di wi − Ei we ] + ∂t ρ i=1 ρ ∂z ρ ∂z (2.2) by referring to Ch. 7, Eq. 5.9. Here, the fractional area σe occupied by the environment is explicitly retained for the purpose of later discussions. The third term on the left-hand side of Eq. 2.2 represents the tendency due to convection. However, this does not play any role in the large-scale vertical velocity budget. Large-scale dynamics is always strongly constrained by the hydrostatic balance: 1 ∂pe + g = 0. ρ ∂z

451

Momentum transfer

The effect of convection is never strong enough to destroy this basic balance. Instead, the environmental circulation responds in such a way with N

 1 ∂ ∂ ¯ H · we ue − 1 σe we + ∇ Me we = 0 [Di wi − Ei we ] + ∂t ρ i=1 ρ ∂z

(2.3)

that the hydrostatic balance is well maintained at any moment. The condition of Eq. 2.3 may be considered an example of a strict equilibrium condition so that convective vertical momentum transport does not play any significant role in the large-scale dynamics. An additional point to consider may be the issue of a distinction between the environmental pressure pe and the grid box-averaged pressure p¯. In order to address this issue, the two terms of the convective tendency may be rewritten as: ′

∂pi 1 ∂ ∂ 1 − [Di wi − Ei we ] = Mi wi + σi wi + σi − gbi , ρ ρ ∂z ∂t ∂z neglecting the horizontal advection term in Eq. 2.1 for now. Here, recall that p′i is a perturbation pressure defined as a deviation from a hydrostatic balance. Substitution into Eq. 2.2 gives: N  1 ∂ ∂ ∂ ¯ H · we ue + 1 ∂ Me we + σe we +∇ Mi wi + σi wi ∂t ρ ∂z ρ ∂z ∂t i=1



 ′ 1 ∂pe ∂pi − gbi +g . +σi = −σe ∂z ρ ∂z An additional term can also be added on the right-hand side that trivially vanishes by hydrostatic balance

1 ∂pe −σi +g ρ ∂z for each convective element i, with a weighting by the fractional area σi . Their sum is:

N  1 ∂pe +g , σi − ρ ∂z i=1 and by further noting that

σe +

N 

σi = 1,

i=1

Eq. 2.2 reduces to:

 ′ 

N  ∂ 1 ∂ ∂pi 1 ∂ ¯ w ¯ + ∇H · we ue + Me we + Mi wi + σi − gbi ∂t ρ ∂z ρ ∂z ∂z i=1

452

Part II: Mass-flux parameterization

1 ∂pe − g, (2.4) ρ ∂z  also noting that w ¯ = i σi wi + σe we . Thus, pe on the right-hand side without weight σe may be equated with the grid box-averaged hydrostatic pressure. Note two major transformations from Eq. 2.2 to Eq. 2.4: first, that the equation is now given in terms of the grid box-averaged vertical velocity w ¯ instead of the environmental vertical velocity we , and second, that the right-hand side now shows more explicitly how the environmental pressure pe contributes to the grid box-averaged pressure without a weight σe of the environmental fraction. By applying the hydrostatic balance again to Eq. 2.4, we obtain an alternative expression for the strict equilibrium:  ′ 

N  1 ∂ ∂pi ∂ ¯ H · we ue + 1 ∂ Me we + w ¯+∇ Mi wi + σi − gbi = 0, ∂t ρ ∂z ρ ∂z ∂z i=1 (2.5) this time in terms of the grid box-averaged vertical velocity tendency. This is a constraint required in order for the large-scale circulation to evolve consistently under hydrostatic balance, and so that the large-scale vertical velocity w ¯ can be evaluated consistently solely from mass continuity. However, pe may still not be equal to the grid box-averaged pressure, which is given by: N  σi p′i . p¯ = pe + =−

i=1

Thus, strictly speaking, the hydrostatic models may not be evaluating the domain-averaged pressure p¯ when parameterized convection is present within a grid box, but only an environmental part of the pressure.

3

Horizontal momentum transfer

The convective transport of horizontal momentum can be formulated in a similar manner. A major difference from the vertical momentum is that now it is possible to apply the standard approximations introduced in Ch. 7, Sec. 6 in full, because the convective-scale horizontal momentum is of order unity in respect of σi in the standard asymptotic limit. By referring to Ch. 7, Eq. 6.8, the diagnostic equation for the convectivescale horizontal velocity ui is given by: ∂ ¯ − Di ui − σi (∇H p)i , (3.1) Mi ui = Ei u ∂z

Momentum transfer

453

where the horizontal pressure gradient over the i-th convective subcomponents may be more explicitly expressed as:  1 (∇H p)i = pi,b dr Si ∂Si

by following the segmentally constant approximation (SCA). Eq. 3.1 can furthermore be rewritten by recalling the mass continuity for the mass flux: ∂ui = Ei (¯ Mi u − ui ) − σi (∇H p)i . (3.2) ∂z The horizontal momentum equation over a grid-box average is given by:   ∂¯ u σe ∂ 1 ∂ ¯ u ¯ + ∇H · u ρw¯ ¯ u = − ∇H pe + ¯u ¯+ , (3.3) ∂t ρ ∂z ρ ∂t c with the convective tendency given by:

  

N N ∂¯ u ∂¯ u  1 ∂ 1 ′ Mc + Mi u′i + σi (∇H p)i , D i ui = − = ∂t c ρ ∂z ρ i=1 ∂z i=1 (3.4) ¯. The Coriolis force may further be added to the above where u′i = ui − u equation for completeness but without modifying the following discussion. By subtracting out the hydrostatic pressure from the last term, the grid box-averaged equation may be rewritten as:

N 1 1 ∂ 1 ∂ ∂ ′ ′ ¯ u ¯ + ∇H · u ρw¯ ¯ u = − ∇H pe − Mi ui + σi (∇H p)i . ¯u ¯+ ∂t ρ ∂z ρ ρ i=1 ∂z (3.5) In the standard presentations, the last term under the summation is dropped, hoping that the sum would vanish, and thus the impact of the convective momentum transfer would reduce to that of the vertical transport represented by the penultimate term. The convective-scale velocity ui , a quantity required in order to evaluate the convective momentum transport, is in turn to be evaluated from a vertical integration of Eq. 3.1. Here, we encounter a major difficulty with convective momentum-transfer parameterization, because in order to integrate Eq. 3.1, the convective-scale pressure appearing in the last term must be known.

4

Simplified approaches

In order to avoid the complexities just described, various simplified approaches have been devised. In this section, some examples are considered.

454

Part II: Mass-flux parameterization

A particularly simple approach is that of Schneider and Lindzen (1976), who proposed to avoid the whole procedure of solving Eq. 3.1 by simply ¯ B . This setting ui to the large-scale value at the convection base, i.e., ui = u procedure is certainly straightforward, but has been found to overestimate the damping tendency of the upper-level momentum by convection. They proposed to call this tendency the “cumulus friction”. Another approach is to actually try to solve Eq. 3.1, but since there is no obvious way to obtain the pressure term, it is simply neglected. Experiments under this simplification suggest that the magnitude of convective velocity ui is overestimated. A simple interpretation of these experiments is that the main role of the convective-scale pressure is to promote mixing between the convective velocity with the environment. Along this line of reasoning, we may hope that the role of convective-scale pressure can be mimicked by simply augmenting the values of entrainment and detrainment. Thus, we replace Eq. 3.1 by: ∂ Mi ui = Ei∗ u ¯ − Di∗ ui , ∂z where Ei∗ and Di∗ are enhanced entrainment and detrainment rates. For example, ECMWF assume the form: Ei∗ = Ei + λDi Di∗ = (1 + λ)Di , with a constant λ currently assumed to be 2 for deep and mid-level convection, and zero for shallow convection.

5

Convective-scale pressure problem

In order to solve the convective momentum transfer problem properly, we have to know the convective-scale pressure gradient (∇H p)i , the last term in Eq. 3.1. As already pointed out in Chs. 7 and 12, the convective-scale pressure may be obtained in a formal sense by solving the Poisson equation given by: ∇2 p′ = −∇ · ∇ · ρvv + ∇ · ρgb

(5.1)

for its non-hydrostatic component, where v is the three-dimensional velocity. In order to solve this Poisson problem, we need some information about the convective-scale circulations, because that is what is represented on the right-hand side.

Momentum transfer

455

Here, it is worth recalling the subtlety of applying SCA to the convective-scale horizontal winds. As emphasized in Ch. 7, for a vertical velocity field satisfying SCA, the horizontal wind field must satisfy the piecewise linear horizontal distribution from mass continuity. Thus, Eq. 3.1 is not quite a consistent equation for the convective-scale winds under the SCA formulation, but only a truncated representation of the full problem. The issues of the convective-scale pressure may be divided into two separate aspects: (1) that of finding a simple solution for the convectivescale pressure; and (2) that of extracting an appropriate component of the convective-scale pressure that contributes to the convective-scale velocity problem of Eqs. 3.1 and 3.2. Towards these goals a very basic analysis is presented in the remainder of the present section. 5.1

Linearized Poisson equation

The first source term on the right-hand side of the Poisson problem in Eq. 5.1 is non-linear. However, if we assume that the background horizontal wind v = (u, w) is dominant in this term, and also assuming that the vertical shear is a dominant term, this  be linearized as:  contribution can v ∂w ∂u ¯ ∂w ∂¯ + . −∇ · ∇ · vv ≃ −2 ∂z ∂x ∂z ∂y Neglecting the second (buoyancy) term from Eq. 5.1 for now, the Poisson problem reduces to:   ∂u ¯ ∂w ∂¯ v ∂w 1 2 ′ ∇ p = −2 + . (5.2) ρ ∂z ∂x ∂z ∂y Suppose now that a simple form is assumed for the vertical velocity, in terms of a constant wavenumber vector (k, l, m): (5.3) w = w0 cos kx cos ly sin mz, where w0 is a velocity amplitude. Here, it is assumed that convection stretches across a rectangular area −πk/2 < x < πk/2, −πl/2 < y < πl/2. Substitution of the expression of Eq. 5.3 into Eq. 5.2 leads to: 2w0 ∂¯ v ∂u ¯ p′ =− 2 sin kx cos ly + l cos kx sin ly sin mz. k 2 2 ρ k +l +m ∂z ∂z Taking the horizontal gradient of the above, and then averaging over the rectangular convective area,  weobtain a final expression: 1 ∂p′ ∂u ¯ − (5.4a) = γk wi ρ ∂x i ∂z   1 ∂p′ ∂¯ v − = γl wi , (5.4b) ρ ∂y i ∂z

456

Part II: Mass-flux parameterization

where 2k 2 + l 2 + m2 2l2 γl = 2 . k + l 2 + m2

γk =

k2

(5.5a) (5.5b)

Here, γk and γl are considered as free parameters to be prescribed. The UK Unified Model (Gregory et al., 1997) adopts this formulation with a choice of γk = γl = 0.7. The effect of buoyancy could be added to the above analysis in a relatively straightforward manner by assuming a similar functional form as in Eq. 5.3 for the buoyancy. The amplitude for the buoyancy can be estimated by solving the convective-scale equation for buoyancy directly, as discussed in considering the closure problem in Ch. 11. Such a buoyancy term is still to be added to an operational implementation. 6

Convective-scale circulation analysis

Before plunging into a full analysis of the convective-scale pressure field, let us first investigate the issue of convective-scale circulations in general. As already emphasized, the mass-flux convection parameterization is best understood in terms of SCA. The basic idea of SCA is to subdivide the grid-box domain into a number of constant segments, each representing a different subgrid-scale component. However, in considering the convective momentum transport problem, the limits of the SCA formulation begin to show up: the horizontal velocity does not satisfy SCA, but it satisfies a piecewise linear distribution. A horizontal distribution of the non-hydrostatic pressure is even more complicated, even if the Poisson equation is solved with an assumption that the vertical velocity and the buoyancy satisfy SCA, and that the horizontal velocity a piecewise linear condition. In short, it is necessary to face the complex flow field around convection in order to fully resolve the convective momentum transport problem. It becomes more questionable to maintain various assumptions behind the standard mass-flux formulation. The non-local nature of the Poisson problem must especially be emphasized: the inversion of the Poisson operator (a Laplacian) is fundamentally non-local. As a result, we have to consider circulations far away from a convective core and to take into account the fully turbulent nature of the subgrid-scale atmospheric flow as a whole.

457

Momentum transfer

In this respect, convection is merely where turbulence is concentrated and intensified, but otherwise has no special status. The most notable issue is a description of the lateral exchange of momentum crossing the convective boundary. Under the standard mass-flux formulation, this term is described by entrainment and detrainment. However, the validity of such a simplified formulation in describing lateral momentum exchange must be questioned. Thus, the main goal of the present section is to address the issue of the convective horizontal momentum exchange by removing the SCA hypothesis. Instead, let us consider a full horizontal momentum equation. The basic strategy is to apply conditional averaging in order to extract the effects of convection. As a full horizontal momentum equation, let us adopt ∂ ∂ (ρu) + ∇H · ρ(uu) + (ρwu) + ρf k × u = −∇H p. (6.1) ∂t ∂z Here, the Coriolis force has been added as the last term on the left-hand side for completeness. By applying a grid-box average, the above reduces to: ∂ ∂ (ρ¯ u) + ∇H · ρ(¯ (ρw¯ ¯ u + ρw′ u′ ) + ρf k × u ¯ = −∇H p¯, uu ¯ + u ′ u′ ) + ∂t ∂z (6.2) where the prime designates a deviation from the grid-box average. By taking the difference between Eqs. 6.1 and 6.2, we obtain the eddy momentum equation: ∂ (ρu′ ) + ∇H · ρ(¯ u u′ + u′ u ¯ + u ′ u′ − u′ u′ ) ∂t ∂ (ρwu ¯ ′ + ρw′ u (6.3) ¯ + ρw′ u′ − ρw′ u′ ) + ρf k × u′ = −∇H p′ . ∂z It is convenient to introduce the horizontal momentum exchange E, defined by: +

uu ¯+u ¯ u ′ + u′ u ¯ + u′ u′ ), E = ∇H · ρ(¯ which is a fully expanded version of the second term of Eq. 6.1. Its grid-box average and the average over the convective domain are, respectively: ¯ = ∇H · ρ(¯ uu ¯ + u ′ u′ ) E ′

(6.4) ′

′ ′

Ec = ∇H · ρ(¯ uu ¯+u ¯u + u u ¯ + u u )c ,

(6.5)

458

Part II: Mass-flux parameterization

in which the subscript c indicates the average over convective subdomain. ¯ and the second Note that the second term in Eq. 6.2 can be replaced by E, ¯ term in Eq. 6.3 by E − E. As inferred from Ch. 7, Eq. 5.7, under the mass-flux formulation, the horizontal momentum flux divergence is given by:   ρ (Di ui − Ei ue ). uib (u∗ib − r˙ ib ) · dr = Ec = S i ∂si i

Assuming that entrainment dominates over detrainment in this flux, and also approximating the environmental wind by the grid box-averaged wind, we obtain:  ∂Mc ¯=− ¯. u (6.6) Ei u Ec ≃ − ∂z i

This is an expression to be compared with a more exact expression that will be derived next. In seeking a more explicit expression of Eq. 6.5, it is first necessary to note a vector identity: u · u + ρk × uζ, (6.7) ∇H · ρuu = (∇H · ρu)u + ρ∇H 2 where ζ = k·∇×u is the vorticity (the vertical component). Mass continuity ∂ρw ∇H · ρu = − (6.8) ∂z is also invoked. The right-hand side of Eq. 6.8 can furthermore be expressed in term of the mass flux M , with appropriate subscripts as required in the following after applying spatial averaging. Substitution of Eq. 6.8 into Eq. 6.7 leads to: u · u ∂ρw + ρk × uζ. (6.9) u + ρ∇H ∇H · ρuu = − ∂z 2 By applying Eq. 6.9 to each term on the right-hand side of Eq. 6.5, we obtain:   ¯ u ¯·u ¯ ∂M u ¯ + ρ∇H [∇H · ρ¯ uu ¯ ]c = − + ρk × u ¯ζ¯ (6.10a) ∂z 2 ¯ ∂M ∂Mc′ ¯ u u′ + u′ u ¯ )]c = − u′c − u ¯ + ρk × (¯ uζc′ + u′c ζ) [∇H · ρ(¯ ∂z ∂z + [ρ∇H (¯ u · u′ )]c (6.10b)  ′ ′ 

′ ∂M ′ u ·u u + ρ∇H [∇H · ρu′ u′ ]c = − + ρk × [u′ ζ ′ ]c . ∂z 2 c c (6.10c)

Momentum transfer

459

After some non-trivial scale analysis, it is found that many of the terms can be neglected to a leading order expansion of the convective fraction σc , and we may approximately set: ¯ ∂M u ¯ (6.11a) uu ¯ ]c ≃ − [∇H · ρ¯ ∂z ∂Mc′ [∇H · ρ(¯ u ¯ + ρk × u ¯ζc′ + [ρ∇H (¯ u · u′ )]c (6.11b) uu ′ + u ′ u ¯)]c ≃ − ∂z  ′ ′ 

u ·u . (6.11c) [∇H · ρu′ u′ ]c ≃ ρ∇H 2 c By substituting Eqs. 6.11a, b and c into Eq. 6.5, and by further noting ¯ , we obtain: Mc ≃ Mc′ , Mc′ ≫ M  ′ ′ 

u ·u ∂Mc u ¯ + ρk × u ¯ ζc′ + [ρ∇H (¯ u · u′ )]c + ρ∇H Ec ≃ − . (6.12) ∂z 2 c The last major step is to find that the penultimate term in Eq. 6.12 can be approximated by: ζ′ ∂Mc′ ζ′ δ′ ¯ − ck ×u ¯=− u ¯ − c kׯ u, (6.13) u · u′ )]c ≃ c u [ρ∇H (¯ 2 2 ∂z 2 where δc is the convective-scale divergence. The derivation of Eq. 6.13 is rather lengthy, and it is given separately in Sec. 10. Substitution of Eq. 6.13 into Eq. 6.12 leads to a final expression:  ′ ′ 

u ·u 1 3 ∂Mc ′ u ¯ + ρk × u ¯ζc + ρ∇H . (6.14) Ec ≃ − 2 ∂z 2 2 c Some efforts may also be made to express the last term in a more explicit form by assuming a certain convective-scale flow structure, but no useful form is easily obtained. For practical purposes, this last term may simply be dropped, because its explicit form sensitively depends on precise flow structures assumed for the sub-convective scale. The precise structures are clearly not of interest for parameterization. The final expression of Eq. 6.14 is to be compared with Eq. 6.6. The most significant difference is a factor 3/2 found in the first term of Eq. 6.14 compared to Eq. 6.6. Furthermore, the full analysis shows that the convective scale-averaged vorticity also contributes to the momentum transport, which is missing in the simpler mass flux-formulation-based formula.

7

Convective momentum transport: A full analysis

By taking the full analysis of the convective-scale circulations from the last section, it now becomes possible to perform a full analysis for the convective

460

Part II: Mass-flux parameterization

momentum transport. In particular, a more precise expression is sought for the right-hand side of Eq. 3.2, or in other words, for ∂uc . ∂z A bulk expression is retained here as in the last section. The starting point is the full convective momentum equation. This equation is found using SCA, by referring to Ch. 7, Eq. 4.6, which is:  ∂ 1 ∂ 1 σc σc uc + Mc uc = − [∇H p]c . uc (u∗c − r˙ cb ) · dr + (7.1) ∂t S ∂Sc ρ ∂z ρ Mc

However, following the spirit of the last section, the entrainment– detrainment hypothesis is not made now, but instead, following the notations of the last section, the above equation is written as:

∂u σc 1 ∂ Mc uc = − [∇H p]c . (7.2) + [∇H · uu]c + ∂t c ρ ∂z ρ

However, again from the last section, to a good approximation we find:    u · u 1 ∂Mc [∇H · uu]c = − . (7.3) u ¯ + ∇H ρ ∂z 2 c By substituting Eq. 7.3 into Eq. 7.2, and after some rearrangements, we find:

  ∂u Mc ∂uc 1 ∂Mc =− (7.4) − u′c − σc [∇H GH ]c , ρ ∂z ∂t c ρ ∂z

where u′c = uc − u ¯, and GH is a two-dimensional Bernoulli function defined by p u·u , GH = + ρ 2 which may furthermore be replaced by its three-dimensional counterpart G to a good approximation: p v·v . GH ≃ G ≡ + ρ 2 By referring to Eq. 5.1, we find that the Bernoulli function G is defined by: ∇2 G +

2 ∂ρ ∂G 1 ∂ 2 ρ G = f, + ρ ∂z ∂z ρ ∂z 2

(7.5)

with the source term f defined by: f=

1 ∂2ρ 1 ∂ρgb 1 1 ∂ρ ∂v · v + . v · v − ∇ · ρ[(∇ × v) × v] + 2ρ ∂z ∂z 2ρ ∂z 2 ρ ρ ∂z

(7.6)

Momentum transfer

461

A full solution is not attempted for this Poisson problem. However, good progress can be made by assuming that the convection takes on a cylindrical shape with a radius R. The Bernoulli function may be, in general, represented in terms of a Fourier–Bessel expansion: ∞  ∞  Gk,n einφ Jn (kr)dk, (7.7) G= n=0

0

where Gk,n is a complex expansion coefficient, which is defined such that Gk,n einφ remains real for all angles, and Jn (x) is the Bessel function of order n. Some useful properties of Bessel functions used in these derivations are given in Sec. 11. Substitution of Eq. 7.7 into the last term of Eq. 7.4 leads to:  1 ∞ (ˆ x + iˆ y)Gk,1 J1 (kr)dk, (7.8) [∇H G]c = R 0

meaning only the first harmonic contributes to the convective-scale pressure gradient term regardless of the details of the convective-scale flows. This result enables us to simplify the computations. The first term on the right-hand side of Eq. 7.4 may be crudely approximated by:

uc (t) − uc (t − τ ) ∂u , (7.9) = ∂t c τ

with τ the timescale for a convective life cycle. The convective wind may be set at the earlier time point simply equal to the large-scale state, i.e., uc (t − τ ) = u ¯ , in order to close this problem. Substitution of Eqs. 7.8 and 7.9 into Eq. 7.4 leads to an expression:    uc (t) − u ¯ 1 ∂Mc σc ∞ Mc ∂uc ′ =− − (ˆ x + iˆ y)Gk,1 J1 (kr)dk. uc − ρ ∂z τ ρ ∂z R 0 (7.10) In order to evaluate the last term of Eq. 7.10, some additional information on the convective-scale circulations is still required. A simplified circulation is considered within the convective area (r ≤ R) given in terms of divergence δc and vorticity ζc by: δc = δ0 + δ1 J1 (kr) cos(φ − φδ )

(7.11a)

ζc = ζ0 + ζ1 J1 (kr) sin(φ − φζ ).

(7.11b)

Here, δ0 , δ1 , ζ0 , and ζ1 are assumed to be functions of height only. Unit ˆ vectors are denoted in the angle directions for φδ and φζ , respectively, as φ δ ˆ and φζ . The following solution is sought by setting ζ0 = 0, and J1 (kR) = 0

462

Part II: Mass-flux parameterization

is also assumed with the zero taken as the smallest value of the argument, so that kR = 3.82. As a result, the vertical velocity is also given by: (7.11c) w = w + w1 J1 (kr) cos(φ − φδ ), where w1 is a non-axisymmetric component of the convective vertical velocity arising from the second term of Eq. 6.11b. After some calculations, the corresponding perturbation wind is also given by:

δ1 ′ δ0 ζ1 J1 (kr) ′ J (kr) cos(φ − φδ ) + ˆ cos(φ − φζ ) + r n u =− k 1 k 2 r2 2

δ1 J1 (kr) ζ1 + sin(φ − φδ ) + J1′ (kr) sin(φ − φζ ) ˆ (7.12) t + u0 k 2 r2 k for r ≤ R. Here, n ˆ and ˆ t are unit normal and tangenetial vectors, and a constant wind component u0 is obtained by continuously extending the convective-scale wind to the outside of the convection, assuming vanishing vorticity and a constant divergence: J ′ (kR) ˆ ˆ ]. [ζ1 φζ + δ1 φ (7.13) u0 = 1 δ 2k The convective momentum flux can be evaluated from the definition  R  2π 1 ρσc [u′c w]c = u′c M rdφdr, πR2 0 0 where M = ρσw. By substituting Eqs. 7.11c and 7.12, and after some lengthy computation, we finally obtain: ˆ w1 ∂Mc J ′ (kR). (7.14) ρσc [u′c w]c = Mc u0 + φ δ 2k ∂z 1 However, under a mass-flux formulation ρσc [u′c w]c = Mc u′c . (7.15) Comparing Eqs. 7.14 and 7.15, also with the help of Eq. 7.13, we obtain a matching condition of:   ′ 1 ∂Mc 1 ˆ ˆ ˆ ] = 2uc − φ [ζ1 φζ + δ1 φ w . (7.16) 1 δ δ k J1′ (kR) Mc ∂z Note that the right-hand side of Eq. 7.16 mostly consists of mass-flux parameterization quantities already known (except for w1 ). Thus, in principle, based on the known values for the right-hand side, it would be possible to evaluate the unknown coefficients ζ1 and δ1 on the left-hand side, given the value for w1 . Once those three coefficients are known, it would be straightforward to evaluate the source f , defined by Eq. 7.6, and then Eq. 7.5 could be solved under the Fourier–Bessel expansion already outlined above. The complete formulation of the problem outlined here remains somewhat involved and is left as an exercise for the readers.

Momentum transfer

8

463

Mesoscale convective momentum transport

Over the last few sections, a complete formulation has been outlined for convective momentum transport parameterization as an extension of a standard mass-flux formulation under SCA. However, this general approach retains some fundamental constraints by assuming that the convective system consists of an ensemble of convective towers. In reality, convection may be organized on the mesoscale, leading, for example, to a squall-line structure, under a background wind sheared in the vertical direction. Such an organization may generally be called a mesoscale convective system (MCS). A schematic for such a mesoscale convective system is shown in Fig. 14.1a, and associated airflows are schematically shown in Fig. 14.1b. The most important feature is an uplifting of the low-level inflow, leading to an upper-level outflow, associated with a jump updraught (A). Such an organized flow would be associated with an enhancement of the upper-level flow, and a decrease of the lower-level flow, leading to an enhancement of the pre-existing horizontal wind shear. In other words, the mesoscale convective system tends to transport horizontal momentum in an upgradient direction, in contrast to the downgradient transport typically found in unorganized convection. This observational fact is well summarized in Wu and Yanai (1994). Thus, in parameterizing the mesoscale convective momentum transport, a qualitatively different strategy than a standard mass-flux based on the hot-tower hypothesis is required. Most importantly, convection no longer consists of an ensemble of isolated updraughts and downdraughts, but as the schematics in Fig. 14.1 show, these are intimately linked to the background flow, so as to enable an upgradient momentum transport. Thus, a key for successfully parameterizing mesoscale convective momentum transport is to explicitly represent such an organized mesoscale flow within the parameterization, albeit in very succinct manner. These considerations lead to a proposal of the archetypal model by Moncrieff (1992). The purpose of the present section is to review this formulation. 8.1

Archetypal formulation

Following Moncrieff (1992), let us assume that a mesoscale convective system is embedded into a two-dimensional domain with a horizontal length Lm , and a vertical depth H. The archetype consists of three flow branches with three different densities (Fig. 14.1b and Fig. 14.2). The major branch

464

Part II: Mass-flux parameterization

Fig. 14.1 (a) Schematic diagram of the relative airflow and physical processes associated with a squall-line mesoscale convective system. (b) Schematic diagram of the airflow in the stationary dynamical model, consisting of three flow branches: (A) jump updraught, (B) downdraught, and (C) overturning updraught. Branches have piecewise constant densities ρB ≥ ρA ≥ ρC with ρA = ρB = ρC in the archetypal model presented in the c Royal Meteorological Society 1992, from Fig. 1 of Moncrieff (1992). Reproduced text.  with permission of the Royal Meteorological Society.

is a jump updraught with a density ρA flowing into the domain at a lower level from the right-hand edge with depth h0 and, after uplifting, flowing out from the left-hand edge at upper levels over a depth H − h. This major branch supports an upgradient momentum transport, as already discussed. The jump updraught is supported by a downdraught with a depth h and a density ρB that occupies the lower-left side of the domain, and a overturning updraught with a depth H − h0 and a density ρC that occupies the upper-right side of the domain. Note that the system contains two lines of stagnation: one near the surface that divides the branches A and B, and one near the top that divides the branches A and C. The stagnation points at the surface and at the domain top are designated by S and T , respectively.

Momentum transfer

465

Fig. 14.2 More details of the flow configuration outlined in Fig. 14.1b: the assumed flow inside the domain is presented in terms of a streamfunction along with the inflow (right) and the outflow (left) profiles indicated at the sides. Note that the coordinate c Royal Meteorological Society 1992, used in the text is shifted by Lm /2 for symmetry.  from Fig. 3 of Moncrieff (1992). Reproduced with permission of the Royal Meteorological Society.

We assume inflow (right edge) and outflow (left edge) profiles given by: ⎧ ⎨−U0 z ≤ h0 Lm ,0 = u 2z − (H + h0 ) ⎩U t 2 z ≥ h0 H − h0 ⎧  ⎨ h − 2z  Us z≤h Lm ,0 = u − h ⎩−U 2 z≥h 1 



(8.1a)

(8.1b)

respectively. Here, U0 , U1 , Us , and Ut are constant winds. The flow is maintained by a pressure difference from the right to the left given by Δp. A rigid lid condition at the top and the bottom is assumed, and so w = 0 at z = 0 and H. For economy of presentation, the system is non-dimensionalized by taking U0 , H, ρa , and ρa U02 as scales for the velocity, the height, the density, and the pressure. Specifically, the pressure difference is measured by: E=

2Δp . ρU02

In general, the non-dimensional variables are indicated by a hat so that, for example, the non-dimensional horizontal wind is u ˆ = u/U0 . A steady solution is sought here, although the problem can easily be generalized into a steadily propagating system with a phase velocity c by introducing a Galilean transformation so that the horizontal advection term is replaced by (u − c)∂/∂x.

466

Part II: Mass-flux parameterization

The system is given by the steady-state momentum equation and by mass continuity: ∂u 1 ∂p ∂u +w + =0 ∂x ∂z ρ ∂x ∂w ∂w 1 ∂p u +w + +g =0 ∂x ∂z ρ ∂z ∂u 1 ∂ρw + = 0. ∂x ρ ∂z u

(8.2a) (8.2b) (8.2c)

In constructing a self-consistent solution, a few constraints play an important role. The first is to notice that the hydrostatic balance 1 ∂p +g =0 ρ ∂z is satisfied at the two edges of the domain where the vertical velocity vanishes. This balance enables us to evaluate the pressure at these two end points as a function of height. 8.1.1

Bernoulli function

The second important constraint is conservation of the Bernoulli function along a fluid trajectory. This is proved by multiplying u and w on Eqs. 8.2a and 8.2b, respectively. Rewriting the sum in vector form: v · ∇G = 0, where the Bernoulli function is defined by: v·v p + + gz. G= 2 ρ

(8.3)

(8.4)

Here, unlike in earlier sections, a contribution of the geopotential is explicitly taken into account as the last term in the definition. Thus, the Bernoulli function G is conserved along any Lagrangian trajectory, and also along streamlines when the flow is steady. This conserved quantity is traced along streamlines in order to define the relative strengths U1 /U0 , Us /U0 , and Ut /U0 of the outflow, downdraught, and updraught relative to the inflow. Note that although a streamline can be traced by crossing a stagnation point from one flow regime to another, the Bernoulli function is discontinuous when crossing the stagnation point. However, importantly, the pressure is continuous when crossing the branch interfaces, and this allows us to link the strength of flow in the branches by crossing the interfaces.

Momentum transfer

467

First, the relative strength of the outflow is determined by tracing a Bernoulli function from the top entry point of the inflow (Lm /2, h0 ), to the top of the outflow (−Lm /2, H): pRT pLT U02 U2 + + gh0 = 1 + + gH, 2 ρA 2 ρA designating the points (±Lm /2, H) by RT and LT . This leads to a relationship: U12 = U02 −

2Δp , ρA

where the pressure difference that drives the inflow is given by: Δp = pLT − pRT + (ρA − ρC )g(H − h0 ).

(8.5)

After non-dimensionalization, we obtain: KA = 1 − E, where KA =



U1 U0

2

(8.6)

.

(8.7)

The downdraught flow strength Us is related to the inflow strength U0 by tracing the streamline along z = 0 crossing the stagnation point S: ρA 2 ρB 2 U + pLB = U + pRB , pS = 2 s 2 0 where pS is the pressure at the stagnation point S, and the points (±Lm /2, 0) are designated by RB and LB, respectively. After nondimensionalization, we obtain: ˆ KB = KA − 2λ2B h

(8.8)

with ρB Us2 , ρA U02

(8.9)

ρB − ρA H g 2 ρA U0

(8.10)

KB = and where λ2B =

is an inverse Froude number. Finally, the jump updraught flow is related to the overturning updraught flow by tracing the streamline from the top entry point of the

468

Part II: Mass-flux parameterization

inflow (Lm /2, h0 ) to the top of the jump updraught outflow RT through the top-surface stagnation point T :   Lm ρC 2 ρA 2 U +p , h0 + ρA g(h0 − H) = U + pRT . pT = 2 0 2 2 t This leads to: ˆ 0) KC = 1 − 2λ2C (1 − h

(8.11)

with KC =

ρC Ut2 , ρA U02

(8.12)

and another inverse Froude number is introduced: λ2C =

8.1.2

ρA − ρC H g 2. ρA U0

(8.13)

Mass conservation

By integrating the mass conservation law of Eq. 8.2c for the whole domain, and noting the vanishing vertical velocity at the top and the bottom of the domain, we obtain:      H  H Lm Lm , z dz = , z dz. ρu − ρu 2 2 0 0 By further noting there is no mass exchange over the domain boundaries within downdraught and updraught branches,      H  h0 Lm Lm , z dz = , z dz ρu − ρu 2 2 h 0 or ρA U0 h0 = ρA U1 (H − h), which is rewritten as: KA =



ˆ0 h ˆ 1−h

2

.

(8.14)

469

Momentum transfer

8.1.3

Momentum conservation

The horizontal momentum equation can be rewritten with the help of mass continuity as:   ∂ p ∂ wu = 0. (8.15) u2 + + ∂x ρ ∂z By integrating it over the whole domain, and using the top and bottom boundary conditions for w, we obtain:  H  H (ρu2 + p)|−Lm /2 dz. (ρu2 + p)|Lm /2 dz = 0

0

Evaluations of both sides lead, after non-dimensionalization, to: 2 2 ˆ + λB h ˆ2 + E . ˆ 0 ) + λC (1 − h ˆ 2 ) = KA (1 − h) ˆ + KB h ˆ 0 + KC (1 − h h 0 3 2 3 2 2 Substitution of Eqs. 8.6, 8.8, 8.11, and 8.14 into the above leads to:

ˆ ˆ h ˆ 2 − 2(2 + λ2 )h ˆ 0 + 1 + λ2 − λ2 h [λ2C + f (h)] 0 C C B = 0,

(8.16a)

where ˆ ˆ = 3 − 4h . f (h) ˆ 2 (1 − h)

(8.16b)

Eq. 8.16a constitutes a characteristic equation that defines a regime of the archetypal flow. 8.1.4

Flow regimes

From now on, the case with no stratification, i.e., λB = λC = 0, will be pursued in order to simplify the following mathematical expressions. The characteristic regime equation, Eq. 8.16a, then reduces to: ˆ 3 − 4h ˆ 0 + 1 = 0. ˆ 2 − 4h h 0 2 ˆ (1 − h)

(8.17)

ˆ 0 , we obtain the two solutions: By solving the above equation for h ˆ ˆ0 = 1 − h , h ˆ 3 − 4h

(8.18)

corresponding to an asymmetric regime with an inflow traversing to the opposite side of the domain by following a jump updraught, and ˆ ˆ 0 = 1 − h, h

470

Part II: Mass-flux parameterization

corresponding to a symmetric regime with no jump flow crossing the domain. For the asymmetric regime, we have: 1 KA = ˆ 2 (3 − 4h) and E = 1 − KA =

ˆ ˆ (1/2 − h)(1 − h) . ˆ 2 (3/4 − h)

For the symmetric regime, KA = 1 and E = 0, and there is no pressure difference across the domain, as expected. Note that Eq. 8.17 can be interpreted as an eigenequation for the downˆ when the pressure difference E is known: draught depth h   1 3 9 2 ˆ ˆ (1 − E)h − (1 − E)h − 1 − E = 0, 2 2 8 whose solutions are ˆ = 1 {3 ± (1 − E)−1/2 }. h 4 ˆ ≤ 1, it can be concluded that it is necessary to take From the condition h the minus sign above. By further substitution into Eq. 8.18, we obtain a pair solution: ˆ 0 = 1 {1 + (1 − E)1/2 }. ˆ = 1 {3 − (1 − E)−1/2 }, h (8.19) h 4 4 ˆ = 0 and E = The maximum for the pressure difference occurs with h ˆ0 = 1 8/9. The minimum for the pressure difference is obtained when h ˆ and E = −8, which also corresponds to the maximum h = 2/3. The characteristic regime diagram is shown in Fig. 14.3. 8.1.5

Non-dimensional inflow and outflow profiles

By referring to the results of Sec. 8.1.1, and setting λB = λC = 0, the non-dimensional inflow and outflow profiles are given by: ⎧  ⎨  ˆ0 −1 zˆ ≤ h Lm   ,0 = (8.20a) u ˆ ˆ0 1+ h ˆ0 ⎩C1 zˆ − 2 2 zˆ ≥ h    ˆ ˆ C2 (ˆ z − h/2) zˆ ≤ h Lm (8.20b) uˆ − ,0 = ˆ −1 zˆ ≥ h, ˆ ˆ 0 (1 − h) 2 −h ˆ 0 ) and C2 = 2h ˆ 0 /h(1 ˆ − h). ˆ where the constants are C1 = 2/(1 − h

Momentum transfer

471

Fig. 14.3 Archetypal regimes identified in the phase space consisting of the normalized ˆ 0 of the jump updraught (horizontal axis) and the normalized downdraught depth h ˆ (vertical axis). Two solutions are available by solving the characteristic regime depth h equation of Eq. 8.16a: the asymmetric regime (solid line) and the symmetric regime (dashed line). Values of the normalized pressure jump E and schematics of the flows are c Royal Meteorological Society 1992, from Fig. 2 of Moncrieff (1992). also indicated.  Reproduced with permission of the Royal Meteorological Society.

8.1.6

Momentum flux divergence

Once the inflow and the outflow profiles are defined, it is relatively straightforward to evaluate the various parameterization outputs. The major point of interest is how this archetype system modifies domain average horizontal wind, which is given by the momentum flux divergence  Lm /2 ∂ ρˆuˆwdx. ˆ Υ= ∂z −Lm /2 By referring to the steady momentum equation of Eq. 8.15, it is rewritten as: %Lm /2 $ (8.21) Υ = − ρˆuˆ2 + pˆ −Lm /2 . By substituting the inflow and the outflow profiles given by Eq. 8.20 as well as the corresponding pressure fields, we obtain an explicit form of the momentum flux divergence.

472

Part II: Mass-flux parameterization

For the asymmetric regimes, we obtain: ⎧ ˆ 2 (2ˆ z − h) E ⎪ ⎪ + −1 ⎪ ⎪ 2 2 ˆ ˆ ⎪ 2 h (3 − 4h) ⎪ ⎪ 2 ⎨ 1 E Υ= + −1 ⎪ ˆ 2 ⎪ ⎪  3 − 4h 2 ⎪ ˆ 0 )2 ⎪ 1 E (2ˆ z−1−h ⎪ ⎪ + − ⎩ ˆ ˆ 0 )2 2 3 − 4h (1 − h

ˆ 0 ≤ zˆ ≤ h ˆ ≤ zˆ ≤ h ˆ0 h

(8.22)

ˆ 0 ≤ zˆ ≤ 1. h

ˆ 0 = 1 − h, ˆ and C1 = Note that for the symmetric regime, E = 0, h ˆ This regime is separated into two sub-regimes depending on the C2 = 2/h. ˆ 0: ˆ 0 . When ˆh ≤ h relative values of ˆ h and h

ˆ ˆ 0 ≤ h: When h

8.1.7

⎧ ˆ 2 ⎪ z − h) ⎪ (2ˆ ⎪ −1 ⎪ ⎪ ˆ2 ⎪ h ⎨ Υ= 0  2 ⎪ ⎪ ˆ0 ⎪ 2ˆ z − 1 − h ⎪ ⎪ ⎪ ⎩1 − 1 − ˆh0

ˆ 0 ≤ zˆ ≤ h ˆ ≤ zˆ ≤ h ˆ0 h

(8.23)

ˆ 0 ≤ zˆ ≤ 1. h

⎧ ˆ 2 (2ˆ z − h) ⎪ ⎪ ⎪ −1 ⎪ ⎪ ˆ2 h ⎨ Υ= 0 ⎪ ⎪ ˆ 2 ˆ 0 )2 ⎪ (2ˆ z − h) (2ˆ z−1−h ⎪ ⎪ − ⎩ ˆ2 (1 − ˆh0 )2 h

ˆ0 0 ≤ zˆ ≤ h ˆ 0 ≤ zˆ ≤ h ˆ h

(8.24)

ˆ ≤ zˆ ≤ 1. h

Mesoscale mass flux

Another key quantity to know is the mass flux Mm associated with this mesoscale archetypal circulation: Mm =



Lm /2

ρˆwdx. ˆ −Lm /2

By invoking the mass continuity, this is rewritten as:  z Lm /2 dz. [ˆ ρu ˆ]−L Mm = m /2

(8.25)

0

Again by substituting the inflow and the outflow profiles, we obtain for

Momentum transfer

the asymmetric ⎧ regime: ˆ ⎪ zˆ(ˆ z − h) ⎪ ˆ ⎪ z ˆ + 0 ≤ zˆ ≤ h ⎪ ⎪ ˆ − 4h) ˆ ⎪ h(3 ⎪ ⎨ ˆ ˆ ≤ zˆ ≤ ˆh0 Mm = 2(1 − 2h) (ˆ z − ˆh) + M1 h ⎪ 3 − 4h ˆ ⎪ ⎪ ⎪ ˆ0 ˆ0 ⎪ zˆ − h zˆ − h ⎪ ⎪ ˆ 0 + 1)] h ˆ 0 ≤ zˆ ≤ 1, ⎩M 2 − + [ˆ z − (2h ˆ ˆ 3 − 4h 1−h ˆ and M2 = M1 + (1/2 − h) ˆ 3 /(3/4 − h) ˆ 2. where M1 = h, ˆ≤h ˆ0: For the symmetric with h ⎧ regime 2 ⎪ zˆ ⎪ ˆ ⎪ 0 ≤ zˆ ≤ h ⎪ ⎪h ˆ ⎨ ˆ ≤ zˆ ≤ h ˆ0 h Mm = M1 ⎪ ⎪ ˆ 0 )(ˆ ˆ 0 − 2) ⎪ (ˆ z−h z−h ⎪ ˆ 0 ≤ zˆ ≤ 1. ⎪ + M1 h ⎩ ˆ0 1−h ˆ 0 ≤ h: ˆ For the symmetric regime with h ⎧ 2 zˆ ⎪ ˆ0 ⎪ 0 ≤ zˆ ≤ h ⎪ ⎪ ˆ ⎪ h ⎨ˆ ˆ 0 ≤ zˆ ≤ h ˆ Mm = h0 (2ˆ z−ˆ h0 ) h ⎪ ˆ ⎪ h ⎪ ⎪ ⎪ ˆ z+h ˆ − 2) + (1 − h)(3 ˆ h ˆ − 1)] h ˆ ≤ zˆ ≤ 1. ⎩ 1 [(ˆ z − h)(ˆ ˆ h 8.2

473

(8.26)

(8.27)

(8.28)

Implementation issues

The archetypal mesoscale system constructed above must still be implemented into a large-scale model as a parameterization. For this purpose, it is necessary to consider two further steps. The first step, which may be less obvious, is to embed this archetype model of domain size Lm into a grid-box domain of size, say, LG (> Lm ). This extra step is necessary, because as suggested by Fig. 14.1 and Fig. 14.2, this mesoscale archetype constitutes an open flow with the inflow and the outflow profiles qualitatively different: a lower-level sheared inflow is translated into an upper-level sheared outflow. From one grid box to next, the resolved-scale wind profile must change only gradually. Thus, the constructed mesoscale open flow must somehow be embedded into a grid-box domain so that a closed circulation is maintained inside the latter. A formal procedure would be to connect the grid-box boundary inflow smoothly to the mesoscale-archetype inflow, and then the mesoscalearchetype outflow to the grid-box boundary outflow. In principle, such a

474

Part II: Mass-flux parameterization

procedure must be possible in an analogous manner as the archetype inflow is smoothly connected to the outflow consistently, as considered in the last subsection. However, the precise procedure is less clear. The second step is a closure: a procedure for defining the free parameters that characterize the mesoscale archetype. These parameters must be modified according to a large-scale state of the system. To some extent, this closure question may be reduced to a part of the problem of connection from a grid-box boundary to an archetype-domain boundary, especially if a unique connection solution can be identified. These two issues, clearly inter-related, must be resolved before this system can be implemented into an operational model. 8.3

Further issues: Numerically generated archetype?

A drastically simplified archetypal model has been introduced as an attempt for parameterizing mesoscale convective momentum transport. The final results were presented only for cases without density stratification. A generalization of the above analysis for cases with stratification by allowing non-vanishing inverse Froude numbers is technically straightforward. By the same token, it is feasible to gradually consider more general flow configurations. However, in considering more and more generalizations of this archetypal model, the problem must be reformulated every time a new type of mesoscale convective system is encountered: this is a shortcoming of an analytical model. Would it be possible, then, to generate a mesoscale archetype model numerically each time under a given large-scale environment? The basic idea of the archetypal model of a mesoscale convective system is to develop an archetypal flow configuration satisfying the basic dynamical constraints of the system. The flow configuration is archetypal in the sense that it can be evaluated in a much more economical manner than the full numerical computations of a CRM. To some extent, the archetype achieves an effective compression of a CRM. In this respect, the NAM-SCA model introduced in Ch. 7, Sec. 4 may provide a key ingredient, by aiming at a compression of a CRM from a different point of view. As already outlined there, NAM-SCA can compress a full CRM system by introducing an adaptive mesh refinement. Yano and Bouniol (2010) tested NAM-SCA with a simple microphysics under an idealized GATE forcing, and demonstrated that it can reproduce a realisticlooking tropical mesoscale squall-line system only with 10% of the total

Momentum transfer

475

mesh number relative to a full-resolution case. Here, a time-dependent adaptive mesh-refinement is crucial in order to simulate propagation of a squall-line system in an efficient manner. This simulation can be taken as a starting point for constructing a mesoscale convective archetype numerically. The first step is to introduce a Galilean transformation so that the mesoscale convective system remains stationary with time. As a result, mesh adaptation is no longer necessary, and NAM-SCA runs with an initial inhomogeneous distribution of meshes (focused on the convective region with less resolution outside) without any time-dependent adaptation. Normally, with NAM-SCA, some extra meshes are always necessary in order to detect a tendency for new convective developments. However, the total mesh number can be reduced by fixing the strategy to generate an archetypal mesoscale circulation rather than an accurate standard CRM simulation. Under these considerations, the total mesh number in the horizontal direction under a two-dimensional configuration (similar to those in Fig. 14.1 and Fig. 14.2) can be reduced even down to four. Figure 14.4 shows snapshots from such a preliminary simulation with four fixed constant segments horizontally distributed in a highly inhomogeneous manner. More specifically, a 512-km horizontal periodic domain is taken as in Yano and Bouniol (2010). In the middle of this domain, a segment of 8 km is placed, which is immediately adjacent to two segments of 16 km. The convective circulation is sustained by adding to the lowest model layer a constant sensible heating of 10−4 Ks−1 at the centre segment and cooling of −0.25 × 10−4 Ks−1 at the two side segments. Although it is very crude, a realistic-looking tropical squall line-like structure is maintained with only the four constant segments, with an updraught– downdraught pair spontaneously generated consistent with the expected propagation direction of the squall line on a cold pool underneath the downdraught. The preliminary result here suggests a potential capacity of NAM-SCA for generating a mesoscale convective archetype numerically. Within this framework, a key question of the archetypal description reduces to that of defining a distribution of constant segments over a large-scale grid box. Importantly, under this formulation, it is neither necessary nor possible to specify the role or function of a given segment (e.g., updraught or downdraught), but the system determines its evolution. In this demonstrative example, the middle and smallest segment ends up as a downdraught despite the fact that heating is added at the surface for this segment, nudging

476

Part II: Mass-flux parameterization

Fig. 14.4 Snapshots of an evolving NAM-SCA system consisting only of four SCA segments in an idealized tropical situation. Shown from the top to the bottom are: (a) vertical velocity (ms−1 ), (b) potential temperature anomaly (deviation from the horizontal average, K), (c) water vapour mixing ratio (gkg−1 ), (d) cloud water (gkg−1 ), and (e) precipitating water (gkg−1 ). Reproduced from Fig. 5 of Yano (2014b), which is c the author, 2014, and published by Elsevier B.V. It is an open access article under  the CC BY-NC-ND license.

Momentum transfer

477

it towards an updraught. 8.4

Further reading

The mesoscale archetypal model is the result of a series of studies by Moncrieff and colleagues. Their studies include: Moncrieff (1981); Moncrieff and Green (1972); Moncrieff and Miller (1976).

9

Bibliographical notes

The analysis in Sec. 5.1 is due to Wu and Yanai (1994). A full analysis for Sec. 6 is found in Cho (1985). Section 7 is based on Zhang and Cho (1991). For full details of the derivation of the archetypal model reviewed in Sec. 8, see Moncrieff (1992); Moncrieff and So (1989).

10

Appendix A: Derivation of Eq. 6.13

The purpose of this appendix is to make a derivation of Eq. 6.13 from the main text. To do so, let us first note the divergence theorem, that allows us to transform an area integral into a line integral:   aˆ ndl ∇H adS = S

C

for any scalar a, where n ˆ is a unit vector normal outward from the line of the integral. More specifically, consider:  1 [ρ∇H (¯ (¯ u · u′ )ˆ ndl (A.1) u · u′ )]c = Sc ∂Sc Here, it is important to keep in mind that the line ∂Sc of the integral mostly likely consists of many separate enclosed curves, corresponding to individual convective elements. However, the following reduction is performed without considering these individual closed integrals separately. In order to make further progress, let us take the background wind to be in the positive x-direction, and so set: ¯x u ¯=U ˆ. The convective-scale eddy wind is also decomposed into components normal and tangential to the line of the integral, with subscripts n and t respectively: u′ = u′n + u′t

478

Part II: Mass-flux parameterization

or t, ˆ + u′tˆ u′ = u′n n where u′n and u′t are scalar values for the normal and the tangential components respectively, and ˆ t is a unit vector in the tangential direction. By substituting these expressions into Eq. A.1, ¯  U uu′ )]c = (u′ x ˆ·n ˆ + u′t x ˆ ·ˆ t)ˆ ndl. [ρ∇H · (¯ Sc ∂Sc n For further reduction, a more explicit form is needed for the two unit vectors n ˆ and ˆ t. For this purpose, let us take the angle of the unit vector n ˆ in respect of the x-axis to be φ. Thus: n ˆ=x ˆ cos φ + y ˆ sin φ ˆ t = −ˆ x sin φ + y ˆ cos φ. For later use, we may further note: 1 [ˆ x(1 + cos 2φ) + y ˆ sin 2φ] 2 1 x sin 2φ + y ˆ(1 − cos 2φ)]. (ˆ x ·ˆ t)ˆ n = −[ˆ x sin φ cos φ + y ˆ sin2 φ] = − [ˆ 2

ˆ sin φ cos φ = (ˆ x·n ˆ)ˆ n=x ˆ cos2 φ + y

For further simplification, it is assumed that only the mean values u′n0 and u′t0 , where  1 ′ un0 = u′ dl lc ∂sc n  1 ′ u′ dl, ut0 = lc ∂sc t contribute to the line integral above. Here, lc is the total length of the line integral. Also note that these mean values are related to the convectivescale average divergence and vorticity, respectively, by:    1 1 lc u′n0 1 ′ ′ ′ δc = ∇H · u dS = u ·n ˆdl = u′n dl = Sc S c Sc ∂Sc Sc ∂Sc Sc    1 1 lc u′t0 ˆ · ∇ × u′ dS = 1 k ζc′ = . u′ · ˆ tdl = u′t dl = Sc S c Sc ∂Sc Sc ∂Sc Sc As a result, uu′ )]c ≃ [ρ∇H · (¯

  ¯ U (ˆ x ·ˆ t)ˆ ndl . (ˆ x·n ˆ)ˆ ndl + u′t0 u′n0 Sc ∂Sc ∂Sc

Momentum transfer

479

In these two line integrals, only constant terms from the integrand remain upon integration, and thus:  lc ˆ (ˆ x·n ˆ)ˆ ndl = x 2 ∂Sc  lc ˆ. (ˆ x ·ˆ t)ˆ ndl = − y 2 ∂Sc Finally, we obtain: [ρ∇H · (¯ uu′ )]c ≃

¯ U 1 ˆ u. (lc u′n0 x ˆ − lc u′t0 y ˆ) = (δc′ − ζ ′ k×)¯ 2Sc 2

¯x ¯y ˆ×u In writing the final expression, the relations U ˆ=u ¯ and U ˆ=k ¯ are recalled. 11

Appendix B: Bessel function

This appendix states properties of Bessel functions that are useful for the analysis of Sec. 6. The Bessel function Jn (kr) is a solution of the differential equation   n2 1 d d 2 r + k − 2 Jn (kr) = 0. r dr dr r A set of Bessel functions satisfy the following orthogonality relation:  ∞ 1 rJn (kr)Jn (k ′ r)dr = δ(k ′ − k), k 0 where δ is the Dirac’s delta.

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Bibliography

Adams, D. K. and Renn´ o, N. O. (2003). Remarks on quasi-equilibrium theory, J. Atmos. Sci. 60, pp. 178–181. Adkins, C. J. (1983). Equilibrium Thermodynamics, 3rd edn. (Cambridge University Press, Cambridge). Ahlgrimm, M. and Forbes, R. M. (2012). The impact of low clouds on surface shortwave radiation in the ECMWF model, Mon. Wea. Rev. 140, pp. 3783– 3794. Ahlgrimm, M., Randall, D. and K¨ ohler, M. (2009). Evaluating cloud frequency of occurence and top height using space-borne lidar observations, Mon. Wea. Rev. 137, pp. 4225–4237. Ambaum, M. H. P. (2010). Thermal physics of the atmosphere, Advancing Weather and Climate Science (John Wiley and Sons, Chichester). Andrews, F. C. (1963). Equilibrium Statistical Mechanics (John Wiley and Sons, Chichester). Anthes, R. A. (1977). A cumulus parameterization scheme utilizing a onedimensional cloud model, Mon. Wea. Rev. 105, pp. 270–286. Arakawa, A. (2004). The cumulus parameterization problem: Past, present and future, J. Climate 17, pp. 2493–2525. Arakawa, A. and Chen, J.-M. (1987). Closure assumptions in the cumulus parameterization problem, in T. Matsuno (ed.), Short- and Medium-range Numerical Prediction (Meteor. Soc. Japan), pp. 107–131. Arakawa, A. and Schubert, W. H. (1974). Interaction of a cumulus cloud ensemble with the large-scale environment. Part I, J. Atmos. Sci. 31, pp. 674–701. Asai, T. and Kasahara, A. (1967). A theoretical study of the compensating downward motions associated with cumulus clouds, J. Atmos. Sci. 24, pp. 487– 496. Austin, J. M. (1948). A note on cumulus growth in a nonsaturated environment, J. Meteor. 5, pp. 103–107. Ayotte, K. W., Sullivan, P. P., Andr´en, A., Doney, S. C., Holtslag, A. A. M., Large, W. G., McWilliams, J. C., Moeng, C.-H., Otte, M. J., Tribbia, J. J. and Wyngaard, J. C. (1996). An evaluation of neutral and convective planetary boundary-layer parameterizations relative to large eddy simulations,

481

482

Bibliography

Bound.-Layer Meteor. 79, pp. 131–175. Bak, P., Tang, C. and Wiesenfeld, K. (1987). Self-organized criticality: An explanation of the 1/f noise, Phys. Rev. Lett. 59, pp. 381–384. Batchelor, G. K. (1954). Heat convection and buoyancy effects in fluids, Quart. J. R. Meteor. Soc. 80, pp. 339–358. Bechtold, P. (2009). Convection parameterization, in Proceedings of the ECMWF Seminar on Parameterization of Subgrid Physical Processes, pp. 63–86, 1–4 September 2008, Reading, UK. Bechtold, P., Bazile, E., Guichard, F., Mascart, P. and Richard, E. (2001). A mass-flux convection scheme for regional and global models, Quart. J. R. Meteor. Soc. 127, pp. 869–889. Bechtold, P., Cuijpers, J. W. M., Mascart, P. and Trouilhet, P. (1995). Modeling of trade wind cumuli with a low-order turbulence model: Toward a unified description of Cu and Sc clouds in meteorological models, J. Atmos. Sci. 52, pp. 455–463. Bechtold, P., Fravalo, C. and Pinty, J. P. (1992). A model of marine boundarylayer cloudiness for mesoscale applications, J. Atmos. Sci. 49, pp. 1723– 1744. Bechtold, P., K¨ ohler, M., Jung, T., Doblas-Reyes, F., Leutbecher, M., Rodwell, M., Vitart, F. and Balsamo, G. (2008). Advances in simulating atmospheric variability with the ECMWF model: From synoptic to decadal time-scales, Quart. J. R. Meteor. Soc. 134, pp. 1337–1351. Bechtold, P., Semane, N., Lopez, P., Chaboureau, J.-P., Beljaars, A. and Bormann, N. (2014). Representing equilibrium and non-equilibrium convection in large-scale models, J. Atmos. Sci. 71, pp. 734–753. Bender, C. M. and Orszag, S. A. (1978). Advanced Mathematical Methods for Scientists and Engineers (McGraw–Hill, New York). Betts, A. K. (1973). Non-precipitating cumulus convection and its parameterization, Quart. J. R. Meteor. Soc. 99, pp. 178–196. Betts, A. K. (1975). Parametric interpretation of trade-wind cumulus budget studies, J. Atmos. Sci. 32, pp. 1934–1945. Betts, A. K. (1982). Saturation point analysis of moist convective overturning, J. Atmos. Sci. 39, pp. 1484–1505. Betts, A. K. (1985). Mixing line analysis of clouds and cloudy boundary-layers, J. Atmos. Sci. 42, pp. 2751–2763. Betts, A. K. (1986). A new convective adjustment scheme. Part I: Observational and theoretical basis, Quart. J. R. Meteor. Soc. 112, pp. 677–691. Betts, A. K. and Dias, M. F. S. (1979). Unsaturated downdraft thermodynamics in cumulonimbus, J. Atmos. Sci. 36, pp. 1061–1071. Bihlo, A. (2010). Symmetry methods in the atmospheric sciences, Ph.D. thesis, University of Vienna. Bluestein, H. B. (1992). Synoptic-Dynamic Meteorology in Midlatitudes, Volume 1: Principles of Kinematics and Dynamics (Oxford University Press, Oxford). Bluestein, H. B. (1993). Synoptic-Dynamic Meteorology in Midlatitudes, Volume 2: Observations and Theory of Weather Systems (Oxford University Press,

Bibliography

483

Oxford). Boatman, J. F. and Auer, A. H. (1983). The role of cloud top entrainment in cumulus clouds, J. Atmos. Sci. 40, pp. 1517–1534. B¨ oing, S. J., Jonker, H. J. J., Nawara, W. A. and Siebesma, A. P. (2014). On the deceiving aspects of mixing diagrams of deep cumulus convection, J. Atmos. Sci. 71, pp. 56–68. B¨ oing, S. J., Jonker, H. J. J., Siebesma, A. P. and Grabowski, W. (2012). Influence of the subcloud layer on the development of a deep convective ensemble, J. Atmos. Sci. 69, pp. 2682–2698. B¨ oing, S. J., Siebesma, A. P., Korpershoek, J. D. and Jonker, H. J. J. (2012). Detrainment in deep convection, Geophys. Res. Lett. 39, doi:10.1029/ 2012GL053735. Bony, S. and Emanuel, K. A. (2001). A parameterization of the cloudiness associated with cumulus convection: Evaluation using TOGA COARE data, J. Atmos. Sci. 58, pp. 3158–3183. Bougeault, P. (1981). Modeling the trade-wind cumulus boundary layer. Part I: Testing the ensemble cloud relations against numerical data, J. Atmos. Sci. 38, pp. 2414–2428. Bougeault, P. (1985). A simple parameterization of the large-scale effects of cumulus convection, Mon. Wea. Rev. 113, pp. 2108–2121. Bourke, W. (1972). An efficient, one-level, primitive-equation spectral model, Mon. Wea. Rev. 100, pp. 683–689. Bourke, W. (1974). A multi-level spectral model. I. Formulation and hemispheric integrations, Mon. Wea. Rev. 102, pp. 687–701. Bradshaw, P. (1994). Turbulence: The chief understanding of our subject, Exp. Fluids 16, pp. 203–216. Bradshaw, P. (1999). The best turbulence models for engineering, in M. D. Sales, J. N. Hefiner and L. Sakell (eds.), Modeling Complex Turbulent Flows, ICASE/LaRC Interdisciplinary Series in Science and Engineering, Vol. 7 (Springer, Dordrecht Netherlands), pp. 9–28. Bretherton, C. S., Blossey, P. N. and Khairoutdinov, M. (2005). An energybalance analysis of deep convective self-aggregation above uniform SST, J. Atmos. Sci. 62, pp. 4273–4292. Bretherton, C. S., McCaa, J. R. and Grenier, H. (2004). A new parameterization for shallow cumulus convectionn and its application to marine subtropical cloud-topped boundary layers. Part I: Description and 1D results, Mon. Wea. Rev. 132, pp. 864–882. Brown, A. R., Cederwall, R. T., Chlond, A., Duynkerke, P. G., Golaz, J.-C., Khairoutdinov, M., Lewellen, D. C., Lock, A. P., Macvean, M. K., Moeng, C.-H., Neggers, R. A. J., Siebesma, A. P. and Stevens, B. (2002). Largeeddy simulation of the diurnal cycle of shallow cumulus convection over land, Quart. J. R. Meteor. Soc. 128, pp. 1075–1094. Brown, J. M. (1979). Mesoscale unsaturated downdrafts driven by rainfall evaporation: A numerical study, J. Atmos. Sci. 36, pp. 313–338. Brown, R. G. and Bretherton, C. S. (1995). Tropical wave instabilities: Convective interaction with dynamics using the Emanuel convective parameterization,

484

Bibliography

J. Atmos. Sci. 52, pp. 67–82. Browning, G., Kasahara, A. and Kreiss, H.-O. (1980). Initialization of the primitive equations by the bounded derivative method, J. Atmos. Sci. 37, pp. 1424–1436. Byers, H. R. and Braham, R. R. (1949). The Thunderstorms (U.S. Government Printing Office, Washington DC). Cannon, W. B. (1929). Organization for physiological homeostasis, Physiol. Rev. 9, pp. 399–431. Cannon, W. B. (1932). The Wisdom of the Body (W. W. Norton Inc., New York). Carnot, S. (1824). Reflections on the motive power of fire, and on machines fitted to develop that power, in E. Mendoza (ed.), Reflections on the Motive Power of Fire (Dover Publications, New York (1960)), pp. 3–59, reprinted. Caro, D., Wobrock, W. and Flossmann, A. I. (2002). A numerical study on the impact of hygroscopic seeding on the development of cloud particle spectra, J. Appl. Meteor. 41, pp. 333–350. Chandler, D. (1987). Introduction to Modern Statistical Mechanics (Oxford University Press, Oxford). Charney, J. G. (1963). A note on large-scale motions in the tropics, J. Atmos. Sci. 20, pp. 607–609. Charney, J. G. (1971). Geostrophic turbulence, J. Atmos. Sci. 28, pp. 1087–1095. Charney, J. G. and Eliassen, A. (1964). On the growth of the hurricane depression, J. Atmos. Sci. 21, pp. 68–75. Cheng, M.-D. (1989). Effects of downdrafts and mesoscale convective organization on the heat and moisture budgets of tropical cloud clusters, J. Atmos. Sci. 46, pp. 1540–1564. Chikira, M. and Sugiyama, M. (2010). A cumulus parameterization with statedepdent entrainment rate. Part I: Description and sensitivity to temperature and humidity profiles, J. Atmos. Sci. 67, pp. 2171–2193. Cho, H.-R. (1985). Rates of entrainment and detrainment of momentum of cumulus clouds, Mon. Wea. Rev. 113, pp. 1920–1932. Clapeyron, E. (1834). Memoir on the motive power of heat, in E. Mendoza (ed.), Reflections on the Motive Power of Fire (Dover Publications, New York (1960)), pp. 73–105, reprinted. Clark, T. L. and Farley, R. D. (1984). Severe downslope windstorm calculations in two and three spatial dimensions using anelastic interactive grid nesting: A possible mechanism for gustiness, J. Atmos. Sci. 41, pp. 329–350. Clauset, A., Shalizi, C. R. and Newman, M. E. J. (2009). Power-law distributions in empirical data, SIAM Review 51, pp. 661–703. Cohen, B. G. and Craig, G. C. (2004). The response time of a convective cloud ensemble to a change in forcing, Quart. J. R. Meteor. Soc. 130, pp. 933– 944. Cohen, B. G. and Craig, G. C. (2006). Fluctuations in an equilibrium convective ensemble. Part II: Numerical experiments, J. Atmos. Sci. 63, pp. 2005– 2015. Cotton, R. W. and Anthes, R. A. (1989). Storm and Cloud Dynamics (Academic Press).

Bibliography

485

Cox, S. M. and Roberts, A. J. (1994). Initialization and the quasi-geostrophic slow manifold, Preprint. Available at: http://arxiv.org/pdf/nlin/0303011.pdf. Craig, G. C. and Cohen, B. G. (2006). Fluctuations in an equilibrium convective ensemble. Part I: Theoretical formulation, J. Atmos. Sci. 63, pp. 1996– 2004. Davies, L., Plant, R. S. and Derbyshire, S. H. (2009). A simple model of convection with memory, J. Geophys. Res. 114, D17202. Davies, L., Plant, R. S. and Derbyshire, S. H. (2013). Departures from convective equilibrium with a rapidly-varying forcing, Quart. J. R. Meteor. Soc. 139, pp. 1731–1746. Davies, R. M. and Taylor, G. K. (1950). The mechanics of large bubbles rising through extended liquids and through liquids in tubes, Proc. Roy. Soc. London A 200, pp. 375–390. Davies, T., Cullen, M. J. P., Malcolm, A. J., Mawson, M. H., Staniforth, A., White, A. A. and Wood, N. (2005). A new dynamical core for the Met Office’s global and regional modelling of the atmosphere, Quart. J. R. Meteor. Soc. 131, pp. 1759–1782. Davoudi, J., McFarlane, N. A. and Birner, T. (2010). Fluctuation of mass flux in a cloud resolving simulation with interactive radiation, J. Atmos. Sci. 67, pp. 400–418. Dawe, J. T. and Austin, P. H. (2011a). The influence of the cloud shell on tracer budget measurements of LES cloud entrainment, J. Atmos. Sci. 68, pp. 2209–2920. Dawe, J. T. and Austin, P. H. (2011b). Interpolation of LES cloud surfaces for use in direct calculations of entrainment and detrainment, Mon. Wea. Rev 139, pp. 444–456. de Rooy, W. C., Bechtold, P., Fr¨ ohlich, K., Hohenegger, C., Jonker, H., Mironov, D., Siebesma, A. P., Teixeira, J. and Yano, J.-I. (2013). Entrainment and detrainment in cumulus convection: An overview, Quart. J. R. Meteor. Soc. 139, pp. 1–19. de Rooy, W. C. and Siebesma, A. P. (2008). A simple parameterization for detrainment in shallow cumulus, Mon. Wea. Rev 136, pp. 560–576. de Rooy, W. C. and Siebesma, A. P. (2010). Analytical expressions for entrainment and detrainment in cumulus convection, Quart. J. R. Meteor. Soc. 136, pp. 1216–1227. Deardorff, J. W. (1970). A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers, J. Fluid Mech. 41, pp. 453–480. Delayen, K. and Yano, J.-I. (2009). Is asymptotic non-divergence of the largescale tropical atmosphere consistent with equatorial wave theories? Tellus A 61, pp. 491–497. Derbyshire, S. H., Beau, I., Bechtold, P., Grandpeix, J.-Y., Piriou, J.-M., Redelsperger, J.-L. and Soares, P. M. M. (2004). Sensitivity of moist convection to environmental humidity, Quart. J. R. Meteor. Soc. 130, pp. 3055–3079. Derbyshire, S. H., Maidens, A. V., Milton, S. F., Stratton, R. A. and Willett, M. R. (2011). Adaptive detrainment in a convective parametrization, Quart.

486

Bibliography

J. R. Meteor. Soc. 137, pp. 1856–1871. Donner, L. J. (1993). A cumulus parameterization including mass fluxes, vertical momentum dynamics, and mesoscale effects, J. Atmos. Sci. 50, pp. 889– 906. Donner, L. J. and Phillips, V. T. (2003). Boundary layer control on convective available potential energy: Implications for cumulus parameterization, J. Geophys. Res. 108, doi:10.1029/2003JD003773. Donner, L. J., Seman, C. J., Helmer, R. S. and Fan, S. (2001). A cumulus parameterization including mass fluxes, convective vertical velocities, and mesoscale effects: Thermodynamic and hydrological aspects in a general circulation model, J. Climate 14, pp. 3444–3463. Donner, L. J., Seman, C. J. and Hemler, R. S. (1993). A cumulus parameterization including mass fluxes, vertical momentum dynamics, and mesoscale effects, J. Atmos. Sci. 50, pp. 889–906. Doswell, C. A. (2001). Severe convective storms, Meteorological Monographs 28, pp. 1–26. Dritschel, D. G. (1989). Contour dynamics and contour surgery: Numerical algorithms for extended, high-resolution modelling of vortex dynamics in twodimensional, inviscid, incompressible flows, Comp. Phys. Rep. 10, pp. 77– 146. Dritschel, D. G. and Ambaum, M. H. P. (1997). A contour-advective semiLagrangian numerical algorithm for simulating fine-scale conservative dynamical fields, Quart. J. R. Meteor. Soc. 123, pp. 1097–1130. Ducrocq, V. and Bougeault, P. (1995). Simulation of an observed squall line with a meso-beta-scale hydrostatic model, Wea. Forecasting 10, pp. 380–399. Durran, D. R. (1989). Improving the anelastic approximation, J. Atmos. Sci. 46, pp. 1453–1461. Durran, D. R. (1999). Numerical methods for wave equations in geophysical fluid dynamics (Springer, Dordrecht). Emanuel, K. A. (1987). An air–sea interaction model of intraseasonal oscillation in the tropics, J. Atmos. Sci. 44, pp. 2324–2340. Emanuel, K. A. (1989). The finite-amplitude nature of tropical cyclogenesis, J. Atmos. Sci. 46, pp. 3431–3456. Emanuel, K. A. (1991). A scheme for representing cumulus convection in largescale models, J. Atmos. Sci. 48, pp. 2313–2335. Emanuel, K. A. (1993a). A cumulus representation based on the episodic mixing model: The importance of mixing and microphysics in predicting humidity, in K. A. Emanuel and D. J. Raymond (eds.), The Representation of Cumulus Convection in Numerical Models, Meteorological Monographs, Vol. 24, chap. 19 (American Meteorological Society, Boston), pp. 185–192. Emanuel, K. A. (1993b). The effect of convective response time on WISHE modes, J. Atmos. Sci. 50, pp. 1763–1775. Emanuel, K. A. (1994). Atmospheric Convection (Oxford University Press, Oxford). Emanuel, K. A. (1995). The behavior of a simple hurricane model using a convective scheme based on subcloud-layer entropy equilibrium, J. Atmos. Sci.

Bibliography

487

52, pp. 3960–3968. Emanuel, K. A. (2000). Quasi-equilibrium thinking, in D. A. Randall (ed.), General circulation model development, chap. 8 (Elsevier, Amsterdam), pp. 225– 255. Emanuel, K. A. (2007). Quasi-equilibrium dynamics of the tropical atmosphere, in T. Schneider and A. H. Sobel (eds.), The Global Circulation of the Atmosphere (Princeton University Press, Princeton), pp. 186–218. Emanuel, K. A. and Bister, M. (1996). Moist convective velocity and buoyancy scales, J. Atmos. Sci. 53, pp. 3276–3285. Emanuel, K. A., Neelin, J. D. and Bretherton, C. S. (1994). On large-scale circulations in convecting atmosphere, Quart. J. R. Meteor. Soc. 120, pp. 1111–1143. Emanuel, K. A. and Raymond, D. J. (eds.) (1993). The Representation of Cumulus Convection in Numerical Models, Meteorological Monographs, Vol. 24 (American Meteorological Society, Boston). Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. 1, 3rd edn. (John Wiley and Sons, New York). Flato, G., Marotzke, J., Abiodun, B., Braconnot, P., Chou, S. C., Collins, W., Cox, P., Driouech, F., Emori, S., Eyring, V., Forest, C., Gleckler, P., Guilyardi, E., Jakob, C., Kattsov, V., Reason, C. and Rummukainen, M. (2013). Evaluation of climate models, Ch. 9, in T. F. Stocker, D. Qin, G.-K. Plattner, M. Tignor, S. K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P. M. Midgley (eds.), Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University Press, Cambridge). Fletcher, J. K. and Bretherton, C. S. (2010). Evaluating boundary layer-based mass flux closures using cloud-resolving model simulations of deep convection, J. Atmos. Sci. 67, pp. 2212–2225. Ford, R. (1994). Gravity wave radiation from vortex trains in rotating shallow water, J. Fluid Mech. 281, pp. 81–118. Ford, R., McIntyre, M. E. and Norton, W. A. (2000). Balance and the slow quasimanifold: Some explicit results, J. Atmos. Sci. 57, pp. 1236–1254. Fowler, L. D., Randall, D. A. and Rutledge, S. A. (1996). Liquid and ice cloud microphysics in the CSU general circulation model. Part 1: Model description and simulated microphysical processes, J. Climate 9, pp. 489–529. Fraedrich, K. (1973). On the parameterization of cumulus convection by lateral mixing and compensating subsidence. Part 1, J. Atmos. Sci. 30, pp. 408– 413. Fraedrich, K. (1974). Dynamic and thermodynamic aspects of the parameterization of cumulus convection: Part II, J. Atmos. Sci. 31, pp. 1838–1849. Fraedrich, K. and McBride, J. L. (1989). The physical mechanism of CISK and the free-ride balance, J. Atmos. Sci. 46, pp. 2642–2648. Frank, W. M. (1983). The cumulus parameterization problem, Mon. Wea. Rev. 111, pp. 1859–1871. Frank, W. M. and Cohen, C. (1987). Simulation of tropical convective systems.

488

Bibliography

Part I: A cumulus parameterization, J. Atmos. Sci. 44, pp. 3787–3799. Fritsch, J. M. and Chappell, C. F. (1980). Numerical prediction of convectively driven mesoscale pressure systems. Part I: Convective parameterization, J. Atmos. Sci. 37, pp. 1722–1733. Fritsch, J. M. and Kain, J. S. (1993). Convective parameterization for mesoscale models: The Fritsch-Chappell scheme, in K. A. Emanuel and D. J. Raymond (eds.), The Representation of Cumulus Convection in Numerical Models, Meteorological Monographs, Vol. 24, chap. 15 (American Meteorological Society, Boston), pp. 159–164. Fritsch, U. (1995). Turbulence: The Legacy of A.N. Kolmogorov (Cambridge University Press, Cambridge). ˇ Gjorgjievska, S. and Raymond, D. J. (2012). Effects of varying the Fuchs, Z., shape of the convective heating profile on convectively coupled gravity waves and moisture modes, J. Atmos. Sci. 69, pp. 2505–2519. ˇ and Raymond, D. J. (2007). A simple, vertically resolved model of Fuchs, Z. tropical disturbances with a humidity closure, Tellus 59A, pp. 344–354. ˇ Sessions, S. and Raymond, D. J. (2014). Mechanisms controlling the Fuchs, Z., onset of simulated convectively coupled Kelvin waves, Tellus A 66, 22107. Gardiner, C. W. (1985). Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd edn. (Springer, Dordrecht). Garratt, J. R. (1992). The atmospheric boundary layer (Cambridge University Press, Cambridge). Geleyn, J.-F., Girard, C. and Louis, J.-F. (1982). A simple parameterization of moist convection for large-scale atmospheric models, Beitr. Phys. Atmosph. 55, pp. 325–224. Genio, A. D. D. and Wu, J. (2010). Sensitivity of moist convection to environmental humidity, J. Climate 23, pp. 2722–2738. Gentine, P., Betts, A. K., Lintner, B. R., Findell, K. L., van Heerwaarden, C. C. and D’Andrea, F. (2013). A probabilistic bulk model of coupled mixed layer and convection. Part II: Shallow convection case. J. Atmos. Sci. 70, pp. 1557–1576. Germano, M. (1986). A proposal for a redefinition of the turbulent stresses in the filtered Navier–Stokes equations, Phys. Fluids 29, pp. 2323–2324. Gill, A. E. (1980). Some simple solutions for heat-induced tropical circulation, Quart. J. R. Meteor. Soc. 106, pp. 447–462. Golaz, J.-C., Larson, V. E. and Cotton, W. R. (2002). A PDF-based model for boundary layer clouds. Part I: Method and model description, J. Atmos. Sci. 59, pp. 3540–3551. Goldstein, H., Poole, C. and Safko, J. (2002). Classical Mechanics, 3rd edn. (Addison Wesley, San Francisco). Gombosi, T. I. (1994). Gaskinetic theory, 3rd edn. (Cambridge University Press, Cambridge). Grabowski, W. W. (2003). Impact of cloud microphysics on convective-radiative quasi equilibrium revealed by cloud-resolving convection parameterization, J. Climate 16, pp. 3463–3475. Grabowski, W. W. (2006). Comments on “Preliminary tests of multiscale model-

Bibliography

489

ing with a two-dimensional framework: Sensitivity to coupling methods”, Mon. Wea. Rev. 134, pp. 2012–2026. Grabowski, W. W. and Smolarkiewicz, P. K. (1999). CRCP: A cloud resolving convection parameterization for modeling the tropical convective atmosphere, Physica D 133, pp. 171–178. Grant, A. L. M. (2001). Cloud-base fluxes in the cumulus-capped boundary layer, Quart. J. R. Meteor. Soc. 127, pp. 407–421. Greenspan, H. P. (1968). The Theory of Rotating Fluids (Cambridge University Press, New York). Gregory, D. (1997). The mass flux approach to the parameterization of deep convection, in R. K. Smith (ed.), The Physics and Parameterization of Moist Atmospheric Convection (Kluwer Academic Publishers, Amsterdam), pp. 297–319. Gregory, D. (2001). Estimation of entrainment rate in simple models of convective clouds, Quart. J. R. Meteor. Soc. 127, pp. 53–72. Gregory, D., Kershaw, R. and Inness, P. M. (1997). Parameterization of momentum transport by convection. II: Tests in single-column and general circulation models, Quart. J. R. Meteor. Soc. 123, pp. 1153–1183. Gregory, D., Morcrette, J.-J., Jacob, C., Beljaars, A. C. M. and Stockdale, T. (2000). Revision of convection, radiation and cloud schemes in the ECMWF integrated forecasting system, Quart. J. R. Meteor. Soc. 126, pp. 1685– 1710. Gregory, D. and Rowntree, P. R. (1990). A mass flux convection scheme with representation of cloud ensemble characteristics and stability-dependent closure, Mon. Wea. Rev 118, pp. 1483–1506. Gregory, P. (2005). Bayesian Logical Data Analysis for the Physical Sciences (Cambridge University Press, Cambridge). Guha, P. (2007). Metriplectic structure, Leibniz dynamics and dissipative systems, J. Math. Anal. Appl. 326, pp. 121–136. Guichard, F., Petch, J. C., Redelsperger, J.-L., Bechtold, P., Chaboureau, J.-P., Cheinet, S., Grabowski, W., Grenier, H., Jones, C. G., Kohler, M., Piriou, J.-M., Tailleux, R. and Tomasini, M. (2004). Modelling the diurnal cycle of deep precipitating convection over land with cloud resolving models and single-column models, Quart. J. R. Meteor. Soc. 130, pp. 3139–3171. Hack, J. J., Schubert, W. H. and Dias, P. L. S. (1984). A spectral cumulus parameterization for use in numerical models of the tropical atmosphere, Mon. Wea. Rev. 112, pp. 704–716. Haltiner, G. J. and Williams, R. T. (1980). Numerical prediction and dynamic meteorology, 2nd edn. (John Wiley and Sons, New York). Haman, K. (1969). On the influence of convective clouds on the large scale stratification, Tellus 21, pp. 41–53. Hayashi, Y. (1970). A theory of large-scale equatorial waves generated by condensation heat and accelerating the zonal wind, J. Meteor. Soc. Japan 48, pp. 140–160. Hayashi, Y. (1971). Large-scale equatorial waves destabilized by convective heating in the presence of surface friction, J. Meteor. Soc. Japan 49, pp. 458–

490

Bibliography

466. Heggem, T., Lende, R. and Løvseth, J. (1998). Analysis of long time series of coastal wind, J. Atmos. Sci. 55, pp. 2907–2917. Hepburn, B. S. (2007). Equilibrium and explanation in 18th century mechanics, Ph.D. thesis, University of Pittsburgh. Heus, T. and Jonker, H. J. J. (2008). Subsiding shells aroud shallow cumulus clouds, J. Atmos. Sci. 65, pp. 1003–1018. Heus, T., Jonker, H. J. J., van den Akker, H. E. A., Griffith, E. J., Koutek, M. and Post, F. H. (2009a). A statistical approach to the life cycle analysis of cumulus clouds selected in a vitual reality environment, J. Geophys. Res. 114, D06208. Heus, T., Pols, C. F. J., Jonker, H. J. J., den Akker, H. E. A. V. and Lenschow, D. H. (2009b). Observational validation of the compensating mass flux through the shell around cumulus clouds, Quart. J. R. Meteor. Soc. 135, pp. 101–112. Heus, T., van Dijk, G., Jonker, H. J. J. and van den Akker, H. E. A. (2008). Mixing in shallow cumulus clouds studied by Lagrangian particle tracking. J. Atmos. Sci. 65, pp. 2581–2597. Hildebrandt, S. and Tromba, A. (1996). The Parsimonious Universe: Shape and Form in the Natural World (Springer-Verlag, New York). Hirons, L., Inness, P., Vitart, F. and Bechtold, P. (2013a). Understanding advances in the simulation of intraseasonal variability in the ECMWF model. Part I: The representation of the MJO, Quart. J. R. Meteor. Soc. 139, pp. 1417–1426. Hirons, L., Inness, P., Vitart, F. and Bechtold, P. (2013b). Understanding advances in the simulation of intraseasonal variability in the ECMWF model. Part II: The application of process-based diagnostics, Quart. J. R. Meteor. Soc. 139, pp. 1427–1444. Hohenegger, C. and Bretherton, C. S. (2011). Simulating deep convection with a shallow convection scheme, Atmos. Chem. Phys. 11, pp. 10389–10406. Holland, J. Z. (1972). Comparative evaluation of some BOMEX measurements of sea surface evaporation, energy flux and stress, J. Phys. Oceanogr. 2, pp. 476–486. Holloway, C. E. and Neelin, J. D. (2007). The convective cold top and quasi equilibrium, J. Atmos. Sci. 64, pp. 1467–1487. Holloway, C. E. and Neelin, J. D. (2009). Moisture vertical structure, column water vapor, and tropical deep convection, J. Atmos. Sci. 66, pp. 1665– 1683. Holloway, C. E. and Neelin, J. D. (2010). Temporal relations of column water vapor and tropical precipitation, J. Atmos. Sci. 67, pp. 1091–1105. Holloway, C. E., Woolnough, S. J. and Lister, G. M. S. (2012). Precipitation distributions for explicit versus parameterized convection in a large-domain high-resolution tropical case study, Quart. J. R. Meteor. Soc. 138, pp. 1692–1708. Holton, J. R. (2004). An introduction to dynamic meteorology, International Geophysics Series, Vol. 88, 4th edn. (Elsevier Academic Press, New York).

Bibliography

491

Horsthemke, W. and Lefver, R. (1984). Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology (Springer-Verlag, New York). Hoskins, B. J., McIntyre, M. E. and Robertson, A. W. (1985). On the use and significance of isentropic potential vorticity maps, Quart. J. R. Meteor. Soc. 111, pp. 877–946. Houghton, H. and Cramer, H. (1951). A theory of entrainment in convective currents, J. Meteor. 8, pp. 95–102. Houze Jr., R. A. (1977). Structure and dynamics of a tropical squall-line system, Mon. Wea. Rev. 105, pp. 1540–1567. Houze Jr., R. A. (1993). Cloud dynamics (Academic Press, Oxford). Houze Jr., R. A. and Betts, A. K. (1981). Convection in GATE, Rev. Geophys. Space Phys. 19, pp. 541–576. Houze Jr., R. A., Cheng, C.-P., Leary, C. A. and Gamache, J. F. (1980). Diagnosis of cloud mass and heat fluxes form radar and synoptic data, J. Atmos. Sci. 37, pp. 754–773. Houze Jr., R. A. and Leary, C. A. (1976). Comparison of convective mass and heat transports in tropical easterly waves computed by two methods, J. Atmos. Sci. 33, pp. 424–429. Igau, R. C., LeMone, M. A. and Wei, D. (1999). Updraft and downdraft cores in TOGA COARE: Why so many buoyant downdraft cores? J. Atmos. Sci. 56, pp. 2232–2245. Ishida, H., Burt, W. V. and Mitsuta, Y. (1984). The effects of mesoscale convective cells on the surface wind field over the ocean, Bound.-Layer Meteor. 29, pp. 75–84. Jakob, C. (2010). Accelerating progress in global atmospheric model development through improved parameterizations: Challenges, opportunities, and strategies. Bull. Amer. Meteor. Soc. 91, pp. 869–875. Jakob, C. and Siebesma, A. P. (2003). A new subcloud model for mass-flux convection schemes; influence on triggering, updraught properties and model climate, Mon. Wea. Rev 131, pp. 2765–2778. Jaynes, E. T. (1978). Where do we stand on maximum entropy? in R. D. Levine and M. Tribus (eds.), The Maximum Entropy Formalism (MIT Press, Cambridge), pp. 15–118. Jaynes, E. T. (2003). Probability Theory, The Logic of Science (Cambridge University Press, Cambridge). Jensen, H. J. (1998). Self-Organized Criticality: Emergent Complex Behavior in Physical and Biological Systems (Cambridge University Press, Cambridge). Jensen, J. B., Austin, P. H., Baker, M. B. and Blyth, A. M. (1985). Turbulent mixing, spectral evolution and dynamics in a warm cumulus cloud, J. Atmos. Sci. 42, pp. 173–192. Johnson, R. H. (1976). The role of convective-scale precipitation downdrafts in cumulus and synoptic-scale interactions, J. Atmos. Sci. 33, pp. 1890–1910. Johnson, R. H. (1980). Diagnosis of convective and mesoscale motions during Phase III of GATE, J. Atmos. Sci. 37, pp. 733–753. Johnson, R. H. and Young, G. S. (1983). Heat and moisture budgets of tropical

492

Bibliography

mesoscale anvil clouds, J. Atmos. Sci. 40, pp. 2138–2147. Jonas, P. R. (1990). Observations of cumulus cloud entrainment, Atmos. Res. 25, pp. 105–127. Jonker, H. J. J., Heus, T. and Sullivan, P. P. (2008). A refined view of vertical mass transport by cumulus convection, J. Geophys. Res. 35, L07810. Jonker, H. J. J., Verzijlbergh, R. A., Heus, T. and Siebesma, A. P. (2006). The influence of the sub-cloud moisture field on cloud size distributions and the consequences for entrainment, in Extended abstract from the 17th Symposium on Boundary Layers and Turbulence, San Diego, USA (American Meteorological Society, Boston). Jonker, S. (2005). Evaluation study of the Kain Fritsch convection scheme, KNMI Technical Report TR275 (KNMI, De Bilt, the Netherlands). Jordan, C. L. (1958). Mean soundings for the West Indies area, J. Meteor. 15, pp. 91–97. Jorgensen, D. P. and LeMone, M. A. (1989). Vertical velocity characteristics of oceanic convection, J. Atmos. Sci. 46, pp. 621–640. Jung, T., Balsamo, G., Bechtold, P., Beljaars, A. C. M., K¨ ohler, M., Miller, M. J., Morcrette, J.-J., Orr, A., Rodwell, M. J. and Tompkins, A. M. (2010). The ECMWF model climate: Recent progress through improved physical parametrizations, Quart. J. R. Meteor. Soc. 136, pp. 1145–1160. Kain, J. S. (2004). The Kain–Fritsch convective parameterization: An update, J. Appl. Meteor. 43, pp. 170–181. Kain, J. S. and Fritsch, J. M. (1990). A one-dimensional entraining/detraining plume model and its application in convective parameterization, J. Atmos. Sci. 47, pp. 2784–2802. Kain, J. S. and Fritsch, J. M. (1992). The role of the convective “trigger function” in numerical forecasts of mesoscale convective systems, Meteor. Atmos. Phys. 49, pp. 93–106. Kain, J. S. and Fritsch, J. M. (1993). Convective parameterization for mesoscale models. The Kain–Fritsch scheme, in K. A. Emanuel and D. J. Raymond (eds.), The Representation of Cumulus Convection in Numerical Models, Meteorological Monographs, Vol. 24, chap. 16 (American Meteorological Society, Boston), pp. 165–170. Kain, J. S., Fritsch, J. M. and Weiss, S. J. (2003). Parameterized updraft mass flux as a predictor of convective intensity, Wea. Forecasting 18, pp. 106– 116. Kamburova, P. L. and Ludlam, F. H. (1966). Rainfall evaporation in thunderstorm downdraughts, Quart. J. R. Meteor. Soc. 92, pp. 510–518. Kasahara, A. (2000). On the origin of cumulus parameterization for numerical prediction models, in D. A. Randall (ed.), General circulation model development (Academic Press, Oxford), pp. 199–224. Kessler, E. (1965). Microphysical parameters in relation to tropical cloud and precipitation distribution and their modifications, Geofis. Int. 5, pp. 79–88. Kessler, E. (1969). On the continuity of water substances in atmospheric circulations, Meteorological Monographs 33, pp. 1–84. Kessler, E. (1985). Thunderstorm Morphology and Dynamics (University of Ok-

Bibliography

493

lahoma Press, Norman). Khain, A., Pokrovsky, A., Pinsky, M., Seifert, A. and Phillips, V. (2004). Simulation of effects of atmospheric aerosols on deep turbulent convective clouds using a spectral microphysics mixed-phase cumulus cloud model. Part I: Model description and possible applications, J. Atmos. Sci. 61, pp. 2963– 2982. Khain, A. P., Ovtchinnikov, M., Pinsky, M., Pokrovsky, A. and Krugliak, H. (2000). Notes on the state-of-the-art numerical modeling of cloud microphysics, Atmos. Res. 55, pp. 159–224. Khairoutdinov, M. F., , Krueger, S. K., Moeng, C. H., Bogenschutz, P. A. and Randall, D. (2009). Large-eddy simulation of maritime deep tropical convection, J. Adv. Model. Earth Syst. 1, doi:10.3894/JAMES.2009.1.15. Khairoutdinov, M. F. and Randall, D. (2006). High-resolution simulation of shallow to deep convection transition over land, J. Atmos. Sci. 63, pp. 3421– 3436. Khouider, B., Biello, J. and Majda, A. J. (2010). A stochastic multicloud model for tropical convection, Comm. Math. Sci. 8, pp. 187–216. Khouider, B. and Majda, A. J. (2006). A simple multicloud parameterization for convectively coupled tropical waves. Part I: Linear analysis, J. Atmos. Sci. 63, pp. 1308–1323. Khouider, B. and Majda, A. J. (2008). Multi-cloud models for organized tropical convection: Enhanced congestus heating, J. Atmos. Sci. 65, pp. 895–914. Kiladis, G. N., Wheeler, M. C., Haertel, P. T., Straub, K. H. and Roundy, P. E. (2009). Convectively coupled equatorial waves, Rev. Geophys. 47, doi:10. 1029/2008RG000266. Kim, D. and Kang, I.-S. (2012). A bulk mass flux convection scheme for climate model, Climate Dyn. 38, pp. 411–429. Knight, C. A. and Miller, L. J. (1998). Early radar echoes from small, warm cumulus: Bragg and hydrometeor scattering, J. Atmos. Sci. 55, pp. 2974– 2992. Knight, C. G., Knight, S. H. E., Massey, N., Aina, T., Christensen, C., Frame, D. J., Kettleborough, J. A., Martin, A., Pascoe, S., Sanderson, B., Stainforth, D. A. and Allen, M. R. (2007). Association of parameter, software, and hardware variation with large-scale behavior across 57,000 climate models, Proc. Natl. Acad. Sci. 104, pp. 12259–12264. Kondepudi, D. and Prigogine, I. (1998). Modern Thermodynamics: From Heat Engines to Dissipative Structures (John Wiley and Sons, New York). Kraichnan, R. H. (1961). Dynamics of nonlinear stochastic systems, J. Math. Phys. 2, pp. 124–148. Kreiss, H. O. (1980). Problems with different time scales for partial-differential equations, Comm. Pure Appl. Math. 33, pp. 399–439. Kreitzberg, C. W. and Perkey, D. J. (1976). Release of potential instability: Part I. A sequential plume model within a hydrostatic primitive equation model, J. Atmos. Sci. 33, pp. 456–475. Krishnamurti, T. N., Kanamitsu, M., Godbole, R., Chang, C. B., Carr, F. and Chow, J. (1976). Study of a monsoon depression II. Dynamical structure,

494

Bibliography

J. Met. Soc. Japan 54, pp. 208–225. Kuang, Z. (2008). A moisture-stratiform instability for convectively coupled waves, J. Atmos. Sci. 65, pp. 834–854. Kuang, Z. (2011). The wavelength dependence of the gross moist stability and the scale selection in the instability of column-integrated moist static energy, J. Atmos. Sci. 68, pp. 61–74. Kuang, Z. and Bretherton, C. S. (2006). A mass-flux scheme view of a highresolution simulation of a transition from shallow to deep convection, J. Atmos. Sci. 63, pp. 1895–1909. Kuo, H.-L. (1962). On the controlling influences of eddy diffusion on thermal convection, J. Atmos. Sci. 19, pp. 236–243. Kuo, H.-L. (1974). Further studies of the parameterization of the influence of cumulus convection on large-scale flow, J. Atmos. Sci. 31, pp. 1232–1240. Kuo, H.-L. and Raymond, W. H. (1980). A quasi-one-dimensional cumulus cloud model and parameterization of cumulus heating and mixing effects, Mon. Wea. Rev. 108, pp. 991–1009. Kurz, M. (1998). Synoptic meteorology, Tech. rep. Deutscher Wetterdienst. Lamb, H. (1932). Hydrodynamics, 6th edn. (Cambridge University Press, Cambridge). Lamontagne, R. G. and Telford, J. W. (1983). Cloud top mixing in small cumuli, J. Atmos. Sci. 40, pp. 2148–2156. Landau, L. D. and Lifshitz, E. M. (1980). Statistical Physics Part 1, 3rd edn. (Elsevier). Lawrence, M. G. and Rasch, P. J. (2005). Tracer transport in deep convective updrafts: Plume ensemble versus bulk formulations, J. Atmos. Sci. 62, pp. 2880–2894. Leary, C. A. and Houze Jr., R. A. (1980). The contribution of mesoscale motions to the mass and heat fluxes of an intensive tropical convective sytem, J. Atmos. Sci. 37, pp. 784–796. Leith, C. E. (1980). Nonlinear normal mode initialization and quasi-geostrophic theory, J. Atmos. Sci. 37, pp. 958–968. LeMone, M. A. and Zipser, E. J. (1980). Cumulonimbus vertical velocity events in GATE. Part I: Diameter, intensity and mass flux, J. Atmos. Sci. 37, pp. 2444–2457. Leonard, A. (1974). Energy cascade in large-eddy simulations of turbulent flows, Adv. Geophys. 418A, pp. 237–248. Lesieur, M. (1987). Turbulence in Fluids (Martinus Nijhoff Publ., Amsterdam). LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems (Cambridge University Press, Cambridge). Levine, J. (1959). Spherical vortex theory of bubble-like motion in cumulus clouds, J. Meteor. 16, pp. 653–662. Liepmann, H. (1979). The rise and fall of ideas in turbulence: Research in turbulence, still the most difficult problem of fluid mechanics, continues to produce technological advances, but the path of progress is anything but straight, Amer. Sci. 67, pp. 221–228. Lighthill, J. M. (1952). On sound generated aerodynamically. I. General theory,

Bibliography

495

Proc. Roy. Soc. London A 211, pp. 564–587. Lighthill, M. J. (1954). On sound generated aerodynamically. II. Turbulence as a source of sound, Proc. Roy. Soc. London A 222, pp. 1–32. Lilly, D. K. (1967). The representation of small-scale turbulence in numerical simulation experiments, in H. H. Goldstine (ed.), Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences (Yorktown Heights, New York), pp. 195–210. Lilly, D. K. (1968). Models of cloud-topped mixed layer under a strong inversion, Quart. J. R. Meteor. Soc. 94, pp. 292–309. Lin, C. and Arakawa, A. (1997). The macroscopic entrainment processes of simulated cumulus ensemble. Part 1: Entrainment sources, J. Atmos. Sci. 54, pp. 1027–1043. Lin, X. and Johnson, R. H. (1996). Heating, moistening, and rainfall over the western Pacific warm pool during TOGA COARE, J. Atmos. Sci. 53, pp. 3367–3383. Lindzen, R. (1974). Wave-CISK in the tropics, J. Atmos. Sci. 31, pp. 156–179. Lindzen, R. S. (1981). Some remarks on cumulus parameterization, in Report on NASA–GISS Workshop: “Clouds in Climate-Modelling and Satellite Observational Studies”, pp. 42–51. Lindzen, R. S. (1988). Some remarks on cumulus parameterization, Pure Appl. Geophys. 126, pp. 123–135. Lintner, B. R., Holloway, C. E. and Neelin, J. D. (2011). Column water vapor statistics and their relationship to deep convection, vertical and horizontal circulation, and moisture structure at Nauru, J. Climate 24, pp. 5454–5466. Lipps, F. B. and Hemler, R. S. (1982). A scale analysis of deep moist convection and some related numerical calculations, J. Atmos. Sci. 39, pp. 2191–2210. Liu, C., Moncrieff, M. W. and Grabowski, W. W. (2001). Explicit and parameterized realizations of convective cloud systems in TOGA COARE, Mon. Wea. Rev. 121, pp. 1689–1703. Liu, P., Satoh, M., Wang, B., Fudeyasu, H., Nasuno, T., Li, T., Miura, H., Taniguchi, H., Masunaga, H., Fu, X. and Annamalai, H. (2009). An MJO simulated by the NICAM at 14- and 7-km resolutions, Mon. Wea. Rev. 137, pp. 3254–3268. Lord, S. J. and Arakawa, A. (1982). Interaction of a cumulus cloud ensemble with the large-scale environment. Part III: Semi-prognostic tests of the ArakawaSchubert cumulus parameterization, J. Atmos. Sci. 39, pp. 88–103. Lord, S. J., Chao, W. C. and Arakawa, A. (1982). Interaction of a cumulus cloud ensemble with the large-scale environment. Part IV: The discrete model, J. Atmos. Sci. 39, pp. 104–113. Lorenz, E. N. (1963). Deterministic nonperiodic flow, J. Atmos. Sci. 20, pp. 130–141. Lorenz, E. N. (1986). On the existence of a slow manifold, J. Atmos. Sci. 43, pp. 1547–1558. Lorenz, E. N. (1992). The slow manifold – what is it? J. Atmos. Sci. 49, pp. 2449–2451. Lovejoy, S. (1982). Area-perimeter relation for rain and cloud areas, Science 216,

496

Bibliography

pp. 185–187. Lovejoy, S. and Schertzer, D. (2010). Towards a new synthesis for atmospheric dynamics: Space-time cascades, Atmos. Res. 96, pp. 1–52. Lovejoy, S. and Schertzer, D. (2012). The Weather and Climate: Emergent Laws and Multifractal Cascades (Cambridge University Press, Cambridge). Lovejoy, S., Schertzer, D. and Allaire, V. (2008). The remarkable wide range spatial scaling of TRMM precipitation, Atmos. Res. 90, pp. 10–32. Lovejoy, S., Tuck, A. F., Hovde, S. J. and Schertzer, D. (2009a). The vertical cascade structure of the atmosphere and multifractal drop sonde outages, J. Geophys. Res. 114, D07111. Lovejoy, S., Tuck, A. F. and Schertzer, D. (2009b). The horizontal cascade structure of atmospheric fields determined from aircraft data, J. Geophys. Res. 115, D13105. Lucas, C., Zipser, E. J. and LeMone, M. A. (1994). Vertical velocity in oceanic convection off tropical Australia, J. Atmos. Sci. 51, pp. 3183–3193. Ludlam, F. H. and Scorer, R. S. (1953). Reviews of modern meteorology: Convection in the atmosphere, Quart. J. R. Meteor. Soc. 79, pp. 317–341. Lumley, J. L. (1965). Interpretation of time spectra measured in high-intensity shear flow, Phys. Fluids 48, pp. 1056–1062. Lumley, J. L. (1983). Atmospheric modelling, Trans. Inst. Eng. Aust. Mech. Eng. 8, pp. 153–159. Madden, R. A. and Julian, P. R. (1971). Detection of a 40-50 day oscillation in the zonal wind in the tropical Pacific, J. Atmos. Sci. 5, pp. 702–708. Majda, A. J. (2007). Multiscale models with moisture and systematic strategies for superparameterization, J. Atmos. Sci. 64, pp. 2726–2734. Majda, A. J., Khouider, B., Kiladis, G. N., Straub, K. H. and Shefter, M. G. (2004). A model for convectively coupled tropical waves: Nonlinearity, rotation, and comparison with observations, J. Atmos. Sci. 61, pp. 2188–2205. Majda, A. J. and Shefter, M. G. (2001a). Models for stratiform instability and convectively coupled waves, J. Atmos. Sci. 58, pp. 1567–1584. Majda, A. J. and Shefter, M. G. (2001b). Waves and instabilities for model tropical convective parameterizations, J. Atmos. Sci. 58, pp. 896–914. Malkus, J. S. (1952). The slopes of cumulus clouds in relation to external wind shear, Quart. J. R. Meteor. Soc. 78, pp. 530–542. Mallat, S. (1998). A wavelet tour of signal processing, 2nd edn. (Academic Press, Oxford). Maloney, E. D. and Esbensen, S. K. (2005). A modeling study of summertime east Pacific wind-induced ocean-atmosphere exchange in the intraseasonal oscillation, J. Climate 18, pp. 568–584. Manabe, S., Smagorinsky, J. S. and Strickler, R. F. (1965). A similarity hypothesis for shallow-cumulus transports, Mon. Wea. Rev. 93, pp. 769–798. Manabe, S. and Strickler, R. F. (1964). Thermal equilibrium of the atmosphere with a convective adjustment, J. Atmos. Sci. 21, pp. 361–385. Mandelbrot, B. B. (1982). The fractal geometry of nature (WH Freeman and Co., New York). Mapes, B. E. (1997a). Equilibrium vs. activation control of large-scale variations

Bibliography

497

of tropical deep convection, in R. K. Smith (ed.), The Physics and Parameterization of Moist Atmospheric Convection (Kluwer Academic Publishers, Amsterdam), pp. 321–358. Mapes, B. E. (1997b). Implications of convective quasi-equilibrium for the largescale flow, in R. K. Smith (ed.), The Physics and Parameterization of Moist Atmospheric Convection (Kluwer Academic Publishers, Amsterdam), pp. 413–446. Mapes, B. E. (1998). The large-scale part of tropical mesoscale convective system circulations: A linear vertical spectrum band model, J. Atmos. Sci. 76, pp. 29–55. Mapes, B. E. (2000). Convective inhibition, subgrid-scale triggering energy, and stratiform instability in a toy tropical wave model, J. Atmos. Sci. 57, pp. 1515–1535. Mapes, B. E. and Neale, R. B. (2011). Parameterizing convective organization to escape the entrainment dilemma, J. Adv. Model. Earth Syst. 3, doi: 10.1029/2011MS000042. Maraun, D., Wetterhall, F., Ireson, A. M., Chandler, R. E., Kendon, E. J., Widmann, M., Brienen, S., Rust, H. W., Sauter, T., Themeβl, M., Venema, V. K. C., Chun, K. P., Goodess, C. M., Jones, R. G., Onof, C., Vrac, M. and Thiele-Eich, I. (2010). Precipitation downscaling under climate change: Recent developments to bridge the gap between dynamical models and the end user, Rev. Geophys. 48, doi:10.1029/2009RG000314. Markowski, P. and Richardson, Y. (2010). Mesoscale Meteorology in Midlatitudes (John Wiley and Sons, New York). Mart´ınez-Alvarado, O. and Plant, R. S. (2014). Parameterised diabatic processes in numerical simulations of an extratropical cyclone, Quart. J. R. Meteor. Soc. 140, pp. 1742–1755. Matsuno, T. (1966). Quasi-geostrophic motions in the equatorial area, J. Meteor. Soc. Japan 44, pp. 25–43. McBride, J. L. (1981). An analysis of diagnostic cloud mass flux models, J. Atmos. Sci. 38, pp. 1977–1990. McIntyre, M. (2001). Balance, potential-vorticity inversion, Lighthill radiation and the slow quasimanifold, in P. F. Hodnett (ed.), Proceedings of the IUTAM/IUGG/Royal Irish Academy Symposium on Advances in Mathematical Modelling of Atmosphere and Ocean Dynamics, (Kluwer Academic Publishers, Amsterdam), pp. 45–68. McKane, A. J. and Newman, T. J. (2005). Predator–prey cycles from resonant amplification of demographic stochasticity, Phys. Rev. Lett. 94, 218102. Mellor, G. L. (1977). Gaussian cloud model relations, J. Atmos. Sci. 34, pp. 356–358. Mellor, G. L. and Yamada, T. (1974). A hierarchy of turbulence closure models for planetary boundary layers, J. Atmos. Sci. 31, pp. 1791–1806. Miller, M. J., Beljaars, A. C. M. and Palmer, T. N. (1992). The sensitivity of the ECMWF model to the parameterization of evaporation from the tropical oceans, J. Climate 5, pp. 418–434. Mironov, D. V. (2009). Turbulence in the lower troposphere: Second-order clo-

498

Bibliography

sure and massflux modelling frameworks, in W. Hillebrandt and F. Kupka (eds.), Interdisciplinary Aspects of Turbulence, Vol. 756 (Springer Berlin Heidelberg, Berlin), pp. 1–61. Molinari, J. (1985). A general form of Kuo’s coumulus parameterization, Mon. Wea. Rev. 113, pp. 1411–1416. Molinari, J. and Dudek, M. (1992). Parameterization of convective precipitation in mesoscale numerical models: A critical review, Mon. Wea. Rev. 120, pp. 326–344. ¨ M¨ oller, F. and Manabe, S. (1961). Uber das strahlungsgleichgewicht in der atmosph¨ are, Z. Meteor. 15, pp. 3–8. Moncrieff, M. W. (1981). A theory of organized steady convection and its transport properties, Quart. J. R. Meteor. Soc. 107, pp. 29–50. Moncrieff, M. W. (1992). Organized convective systems: Archetypal dynamical models, mass and momentum flux theory, and parametrization, Quart. J. R. Meteor. Soc. 118, pp. 819–850. Moncrieff, M. W. and Green, J. S. A. (1972). The propagation and transfer properties of steady convective overturning in shear, Quart. J. R. Meteor. Soc. 98, pp. 336–352. Moncrieff, M. W. and Miller, M. J. (1976). The dynamics and simulation of tropical cumulonimbus and squall lines, Quart. J. R. Meteor. Soc. 102, pp. 373–394. Moncrieff, M. W. and So, D. W. K. (1989). A hydrodynamical theory of conservative bounded density currents, J. Fluid Mech. 198, pp. 177–197. Moorthi, S. and Suarez, M. J. (1992). Relaxed Arakawa–Schubert: A parameterization of moist convection for general circulation models, Mon. Wea. Rev. 120, pp. 978–1002. Morton, B. R. (1968). On Telford’s model for clear air convection, J. Atmos. Sci. 25, pp. 135–138. Morton, B. R., Taylor, G. I. and Turner, J. S. (1956). Turbulent gravitational convection from maintained and instantaneous sources, Proc. Roy. Soc. London A 234, pp. 1–23. Nastrom, G. and Gage, K. S. (1985). A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft, J. Atmos. Sci. 42, pp. 950–960. Neelin, J. D. and Held, I. M. (1987). Modeling tropical convergence based on the moist static energy budget, Mon. Wea. Rev. 115, pp. 3–12. Neelin, J. D., Peters, O. and Hales, K. (2009). The transition to strong convection, J. Atmos. Sci. 66, pp. 2367–2384. Neelin, J. D., Peters, O., Lin, J. W.-B., Hales, K. and Holloway, C. E. (2008). Rethinking convective quasi-equilibrium: Observational constraints for stochastic convective schemes in climate models, Phil. Trans. R. Soc. A 366, pp. 2579–2602. Neelin, J. D. and Yu, J.-Y. (1994). Modes of tropical variability under convective adjustment and the Madden-Julian oscillation. Part I: Analytical theory, J. Atmos. Sci. 51, pp. 1876–1894. Neelin, J. D. and Zeng, N. (2000). A quasi-equilibrium tropical circulation model

Bibliography

499

– formulation, J. Atmos. Sci. 57, pp. 1741–1766. Neggers, R. A. J., K¨ ohler, M. and Beljaars, A. C. M. (2009). A dual mass flux framework for boundary layer convection. Part I: Transport, J. Atmos. Sci. 66, pp. 1464–1487. Neggers, R. A. J., Siebesma, A. P. and Jonker, H. J. J. (2002). A multiparcel method for shallow cumulus convection, J. Atmos. Sci. 59, pp. 1655–1668. Neggers, R. A. J., Siebesma, A. P., Lenderink, G. and Holtslag, A. A. M. (2004). An evaluation of mass flux closures for diurnal cycles of shallow cumulus, Mon. Wea. Rev 132, pp. 2525–2536. Newton, C. W. (1950). Structure and mechanism of the prefrontal squall line, J. Meteor. 7, pp. 210–222. Nitta, T. (1975). Observational determination of cloud mass flux distributions, J. Atmos. Sci. 32, pp. 73–91. Nitta, T. (1977). Response of cumulus updraft and downdraft to GATE A/B-scale motion systems, J. Atmos. Sci. 34, pp. 1163–1186. Nitta, T. and Esbensen, S. (1974). Heat and moisture budget analyses using BOMEX data, Mon. Wea. Rev. 102, pp. 17–28. Nober, F. J. and Graf, H. F. (2005). A new convective cloud field model based upon principles of self-organization, Atmos. Chem. Phys. 5, pp. 2749–2759. Nordeng, T. E. (1994). Extended versions of the convective parameterization scheme at ECMWF and their impact on the mean and transient activity of the model in the tropics, Technical Memorandum 206 ECMWF. North, G. R. and Erukhimova, T. L. (2009). Atmospheric Thermodynamics (Cambridge University Press, Cambridge). Ogura, Y. (1963). A consequence of the zero fourth cumulant approximation in the decay of isotropic turbulence, J. Fluid Mech. 16, pp. 33–40. Ogura, Y. and Cho, H.-R. (1973). Diagnostic detrainment of cumulus populations from large-scale variables, J. Atmos. Sci. 30, pp. 1276–1286. Ogura, Y. and Phillips, N. A. (1962). Scale analysis of deep and shallow convection in the atmosphere, J. Atmos. Sci. 19, pp. 173–179. Olver, F. W. J. (1974). Asymptotics and Special Functions (Academic Press, Oxford). Ooyama, K. V. (1971). A theory on parameterization of cumulus convection, J. Meteor. Soc. Japan 26, pp. 3–40. Ooyama, K. V. (1972). On parameterization of cumulus convection, in Dynamics of the tropical atmosphere (National Center for Atmospheric Research, Boulder, Colorado), pp. 496–505, notes from a colloquium: Summer 1972. Ooyama, K. V. (1982). Conceptual evolution of the theory and modeling of the tropical cyclone, J. Meteor. Soc. Japan 60, pp. 369–380. Orlanski, I. (1975). Rational subdivision of scales for atmospheric processes, Bull. Amer. Meteor. Soc. 56, pp. 527–530. Orszag, S. A. (1970). Analytical theories of turbulence, J. Fluid Mech. 41, pp. 363–386. Paluch, I. R. (1979). The entrainment mechanism in Colorado cumili, J. Atmos. Sci. 36, pp. 2467–2478. Pan, D.-M. and Randall, D. A. (1998). A cumulus parameterization with prog-

500

Bibliography

nostic closure, Quart. J. R. Meteor. Soc. 124, pp. 949–981. Parodi, A. and Emanuel, K. (2009). A theory for buoyancy and velocity scales in deep moist convection, J. Atmos. Sci. 66, pp. 3449–3463. Paul, W. and Baschnagel, J. (1999). Stochastic Processes: From Physics to Finance (Springer–Verlag). Pedlosky, J. (1987). Geophysical Fluid Dynamics, 2nd edn. (Springer-Verlag, New York). Peters, M. E. and Bretherton, C. S. (2006). Structure of tropical variability from a vertical mode perspective, Theor. Comput. Fluid Dyn. 20, pp. 501–524. Peters, O., Hertlein, C. and Christensen, K. (2002). A complexity view of rainfall, Phys. Rev. Lett. 88, 018701. Peters, O. and Neelin, J. D. (2006). Critical phenomena in atmospheric precipitation, Nature Physics 2, pp. 393–396. Peters, O., Neelin, J. D. and Nesbitt, S. W. (2009). Mesoscale convective systems and critical clusters, J. Atmos. Sci. 66, pp. 2913–2924. Piriou, J.-M., Redelsperger, J.-L., Geleyn, J.-F., Lafore, J.-P. and Guichard, F. (2007). An approach for convective parameterization with memory: Separating microphysics and transport in grid-scale equations, J. Atmos. Sci. 64, pp. 4127–4139. Planck, M. (1922). Treatise on Thermodynamics, 7th edn. (Dover Publications, New York), translation by Ogg, A. (1926), 3rd edition translated from the seventh German edition. Plant, R. S. (2009). Statistical properties of cloud lifecycles in cloud-resolving models, Atmos. Chem. Phys. 9, pp. 2195–2205. Plant, R. S. (2010). A review of the theoretical basis for bulk mass flux convective parameterization, Atmos. Chem. Phys. 10, pp. 3529–3544. Plant, R. S. (2012). A new modelling framework for statistical cumulus dynamics, Phil. Trans. Roy. Soc. A 370, pp. 1041–1060. Plant, R. S. and Craig, G. C. (2008). A stochastic parameterization for deep convection based on equilibrium statistics, J. Atmos. Sci. 65, pp. 87–105. ˇ and Yano, J.-I. (2014). Basic theory behind parameterPlant, R. S., Fuchs, Z. izing atmospheric convection. COST Action ES0905 final training school: Convection parameterization from theory to operations Brac, Croatia, 29 September – 9 October 2014, Eos 95, p. 137. Plant, R. S. and Yano, J.-I. (2013). The energy-cycle analysis of the interactions between shallow and deep atmospheric convection, Dyn. Atmos. Ocean. 64, pp. 27–52. Pop, S. B. (2000). Turbulant Flows (Cambridge University Press, Cambridge). Posselt, D. J., Heever, S. V. D., Stephens, G. and Igel, M. R. (2012). Changes in the interaction between tropical convection, radiation, and the large-scale circulation in a warming environment, J. Climate 25, pp. 557–571. Raga, G. B., Jensen, J. B. and Baker, M. B. (1990). Characteristics of cumulus band clouds off the coast of Hawaii, J. Atmos. Sci. 47, pp. 338–355. Randall, D. A. and Huffman, G. J. (1982). Entrainment and detrainment in a simple cumulus cloud model, J. Atmos. Sci. 39, pp. 2793–2806. Randall, D. A., Khairoutdinov, M., Arakawa, A. and Grabowski, W. (2003).

Bibliography

501

Breaking the cloud parameterization deadlock, Bull. Amer. Meteor. Soc. 84, pp. 1547–1564. Randall, D. A. and Pan, D.-M. (1993). Implementation of the Arakawa–Schubert cumulus parameterization with a prognostic closure, in K. A. Emanuel and D. J. Raymond (eds.), The Representation of Cumulus Convection in Numerical Models, Meteorological Monographs, Vol. 24, chap. 11 (American Meteorological Society, Boston), pp. 137–144. Raymond, D. J. (1995). Regulation of moist convection over the warm tropical oceans, J. Atmos. Sci. 52, pp. 3945–3959. Raymond, D. J. and Blyth, A. M. (1986). A stochastic mixing model for nonprecipitating cumulus clouds, J. Atmos. Sci. 43, pp. 2708–2718. ˇ (2007). Convectively coupled gravity and moisture Raymond, D. J. and Fuchs, Z. modes in a simple atmospheric model, Tellus A 59, pp. 627–640. ˇ (2009). Moisture modes and the Madden–Julian Raymond, D. J. and Fuchs, Z. oscillation, J. Climate 22, pp. 3031–3046. Raymond, D. J. and Herman, M. J. (2011). Convective quasi-equilibrium reconsidered, J. Adv. Model. Earth Syst. 3, doi:10.1029/2011MS000079. ˇ (2007). A theory for the spinup of Raymond, D. J., Sessions, S. L. and Fuchs, Z. tropical depressions, Quart. J. R. Meteor. Soc. 133, pp. 1743–1754. ˇ (2009). The meRaymond, D. J., Sessions, S. L., Sobel, A. H. and Fuchs, Z. chanics of gross moist stability, J. Adv. Model. Earth Syst. 1, 9, doi: 10.3894/JAMES.2009.1.9. Redelsperger, J. L., Parsons, D. B. and Guichard, F. (2002). Recovery processes and factors limiting cloud-top height following the arrival of a dry intrusion observed during TOGA COARE, J. Atmos. Sci. 59, pp. 2438–2457. Reuter, G. W. (1986). A historical review of cumulus entrainment studies, Bull. Amer. Meteor. Soc. 67, pp. 151–154. Reynolds, W. C. (1990). The potential and limitations of direct and large eddy simulations, in J. L. Lumley (ed.), Whither Turbulence? Turbulence at the Crossroads (Springer-Verlag, New York), pp. 313–343. Richard, J. L. and Royer, J. F. (1993). A statistical cloud scheme for use in an AGCM, Ann. Geophys. 11, pp. 1095–1115. Riehl, H. (1954). Tropical Meteorology (McGraw–Hill, New York). Riehl, H. and Malkus, J. S. (1958). On the heat balance in the equatorial trough zone, Geophysica (Helsinki) 6, pp. 503–538. Riehl, H. and Malkus, J. S. (1979). The heat balance of the equatorial trough zone, revisited, Contrib. Atmos. Phys. 52, pp. 287–305. Ring, D. (2009). Non-linear wave interactions in rotating stratified fluid flow, Ph.D. thesis, The Johns Hopkins University. Rodts, S. M. A., Duynkerke, P. G. and Jonker, H. J. J. (2003). Size distributions and dynamical properties of shallow cumulus clouds from aircraft observations and satellite data, J. Atmos. Sci. 60, pp. 1895–1912. Roff, G. L. and Yano, J.-I. (2002). Tropical convective variability in the CAPE phase space, Quart. J. R. Meteor. Soc. 128, pp. 2317–2333. Romps, D. M. (2010). A direct measurement of entrainment, J. Atmos. Sci. 67, pp. 1908–1927.

502

Bibliography

Romps, D. M. and Kuang, Z. (2010a). Do undiluted convective plumes exist in the upper tropical troposphere? J. Atmos. Sci. 67, pp. 468–484. Romps, D. M. and Kuang, Z. (2010b). Nature versus nurture in shallow convection, J. Atmos. Sci. 67, pp. 1655–1666. Romps, D. M. and Kuang, Z. (2011). A transilient matrix for moist convection, J. Atmos. Sci. 68, pp. 2009–2025. Rossby, G.-G. (1938). On the mutual adjustment of pressure and velocity distribution in certain simple current systems. II, J. Mar. Res. 1, pp. 239–263. Sahany, S., Neelin, J. D., Hales, K. and Neale, R. B. (2012). Temperature– moisture dependence of the deep convective transition as a constraint on entrainment in climate models, J. Atmos. Sci. 69, pp. 1340–1358. Saito, M. and Asanuma, J. (2008). Eddy covariance calculation revisited with wavelet cospectra, SOLA 4, pp. 49–52. Salmon, R. (1998). Lectures on Geophysical Fluid Dynamics (Oxford University Press, Oxford). Schneider, E. K. and Lindzen, R. S. (1976). A discussion of the parameterization of momentum exchange by cumulus convection, J. Geophys. Rev. 81, pp. 3158–3160. Schubert, W. H. (2000). A retrospective view of Arakawa’s ideas on cumulus parameterization, in D. Randall (ed.), General Circulation Model Development, chap. 6 (Elsevier, New York), pp. 181–198. Schubert, W. H., Wakefield, J. S., Steiner, E. J. and Cox, S. K. (1979). Marine stratocumulus convection. Part I: Governing equations and horizontally homogeneous solutions, J. Atmos. Sci. 36, pp. 1286–1307. Sessions, S. L., Sugaya, S., Raymond, D. J. and Sobel, A. H. (2010). Multiple equilibria in a cloud-resolving model using the weak temperature gradient approximation, J. Geophys. Res. 115, D12110. Shapiro, M. A., Hampel, T., Rotzoll, D. and Mosher, F. (1985). The frontal hydraulic head: A meso-α scale (∼ 1 km) triggering mechanism for mesoconvective weather systems, Mon. Wea. Rev. 113, pp. 1166–1183. Shutts, G. J. and Gray, M. E. B. (1999). Numerical simulations of convective equilibrium under prescribed forcing, Quart. J. R. Meteor. Soc. 125, pp. 2767–2787. Shutts, G. J. and Palmer, T. N. (2007). Convective forcing fluctuations in a cloudresolving model: Relevance to the stochastic parameterization problem, J. Climate 20, pp. 187–202. Siebesma, A. P. (1998). Shallow cumulus convection, in E. J. Plate, E. E. Fedorovich, X. V. Viegas and J. C. Wyngaard (eds.), Buoyant Convection in Geophysical Flows, Vol. 513 (Kluwer Academic Publishers, Amsterdam), pp. 441–486. Siebesma, A. P., Bretherton, C. S., Brown, A., Chlond, A., Cuxart, J., Duynkerke, P. G., Jiang, H., Khairoutdinov, M., Lewellen, D., Moeng, C.-H., Sanchez, E., Stevens, B. and Stevens, D. E. (2003). A large eddy simulation intercomparison study of shallow cumulus convection, J. Atmos. Sci. 60, pp. 1201–1219. Siebesma, A. P. and Cuijpers, J. W. M. (1995). Evaluation of parametric assump-

Bibliography

503

tions for shallow cumulus convection, J. Atmos. Sci. 52, pp. 650–666. Siebesma, A. P. and Holtslag, A. A. M. (1996). Model impacts of entrainment and detrainment rates in shallow cumulus convection, J. Atmos. Sci. 53, pp. 2354–2364. Siebesma, A. P. and Jonker, H. J. J. (2000). Anomalous scaling of cumulus cloud boundaries, Phys. Rev. Lett. 85, pp. 214–217. Simpson, J., Simpson, R. H., Andrews, D. A. and Eaton, M. E. (1965). Experimental cumulus dynamics, Rev. Geophys. 3, pp. 387–431. Simpson, J. and Wiggert, V. (1969). Models of precipitating cumulus towers, Mon. Wea. Rev. 97, pp. 471–489. Sivashinsky, G. I. (1985). Weak turbulence in periodic flows, Physica D 17, pp. 243–255. Soares, P. M. M., Miranda, P. M. A., Siebesma, A. P. and Teixeira, J. (2004). An eddy-diffusivity/mass-flux parameterization for dry and shallow cumulus convection, Quart. J. R. Meteor. Soc. 130, pp. 3365–3384. Sobel, A. H. and Maloney, E. D. (2013). Moisture modes and the eastward propagation of the MJO, J. Atmos. Sci. 70, pp. 187–192. Sobel, A. H., Nilsson, J. and Polvani, L. M. (2001). The weak temperature gradient approximation and balanced tropical moisture waves, J. Atmos. Sci. 58, pp. 3650–3665. Sol´e, R. V. and Bascompte, J. (2006). Self-Organization in Complex Ecosystems (Princeton University Press, Princeton). Sommeria, G. and Deardorff, J. W. (1977). Subgrid-scale condensation in models of nonprecipitating clouds, J. Atmos. Sci. 34, pp. 344–355. Squires, P. (1958). The microstructure and colloidal stability of warm clouds. Part I. The relation between structure and stability, Tellus 10, pp. 256–261. Squires, P. and Turner, J. S. (1962). An entraining jet model for cumulo-nimbus updraughts, Tellus 16, pp. 422–434. Stanley, H. E. (1972). Introduction to Phase Transition and Critical Phenomena (Oxford University Press, Oxford). Starr, V. P. (1968). Physics of negative viscosity phenomena (McGraw–Hill, New York). Stensrud, D. J. (2007). Parameterization schemes: Keys to understanding numerical weather prediction models (Cambridge University Press, Cambridge). Stephens, G. L., van den Heever, S. and Pakula, L. (2008). Radiative-convective feedbacks in idealized states of radiative-convective equilibrium, J. Atmos. Sci. 65, pp. 3899–3916. Stevens, B., Ackerman, A. S., Albrecht, B. A., Brown, A. R., Chlond, A., Cuxart, J., Duynkerke, P. G., Lewellen, D. C., Macvean, M. K., Neggers, R. A. J., Sanchez, E., Siebesma, A. P. and Stevens, D. E. (2001). Simulations of trade-wind cumuli under a strong inversion, J. Atmos. Sci. 58, pp. 1870– 1891. Stirling, A. and Stratton, R. A. (2012). Entrainment processes in the diurnal cycle of deep convection over land, Quart. J. R. Meteor. Soc. 138, pp. 1135–1149. Stommel, H. (1947). Entrainment of air into a cumulus cloud, J. Meteor. 4, pp. 91–94.

504

Bibliography

Straub, K. H. and Kiladis, G. N. (2002). Observations of a convectively coupled Kelvin wave in the eastern Pacific ITCZ, J. Atmos. Sci. 59, pp. 30–53. Stull, R. B. (1988). An introduction to boundary layer meteorology (Kluwer Academic Publishers, London and High Wycombe). Sud, Y. C., Chao, W. C. and Walker, G. K. (1991). Contributions to the implementation of the Arakawa–Schubert cumulus parameterization in the GLA GCM, J. Atmos. Sci. 48, pp. 1573–1586. Sud, Y. C. and Walker, G. K. (1999). Microphysics of clouds with relaxed Arakawa–Schubert scheme (McRAS). Part I: Design and evaluation with GATE Phase III data, J. Atmos. Sci. 56, pp. 3196–3220. Sugiyama, M. (2009a). The moisture mode in the quasi-equilibrium tropical circulation model. Part I: Analysis based on the weak temperature gradient approximation, J. Atmos. Sci. 66, pp. 1507–1523. Sugiyama, M. (2009b). The moisture mode in the quasi-equilibrium tropical circulation model. Part II: Nonlinear behavior on an equatorial beta plane, J. Atmos. Sci. 66, pp. 1525–1542. Sui, C. H., Lau, K. M., Tao, W. K. and Simpson, J. (1994). The tropical water and energy cycles in a cumulus ensemble model. Part I: Equilibrium climate, J. Atmos. Sci. 51, pp. 711–728. Sui, C.-H. and Yanai, M. (1986). Cumulus ensemble effects on the large-scale vorticity and momentum fields of GATE. Part I: Observational evidence, J. Atmos. Sci. 43, pp. 1618–1642. Su˘selj, K., Teixeira, J. and Chung, D. (2013). A unified model for moist convective boundary layers based on a stochastic eddy-diffusivity/mass-flux parameterization, J. Atmos. Sci. 70, pp. 1929–1953. Swann, H. (2001). Evaluation of the mass-flux approach to parametrizing deep convection, Quart. J. R. Meteor. Soc. 127, pp. 1239–1260. Tatsumi, T. (1980). Theory of homogeneous turbulence, Adv. Appl. Mech. 17, pp. 39–133. Taylor, G. R. and Baker, M. B. (1991). Entrainment and detrainment in cumulus clouds, J. Atmos. Sci. 48, pp. 112–121. Telford, J. W. (1966). The convective mechanism in clear air, J. Atmos. Sci. 23, pp. 652–666. Telford, J. W. (1968). Reply, J. Atmos. Sci. 25, pp. 138–139. Thompson Jr., R. M., Payne, S. W., Recker, E. E. and Reed, R. J. (1979). Structure and properties of synoptic-scale wave disturbances in the intertropical convergence zone of eastern Atlantic, J. Atmos. Sci. 36, pp. 53–72. Tiedtke, M. (1989). A comprehensive mass flux scheme for cumulus parameterization in large-scale models, Mon. Wea. Rev 177, pp. 1779–1800. Tiedtke, M. (1993). Representation of clouds in large-scale models, Mon. Wea. Rev. 121, pp. 3040–3061. Tokioka, T., Yamazaki, K., Kitoh, A. and Ose, T. (1988). The equatorial 30–60 day oscillation and the Arakawa–Schubert penetrative cumulus parameterization, J. Meteor. Soc. Japan 66, pp. 883–900. Tompkins, A. M. (2002). A prognostic parameterization for the subgrid-scale variability of water vapor and clouds in large-scale models and its use to

Bibliography

505

diagnose cloud cover, J. Atmos. Sci. 59, pp. 1917–1942. Treut, H. L. and Li, Z. X. (1991). Sensitivity of an atmospheric general circulation model to prescribed SST changes: Feedback effects associated with the simulation of cloud optical properties, Climate Dyn. 5, pp. 175–187. Tuck, A. (2008). Atmospheric Turbulence: A Molecular Dynamics Perspective (Oxford University Press, Oxford). Tulich, S. N., Randall, D. A. and Mapes, B. E. (2007). Vertical-mode and cloud decomposition of large-scale convectively coupled gravity waves in a twodimensional cloud-resolving model, J. Atmos. Sci. 64, pp. 1210–1229. Turner, J. S. (1963). The motion of buoyant elements in turbulent surroundings, J. Fluid Mech. 16, pp. 1–16. Van der Hoven, I. (1956). Power spectrum of horizontal wind speed in the frequency range from 0.0007 to 900 cycles per hour, J. Meteor. 14, pp. 160– 164. van Kampen, N. G. (1985). Elimination of fast variables, Phys. Rep. 124, pp. 69–160. van Kampen, N. G. (2007). Stochastic Processes in Physics and Chenistry, 3rd edn. (Elsevier, New York). Verzijlbergh, R. A., Heus, T., Jonker, H. J. J. and de Arellano, J. V.-G. (2009). Turbulent dispersion in cloudy boundary layers, Atmos. Chem. Phys. 9, pp. 1289–1302. Vinnichenko, N. K. (1970). The kinetic energy spectrum in the free atmosphere – 1 second to 5 years, Tellus 22, pp. 2–166. Wang, Y., Geerts, B. and French, J. (2009). Dynamics of the cumulus cloud margin: An observational study, J. Atmos. Sci. 66, pp. 3660–3677. Warner, J. (1955). The water content of cumuliform clouds, Tellus 7, pp. 449–457. Webster, P. and Lukas, R. (1992). TOGA COARE: The Coupled Ocean– Atmosphere Response Experiment, Bull. Amer. Meteor. Soc. 73, pp. 1377– 1416. Wedi, N. P. and Smolarkiewicz, P. K. (2010). A nonlinear perspective on the dynamics of the MJO: Idealized large-eddy simulations, J. Atmos. Sci. 67, pp. 1202–1217. Wei, D., Blyth, A. M. and Raymond, D. J. (1998). Buoyancy of convective clouds in TOGA COARE, J. Atmos. Sci. 55, pp. 3381–3391. Weisman, M. L. and Rotunno, R. (2004). “A theory for strong long-lived squall Lines” Revisited, J. Atmos. Sci. 61, pp. 361–382. Wheeler, M. and Kiladis, G. N. (1999). Convectively coupled equatorial waves: Analysis of clouds and temperature in the wavenumber-frequency domain, J. Atmos. Sci. 56, pp. 374–399. Wheeler, M., Kiladis, G. N. and Webster, P. (2000). Large-scale dynamical fields associated with convectively coupled equatorial waves, J. Atmos. Sci. 57, pp. 613–640. Wikle, C. K. and Cressie, N. (1999). A dimension-reduced approach to space-time Kalman filtering, Biometrika 86, pp. 815–829. Willett, M. R. and Milton, S. F. (2006). The tropical behaviour of the convective parameterization in aquaplanet simulations and the sensitivity to timestep,

506

Bibliography

Forecasting Research Technical Report 482 (Met Office, UK). Wilson, J. W. and Schreiber, W. E. (1986). Initiation of convective storms at radar-observed boundary-layer convergence lines, Mon. Wea. Rev. 114, pp. 2516–2536. Wilson, K. G. (1983). The renormalization group and critical phenomena, Rev. Mod. Phys. 55, pp. 583–600. Woodward, B. (1959). The motion in and around isolated thermals, Quart. J. R. Meteor. Soc. 85, pp. 144–151. Wu, X. and Yanai, M. (1994). Effects of vertical wind shear on the cumulus transport of momentum: Observations and parameterization, J. Atmos. Sci. 51, pp. 1640–1660. Wyngaard, J. C. (2010). Turbulence in the Atmosphere, Vol. 774 (Cambridge University Press, Cambridge). Xu, K.-M. and Emanuel, K. A. (1989). Is the tropical atmosphere conditionally unstable? Mon. Wea. Rev. 117, pp. 1471–1479. Yanai, M., Chu, J.-H., Starx, T. E. and Nitta, T. (1976). Response of deep and shallow tropical maritime cumuli to large-scale processes, J. Atmos. Sci. 33, pp. 976–991. Yanai, M., Esbensen, S. and Chu, J.-H. (1973). Determination of bulk properties of tropical cloud clusters from large-scale heat and moisture budgets, J. Atmos. Sci. 30, pp. 611–627. Yanai, M. and Johnson, R. H. (1993). Impacts of cumulus convection on thermodynamic fields, in K. A. Emanuel and D. J. Raymond (eds.), The Representation of Cumulus Convection in Numerical Models, Meteorological Monographs, Vol. 24, chap. 4 (American Meteorological Society, Boston), pp. 39–62. Yano, J.-I. (1992). Asymptotic theory of thermal convection in the rapidly rotating systems, J. Fluid Mech. 243, pp. 103–131. Yano, J.-I. (1998). Planetary-scale coherent structures of tropical moist convection, Aust. J. Phys. 51, pp. 865–874. Yano, J.-I. (1999). Scale-separation and quasi-equilibrium principles in Arakawa and Schubert’s cumulus parameterization, J. Atmos. Sci. 56, pp. 3821– 3823. Yano, J.-I. (2001). Residual cumulus parametrization, Quart. J. R. Meteor. Soc. 127, pp. 1261–1276. Yano, J.-I. (2003a). Comments on “Remarks on quasi–equilibrium theory”, J. Atmos. Sci. 60, pp. 2342–2343. Yano, J.-I. (2003b). The cumulus parameterization problem in the context of MJO simulations, in Proceedings for the MJO workshop, ECMWF, pp. 115–127, 2–5 November 2003, Reading, UK. Yano, J.-I. (2009). Deep-convective vertical transport: What is mass flux? Atmos. Chem. Phys. Discuss. 9, pp. 3535–3553. Yano, J.-I. (2010). Downscaling, parameterization, decomposition, compression: A perspective from the multiresolutional analysis, Adv. Geophys. 23, pp. 65–71. Yano, J.-I. (2011). Interactive comment on “Simulating deep convection with a

Bibliography

507

shallow convection scheme” by C. Hohenegger and C. S. Bretherton, Atmos. Chem. Phys. Discuss. 11, pp. C2411–C2425. Yano, J.-I. (2012). Mass-flux subgrid-scale parameterization in analogy with multi-component flows: A formulation towards scale independence, Geosci. Model Dev. 5, pp. 1425–1440. Yano, J.-I. (2014a). Basic convective element: Bubble or plume? A historical review, Atmos. Chem. Phys. 14, pp. 7019–7030. Yano, J.-I. (2014b). Formulation structure of mass-flux convection parameterization, Dyn. Atmos. Ocean. 67, pp. 1–28. Yano, J.-I. and Baizig, H. (2012). Single SCA-plume dynamics, Dyn. Atmos. Ocean. 58, pp. 62–94. Yano, J.-I. and Bechtold, P. (2009). Toward physical understanding of cloud entrainment-detrainment processes, Eos 90, p. 258. Yano, J.-I., Bechtold, P., Redelsperger, J.-L. and Guichard, F. (2004). Waveletcompressed representation of deep moist convection, Mon. Wea. Rev. 132, pp. 1472–1486. Yano, J.-I., Benard, P., Couvreux, F. and Lahellec, A. (2010a). NAM-SCA: Nonhydrostatic anelastic model under segmentally-constant approximation, Mon. Wea. Rev. 138, pp. 1957–1974. ˇ Gerard, L., Phillips, V. T. J., Barkidija, S. Yano, J.-I., Bister, M., Fuchs, Z.., and Piriou, J.-M. (2013a). Phenomenology of convection-parameterization closure, Atmos. Chem. Phys. 13, pp. 4111–4131. Yano, J.-I., Blender, R., Zhang, C. and Fraedrich, K. (2004). 1/f noise and pulselike events in the tropical atmospheric surface variabilities, Quart. J. R. Meteor. Soc. 130, pp. 1697–1721. Yano, J.-I. and Bonazzola, M. (2009). Scale analysis for large-scale tropical atmospheric dynamics, J. Atmos. Sci. 66, pp. 159–172. Yano, J.-I. and Bouniol, D. (2010). A minimum bulk microphysics, Atmos. Chem. Phys. Discuss. 10, pp. 30305–30345. Yano, J.-I., Chaboureau, J.-P. and Guichard, F. (2005a). A generalization of CAPE into potential-energy convertibility, Quart. J. R. Meteor. Soc. 131, pp. 861–875. Yano, J.-I. and Emanuel, K. A. (1991). An improved model of the equatorial troposphere and its coupling with the stratosphere, J. Atmos. Sci. 48, pp. 377–389. Yano, J.-I., Fraedrich, K. and Blender, R. (2001). Tropical convective variability as 1/f noise, J. Climate 14, pp. 3608–3616. Yano, J.-I., Geleyn, J.-F., K¨ ohler, M., Mironov, D., Quaas, J., Soares, P. M. M., Phillips, V. T. J., Plant, R. S., Deluca, A., Marquet, P., Stulic, L. and Fuchs, Z. (2015). Basic concepts for convection parameterization in weather forecast and climate models: Cost action es0905 final report, Atmosphere 6, pp. 88–147. Yano, J.-I., Geleyn, J.-F. and Malinowski, S. (2010b). Challenges for a new generation of regional forecast models: Workshop on concepts for convective parameterizations in large-scale models III: Increasing resolution and parameterization; Warsaw, Poland, 17-19 March 2010, Eos 91, p. 232.

508

Bibliography

Yano, J.-I., Grabowski, W. W., Roff, G. L. and Mapes, B. E. (2000). Asymptotic approaches to convective quasi-equilibrium, Quart. J. R. Meteor. Soc. 126, pp. 1861–1887. Yano, J.-I., Graf, H.-F. and Spineanu, F. (2012a). Meeting report: Theoretical and operational implications of atmospheric convective organization, Bull. Amer. Meteor. Soc. 93, pp. ES39–ES41. Yano, J.-I., Guichard, F., Lafore, J.-P., Redelsperger, J.-L. and Bechtold, P. (2004). Estimations of mass fluxes for cumulus parameterizations from highresolution spatial data, J. Atmos. Sci. 61, pp. 829–842. Yano, J.-I., Kumar, S. and Roff, G. L. (2012b). Towards compressed superparameterization: Test of NAM-SCA under single-column GCM configurations, Atmos. Chem. Phys. Discuss. 12, pp. 28237–28303. Yano, J.-I., Machulskaya, E., Bechtold, P. and Plant, R. S. (2013b). Bells and whistles of convection parameterization: Workshop report, Bull. Amer. Meteor. Soc. 94, pp. ES5–ES7. Yano, J.-I., Moncrieff, M. W. and McWilliams, J. C. (1998). Linear stability and single-column analyses of several cumulus parametrization categories in a shallow-water model, Quart. J. R. Meteor. Soc. 124, pp. 983–1005. Yano, J.-I., Mulet, S. and Bonazzola, M. (2009). Tropical large-scale circulations: Asymptotically non-divergent? Tellus A 61, pp. 417–427. Yano, J.-I. and Plant, R. S. (2012a). Convective quasi-equilibrium, Rev. Geophys. 50, doi:10.1029/2011RG000378. Yano, J.-I. and Plant, R. S. (2012b). Finite departure from convective quasiequilibrium: Periodic cycle and discharge-recharge mechanism, Quart. J. R. Meteor. Soc. 138, pp. 626–637. Yano, J. I. and Plant, R. S. (2012c). Interactions between shallow and deep convection under a finite departure from convective quasi-equilibrium, J. Atmos. Sci. 69, pp. 3463–3470. Yano, J.-I., Quass, J., Wagner, T. M. and Plant, R. S. (2008). Toward statistical descriptions of convective cloud dynamics: Workshop on concepts for convective parameterizations in large-scale models, Eos 88, p. 212. Yano, J.-I., Redelsperger, J.-L., Bechtold, P. and Guichard, F. (2005b). Mode decomposition as a methodology for developing convective-scale representations in global models, Quart. J. R. Meteor. Soc. 131, pp. 2313–2336. Yano, J.-I., Soares, P., K¨ ohler, M. and Deluca, A. (2014a). Convection parameterization problem: Breadth and depth, Bull. Amer. Meteor. Soc. (in press). Yano, J.-I. and Takeuchi, Y. (1987). The self-similarity of horizontal cloud pattern in the intertropical convergence zone, J. Meteor. Soc. Japan 65, pp. 661– 667. Yano, J.-I., Vlad, M., Derbyshire, S. H., Geleyn, J.-F. and Kober, K. (2014b). Generalization, consistency, and unification in the parameterization problem, Bull. Amer. Meteor. Soc. 95, pp. 619–622. Yeomans, J. (1992). Statistical Mechanics of Phase Transitions (Oxford University Press, Oxford). Yu, J.-Y. and Neelin, J. D. (1994). Modes of tropical variability under convective adjustment and the Madden–Julian Oscillation. Part II: Numerical results,

Bibliography

509

J. Atmos. Sci. 51, pp. 1895–1914. Yuter, S. E. and Houze Jr., R. A. (2000). The 1997 Pan American climate studies tropical eastern Pacific process study. Part I: ITCZ region, Bull. Amer. Meteor. Soc. 81, pp. 451–481. ˇ Zagar, N., Tribbia, J., Anderson, J. L. and Raeder, K. (2009). Uncertainties of estimates of inertia-gravity energy in the atmosphere. Part I: Intercomparison of four analysis systems, Mon. Wea. Rev. 137, pp. 3837–3857. Zhang, C. (2005). Madden–Julian oscillation, Rev. Geophys. 43, pp. 1–36. Zhang, D.-L., Chang, H.-R., Seaman, N. L., Warner, T. T. and Fritsch, J. M. (1986). A two-way interactive nesting procedure with variable train resolution, Mon. Wea. Rev. 114, pp. 1330–1339. Zhang, G. J. (2002). Convective quasi-equilibrium in midlatitude continental environment and its effect on convective parameterization, J. Geophys. Res. 107, doi:10.1029/2001JD001005. Zhang, G. J. (2003). Convective quasi-equilibrium in the tropical western Pacific: Comparison with midlatitude continental environment, J. Geophys. Res. 108, doi:10.1029/2003JD003520. Zhang, G. J. (2009). Effects of entrainment on convective available potential energy and closure assumptions in convection parameterization, J. Geophys. Res. 114, doi:10.1029/2008JD010976. Zhang, G. J. and Cho, H.-R. (1991). Parameterization of the vertical transport of momentum by cumulus clouds. Part I: Theory, J. Atmos. Sci. 48, pp. 1483–1492. Zhang, G. J. and McFarlane, N. A. (1995). Sensitivity of climate simulations of the parameterization of cumulus convection in the Canadian Climate Centre general circulation model, Atmos. Ocean 33, pp. 407–446. Zhang, J., Lohmann, U. and Stier, P. (2005). A microphysical parameterization for convective clouds in the ECHAM5 climate model: Single-column model results evaluated at the Oklahoma Atmospheric Radiation Measurement program site, J. Geophys. Res. 110, doi:10.1029/2004JD005128. Zhao, M. and Austin, P. H. (2005a). Life cycle of numerically simulated shallow cumulus clouds. Part I: Transport, J. Atmos. Sci. 62, pp. 1269–1290. Zhao, M. and Austin, P. H. (2005b). Life cycle of numerically simulated shallow cumulus clouds. Part II: Mixing dynamics, J. Atmos. Sci. 62, pp. 1291– 1310. Zimmer, M., Graig, G. C., Wernli, H. and Keil, C. (2011). Classification of precipitation events with a convective response timescale, Geophys. Res. Lett. 38, L05802. Zipser, E. J. (1969). The role of unsaturated convective downdrafts in the structure and rapid decay of an equatorial disturbance, J. Appl. Meteor. 8, pp. 799–814. Zipser, E. J. (1977). Mesoscale and convective-scale downdrafts as distinct components of squall-line circulation, Mon. Wea. Rev. 105, pp. 1568–1589.

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Index

Entries in bold indicate the definition of a term. 1/f -noise, 77

Bernoulli function, 16, 409, 460, 466–468 Bessel function, 461, 479 BOMEX (Barbados Oceanographic and Meteorological EXperiment), 281, 294 boundary-layer quasi-equilibrium, 382–383, 386–388, 390 boundary-layer turbulence, 55, 69, 202 Brunt–V¨ ais¨ al¨ a frequency, 19 bubble, 12–14, 20, 224, 274, 404, 407, 409, 416, see also thermal bulk plume, 253, 261 buoyancy, 7, 10–11, 233, 405, 416 buoyancy sorting, 278, 288, 292, 295–298

activation control, 113, 135–136, 142, 390–392 & CQE, 139–140 adaptive detrainment, 303 adjustment process, 43, 46, 127, see also relaxation process adjustment scheme, 44, 64, 127–129 alchemy, 70 analytic limit, 91 anelastic approximation, 9, 80, 197, 224 apparent source, 36, 36–42, 56, 82, 86, 88, 90, 190, 230–231, 243 archetype, 463–477 closure, 474 numerically generated, 474–477 assumed PDF/DDF, 58–59 asymptotic analysis, 74, 114 expansion, 37, 40–44, 44–48, 79, 80, 83–88, 90–91, 98, 99, 196 limit, 91–94, 96, 186, 210–219, 450 ATEX (Atlantic Trade wind EXperiment), 281

CAPE, see convectively available potential energy Clausius–Clapeyron equation, 17, 18 closure, 50, 56, 103, 192, 216, Ch. 11, 325 boundary-layer quasi-equilibrium, 386–388 CAPE based, 331, 379–381, 392, 394 in mesoscale archetype, 474 moisture based, 352–354 of downdraughts, 432

balance, 102, 115–118 Bayesian, 56, 66 511

512

of mesoscale downdraught, 439 parcel-environment-based, 381–386 PBL-based, 392 quasi-equilibrium, 355–363 cloud model, 192, 232, 239, 275, 283, 326–328, 395 cloud-radiation interaction (CRI), 161 cloud-resolving model (CRM), 39, 86, 197–199 cloud-top entrainment, 279 cloud-work function, 104, 113, 128, 141, 355, 349–380 cold pool, 307 compression, 60 conditional instability of the second kind (CISK), 63, 153 convective adjustment timescale, 95 convective available potential energy (CAPE), 23–24, 104, 113–114, 128, 135–137, 141, 153, 351, 393 convective core, 417 convective energy cycle, 369–379 convective inhibition (CIN), 16, 113, 136, 159, 163, 390 convective neutrality, 112 convective quasi-equilibrium, 95, 101, 103–104, 187, 334, 355–363 convective response, 103 convective tower, 199, 463 convective vertical velocity, 219, Ch. 12, 403 convective-scale circulation, 456–459 convective-scale collective balance, 212–213, 391 convective-scale horizontal velocity, 452 convective-scale pressure, 454–456 convectively coupled equatorial waves (CCEW), 150–154 convectively coupled Kelvin waves (CCKW), 154–167 CQE, see convective quasi-equilibrium critical mixing fraction, 295, 302 cumulonimbus, 21 cumulonimbus arcus, 22

Index

cumulonimbus calvus, 22 cumulonimbus capillatus, 22 cumulonimbus mammatus, 22 cumulonimbus pileus, 22 cumulus, 14 cumulus condensation level, see lifted condensation level (LCL) cumulus congestus, 20 cumulus friction, 454 cumulus humilis, 14, 16 detrainment, 189, Ch. 10, 273 fractional, 189 detrainment level, 261, 433 dimensional analysis, 404–405 distribution density function (DDF), 56 double-counting, 73 downdraughts, 199, Ch. 13, 419 discoveries, 421 entrainment and detrainment in, 432 importance, 428 mass-flux formulation, 429 saturated and unsaturated, 425 downscaling, 38 drag force, 410, 414 dry adiabatic, 7, 14 dry static energy, 228 dynamical detrainment, 276 dynamical entrainment, 275, 318 eddy diffusion, 38–39, 47–48, 64, 202–203, 441 eddy-damping quasi-normal approximation (EDQN), 51 eddy-diffusion and mass-flux approach (EDMF), 202–203 ensemble, 49 ensemble average, 55, 98 entraining plume, 231, 235, 239–243, 275–283, 316–317, 405–407 entrainment, 24, 189, Ch. 10, 273 fractional, 189, 275 entrainment and detrainment, 205–210, 232, 239, Ch. 10, 273,

513

Index

275, 283 differences between deep and shallow convection, 305 in downdraughts, 432 in parameterization, 295–313 LES diagnosis, 283–294, 320–322 without environment hypothesis, 220–222 entrainment by organized flow, 318 entrainment dilemma, 323 entrainment–detrainment hypothesis, 209–210, 216, 415 entropy, 178 environment, 7, 183, 199, 405 environment hypothesis, 220, 278 epicycle, 70 equatorial waves, 151–152 equilibrium, 102–108 equilibrium control, 112–113, 114, 135 equilibrium level (EL), 21 equivalent potential temperature, 18, 19 evaporative cooling, 435, 440–441

grid-box size, 90 gross moist stability (GMS), 161 Hadley cell, see Hadley circulation Hadley circulation, 148, 176 heat equation, 45 Hill’s spherical vortex, 407–409 homeostasis, 44, 108 hot tower, 199, 404 hot-tower hypothesis, 64, Ch. 6, 175, 178–182, 195 hydrostatic balance, 450, 466 immediate environment, 278 IPCC, 69 kinetic energy spectrum, 49

fast process, 40, 212 filtering, 37, 87–89, 91, 99 finite volume, 37, 83, 203 forced convection, 7 forced detrainment, 319 fractality, 78–80 fractional area occupied by convection, 96–97, 182–183, 186, 201, 203, 210–219, 221, 451 fractional entrainment rate, 214, 238 free ride principle, 121–122, 140, 145, 166, 248, 398, 399 frozen moist static energy, 305

large scale, 40, 74, 90, 180 large-eddy simulation (LES), 203, 281–294 large-scale average, 84 large-scale forcing, 103, 330, 340, 368, 397 law of large numbers, 124–125 level of free convection (LFC), 14, 21 level of free sinking (LFS), 432 level of neutral buoyancy (LNB), 233, 254–257, 261, 270 level of the maximum cloud top (LMCT), 22 life cycle, 11–25, 43, 94, 213 lifted condensation level (LCL), 14, 426, 427 Lighthill’s theorem, 119–121 linear stability analysis, 168 Liouville equation, 57–58 local mixing, 64

GATE (GARP Atlantic Tropical Experiment), 39, 267, 419, 436, 439 Gaussian distribution, 51, 53, 58 gravity mode, 154, 166 grid box, 37, 83 grid-box average, 36 grid-box averaging, 81

Madden–Julian oscillation (MJO), 63, 69, 150, 323, 429 Markovian random coupling model, 53 Markovization, 52 mass flux, 182–192 mass-flux approximation, 202

514

mass-flux convection parameterization, 55, 61, 63, 182–192, see Part II bulk, 224, 277 generalization, 219–223 spectrum and bulk, 199 hybrid approaches, 267–268 mass-flux formulation, see mass-flux convection parameterization mass-flux parameterization, see mass-flux convection parameterization mass-flux spectrum, 232 mechanics, 115–122 mesoscale convective system (MCS), 463 mesoscale downdraught, 436 mesoscale gap, 76 microphysics, 36, 59, 218, 422, 435 bin, 57 mixed-layer model, 65 mixing detrainment, 254, 265, 319 mode decomposition, 60–63 moist static energy, 179, 228 moisture mode, 154, 162, 164 moment, 55 moment expansion, 49 moment-based parameterization, 55 momentum transfer, Ch. 14, 449 momentum transport mesoscale convective, 463–477 multi-scale asymptotic expansion, see asymptotic expansion NAM-SCA, 204, 474–477 nesting, 39 non-hydrostatic anelastic model (NAM), 197 non-local mixing, 64 numerical algorithm, 83, 89 Occam’s razor, 65–66 organization parameter, 322 organized detrainment, 276, 320 organized entrainment, 399

Index

parcel, 7 parcel method, 24, 417 PCAPE (pressure-integrated convective available potential energy), 379 plume, 199, 274, 404 Poisson problem, 198–199, 416, 454 potential energy convertibility (PEC), 391 potential temperature, 13 power-law spectra, 77 precipitation, 237–239, 263–265 precipitation forcing, 363–365 pressure perturbation, 415 probability density function (PDF), 56 pseudo-adiabatic, 14 pseudo-wet adiabatic, see pseudo-adiabatic Q1 , Q2 analysis, 39 quasi-equilibrium, 35, 43, 46, Ch. 4, 101, 153, 327, 331 quasi-normal approximation (QN), 50 radiative-convective equilibrium (RCE), 110–112 rain evaporation, 422–428, 431, 433 relaxation process, 127–129, see also adjustment process renormalization, 38, 41, 47 renormalization group (RNG), 80 residual scale, 87 reverse entraining plume, 432 Reynolds stress subgrid-scale, 89 scale gap, 75 scale separation, 35, 38, 42, 44, 45, Ch. 3, 73, 125–126, 186–187, 222 limits, 93–94 space and time, 96 scaling, 78–80 segmentally constant approximation (SCA), 196, 199–205, 223, 277, 414–415, 455, 456

515

Index

self-organized criticality (SOC), 44, 137–139 & CQE, 139–140 separation of variables, 216–218, 326 slow manifold, 118–121 slow quasi-manifold, 120 spatial and ensemble averages, 55 squall line, 420, 463 statistical, 49, 97–98, 122–130 statistical cumulus dynamics, 130–134 statistical mechanics, 55, 131–134 statistics, 98 steady-plume hypothesis, 43, 212–213, 277 stochastic entrainment, 312 stochastic mixing, 298–299 stochastic parameterization, 52–53 stochasticity, 52–53, 56 strange attractor, 60 strict equilibrium, 451 sub-grid parameterization, see Ch. 2 subgrid, 90 subgrid scale, 74 subgrid-scale distribution, 56–60 subgrid-scale parameterization problem, formal statement of, 36–40 super-parameterization, 39–40, 66 switch, 394–396 terminal detrainment, 254–262 thermal, 7, 11, 274, 275, 416, see also bubble thermodynamic equilibrium, 102, 104–106 thermodynamics, 104–115 time splitting, 59 timescales, 94–96, 106–136, 329–331 TOGA-COARE (Tropical Ocean Global Atmosphere-Coupled Ocean-Atmosphere Response Experiment), 39, 121, 391, 420 Tokioka parameter, 322 transient, 43, 213 trigger, 43, 213, 395–397, 434 trigger function, 136, 395–396

tropical cyclone, 429 turbulence, 80 turbulence parameterization, 48–54 turbulent detrainment, 308, 319 turbulent entrainment, 275, 317 two-scale systems, 80 universality, 66–71 variational principle, 106–107 verification, 232 virtual temperature, 10 vorticity, 24, 458–478 water mixing ratio, 10, 198, 228 water vapour deposition, 422, 443–447 wave-CISK, 153, 168 wavelet, 60 weak temperature gradient (WTG) approximation, see free ride principle weather modification experiments, 413 wet potential temperature, 18, 20 wind shear, 6, 20, 463 wind-induced surface heat exchange (WISHE), 153

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Series on the Science of Climate Change ISSN: 2045-9726 Editor: Hans-F Graf (University of Cambridge, UK)

Published Vol. 1

Parameterization of Atmospheric Convection (In 2 Volumes) Volume 1: Theoretical Background and Formulation Volume 2: Current Issues and New Theories edited by Robert S Plant and Jun-Ichi Yano

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Library of Congress Cataloging-in-Publication Data Plant, Robert S. author Parameterization of atmospheric convection / Robert S. Plant, Jun-Ichi Yano. pages cm -- (Series on the science of climate change ; volume 1) Includes bibliographical references and index. ISBN 978-1-78326-693-7 (hardcover, v. 1 : alk. paper) -ISBN 978-1-78326-694-4 (hardcover, v. 2 : alk. paper) -ISBN 978-1-78326-690-6 (set : alk. paper) 1. Convection (Meteorology). 2. Atmospheric physics. 3. Weather forecasting. I. Yano, Jun-Ichi. II. Title. QC880.4.C64P595 2015 551.51'5--dc23 2015014333

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Preface to Volume 2

Volume 1 of this set was devoted to introducing important basic concepts as well as the formulation of mass-flux convection parameterization. Volume 2, in turn, is devoted to more advanced issues, possible extensions of the mass-flux framework, alternatives to the mass-flux framework, and the interactions of the convection parameterization with other aspects of weather and climate models. The first part of Volume 2, Part III, is devoted to operational issues. Not only do we discuss directly relevant operational issues such as important aspects of practical implementations, testing, and performance, but also we consider more general issues including the use of novel observations, and new approaches to model verification. Furthermore, issues arising from improved model resolutions are addressed from operational perspectives and possibilities for stochastic parameterization are discussed. Building upon both the basic formulation presented in Part II and the wider operational perspectives in Part III, we are well placed in Part IV to consider further generalization of convection parameterization, including alternative possibilities such as the similarity theory and the subgrid-scale distribution density function (DDF). A major theme of Part IV is the relationship of convective parameterization to other parameterization formulations originally developed to respresent other processes in numerical models of the atmosphere. This is important partly in providing other perspectives and possibilities, but also in considering how the convective parameterization does or does not fit alongside other parts of the model. Questions of self-consistency across parameterizations naturally emerge, and these are also presented and discussed. Chapters 18 and 23 are exclusively devoted to the issues of cloud microphysics. Proper implementation of cloud microphysics into convection v

vi

Preface to Volume 2

parameterization is still an unresolved issue. Although these two chapters do not offer a direct solution, they do provide basic materials required in order to address this problem systematically. Part V is devoted to more fundamental issues that may critically influence future convection parameterization development. The key topics are taken from the contexts of theoretical physics and applied mathematics. This part is intended to suggest how such apparently unrelated fields may ultimately contribute a great deal to a practical problem such as subgridscale parameterization. Volume 2 assumes a basic familarity with the ideas behind sub-grid parameterization generally, and a solid background knowledge of mass-flux convection parameterization ideas more specifically. Naturally, we would recommend Volume 1 as the ideal foundation, and readers already well acquainted with the earlier volume should find no difficulty in following the discussions herein. However, Volume 2 may also be read relatively independently by those with some previous experience in numerical weather prediction or climate modelling.

R.S. Plant and J.-I. Yano Reading, UK and Toulouse, France October 2014

Acknowledgments

This volume is an outcome of the COST Action ES0905, as was Volume 1. RSP and JIY would like to repeat the same acknowledgments as in Volume 1 for this volume for obvious reasons. For Ch. 16, JQ and PS acknowledge funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/20072013), specifically ERC grant agreements no. 306284 (“QUAERERE”) and FP7-280025 (“ACCLAIM”), respectively. For Ch. 18, VJTP was supported by two awards from the National Science Foundation (NSF) (ATM-0427128, which later became ATM-0852620) and from the Office of Science (BER) of the US Department of Energy (DESC0007396). For Ch. 21, DR acknowledges support from project LD11044 MSMT CR, which supported verification activities during the period of the COST Action. For Ch. 23, APK acknowledges support from grants from US Department of Energy’s (DoE) Office of Biological and Environmental Research (BER), (DE-S0006788; DE-SC0008811) and from the Binational US-Israel Science foundation (grant 2010446). APK would also like to thank Klaus Beheng, Andy Heymsfield, Alexei Korolev, Simon Krichak, Zev Levin, Mark Pinsky, Vaughan Phillips, Thara Prabhakaran, Amit Teller, Sue C. van den Heever, and Jun-Ichi Yano for contributing to the drafting of this chapter. For Ch. 25, EM is grateful to Ulrich Blahak, Dmitrii Mironov, Robert S. Plant, Axel Seifert, and Jun-Ichi Yano for fruitful discussions. Special thanks are due to Dmitrii Mironov and Robert S. Plant for their numerous suggestions that helped to considerably improve the chapter. For Ch. 27, the authors acknowledge that their work has been partly vii

viii

Acknowledgments

supported by the grant ERC-Like nr.4/2012 of UEFISCDI Romania. For Ch. 28, AB acknowledges that he is a recipient of an APART Fellowship of the Austrian Academy of Sciences. The contributions of ECB and RP were supported by project P25064 of the Austrian Science Fund (FWF).

Contents

Preface to Volume 2

v

Acknowledgments

vii

List of Contributors

xiii

Part III

Operational issues

1

Introduction to Part III

3

15. Convection in global numerical weather prediction

5

P. Bechtold 16. Satellite observations of convection and their implications for parameterizations

47

J. Quaas and P. Stier 17. Convection and waves on small planets and the real Earth

59

P. Bechtold, N. Semane, and S. Malardel 18. Microphysics of convective cloud and its treatment in parameterization V.T.J. Phillips and J.-I. Yano ix

75

x

Contents

19. Model resolution issues and new approaches in the convection-permitting regimes

113

L. Gerard 20. Stochastic aspects of convective parameterization

135

R.S. Plant, L. Bengtsson, and M.A. Whitall 21. Verification of high-resolution precipitation forecast with radar-based data

173

ˇ aˇcov´ D. Rez´ a, B. Szintai, B. Jakubiak, J.-I. Yano, and S. Turner

Part IV

Unification and consistency

215

Introduction to Part IV

217

22. Formulations of moist thermodynamics for atmospheric modelling

221

P. Marquet and J.-F. Geleyn 23. Representation of microphysical processes in cloudresolving models

275

A.P. Khain 24. Cumulus convection as a turbulent flow

359

A. Grant 25. Clouds and convection as subgrid-scale distributions

377

E. Machulskaya 26. Towards a unified and self-consistent parameterization framework J.-I. Yano, L. Bengtsson, J.-F. Geleyn, and R. Brozkova

423

Contents

Part V

Theoretical physics perspectives

xi

437

Introduction to Part V

439

27. Regimes of self-organized criticality in atmospheric convection

441

F. Spineanu, M. Vlad, and D. Palade 28. Invariant and conservative parameterization schemes

483

A. Bihlo, E. Dos Santos Cardoso-Bihlo, and R.O. Popovych

Part VI

Conclusions

29. Conclusions

525 527

R.S. Plant and J.-I. Yano Bibliography

539

Index

607

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List of Contributors

P. Bechtold European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading RG2 9AX. United Kingdom. [email protected] L. Bengtsson Swedish Meteorological and Hydrological Institute, Folkborgsv¨agen 1, 60176 Norrk¨oping. Sweden. [email protected] A. Bihlo Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s (NL), A1C 5S7. Canada. [email protected] E.M. Dos Santos Cardoso-Bihlo Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien. Austria. [email protected] J.-F. Geleyn ´ Groupe d’Etude de l’Atmosph`ere M´et´eorologique/Centre National de Recherches M´et´eorologiques, and Centre National de la Recherche Scientifique, M´et´eo-France, 42 Av. Coriolis, 31057 Toulouse. France. [email protected]

xiii

xiv

List of Contributors

L. Gerard Royal Meteorological Institute of Belgium, Dept R&D, 3 Av. Circulaire, B 1180 Brussels. Belgium. [email protected] A. Grant Department of Meteorology, University of Reading, Earley Gate, PO Box 243, Reading, RG6 6BB. United Kingdom. [email protected] B. Jakubiak Interdisciplinary Centre for Mathematical and Computational Modelling, Warsaw University, Pawinskiego 5a, 02-106 Warsaw. Poland. [email protected] A.P. Khain Department of Atmospheric Sciences, The Hebrew University of Jerusalem, Jerusalem, Givat Ram 91904. Israel. [email protected] E. Machulskaya Deutscher Wetterdienst, Forschung und Entwicklung, FE14, Frankfurter Str. 135, D-63067 Offenbach am Main. Germany. [email protected] S. Malardel European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading RG2 9AX. United Kingdom. [email protected] P. Marquet GMAP/Proc. CNRM/GAME UMR 3589. M´et´eo-France and CNRS, 31057 Toulouse Cedex. France. [email protected]

List of Contributors

xv

D. Palade National Institute of Laser, Plasma and Radiation Physics, Magurele, Bucharest 077125. Romania. [email protected] V.T.J. Phillips Department of Physical Geography and Ecosystem Science, Lund University, Solvegatan 12, 223 62 Lund. Sweden. [email protected] R.S. Plant Department of Meteorology, University of Reading, Earley Gate, PO Box 243, Reading, RG6 6BB. United Kingdom. [email protected] R.O. Popovych Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien. Austria. Also: Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., 01601 Kyiv. Ukraine. [email protected] J. Quaas Institute for Meteorology, Universit¨at Leipzig, Stephanstr. 3, 04103 Leipzig. Germany. [email protected] ˇ aˇ D. Rez´ cov´ a Institute of Atmospheric Physics, Prague, ASCR, Bocni Str. II/1401, Prague. Czech Republic. [email protected] N. Semane European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading RG2 9AX. United Kingdom. [email protected]

xvi

List of Contributors

F. Spineanu National Institute of Laser, Plasma and Radiation Physics, Magurele, Bucharest 077125. Romania. spineanu@ifin.nipne.ro P. Stier Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU. United Kingdom. [email protected] B. Szintai Hungarian Meteorological Service, Kitaibel P´ al u. 1. H-1024, Budapest. Hungary. [email protected] S. Turner ATMOSPHERE, Syst`emes et Services, Parc Technologique du Canal, Villa Sacramento, 14 avenue de l’Europe, 31520 Toulouse. France. [email protected] M. Vlad National Institute of Laser, Plasma and Radiation Physics, Magurele, Bucharest 077125. Romania. madi@ifin.nipne.ro M.A. Whitall Department of Meteorology, University of Reading, Earley Gate, PO Box 243, Reading, RG6 6BB. United Kingdom. michael.whitall@metoffice.gov.uk J.-I. Yano ´ Groupe d’Etude de l’Atmosph`ere M´et´eorologique/Centre National de Recherches M´et´eorologiques, and Centre National de la Recherche Scientifique, M´et´eo-France, 42 Av. Coriolis, 31057 Toulouse. France. [email protected]

PART III

Operational issues

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Introduction to Part III

Part III places convection parameterization into its operational context within numerical weather prediction and climate modelling. For example, it considers how observational data can be exploited in order to verify forecasts based on these parameterizations. Another important aspect addressed is the issue of implementing cloud microphysics into convection parameterization. A goal of Part III is, by expanding the discussions of Part II into much wider contexts, to prepare the way for introducing various alternative approaches other than mass flux in Part IV. We begin with a review of how a modern convection parameterization is implemented into a state-of-the-art operational model, and how it actually performs. Both strengths and weaknesses are presented and the scope for improving model performance through rethinking aspects of the parameterization is described. More specifically, a major aspect of Ch. 15 is how the model performance changes by switching convection closure, the main closure concepts used having been introduced in Vol. 1, Ch. 11. In considering the exploitation of observations in order to verify and improve convection parameterizations, the dramatic increase in the volumes and types of satellite observations of clouds over recent years has been an important development. Chapter 16 provides an overview of currently available satellite observations. The chapter also suggests how they can best be exploited in the context of convection parameterization development. Chapter 17, in turn, addresses the issue of the interactions between parameterized physics and resolved model dynamics, which are the key to the performance of a complete model. The discussion is based around some ongoing experiments at ECMWF, including tests for which the radius of the Earth is artificially reduced in order to telescope the interactions between convective-scale and planetary-scale motions. 3

4

Introduction to Part III

A particular challenge to be considered is the inclusion of cloud microphysical processes into convection parameterization. Chapter 18 provides a self-contained introduction to those processes within cloud microphysics that are most important for convective cloud. It also describes the implementation problem and some specific examples of ongoing efforts are discussed. We are also facing major and increasingly pressing challenges arising from the increasing horizontal resolutions of operational forecast models. Chapters 19 to 21 consider some of the implications. Some current and some planned operational model configurations have increased the horizontal resolution such that the numerical grid almost begins to resolve deep convection, but not quite. Such scales are commonly, if rather loosely, referred to as the “grey zone” and Ch. 19 is devoted to the issues of convection parameterization for this convection-permitting, but not convectionresolving, regime. Various modifications required for the mass-flux formulation are discussed, and some examples of operational implementations based on these needs are presented. Chapter 20 introduces stochasticity. Stochastic aspects of sub-grid parameterization have received increasing attention over the past decade for both theoretical and practical reasons. Parameterization requires that relationships exist between the instantaneous state of the atmosphere in a vertical grid-box column and the sub-grid convective activity. The motivation for stochastic parameterization is a recognition that while suitable relationships may hold in some statistical sense, the sub-grid activity may not be fully predictable from the resolved-scale state alone. Such predictability considerations are related to the resolution of the model and become increasingly apparent at higher resolutions. Various methods for introducing a stochastic element to a parameterization have been proposed and the chapter focuses on comparing their physical basis: what physical aspects of a convective system are considered to lead to what forms of uncertainty and with what implications for the parameterization. The verification of forecasts of convection requires some special considerations, even within the context of the general forecast verification problem, and the particular issues become increasingly more important as individual convective elements begin to be resolved in forecast models. Convective systems tend to be localized, and thus we need to distinguish between the predicted location and the intensity in order to verify those two aspects appropriately. These issues are surveyed in Ch. 21.

Chapter 15

Convection in global numerical weather prediction

P. Bechtold Editors’ introduction: We begin Volume 2 by considering the actual performance of a state-of-the-art mass-flux convection parameterization within the context of global numerical weather prediction (NWP). A single parameterization is considered within a single host model, specifically that of the European Centre for Medium-Range Weather Forecasts (ECMWF). Although particular parameterizations do have strengths and weaknesses in particular contexts, it is fair to say that many of the behaviours and issues identified here have a certain commonality. Thus, it is valuable to study a particular scheme in some detail: partly to obtain a feeling for the operational implementation of many of the ideas of Volume 1, and partly also because there is much that is general to be learnt. Another feature of the chapter is the author’s discussions regarding some recent improvements to the parameterization, particularly a new closure formulation. An important point to stress is that these improvements arose not from some tuning or engineering fix, but by revisiting and rethinking the physical basis of the closure. Our conviction that this approach towards model development is the most fruitful and robust basis for the future of convection modelling underpins much of this volume.

1

Forecast errors and analysis uncertainty

Analysing the convection in a forecast system is a huge and difficult task. This is particularly true for the tropical regions, where due to the small Coriolis force and large Rossby radius of deformation, convection affects a vast variety of space and timescales, from the individual convective cloud 5

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Part III: Operational Issues

ˆ to the large-scale convectively coupled waves (Simmons, 1982; Zagar et al., 2005): for example, the intraseasonal oscillations and the monsoon circulations (see Vol. 1, Ch. 5). Furthermore, it is difficult to observe convection directly, or all of the convective transport processes. It is therefore appropriate to describe convection by measuring the quality of the forecast system in terms of convection- and cloud-related quantities such as surface precipitation, outgoing longwave radiation (OLR), and observables such as the temperature, moisture, and wind fields. Given these measures, it is possible to reach a conclusion on the overall forecast quality in representing convective phenomena, but not necessarily to conclude on the quality of the convection parameterization scheme itself as employed in large-scale models with horizontal resolutions larger than, say, 5 km. This is especially true in the tropics where, on larger space and timescales, the atmosphere is in radiativeconvective equilibrium (Held et al., 1993, see also Vol. 1, Ch. 4, Sec. 3.7), so that heating processes like upper-tropospheric stratiform condensational heating and cloud radiation interaction play an important role. On synoptic scales a balance between the large-scale dynamical forcing and the convection damping prevails, while both processes strongly interact. The momentum budget is also critical, depending on an equilibrium between the large-scale pressure gradient (vertically integrated temperature anomaly), turbulent dissipation, and friction due to cumulus momentum transport (cf., Vol. 1, Ch. 14). A major difficulty in evaluating tropical convection and forecasts in the tropics is in the sparseness of upper-air in situ data over tropical oceans, making satellite data products the main observational information source in these regions (cf., Ch. 16). The impact of conventional and satellite data in the analysis is defined by data density and the assigned observation errors, and thus in areas with extended cloud coverage and convection, the analysis is more strongly driven by the forecast model than by observations, and is therefore more affected by model errors. At present there is not much in the literature assessing the uncertainty of atmospheric analyses (Langland et al., 2008; Wei et al., 2010) or studying the predictability of forecast systems (Kanamitsu, 1985; Simmons and Hollingsworth, 2002) for the tropics, but this should change with the availability of the TIGGE archive (THORPEX Interactive Grand Global Ensemble: Bougeault et al., 2010; Park et al., 2008). At ECMWF, satellite observations sensitive to temperature, moisture, clouds, and wind are assimilated over the tropics.

Convection in global numerical weather prediction

7

The impact of the satellite observing system on the analysis and forecasts over tropical oceans has been outlined in Andersson et al. (2007), Bauer et al. (2011) and Kelly et al. (2007). All the available studies agree in that the tropical East Pacific and East Atlantic, as well as the equatorial Indian Ocean, stand out as key areas sensitive to observations. However, the impact of moisture-related observations on the dynamics is generally weaker and dissipates fairly quickly into the forecast while temperaturesensitive observations are difficult to make use of in cloudy areas. In order to illustrate the uncertainty in tropical analysis, in Fig. 15.1 the zonal-mean analysis difference during 2011 between ECMWF and the United Kingdom Met Office (hereafter refered to as UKMO) has been plotted for temperature, zonal wind, and relative humidity. It is evident that large differences occur in the tropics with the tropical troposphere being around 0.5 K colder in ECMWF compared to UKMO. This difference is significantly larger than expected with random errors in satellite observations, which are of order 0.1–0.2 K. Wind differences amount to roughly 0.5 ms−1 , whereas upper-tropospheric relative humidities differ by up to 20%. Analysis differences express model, data assimilation system, and observational data usage differences. Disentangling these contributions can be difficult. However, not only analysis differences are relatively large in the tropics, also forecast errors are significant, as illustrated in Fig. 15.2 by the annualmean differences for 2011 between the ECMWF operational forecasts at lead times of 24h and 120h compared to the ECMWF operational analysis. Lower and upper-tropospheric tropical zonal-mean errors at lead time 120h are similar in magnitude to the analysis differences seen in Fig. 15.1 and amount to 0.5 K for temperature and to 0.5 ms−1 for wind, while midtropospheric errors are small. The humidity errors, however, are of order 2% and therefore much smaller than the analysis difference in Fig. 15.1. The errors shown point to errors in the model’s physical parameterizations of which the convection is an important part. Representative of all global weather forecasting systems, the ECMWF model, considered one of the leading models, will serve as the numerical testbed in this chapter.

2

Global distribution of convection

The global annual distribution of rainfall for 2000/2001 from the Global Precipitation Climatology Project (GPCP version 2.2, a combination of

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Part III: Operational Issues

Diference in analyses -10 -6 -2 2 6 10

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Fig. 15.1 Pressure (hPa) versus latitude cross section of 2011 annual- and zonal-mean difference in analysis between ECMWF and UKMO: (a) temperature (K), (b) U-wind (ms−1 ), and (c) relative humidity (%). Significant differences at the 95% level are marked by dark colours; pale shading is used otherwise.

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Fig. 15.2 Pressure (hPa) versus latitude cross-section of annual-mean ECMWF forecast errors from verification against the model’s own analysis for forecast lead times of 24h (upper row) and 120h (lower row): left-hand column temperature (K), middle column U-wind (ms−1 ), and right-hand column relative humidity (%). Note the different scaling for the different lead times.

satellite-derived rainfall rates and surface observations) is depicted in Fig. 15.3a. The results from an ensemble of one-year integrations with the ECMWF operational forecasting system in 2013 using analysed sea surface temperatures are depicted in Fig. 15.3b,c. The global average daily rainfall rate from GPCP2.2 is around 2.7 mmday−1 , but most of the precipitation occurs in the tropical belt with a rate of 5–8 mmday−1 . The dry subtropical anti-cyclonic areas to the west of the continents also stand out, as well as the mid-latitude storm tracks. With 2.8 mmday−1 , the model reason-

Convection in global numerical weather prediction

9

ably reproduces the observed global mean precipitation and its distribution (Fig. 15.3b), but overestimates rainfall rates in India and South-East Asia, and in particular during the summer monsoon season. The model’s convective rain rates are depicted in Fig. 15.3c. Globally, about 60% of the precipitation is of the convective type, while in the tropics the convective rainfall contributes 75% of the total. These numbers are certainly model and resolution dependent. However, they turned out to be optimal for the ECMWF forecast system in order to minimize errors in the state variables such as temperature, wind, moisture, and cloudiness. The global distribution of convective clouds can be inferred from satellite observations such as CLOUDSAT/CALIPSO. As a proxy for these observations, the annual frequency of occurrence of deep convective clouds (defined as having a thickness exceeding 200 hPa and positive buoyancy) and shallow convective clouds (having vertical extent < 200 hPa) as obtained from the ECMWF model’s convection parameterization is illustrated in Fig. 15.4. The observed convective cloud distribution is actually tri-modal (Johnson et al., 1999), but the deep cloud type includes also the cumulus congestus clouds that detrain in the middle troposphere around the melting level. As shown in Fig. 15.4a, deep convective clouds are a prominent feature of the tropical belt, but also frequently occur in the mid-latitude storm tracks, the Gulf Stream and Kuroshio region in particular. With a frequency of occurrence of up to 90%, shallow convective clouds (Fig. 15.4b) are a ubiquitous feature of the subtropical anti-cyclonic regions. The climatological cloud distribution is assessed indirectly in Fig. 15.4c by comparing the model’s shortwave radiative flux at the top of the atmosphere to the shortwave flux from the Clouds and Earth’s Radiant Energy Systems (CERES) Energy Balanced and Filled (EBAF) data product. The global-mean model bias is around 10 Wm−2 . However, the main bias is in the stratocumulus areas off the western coasts of the continents, where clouds are underestimated and also in the southern hemisphere storm track. The convective cloud regions are reasonably reproduced but the trade cumulus clouds are too reflective due to a too-high water content (Ahlgrimm and K¨ohler, 2010). The cloud and radiation errors strongly depend on the other physical parameterizations such as the cloud parameterization, the boundary-layer diffusion, and the radiation. However, in tropical regions convection is the major source term for the production of clouds. The zonal-mean total (deep+shallow) and shallow convective tendencies for temperature, specific humidity, and mass fluxes are depicted in Fig. 15.5 for July 2013. Convective tendencies and mass fluxes are important in the

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Part III: Operational Issues

(a) GPCP2.2 (mm/day) 80°N 60°N 40°N 20°N 0° 20°S 40°S 60°S 80°S 135°W

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Fig. 15.3 Annual-mean daily global precipitation (mmday−1 ) from (a) the GPCP2.2 precipitation climatology dataset, and (b) and (c) total and convective precipitation from an ensemble of one-year integrations at spectral truncation T159 (125 km resolution) with the ECMWF operational model version in 2013.

11

Convection in global numerical weather prediction

(a) Deep (%) 80°N 60°N 40°N

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Fig. 15.4 As in Fig. 15.3 but for the annual mean frequency (%) of (a) deep and (b) shallow convective clouds. (c) Difference in climatological net shortwave radiative flux at the top of the atmosphere between the model and the CERES EBAF data product. The sign convention is that positive values correspond to an excess in reflection (too many clouds or optically too-thick clouds), and negative values correspond to an underestimation.

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Part III: Operational Issues

tropics, the mid-latitude storm tracks, and the summer (here, the northern) hemisphere in particular. Zonal-mean deep convective heating and drying rates reach around 5 Kday−1 in the middle troposphere. In the lowest 300 hPa of the atmosphere, shallow convective transport is important (Fig. 15.5d–f) as it dries the subcloud layer by up to 10 Kday−1 in the zonal mean, and moistens and heats the cumulus cloud layer. Shallow convection also decelerates the mean flow in the cumulus layer (not shown). The shallow convective mass fluxes (Fig. 15.5f) are likely to be larger than those obtained from CRM data (Grabowski et al., 2006). However, large mass fluxes are needed to assure a good forecast performance in the mid-latitude storm tracks. From the above picture it becomes clear that convection parameterization remains a challenging and important task, even if deep convective motions will become gradually more resolved in the coming decade with the next generation of high-resolution global models (see Ch. 19). a

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Fig. 15.5 Pressure (hPa) versus latitude zonal-mean cross-sections of convective tendencies for temperature and specific humidity (Kday−1 ) from (a) and (b) deep+shallow convection, and (d) and (e) shallow convection only, as well the convective mass fluxes (kgm−2 s−1 ) from (c) deep+shallow convection and (f) shallow convection only. Data is obtained for July 2013 from the ECMWF operational model at spectral truncation T511 (40 km resolution).

Convection in global numerical weather prediction

3 3.1

13

Prediction of tropical modes Wave analysis

Convectively coupled large-scale disturbances that propagate horizontally along the equator and vertically into the stratosphere are the major contributor to intraseasonal variability in the tropics. It is therefore important in a forecast system to correctly reproduce the amplitude and phase of these disturbances. As shown, for example by Cho and Pendlebury (1997), Holton (2004), and Verkley and van der Velde (2010), the dominant tropical modes can be derived from the linearized shallow water equation system on the equatorial plane. Following the method developed by Kiladis et al. (2009) these modes are illustrated in Fig. 15.6a with the aid of wavenumber frequency diagrams of the OLR as observed by National Oceanic and Atmospheric (NOAA) administration satellites. The dominant wave types are westward-propagating Rossby waves, eastward-propagating Kelvin waves, and eastward- and westward-propagating inertia-gravity waves. The Madden–Julian oscillation (MJO) (Madden and Julian, 1971) is not a wave and appears as a strong amplitude signature in the wavenumber 1–2 and frequency 20–60 days range. The theoretical dispersion relations have also been superposed in Fig. 15.6a using different values for the equivalent shallow water depth. In general, high phase speeds correspond to dry waves, and reduced phase speeds to an increased coupling between the waves and the convection (see Vol. 1, Ch. 5). In order to describe the propagation of the large-scale waves and their effect on the stratospheric circulation we make use of the Eliassen–Palm fluxes. The generalized Eliassen–Palm theorem as outlined in Andrews and McIntyre (1976) is a powerful tool used to describe the wave action on the mean flow and to indicate the net wave propagation for stationary planetary waves. The Eliassen–Palm flux vectors and their divergence (shaded) have been computed in Fig. 15.7 from the zonal-mean meridional momentum and heat fluxes following Peixoto and Oort (1992, p. 338). Negative (positive) values of Eliassen–Palm divergence in the stratosphere indicate a reduction (acceleration) of zonal-mean zonal winds by breaking stationary planetary waves. The two European reanalysis products ERA-40 and ERA-Interim show that tropospheric stationary planetary waves propagate upwards into the stratosphere where they break and slow down the stratospheric polar vortex. Also shown are model results from the ECMWF model version in 2006 (same as Fig. 15.6b), i.e., before the revision of the convection

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Part III: Operational Issues

a

Observations from NOAA satellites 2 90 25 50 0.4 25 12 8 12 3 0.3 IG 8 4 0.2 Kelvin 5 ER n=1 0.1 10 20 0 60 -15 -10 -5 0 5 10 15 Wavenumber Westward Eastward b ERA-Interim = Oper 2006 0.5 2 90 25 50 0.4 25 12 3 8 0.3 12 IG 8 4 0.2 Kelvin 5 ER n=1 0.1 10 20 0 60 -15 -10 -5 0 5 10 15 Wavenumber Westward Eastward c Oper 2013 0.5 2 90 25 50 0.4 25 12 3 8 0.3 12 IG 8 4 0.2 Kelvin 5 ER n=1 0.1 10 20 0 60 -15 -10 -5 0 5 10 15 Wavenumber Westward Eastward

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Fig. 15.6 Wavenumber frequency spectra of the OLR with data from (a) the NOAA satellites, (b) the ECMWF model version in 2006, as also used to produce the European Reanalysis Interim project, and (c) the operational model version in 2013. The data has been averaged between 10◦ S and 10◦ N, a background spectrum corresponding to rednoise has been subtracted, and only the symmetric part of the spectrum is displayed. Also shown are the theoretical dispersion relations with external gravity wave phase speed c = (gH)1/2 as a function of equivalent depth H.

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Convection in global numerical weather prediction

parameterization, and after the revision (same as Fig. 15.6c). It is possible to observe, consistent with Fig. 15.6b, a lack of stationary planetary wave breaking in the middle and lower stratosphere in Fig. 15.7c. The consequence of the lack of wave breaking is an overestimation of the strength of the stratospheric polar vortex (not shown) in the middle and lower stratosphere. Zonal-mean zonal winds and Eliassen–Palm flux diagnostics with the revised convection in Fig. 15.7d show better agreement with the two reanalysis products throughout the stratosphere and an improvement in the stationary planetary wave structure throughout the troposphere, though the Eliassen–Palm flux divergence in the middle to upper troposphere between 45◦ N and 70◦ N is somewhat overestimated. (a) ERA40 (1988-2001 DJFM)

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ms−1 )

Fig. 15.7 Zonal-mean zonal wind (contour interval is 2.5 and Eliassen–Palm flux vectors (m2 s−1 day−1 ) and divergence (shaded in ms−1 day−1 ) associated with stationary planetary waves for winters (December–March) of the period 1988–2001: (a) ERA40, (b) ERA Interim, (c) and (d) from seasonal integrations with model versions as in Fig. 15.6b,c. The Eliassen–Palm flux vectors are defined as in Peixoto and Oort (1992, p. 388) as [−R cos φ u∗ v∗ , f R cos φ v∗ θ ∗ (∂θ/∂p)−1 ], where f is the Coriolis parameter, φ is latitude, θ the potential temperature, and stars denote anomalies from the zonal mean.

Recent model intercomparison studies (Benedict et al., 2013; Blackburn et al., 2013; Hirons et al., 2013a; Kim et al., 2011; Lin et al., 2006) showed

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Part III: Operational Issues

that models produce very different tropical wave spectra depending on the formulation of the convection parameterization. In particular, these studies demonstrated the sensitivity of the Kelvin waves and MJO mode to the formulation of the entrainment, with stronger entrainment rates producing stronger amplitudes with more realistic (slower) propagation speeds. The ECMWF model before 2008 had been unable to produce realistic Kelvin waves and MJO as illustrated in Fig. 15.6b using the same model version as used to produce the European Reanalysis Interim project (Dee et al., 2011). However, with a revised convection parameterization using realistic strong entrainment rates as discussed in Bechtold et al. (2008) and de Rooy et al. (2013) (see also Vol. 1, Ch. 10), the model now reproduces the large-scale tropical modes (Fig. 15.6c). 3.2

The MJO

The MJO and the role of convection in predicting the MJO deserves some further discussion. The MJO is the dominant mode in intraseasonal variability (Zhang, 2005) and provides the major source of predictability in the mid-latitudes on monthly timescales (Vitart and Molteni, 2009). In spite of that, models have difficulties in predicting the MJO and in particular its initiation and propagation from the Indian Ocean to the West Pacific (Kim et al., 2011; Lin et al., 2006). All studies agree that the convection parameterization and not the model resolution is the dominant factor in the prediction of the MJO, which can be seen as a projection of convection on a planetary scale. To document the evolution of the quality of the MJO forecasts in the ECMWF model, hindcasts have been run for the period 1995–2001 with all model cycles since 2002. As a generally accepted quality measure, Fig. 15.8 shows the linear correlation coefficient between the forecasted and analysed OLR and wind field projected onto the two leading Empirical Orthogonal Functions (EOFs). Assuming a meaningful correlation threshold of 0.6, the predictability of the MJO has improved from about 15 days in 2002 to 26 days in 2011. Note that the National Centers for Environmental Prediction (NCEP) hosts a service collecting and evaluating the real-time MJO forecasts from all major centres. The current predictability limit of the ECMWF model, which is ahead of the other centres, has to be compared to the theoretical predictability estimate of the MJO, which is one cycle or roughly 50 days (Ding et al., 2010). Judging from this, there is still room for further improvement. However, between 2007 and 2008 a strong increase in MJO predictability was observed, which

Convection in global numerical weather prediction

17

Forecast day

was due to a revision of the convection parameterization (Bechtold et al., 2008). 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

MJO Bivariate Correlation

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2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 Year Fig. 15.8 Time series of different thresholds of correlation coefficient between forecasted and analysed OLR and wind field projected onto the two leading EOFs. Dots correspond to all operational model cycles between 2002 and 2011 that have been run for the hindcast period 1995 to 2001; triangles denote the operational model version in 2013.

To illustrate the evolution of individual MJO events and their forecasts, in Fig. 15.9 time-longitude diagrams of OLR anomalies (Wm−2 ) during May 2008 to June 2009 have been plotted, with a 10-day low-pass filter applied to the data. This period also corresponds to the declared Year of Tropical Convection (Waliser et al., 2012) during which the international community focused on all aspects of tropical convection and the MJO in particular. The OLRs in Fig. 15.9 have been averaged between ±10◦ latitude. Figure 15.9a shows the observations from the NOAA satellites. The individual MJO events are characterized by negative OLR anomalies propagating from the Western Indian Ocean to the Central Pacific in about 15–20 days. Also displayed are the day+1 and day+10 operational ECMWF forecasts during 2008 and 2009 (Fig. 15.9b,c), the day+10 forecasts from the ERA-Interim version, and the experiment labelled CONV that uses the operational model version, but with the convection parameterization reverted to that used in the ERA-Interim, i.e., the version used operationally before November 2007. Clearly, the day+1 operational forecasts closely fit the observations. The operational day+10 forecasts are able to maintain the propagating

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Part III: Operational Issues

anomalies, but the amplitude of the convective signals is now overestimated, leading through teleconnections to too-strong positive OLR anomalies in the Eastern Pacific. In contrast, in both ERA-Interim and CONV forecasts, the amplitude of the anomalies has become rather weak by day+10 and their propagation is compromised. a NOAA

b Oper t+1d

c Oper t+10d

d ERA-I t+10d

e CONV t+10d

50°E

50°E

50°E

50°E

Jul Jun May Apr Mar Feb 2009 Jan Dec Nov Oct Sep Aug Jul 2008 Jun 50°E

150°E -90

-75

150°E -60

-45

150°E -30

-15

15

150°E 30

45

150°E 60

Fig. 15.9 Time versus longitude diagram of 10-day low-pass filtered anomalies of the OLR (Wm−2 ), averaged over ±10◦ latitude for the period May 2008 to July 2009 as observed by (a) NOAA satellites, and from operational (b) day+1 and (c) day+10 forecasts, as well as from (d) day+10 forecasts from the ERA-Interim, and (e) the CONV experiment that uses the operational cycle but with the old convection.

It was shown by Hirons et al. (2013a,b) that the loss of MJO amplitude in versions of the ECMWF model before 2008 was due to too-active deep convection that did not depend on the phase of the MJO. The authors also showed that realistic strong entrainment rates and convective adjustment are necessary to represent the different phases of the MJO, i.e., a suppressed phase with lower-tropospheric moistening mainly through shallow and congestus clouds, followed by an active phase with large-scale lifting and deep convective heating and drying. Concerning the interaction between heating and the MJO, it was probably Yanai et al. (2000) who first demonstrated that for the MJO with its vertical westward-tilted structure, the amplification occurs in the upper troposphere through: (a) generation of potential energy; and, (b) conversion of potential energy into kinetic energy. This

Convection in global numerical weather prediction

19

principle has also been demonstrated for the Kelvin wave by Shutts (2008), and for the global energy cycle by Steinheimer et al. (2008). It is therefore crucial for a realistic simulation of the MJO that the model’s convection parameterization produces a realistic heating profile in the right phase of the background or dry wave. Or, in other words, for the generation of potential energy heating must occur in warm regions, and for the kinetic energy conversion −T˙ ω > 0, i.e., the correlation between the heating anomaly and vertical velocity in pressure coordinates must be negative. This explains the decrease in MJO amplitude through negative energy conversion in the CONV experiment where convection is active irrespective of the phase of the large-scale wave.

4

Deep and shallow convective scaling

Today, most global numerical weather prediction and climate models employ a convection parameterization scheme based on the mass-flux concept. A non-exhaustive list of basic parameterization schemes used in global weather prediction models includes Arakawa and Schubert (1974), Bougeault (1985), Emanuel (1991a), Gregory and Rowntree (1990), Kain and Fritsch (1993), Tiedtke (1989), and Zhang and McFarlane (1995), although many of these schemes have since been substantially modified and improved. Despite employing a similar basic framework, these models can produce substantially different results in terms of large-scale tropical wave spectra, intraseasonal variability and the diurnal variation of convection. However, Bechtold et al. (2008) and Jung et al. (2010) demonstrated with the ECMWF model that the basic mass-flux framework under the quasiequilibrium assumption can provide a realistic reproduction of the observed mid-latitude synoptic variability, as well as intraseasonal tropical variability. In order to achieve this, two important properties of the convection scheme were required: an adaptive convective adjustment or mass-flux scaling, and a realistic strong entrainment rate (see Vol. 1, Ch. 10). The latter represents the observed heating modes from shallow, congestus, and deep clouds in the tropics (Lin et al., 2012). Following Bechtold et al. (2014) let us now derive the convective scaling that has been used to produce Fig. 15.5 and has proven to give optimal forecast performance in the ECMWF system. The convective scaling can be regarded as the application in a numerical weather prediction model of the ideas presented in Vol. 1, Chs. 2, 3, and 11.

20

Part III: Operational Issues

The convective available potential energy (CAPE: Jkg−1 ) is defined as the buoyancy integral  ztop upad  ztop upad ¯ θe (T, q) − θesat (T¯ ) Tv − Tv dz, (4.1) dz ≈ g CAPE = g ¯ Tv θesat (T¯) zbase zbase where the integration in height coordinates z is between cloud base and cloud top, Tv is the virtual temperature, and g gravity; the superscript upad denotes values of an air parcel lifted pseudo-adiabatically (i.e., without considering mixing with environmental air), and bars denote environmental or grid-mean values. For diagnostic purposes, CAPE can be reasonably approximated by using the saturated equivalent potential temperature θesat instead of T¯v , and the equivalent potential temperature θeupad , depending on temperature T and specific humidity1 q, instead of Tvupad. As θe is conserved during moist adiabatic ascent, the right-hand side of Eq. 4.1 shows that the updraught thermodynamic properties are determined by the temperature and moisture in the departure layer of the rising air parcel that predominantly roots in the boundary layer. In the context of convection parameterization, we use integration over pressure and define PCAPE (pressure-integrated convective available potential energy: Jm−3 ) as the density-weighted buoyancy integral of an entraining ascending air parcel:  ptop up ¯ Tv − Tv dp. (4.2) PCAPE = − T¯v pbase The entrainment rates used to compute Tvup are those given in Vol. 1, Ch. 10. The advantage of PCAPE over an entraining CAPE is the density scaling that more readily relates the time derivative of PCAPE to the convective mass flux. Under the assumption of vanishing updraught temperature excess at cloud top, and using Tvup − T¯v ≪ T¯v , the time derivative of PCAPE is obtained as:   ptop  ptop ∂pbase ∂PCAPE 1 ∂ T¯v 1 ∂Tvup Tvup − T¯v  = dp − dp + .  ¯ ¯ ¯ ∂t ∂t ∂t T T T v v v pbase pbase base ∂t       LS+CONV

BL+CONV

(4.3) The evolution of PCAPE includes its production by radiative and advective large-scale processes (LS), and its destruction by cumulus convection

1 For most of this set, q denotes a mixing ratio, but it is used in this chapter to denote specific value.

Convection in global numerical weather prediction

21

(CONV), both affecting T¯v . Furthermore, there is production of PCAPE by boundary-layer (BL) processes other than convection, removal by convective boundary-layer venting, and cooling by downdraughts and subcloud rain evaporation, all affecting Tvup . The prognostic equation for PCAPE can then be formally rewritten as:    ∂PCAPE  ∂PCAPE ∂PCAPE  ∂PCAPE  = .  +  +  ∂t ∂t ∂t ∂t LS BL CONV=shal+deep (4.4) Note that the CONV term contains both the convective stabilization of the free troposphere (LS) and the boundary-layer (BL), and that it is the sum of the contributions from shallow and deep convection. Similar prognostic equations for CAPE were also derived in Donner and Phillips (2003) and Zhang (2002) (see also Vol. 1, Ch. 11, Sec. 11.1). The LS production term includes the tendencies due to mean vertical and horizontal advection and radiation, and is given by:    ptop 1 ∂ T¯v  ∂PCAPE  = dp. (4.5)   ¯ ∂t pbase Tv ∂t adv+rad LS The tendency due to convection can either be approximated assuming that cumulus convection acts to remove PCAPE over a convective timescale τ (Betts and Miller, 1986; Fritsch and Chappell, 1980; Nordeng, 1994) thus:  PCAPE ∂PCAPE  , (4.6) =−  ∂t τ CONV,1

or by approximating the convective tendency by the heating through compensating environmental subsidence, so that the convective mass flux M (kgm−2 s−1 ) becomes apparent. This relation is more easily written using the height coordinate instead of pressure:   ¯  ztop g ∂ Tv g ∂PCAPE  ≈− (4.7)  ¯ M ∂z + cp dz ∂t zbase Tv CONV,2  ¯  ∂ Tv g Mbase ztop g ∗ =− ∗ + M dz, (4.8) M ∂z cp T¯v z base

base

with cp the specific heat at constant pressure. The initial mass-flux profile ∗ M ∗ and initial cloud-base mass-flux Mbase are known from the updraught computation. The ratio between the actual (final) cloud-base mass-flux, ∗ is the convective and the unit (initial) cloud-base mass-flux Mbase /Mbase scaling or closure factor with which the initial mass-flux profile is rescaled.

22

Part III: Operational Issues

Different convective closures can then be formulated on the basis of Eq. 4.4, keeping in mind that a mass-flux scheme requires a closed expression for the cloud-base mass-flux Mbase , and therefore it is necessary to retain Eq. 4.8. If we know the boundary-layer (BL) term, PCAPE can be determined prognostically from Eq. 4.4 using Eqs. 4.5 and 4.6. The convective mass flux is then obtained diagnostically from   ∂PCAPE  ∂PCAPE  = . (4.9)   ∂t ∂t CONV,2 CONV,1

Alternatively, in a purely diagnostic scheme PCAPE can be computed from Eq. 4.2, and again Eq. 4.9 used to compute the convective mass flux. Note that in this diagnostic formulation PCAPE implicitly contains the production from BL and LS. Another diagnostic closure is obtained from Eq. 4.4 if Eqs. 4.5 and 4.8 are used, the left-hand side is neglected, and a boundary-layer in equilibrium is assumed, thus:   ∂PCAPE  ∂PCAPE  . (4.10)  =−  ∂t ∂t LS CONV,2 This relation is another formulation of the quasi-equilibrium between the large-scale destabilization and the convection, but as defined by the integral bounds, it is the quasi-equilibrium for the free troposphere. No timescale τ has to be specified, as it is implicitly contained in the LS tendency. However, experimentation shows that this closure is not general enough, as it underestimates convective activity in situations where the LS forcing is weak, and where convective heating precedes the dynamic adjustment. Finally, even a suitable moisture convergence closure (cf., Vol. 1, Ch. 11, Sec. 7) can be formulated that is consistent with Eq. 4.4, using Eq. 4.8: 

ptop psurf

  ∂ q¯ ∂PCAPE  dp = ,  ∂t adv+BL ∂t CONV,2

(4.11)

where the integration is from the surface to the top of the atmosphere including LS and BL contributions. This closure, despite assuming moisture to be the source of convection instead of instability, has properties of both Eqs. 4.10 and 4.9. It is still applied in NWP (Bougeault, 1985), but tests with the ECMWF Integrated Forecast System (IFS) did not lead to optimal model performance.

23

Convection in global numerical weather prediction

4.1

Diagnostic CAPE closure

As outlined above, a convenient diagnostic CAPE closure can be defined using Eq. 4.9 and substituting with Eqs. 4.8 and 4.6, and computing PCAPE from Eq. 4.2. The cloud-base mass-flux is then obtained as: −1  ¯

 ztop g g PCAPE ∂ Tv ∗ ∗ + . (4.12) dz Mbase = Mbase ¯ M τ ∂z cp zbase Tv Apart from using a density-weighted PCAPE instead of an entraining CAPE, this is the closure for the deep convective mass fluxes that has been used in the IFS since Gregory et al. (2000). With this formulation, the convective mass flux closely follows the large-scale forcing and/or the surface fluxes when the CIN (convective inhibition) is small and the adjustment timescale is reasonably short. The closure of Eq. 4.12 is complete with a definition of the convective adjustment timescale τ following Bechtold et al. (2008): τ=

Hc f (n) = τc f (n); w ¯up

f (n) = 1 +

264 . n

(4.13)

Here, τc is the convective turnover timescale with Hc the convective cloud depth, w ¯ up the vertically averaged updraught velocity as obtained from the updraught kinetic energy equation (see Ch. 12), and f is an empirical scaling function decreasing with increasing spectral truncation (horizontal resolution) n. The minimum allowed value for τ is set to 12 min. Note ¯ up , which is consistent with that τc itself depends on PCAPE through w the observations by Zimmer et al. (2011). In the following, the closure described by Eqs. 4.12 and 4.13 is referred to as CTL. 4.2

Diagnostic equilibrium

CAPE

closure

with

boundary-layer

As the above closure of Eq. 4.12 does not reproduce the observed diurnal cycle (as shown later), even when employing large entrainment rates in the convection scheme that are consistent with CRM data (de Rooy et al., 2013; Genio and Wu, 2010, see also Vol. 1, Ch. 10), it is suggested that it does not reproduce the observed non-equilibrium between the boundary-layer forcing and the deep convection. Donner and Phillips (2003) and Zhang (2002) have shown through an analysis of observational data of mid-latitude and tropical convection that the assumption that ∂PCAPE/∂t is small compared to the individual terms on the right-hand side of Eq. 4.4 is not valid

24

Part III: Operational Issues

if the boundary layer is not in equilibrium. Indeed the boundary-layer production term is the dominant term in surface-driven convection under weak large-scale forcing. In most parameterizations using a CAPE-type closure, the imbalance between the deep convection and the BL production is not explicitly taken into account. However, some authors (e.g., Raymond, 1995) have taken an alternative approach by proposing separate boundary-layer equilibrium closures. The total boundary-layer production in Eq. 4.4 is defined as proportional to the surface buoyancy flux   pbase ¯  ∂ Tv  1 ∂PCAPE  =− dp, (4.14)  ∂t T⋆ ∂t  BL

psurf

BL

where ∂ T¯v /∂t|BL includes the tendencies from mean advection, diffusive heat transport, and radiation. In the model context, these tendencies must be available before the convection is taken into account. The temperature T⋆ scales as T⋆ = c−1 p gH with H a characteristic height. T⋆ = 1K has been set and the scaling cast into the coefficient α below. In a prognostic scheme one could in principle formulate the boundary-layer contribution to be formally consistent with the second term on the right-hand side of Eq. 4.3, the third term being generally small. However, the BL contribution in Eq. 4.3 is the sum of the convective contribution and the forcing. In a model, the non-convective BL forcing could be isolated by calculating the BL temperature tendency due to non-convective terms. Furthermore, the tendency of the updraught virtual temperature can be rather discontinuous in space and time, and even become negative while there is surface heating. Therefore, Eq. 4.14 is the preferred formulation of the boundary-layer contribution to PCAPE taking into account all relevant forcings. In order to account for the imbalance between boundary-layer heating and deep convective overturning we write the convective tendency as the relaxation of an effective PCAPE:   ∂PCAPE  PCAPE τbl ∂PCAPE  + α , (4.15) = − α=   ; ∂t τ ∂t τ CONV=deep BL

with α the fraction of boundary-layer forcing consumed by shallow convection. α is given as the ratio of the boundary-layer timescale τbl to the deep convective adjustment timescale τ , and can also be interpreted as a convective coupling coefficient between the free troposphere and the boundary layer, with α = 0 corresponding to a perfect coupling regime and α = 1 to decoupling. The boundary-layer timescale τbl should satisfy the dimensional form [HU⋆−1 ] with U⋆ a characteristic speed. It is set equal to

Convection in global numerical weather prediction

25

the convective turnover timescale τc over land, assuming that the boundary layer adjusts to deep convective heat transport through the updraughts and downdraughts. Over water it is set to the horizontal advective timescale, assuming a quasi-homogeneous oceanic boundary layer in equilibrium τbl = τc τbl = Hu¯base bl

land water,

(4.16)

where Hbase is the cloud-base height and u¯bl the average horizontal wind speed in the subcloud layer. Setting ∂PCAPE/∂t = 0 in Eq. 4.4 enforces essentially a balance between the second and third terms of the right-hand side when the boundary layer forcing dominates, and an equilibrium between the first and third terms when the boundary layer is in equilibrium and the large-scale forcing dominates. Using Eq. 4.15 for the PCAPE consumption by deep convection and following the same procedure as used for deriving Eq. 4.12, the scaling for the deep convective cloud-base mass flux can be written as: PCAPE − PCAPEbl Mbase = ∗ Mbase τ



ztop

g M∗ T¯v

zbase

provided that Mbase ≥ 0, and where PCAPEbl = −τbl

1 T⋆



pbase

psurf

−1  ¯ ∂ Tv g + dz (4.17) ∂z cp

 ∂ T¯v  dp ∂t BL

(4.18)

for convection rooting in the boundary layer. For convection rooting above the boundary layer, PCAPEbl is set to zero. The closure is equivalent to relaxing PCAPE towards a value PCAPEbl instead of zero. It considers only the part of PCAPE that is due to free tropospheric production as long as the boundary layer is not in equilibrium (i.e., parcel environment-based closure: see Vol. 1, Ch. 11, Sec. 11.1). The closure might also be interpreted as providing a correction to the prediction of convective ensemble properties (mass flux) by simple parcel theory (CAPE). Importantly, the different factors in Eq. 4.17 mutually interact, and it will be shown that when integrated over a diurnal cycle, Eq. 4.17 roughly produces the same daily averaged mass flux and precipitation as Eq. 4.12. The scaling of Eq. 4.18 is consistent with the free tropospheric and energy conversion scaling suggested in Shutts and Gray (1999) when using the surface buoyancy flux instead of the integrated tendencies. In the following, the closure specified by Eq. 4.17 is referred to as NEW.

26

4.3

Part III: Operational Issues

Closure for shallow convection

A distinction between deep and shallow convection is made on the basis of the first-guess convective cloud depth. If the cloud extends over more than 200 hPa then convection is classified as deep, otherwise it is classified as shallow. This distinction is only necessary for the closure and the specification of the entrainment rates that are a factor of two larger for shallow convection (see Vol. 1, Ch. 10). In the case of very shallow convection, both PCAPE and the denominator in Eq. 4.17 tend to zero, and a closure based solely on boundary-layer equilibrium becomes appropriate. A closure for shallow convection is obtained by assuming a balance between the second and third terms on the right-hand side of Eq. 4.4 (i.e., a balance between the convection and the mean advection and other physical processes in the boundary layer), and replacing the tendency for PCAPE by the vertically integrated tendency of the moist static energy h:  pbase ¯   pbase ¯  ∂ h  ∂ h  pbase dp = g[Fh ]psurf = − dp, (4.19)  ∂t ∂t  psurf

CONV

psurf

BL

where Fh is the convective moist static energy flux. Assuming zero convective mass flux at the surface, the cloud-base mass flux is then obtained as: 

  1 pbase ∂ ¯h  up ¯ dp, (4.20) Mbase hbase − hbase = − g psurf ∂t BL

provided that Mbase ≥ 0. Instead of Eq. 4.20, a closure based on Eq. 4.14 could also have been used, although the operational performance of the model would be slightly degraded. The deep and shallow convective closures, Eqs. 4.12, 4.17, and 4.20 together with the entrainment and detrainment rates as given in Vol. 1, Ch. 10 take into account the vertical stratification and/or the boundarylayer tendencies. Together with the horizontally variable timescales τc and τbl , the closures provide a flexible framework so that the convective fluxes can adjust to varying synoptic and boundary conditions. 5

Diurnal cycle of convection

In contrast to equilibrium convection (see Vol. 1, Ch. 4), the forcing of non-equilibrium convection varies typically on timescales of a few hours (Davies et al., 2013; Jones and Randall, 2011; Yano and Plant, 2012). Nonequilibrium convection under rapidly varying forcing typically occurs when

Convection in global numerical weather prediction

27

either the upper-tropospheric forcing is strong and the convection is inhibited by a capping inversion, or the upper-level forcing is weak and the precipitating convection is driven by rapidly varying and strong surface heat fluxes. Forecasting non-equilibrium convection is challenging for models, and this is particularly true for surface-forced convection where the mesoscale adiabatic lifting/sinking couplet in the free troposphere is the response to and not the source of convective heating. The diurnal cycle of convection is probably the most prominent manifestation of non-equilibrium convection driven by the boundary layer. Numerous observational studies (e.g., Dai et al., 1999; Tian et al., 2005; Yang and Slingo, 2001; Zhang and Klein, 2010) and those based on cloud-resolving models (CRMs) (e.g., Chaboureau et al., 2004; Khairoutdinov and Randall, 2006; Schlemmer et al., 2011) have been devoted to the diurnal cycle of convection over land. The phase of the diurnal cycle can strongly vary on regional scales, though the general picture is that of a morning shallow convective phase, followed by a gradual onset of deeper convection, with rain rates peaking in the late afternoon to early evening. It has been found that the phase and intensity of precipitation mainly depends on the surface fluxes and lower to mid-tropospheric stability and moisture, but boundarylayer processes such as convergence, gravity waves, and cold pools also play a role in the onset and propagation of deep convection. It has been shown that CRMs with resolutions of order 2.5 km or higher are able to reproduce the observed diurnal cycle (e.g., Petch et al., 2002; Sato et al., 2008; Stirling and Stratton, 2012), but a strong resolution sensitivity exists in respect of both amplitude and phase for coarser horizontal resolutions when no convection parameterization is employed. However, Marsham et al. (2013) and Sato et al. (2009) have reasonably reproduced the observed phase in CRM-type simulations at 7 and 12 km horizontal resolutions, respectively. The same success in reproducing the observed diurnal cycle can generally not be reported for large-scale models. Indeed, numerous global and regional model studies (Bechtold et al., 2004; Betts and Jakob, 2002; Brockhaus et al., 2008; Clark et al., 2007; Dai et al., 1999; Langhans et al., 2013; Marsham et al., 2013; Slingo et al., 1992; Stratton and Stirling, 2012) and comparisons of CRMs with single-column models (Grabowski et al., 2006; Guichard et al., 2004) point to systematic errors in the diurnal cycle of precipitation when a convection parameterization scheme is employed, namely a too-early onset of deep convection with a diurnal cycle of precipitation that is roughly in phase with the surface fluxes. A notable exception is the successful simulations reported by Takayabu and Kimoto (2008). The

28

Part III: Operational Issues

diurnal cycle of non-precipitating shallow convection, however, can be realistically represented with a quasi-equilibrium closure for the boundary layer and a prognostic cloud scheme, as demonstrated for the IFS by Ahlgrimm and Forbes (2012). Various approaches have been taken to improve the representation of convection driven by surface fluxes. While Genio and Wu (2010), Piriou et al. (2007), and Stratton and Stirling (2012) focused on the entrainment rates, important work has also been undertaken on convective closure as reviewed in Yano et al. (2013). In particular, Gerard et al. (2009) and Pan and Randall (1998) accounted for convective memory through a prognostic closure for the updraught kinetic energy and/or updraught area fraction. Fletcher and Bretherton (2010), Mapes (2000) and Rio et al. (2009) proposed convective closures involving the convective inhibition (CIN) and/or lifting by cold pools, while a humidity-dependent closure was adopted in Fuchs and Raymond (2007). However, none of the above methods have thus far proved to be general and robust enough to replace, at least in the global numerical weather prediction context, the standard equilibrium closures for the CAPE or cloud-work function. The studies by Donner and Phillips (2003) and Zhang (2002) evaluated the quasi-equilibrium assumption for CAPE against observations, whilst recognizing findings by Raymond (1995) on different adjustment timescales for the free troposphere and the boundary layer. From those studies it was concluded that it should be possible to formulate a CAPE closure for the free troposphere under a quasi-equilibrium assumption that also holds for rapidly varying boundary-layer forcing. In the following it will be shown that the convective scaling of Eqs. 4.20 and 4.17 is indeed appropriate for global numerical weather prediction as it captures not only large-scale synoptically driven convection, but also non-equilibrium boundary-layer driven convection with its characteristic diurnal cycle, and the inland advection of winter-time convective showers. 5.1

Climatology

The global diurnal cycle of precipitation in the tropics is obtained from from a ten-year precipitation climatology from TRMM (Nesbitt and Zipser, 2003; Takayabu and Kimoto, 2008), while the climatological diurnal cycle of precipitation in the ECMWF model is obtained from an ensemble of one-year integrations. The simulations are forced by analysed sea surface temperatures, and use spectral truncation n = 159 (Δx = 125 km) with

29

Convection in global numerical weather prediction

91 vertical levels, and a timestep of 1 hr. Precipitation data from both the simulations and the observations are composited in hourly bins, and the diurnal amplitude and phase are computed from the first harmonic of a Fourier series. The diurnal amplitude (mmday−1 ) of the precipitation in the tropical belt from the TRMM radiometer is displayed in Fig. 15.10a. Maximum amplitudes reach around 10 mmday−1 over tropical land. Amplitudes from the model integrations using the CTL and NEW closures are displayed in Fig. 15.10b,c. Overall, the spatial distribution of the amplitudes is reasonably reproduced in the simulations, but the simulated amplitudes reach higher values, particularly over northern Amazonia. However, the simulated total rainfall over Amazonia appears realistic when compared to the GPCP2.2 dataset (not shown).

a

Amplitude (mm/day) TRMM radiometer

b

Amplitude (mm/day) CTL

c

Amplitude (mm/day) NEW

JJA climate

20N 0 20S 150W 120W

90W

60W

30W

0

30E

60E

90E

120E

150E JJA climate

20N 0 20S 150W 120W

90W

60W

30W

0

30E

60E

90E

120E

150E JJA climate

20N 0 20S 150W 120W

90W

60W

30W

0

30E

60E

90E

120E

150E

20 15 10 5 2.5 1 0.5 0.2 20 15 10 5 2.5 1 0.5 0.2 20 15 10 5 2.5 1 0.5 0.2

Fig. 15.10 Diurnal amplitude (mmday−1 ) of the precipitation in the tropical band during JJA as obtained (a) from a ten-year climatology of TRMM radiometer data (courtesy Yukari Takayabu and colleagues), and from an ensemble of annual IFS integrations at truncation n = 159 (Δx = 125 km) with (b) the CTL, and (c) the NEW closure.

30

Part III: Operational Issues

The corresponding phase of the diurnal cycle (LST) is displayed in Fig. 15.11. Maximum precipitation in the TRMM radar data (Fig. 15.11a) occurs over tropical land roughly in the late afternoon to early evening, though strong regional variations are present. In particular, in the TRMM climatology, convective rainfall over Amazonia occurs during the early afternoon, but may peak as early as 12:00 local time due to the high relative humidity and low stability in the lower troposphere (Betts and Jakob, 2002). In contrast, maximum precipitation over the tropical oceans occurs during the early morning. The CTL (Fig. 15.11b) provides a reasonable reproduction of the diurnal phase over water, but the convective precipitation over land generally peaks around 12:00 local time, except over Amazonia where it peaks during late morning. This systematic model error has not improved significantly in the IFS over recent decades (Bechtold et al., 2004; Slingo et al., 1992). However, a marked improvement is obtained with the NEW closure that shifts the diurnal cycle over land by 4–5 hours compared to CTL, and also improves the diurnal cycle in coastal regions (e.g., off the Central American and West African coasts, as well as off the Indian subcontinent, and over the Maritime Continent). Experimentation shows that the improvements over coastal regions are primarily due to a better representation of the convection generated over land and advected over sea, along with the associated subsiding motions, but the modified adjustment over sea via τbl also contributes. 5.2

High-resolution integrations

In addition to seasonal integrations, higher resolution daily three-day forecasts were performed for June, July, and August (JJA) 2011 and 2012 using n = 511 (Δx = 40 km) with 137 vertical levels and a timestep of 900 s. The forecasts were initialized from ECMWF’s operational analyses at n = 1279 (Δx = 16 km) with 91 levels. The forecasts are compared to the NCEP Stage IV composites (Lin and Mitchell, 2005) obtained from the combination of radar and rain gauge data (NEXRAD, hereafter) over the continental United States during summer 2011 and 2012, and German radar composites from the Deutsche Wetterdienst (DWD) for summer 2011. All forecast days have been used to compute the diurnal composites, amounting to 3 × 90 days of data for each JJA season. The amplitude and phase of the diurnal cycle of precipitation averaged over the summers 2011 and 2012 are depicted in Figs. 15.12 and 15.13 for the continental United States. Numerous previous studies have already

31

Convection in global numerical weather prediction

a

Phase (LST) TRMM radiometer

JJA climate

24 20

20N

16

0

12 8

20S

4 150W 120W

b

90W

60W

30W

0

30E

60E

90E

120E

Phase (LST) CTL

150E JJA climate

0 24 20

20N

16

0

12 8

20S

4

c

150W 120W 90W Phase (LST) NEW

60W

30W

0

30E

60E

90E

120E

150E JJA climate

0 24 20

20N

16

0

12 8

20S

4 150W 120W

90W

60W

30W

0

30E

60E

90E

120E

150E

0

Fig. 15.11 Same as Fig. 15.10, but for the diurnal phase (LST) of the precipitation. TRMM radar data has been used instead of the radiometer data. White shading is applied for areas where the amplitude of precipitation is below 0.2 mmday−1 .

described the diurnal cycle over this region (e.g., Dai et al., 1999; Tian et al., 2005). In summary, as is also evident from the NEXRAD data (Fig. 15.12a and Fig. 15.13a), the diurnal cycle over the continental United States is characterized by three distinctive regions: the Rocky Mountains, where convective activity peaks during the late afternoon; the Central Plains with predominantly night-time convection from propagating mesoscale convective systems triggered over the Rocky Mountains; and, finally, the Eastern United States and coastal regions with predominantly late afternoon convection and a particularly strong diurnal amplitude over the Florida peninsula. The CTL forecasts have quite a reasonable representation of the spatial variations in the amplitude (Fig. 15.12b), but underestimate the amplitude east of the mountain ridge and somewhat overestimate the amplitude in the coastal regions. The results with the NEW forecasts are rather

32

Part III: Operational Issues

a

Amplitude (mm/day) NEXRAD

JJA 2011-12 20 15 10 5

40°N 2.5 1 0.5 0.2 90°W

b

Amplitude (mm/day) CTL

JJA 2011-12 20 15 10 5

40°N 2.5 1 0.5 0.2 90°W

c

Amplitude (mm/day) NEW

JJA 2011-12 20 15 10 5

40°N 2.5 1 0.5 0.2 90°W

Fig. 15.12 Amplitude (mmday−1 ) of the precipitation averaged over JJA 2011 and 2012 for the continental United States from (a) NEXRAD, and from daily 72-hour forecasts at truncation n = 511 (Δx = 40 km) with (b) CTL and (c) NEW closure.

33

Convection in global numerical weather prediction

a

Phase (LST) NEXRAD

JJA 2011-12 24 22 20 18 16 14 12 10 8 6 4 2 0

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24 22 20 18 16 14 12 10 8 6 4 2 0

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Fig. 15.13

Same as for Fig. 15.12, but for the diurnal phase of the precipitation (LST).

34

Part III: Operational Issues

similar though slightly improve on the CTL. However, concerning the phase (Fig. 15.13) the NEW forecasts substantially delay the diurnal cycle by 4–5 hours compared to CTL so that the results more closely match the observations, though over the Eastern United States the diurnal cycle in NEW still precedes the observed cycle by up to 2 hours. To give an overview of the diurnal cycle in the high-resolution shortrange forecasts, the area-averaged diurnal rainfall composites are depicted in Fig. 15.14 for the Eastern United States and Germany and also for the central Sahel region, which has TRMM climatological data for comparison. The area-averaged representation shows that NEW has quite a good fit to the daytime and evening diurnal cycle of precipitation, shifting it by up to 6 hours compared to CTL. The late-night precipitation, however, remains underestimated in both NEW and CTL in spite of having the convection parameterization coupled to a five-species prognostic cloud scheme via the detrainment of convective condensate. The late-night precipitation deficit might be due to the missing representation of convective system dynamics, including spreading surface cold pools and predominantly upper-level mesoscale lifting during the night. Finally, over the Sahel (Fig. 15.14c), NEW realistically increases the precipitation in respect of CTL. As shown by Marsham et al. (2013), a correct phase representation of the diurnal cycle is particularly important in this region where the convective heating is a key driver of the meridional pressure gradient and the large-scale dynamics.

Fig. 15.14 Diurnal composites of area-averaged total precipitation (mmday−1 ) from CTL (black solid lines), and NEW (dashed lines) against observations (grey solid lines) for JJA 2011 (Europe) and JJA 2011 and 2012 for the other areas: (a) Germany [48◦ – 52◦ N,7◦ –14◦ E] using DWD radar, (b) eastern United States [30◦ –45◦ N,100◦ –80◦ W] using NEXRAD, and (c) central Sahel region [5◦ –20◦ N,10◦ –30◦ E] using TRMM climatological radiometer data.

Convection in global numerical weather prediction

6

35

Discussion

In the following the focus is on the central Sahel region (roughly a 2, 200 by 1, 700 km domain, as defined in Fig. 15.14c) for the analysis of the convective closure, and also to provide further evaluation of the convective heating and its dynamical response, using CRM and complementary satellite data. All model results are based on the high-resolution short-range forecasts discussed in the previous section. In addition to the forecasts, data assimilation cycles have been run with the IFS, providing a more direct comparison of model and data in space and time. The CRM data is from the Meso-NH limited-area model (Lafore et al., 1998) that was run between 10–25 June 2012 at 2.5 km grid-spacing daily for 24 hours over the central Sahel region. The CRM uses the same ECMWF n = 1279 analyses as initial conditions, as with CTL and NEW. In addition, the CRM open boundaries are updated every 6 hours from the analyses. 6.1

Diagnostics on closure

Diurnal composites of quantities related to the convective closure are illustrated in Fig. 15.15 for the period 10–25 June 2012; shown are the total-area averages (dashed lines) and averages only over the convectively active grid columns (solid lines) which are also labelled by the suffix c (convective). The quantities considered in Fig. 15.15 include the surface convective precipitation rate, the CAPE (Eq. 4.1), and the various terms involved in the closures of Eqs. 4.12 and 4.17. The surface convective precipitation rate is proportional to the product of the convective mass flux at cloud base and the updraught rain/snow content, although over land it is also strongly affected by evaporation in the subcloud layer. It is also approximately equal to the total surface precipitation as most stratiform precipitation evaporates before reaching the ground. Concerning the total area averages, one notices that for both CTL and NEW the convective precipitation, mass flux, and PCAPE′ are in phase. The forecasts barely differ during the night, but there is a clear 5-hour shift in the maxima in NEW in respect of CTL. CAPE (Fig. 15.15b) has been computed diagnostically for all grid columns from the mean thermodynamic profiles, while PCAPE is computed inside the convection scheme and therefore is non-zero only in grid columns with active convection. Though CAPE and PCAPE have different units and cannot be directly compared, their difference in magnitude mainly reflects the entrainment that is accounted

36

Part III: Operational Issues

a

b CAPE (J kg−1)

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0

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24

0.03 0.02 0.01 0

0

Fig. 15.15 Diurnal composites of convective closure-related diagnostics between 10–25 June 2012 over the central Sahel region: (a) convective precipitation; (b) CAPE; (c) PCAPE′ =PCAPE for CTL and PCAPE−PCAPEbl for NEW; (d) the compensating mass-flux term (Eq. 4.8); (e) the convective adjustment timescale τ , and (f) the cloudbase mass flux. Dashed lines denote total-area averages and solid lines and legends with suffix c denote averages over the regions with convective precipitation. Black and darkgrey lines correspond to CTL and NEW respectively, and light-grey lines correspond to the CRM. Precipitation statistics from the CRM are included in (a), with the precipitation rates per rain event scaled to account for the difference in resolution between the CRM and the IFS.

for in PCAPE. The main conclusion here is that CAPE shows an unphysical maximum at 10 LST in CTL, if taken as either a domain average or averaged over the convective regions, while its evolution in NEW roughly follows the evolution of the surface heat fluxes. The evolution of the convective-area averages CTLc, NEWc and CRMc (solid lines) is more revealing. Recall that the cloud-base mass flux (or convective precipitation) is proportional to PCAPE′ /τ divided by the subsidence term. In CTLc most closure-related quantities peak around 10 LST, vary only weakly during daytime, and precede the peak in domain-mean

Convection in global numerical weather prediction

37

mass flux and precipitation by about 2 hours. In contrast, the daytime amplitudes are important in NEWc, and the total-domain and convectivedomain averages are in phase. It will be shown later that the reason for this is that the convection in NEW is strongest at the end of the lowertropospheric moistening phase, while in CTL the convection is already active during the strong moistening phase. Interestingly, the convective precipitation rate per event (solid lines in Fig. 15.15a) is at its minimum during the day in CTLc, while NEWc produces precipitation rates per event that peak at around 30 mmday−1 during late afternoon, which is more in line with the observed rain rates from mesoscale convective systems in the Sahel region (Mathon et al., 2003). For comparison, in Fig. 15.15a the total-area mean (dashed grey line) and resolution-scaled rainy area mean precipitation (solid grey line) from the CRM have also been plotted, though data on the diurnal cycle from the CRM has to be interpreted with care (Langhans et al., 2012). The evolution of the total-area mean precipitation in the CRM during daytime is comparable to that of NEW, but peaks 1– 2 hours later. This shift is attributable to the growth in number and size of convective systems in the CRM during the late afternoon and their tendency to produce more surface precipitation through reduced evaporation; these features are more difficult to represent with a diagnostic convection formulation. The CRM also produces more precipitation during the night, which is consistent with radar observations (Fig. 15.14). Interestingly, the onset of convection around 12:00 local time, the average intensity, and its evolution during the afternoon (as measured by the rainy-area mean precipitation) compare reasonably well between NEW and the CRM. The low early-morning rain rates in the CRM are related to boundary-layer spin-up processes (to be discussed later). The low total-area mean precipitation rates in NEW in the late morning and early afternoon are the consequence of low values of PCAPE′ in connection with long adjustment times and moderate subsidence stabilization (Fig. 15.15a,c,d,e). It will be illustrated in the next section that the resulting convective heating keeps the free troposphere in a marginal stability regime. The increase in the convective adjustment time during late morning is produced by an increase in the cloud depth, while its decrease in the afternoon is caused by an increase in the mean updraught velocities. In conclusion, in non-stationary or non-equilibrium convection the various contributions to the forcing and stabilization interactively adjust. A successful simulation of the diurnal cycle requires most importantly a realistic formulation of the evolution of PCAPE′ which is dependent on the

38

Part III: Operational Issues

entrainment rates. The adjustment timescale (Eq. 4.13), which depends on PCAPE, is also an important factor for the representation of the spatial and temporal variability of convection. 6.2

Heating and moistening profiles

Composite diurnal cycles of the vertical distribution of the total heating rate (but excluding the radiative heating) and the total moistening rate are illustrated in Fig. 15.16. Using units of Kd−1 , these quantities are usually referred to as Q1 − Qrad and −Q2 , respectively. The heating and moistening rates due to adiabatic motions have also been added as contours in Fig. 15.16 in order to distinguish convective and dynamical forcings. One recognizes for both CTL and NEW (Fig. 15.16a,c) a distinctive phase with deep boundary-layer heating from 06:30 to 12 LST, followed by boundary-layer cooling and more elevated dry and shallow convective heating lasting until 17 LST. Boundary-layer moistening lasts until roughly 09 LST, followed by strong drying of the lower boundary layer, and dry convective and shallow convective moistening of the lower troposphere extending to or exceeding the 600 hPa level at 15–16 LST. In both CTL and NEW, during the afternoon, there is also a strong drying by mean advection around 850 hPa that has also been noticed in observational studies (Zhang and Klein, 2013). During the strong growth phase of the boundary layer from 10–17 LST, corresponding to a continuous growth of PCAPE′ in NEW (Fig. 15.15), the heating in the upper part of the boundary layer is in balance with the cooling due to adiabatic motions, but the upper troposphere is not in equilibrium. Indeed, the evolution of the upper-tropospheric heating profiles differs strongly between CTL and NEW. Whereas in CTL the mid- to upper-tropospheric heating of 5–10 Kd−1 from precipitating deep convection occurs around 13 LST, and therefore during the growth of the boundary layer, the strong deep convective heating in NEW occurs when the lower- to middle-troposphere has reached its maximum total heat content. Note that in NEW, modest midtropospheric heating and therefore stabilization occurs from around 11 LST onwards, and is due to cumulus congestus reaching heights of 500–400 hPa. The dynamic response to deep convective heating is a couplet of uppertropospheric cooling (lifting) and lower-tropospheric warming (subsidence), often called the stratiform mode. Through the quasi-geostrophic adjustment process it becomes effective a few hours after the convective heating. This dynamic cooling/heating couplet is particularly important for

39

Convection in global numerical weather prediction

a 200

b

Q1-Qrad (K/day) CTL 1

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Fig. 15.16 Diurnal composites of heating and moistening rates (Kd−1 ) between 10–25 June 2012 over the central Sahel region for (a) and (b) CTL, (c) and (d) NEW, and (e) and (f) from the CRM. Total heating rates minus radiation, Q1 − Qrad , and total moistening rates −Q2 are shaded. Solid contour lines (interval 1 Kd−1 ) denote cooling and drying rates due to adiabatic motions; dashed contour lines denote adiabatic heating and moistening.

the formation of mesoscale stratiform rain during the night. The uppertropospheric response in NEW is clearly delayed, and is stronger than in CTL, attaining values of −4 Kd−1 . Nevertheless, NEW still underestimates the night-time precipitation in respect of the observations (Fig. 15.15). A comparison of the heating and moistening profiles with CRM data (Fig. 15.16e,f) reveals that NEW produces a realistic diurnal cycle in phase and amplitude, including the shallow and congestus heating phase,

40

Part III: Operational Issues

though the latter is less pronounced in the CRM. The heating profiles in Fig. 15.16c,e are also in fair agreement with the observed cloud evolution during days with late afternoon convection, as reported by Zhang and Klein (2010). Interestingly, both in NEW and the CRM the maximum uppertropospheric heating of up to 10 Kd−1 occurs around 17 LST. However, the heating peaks at higher altitudes in the CRM (400 hPa compared to 500 hPa in NEW), extends over a larger depth, and maintains its amplitude during the early night-time hours as does the surface precipitation. The moistening rates −Q2 (Fig. 15.16d,f) are also in good agreement during the daytime. However, larger differences in the heating profiles between the CRM and the IFS exist in the early morning hours, which can be partly attributed to boundary-layer spin-up processes in the CRM. The dynamic response to the convective heating is also comparable in structure and intensity between the CRM and NEW. The main difference is that the dynamical cooling is somewhat weaker in the CRM, but occurs earlier (i.e., shortly after the maximum heating). The phase lag in the dynamical response between NEW and CRM becomes even more evident for the moistening profile (Fig. 15.16f). The reason for this phase difference is a tight coupling between resolved microphysics (condensation) and resolved dynamics (lifting) in the CRM, whereas with parameterized convection: (a) the heating rates Q1 and Q2 already contain a contribution from subgrid transport; and, (b) the resolved flow has to adjust in response to a subgrid heat source. Furthermore, the dynamical drying in the CRM extends down to the surface between 15 and 18 LST when also strong dynamical cooling occurs. This dynamical feature is a signature of resolved downdraughts and cold pools in the CRM. Generally, it appears that the structure and evolution of the convective heating and its dynamical response compares fairly well between the CRM and NEW, given the limited domain size of the CRM and its sensitivity to the parameterization of horizontal mixing.

7

Comparison with satellite data

A global picture of the improvement in the heating structure of NEW compared to CTL is presented in Fig. 15.17 using July 2012 as an illustration. This shows a reduction in root-mean-square (rms) error of the brightness temperatures when evaluating the short-range (first-guess) forecasts during the 12-hour assimilation window against the clear-sky brightness temperatures from the Advanced Microwave Sounder Unit A on board

Convection in global numerical weather prediction

41

sun-synchronous NOAA satellites. The satellites have different twice-daily overpass times, and the results are shown for two channels, sensitive to temperature over broad atmospheric layers around 500-1000 hPa and 250600 hPa. Clearly, NEW provides an improvement over CTL over most land regions with persistent active convection, and in particular in the middle to upper troposphere where the convective heating is strongest. The improvement of order 0.1 K is primarily a result of a reduction in the bias for the daytime overpasses. It is small in absolute values, but it is statistically significant, and has to be compared to the absolute rms error of the 12-hour forecasts that does not exceed 0.3 K. The areas of reduction in the shortrange forecast errors are consistent with the improvements in the diurnal cycle seen in the long integrations (Fig. 15.10). a

NOAA−18, channel 5

500−1000 hPa

b

NOAA−19, channel 5

500−1000 hPa

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250−600 hPa

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NOAA−19, channel 6

250−600 hPa

−0.08

−0.04

−0.01

0

0.01

0.04

0.08

Fig. 15.17 Root-mean-square error differences in clear-sky brightness temperatures (K) for July 2012 between NEW and CTL during the 12-hour window of the fourdimensional variational analysis, when evaluated against Advanced Microwave Sounding Unit (AMSU)-A sounding channels on board NOAA sun-synchronous satellites. The channels are representative for different atmospheric layers: (a) and (b) NOAA-18 and 19 channel 5 for the 500–1000 hPa layer, and (c) and (d) NOAA-18 and 19 channel 6 for the 250–600 hPa layer. The twice-daily overpass times are 03 LST and 15 LST for NOAA-18, and 01:30 LST and 13:30 LST for NOAA-19.

The impact of the convective closure on the forecasts can also be easily identified on infrared satellite imagery. As an illustrative example, Fig. 15.18 shows the observed 10.8 μm infrared satellite images over Europe

42

Part III: Operational Issues

at 12 and 18 UTC on 1 July 2012 from Meteosat 9 and the synthetic forecast images from the CTL and NEW short-range forecasts at T1279 (grid length 16 km). The satellite images show synoptically forced convection over western and central Europe, and surface-forced convection over the Balkans and the Atlas Mountains, a situation that is frequently observed during summer. Indeed, CTL overestimates the convection over Eastern Europe and the Balkans at 12 UTC and underestimates the convection in these areas and the Atlas Mountains at 18 UTC. In contrast, NEW clearly better reproduces the daytime convective evolution and also better represents the localization and structure of convection. Due to the higher CAPE values in NEW, the convection is more strongly localized with higher local convective rain intensities during late afternoon. A separate comparison of NEW with CRM data revealed that the higher daytime convective rain intensities in NEW are more realistic than the larger spread, but weaker rain events in CTL.

a

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Fig. 15.18 Infrared 10.8 μm satellite image over Europe on 5 June 2012 12 UTC from (a) Meteosat 9 channel 9, and from the 12-hour forecasts at n = 1279 (Δx = 16 km) with (b) CTL, and (c) NEW. (d) as (a) but for 18 UTC on 5 July 2012. (e), (f) as (b), (c) but from the 18-hour forecasts. All images are at resolution 0.2◦ .

Convection in global numerical weather prediction

8

43

Convection over the oceans and wintertime showers

So far, there has been little discussion on the effect of convective closure over the oceans. Comparing the CTL and NEW CAPE closures, the overall synoptic impact can be described as largely neutral, including the mediumrange forecasts of tropical cyclones and the representation of the Madden– Julian oscillation in seasonal integrations. However, there is a positive impact on the representation of convection and the diurnal cycle in nearcoastal areas. Of particular concern in numerical weather prediction is the inland advection of wintertime showers forming over the relatively warm sea. This is illustrated by Fig. 15.19, which shows the 24-hour precipitation accumulations over the British Isles and the near European mainland on 1 December 2010 as observed from ground-based radar, along with the 24-hour forecasts for CTL and NEW with n = 1279. The radar roughly extends to 2◦ off the coast. Nearly all precipitation accumulated as snow on the ground, reaching up to 20 cm, and was predominantly convective. Clearly, NEW reduces the unrealistically strong snowfall along the coast by up to 50% compared to CTL and more realistically moves the convective snowfall inland, bringing an extra 10 cm of snow (Fig. 15.19d). This is possible even with a diagnostic formulation of convection as the moist unstable air is advected inland, and the simulated convection is formulated so that it is allowed to depart from elevated layers. The main difference between NEW and CTL is the slower convective adjustment, avoiding a too-strong response at the resolved scales that leads to coastal convergence. An improved version of CTL for this particular case could also be obtained by increasing the convective adjustment time. However, tests showed that this would significantly degrade the general model performance, highlighting again the need for a more flexible and dynamically targeted formulation of the convective adjustment in numerical weather prediction.

9

Challenges

The studies on tropical variability and the diurnal cycle have clearly given an indication of the strong impact of the convection parameterization, and in particular its entrainment and closure formulations, on the performance of a global forecast system. However, these are only a few aspects, as for the sake of brevity it was impossible to address other meteorological aspects

44

Part III: Operational Issues

a

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Fig. 15.19 24-hour precipitation accumulations (mm) for 1 December 2010 over the British Isles and near European mainland from (a) radar observations on a 0.25◦ grid, and 24-hour n = 1279 (Δx = 16 km) forecasts with (b) CTL, (c) NEW, and (d) difference between NEW and CTL. The advection is represented in (d) by the mean 500–850 hPa wind. NEW improves the root-mean-square error against observations by 2% compared to CTL.

such as biases in the temperature and moisture fields and mean circulation, mid-latitude and tropical cyclone forecasts, and the effect of convectively generated waves on the stratospheric circulation. The positive message is that a convection parameterization using (i) realistic strong entrainment rates; and, (ii) a CAPE-dependent diagnostic closure under the assumption of free-tropospheric quasi-equilibrium that is subject to boundary-layer forcing, is able to reasonably represent convective systems and their impact on larger scales. However, it is also clear that a convective parameterization can only estimate the ensemble effect of convective transports and not realistically represent three-dimensional non-stationary subgrid-scale convective effects such as from propagating cold pools or up-gradient momentum transport from organized convection (Zhang and Wu, 2003). Experience shows that by efficiently remov-

Convection in global numerical weather prediction

45

ing grid-scale convective instability, convective parameterizations provide forecast skill and are still essential in current deterministic high-resolution medium-range weather forecasts and lower-resolution ensemble forecasts on the monthly and seasonal timescales. Taking the ECMWF high-resolution forecast and analysis system as a benchmark, the plans foresee a resolution upgrade from the current n = 1279 (16 km) operational resolution to n = 3999 (5 km) resolution in 2020 and n = 7999 (2.5 km) resolution in 2030. However, some experimental models like the Japanese NICAM model can already be run at 1 km resolution globally, though these simulations cannot be run in quasi-realtime and the results have not been optimized. The convective closure described here should enable a smooth transition from parameterized to more resolved deep convection as there is no longer a substantial discrepancy in the phase and location between parameterized and resolved convection. However, the intensity of the parameterized deep convection does not naturally diminish as resolution increases. For reasons of forecast performance (i.e., stronger stabilization with increasing forcing), the current adjustment timescale τ in Eq. 4.13 converges to the convective turnover timescale as resolution increases. One possible way of achieving vanishing parameterized tendencies for deep convection at high resolution would be to increase the resolution dependent factor f (n) in Eq. 4.13 to infinity from some resolution onwards, say, n = 2000 (10 km). By doing so, the so-called “small-area approximation” (cf., Vol. 1, Ch. 7, Sec. 6) in the mass-flux formulation is indirectly corrected for by a scaling, and the shape of the convective profiles is conserved. Instead, it might be necessary to recognize the limits of the small-area approximation more thoroughly, and replace the grid-mean values in the computation of CAPE by the actual values in the environment (cf., Ch. 19).

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Chapter 16

Satellite observations of convection and their implications for parameterizations J. Quaas and P. Stier Editors’ introduction: The increasing use of satellite data has been a major trend in atmospheric science over recent decades, and studies of convection are no exception. Moreover, further and new uses of satellite data are easy to anticipate in the coming decades. This chapter presents a balanced introduction and overview of this important area, focusing on the evaluation of models, especially through the use of satellite simulators. While rightly stressing the value of such evaluations, which can hardly be denied, the authors also point to various limitations of the approach, which can hardly be denied either.

1

Introduction

Parameterization development and evaluation ideally takes a two-step approach (Lohmann et al., 2007). Insight into new processes, and initial parameterization formulation should be guided by theory, process-level observations (laboratory experiments or field studies) or, if these are unavailable, by high-resolution modelling. However, once implemented into large-scale atmospheric models, a thorough testing and evaluation is required in order to assure that the parameterization works satisfactorily for all weather situations and at the scales the model is applied to. Satellite observations are probably the most valuable source of information for this purpose, since they offer a large range of parameters over comparatively long time series and with a very large (even global) coverage. However, satellites usually retrieve parameters in a rather indirect way, and some quantities (e.g., vertical wind velocities) are unavailable. It is thus essential for model eval47

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Part III: Operational Issues

uation (1) to assure comparability; and, (2) to develop and apply metrics that circumvent the limitations of satellite observations and help further learning about parameterizations. In terms of comparability, the implementation of so-called “satellite simulators” has emerged as the approach of choice, in which satellite retrievals are emulated, making use of model information about the subgrid-scale variability of clouds, and creating summary statistics (Bodas-Salcedo et al., 2011; Nam and Quaas, 2012; Nam et al., 2014). In terms of process-oriented metrics, a large range of approaches has been developed; for example, investigating the life cycle of cirrus from convective detrainment (Gehlot and Quaas, 2012), or focusing on the details of microphysical processes (Suzuki et al., 2011). Besides such techniques focusing on individual parameterizations, the data assimilation technique might be exploited, by objectively adjusting convection parameters and learning about parameter choices and parameterizations (Schirber et al., 2013). In this chapter, after introducing the available satellite data, considering their limitations and the approaches to account for these, then there will follow a discussion of observations-based process-oriented metrics that have been developed so far.

2

Satellite data

Satellite instruments measure radiances and obtain information about the Earth’s surface and the atmosphere by inverse radiative transport modelling (known as “retrieval”). Information is contained in the wavelength dependency of scattering or absorption by the Earth’s surface, or by particles or gases within the atmosphere. Further information is also available from changes in polarization. The radiation measured by satellites fundamentally can be of three types: (1) sunlight reflected by the Earth’s surface or by particles in the atmosphere (passive remote sensing in the solar spectrum); (2) radiation emitted by the Earth’s surface or by gases or particles in the atmosphere (passive remote sensing in the infrared or microwave spectrum); or, (3) radiation emitted by the satellite instrument and scattered back by the Earth’s surface or by particles in the atmosphere (active remote sensing).

Satellite observations of convection and their implications for parameterizations

49

Passive methods usually have the advantage that they may scan the atmosphere covering a broad swath, and that radiation in multiple wavelength bands may be measured, while active methods have the advantage that by measuring the time elapsed between emission and reception of the signal, a precise vertical sounding is possible. In terms of viewing angle, there is fundamentally a distinction between nadir viewing instruments that measure radiation from a column perpendicular to the orbit directly below the satellite, and limb viewing instruments that measure radiation perpendicular to the orbit, tangential to the Earth’s surface. Between these two types, all inclined angles are also possible and are in use for atmospheric science applications. Angles close to the limb are rarely applied to observe the troposphere due to the long paths sunlight takes through the atmosphere in such cases, which hampers localization of scatterers or emitters. As such, angles close to the nadir are usually used, and passive instruments often scan perpendicular to the orbit to cover a wide swath, with more oblique viewing angles at the edges, and viewing nadir at its centre. Several satellites in use or in development for atmospheric sciences use cameras that observe at different angles along the orbit, measuring radiation from each column at different angles and thus allowing for a stereoview and subsequently retrieval of additional information such as of the altitude of cloud and aerosol layers. Three kinds of orbits are in use for atmospheric science applications: (1) geostationary orbits at about 36,000 km above the equator. Such a satellite always sees one hemisphere, albeit at oblique angles for the remote parts of it. The most important advantage is that a column above a point on the Earth may be observed with high temporal resolution; (2) sun-synchronous polar orbits at lower altitudes (typically 600 to 800 km), observing each point on the Earth at approximately the same local time of the day for each overpass. The most important advantage is that almost the full globe is covered regularly, depending on the scanning across the track (swath width); and, (3) inclined low-Earth orbits, where, for example, the tropics are covered with higher frequency but polar regions are never overpassed. In the context of this set, the interest is mostly in remote sensing of clouds and their environment. In the following, the focus will thus be on retrieval methods for information about clouds. Firstly, a retrieval has to decide whether or not a cloud is present in the pixel. In the solar spectrum, this is usually based on the assumption

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that a cloud scatters more radiation than the Earth’s surface or a cloudfree atmosphere. The distinction is difficult over bright surfaces (desert, snow, sea-ice, sun-glint over ocean) or in the presence of thick, possibly deliquesced, aerosol layers. In the terrestrial spectrum, clouds usually emit less infrared radiation than the surface below a clear atmospheric column, since the temperature in the troposphere usually decreases with altitude. The exceptions occur in cases of clouds or within an inversion layer, and the distinction between these two is generally difficult within the atmospheric boundary layer. In any event, the detection of clouds becomes difficult if clouds are thin (small liquid- or ice-water paths), and the distinction from haze is also ambiguous (Koren et al., 2007; Stubenrauch et al., 2013). Beyond the detection of clouds, or equivalently the retrieval of cloud fraction1 , several other characteristics of clouds are of interest. Since the effect of clouds on radiation is so important for weather and climate, and since at the same time radiation is what satellites measure, often the radiative properties are characterized. The radiative effect in the solar spectrum is determined by the cloud albedo, which in turn is a function of cloud optical depth τc . For a layer in which the cloud particle size spectrum remains constant, the cloud optical depth can be written in terms of the liquidwater path2 (LWP) and the effective radius3 re , as τc = 3LWP/2re . The radiative effect in the terrestrial spectrum is determined by the emission, or brightness temperature TB . Since temperature is usually monotonically decreasing with altitude (increasing with pressure), cloud-top height ztop or cloud-top pressure ptop can be used to characterize TB . Cloud optical depth can be retrieved by the reflectance measured in a pixel, with a priori information about the reflectance of the Earth’s surface. If radiation is measured in narrow wavelength bands in the visible, near-infrared spectrum, then by choosing one wavelength where water has little effect and another where it is more absorbing, τc and re can jointly be retrieved if they are within certain ranges (Nakajima and King, 1990). Brightness temperature can directly be inferred from measured terrestrial radiation, and can then be converted into cloud-top altitude or pressure using ancillary information about temperature profiles. A classic example where the three quantities cloud fraction, cloud optical depth, and cloud1 Defined at a larger spatial or temporal scale as the ratio of pixels detected as cloudy to the total number of pixels. 2 The mass-weighted vertical integral of the liquid-water mixing ratio; similar relationships can be established for ice clouds. 3 The ratio of the third moment to the second moment of the cloud particle size distribution.

Satellite observations of convection and their implications for parameterizations

51

top pressure are used to characterize clouds is the International Satellite Cloud Climatology Project (ISCCP). ISCCP produces joint histograms of the cloud-top pressure and cloud optical depth (Rossow and Schiffer, 1991, 1999, see also Fig. 16.1).

Fig. 16.1 ISCCP joint histogram of cloud optical depth and cloud-top pressure defining 42 cloud classes that may be grouped into nine cloud classes loosely referring to the WMO-defined cloud types. If diagnosed by models, often a sub-visible cloud class (optical depths below 0.3) is additionally defined leading to 49 cloud classes in total. Note that the cloud optical depth is a measure for the solar-spectrum cloud radiative effect, while the cloud-top pressure is a measure for the terrestrial-spectrum cloud radiative effect. When analysing satellite or model data, the histogram is normalized such that the integral over it yields the total cloud fraction in the region and time period considered (after Rossow and Schiffer, 1991).

In the visible or infrared spectrum, radiation is scattered or emitted from the cloud top, or at least from shortly beneath it. In contrast, microwave radiation can penetrate clouds and thus yield information about the full cloud depth (e.g., a liquid-water path retrieval) and also information about precipitation. This is particularly used in active remote sensing with radar instruments (Stephens et al., 2002). When a large range of wavelengths is measured, such as by infrared sounders, height information can be retrieved by exploiting the exponential decrease of density with height in the atmosphere (Susskind et al., 2003, 2006). 3

Sampling of clouds by satellite observations

While sampling biases are a general issue in observational studies, the near global coverage of most satellite-based remote sensing instruments provides the advantage of large sample sizes, reducing random sampling errors.

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However, satellite resolution, overpass times, viewing geometry and collocation of instruments all contribute to sampling biases that often remain unquantified. 3.1

Orbital considerations

The majority of satellite-based cloud retrievals from space are based on instruments aboard sun-synchronous polar orbiting satellites. While this has retrieval advantages, such as consistent solar illumination angles and comparability due to near-identical local sampling times, such observations generally provide only one snapshot of the cloud or convective life cycle. In regions with distinct diurnal variability this can introduce significant sampling biases. For example, cloud-top heights retrieved from the Moderate resolution Imaging Spectroradiometer (MODIS) on board the Aqua satellite (early afternoon equatorial crossing time) are about 100 hPa higher over land as compared to MODIS aboard the Terra satellite (morning overpass time) (King et al., 2013). Tilted orbits with lower inclination, such as the 35◦ inclination orbit of the Tropical Rainfall Measuring Mission (TRMM), are designed to statistically sample the diurnal cycle of convection. However, it should be noted that such orbits provide a spatially inhomogeneous sampling. Depending on the swath width of the instrument, this can result in insufficient coverage to determine, for example, the diurnal cycle of precipitation from even three years of TRMM Precipitation Radar (footprint of approximately 5 km) data on spatial scales smaller than 12◦ and 4-hour temporal averaging (Negri et al., 2002). Such sampling issues are common to all active space-borne instruments, characterized by the gain of vertical resolution at the price of a narrow horizontal footprint. The high temporal resolution of state-of-the-art instruments in geostationary orbits, such as the Spinning Enhanced Visible and Infrared Imager (SEVIRI) and Geostationary Earth Radiation Budget experiment (GERB) instruments with 15 min resolution aboard the Meteosat Second Generation satellite centred over Africa, provide novel constraints on the observation of convection from space. In a Eulerian framework, the high time resolution allows detailed investigation of the diurnal cycle of convection. For example, Comer et al. (2007) used GERB to quantify the diurnal cycle of outgoing longwave radiation, which they decomposed into surface radiation and convective development. However, the high time resolution of geostationary instruments also allows the investigation of convection in a Lagrangian

Satellite observations of convection and their implications for parameterizations

53

framework. Williams and Houze Jr. (1987) developed algorithms to identify and track the evolution of convective systems from brightness temperature observations from geostationary satellites. Such tools have been extended to track and analyse stages of storm development (Zinner et al., 2008) and applied to derive long-term statistics of deep convection (Schr¨ oder et al., 2009). However, the superior temporal sampling of geostationary orbits comes at the price of inferior horizontal resolution due to the large distance from the Earth’s surface (36,000 km as compared to hundreds of km for a typical sun-synchronous polar orbit). SEVIRI has a resolution of 3 km for the standard channels at the sub-satellite point and one high-resolution 1 km channel in the visible. In comparison, the widely used polar-orbiting MODIS instrument has two bands imaged at a 250 m resolution at the subsatellite point with five bands at 500 m. Resolution has direct implications for the detection of convection: at resolutions larger than 1 km or so, individual convective cells frequently may not completely fill the instrument’s pixels, which causes the retrieval to be either rejected, introducing sampling biases, or to be erroneous due to the contribution of surface radiances. This effect is enhanced for slanted paths at higher solar and viewing zenith angles, introducing further sampling biases depending on the viewing geometry. Even for scenes with 100% cloud cover and fully filled instrument pixels the effects of 3D radiative transfer and preferred cloud field orientations can affect the retrieval of cloud properties (Kato and Marshak, 2009). Another important source of sampling errors is introduced in studies relating multiple satellite-retrieved cloud or aerosol properties. It is generally not possible to retrieve aerosol and cloud properties at the same location and time. Aerosol retrievals are strongly affected by cloud contamination, while cloud retrievals are strongly affected by partially cloud-filled pixels. As a consequence, both aerosol and cloud retrievals apply very conservative masks for their specific purpose, leaving an unsampled zone that is rejected as potential cloud by the aerosol retrieval and as potential clear sky by the cloud retrievals. This zone is estimated to cover about 20% of the globe, introducing significant sampling biases in both cloud and aerosol retrievals. 3.2

Satellite simulators

Satellite instruments measure radiances, which are not considered directly as model variables. Based on the measured radiances, and applying inverse radiative transfer modelling with ancillary information, cloud quantities

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are retrieved to enable comparison with models (see Sec. 2). However, as discussed above, assumptions, resolutions, and definitions are different between models and retrieval algorithms. Thus, the quantities as simulated in models and as retrieved from satellite observations differ not only due to model errors or measurement errors, but also due to inconsistencies. The aim of satellite simulators is to minimize these inconsistencies. The pioneering ISCCP simulator (Klein and Jakob, 1999; Webb et al., 2001) and later the comprehensive Cloud Feedback Model Intercomparison Project (CFMIP) Observational Simulator Package (COSP, Bodas-Salcedo et al., 2011) in three steps yield diagnostics comparable between retrievals and models: (1) A subcolumn sampler to bridge the gap in horizontal resolution. Since general circulation models (GCMs) usually work at resolutions of the order of 100 km, much coarser than satellite retrievals (1–10 km resolution), subgrid-scale variability needs to be taken into account for the cloud information. In most GCMs, the only subgrid scale information available stems from the fractional cloud cover and its vertical overlap, although more recent developments also take into account more information about cloud variability (R¨ais¨ anen et al., 2004). Based on this, subcolumns are generated that at each level are either cloud free or overcast, with homogeneous clouds (Klein and Jakob, 1999). (2) A mimicking of the satellite retrieval. This takes into account instrument limitations and sensitivities such as lower bounds for cloud detectability or the indistinguishability of multi-level clouds by passive instruments that retrieve a mid-tropospheric cloud-top pressure (e.g., for a thin cirrus overlying a boundary-layer cloud). It also generates, based on model assumptions, and possibly further ancillary assumptions, observables such as the lidar-scattering ratio or radar reflectivity. (3) The diagnostics of summary statistics such as the ISCCP joint histogram that contain relevant information at arbitrary scales. Other examples are the Contoured Frequency by Altitude Diagrams (CFADs) for lidar scattering ratio or radar reflectivity as a function of altitude. Often, the retrievals have to be prepared in such a way that they are consistent with what the satellite simulator produces: for example, for the lidarscattering ratio a vertical rather than horizontal averaging is used (Chepfer et al., 2010). Depending on the application, simplified simulators that just sample cloud-top quantities using an overlap assumption and take into account detection limits by satellite instruments may be sufficient (Quaas

Satellite observations of convection and their implications for parameterizations

55

et al., 2004). Note that the issue of vertical overlap is introduced alongside the computation of a fractional cloud cover since for partly cloudy skies in two vertical layers, a cloud might overlap with either other clouds or clear sky, which affects radiation and precipitation processes. Standard overlap assumptions are maximum overlap, in which the smaller cloud always is completely covered by the larger cloud, and random overlap in which the probability of overlap is just the cloud fraction percentage itself. Combinations of the two assumptions are also in use, possibly taking into account the separation of cloudy layers.

4 4.1

Observational constraints on models Convection and cloud-type identification

It has become clear that for a thorough observational constraint on cloud parameterizations, a separation of weather situations, or cloud regimes, is essential (Stephens, 2005). The regional separation of predominantly convective and stratiform regimes has often been used to investigate convection. In these studies, tropical clouds are frequently used synonymously with convection and no specific convective identification is applied (e.g., Yang and Slingo, 2001). However, convection also plays an important role in the extra-tropics and even in the tropics it is desirable to gain a better understanding of different convective elements, such as cores and anvil areas. Jakob and Tselioudis (2003) introduced an objective identification of cloud regimes based on clustering, making use of ISCCP histograms (Tselioudis et al., 2000). The k-means clustering algorithm provides objective mean histograms for each cluster (cluster centroids), which correspond (subjectively) to cloud regimes in the conventional sense, including shallow cumulus, anvil cirrus and deep convective systems. Such objective clustering of cloud regimes has been used to investigate the performance of climate models (Williams and Webb, 2009), highlighting limitations of current climate models in simulating specific cloud regimes such as a significant underestimation of the solar cloud radiative effect of deep convective clouds. It has also been used to assess aerosol–cloud interactions in a regime-based context (Gryspeerdt and Stier, 2012; Gryspeerdt et al., 2014).

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Part III: Operational Issues

Convective life cycles

Geostationary data, such as that available in the ISCCP dataset, are valuable for tracking clouds along their life cycle. Luo and Rossow (2004) used this data for tracking cloud systems, starting from each occurrence of a cloud characterized as “deep convective” by the ISCCP classification and following that through to convective outflow and anvil cirrus, which decays over time. This approach has proven useful in model analysis (Gehlot and Quaas, 2012) to identify deficiencies such as too little mid-tropospheric detrainment and too long lifetimes of cirrus from convective origins, possibly related to the ice crystal sedimentation parameterization. 4.3

Convective environment: Relative humidity variability parameterization

Clouds and convection are embedded in a thermodynamic environment, in which the distribution of relative humidity is one of the most important characterizations. From the perspective of a GCM, important fluctuations of relative humidity take place at subgrid scales. This subgrid-scale variability of relative humidity is in fact the basis of cloud parameterizations in GCMs (Smith, 1990; Sommeria and Deardorff, 1977; Sundqvist et al., 1989, see also Ch. 25) and may be coupled to the convection parameterization (Klein et al., 2005; Tompkins, 2002). Usually the temperature variability is neglected, and in clouds, saturation is assumed, so that the variability in relative humidity translates into variability in the total-water specific humidity. Unfortunately, vertically resolved satellite retrievals of humidity at high resolutions do not exist. However, from proxies such as the spatial distribution of the vertically integrated total-water path (Weber et al., 2011) or the critical relative humidity (Quaas, 2012), evaluations of available parameterizations of the subgrid-scale variability indicate shortcomings of current schemes. This is particularly relevant since processes such as precipitation formation strongly depend on subgrid-scale variability (Boutle et al., 2014; Weber and Quaas, 2012). 4.4

Convective microphysics

In convective parameterizations, microphysical processes are often very crudely parameterized (Tiedtke, 1989, see also Ch. 18). However, ample observations exist that may lead to more realistic parameterizations. This is particularly relevant in light of the multiple effects aerosols might have

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57

on convection, implying possible radiative forcings (Khain, 2009; Rosenfeld et al., 2008). The representation of such effects requires, as a necessary but probably not sufficient condition, that precipitation formation is formulated as dependent on droplet number concentrations, and that ice microphysics is included. In convective clouds, droplets first grow by water-vapour deposition onto them, leading to increased liquid-water content but constant droplet number concentrations. Once precipitation formation sets in, collision-coalescence processes lead to decreasing droplet concentrations at constant cloud-water content. Suzuki et al. (2011) demonstrated from satellite data that this transition between microphysical processes may be identified by investigating the radar reflectivity as a function of cloud droplet-effective radius, which follows a power law to the sixth power for condensational growth of particles, and to the third power for collision-coalescence growth. The transition between the processes is not well captured by many models. Aircraft observations imply that accounting for the non-linearity and droplet-size dependency of the precipitation formation process may be as simple as introducing a threshold in the cloud droplet-effective radius, by a critical effective radius of re ≈ 14 μm (Freud and Rosenfeld, 2012). Conditioning the frequency of occurrence of precipitation as retrieved by cloud radar on the cloud liquid-water path (Suzuki et al., 2010b), or comparing the frequency of occurrence of large radar reflectivities at low altitudes between models and observations (Nam et al., 2014), it is evident that models tend to produce light, warm rain far too often. One reason for this is that models often do not account for cloud variability. Once it is taken into account, the problem is much less severe (see above).

5

Outstanding issues and future perspectives

The purpose of the present chapter has been to outline the use of satellite data for convection studies, and to suggest its huge potential. Given the vast amount of data available from satellite measurements, its potentials can hardly be overemphasized. There are already scientific achievements in process understanding and advances in the modelling of clouds and convection based on satellite observations. However, there are also several limitations that we still need to overcome. Although satellite data is available at high resolution, it is usually only high in some particular sense: either temporally or spatially, and only hori-

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zontally or vertically. High temporal resolution from geostationary satellites comes at the expense of relatively coarse horizontal and virtually no vertical resolution. Active remote sensing allowing for vertical information comes at the expense of small horizontal and temporal coverage. No detailed, vertically resolved information at the horizontal resolution of O(100 m) is yet available that would be ideal for cloud and convection observations. Also, no vertically resolved information about relative humidity is available at horizontal resolutions better than O(20 km). We should also realize that satellites do not measure everything of interest. Mass flux information is only very indirectly available from satellites, and even future Doppler radar instruments as on the Earth Clouds, Aerosols and Radiation Explorer (EarthCARE) will only provide very coarse information (Kollias et al., 2014). In this respect, we should not naively expect that satellite technology will replace the conventional sounding network in the near future. The important role of conventional soundings has already been emphasized in several places in this set (e.g., Vol. 1, Ch. 5, Sec. 2.2). As such, development of methods and metrics that permit insights into processes and allow for model evaluation and improvement remains a scientific topic in its own right, in order to exploit the wealth of satellite data that is available. 6

Further reading

For further reading on the recent development of satellite technologies, Rosenfeld et al. (2014) and Stephens et al. (2002) provide good starting points on microphysics and cloud structures, respectively.

Chapter 17

Convection and waves on small planets and the real Earth

P. Bechtold, N. Semane, and S. Malardel Editors’ introduction: The relentless increase of available computing power now allows for simulations to be performed (at least in a research context) in which both large-scale dynamics and convective dynamics can be simulated. Although often fascinating, such simulations do have their limitations, and remain far from able to properly resolve the dynamics and microphysical processes within individual convective clouds. Since the integrations are feasible but computationally expensive, the number of simulations which can be performed is very limited and it is not possible to explore the full range of behaviours and the complexity of these large-domain explicit-convection models, which include many competing processes, leading to difficulties in identifying causal links between the convection and the large-scale circulation. Therefore, a relatively cheap simulation framework which nonetheless allows for explicit-convection/large-scale interactions may prove a valuable tool. This chapter describes ongoing work to develop such a tool at ECMWF.

1

Introduction

Global weather forecasting and climate models employ increasingly high horizontal and vertical resolutions. For example, the European Centre for Medium-Range Weather Forecasts (ECMWF) plans foresee a horizontal resolution upgrade to T2047 (10 km) in 2015, moving towards T3999 (5 km) around 2020. Exploratory forecasts at T7999 resolution (equivalent to about 2.5 km horizontal grid length) have already been presented by Wedi et al. (2012). The particular challenge at such high resolutions 59

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is that deep convective motions become gradually more resolved. Consequently, the optimal partitioning in the model between resolved and subgrid vertical motions and condensation processes has to be reconsidered (cf., Ch. 19). It is also desirable to estimate the horizontal resolution beyond which the global forecasting system could eventually be run without a deep-convection parameterization whilst improving the forecast skill on the medium to monthly timescales for both mid-latitude and tropical regions. At the high resolutions being envisaged, not only the simulations but also the data storage and processing are extremely costly, rendering extensive experimentation and developments impractical on current computer systems. In order to overcome these limitations, Wedi and Smolarkiewicz (2009) set up a prototype reduced-planet version of the atmospheric model used in the Integrated Forecasting System (IFS) without model physics. This enabled the development and efficient evaluation of the non-hydrostatic dynamical core of the model. The small-scale prototype approach is extensively used in the engineering community where, for example, the aerodynamic properties of a small-scale object are experimentally determined in a wind tunnel, and the performance data can then be up-scaled to the true-sized object via a numerical model that is written in non-dimensional form. The goal is to develop a scaled version of the IFS with full physics that not only allows the model to be applied to planets of different size and gravity, but also faithfully reproduces the general circulation of the Earth. The final goal will be to have a system that permits deep convective motions while maintaining a realistic large-scale circulation. Such a system, which necessarily involves some approximations, was pioneered by Garner et al. (2007), Kuang et al. (2005), and Pauluis et al. (2006), but has never been developed for a complex numerical weather prediction system. This chapter presents step by step, with increasing complexity, the various reduced-planet configurations that have been used, and illustrates what each configuration can and cannot do.

2

Aqua planet and scaling laws

An ideal prototype is the aqua planet, which is a standard configuration used in the present chapter. In this configuration, as proposed by Neale and Hoskins (2000), the whole planet is covered by water and the sea surface temperatures (SSTs) are specified. For an Earth-like simulation, typ-

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ically a zonally symmetric SST distribution is chosen peaking at 27◦ C at the equator and decreasing to 0◦ C beyond 60◦ latitude. Furthermore, the Sun is fixed over the equator, and thus there is no seasonal cycle. The advantage of the aqua planet is that it removes complications due to landsurface/atmosphere interactions such as orographic drag, soil hydrology, and the diurnal cycle over land. Also, it avoids the complications of scaling the orography and height of the vegetation for planets of different size. The reduced-planet system has the following scalings: (1) Horizontal length scale. The Earth’s radius Ra is divided by a factor γR , and consequently the horizontal length scale L is also expected to be reduced by a similar factor. (2) Vertical length scale. The vertical length scale Z is conveniently reduced by a factor γg by increasing the gravity by the same factor, recognizing that the scale height H of the atmosphere is given by H ≈ RT /g involving the mean temperature T , the gas constant R, and the gravity g. (3) Timescale. The timescale t is reduced by a factor γΩ by increasing the rotation rate Ω of the planet by a factor γΩ , which is equivalent to reducing the length of the day. 2.1

Scaling parameters

Scaling of the external planetary parameters, planetary radius Ra , gravity g, and rotation speed Ω by a factor γ leads to the scaling of horizontal length L, height scale Z, and timescale t as follows: Ra′ = Ra /γR g ′ = gγg

→

→ L′ = L/γR RT → Z ′ = Z/γg H′ = gγg

Ω′ = ΩγΩ

→

(2.1)

t′ = t/γΩ .

Here, the aspect ratio r = H/L is also introduced. 2.2

Non-dimensional characteristic numbers

Non-dimensional numbers can be derived that include the Rossby number Ro, the Richardson number Ri, and the Lamb parameter La, involving also the Brunt–V¨ ais¨ al¨ a frequency N and the internal gravity wave phase speed c.

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g Δθ ; c = NH θ H U N 2H 2 Ro = ; Ri = 2ΩRa U2 2 2 4Ω Ra 1 , = La = Ro2 Ri c2 N2 =

(2.2)

where θ is a reference potential temperature and Δθ is the change of the potential temperature over the height H. Consequently, choosing various values for the factors γR , γg , and γΩ allows a choice of configurations. The full aqua planet has γR = γΩ = γg = 1. While applying these scalings, the atmospheric motions on the reduced planet can be faithfully reproduced in respect of the full planet if two nondimensional numbers, namely the Rossby and Richardson numbers, are kept constant, and if the system is adiabatic. The Rossby number measures the ratio between the acceleration and the Coriolis force or rotational acceleration, while the Richardson number measures the relative importance of the buoyancy acceleration to the acceleration due to vertical advection (i.e., the importance of convection through the Brunt–V¨ ais¨ al¨ a frequency). Finally, the Rossby and Richardson numbers can be cast into one single parameter, the Lamb parameter, which can be interpreted as the ratio between the planet’s rotational speed and the internal gravity wave phase speed. Interestingly, the Lamb parameter does not involve the horizontal wind speed, which is an internal parameter. This means that when conserving the Lamb parameter, the wind field on the reduced planet is the same as on the full planet. More precisely, the Lamb parameter characterizes all of the linear global atmospheric waves that are described by Laplace’s tidal equation under hydrostatic balance. The Rossby number, in turn, measures the non-linearity of the system. When the system is diabatic, it is also necessary to take into account the characteristic timescale associated with diabatic processes, as discussed below.

3

Isolated thunderstorm on a reduced planet

The first example is the simulation of an isolated split thunderstorm that evolves into a supercell thunderstorm. This is achieved by initializing the model with a convectively unstable sounding with rotational wind shear

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that is the same everywhere. The non-hydrostatic version of the IFS is run for two hours with full physics, but without deep-convection parameterization at truncation T511. For this experiment, the Earth’s radius is scaled by a factor γR = 12.5, giving an effective horizontal resolution of about 3 km, but the vertical length scale and the timescale are not changed (i.e., γg = γΩ = 1). This means that convective systems have the same size as on the real planet, while occupying a larger portion of the reduced planet. Figure 17.1a shows the temperature and wind at the first model level along with the three-dimensional distribution of hydrometeors (i.e., cloud droplets, cloud ice, rain, and snow). Also the accumulated surface precipitation and low-level wind is shown in Fig. 17.1b. The results can be directly compared to corresponding simulations with limited-area models because, given the small size of convective systems, the Coriolis effects due to the planet’s rotation can be neglected. From these results it can be concluded that the reduced planet version of the IFS can be used to simulate intense isolated thunderstorms. a

b

Fig. 17.1 Non-hydrostatic simulation of a split thunderstorm after 2 hr at resolution T511 with γR = 12.5 (3 km resolution) and γΩ = γg = 1 in an environment with rotational wind shear. The 3D box dimensions are 200 × 200 × 20 km. (a) Temperature (colour shading) and wind at the first model level, as well as the 0.5 gkg−1 iso-surfaces of cloud droplets (blue), cloud ice (cyan), rain (green), and snow (white). (b) Accumulated surface precipitation during the first 2 hr, and low-level winds. The contour interval is 0.5 mm. The vortex to the right (right mover) is characteristic of vortices that in nature favour the formation of tornadoes.

In this somewhat idealized study there has been a saving of roughly 2 (about 156) in computer time and data storage in respect of a factor γR the actual 3 km full-planet version of the model. However, by only scaling

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the planet radius, it is not possible to realistically represent the large-scale circulation or the interaction between the convective and synoptic-scale motion systems. 4

Dry baroclinic waves on the super-rotating planet

The synoptic scales on the reduced planet can be faithfully reproduced (in a linear regime) in respect of the full planet if the Lamb parameter is conserved. This means that the rotation rate of the reduced planet has to be increased accordingly, so that both the characteristic wavenumber and frequency of the planetary Rossby waves remain unchanged. To illustrate the scaling of dry synoptic waves, a dry baroclinic test case following Jablonovski and Williamson (2006) is employed, where the model is initialized with a meridional temperature gradient and a geostrophically balanced wind field with a strong zonal upper-level jet. The hydrostatic dynamical core of the model is then run without moist physics for ten planetary rotations at truncation T159, with the vertical length scale unchanged (γg = 1) and the other scaling factors as follows: (1) γR = γΩ = 1, which correspond to a 125 km horizontal resolution and rotation rate of 24 hr or 86400 s (Fig. 17.2a). (2) γR = γΩ = 1000, which correspond to a 125 m horizontal resolution and rotation rate of 86.4 s (Fig. 17.2b). The results in Fig. 17.2 show that the synoptic waves are quasi-identical in these simulations, indicating that the scaling is correct. However, nothing has been gained from the reduced planet simulations in terms of computer time and data storage. In respect of the full-planet simulations the length of the integrations is still ten rotations, but the length of the day and the model timestep have to be reduced accordingly. Furthermore, no model physics has been used in these planetary simulations. Had it been so, Figs. 17.2a and 17.2b would no longer be identical. The reason being that, as the timescale of the synoptic forcing has been shortened, the physics also has to be accelerated. This is discussed in the next section. 5

The global circulation and the shallow atmosphere

The most effective way to accelerate the physics consistently with the shortening of the synoptic-flow timescale is by scaling the gravity. This leads to

Convection and waves on small planets and the real Earth

65

a γR = γΩ =1 50.0 m/s 70°N 60°N 50°N 40°N 30°N 160°E 180° 160°W140°W120°W100°W 80°W 60°W

b γR = γR =1,000 70°N 60°N 50°N 40°N 30°N 160°E 180° 160°W140°W120°W100°W 80°W 60°W 932 940 950 960 970 980 990 1000 1010 1020 1025 Fig. 17.2 Baroclinic wave train with surface pressure contours (interval 10 hPa) and wind field, in hydrostatic simulations at resolution T159 with γg = 1 and (a) γR = γΩ = 1 (125 km resolution and 86400 s rotation rate) and (b) γR = γΩ = 1000 (125 m resolution and 86.4 s rotation rate) after ten rotations of the planet. (a) corresponds to a real time of 864000 s and (b) to a real time of 864 s.

a system where the horizontal length, vertical length, and timescales are all reduced by the same factor (i.e., γR = γΩ = γg = γ). This is referred to as SASE (Shallow Atmosphere Small Earth). In this system with reduced scaled height, the diabatic forcing is naturally increased through radiative and surface heating as is the response through stratiform heating and convective heating and transport. However, a few precautions have to be taken in the physics concerning the internal constants that have been given absolute values instead of generally scaled

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values. In the model this affects the microphysical timescales which have to be scaled by γΩ . Furthermore, the turbulent length scale, and the entrainment and detrainment rates in the convection parameterization have to be rescaled by a factor γg as the scale height of the atmosphere is changed. With these scalings the system not only reproduces the dry baroclinic wave case in Fig. 17.2 but also the full moist circulation as discussed next. In order to study the general circulation and climate for the different aqua planet configurations, it is first necessary to create a balanced state that is obtained from a six-month integration starting with arbitrary initial conditions (e.g., any date from the European Interim reanalysis project). A four-member ensemble is then generated by perturbing the balanced state that serves as the initial condition and integrating the model for one year at spectral truncation T159. Figure 17.3a shows the annual mean precipitation (mmday−1 ) on the real Earth as obtained from the Global Precipitation Climatology Project (GPCP) version 2.2 dataset. The other panels in Fig. 17.3 show the results for the full and reduced aqua planet using the deep-convection parameterization but different scalings. The full aqua planet with γR = γΩ = γg = 1 (Fig. 17.3b) exhibits a distinctive tropical band with a precipitation rate of about 11 mmday−1 , and equatorially symmetric mid-latitude storm tracks. Its climate is in qualitative agreement with what is observed for the real Earth (Fig. 17.3a). The reduced aqua planet with γR = γΩ = 8 and γg = 1 (Fig. 17.3c) produces a split inter-tropical convergence zone and mid-latitude storm tracks that are shifted too far poleward. The scaling in Fig. 17.3c is similar to that in Fig. 17.2b in that only Ro is conserved but not Ri or La. It is therefore not an accurate scaled version of the full aqua planet. Using the SASE system with γR = γΩ = γg = 8 (Fig. 17.3d), an accurate small-scale version of the climate on the full planet is obtained. The convectively-coupled waves in the tropical band are analysed in Fig. 17.4 using wavenumber frequency diagrams of the outgoing longwave radiation. Satellite observations (Fig. 17.4a) reveal the dominant tropical wave types which are the eastward-propagating Kelvin waves, the westward-propagating equatorial Rossby waves and the east- and westwardpropagating inertia-gravity waves. The Madden–Julian oscillation is also apparent as a distinct mode in the wavenumber 1–2 and period 20–60 days band. The dominant wave types are reasonably reproduced for the full aqua planet simulations (Fig. 17.4b), but the amplitudes are larger than those observed because these waves can freely circumnavigate the equator without being disturbed by land effects. However, the tropical wave spectra,

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Fig. 17.3 Annual-mean daily global precipitation (mm). (a) Observations from the GPCP2.2 precipitation climatology dataset. Also shown is a one-year integration at T159 of a four-member ensemble with deep-convection parameterization using: (b) the full aqua planet with γR = γΩ = γg = 1; (c) the reduced aqua planet with γR = γΩ = 8 and γg = 1; and, (d) the SASE system with γR = γΩ = γg = 8.

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and in particular the Kelvin waves, are heavily distorted for the reduced aqua planet with γR = γΩ = 8 and γg = 1 (Fig. 17.4c), but faithfully reproduced with the SASE system using γR = γΩ = γg = 8 (Fig. 17.4d). The SASE system is shown to provide the correct scaling of the full planet and therefore allows the model to be applied to planets of different size and gravity. Unfortunately, nothing has been gained in terms of computer time, and the final goal of resolving deep convection has still not been achieved. Indeed, in the SASE system the scale of the convection has also been reduced. This follows from observational evidence showing that the horizontal scale of convective clouds is related to the scale height. The timescale of convection scales as τ = H/W = H/(rU ) = L/U , with aspect ratio r = H/L and vertical velocity W = rU . It follows that for SASE the time and length scales of convection (τ and H) have been reduced, but the aspect ratio r and the vertical velocity W are the same as for the full aqua planet simulation.

6

Towards global resolved convection

The strategy to resolve deep convection on the reduced planet, while maintaining a realistic interaction between the convection and the large-scale circulation, is to reduce the gap between the convective and large-scale motions. This can be achieved either by reducing the scale of the synoptic circulations, therefore bringing them closer to the convective scales, or by increasing the scale of the convective motions. The former approach was proposed by Kuang et al. (2005) and called DARE (Diabatic Acceleration and Rescaling). In this system γR = γΩ = γ and γg = 1 with rescaling of the external forcings. In addition, there is also the alternative Deep Atmosphere Small Earth (DASE) approach, where DARE is combined with the scaling of the SASE system giving γR = γΩ = γ and γg = γ −1 . This means that the scale height of the atmosphere is increased with a consequent increase in the horizontal scale of convective motions. DARE requires the rescaling (acceleration) of the external forcing including radiation and surface fluxes in order to increase the forcing of convective-scale motions. A key aspect of DARE is that the vertical scale remains unchanged. Thus, even though the convection is driven more strongly by a factor γ, the natural horizontal scale of the convection (H) is unchanged, and hence is a factor γ closer to the synoptic scale (L). Yet, DARE requires non-hydrostatic simulations to take advantage of the

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reduced synoptic/convective scale separation. MacDonald et al. (2000) defined a quasi-non-hydrostatic parameter r2 , which is the square of the aspect ratio. The non-hydrostasy increases by increasing r2 (Garner et al., 2007). DARE increases r2 by a factor γ 2 . Yet, this increase is not large enough to allow a transition into the resolved range. While DASE increases the convective scales, and increases the aspect ratio and vertical motions by a factor γ 2 , it also reduces the synoptic scales. The convective and synoptic motions are therefore closer in scales, and can be resolved simultaneously over a timestep equivalent to the one used in a simulation γ times wider in the horizontal direction, γ times narrower in the vertical direction and γ times longer in time. The quasi-non-hydrostatic condition becomes sufficiently large to allow a transition into the resolved range. The DARE and DASE systems include the physics scaling of the radiative flux, surface heat and momentum fluxes, microphysical timescales, and fall velocity of precipitation. Figure 17.5 displays the annual-mean precipitation rate, similar to that shown in Fig. 17.3, but for hydrostatic integrations without convection parameterization for: (1) The full aqua planet at truncation T159 and T1279 (Figs. 17.5a and 17.5b respectively). (2) The reduced aqua planet at T159 with γ = 8 using the DARE and DASE configurations (Figs. 17.5c and 17.5d respectively). The T159 integration without convection parameterization (Fig. 17.5a) greatly overestimates the equatorial precipitation compared to the control run with deep-convection parameterization (Fig. 17.3b). But, when increasing the resolution to T1279 (Fig. 17.5b), the results without convection parameterization become comparable to the control. If the DARE and/or DASE configurations at T159 (with γ = 8 using the same timestep as the T1279 integration) are able to reproduce the T1279 results, then we have a system that allows a saving in computer time of order γ 3 (γ 2 from the size of the horizontal grid and γ from the timestep). However, DARE (Fig. 17.5c) essentially only reproduces the results of the T159 integration (Fig. 17.5a). This was expected as we are not yet in the non-hydrostatic regime. In contrast, the results with DASE (Fig. 17.5d) become more comparable to the T1279 full planet integration (Fig. 17.5b) and also to the T159 integration preformed with deep-convection parameterization (Figs. 17.3b and 17.3d).

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Fig. 17.5 Annual-mean daily global precipitation (mmday−1 ) for hydrostatic integrations with deep-convection parameterization for the full aqua planet (γR = γΩ = γg = 1) at (a) T159 and (b) T1279, and the reduced aqua planet at T159 with (c) DARE (γR = γΩ = γ = 8 and γg = 1) and (d) DASE (γR = γΩ = γ = 8 and γg = γ −1 ).

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As well as considering the distribution of precipitation, it is worthwhile examining whether DASE can produce realistic tropical wave spectra. Figure 17.6 displays the tropical wave spectra for the simulations illustrated in Fig. 17.5. In the T159 integration without convection parameterization, as well as in DARE, the wave spectra are broad and noisy (Figs. 17.6a and 17.6c). However, at T1279 (Fig. 17.6b) the wave spectra become comparable to the integration with deep convection (Fig. 17.4b). The dominant wave types are also reasonably reproduced with DASE (Fig. 17.6d). The results given in Figs. 17.5 and 17.6 show that the new, fully scaled system DASE is indeed a step forward in resolving convection as it is able to mimic both the mean climate and the wave motions of the T1279 full planet integrations. 7

Perspectives

A scaled version of the IFS has been developed, which can be applied to planets of different size and gravity. Also, alternative system DASE offers the potential to efficiently mimic resolved deep convection. This constitutes ground-breaking research, and the prototype version will be available to users of the public version of the IFS (OpenIFS). There are, however, evident limitations in the method in that it is not possible to rescale the microphysical processes in a way that is consistent both with the small- and large-scale processes. It is estimated that values of γ in the range of 4 to 10 might provide sufficient physical realism, while allowing efficient high-resolution experimentation. In particular, the intention is to study the transition from parameterized to resolved convection in the 10 km to 1 km resolution range, and the effects of convectively generated gravity waves on the circulation in the stratosphere and mesosphere.

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Chapter 18

Microphysics of convective cloud and its treatment in parameterization

V.T.J. Phillips and J.-I. Yano Editors’ introduction: The description of microphysical processes in convection can be considered as encompassing three broad areas: the initiation of particles, the growth of particles, and the dependencies of these processes on radiative and dynamical properties. This chapter reviews the key processes in the context of convective clouds, drawing on laboratory, observational, and cloud-modelling studies. This overview of the important processes may be contrasted with the actual microphysical representations that have been used within mass-flux parameterizations of deep convection to date. The descriptions within convection parameterizations are relatively simple; the authors focus on describing which processes have been treated so far, and some of the prerequisites that would have to be met in order to incorporate more complex microphysical descriptions.

1

Overview

Ascent is faster in vertically developed clouds created by the action of the buoyancy force (convective clouds) than in layer clouds (stratiform or cirriform clouds). In fact, vertical velocity typically differs by orders of magnitude between stratiform and convective clouds. This in turn implies contrasting timescales of ascent of cloudy parcels through the clouds: several hours for deep stratiform clouds as compared to a few tens of minutes for deep convective clouds. Convective clouds have sometimes been defined as cloudy regions where maximum vertical velocities aloft exceed about 3 ms−1 (e.g., Xu, 1995), such that they are capable of supporting large precipitation particles that fall with similarly large fall speeds. In reality 75

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there is no single universal threshold of vertical velocity separating convective and stratiform clouds, with the much weaker buoyancy force on the large scale causing stratiform cloudy ascent in systems of baroclinic or symmetric instability. The convective ascent, being faster, causes convective precipitation to be more intense, but also more short-lived, than stratiform precipitation. It also causes turbulence on smaller scales, and drives the three-dimensional downscale cascade of turbulent kinetic energy. The microphysics of convective clouds is quite distinct from that of stratiform and cirriform clouds. The supply of cloud liquid from condensation is proportional to vertical velocity generally, while its removal occurs on a timescale governed by the precipitation burden. Hence, the average cloud liquid is typically greater in convective clouds than in stratiform ones. This causes precipitation production to be more intense. Ice precipitation > 0.2 mm) in convective clouds tends to be heavily rimed, such as hail or (∼ graupel. “Rime” is the layer or layers of ice from supercooled cloud liquid frozen onto large ice particles, and rimed precipitation is favoured by the large values of supercooled cloud liquid typical of convective updraughts. For the same reason, larger raindrops fall from convective clouds and the total rain rate (mmh−1 ) at the ground is greater than for stratiform clouds. Much rain is commonly formed by accretion of cloud liquid. > 0.2 mm) without much rime, By contrast, snow, ice precipitation (∼ reaches the ground after being generated by long-lived mixed-phase nimbostratus (deep stratiform) clouds. For example, one almost never observes lightning initiated in any cloud producing heavy snowfall on the ground. Lightning is generally initiated only in convective clouds and is caused by charge separation in collisions between large (rimed) and small ice particles in mixed-phase conditions. Severe weather involves deep convective clouds, which cause floods, hail, lightning, and strong gusty winds. The greater rates of production of condensate by the faster ascent in convection causes greater rates of phase change in the cloud between the vapour, liquid, and ice phases than in layer clouds. Thus, rates of latent heating and cooling are stronger in convection, invigorating downdraughts and updraughts through impacts on the buoyancy. Microphysical processes involving diffusional growth and coagulation of hydrometeors are at the nexus of this set of linkages with the cloud dynamics. Contrasts in buoyancy between different parts of the updraught and downdraught cause turbulence and gustiness. For example, the gust front is at the edge of the cold pool of downdraught air near the ground below cloud, being especially turbulent. Indeed, the most detailed cloud

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models are able to analyse the impacts from changes in aerosol loadings on cloud dynamics through the response of profiles of latent heating and cooling of the various phase changes (e.g., Benmoshe and Khain, 2014; Khain et al., 2005; Phillips et al., 2007b). When the cloud base of convection is cold near the freezing level and aerosol concentrations are high (e.g., over the US High Plains in summer when the troposphere is dry), then coalescence may be inactive due to scarcity of vapour, which restricts condensation onto cloud droplets: they are too small to collide. Consequently, cloud ice grows by vapour diffusion to become snow crystals. This is the “ice-crystal process” for precipitation production. The snow is then converted to heavily rimed precipitation (graupel, hail). In deep stratiform clouds, such as mixed-phase nimbostratus cloud, the ice-crystal process leads to aggregation of crystals and snow to form larger snowflake aggregates, instead of intense riming. However, for convective clouds globally, it is more common for the cloud base to be warm. This is because any deep moist convection always requires a level of free convection (LFC: cf., Vol. 1, Ch. 1) somewhere in the cloud not too far above the cloud base where parcels lifted from the surface become positively buoyant. Both the adiabatic liquid-water content and the related latent heating from condensation are greater for a warmer cloud base than for a colder one, making such an LFC more likely. In warmbased deep convective clouds, coalescence is usually the dominant pathway for the surface precipitation (warm rain process), even though they may be glaciated aloft. In such convection, the glaciation is more a side effect from precipitation generation, rather than a major cause of it, and much of the ice mass arises from freezing of raindrops formed by coalescence (Phillips et al., 2001, 2002). Thus, in convection globally, the most important microphysical processes usually tend to be condensation, coalescence, and droplet activation from soluble aerosols. Nevertheless, the ice phase from other microphysical processes can modify the dynamics and precipitation production. Raindrop freezing generates hail and graupel. Ice initiation can trigger glaciation that accelerates the updraughts through latent heating. Important areas of uncertainty include turbulent and electrical effects on ice nucleation, collision efficiency and orientation of ice crystals. This chapter focuses on key problems in the representation of the microphysics of convective clouds with schemes used in large-scale models. The emphasis is on the cloud microphysics of convection and the dynamical framework for representing it. Chapter 23 will, in turn, address a

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comparison of two types of microphysical description (bin and bulk). In the following two sections, processes for initiation and growth of hydrometeors that are most important for convective clouds are described, and progress in representing the processes in detailed cloud models is discussed. In the following section, ways to introduce cloud microphysics into massflux convection parameterizations are discussed, with specific examples of such microphysical implementations presented. 2

Initiation of particles in convection

The microphysical properties of convective clouds are related to the mean sizes of cloud droplets and ice crystals. Such sizes determine the collision efficiencies for collisions between cloud particles. The sizes also determine the cross section for interception of shortwave radiation. This in turn determines the reflectivity of convective clouds and their stratiform or cirriform outflow. Cloud droplet-number concentrations control these mean sizes. A pivotal problem is how to predict the number concentrations of cloud particles in the descriptions of microphysics within parameterizations of convective clouds. This problem has two aspects: one thermodynamical and the other microphysical. First, there is the task of predicting the supersaturation from the rate of ascent and the ascent from the dynamical component of the scheme. There is the variability of ascent within mesoscale systems of deep convection, which controls that of supersaturation. Second, there is the task of predicting the numbers of cloud droplets and ice crystals from aerosol conditions for given thermodynamical conditions of humidity and temperature. While the first aspect of the problem has been solved by theories (e.g., Korolev, 2007) and numerical schemes for the ascent at cloud base (e.g., Morrison et al., 2005a) and in-cloud (Khain et al., 2005, 2012; Ochs, 1978; Phillips et al., 2005, 2007a, 2009) and is well understood, the second part of the problem remains more uncertain. The purpose of this section is to deal with this uncertain microphysical aspect of the problem, involving prediction of numbers of cloud particles. Recent progress in observing initiation of droplets and ice crystals for given aerosol and thermodynamical conditions is discussed.

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Recent progress in observational studies of heterogeneous nucleation of ice and droplets

Aerosol particles are mostly sub-micron in size and contain substances of foreign material (not water). They are ubiquitous in the atmosphere at concentrations of typically 102 to 104 cm−3 , or more. All clouds consist of aerosol particles made bigger by becoming cloud particles. Cloud particles are small enough to have a fall speed sufficiently low for them to be effectively suspended in the air. Cloud particles, either cloud droplets or cloud-ice particles, constitute all natural clouds. Heterogeneous nucleation is the process of conversion of an aerosol particle to a cloud particle. Observational studies of the initiation of ice and droplets by solid and liquid aerosols have progressed to the point where high-resolution cloud models and large-scale models can predict cloud radiative and microphysical properties from the chemistry, sizes, and loadings of the most common aerosol species (e.g., Barahona et al., 2013; Lauer et al., 2009; Ming et al., 2007; Phillips et al., 2008, 2009). This has been achieved by incorporating recent laboratory data into descriptions of nucleation. 2.1.1

Initiation of ice by solid aerosol material

Heterogeneous ice nucleation involves action at the surface of solid aerosol material to initiate ice. A rare minority of aerosols (ice nuclei: IN) have enough solid material to nucleate ice. Atmospheric observations show that the size of solid aerosol particles must exceed about 0.1 μm to nucleate ice. Supercooling and humidity, as well as composition, also promote initiation of an ice crystal by such IN activity. Concentrations of active IN nucleating primary ice are quite low in the atmosphere, and are observed to have a range of about 0.1 to 1 l−1 (DeMott et al., 2003). There has been a resurgence of observations regarding ice nucleation recently (DeMott et al., 2011). Early laboratory observations (DeMott, 1990) showed how the active fraction of black carbon particles is proportional to the surface area. Approximate proportionality between the number of active IN and total surface area of IN aerosols was assumed by a description of ice nucleation (Phillips et al., 2008) and this was confirmed by subsequent laboratory observations (Niemand et al., 2012; Welti et al., 2009). Extensive efforts have been made to identify the chemical species of solid aerosol material that nucleates ice. Carbonaceous IN in particular have been a focus of attention. Until recently, only insoluble materials were known to nucleate ice. How-

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ever, soluble organic aerosols are prolific in the upper troposphere and may act as IN at the extremely cold temperatures there (Murray et al., 2010). Normally, soluble organics are dissolved in the aerosol particles, which are solution droplets. The cold temperatures cause them to be glassy, however. Once solid, they may nucleate ice. Controversy engulfs the topic of the IN activity of black carbon, which is potentially an important species. The activity of soot is pertinent in climate change questions, because soot is emitted by anthropogenic activity. Certain types of black carbon (e.g., hydrophobic soot) were seen to be very poor at nucleating ice (Crawford et al., 2011; Koehler et al., 2009), in contrast with earlier experiments by DeMott (1990) and Diehl and Mitra (1998). For example, IN activity was seen to be scarce in carbonaceous plumes freshly emitted from fossil fuel combustion (Koehler et al., 2009). By contrast, efficient IN activity was seen in some samples of biomassburning particles from diverse plumes (Petters et al., 2009), though not from all of them. It was unclear if the black carbon in the plumes was causing the observed ice nucleation. Field observations of other plumes with modern techniques to identify IN composition confirmed that black carbon does act as IN in the atmosphere. Bingemer (2012) observed the composition of active IN sampled from the troposphere over Germany and then activated by exposure to subzero temperatures. From 1 to 15% of active IN at −18◦ C were apparently pure particles of black carbon. The sampled soot IN were anthropogenic and were not emitted by the volcanic eruptions being studied (Bingemer, 2012). Moreover, atmospheric cloud ice residues were seen by Cozic et al. (2008) to be enriched with black carbon in biomass-burning plumes. For many decades it has been known that biological particles can nucleate ice (Mohler et al., 2007; Morris et al., 2004). Bacteria, pollen, and leaf litter can nucleate ice, as observed in laboratory experiments by both the microbiological and meteorological communities over many years (e.g., Levin and Yankofsky, 1983; Lindow, 1982; Lindow et al., 1982, 1989; Maki et al., 1974; Schnell and Vali, 1976; Vali et al., 1976). Recent developments have probed the influence of biological IN in the real atmosphere. Field studies over the Amazon (Prenni et al., 2009), Wyoming (Pratt et al., 2009), and Colorado and Nebraska (Bowers et al., 2009; Garcia et al., 2012) have shown that primary biological particles act as IN. Over the Amazon, primary biological aerosol particles (PBAPs) were seen to act as IN, dominating ice nucleation at temperatures warmer than −25◦ C (Prenni et al., 2009). Laboratory observations have quantified the IN activity of bacteria

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cultivated from cloud water samples (Joly et al., 2013). The PBAP material need not be alive or culturable in order to nucleate ice, and its fragments may remain suspended in the atmosphere for long periods, especially if small. Non-biological insoluble organics may nucleate ice, but their activity is not yet fully empirically quantified. An organic IN source from decomposing vegetation has been observed (Conen et al., 2011; Schnell and Vali, 1972). Predominantly organic particles from fossil fuel pollution freshly emitted in urban areas were seen by Knopf et al. (2010) to nucleate ice by freezing at temperatures colder than about 230 K. Marcolli et al. (2007) simulated the observed stochastic behaviour of rare but very efficient active sites for ice nucleation on the dust surface. The modified singular approximation assumes that ice nucleation happens almost immediately when an IN particle’s characteristic temperature (or humidity) is reached. The freezing probability per unit time is assumed to increase very steeply from almost zero to a high value with supercooling (Vali, 2008). Marcolli et al. (2007) obtained agreement with the observed active fraction of dust from laboratory studies by assuming a probability distribution of nucleating efficiencies among the dust particles, with a rare minority of very efficient IN. By contrast, other schemes were based on classical theory that neglected the probability distribution of nucleation efficiencies. Hence, they had to threshold artificially the maximum number that are active (e.g., ice nucleation of soot is assumed to stop somehow when 1% are activated). For each species of IN, only one value of nucleating efficiency (measured by contact angle) was often assumed, quite unrealistically. The nature of the time dependence of heterogeneous ice nucleation has been observed in some studies. The relaxation time of the active fraction of illite-rich dust was seen to increase with time when conditions of humidity and temperature were held constant (Niedermeier et al., 2011, their Fig. 7b). Widely varying efficiencies of multiple sites on each IN particle arose from lack of purity of composition. For Arizona Test Dust (ATD) at constant temperature, the active fraction after 10 min was seen to be only about twice that found after 10 s. This agreed with the modified singular hypothesis, as treated by Marcolli et al. (2007). Real dust found in the atmosphere is better represented as mixtures, rather than as pure minerals. A problem with such theoretical approaches is that the probability distributions of nucleation efficiencies among natural atmospheric IN of a given species have scarcely been measured. An alternative approach to treating

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the ice nucleation in models has been to use an empirical description, neglecting all time dependence (Phillips et al., 2008, 2013). This does not use any classical theory and is based on field observations of the IN activity of natural aerosols. A variety of different species of IN have been treated within this framework. In principle, it would be possible to develop the scheme further with inclusion of an empirical time dependence for each IN species. In addition to the topic of the efficiency and identity of atmospheric IN, there are some other areas of controversy. For example, electric fields might affect ice nucleation. Electrical fields have been observed to enhance the likelihood of ice nucleation. Drops (0.2–0.7 mm in diameter) in free fall in a wind tunnel were observed to freeze at warmer temperatures when contacted by charged sulphur particles than when these were uncharged (Pruppacher, 1973). Drops in free fall, larger than about a millimetre, were observed to freeze between −5◦ and −12◦ C if there were both intense electric fields and small filaments drawn from the drops (Abbas and Latham, 1969; Smith et al., 1971). However, some other studies observed no effect from electric fields of strengths found in the atmosphere (Dawson and Cardell, 1973). 2.1.2

Initiation of droplets by soluble aerosol material

Initiation of droplets in natural clouds usually involves the action of soluble aerosol material (cloud condensation nuclei: CCN) in the atmosphere. The chemical composition of the material is complex, even in remote regions. Long-range transport of aerosol pollution in the free troposphere causes remote regions to have an aerosol composition combining both natural and anthropogenic sources, with the pollution entraining into the boundary layer again (Clarke et al., 1999). The background free troposphere has been observed by aircraft in many field campaigns over recent decades (e.g., Clarke and Kapustin, 2002). In remote parts of the atmosphere, aerosol particles typically contain a mixture of various soluble compounds (internally mixed aerosols). Volatile compounds such as soluble organic species and sulphate have been observed to be coated onto refractory cores such as sea salt, dust, and soot in the background troposphere. Organics, black carbon, other biomassburning components and soluble species (e.g., sulphate) are typically internally mixed together in the same particle. During long-range transport across the oceans, aerosols tend to be processed microphysically by clouds,

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and chemically by condensation of trace gases from the ambient air (Phillips et al., 2013). Some anthropogenic sources of aerosol, such as biomass burning or combustion of fossil fuels, generate internally mixed aerosol at the moment of actual emission into the atmosphere, or just afterwards. Pollution consists partly of numerous organic species, as external mixtures of internally mixed multi-component aerosol populations (Kanakidou et al., 2005). For simple salt particles, the theory of CCN activation has been well understood for many decades (e.g., as included in the cloud model by Ochs, 1978) and is as follows. The vast majority of aerosols are water soluble. A soluble aerosol particle exists as a solution droplet with a radius r less than a critical value r∗ . Its size has a stable equilibrium because the vapour pressure at the surface of the aerosol is an increasing function of r (due to the solute concentration) and the condensation rate is proportional to the difference in vapour pressures between the ambient air and the surface. The size continually adjusts by diffusion of vapour to maintain this difference at zero, as the ambient humidity changes. At a critical supersaturation of the ambient air, then r = r∗ , and there is activation. Uncontrollable growth of the solution droplet occurs until it becomes a cloud droplet. Just above the cloud base, the supersaturation increases from zero in updraughts during ascent. First, the largest aerosols activate to form cloud droplets. Their values of critical size and critical supersaturation for activation are smaller. Next, successively smaller sizes of aerosols activate and eventually the supersaturation reaches a maximum. At that point activation stops. The supersaturation then declines during continued ascent and reaches a positive equilibrium value. Controversy has surrounded the role of the largest CCN. Such large or giant aerosol particles, by activating and removing vapour in growth, limit the cloud supersaturation. This in turn reduces the numbers of cloud droplets activated from smaller particles (Feingold et al., 2001; Ghan et al., 1998; O’Dowd et al., 1999). Yet, as noted above, the atmosphere is far removed from such a simple situation. Internal mixing of different chemical species, both volatile and refractory, in the same aerosol particle is the norm and not the exception. For decades, the microphysical modelling community was challenged by the problem of how to treat the CCN activity of internal mixtures in a simple way (reviewed by McFiggans et al., 2006). One method was to approximate the critical supersaturation of activation of an internal mixture by that of a simple salt CCN with the same mass of soluble material as the mixture. The

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problem with that approach is that there may be several soluble species in a single mixed particle, each with a different propensity for CCN activation. Then in 2007, the traditional theory for activation of aerosols as cloud droplets was reformulated in terms of a hygroscopicity parameter κ (Petters and Kreidenweis, 2007). Hygroscopy is the ability of a substance to attract and hold water molecules from the ambient air either by absorption or adsorption. κ is related to the water activity aw , which is the ratio between the actual vapour pressure at the surface of a solution droplet and that of a pure water drop of the same size. The relation is 1/aw = 1+κVs /Vw , where Vs and Vw are the volumes of dry solute and water. This new formulation has enabled prediction of droplet activation by an aerosol particle that is an internal mixture of different compounds. The method assumes that the hygroscopicity parameter of the mixture is the (dry) volume-weighted mean of the individual hygroscopicity parameters of the component compounds. The method has proven to be realistic for internal mixtures of many compositions. However, the role of organic species, dissolved in CCN aerosols, for droplet activation remains complex and uncertain. On the one hand, their reduction of surface tension tends to make CCN aerosols activate as droplets at a lower supersaturation. On the other, organics in CCN reduce the hygroscopicity of the internal mixture, because of their low solubility, which tends to make the CCN activate at a higher supersaturation. Although film-forming compounds that are almost insoluble will reduce the surface tension, their net effect appears to impede the CCN activity and reduce the number of activated droplets (McFiggans et al., 2006). Moreover, activation takes a finite time. While for soluble aerosols of simple salts, the activation is practically instantaneous, low-solubility substances such as organics can have an appreciable time dependence for activation. Organic content of internal mixtures and their composition reduce the rate of water uptake. This slows their approach to equilibrium size. Such kinetic limitations on condensational growth of CCN, delaying their droplet activation, are often significant. Film-forming compounds were predicted to inhibit the growth of giant CCN, which in turn can alter the formation of drizzle (Medina and Nenes, 2004). Smaller uptake of liquid water due to the slowing down of activation can result in larger activated fractions eventually, as the peak supersaturation at cloud base is raised and intensified, being reached later during ascent. Normally after reaching cloud base, it takes about 10 s for parcels to attain their peak supersaturation, and organics can appreciably delay the CCN growth on this timescale.

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Thus, the atmospheric chemistry can modify the activation in convective updraughts. This can also happen in the gas phase. Most water-soluble trace gases in the atmosphere can condense into aqueous solution droplets before the maximum supersaturation in a cloud has been reached. However, it may affect the activation behaviour and subsequent properties of the cloud droplets in an uncertain way. The effect depends strongly on the kinetics of droplet activation. For example, nitric acid was found to condense onto solution droplets, reducing the critical supersaturation at which activation occurs and boosting the numbers of cloud droplets. This effect is more important at lower updraught velocities (Nenes et al., 2002) since the fraction of all aerosols activated as cloud droplets (activated fraction) decreases with decreasing updraught speed. Hence, the trace gas causes a relatively larger enhancement in the activated fraction in slower updraughts.

2.2

In-cloud droplet activation simulated with models

Owing to the increase in vertical velocity with height and intense depletion of droplets by precipitation, typically many droplets are initiated far above the cloud base in convective clouds (in-cloud droplet activation). Incloud droplet activation was first discovered by Ochs (1978) with a parcel model that had a bin microphysics scheme. The model treated CCN activation, condensational growth, and coalescence. During deep ascent at constant velocity in the updraught of a Hawaiian convective cloud, the supersaturation was predicted to rise once precipitation formed and clouddroplet concentrations were reduced, exceeding the peak at cloud base. At this point, the previously unactivated CCN were predicted to be activated as cloud droplets. Although the corresponding simulation for continental aerosol conditions did not produce such in-cloud droplet activation, the lack of vertical acceleration of ascent must have artificially suppressed any increase of supersaturation in all simulations by Ochs. In reality, the buoyancy force vertically accelerates updraught parcels and there is a steady increase with height of vertical velocity (e.g., Blyth and Latham, 1997). This seminal paper by Ochs revealed that the cloud droplets of a deep convective cloud exist in a balance: although rapidly removed by intense precipitation after a few km of ascent, they are replenished by activation of small aerosols not activated at the cloud base. However, this discovery by Ochs was overlooked for a while. It was often believed that convective

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clouds do not have any significant supercooled cloud-liquid content at altitudes of about 8–10 km, or any significant concentration of ice crystals (see reviews by Khain, 2009; Rosenfeld et al., 2008). According to this view, the cloud droplet concentrations at levels of homogeneous freezing (about 10 km altitude) were assumed somehow to be negligible, so these clouds were supposed to contain very low concentrations of ice crystals in the anvils of deep convective clouds and the related cirrus clouds. Aircraft observations of cirriform cloud from the last couple of decades show that in fact the opposite is generally true. Pinsky and Khain (2002) predicted a major role for in-cloud droplet activation if the following conditions are met: (1) the supersaturation within a rising cloud parcel exceeds the maximum it reaches at the cloud base; and, (2) the reservoir of potential CCN is large enough to allow new droplet nucleation if condition (1) is met. They showed that both conditions are generally met for deep convective clouds in a wide range of environments. Increasing realization of the importance of in-cloud droplet activation arose when subsequent modelling studies of convective clouds predicted it and when the technology had progressed to allow for in-situ field observations of cloud properties far above the cloud base. Aircraft observations showed high concentrations of ice in the anvil outflow from deep convective updraughts and of droplets in the convective updraught (e.g., Phillips et al., 2005; Rosenfeld and Woodley, 2000). The formation of small droplets can be attributed to in-cloud activation of CCN ascending from the cloud base together with growing droplets. In-cloud nucleation in stratocumulus and cumulus clouds was observed and discussed in a number of studies (e.g., Korolev, 1994; Ludlam, 1980; Ochs, 1978; Phillips et al., 2005; Pinsky and Khain, 2002; Prabha et al., 2011; Segal et al., 2003). In the case of a vigorous Florida thunderstorm simulated by Phillips et al. (2005), switching off the process of in-cloud droplet activation caused most of the cloud droplets above the freezing level to be eliminated from the simulated convective updraught. The droplet concentration just below the level of homogeneous freezing (about −36◦C) was reduced by about two orders of magnitude. There was a similar reduction of the concentration of ice crystals above this level in the updraught and rain formation in the updraught. Similarly, Pinsky and Khain (2002) found that in-cloud droplet activation accelerated raindrop formation by broadening the droplet size

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distribution. Entrainment of aerosol particles (AP) aloft from the free troposphere into deep convective clouds was found to produce many small droplets in simulations by both Fridlind et al. (2004) and Phillips et al. (2005). Bimodal droplet spectra in natural convective clouds were observed by Warner (1969) and then again by Prabha et al. (2011). These are consistent with in-cloud droplet activation aloft. 2.3

Ice multiplication and homogeneous freezing

Ice is inherently difficult to form in the mixed-phase region of temperature (0 to −36◦ C). At such temperatures, water tends to persist in the supercooled liquid state without freezing, or just evaporates. Aircraft observations of glaciated clouds began a few decades ago. It was soon observed (e.g., Auer et al., 1969; Braham Jr., 1964; Hobbs, 1969; Hobbs et al., 1980; Koenig, 1963, 1968; Mossop, 1968) that (number) concentrations of ice particles typically exceed those of active IN at the cloud top by orders of magnitude in glaciated clouds. This observation was made for clouds deep enough to precipitate but with tops in the mixed-phase region. The ice-enhancement (IE) ratio characterizes the discrepancy and is defined as the ratio of total ice to active IN. Multiplication of ice particles somehow involving fragmentation of ice is implied by such observations, since homogeneous freezing cannot occur at such warm sub-zero temperatures. The set of mechanisms of fragmentation of ice and their occurrence in nature has remained a mystery. Some of the high concentrations of ice observed (e.g., Rauber, 1987) are explicable in terms of biases of ice shattering on impact with optical probes deployed on aircraft. Yet, concentrations of ice an order of magnitude higher than for the coincident IN concentrations have been observed using handheld instruments without any aircraft (Hobbs, 1969). Other techniques have identified ice crystals in Arctic mixed-phase clouds as being mostly naturally fragmented before sampling, using a modern probe (cloud particle imager) with laser holometry flown on an aircraft (Schwarzenboeck et al., 2009). Concentrations higher by up to a factor of 104 relative to the available active IN number are often observed (e.g., Hobbs et al., 1980). The most likely explanation to reconcile this discrepancy is to assume that the ice crystals multiply by fragmenting. The most accepted theory is the Hallett– Mossop (HM) process (Hallett and Mossop, 1974): small ice splinters break away from a graupel particle as the latter grows by accretion (riming) of

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supercooled cloud droplets. Yet the process works only within a limited temperature range (−3◦ to −8◦ C) and only when there are supercooled cloud droplets larger than 24 μm at these temperatures. From statistical analysis of aircraft data, this explanation was proven to apply to many warm-based convective clouds with rather copious ice particles and extending above the freezing level yet without any homogeneous freezing (Blyth and Latham, 1993; Harris-Hobbs and Cooper, 1987). Equally, ice particles multiply in number by fragmentation as they collide with each other in clouds and consequently break up mechanically. Although this possibility received serious attention several decades ago (Hobbs and Farber, 1972), it has received less attention in recent years. Pioneering laboratory experiments were performed by Vardiman (1978) but not much work has followed since. Another mechanism observed in lab studies has been the fragmentation of ice during freezing of large drops (Kolomeychuk et al., 1979). This is particularly likely for the larger drops. Such fragmentation during raindrop freezing was simulated with a spectral-bin microphysics model of a New Mexican cumulus cloud (Phillips et al., 2001). Assuming ten raindrop freezing (RF) splinters per drop was justified as an upper limit after considering the literature of laboratory studies (reviewed by Pruppacher and Klett, 1997). Phillips et al. (2001) found this assumed generation rate caused graupel concentrations (0.1–1 l−1 ) to be doubled in the simulation, while rain content was halved. RF splinters had appreciable concentrations of 1–10 l−1 . Since the HM process was active in the simulated cloud, the impact from RF splinters was obscured. It would have been more evident if the HM process had been inactive. Analysis of images from the Cloud Particle Imager probe in aircraft observations of shallow stratiform frontal clouds confirmed that shattering of freezing raindrops indeed occurs in nature (Rangno, 2008). In another modelling study of ice-formation mechanisms in Arctic clouds, various hypothesized mechanisms for ice formation were quantified. Results were documented by Fridlind et al. (2007). Mechanical fragmentation in ice–ice collisions was found in some runs to cause runaway glaciation (Fridlind et al., 2007, their Sec. 4.4). Such multiplication may have been possible by the freezing of drizzle to yield small graupel particles. The model used in that study has since been updated to include a better representation of ice crystal habit, depositional growth, fall velocity, and ice–ice collision efficiency. Yano and Phillips (2011) showed that a mostly overlooked process of

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mechanical break-up of ice in collisions between small and large graupel particles contributes to ice multiplication. An idealized, zero-dimensional analytical model of a mixed-phase cloud was created, with three species of ice, described by three evolution equations for ice crystals, and small and large graupel. It was drastically simplified. A regime for explosive multiplication was identified in the model’s phase space of ice-multiplication efficiency and number concentration of ice particles. Many natural mixedphase clouds, if they have copious millimetre-sized graupel, were found to fall into this explosive regime. For deep clouds with a cold base temperature where the usual HM process is inactive, it was shown that the ice break-up mechanism should play a critical role. Supercooled rain, which may freeze to form graupel directly in only a few minutes, was predicted by Yano and Phillips (2011) to accelerate such ice multiplication by mechanical breakup, with an ice enhancement ratio exceeding 104 about 20 min after small graupel first appear. All of the above mechanisms of ice multiplication involve ice precipitation. Convective clouds are particularly susceptible to fragmentation of ice because they generate large precipitation particles and have high liquid-water contents. Hail or graupel with much kinetic energy causing fragmentation in collisions, or freezing drops that also are large enough to fragment, are produced in deep convection. By contrast, there appears to be no ice multiplication in shallow glaciated layer clouds such as wave clouds at about −30◦C (Eidhammer et al., 2009), because these have little precipitation. The HM process was predicted to produce most of the ice splinters in the mixed-phase region of a deep updraught of a deep convective cloud over Florida, in a positive feedback with collisional raindrop freezing (Phillips et al., 2005). Homogeneous freezing of cloud droplets near −36◦ C predicted most of the ice particles in the entire convective cell. In the convective updraught, homogeneous aerosol freezing was unimportant. Similarly, in a modelling study of nucleation processes in mesoscale convective systems (MCSs), homogeneous freezing of aerosols was found to produce most of the ice crystals in the weak ascent of cirriform cloud, while that of cloud droplets was restricted to the convective ascent (Phillips et al., 2007a). If the cirriform ascent is very weak, then the supersaturation cannot reach the threshold for homogeneous aerosol freezing and so heterogeneous aerosol freezing prevails. In summary, the mechanisms of homogeneous freezing are organized by vertical velocity variability on the mesoscale in the upper troposphere.

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Thus, accuracy of ice nucleation in convection parameterization requires some representation of the vertical velocity variability in the convective core and mesoscale stratiform regions (Sec. 4). 3 3.1

Growth of particles in convection Diffusional growth and related uncertainties

It is well known that condensational growth of droplets tends to cause a narrowing of the droplet size distribution during ascent (Rogers and Yau, 1989). Thus, a classic problem in cloud physics has been to explain the broad droplet spectra usually observed, even in updraughts that have not ascended far enough above the cloud base for precipitation production. Spatial variability in rates of condensation (diffusional growth of drops) in the same cloudy parcel has been implicated by some theories for the broad droplet spectra. Shaw et al. (1998) proposed that long-lived turbulent tubes have a different supersaturation than their surroundings, causing their droplets to be different in size. However, the size of a drop during deep ascent is governed by the difference in (approximately saturated) vapour pressures between the level of activation (e.g., the cloud base) and the actual level of the drops aloft, which depends mostly on the corresponding difference in temperature, and hence, in altitude. Another explanation for the broadness of droplet size distributions is that mixing between cloudy parcels with different ascent rates at the cloud base or different aerosol loadings will lead to differences in droplet number concentration among parcels at the same level. Those parcels therefore have different sizes. When they mix, a broad size distribution is created. While diffusional growth is well understood theoretically for pure water drops, there are greater uncertainties for ice crystals. There are many different shapes of ice particles grown by diffusion of vapour, depending on the temperature. In the theoretical formula for such vapour growth of ice, a constant of proportionality linking the mass growth rate to the supersaturation in respect of ice is the capacitance. For a perfectly spherical particle, this equals the radius. The capacitance is uncertain and variable for dendrites because they consist of many branches. Fukuta (1969) found that observed growth rates of small ice crystals are about half of those predicted using the theory, which involves an analogy with electrostatics. A problem with any simple theory of diffusion is that kinetic effects become important for diffusional growth of small particles, if

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the mean free path of the vapour molecules between collisions is not much smaller than the particle size. In that case, the vapour mass field cannot be regarded as a continuum. A measure of such kinetic effects for ice is the deposition coefficient (net mass flux onto an ice surface measured relative to that predicted), which is analogous to the condensation coefficient for drops (Rogers and Yau, 1989). For ice crystal growth, the deposition coefficient is uncertain, and was recently inferred from laboratory observations to be about 0.5 (Haag et al., 2003), in agreement with Haynes et al. (1992). Equally, Jensen et al. (2005) observed in field experiments a remarkable absence of homogeneous freezing at cold temperatures for humidities for which it is theoretically expected. Jensen et al. (2005) hypothesized that a possible cause was kinetic effects on condensational growth arising from organics inside aerosol particles. It may have caused them to be smaller, concentrating their solute and inhibiting freezing. Another complex issue for convection is the evaporation of a fraction of the cloud liquid during massive homogeneous freezing near about −36◦C. The critical temperature for freezing of drops decreases with decreasing drop size, if the freezing is assumed to occur instantaneously. Thus, homogeneous freezing occurs over a range of temperatures of −36◦ to −38◦C, a layer of about 200–300 m, which is too narrow to resolve adequately in most cloud models. In nature, depending on the ascent rate and initial size distribution of the supercooled droplets, a fraction of the drops evaporate without freezing. This was treated by Phillips et al. (2007b) in a cloud model with a bulk description of microphysics, using a lookup table of the fraction of cloud droplets evaporated. Here, bulk microphysics descriptions make assumptions about the mathematical form of the size distribution of particles for each species of hydrometeor, with prediction only of bulk properties of the entire distribution at each grid point. It was inferred from a detailed parcel model with a spectral-bin description of the microphysics. (Both approaches for describing microphysics, namely bulk and bin, are described in Ch. 23.) The lookup table has dependencies on updraught speed, mean droplet size, and supersaturation before any homogeneous freezing starts during ascent. Phillips et al. (2007b) found an order-of-magnitude impact from this preferential evaporation of smaller cloud droplets during freezing on the domain-wide ice concentrations in mesoscale simulations of deep convection.

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Coalescence

Hitherto, convection parameterizations used in large-scale models have applied highly simplified schemes of bulk microphysics. The weaknesses of bulk schemes will be discussed in detail in Ch. 23. In essence, to treat it in such bulk schemes, coalescence is separated into two processes: (1) autoconversion to generate the initial raindrop embryos; and, (2) accretion of cloud liquid by raindrops. The earliest autoconversion schemes (e.g., Kessler, 1969) treated raindrop initiation as a simple increasing function of cloud liquid mass. They include no dependency of raindrop production on the cloud droplet size or the width of the droplet size spectrum. Both neglected dependencies are now seen as of paramount importance for controlling rain formation in natural clouds. As shown by Khain et al. (2000), the rate of precipitation formation predicted by the Kessler scheme may be up to a factor of 10 in error. To obtain the correct sensitivity of rain production in respect of changes in aerosol loading, it is necessary to use more recent autoconversion schemes. A new approach has been to apply lookup tables of autoconversion rates predicted by bin models (Feingold et al., 1998). Khairoutdinov and Kogan (2000) simulated marine stratocumulus clouds with a 3D spectralbin microphysics model and created an autoconversion scheme appropriate for two-moment bulk schemes to treat drizzle formation. Nevertheless, a modified version of the Kessler scheme has proved popular. The modification expresses the threshold of cloud liquid mixing ratio in terms of the product of a particle number mixing ratio and a critical mass per droplet. This makes the scheme switch on when the average cloud droplet size exceeds a threshold of about 20 microns (e.g., Swann, 1996, 1998). In terms of its threshold behaviour the scheme resembles qualitatively the accurate scheme by Khairoutdinov and Kogan (2000), who expressed the autoconversion rate as a function of a high power (about 5.6) of the droplet mean size. Such schemes involving a dependency on size are physically realistic since the underlying physical quantities, namely collision efficiencies and fall speeds of droplets, are functions of droplet size. Coalescence was implicated as the major cause for the mass of ice in the glaciation of warm-based convective clouds over New Mexico in detailed microphysical simulations (Phillips et al., 2001, 2002). A positive feedback between the HM process of ice multiplication, and collisions between supercooled raindrops and ice crystals to yield graupel (or hail), was predicted

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to dominate the glaciation in simulations with a spectral-bin microphysics model. The HM ice splinters were predicted to produce most of the ice crystals in the mixed-phase part of the updraught. A smaller yet significant impact on the ice concentration was predicted from RF splinters, as noted above (Sec. 2.3). Detailed simulations by another cloud model, with a non-hydrostatic dynamical core and spectral-bin microphysics, confirmed a strong role for coalescence and drop freezing (Khain et al., 2005). Latent heat release from such freezing was predicted to invigorate updraughts in deep continental convection. 3.3

Controversy about turbulent effects on collision efficiency

Convective clouds are turbulent, with high rates of production of turbulent kinetic energy from buoyancy and velocity shears. Turbulence accelerates coagulation by three mechanisms: (1) Increased variance of swept volume, due to horizontal and vertical accelerations of colliding particles in eddies; (2) Increase in collision efficiency due to turbulence, with colliding particles having a higher relative velocity from (1); and, (3) Clustering of particles due to inhomogeneities in turbulent flows (Khain et al., 2007; Pinsky et al., 2008a). Especially by the action of mechanisms (1) and (2), the collision kernel (product of rate of sweep-out of volume per particle and the collision efficiency) is boosted by up to about an order of magnitude. Raindrops are predicted to form first in the most turbulent regions of convective clouds, and so an effect of turbulence is to lower the height at which raindrops first form during ascent. Particular uncertainty is associated with such effects on ice particles. The motion of non-spherical particles in sheared flows was analysed by Gavze et al. (2012). Ice precipitation of bulk density much lower than water, such as snow or graupel, is especially susceptible to such turbulent enhancement when collecting supercooled droplets. This is because the variance of relative velocity (as a fraction of the mean relative velocity) in ice–droplet collisions is greatly boosted, relative to corresponding droplet– droplet collisions for particles of the same mass. That in turn is explicable in terms of a greater cross-sectional area and drag force, for a given mass, due to the lower bulk density. Snow particles are very readily deflected hor-

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izontally by eddies, increasing their volume of air swept out during fallout. Indeed, Rosenfeld and Woodley (2000) reported such enhanced riming of graupel in turbulent clouds. Benmoshe and Khain (2014) found that, although the microphysical structure of the clouds is altered by turbulence, the accumulated surface precipitation is unaltered by it. They found that formation of aggregates is caused by turbulent enhancement of collisions between ice particles. Formation of graupel and hail by riming is greatly accelerated by turbulence, accelerating formation of ice precipitation. Clouds in continental or polluted aerosol conditions were more influenced by turbulence than those in cleaner conditions. Timing and height of precipitation initiation were affected. In polluted clouds, turbulent enhancement of riming produced rapid growth of graupel and hail. In maritime clouds, however, coalescence and raindrop freezing were boosted by turbulence, reducing the extent of hail and graupel aloft. An issue of great uncertainty is the provision of autoconversion and accretion formulae for bulk microphysics models that include dependencies on the intensity of turbulence. That intensity is measured by the turbulent dissipation rate or by the turbulent kinetic energy. Hsieh et al. (2009) found that turbulence boosted the autoconversion rate by a factor of up to about 2 in vigorous deep convective clouds, when analysing aircraft observations of droplet size distributions using coalescence schemes. 3.4

Time-dependent freezing: Wet growth and drop freezing

Most hail particles in the wet-growth regime are observed to be spheroidal (Knight, 1986) and have both an inhomogeneous depth of the outer water film and an inhomogeneous surface temperature (e.g., Garc´ıa-Garc´ıa and List, 1992). Recently, List (2013) proposed a theory that applies to wet growth of spherical hail. The scheme was not validated by any simulation of observed hail or hail models. Yet an elegant reduction of the number of independent variables to only four and simplifying assumptions (e.g., approximate constancy of liquid skin depth) provided a concise description. Phillips et al. (2014) developed a more detailed theory for spheroidal hail that treats the inhomogeneity of surface temperature, ice density, and wetness of the surface over any given particle. By simulating laboratory experiments performed in a wind tunnel with hailstone replicas, Phillips et al. (2014) were able to validate the new scheme. The chief premise of the

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scheme was that both dry and wet regions coexist simultaneously on the surface of a hail particle. Also, the effective depth of liquid was assumed to be shallower than the actual average depth, especially at the onset of wet growth. Internal heat flow through the stone was simulated, boosting the freezing of the wet part of the stone. Better agreement with the laboratory observations was obtained with the new theory. The new theory, applied in a cloud model with bin microphysics, showed that time-dependent freezing has a major impact on surface precipitation from a hailstorm, updraught strength, and on hail amounts aloft (Phillips et al., 2015). Such advanced treatments of wet growth have not yet been introduced into convection parameterizations, and have only just been introduced into bin microphysics schemes for high-resolution models resolving convection. 3.5

Electrical effects on collisions

There is controversy about the impacts on collisions from electric fields. On the one hand, the most detailed cloud-microphysical models appear to reproduce the observed microphysical properties of electrified deep convective clouds without representing their electrical properties. On the other, laboratory experiments show strong sensitivities of coagulation processes in respect of electric fields. Fall speed and orientation of particles are known to be affected by electric fields and can influence the rates of collisions. Charged drops of up to 0.6 mm in diameter in free fall in an electric field were observed and simulated by Gay et al. (1974). Their terminal velocity decreased with increasing electric field. Weinheimer and Few (1987) observed that snow particles (0.2–1 mm in major diameter) and ice crystals (less than 0.05 mm) would align in electric fields of about 1 and 0.1 kVcm−1 respectively. The growth rate of drops due to collisions with smaller drops was increased by 20% to 100% after application of electric fields of between 0.5 and 1.6 kVcm−1 (Latham, 1969). The effect of course increased with increasing electric field. These observations were qualitatively corroborated by Phan-Cong and Dinh-Van (1973). Aggregation of ice crystals in collisions was observed to be enhanced by electric fields by Crowther and Saunders (1973), Latham (1969), Latham and Saunders (1970), and Saunders and Wahab (1975). In these experiments, inclusion of an electric field of 1.5 kVcm−1 caused all ice crystals to form aggregates, each consisting of ten crystals, when otherwise only 10% of crystals formed aggregates. Riming of cloud droplets onto ice was sim-

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ilarly enhanced by electric fields, with 200% greater growth at 2 kVcm−1 than without an electric field. Such observations imply that electric fields, of strengths similar to those occurring in thunderstorms (1–4 kVcm−1 ), can modify the ice–ice collisions that separate charge and generate them. Electric fields can also alter the coalescence and raindrop freezing that control the glaciation of warm-based deep convection. Thus, there may be a feedback. Since latent heat release is altered by changes in coagulation, the vertical velocities and turbulence would be expected to be part of this feedback. For instance, one can hypothesize a positive feedback loop from turbulent enhancement of ice–ice collisions, leading to charge separation and electric fields, leading to more coalescence and raindrop freezing in dropice collisions, leading to more latent heat release and hence faster ascent and more turbulence. The most detailed cloud models are almost capable of representing and investigating such feedbacks. Mass-flux parameterizations of convective clouds in climate models are far from being able to treat them, however.

4

Representation of microphysics in mass-flux convection schemes of large-scale models

Historically, over the last few decades of global modelling, there has been a difficulty to introduce cloud microphysics into convection parameterizations in any straightforward manner. The earliest convection schemes (moist convective adjustment schemes) implicitly assumed that the only microphysical processes were condensation, which maintains saturated adiabatic lapse rates, and the instantaneous removal of condensate as precipitation. The lapse rate was simply adjusted to some critical value (related to the saturated adiabatic lapse rate) while conserving moist static energy whenever convection was supposed to occur. In any parameterization, the many dependencies of the subgrid-scale process in reality are drastically simplified so as to include only a few approximate dependencies on just a few important parameters of the largescale flow. The parameterization does not necessarily need to be precise or to include all the contributing processes that might appear in a highresolution model, such as a CRM or LES. A moist convective adjustment scheme is a good example of how any parameterization must be created. It relies on an intuitively defensible physical principle.

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However, a consequence of this approach of simplification is that it may quickly face difficulties if the representation of more detailed (e.g., microphysical) aspects of the process is also sought. Such extra physical details may need to be approximated by a parametric representation too, in order to be incorporated. Fundamentally, what is missing in the literature are approximate parametric descriptions of extensively developed and detailed phenomenological descriptions of microphysics. The next generation of schemes were mass-flux schemes, which were more detailed dynamically, enabling more accurate treatment of the microphysics in convective plumes. Problems with the dynamical realism of mass-flux schemes have persisted, however. Mass-flux schemes simulate only a very few representative instances of all of the convective plumes that coexist in a mesoscale area. Missing dynamical features of convection have impoverished the microphysical realism of mass-flux schemes: the continuum of properties of the ascent and descent among many plumes; the threedimensional momentum transport by the convection; downdraught speeds governing precipitation loading; and, interactions with the turbulence of entrainment and overturning in the boundary layer, are not treated in much, if any, detail by mass-flux schemes. Moreover, the non-linear dependence of the microphysical properties of any plume on its depth and ascent rate creates a bias when the microphysical properties of some average bulk plume are treated. The average of all the microphysical properties of many elements of ascent is not equal to the microphysical property of the element of average ascent, in view of such non-linearity (see also Vol. 1, Ch. 9). Cloud dynamics provides the forcing for the cloud microphysics yet is treated simplistically in most convection parameterizations. The main issues for incorporating microphysical details into convection parameterizations have been regarding the treatment of the following phenomena (cf., Vol. 1, Chs. 9, 10, 12). • Vertical velocity variability in an ensemble of convective plumes, since the probability distribution of entrainment rates is uncertain, which is the forcing for cloud microphysics. • The evolving and maximum cloud-top levels of convective plumes, which depend in nature on the turbulent dynamics underlying entrainment and in-cloud buoyancy (e.g., Warner, 1970). • Probabilistic mixing of environmental air with undilute updraught air, determining cloud liquid concentrations of mixed draughts and their detrainment levels, with the probability distribution of mixing fractions in

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mixing events being largely unknown. • Coalescence, which depends on cloud droplet properties at cloud scale, but is more difficult to predict from the simple averaged quantities characterizing cloud liquid in the schemes. • Precipitation and condensation in the stratiform component of an MCS and the resulting latent heating, which affects the thermodynamics of the coupling between the stratiform cloud and the convective core. • The ascent-dependent linkage between aerosol conditions and convective cloud properties. The multi-scale nature of convective dynamics and the related turbulence has been a challenge to represent in any parameterization. The problem is that parameterizations generally assume an artificial separation between the resolved and unresolved scales of convection, with an approximate parametric representation of the (usually many, but cf., Ch. 20) unresolved convective cells. Such a scale separation is only conceptual and is not strictly observed in the real atmosphere (cf., Vol. 1, Ch. 3 for more details). Quasi-geostrophic two-dimensional turbulence characterizes the synoptic-scale flow, while three-dimensional turbulence characterizes scales smaller than hundreds of metres, with mesoscale or convective-scale eddies also existing in between. Transfer of energy between scales is a fundamental property of turbulence and various types of waves in nature. The supposed mesoscale gap in the power spectrum is in reality observed to be quite weak. In mass-flux parameterizations, convective dynamics are treated with analytic formulae that are often linearized (e.g., neglecting vertical perturbation pressure-gradient forces), when in nature there is a continuum of buoyancy-driven motions interacting on multiple scales. The aggregate effect of the microphysics on motions on one scale can cause non-hydrostatic pressure perturbations and latent heating that drive additional motions on other scales. Generally, improvements to microphysical realism have tended to follow improvements to the representation of the dynamics of convection during parameterization development. In the present section, there is a focus on such breakthroughs in treating the microphysics of convection in massflux parameterizations. Early advances involved simplified approaches, for example, using average conversion factors for transferring vapour to precipitation. More recent breakthroughs have involved embedding a onedimensional parcel model inside the parameterization, tracing the parcel’s trajectory.

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The gap has been vast between the advanced knowledge of microphysics (reviewed by Pruppacher and Klett, 1997) as treated in detailed cloud models, and the convection schemes of large-scale models. For almost all of the history of convection parameterization development, very few of the many microphysical processes with their uncertain dependencies (as discussed in Secs. 2 and 3) have been represented. 4.1

Literature on mass-flux schemes

Part II in the first volume of this set is devoted to discussion of the mass-flux convection parameterization formulation in detail. The present subsection summarizes key aspects of the formulation from the point of view of a microphysical representation. In the broadest sense, moist convection is driven by the thermal buoyancy force. Latent heat release from condensation, from vapour deposition onto ice, or from freezing all influence the temperature, density, and buoyancy of cloudy updraughts. Equally, in nature, the evaporation of precipitation in fall out also controls the heating of the troposphere. The height of precipitation formation and the updraught tilt are key for determining the intensity of evaporative cooling. The fine details of dynamics are important for the overall convective heating of the large-scale flow by microphysical processes. Rain drives downdraughts, creating cold pools influencing organization and longevity of MCSs. Precipitation only forms in updraughts deeper than a critical depth and depends on the convective ascent wc . The non-linear dependencies of microphysical properties on the rate of ascent implies a need to simulate the spectrum of wc when parameterizing convection. The first mass-flux convection schemes were pioneered by Ooyama (1971) and Fraedrich (1973). Convection is assumed to exist in a fixed number of bulk plumes, each at steady state, with a profile of vertical mass flux. Mass-flux schemes (e.g., Bechtold et al., 2001; Fritsch and Chappell, 1980; Tiedtke, 1989) are applied in global and regional models for forecasts and climate studies and usually involve only one bulk plume per grid box. Emanuel’s (1991a) scheme uses a single bulk updraught which detrains and entrains at all levels, with a spectrum of mixed draughts formed at each level of entrainment. The mixed draughts are supposed to detrain at their respective dilute levels of neutral buoyancy, with contents of heat and moisture that are determined by the mixing fraction from the original mixing event. In turn, the mass flux of each draught is governed by

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the probability distribution of the mixing fraction. However, a paradoxical quality of Emanuel’s scheme is that, despite treating the diversity of possible mixing events at each level, the cloud-top height is entirely determined by the adiabatic ascent of the undiluted updraught. Thus, the scheme could not simulate the shallow/deep regime transition seen to occur at relative humidities of the environment approaching 50–60% (Grandpeix et al., 2004). Partly, the problem is that dynamically the single bulk plume does not resemble a mesoscale ensemble of many plumes, which have a wide range of depths. Arakawa and Schubert (1974) considered a spectrum of plumes of many different entrainment rates within each grid box, spanning a wide range of depths. The scheme was dynamically more realistic, but lacked any microphysical detail. Two decades later, the convection parameterization by Donner (1993) and Donner et al. (2001) provided more microphysical realism in several ways. It included some of the best features of earlier schemes, such as those of Anthes (1977) for microphysics and Arakawa and Schubert (1974) for the spectrum of plumes. The distinctive philosophy underpinning the Donner scheme is that, in nature, microphysics at the scales of convective cells depends on vertical velocities also on the cell scale. Hence, prediction of mass fluxes alone in the dynamics of any scheme is not enough for microphysical realism. The relation between convective cores, the stratiform (or mesoscale) component and (cold pools) density currents in MCSs, such as squall-lines, was elucidated following results from field campaigns (e.g., Barnes and Sieckman, 1984) and numerical modelling (e.g., Rotunno et al., 1988; Thorpe et al., 1982) at around the same time. A loop of interactions between three branches of MCS flow was proposed, namely the convective and stratiform branches and the density current (e.g., Moncrieff, 1992, see Vol. 1, Ch. 13). From such studies, the following picture emerged. Convective cells generate precipitation that intensifies downdraughts by evaporation and by inertia. The downdraughts from each cell create a cold pool near the Earth’s surface, which spreads out at the surface as a density current, triggering new convective cells and limiting the lifetime of the parent cells. Mesoscale subsidence is driven by evaporation of precipitation from the stratiform part (i.e., the anvil shield) and this reinforces the density current, which in turn helps to initiate a rear-to-front mid-level flow and the convective ascent with a jump updraught driven by a propagating positive pressure jump near the edge of the cold pool. The jump updraught is sensitive to the low-

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level vertical shear of the environment. Other studies have shown that the stratiform cloud of the mesoscale component may insulate the convective core from entrainment of cool dry air, invigorating convective updraughts. In this schematic model of MCSs, the latent cooling from evaporation of precipitation in mesoscale and convective downdraughts is crucial to the feedback loop. A major attempt to incorporate this new understanding into mass-flux schemes was made by Donner (1993), who included a mesoscale component so as to treat the convective source of condensate for large-scale layer clouds resolved by global models. Satellite measurements show strong cloud radiative forcing in convectively active regions such as the west Pacific or inter-tropical convergence zone (Harrison et al., 1990), consistent with convection being a source of large-scale stratiform or cirriform cloud. The mesoscale component is dynamically active. Donner (1993) also predicted the vertical velocity in the convective plumes explicitly. This was done by solving the vertical momentum equation with a Lagrangian technique: the evolution of microphysical and thermodynamical properties of an entraining parcel were predicted by tracing its ascent through each updraught plume. The vertical momentum equation represents thermal buoyancy, the gravitational burden of condensate, and frictional turbulent drag. The evolution of the mass mixing ratios of cloud condensate and precipitation during ascent then determine the properties of the outflow to the mesoscale component. The Donner scheme was an improvement on previous schemes but its convective dynamics were necessarily simplified. For example, the perturbation pressure-gradient force was neglected in the vertical momentum equation. Microphysics is vital for convective downdraughts too, yet downdraughts are treated with little detail by Donner. Vapour and condensate enter the mesoscale component from the convective core, and freeze or sublime. In nature, this process must depend on the mesoscale ascent rate, which is artificially prescribed by the Donner scheme. Also, the range of temperatures over which the freezing occurs is artificially prescribed. The fine details of individual microphysical processes are neglected, only treated in an approximate sense. A mass-flux scheme with an explicit cold-pool representation was created by Grandpeix and Lafore (2010). Circular cold pools (the wakes) are treated parametrically with cooling by the precipitating downdraughts, while the outside area is warmed by the subsidence induced by the saturated draughts. A closure is included for determining the overall intensity of convection by treating the triggering of new cells at the upwind edge of

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the cold pool. The closure is based on available lifting energy (ALE) due to the lifting of air over the cold pools and this contributes to the available lifting power (ALP), determining the intensity of convection. This more realistic dynamical framework enabled microphysical impacts on the occurrence of convection to be parameterized in a mass-flux scheme, through precipitation, evaporation, cold pools, and triggering of new cells, albeit inexactly and simply. However, the self-consistency of its derivation was criticized (Yano, 2012) and it was applied as an addition to the Emanuel scheme, and so still treats the convection with the less realistic single-plume approach. A major advance in microphysical realism is offered by the segmentally constant approximation (SCA) approach, which bridges the gap between conventional mass-flux schemes and super-parameterizations. Consistent with the premise of the Donner scheme in terms of vertical velocity being the crucial forcing of the microphysics, a key feature of the SCA proposed by Yano et al. (2005, 2010); Yano et al. (2012); Yano and Baizig (2012) is that it provides a realistic evolution over time of the vertical velocity field. The geometrical constraint of SCA is applied to simplify the nonhydrostatic evolution equations. The SCA approach can represent the time evolution at almost any level of geometrical detail in order to represent convective and mesoscale motions by bulk plumes, stratiform clouds, and organized structures in the boundary layer, such as cold pools. Multiple convective plumes can also be considered. It solves the evolution equations numerically, and may be likened to a simplified CRM. Recent advances in treating the microphysics realistically within more traditional convection schemes are presented in Song and Zhang (2011) and Song et al. (2012). The fundamental dependency on aerosol conditions was represented using information about vertical velocity in the convective plumes. A detailed description is given below (see Sec. 4.3.2).

4.2 4.2.1

Theoretical framework for treating microphysics Continuity equations

Under a standard mass-flux formulation (e.g., Arakawa and Schubert, 1974), the continuity equation for water components (e.g., vapour, cloud water, cloud ice) with mass mixing ratios designated by qi for the i-th

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convective element is given by: 1 ∂ 1 (Di qi − Ei qe ) + Mi qi = σi Fi ρ ρ ∂z

(4.1)

(cf., Vol. 1, Ch. 7, Eq. 6.8). Here, σi = Si /S

(4.2)

is the fractional area occupied by the i-th subgrid-scale segment (Si is the area occupied by the i-th subgrid-scale segment in a grid box of area S). Ei and Di are the entrainment and the detrainment rates respectively, qe is the corresponding environmental mixing ratio of the water component, Mi = ρσi wi is the i-th segment mass flux, with ρ the density and wi the vertical velocity. Finally, Fi designates any source term associated with this water component, including any conversion rate associated with microphysics. It is important to recall that variables subscripted by a segment label indicate averages over that segment:  1 ϕdxdy. (4.3) ϕi = Si S i Recall also that the mass flux itself is determined by a continuity equation: ∂ Mi = Ei − Di . (4.4) ∂z The beauty of the mass-flux parameterization formulation resides in the fact that, insofar as any given physical variable is a conserved quantity (i.e., Fi = 0 in the above), everything can be written down in terms of the mass flux Mi without knowing the value of either wi or σi . In short, the above equation reduces to a matter of vertical integration for obtaining qi with given Di , Ei , qe , and Mi . However, once a representation of microphysics is introduced, we have a non-vanishing term involving Fi = 0 on the right-hand side of the water component continuity equation above (Eq. 4.1), which also has a multiplication factor of σi . Thus, it is necessary to know σi in order to account for any effect from the microphysics. While a formulation for defining σi could be developed directly, the strategy of evaluating the convective vertical velocity wi is more often taken, from which σi can be estimated from σi = Mi /ρwi . This strategy is preferred because Fi represents the net effect of all microphysical conversions, many of which depend on vertical velocity inside convective plumes, either directly (e.g., for activation of cloud particles from aerosols, via the supersaturation) or indirectly (e.g., by the timescale for

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ascent of cloudy parcels determining accretional losses of condensate and average mass contents of cloud liquid). Thus, there are two problems to face in order to implement microphysics into mass-flux convection parameterizations: (1) how to evaluate the convective vertical velocity; and, (2) how to deal with fast timescale microphysics without increasing the computational cost enormously. Regarding the second issue, note that the above convective-scale equations are obtained by assuming the steadiness of a given convective plume, neglecting the term ∂/∂t(σi qi ) (see Vol. 1, Ch. 7, Sec. 6.4). 4.2.2

Evaluation of wi or σi

Convective vertical velocity equation: The most formal statement for the vertical velocity of the i-th convective element is:

 1 ∂p′i 1 ∂ 1 ∂ wi (v∗H − r˙ i,b ) · dr + σi wi + ρσi wi2 = σi − + Bi − FD,i ∂t S ∂Si ρ ∂z ρ ∂z (4.5) as given by Eq. 7.2 in Vol. 1, Ch. 12. The three terms on the right-hand side are the sources of vertical mass flux due to the vertical pressure perturbation gradient force, buoyancy (thermal and gravitational burden of condensate) and the aerodynamic drag force. The pressure perturbation force arises from the non-hydrostatic nature of the fluid flow, with the pressure in the vicinity of the convective cell being altered (a dynamic pressure perturbation p′ ). Aloft there tends to be a meso-high around the outflow from the updraught and a meso-low near the inflow below it. Note that the dynamic pressure can be solved by: △p′ = −∇ · (ρv · ∇v) + ∇ · ρ(B − FD )

(4.6)

(cf., Eq. 8.2 of Vol. 1, Ch. 12). However, the standard formulation (as presented in Vol. 1, Ch. 12), relies on a simplification to: ∂ ∗ , (4.7) wi wi = Bi − FD,i ∂z where an effective buoyancy force (per unit mass) is: B=

g Tv − Tv,e 1 + γ Tv,e

(4.8)

and the turbulent (frictional) drag force (per unit mass) is: ∗ ∗ = CD FD,i

wi2 , Ri

(4.9)

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with ∗ CD

3 = 8



2 K2 + CD . 3

(4.10)

The coefficient γ is introduced to parameterize the effects from the vertical pressure perturbation gradient force (e.g., the virtual mass coefficient of γ = 0.5 from Anthes, 1977) and from the gravitational burden of condensate. Fractional area of convection: Zhang et al. (2005) took an alternative approach and introduced a formula: σc = 0.035 ln(1 + Mc )

(4.11)

by invoking Kiehl et al. (1996) and Xu and Krueger (1991). Here, Mc is the total mass flux of all the plumes of convection, while σc is the convective fractional area. 4.3

Examples of microphysical descriptions in mass-flux schemes

Here follows a discussion of selected examples of representations of what may be the most important microphysical process, namely the formation of rain by coalescence of cloud droplets. 4.3.1

Formulations not invoking σc or wc

In most convection schemes, some fraction of the condensate is assumed to be instantly removed as rain. These schemes generally do not predict the fallout of rain realistically. Hence, it has been considered appropriate for the same level of detail to be applied to the generation of precipitation, mostly without any detailed treatment of the sizes of cloud particles. In reality, the sizes of cloud particles are critical for the efficiency of collisions between drops that result in precipitation. Precipitation can only form in natural clouds when the maximum of the mean cloud droplet size exceeds about 20 microns near the cloud top. Arakawa and Schubert (1974) set the rate that cloud water is converted into precipitating water as: r(z) = C0 qc .

(4.12)

Here, qc is the cloud liquid mixing ratio and r is a rain rate defined as the change, due to precipitation production, in the vertical gradient (∂qc /∂z) of qc . There is also an empirical constant C0 . The premise is that for

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a given depth of ascent, a constant fraction of the cloud liquid mass is removed, irrespective of the ascent rate. Thus, accretion is assumed to be independent of time, which seems an unrealistic assumption in view of the fact that the process involves collisions among particles with contrasting fall speeds. In reality, the conversion from cloud to rain consists of two processes: autoconversion and accretion. Note that physically speaking, the autoconversion rate depends on (the fifth or sixth power of) the average size of cloud droplets, and also on their number concentration. Accretion increases both with qc and with the precipitation concentration. Most of these underlying fundamental quantities are not predicted by the Arakawa and Schubert (1974) scheme. This is why Eq. 4.12 is necessarily simplistic and approximate. Fritsch and Chappell (1980) set the convective precipitation rate as: Pr = ES,

(4.13)

where E = E(dw/dz) ≤ 0.9 is a fraction characterizing the precipitation efficiency. S is a rate of supply of moisture to the updraught, given by the sum of the supply from the cloud base and condensation within updraught:  zT Conddz. (4.14) S = (σc ρwc qc )z=zB + zB

The first term is close to zero. The second term is the column-average of the condensation rate per unit volume (Cond). Although Eq. 4.14 might seem to include a dependency on convective vertical velocity, in fact the fundamental dependency is on vertical mass flux. The underlying concept is that precipitation formation is proportional to the rate of condensation, which is proportional to the ascent. Thus, Eqs. 4.13 and 4.14, by including proportionality to vertical velocity in convection, qualitatively represent the fact that convective clouds (fast ascent) have stronger precipitation rates than stratiform clouds (weak ascent). However, it is unclear what the real dependencies should be of the precipitation efficiency. One would expect E to depend on cloud droplet mean size, as noted above, and hence on cloud depth. Emanuel (1991a) introduced a height-dependent precipitation efficiency ǫp (z) that defines the fraction of condensed water converted into precipitation during ascent of updraught parcels from cloud base up to some pressure level p: AUTΔt = ǫp qa,w .

(4.15)

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Here, qa,w is the adiabatically lifted condensed water and Δt is the model timestep, while AUT is the conversion rate averaged over the timestep for that ascent. As a result, the retained cloud water in the layer is given by: qc = (1 − ǫp )qa,w

(4.16)

at the same vertical level. The quantity AUTΔt is added to the precipitating water budget within the unsaturated downdraught (cf., Emanuel, 1991a, his Eq. 9). 4.3.2

Formulations invoking wc

A framework involving convective-scale vertical velocity wc for representing microphysical processes has been followed ever since the first numerical models of clouds were run on computers. Generally, by substituting the definition of the mass flux Mi = σi ρwi into Eq. 4.1, we obtain: 1 ∂ Ei (qi − qe ) + σi wi qi = σi Fi . ρ ∂z

(4.17)

This equation can, furthermore, be rewritten in the form: Ei Fi ∂ − (qi − qe ) qi = ∂z wi Mi

(4.18)

and can be integrated vertically once Fi , wi , and the fractional entrainment rate Ei /Mi are known. This general method was adopted by Simpson et al. (1965) and Simpson and Wiggert (1969) for their cloud-seeding experiment verification. Bulk microphysics developed by Kessler (1965, 1969) are introduced here. The next advance in the formulation of microphysical conversions was from Anthes (1977). The standard formulation for the vertical momentum budget was applied, using the prediction of condensate in the column to modify the effective buoyancy, and using an entrainment rate, estimated ∗ /Ri . Vapour and cloud liquid confrom cloud model output, instead of CD tents in a single one-dimensional convective bulk plume could be predicted from wc and entrainment of environmental air. At each timestep at each level the following changes for the mass contents of cloud liquid (Qcw ) and rain (Qrw ) were computed, while the ascent of a cloudy parcel was tracked from level to level: ΔQcw = (ΔQcw )cond − (ΔQcw )auto − (ΔQcw )coll

(4.19a)

ΔQrw = (ΔQcw )auto + (ΔQcw )coll − (ΔQrw )fall .

(4.19b)

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The terms on the right-hand side consist of condensation for cloud liquid (denoted by subscript cond), autoconversion from cloud liquid to rain (subscript auto), accretion of cloud liquid to produce rain (subscript coll), and fallout of rain (subscript fall). The rates of condensation, accretion, and fallout are time dependent, and so depend in various ways on vertical velocity. For example, the mass of vapour condensed for a given depth of ascent is a function of the temperature change inside ascending parcels, and the rate of their condensation thus depends on wc . Autoconversion in Eq. 4.19a,b is performed using the Kessler (1965) scheme:

K1 Δt(Qcw − a), if Qcw ≥ a (ΔQcw )auto = (4.20) 0, if Qcw < a. Here, K1 and the threshold a = 0.5 gm−3 together represent the effect of collision efficiencies being negligible for small cloud droplets (low Qcw ), as in shallow clouds, and significant otherwise. It is remarkable that such a simple scheme for autoconversion by Kessler often produces adequate results. The Kessler scheme has been used in a similar form in various cloud models even relatively recently, for example after modification to make the threshold a dependent on droplet number concentration (see Sec. 3.2). The reason for the Kessler scheme’s wide applicability is that, in real clouds, autoconversion acts like a switch to start the rain formation (cf., Yano and Bouniol, 2010), but soon becomes less important after the first rain has appeared. The actual rate of conversion of mass from cloud liquid to rain is dominated by accretion after rain has appeared. Therefore, as soon as the predicted autoconversion rate becomes non-zero, its accuracy is unimportant. In other words, a is crucial for any simulation but K1 not so much, and this is the more uncertain quantity of the two. More recently, Bechtold et al. (2001) and Kain and Fritsch (1990) assumed a rate of conversion of cloud liquid (of mixing ratio qc ) to precipitation of: Pr = Cpr qc

(4.21)

(cf., Ogura and Cho, 1973), where Pr is the mixing ratio tendency. Equation 4.21 assumes that precipitation growth occurs on a characteristic timescale, 1/Cpr . This is easily seen if it is substituted into a simplified cloud-water budget: wc

∂ qc = −Pr ∂z

(4.22)

Microphysics of convective cloud and its treatment in parameterization

yielding

109

∂ Cpr qc = − qc . (4.23) ∂z wc For the convective updraught, Donner (1993) applied the microphysics scheme from Anthes (1977) described above (see also Kuo and Raymond, 1980). It involves ascent of a cloudy parcel in which there are three scalars: mixing ratios of vapour, cloud liquid, and rain. However, the downdraught of Donner is treated in a very simple fashion. The Wagner and Graf (2010) scheme is similar to the Donner scheme in some ways. It has a spectrum of convective plumes in each grid box. Initial widths of the bubbles ascending through each plume are predicted with a population dynamics model. Each cumulus cloud type is represented by a one-dimensional entraining parcel model, which includes a description for temperature, water vapour, and four species of hydrometeors, as well as vertical velocity. However, the Wagner–Graf scheme has no mesoscale component and the microphysics in the parcels are quite simple. The selfconsistency of the spectrum calculation in the Wagner–Graf scheme has been criticized (Plant and Yano, 2011). Zhang et al. (2005) and Sud and Walker (1999) proposed convection schemes that include extensive cloud physics. Zhang et al. (2005) achieved this goal under a bulk mass-flux formulation. However, there are some ambiguities in their paper. The cloud fraction, b in their notation, is defined in two different ways (Zhang et al. 2005, their Subsec. 2.2 and towards the end of their Sec. 3). Also, the microphysical relations of Zhang et al. (2005) are not always well stated: the source terms in their equation corresponding to Eq. 4.1 must be multiplied by σc , but that factor is missing. It is not clear whether the definition of the source term itself already includes this factor or not. The plan of Sud and Walker (1999) is far more ambitious: as summarized by their Fig. 1, they considered not only the problem of the convection–environment interaction in a narrow sense, but also included various types of clouds: anvil cirrus, large-scale cloud, and some shallow boundary-layer clouds. For this purpose, various additional fractional areas are introduced under a non-mass-flux framework. It is unclear whether everything is formulated in a fully consistent manner. The most ambitious treatments are from Song and Zhang (2011) and Song et al. (2012) with the inclusion of mixing ratios of both mass and number for each of several microphysical species: ∂Mc qx (4.24a) = −Dqx + σc Sxq ∂z

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∂Mc Nx = −DNx + σc SxN . ∂z

(4.24b)

Here, mass and number mixing ratios of the x-th microphysical species are qx and Nx with source terms Sxq and SxN respectively, while Mc is the upward vertical mass flux. The detrainment rate is D. These continuity equations for mass and number of particles of each species are similar to those above (Eq. 4.1), with environmental concentrations set to zero. The number mixing ratio is treated similarly to the mass mixing ratio. A twomoment bulk microphysics scheme for four species, namely cloud liquid, ice, rain, and snow, is embedded for a parcel of the Zhang and McFarlane (1995) scheme. Despite having more microphysical realism than previous schemes, there are some imperfections of the Song–Zhang scheme. As with the Wagner– Graf scheme, graupel and hail are omitted entirely; in real convective clouds such heavily rimed precipitation is generally more important than snow (see Sec. 1). Vertical velocity is diagnosed in a similar manner to Donner (1993) by solving the vertical momentum equation. The ascent then determines the cloud droplet activation. However, no separate representations of the supersaturation both in-cloud and at cloud base are performed. Rather, there is only a scheme for cloud-base droplet activation. Khain et al. (2012) have shown that in-cloud droplet activation is a separate process. This problem was partly rectified by Song et al. (2012), who attempted to treat in-cloud droplet activation by making simple links to aerosols.

5

Summary

Improvements to representations of two aspects of convection schemes, namely the dynamics of convection and the microphysics, have both been closely interconnected during parameterization development. Each advance of dynamical representation has enabled an advance in the microphysics. Yet generally, the gap has always been vast between the advanced state of knowledge of the microphysics of convective clouds (Pruppacher and Klett, 1997) and its application in convection parameterizations. The problem has been the simplistic treatment of cloud dynamics, which is the forcing for the microphysics in nature. Very few of the many microphysical processes known to be active in natural convection have been represented by most convection schemes. Perhaps the most promising approaches have involved the embedding

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of detailed microphysics schemes inside a one-dimensional Lagrangian parcel model, which is embedded in turn inside the convection scheme. The scheme somehow solves the vertical momentum equation to predict vertical velocity. Donner (1993); Donner et al. (2001), Song et al. (2012), and Wagner and Graf (2010) have taken this path. A blot on the history of parameterization development is that lessons regarding the necessity for prediction of the probability distribution of vertical velocity in a grid box learnt decades ago in seminal papers by Arakawa and Schubert (1974) and Donner (1993) have not yet been applied to all modern convection schemes. Such vertical velocity statistics may be predicted by the more realistic parameterizations in terms of the interactions between the microphysics of condensation and coalescence, entrainment and latent heating. In the future, a myriad of microphysical processes may be included in convection parameterizations. Challenges will persist around how to treat with realism the microphysically sensitive evaporation of cloud/liquid aloft, driving convective and mesoscale downdraughts, and causing triggering of new cells by cold pools. Other challenges now being tackled include the aerosol–cloud linkage in convection schemes, which has the potential to improve the simulated distribution of precipitation over continental regions. Super-parameterizations of convection, or an SCA or SCA-like scheme, offer the best platform for treating the impact from turbulence and electrification on the microphysics of convective clouds, because of their potential for inclusion of non-hydrostatic dynamics. However, detailed super-parameterizations are not a perfect panacea for solving all problems in global modelling. Held (2005) argues that there are two types of science: one aimed at quantifying processes in the atmosphere using realistic simulation, and the other striving to gain understanding of aspects of the atmosphere by constructing highly simplified models. Held (2005) laments the wide gap in progress between both approaches in atmospheric science. Super-parameterizations offer the prospect of more accuracy in the future, especially the accuracy of treatments of microphysical processes through better dynamical forcings of them. Yet the act of creation of conventional convection parameterizations using analytical models (e.g., one-dimensional Lagrangian parcels) encapsulates knowledge and is necessary to advance understanding about how convection and microphysics function. New tools for checking conventional convection parameterizations before they are implemented now exist. Checks on the self-consistency of the

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equations used by conventional schemes can now be done using the SCA framework. The latest field campaigns are including coincident observations of aerosol conditions of chemistry and loading, and of cloud-microphysical and thermodynamic observations with which to validate the latest convection schemes and detailed cloud models. The new generation of convection schemes, as with the CRMs, can now predict cloud-microphysical properties from the aerosol conditions of the environment.

Chapter 19

Model resolution issues and new approaches in the convection-permitting regimes L. Gerard Editors’ introduction: With increasing horizontal resolutions of numerical weather prediction (NWP) models, we are facing new challenges for convection parameterization. At the time of writing, regional forecast models have begun to operate at horizontal resolutions of ∼ 1–10 km in which deep convection can be simulated to some degree, but is not resolved by the models. (In this chapter, the focus is on deep convection, and the adjective “deep” is omitted in most parts for economy of presentation. Similar arguments and issues, however, would apply for shallow convection at grid lengths of 100 m to 1 km). Although it is tempting to turn off convection parameterization under such a situation, our operational experiences tell us that it is not necessarily a good choice, as it may lead to a failure to predict some convective events or to erroneous runaway growth of convection at a single grid point. In this chapter, this scale range will be referred to as the “convectionpermitting resolution”. The term “grey zone” is also a popular term for this resolution range, but this term will be avoided here due to its ambiguity (it would be natural to consider that there is also a shallow convection grey zone across another range of scales). Issues for convection parameterization at scales which are then considered to be “high-resolution” already have a long history. A similar issue was faced when regional models began to resolve mesoscale convection, then described as the “mesoscale limit” (cf., Molinari and Dudek, 1992). To some extent, processes close to the grid scale of the model will always be problematic, and one may even say that we are perpetually within a grey zone. It has been shown in Vol. 1, Ch. 7 that the standard formulation of the mass-flux convection parameterization consists of two major steps. First, 113

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the evaluation of convective mass flux Mi using Eq. 6.9 of Vol. 1, Ch. 7: ∂ Mi = Ei − Di , ∂z and second, the evaluation of the convective-scale variables ϕi using Eq. 6.8 of Vol. 1, Ch. 7: ∂ Mi ϕi = Ei ϕ¯ − Di ϕi + ρσi Fi . ∂z Here, the suffix i indicates the convection type under consideration. Recall that all these formulae are developed under an asymptotic limit of vanishing fractional area occupied by convection, i.e., σi → 0. However, in the high-resolution limit, this approximation is no longer valid. Thus, we have to follow a more general formulation before the asymptotic limit is taken, as presented in Sec. 5.2 of Vol. 1, Ch. 7. These generalizations modify the above equations into: ∂ ∂ Mi = Ei − Di ρ σi + ∂t ∂z 1 1 ∂ ∂ σi ϕi + [Di ϕi − Ei ϕe ] + ρσi (wϕ)i = σi Fi ∂t ρ ρ ∂z (cf., Eqs. 5.13 and 5.10 of Vol. 1, Ch. 7). As a major modification, a completely diagnostic set of equations now becomes completely prognostic: i.e., the equations must be integrated in time. Moreover, under such a high-resolution limit, the entrainment– detrainment hypothesis may no longer be valid, and terms in the above involving the entrainment and detrainment parameters may be better considered as shorthand expressions for complex lateral exchange processes. By the same token, it is no longer straightforward to maintain a simple concept of the “environment” at high resolutions, and thus more involved interactions between different subgrid-scale subcomponents may also need to be considered. For both aspects, see Vol. 1, Ch. 7. Unfortunately, all of these aspects are still to be fully investigated. This chapter presents a stateof-the-art approach towards modifying the standard mass-flux formulation from current operational perspectives.

1

Introduction

In this chapter the focus is on the problem of parameterization when the grid-box length approaches the dimension of convective systems or convective cells: i.e., between a few hundred metres and about ten kilometres. At

Model resolution issues and new approaches in the convection-permitting regimes 115

convection-permitting resolutions, a parameterization still appears necessary (Bryan et al., 2003), while some specific issues need to be addressed as presented below. Coarser resolution models usually represent deep convection with a separate parameterization, while the non-convective clouds are represented by the so-called “cloud scheme” (more discussion of cloud schemes can be found in Ch. 25). At the other end of the resolution scale, convection-resolving models (CRM) or large-eddy simulations (LES) with a grid spacing of a few hundred metres or less usually assume that the convective cells are completely resolved by the model grid (their condensation is directly estimated by the cloud scheme, while the convective transports are represented explicitly on the model grid). At convection-permitting resolutions, a choice has to be made either to keep a maximum separation of deep convective and non-convective clouds as in coarser resolution models, or on the contrary to try to converge gradually to the CRM case with no deep-convection parameterization by producing a gradual extinction of the deep convection scheme when the grid spacing is decreased. This choice has implications for the input provided to the two schemes, and on the method of organizing the calls to the parameterizations and the interactions between them. A practical example is presented below. A completely different path is followed by super-parameterization (see, e.g., Arakawa and Jung, 2011, and also Vol. 1, Ch. 2, Sec. 2.3), where a convection-resolving model is run in a network of cloud-resolving grids with gaps superposed to the much larger main grid boxes. This configuration, where the resolution of the main model is clearly outside the convectionpermitting resolution range, is not considered here. Deep-convection parameterizations developed for coarse resolution models commonly make various assumptions that no longer hold when the resolution is increased. At convection-permitting resolutions, the formulation of the parameterization must be adapted, accounting, for example, for evolution in time and direct contributions of the updraught to the mean grid-box properties, which can substantially depart from the updraught environment. Finally, attention should be paid to some other aspects at high resolution, such as triggering and the ability to represent mesoscale organization.

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Organization of the parameterizations

When moving to high resolution, two opposite paths may be chosen: either to stick to the separation existing at coarser resolution, where convective clouds and other clouds are treated by two different schemes; or to join up with the LES, where all clouds are treated by a single scheme, meaning that the deep convection (DC) scheme contribution should gradually die out when the grid spacing is decreased. These two approaches, which will be referred to respectively as “separation” and “subgrid complement”, are outlined below. Prior to this, another distinction has to be made between parallel and sequential physics. When both the cloud and the DC schemes compute condensation (and sometimes precipitation) in their own way, a parallel organization of the physics (all parameterizations starting from the same state as if they were acting alone) would lead to some double counting, in the sense that they could both (partly or fully) represent the same condensation event: adding their results could lead to an exaggerated condensation or condensing more than the available water vapour. A sequential organization of the moist parameterizations must be taken where one of the two schemes is computed first and modifies the state passed to the other. In this concept of sequential physics or fractional step method (Dubal et al., 2003), the ordering becomes important, and it is desirable that all processes should have the opportunity to influence each other. This can be achieved by using a symmetrized sequential split method, where the interacting parameterizations are called twice, increasing the cost of the calculation. When the timestep is short, a cheaper solution is to let the interaction occur from one timestep to the next, after having decided a satisfying order of the parameterizations, and taken some protective measures against pathological behaviour (e.g., preventing the cloud scheme re-evaporating what has just been condensed by the DC scheme). In principle, this will mean that either the DC or the cloud scheme takes the lead, while the other has to adapt. 2.1

The path of separation: Specialized parameterizations

This path maintains a maximum separation between the convective clouds, treated by the DC scheme, and the non-convective ones, treated by the cloud scheme. This requires evaluation of the properties of different parts of a single grid box, identified as “convective” and “non-convective” areas

Model resolution issues and new approaches in the convection-permitting regimes 117

(Fig. 19.1). The respective sizes of these areas (mesh fractions) must also be known.

Vu Nc

Ns clear

Fig. 19.1 Bulk representation of condensed-water distribution in a grid box. σu is the updraught fraction, Nc is the convective cloud fraction, and Ns is the non-convective cloud fraction. The non-convective part (1 − Nc ) is the sum of Ns and the clear part.

The convective region includes the updraught, but also part of the detrained material and the anvils. Its area depends on the activity of the DC scheme, which needs to be computed first. The gradual transformation of the anvil clouds into stratiform clouds should be accounted for. In the scheme used below, this transformation has been represented by advecting a memory of the convective cloud fraction from one timestep to the next. A prognostic equation in the deep-convection parameterization makes the cloud fraction evolve, following the budget between new detrainment and transformation of existing detrained material into stratiform cloud, represented as a relaxation process in time. Note, however, that it is difficult to trace detrained materials after they have left the model column where the updraught is situated. This gradual change of affiliation of the condensates also means that the separation between the two kinds of processes cannot be total. Once the convective area Nc is delimited, the cloud scheme can be applied to the remaining part (1 − Nc ) of the grid box. In principle, geometrical considerations can help to evaluate the mean properties in the non-convective area, knowing the values for grid-box means and for the convective area. However, the properties over the convective area do not only result from the updraught scheme, but also from the history and other processes like precipitation, phase changes in the microphysics, or radiative effects. Note also that the cloud condensate contents could differ between

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the updraught itself and the convective detrainment area. To know their initial value (i.e., the value after advection by the mean flow, before the calls to the physics parameterizations), the distribution of cloud condensates between convective and non-convective areas must be known, either by parameterizing, by making reasonable assumptions, or by advecting an additional model variable from the previous timestep; in the last case, as well as the difficulties of advecting a positive-definite quantity, the effects of the conversion process from convective to stratiform do not ease the treatment. In the test presented below, the condensate distribution is treated by simple assumptions when computing the cloud scheme, namely considering the same wet-bulb properties and intensive cloud condensates within convective and non-convective clouds. The convective condensates are protected while computing condensation in the non-convective area. This technique was applied in the cloud scheme used in Gerard et al. (2009), but there its purpose was mainly to prevent re-evaporation by the cloud scheme of the convective condensate, while the whole package followed the path of complementarity. Figure 19.2 presents results of total and convective precipitation, accumulated over a period of 2 hours from an idealized experiment at two resolutions, with grid spacings of 2 and 1 km. The simulations were started with horizontally homogeneous conditions over a cyclic domain (larger than shown) with a vertical profile including CAPE and CIN and a vertical shear of the zonal wind (Weisman and Klemp, 1982). A spherical potential temperature perturbation with a radius of 3 km was applied near the surface at the centre of the domain. The 1 km run gives more intense precipitation locally, but the domain average is similar. For the convective part, it can also be observed that the domain-averaged precipitation is sensibly the same, while the local maximum increases together with resolution. The figure also shows the increase of the updraught mesh fraction with increasing resolution.

2.2

The path of the subgrid complement

In this case the goal is rather to gradually join up with the very high resolution simulations where the DC scheme could be switched off. Models with grid-box lengths of a few hundred metres can be qualified as “deep convection-resolving models”, in the sense that the convective

Model resolution issues and new approaches in the convection-permitting regimes 119

Fig. 19.2 Path of separation. Central part of the domain of an idealized simulation with grid spacing of 2 km (left) and 1 km (right). 2-hr accumulated surface precipitation (mm, top row) and DC scheme-related precipitation (lower row). The maxima and the averages over the drawn area are indicated on the left. Black contours: isolines at 0.1, 0.5, and 0.9 of the updraught-base mesh fraction σB at +2 hr. The maximum σB is 0.699 at 2 km and 0.999 at 1 km spacing.

Fig. 19.3 Same as Fig. 19.2, but following the path of complementarity. Here, the maximum updraught-base fraction is 0.654 at 2 km and 0.994 at 1 km spacing.

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cells most of the time stretch over several grid columns.1 They can also be treated realistically by the dynamics and a single cloud scheme, based as usual on an instantaneous diagnostic calculation, making assumptions about the distribution of water and temperature. At coarse resolution, the cloud scheme essentially represents stratiform clouds; when gradually refining the resolution from several kilometres down to convection-resolving scales, the resolved vertical velocity increases together with the updraught mesh fraction so that the convective systems contribute more and more to the cloud scheme condensation, as well as to the resolved vertical transports. To make a smooth transition from completely subgrid to completely resolved convection, the deep-convection parameterization should produce a contribution that decreases as the resolution increases, and which complements the cloud scheme. In a sequential organization, this implies that the DC scheme should be provided with an input state updated after the condensation in the cloud scheme, and that the condensates from both schemes should be combined to feed a single microphysical package after the updraught calculation. The downdraught parameterization can be computed after the microphysics, once the precipitation flux is known. As an example, Fig. 19.4 shows the sequential organization used by Gerard et al. (2009). Figure 19.3 shows the same experiment as Fig. 19.2 but for the subgridcomplement choice. We see that at both of the resolutions the subgrid convective part is smaller than convective part obtained with separation, and its share of the precipitation decreases together with the grid spacing (extinction behaviour).

3

Assumptions in the deep convective parameterization

While the organization of the parameterizations addresses the complementarity of their outputs, the assumptions made for representing the deep convective updraught must be appropriate for the targeted resolutions.

1 A phenomenon is generally considered resolved when its length is greater than six grid lengths.

Model resolution issues and new approaches in the convection-permitting regimes 121

(initial state from advection) [qv∗ , qi∗ , qℓ∗ qr∗ , qs∗ ]

ւ

Nc− −→ Stratiform cloud fraction −→ Ns

(Ns , Nc− ) → N ∗

−→ (Radiation)

→ (Tsurf , Turbulent diffusion) −→ turbulent diffusion fluxes [qv∗ , qi∗ , qℓ∗ , T ∗ ]

ւ

(Nc− , Ns ) → Stratiform condensation/evaporation −→ stratiform condensation fluxes [qv∗ , qi∗ , qℓ∗ , T ∗ ]

ւ

 moisture convective condensation fluxes conver- → Deep convective updraught −→ convective transport fluxes gence [qv∗ , qi∗ , qℓ∗ , T ∗ ] ւ

detrainment fraction δσD updraught fraction σu



−→ σD , Nc+ ⎧ ⎨ precip. mesh fraction σP + convective precip. fraction σP c Ns , Nc −→ ⎩ equiv. cloud fractionfeq ⎧ ⎨ cloud to precipitation conversion fluxes precipitation fluxes Microphysics −→ ⎩ precipitation evaporation fluxes

precipitation

ւ

enthalpy flux

[qv∗ , qi∗ , qℓ∗ , qr∗ , qs∗ , T ∗ ] ց

correction to condensates

→

ւ

Downdraught −→

[qv∗ , qi∗ , qℓ∗ , qr∗ , qs∗ ]ւ

ratio of condensation fluxes

−→

⎧ ⎨ ⎩ 

precipitation fluxes precipitation evaporation fluxes transport fluxes

convective rain and snow non-convective rain and snow

Fig. 19.4 Example of sequential organization for the subgrid-complement approach. c American Meteorological Society (Adapted from Fig. 1 of Gerard et al. (2009) which is  2009. Reproduced by permission of the AMS.) Starred variables are updated after each parameterization. The convective cloud fraction Nc− advected from the previous timestep is used at the beginning, until the final new value Nc+ has been obtained.

3.1

Reconsideration of the quasi-equilibrium and steady-plume hypotheses

As long as the grid-box length is greater than a few tens of kilometres, it can be assumed that it contains a sufficient variety of convective cells of all ages and extensions to allow the replacement in time of the cells changing category, so that a kind of equilibrium or steady state can be assumed (see Vol. 1, Chs. 4 and 7 for discussions of equilibrium and the steady-plume

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hypothesis). It is also helped by the use of quite long timesteps, implying the representation of time-averaged conditions. While these hypotheses are acceptable approximations at coarse resolution, a steady state becomes hard to assume when the grid-box length is reduced to a few kilometres, and when shorter integration timesteps do not leave time for a cell to reach its full vertical extension within a single timestep. The quasi-equilibrium hypothesis (Arakawa and Schubert, 1974, see also Vol. 1, Chs. 4 and 11) widely used in coarse-resolution models requires such a steady state, together with an additional assumption of a relatively fast reaction of the convective system to a slower “larger-scale” forcing. Short timesteps and feedback of the convective system onto the so-called “external” forcing (e.g., effects of convective anvils on radiation, or of precipitation on the boundary layer) with various timescales make this assumption become unrealistic. Considering the evolving character of the subgrid updraughts requires that some information about them (e.g., the mass flux) is propagated and possibly advected from one timestep to the next. A bulk representation (cf., Vol. 1, Ch. 9) may then be considered advantageous in terms of limiting the amount of information to advect, given the imprecision associated with the numerical advection of essentially discontinuous fields. Gerard and Geleyn (2005) used the updraught vertical velocity together with an updraught mesh fraction as prognostic variables. Despite the inevitable smoothing required, the evolution could generally be dealt with sufficiently. The gradual increase in time of the bulk updraught mass flux is a first aspect of evolution. To better represent a life cycle of a reduced set of subgrid cells, the question of the vertical extension is relevant. If the grid column contains cells that are at very different stages of evolution, their vertical velocities and their vertical extensions can differ substantially. The vertical exchanges may be underestimated by the bulk updraught velocity. If the population of cells is rather homogeneous (which may be more likely as the resolution is increased), then the bulk vertical velocity makes more direct physical sense and it can be used to represent the gradual rise of the mean updraught top. The growth to the mature cumulonimbus stage with a height of about 10 km can take several timesteps, so that it may be important to represent this evolution.

Model resolution issues and new approaches in the convection-permitting regimes 123

3.2

Formulation of the updraught effects

The budget equation for a quantity ϕ can be written as follows, in terms of the hydrostatic pressure vertical coordinate p, noting that ω = p: ˙ ∂ϕ ∂ϕ′ u′ ∂ϕ′ v ′ ∂ϕ + u · ∇ϕ + ω = fϕ + + + ∂t ∂p ∂x ∂y     parameterized source

horizontal diffusion

∂ϕ′ ω ′ ∂p   

(3.1)

parameterized vert. sg. transp.

Here, a simple translation between a geometrical height (z) coordinate to the pressure coordinate can be performed if desired by noting the relations: w

∂ ∂ =ω ∂z ∂p ∂ ∂ = −ρg . ∂z ∂p

The latter is obtained directly from the hydrostatic balance dp = −ρgdz. The horizontal turbulent terms are often represented by the horizontal diffusion acting in the dynamical package of the model. In this case, the physical parameterizations are computed over individual vertical model columns, and provide the vertical turbulent transport fluxes together with the sources for the energy and water equations. The bulk mass-flux formulation considers a single equivalent updraught covering a fraction σu of the grid-box area, while the remaining part (1−σu ) is considered as its environment. The mean grid-box value of a variable ϕ and the vertical turbulent flux of that variable are given by: ϕ = σu ϕu + (1 − σu )ϕe ,

(3.2) u

ϕ′ ω ′ = σu (1 − σu )(ωu − ωe )(ϕu − ϕe ) + σu ω ′ ϕ′ + (1 − σu )ω ′ ϕ′ ≈ σu ω(ϕu − ϕ),

e

(3.3)

where it can generally be assumed that the updraught environment has a much smaller vertical velocity, and hence ωe ∼ 0 and ω ∼ σu ωu . The final two terms on the right-hand side, associated with the sub-plume variability, are usually neglected; as pointed out by Yano et al. (2004), this results in underestimation of the vertical transports. A common practice is to transform the budget equation of Eq. 3.1 following the formulation of Yanai et al. (1973), by introducing entrainment Eu and detrainment Du fluxes. A full formulation is presented in Vol. 1, Ch. 7, Sec. 5.2. Most often, the asymptotic limit of σu → 0 is also taken, as

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in Vol. 1, Ch. 7, Sec. 6, so that the effect of the parameterized convection on the mean grid-box variable is expressed by Vol. 1, Ch. 7, Eq. 6.14:  ∂ϕ ∂ϕ  . (3.4) = Du (ϕu − ϕ) + σu ωu ∂t ud ∂p u

This method allows us to get rid of the mean source fϕ in the updraught area, and the updraught acts on its environment through detrainment and through a pseudo-subsidence term2 , representing a downwards transport of ϕ by the updraught mass flux. The formulation replaces the environmental value ϕe by ϕ, which again assumes that the mesh fraction σu ≪ 1. In addition, the time variation of the storage of water vapour or heat in the updraughts by an increase (or decrease) of σu is neglected. The estimation of the detrainment term in the parameterization is delicate and requires some arbitrary assumptions. Because of this, and because of the inadequacy of the asymptotic limit at high resolution and of the necessity to account for evolution in time, it appears simpler and more accurate (as suggested by Piriou et al., 2007, under the name “microphysics and transport convective scheme” or MTCS), to directly estimate the two terms in Eq. 3.1, i.e., a source term (net condensation in the updraught) and the vertical transport. Note, however, that some schemes (e.g., Bougeault, 1985), while expressing their final interaction with the resolved model variables through convective transport and the release of latent heat, still use internally the detrainment and pseudo-subsidence formulation, with all the above-mentioned approximations. 3.3

Mesh fraction

Coarse resolution models usually assume that the convective cells cover a negligible fraction of the grid-box area, σu ∼ 0, so that Eq. 3.2 reduces to ϕ ≈ ϕe . This is no longer valid for small grid boxes, for which the distinction ϕe = ϕ must be accounted for. If we have an estimate of σu , and given the updraught value ϕu , then we can compute a mean environmental ϕe that can be used to express the properties of the air that is entrained into the updraught (Gerard and Geleyn, 2005). It should be recalled, however, that the actual local updraught environment properties could depart significantly from ϕe . 2 At the scale of a single mesh, this subsidence represents a correction to the bulk apparent motion associated with the mean grid-box vertical velocity; it differs from the physical process of compensating subsidence occurring over a wider area.

Model resolution issues and new approaches in the convection-permitting regimes 125

An estimation of the updraught area fraction is also useful for other purposes (see, e.g., Sec. 2). If the updraught velocity is expressed with a vertical momentum equation, a mass-flux closure at the updraught base can deliver the updraught-base mesh fraction. 3.4

The closure problem

The primary purpose of the closure is to scale the subgrid scheme response as a function of the resolved model state and fulfil some physical budgets valid for a complete model column. Writing such budgets in a transient situation where the cloud top is rising, the clouds are extending their horizontal area, and the updraught velocity is evolving presents some difficulties. The choice of the closure criterion has been subject to many discussions (Yano et al., 2013, see also Vol. 1, Ch. 11). The context of the NWP model has however some aspects specific beyond the physical-world phenomenologies, especially when going to high resolution where the expression of the reference quantities used in the closure is increasingly affected by the grid-box length. 3.4.1

Closing value

The closure provides the value of a scalar quantity, either the base mass flux MB or the base mesh fraction σB , the mass-flux profile being expressed as: σB (t)ν(z)ωu (z, t) σu ωu = MB (t)η(z) = − . (3.5) Mu (z, t) = − g g When using a prognostic updraught velocity ωu , the closure has to determine σB , assuming a normalized mesh fraction profile ν(z). The use of a closure relation to obtain σB has another implication, as compared to a closure designed only to obtain MB . In the situation where the updraught covers the entire grid box, σB → 1; the mass flux of the absolute updraught MB → −ω/g, while the subgrid part of it must tend to zero. 3.4.2

Steady-state vs prognostic closure

A prognostic formulation of mass-flux parameterization closure was first proposed by Pan and Randall (1998) as an improvement of Arakawa and Schubert’s (1974) quasi-equilibrium (QE) formalism (cf., Vol. 1, Ch. 11, Sec. 10). While it was developed for wide grid boxes (with negligible updraught mesh fractions), the authors did consider that the cloud system

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does not adapt instantaneously to changes in the large-scale forcing. A prognostic closure has the substantial advantage of being more flexible than the QE hypothesis, and at the same time it avoids the heavy calculation of the kernel integral, which in the original scheme relates the tendency of the cloud-work function to the updraught mass flux (cf., Vol. 1, Ch. 11, Sec. 8). In this context, the prognostic character of the closure focuses on the evolution of the updraught-base mass flux or updraught-area fraction. The other characteristics of the updraught (or of the subgrid ensemble) are taken at steady state. With much smaller grid boxes, the steady state of the updraught properties appears less guaranteed, especially the updraught velocity and vertical extension. While convection is growing, the forcing contributes not only to increase the base mesh fraction, but also to create new cloud above the rising top, or to accelerate the updraught where it already exists. Some closure relations, such as CAPE relaxation, are valid when the updraught properties have reached a steady state. With a moistureconvergence closure, a transient state can more easily be envisaged (moreover, current moisture convergence can be partly a consequence of the current level of updraught activity and not an extrapolated steady state), but in this case we should find a way to estimate the part of the forcing absorbed (during growth) in accelerating the updraught, promoting the rising top, and the part involved in increasing the mesh fraction. A pragmatic solution is to begin by writing a closure for the steadystate value of the updraught profile, with the final vertical extension of the cloud. The steady-state closure assumes that both the mesh fraction and the updraught velocity no longer vary in time. This allows us to apply a CAPE closure, or to get rid of unknown terms in a moisture-convergence closure. In a second step, a prognostic closure can be written on top of the steady-state one, allowing the final mesh fraction to evolve towards the steady-state value. We remark that in the decaying phase, the newly computed steady-state fraction can be smaller than at previous timesteps, and the prognostic fraction gradually decreases while trying to approach it. Hence, the complex problem of both the updraught properties and the mesh fraction evolving in time is replaced by computing a steady-state mesh fraction from the extrapolated steady-state properties and the forcing, and then allowing the current mesh fraction to evolve towards it under the same forcing. The method is presented in Sec. 3.4.6.

Model resolution issues and new approaches in the convection-permitting regimes 127

3.4.3

CAPE closure

It is common practice to express the relaxation of the CAPE (e.g., Gregory et al., 2000) as:  CAPE ∂CAPE  =− (3.6) ∂t ud τ

(cf., Vol. 1, Ch. 7, Sec. 11). The tendency on the left-hand side considers only the effect of updraughts on CAPE evolution. The relaxation time τ can be related to the adjustment time defined by Arakawa and Schubert (1974) as the “time required for convective processes to produce a neutral state, reducing the CAPE (or cloud-work function) to zero, in the absence of large-scale forcing”. In the presence of a large-scale forcing tending to increase the cloud-work function, this neutral state will not be reached, and cumulus activity will be maintained, leading to the notion of quasiequilibrium when the large-scale forcing varies sufficiently more slowly than this adjustment time (cf., Vol. 1, Ch. 4). One difficulty of this definition is that the extinction of the convective activity could imply a gradual change of the updraught properties in time. In Arakawa and Schubert’s (1974) context of large grid boxes, one can assume that the reduction of convective activity associated with a decrease of mean CAPE over the area corresponds primarily to a decrease of the mesh fraction occupied by the updraughts. The existence of an adjustment time does not provide information about the trajectory of the system to reach the neutral state. If the updraught properties are unchanged by the decrease of CAPE, the relaxation relation expresses that the base mesh fraction, and hence the total mass of active elements, is proportional to the mean CAPE over the area. The CAPE can be expressed as: pT dp CAPE = −Rd (Tvu − Tv ) , p

(3.7)

pB

where pB and pT are, respectively, the pressure at the level of free convection and the equilibrium level of the lifted parcel (further replaced by the updraught base and top in a modelling context), and Rd is the gas constant of dry air. This definition refers to an unperturbed environment, through which a parcel ascent Tvu (z) is computed representing the virtual temperature of the lifted parcel. Below this real CAPE computed in the unperturbed environment independently of any model grid (the environmental conditions correspond to a sufficiently wide area to determine the

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mean vertical pressure gradient intervening in the buoyancy of the parcel) is distinguished from the estimation that can be done in the model context. In deep-convection parameterization, diluted CAPE is frequently used, Tvu being derived from the parameterized updraught properties that include the effect of entrainment and mixing with environmental air. Assuming steady-state updraught properties, the tendency of the CAPE only contains the tendency of the mean grid-box virtual temperature due to the action of the updraughts. When the mesh fraction cannot be neglected, the updraught properties affect the mean grid-box properties, as stated by Eq. 3.2. This can be interpreted as the updraught being partly resolved by the mesh. In this case, the grid-column CAPE (gc-CAPE) computed using the mean grid-box properties starts to depart significantly from the real CAPE, so that when the mesh fraction tends to unity, the buoyancy and the gc-CAPE tend to zero. If gc-CAPE is wrongly assumed to represent real CAPE, this decrease of the gc-CAPE would be interpreted as a decrease to zero of the mesh fraction or absolute updraught mass flux, while in reality these should reach their maximum value. One cannot consider in this case that the DC scheme represents the absolute updraught. Note, however, that a gc-CAPE closure can still work when considering a perturbation updraught that complements the mean resolved updraught: the perturbation-updraught velocity vanishes when the mesh fraction tends to unity, the resolved velocity becoming equal to the updraught velocity (this technique has been used in the idealized tests presented above). If we want to close the absolute updraught, we should obtain an estimate of the real CAPE considering a wider environment than the individual, narrow grid columns. As long as the mesh fraction is smaller than 1, it is possible to estimate the environmental CAPE by referring the buoyancy to the still updraught environment, assuming it to be closer to the mean properties over a wider area. Using Eq. 3.2, the steady-state relaxation relation is written:  pT pT Rd ∂Tv  dp dp =− (3.8) (Tvu − Tv ) , Rd  ∂t ud p τ (1 − σB ) p pB

pB

and it can be seen that the effect is to multiply the relaxation time by (1 − σB ), where σB is now the real mesh fraction. It was explained above that in small grid boxes, the updraught properties can no longer be considered at steady state when the mean updraught may be accelerating or rising its top. It is then unlikely that the evolution

Model resolution issues and new approaches in the convection-permitting regimes 129

of CAPE can be represented with a constant relaxation time. For this reason, the CAPE relaxation closure has to be based on a steady state, present or extrapolated. Out of the transient phases, the steady state of the updraught properties requires that the latent heat released by the updraughts is continuously transferred to the environment. When the mesh fraction becomes large, this transfer cannot be limited to a single grid column. The problem of representing the complete updraught appears here a second time, with the difficulty of gathering the conjugated effects of the resolved and subgrid parts of the updraught on CAPE. 3.4.4

Moisture convergence closure

When studying the phenomenology of deep convection, moisture convergence may appear as a consequence of convective activity, and then provides a source for further activity. The local moisture budget is written:  ∂q ∂ω ′ q ′  ∂q = fq + V · ∇q + ω +∇V′ q ′ + ∂t ∂p ∂p ud+vdif         source accumulation

−MoC

subgrid transport

(cf., Vol. 1, Ch. 11, Sec. 7), where MoC is the resolved moisture convergence and the sources include the updraught condensation, the cloud scheme’s condensation/evaporation, and the evaporation of precipitation. Dropping the horizontal subgrid transport and integrating over the entire grid-column height, we can obtain:   pT pT ∂q  ∂qu  dp = − σu ωu dp − ∂t ud ∂p uc pB

pB

=

pT

pB

−

  pT ∂Jqtur ∂q  dp MoC − g dp + ∂p ∂t cs

pT

pB

pB

  pT ∂q  ∂qe  (1 − σ ) dp + dp. u ∂t accum ∂t evap

(3.9)

pB

On the right-hand side, the vertical divergence of the turbulent moisture flux Jqtur = −g ω ′ q ′ vdif complements the moisture convergence. The vertical budget of the updraught transport flux is zero (simple vertical reorganization). Index cs is the cloud-scheme condensation and uc the condensation in the updraught. The two final terms represent the accumulation of vapour

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within the grid column and the evaporation of the precipitation or cloud condensates outside of the updraught. At the steady state, one commonly assumes that the moisture convergence balances the updraught moisture sink. However, Eq. 3.9 makes it clear that at least one other moisture sink, the cloud-scheme condensation, can compete with the updraught scheme. This information is not yet available if the cloud scheme is called after the DC scheme, but will be if it is called beforehand, such as in the subgridcomplement approach outlined above. The additional source by precipitation evaporation remains difficult to include, however, since evaporation is obtained only after the microphysics computation. Finally, the storage term can be accounted for by writing a prognostic closure (Sec. 3.4.6). An important feature of moisture-convergence closure is that the measuring parameter increases together with updraught activity, unlike areaaveraged CAPE, which decreases with increasing mesh fraction. Therefore, a moisture-convergence closure can more easily produce a mesh fraction reaching unity. If the feedback from MoC to activity and back again is all positive, the situation could get out of control, so it is important that the feedback can be capped; this is the case if the updraught buoyancy is reduced by an increased mesh fraction, and because of the bound on the mesh fraction. Reasoning in the opposite direction, the closure relation scales the steady-state mass flux in respect of the (current) forcing: because the target value increases with updraught activity, the steady-state mesh fraction or mass flux remains underestimated in the growing phase, slowing down the growth and the above-mentioned feedback.

3.4.5

Mixed closure

CAPE is a necessary condition for deep convection to start; gc-CAPE departs substantially from real CAPE (cf., Vol. 1, Ch. 1, Eq. 2.11) when the mesh fraction becomes large, but it remains a manageable criterion for scaling a perturbation updraught. CAPE relaxation requires extrapolating a steady state of the updraught. On the other hand, moisture convergence is not a prerequisite for the start of convection, but it occurs anyway after the convection has started, and it is not reduced by the convective activity. It is possible to combine both closures by writing the steady-state mesh fraction to be a function of the values obtained by CAPE and by moisture convergence closure, the ratio of the two contributions depending on the situation and/or the evolution stage.

Model resolution issues and new approaches in the convection-permitting regimes 131

3.4.6

Prognostic closure

Equation 3.9 can be used in a prognostic manner by considering energy. The idea is that the latent heat brought by convergence of moisture (both resolved and by turbulent diffusion from the surface and in the boundary layer) is in balance with the condensation and also with the extension/shrinking of the updraught area. Still assuming steady updraught properties, the creation of updraught area implies storing more moist static energy and (mainly) convective kinetic energy. This can be expressed as: L



pT

pB

 pT  ∂σB  1 ∂q  (h (k dp = − h ) + − k ) dp, u e vu ve ∂t accum ∂t ǫ

(3.10)

pB

where h and kv represent the specific moist static energy and the vertical convective (turbulent) kinetic energy (CKE). The coefficient ǫ is the ratio of the vertical CKE to the total CKE. Pan and Randall (1998) estimate from CRM experiments that the horizontal part of the deep-convective circulation is predominant. They evaluate the ratio of vertical to total eddy kinetic energy to be between 0.02 and 0.1. Here, we rather need the ratio of vertical (or upwards) CKE to total CKE. The results of Xu et al. (1992) may suggest a value of 0.33, corresponding to an equal share of CKE in the three directions. The difference between the two approaches may reside in the turbulent eddy energy remaining in the absence of convection; given the current uncertainties, it seems reasonable to take ǫ as a tunable parameter. Equation 3.10 states that the excess latent heat brought to the column is used to increase the updraught area in terms of moist static energy and kinetic energy. Conversely, if the budget of latent heat is negative, the missing energy is produced by reducing the updraught area. The prognostic relation can further be rewritten in terms of the steadystate mesh fraction instead of the moisture forcing, and this fraction can be obtained as well by a CAPE closure or a mixed closure. 4 4.1

Ancillary aspects of the parameterization Triggering

Many triggering criteria used in NWP (cf., Vol. 1, Ch. 11, Sec. 13.2) present a sensitivity to the model resolution. The method of triggering must be adapted to the choice made either to separate deep convection and nonconvective clouds at all resolutions, or to produce a complement to a cloud

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scheme representing an increasing part of deep convection at high resolution (Sec. 2). The presence in nature of a barrier of convective inhibition energy (CIN) results in delayed (cf., Vol. 1, Ch. 7, Sec. 11.5), but substantially stronger, updraught activity when sufficient CAPE has been accumulated and some localized trigger mechanism allows the CIN barrier to be passed. In a deep-convection parameterization with a moisture-convergence closure, the absence of moisture convergence is somehow able to delay the DC scheme initiation as it starts to appear when resolved motions following CAPE accumulation eventually lead to condensation in the cloud scheme, or when local evaporation starts to develop shallow clouds. With a CAPE closure, a triggering condition is essential to prevent small amounts of CAPE being released as soon as they appear, producing weak precipitation over large areas. The trigger model proposed by Fritsch and Chappell (1980) introduced the notion of an “updraught source layer” (USL), obtained by mixing a layer of a given pressure thickness (e.g., 60 hPa) at the bottom of the atmosphere. The first USL is chosen starting from the lowest model level. The mixed parcel is raised to its lifting condensation level (LCL); in the presence of a CIN barrier, it may not be buoyant there, and a further boost would be necessary to allow the parcel to reach its level of free convection (LFC). For this purpose, a local perturbation is attributed to the parcel, either in the form of a buoyancy kick or through an initial kinetic energy of the ascent at the LCL. If the parcel is able to continue its ascent over a minimum height above its LCL (e.g., 300 hPa), the triggering is confirmed; otherwise another USL is tried, starting from the second model level above the surface, and so on. When moving to high resolution, the local perturbation given at the LCL should be prevented from becoming excessive, either by construction or by modifying its computation. In the separation approach, the triggering is directly related to that of the physical updraught: physical criteria estimating CIN and available upwards turbulent energy may continue to work. More heuristic criteria, like Kain’s (2004) method based on a threshold of resolved vertical velocity w at the LCL, need to be adapted to prevent the amplitude of the kicks from increasing unduly when w becomes large at high resolution. Such an adaptation was used for the tests in Fig. 19.2. In the subgrid-complement approach, the question is less the triggering of the real updraught than that of the subgrid complement. The tests

Model resolution issues and new approaches in the convection-permitting regimes 133

with a triggering following the cloud scheme condensation appear to give encouraging results in this case; this method was used for Fig. 19.3. 4.2

Mesoscale organization

It has been noted that convection-permitting models that do not make any use of a deep-convection parameterization scheme often tend to produce more sparse, less structured precipitation areas. An illustration is given in Fig. 19.5, for a 2-km grid-spacing simulation with and without deepconvection parameterization. In this case, the run with parameterization shows more organized structure, and is closer to the observations. It must be noted that this behaviour can depend on various tunings of the model, in particular the diffusion and the cloud scheme. An idealized study by Piotrowski et al. (2009) showed a relation between the degree of organization and the ratio of the vertical to the horizontal diffusion. A deep-convection parameterization strongly affects the horizontal and vertical transports, which supports its positive role in the flow organization and the interest of its use at high resolution.

Fig. 19.5 1 hr of accumulated precipitation at 1400 UTC, for thunderstorms on 29 June 2009 over Czech Republic. Left: observations (radar and rain gauges, source: CHMI); centre: 2 km run with 3MT convective scheme; right: 2 km run with no convective scheme.

4.3

Stochasticity

For small grid boxes (and short timesteps), the subgrid ensemble departs from a larger statistical sample. Lacking the information about this departure, the behaviour of the model can become too smooth if we apply budgets valid for wider ensembles everywhere. It then becomes interesting to relax some budget relations at the scale of individual grid columns by

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introducing a stochastic component. This can quite easily be done when expressing the DC scheme closure; recent tests have shown that it could improve the representation of organized structures (see also Ch. 20). 5

Conclusions

This chapter discussed the specific problems encountered in order to apply a deep-convective parameterization at convection-permitting resolution, down to grid-box lengths that can completely resolve the convective cells. The interests of studying parameterization in this resolution range are multiple. Keeping a scheme that is not appropriate, or else working with no deep-convection scheme both introduce forecast errors, albeit errors of different types. The latter approach, for example, often misses important features of the mesoscale organization. When working with variable grid spacing as well as when nesting models of different resolutions, it is more consistent to keep the same parameterization, but it then needs to be configured to behave correctly over a whole range of resolutions. Finally, the parameterization can be an easy way of introducing stochasticity.

Chapter 20

Stochastic aspects of convective parameterization

R.S. Plant, L. Bengtsson, and M.A. Whitall Editors’ introduction: The following two chapters deal with the issues of uncertainties in convective modelling. This chapter attempts to account for uncertainties within the model simulation by means of stochastic parameterization, while Ch. 21 attempts to quantify model uncertainties in a diagnostic manner by comparing model outputs with observations without necessarily requiring point-by-point comparison. A close link between the uncertainties and the stochasticity must be emphasized, as discussed in Sec. 2. However, we also have to distinguish conceptually between the uncertainty and the stochasticity: the former can exist without the latter. A subtle contrast between Ch. 25 and the present chapter is also worth noting. Chapter 25 considers the distributions of physical variables at the subgrid scale (i.e., distribution density function: DDF). This chapter, in contrast, deals with the distribution of physical variables in probability space (i.e., probability density function: PDF). Thus, it considers the probability of finding a specific value for a physical variable at a specific spatial point. The physical variables must be considered after a filtering of the subgrid-scale fluctuations. However, at issue for now is the filtered value, rather than a deviation from it as in Ch. 25. In this respect, a subtle difference between DDF and PDF must carefully be recognized. Both distributions may be described by the Liouville equation, whose derivation in Ch. 25 is equally valid for the PDF problem as long as the system is deterministic. When a system is stochastic, a generalization of this equation become necessary, and for certain forms of stochastic forcing, the generalization leads to the Fokker–Planck equation. However, instead of addressing this formal approach for the PDF, this chapter presents a more explicit approach based on a direct time integration of a physical 135

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system. Stochastic parameterization is currently a very popular approach. The present chapter tries to convey this current enthusiasm. It is commonly believed that the best way to represent model uncertainty is to include a stochastic element directly within the model. However (as suggested by the quotation below), stochasticity is not an intrinsic property of the atmosphere, but rather a matter of interpretation and a useful or convenient way of formulating some problems. The current enthusiasm for stochasticity may sometimes seem to verge on over-enthusiasm, as if it were a panacea for all geophysical problems. One of the editors (JIY) will never forget his personal experience at a conference of hearing a graduate student say “stochastic modelling is inevitable, am I right?” The present chapter tries to take a balanced position. For all that stochasticity has proved a valuable treatment for many studies, there are also many studies in which it is introduced into a parameterization in a rather naive manner without carefully considering the real need for doing so. Readers should be careful not to miss various notes of caution sounded throughout this chapter. Rainfall ... according to Aristotle, happens ‘naturally’ at a certain time of year (winter), but occurs by chance or coincidence during the summer. (Taub, 2003, p. 89).

1

Introduction

The issue of stochastic parameterization for numerical models of the atmosphere is far from new (e.g., Lorenz, 1975) but has attracted increasing attention in recent years, both for practical and theoretical reasons. On the practical side, there is a need to deliver weather forecasts and climate projections which account for uncertainty in the model formulation as well as in the forcings or the initial and boundary conditions (e.g., Buizza et al., 2005). Clearly this is an important issue in order for the models to deliver results with greater practical utility. Some of the motivations and possible benefits arising from practical considerations will be alluded to in the following, although the assessment and calibration of ensembles will not be the focus here. On the theoretical side, the discussions need to recognize the basis of a model’s filter operation, a key issue being the fact that a space–time average is not necessarily a good approximation of the average effect of many realizations of the subgrid-scale processes that might occur

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within a grid box. The main purpose of this chapter is to provide an overview of the methods that have been proposed for the inclusion of stochastic effects, and which are relevant for convection parameterization. There is a focus on the physical basis behind the methods rather than an attempt to compare their utility. There is much good evidence to suggest that introducing some stochastic component to an ensemble weather forecasting model may be helpful in that aspects of the spread–error relationship (e.g., Jung and Leutbecher, 2008) of the ensemble can be improved without any undue damage to skill measures (e.g., many of the references within Palmer, 2012). The relative practical value of different approaches is less clear, not least because the different methods have been designed for different reasons and to capture different effects. Moreover, there are few tests directly comparing different stochastic methods within the same simulation framework. Exploratory studies along these lines were reported by Ball and Plant (2008); Whitall (2012) and some comparisons are also made by Sanchez (2014). Suggestions have also been made that the time may be right for making more systematic comparisons (e.g., Pincus et al., 2011). To clarify the scope, attention will be restricted to methods that have been applied in full-scale numerical weather prediction (NWP) and general circulation models (GCM). There is a separate strand of the literature exploring stochastic parameterizations within simplified or idealized models of the atmosphere (e.g., Monahan and Culina, 2011; Sardeshmukh and Sura, 2007; Stechmann and Neelin, 2011). For example, various explorations into methods for formulating a stochastic parameterization have been made using the Lorenz (1996) system as a toy model. This describes the dynamics of a set of slow variables, each of which is coupled to a sub-system of rapidly evolving variables. Various authors have attempted to construct parameterizations for the effects of the rapid variables on the slow variables. Wilks (2005), for example, demonstrated that a simple, deterministic parameterization based on polynomial fitting is improved by the inclusion of a stochastic component based on first-order auto-regression. Crommelin and Vanden-Eijnden (2008), on the other hand, developed a data-driven approach using outputs from the full model to train a Markov chain formulation. More recently, Laine et al. (2012) addressed parameter uncertainty by showing how parameters within a stochastic parameterization of this system might be systematically tuned using the results of an ensemble of forecasts. Alternatively, if the fast variables can be considered to be at quasi-equilibrium for given values of the slow variables, then a PDF for

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the former may be deduced from a maximum entropy principle (Verkley, 2011). Several simple stochastic approaches for Lorenz (1996) were recently compared by Arnold et al. (2013). These studies and others have echoes in the various approaches to the stochastic parameterization of convection to be discussed below. Another strand of the literature has explored the use of stochastic parameterizations of convection (e.g., Khouider et al., 2010; Majda and Khouider, 2002) for theoretical models of tropical dynamics. A recent overview is provided by Khouider et al. (2013). Such parameterizations might be further developed for operational implementation in the future. Some necessary first steps have been attempted by Deng et al. (2015) and Peters et al. (2013), while an alternative, simplified approach has been considered by Ragone et al. (2015), but these studies lie outside the scope of this chapter. A key point is that there are distinct sources of stochasticity with different physical (or numerical) origins, and that they cannot all be treated in the same way. Let us begin therefore with a brief overview of parameterization uncertainties. 2

Sources of parameterization uncertainty

The uncertainties associated with parameterization can be considered under the following general categories: (1) Structural errors. Clearly there are some fundamental issues, and strong assumptions in the basic formulation of many parameterizations. The convective parameterizations in operational use are often based on the mass-flux formulation, as discussed extensively in Vol. I, Part II. However, other structures are certainly possible, such as adjustmentbased schemes (e.g., Betts, 1986) and turbulence-based schemes (see Ch. 24). Structural uncertainties are particularly hard to assess, short of a brute-force approach of rerunning simulations using different parameterizations with different underlying structures. Even then, it is undoubtedly the case that the range of structures available from preexisting schemes in the literature will not span the true structural uncertainty. (2) Physical processes which have been excluded. These may arise from decisions taken in the construction of the parameterization, in the belief (or perhaps merely the hope) that the excluded processes do not

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have important systematic effects on the evolution of the filtered flow. Judgements about the appropriate level of complexity for the microphysics in a convective parameterization could be considered from this perspective (see Ch. 23). However, other uncertainties may arise from physical processes inadvertently excluded or misrepresented because of the interactions between model parameterizations. Further discussions can be found in Chs. 25 and 26. (3) Parameter uncertainties. The results of various massive multiparameter experiments have highlighted the entrainment rate in bulk mass-flux schemes as being the largest source of parameter uncertainty for climate projections. Good examples are the studies of Knight et al. (2007) and Zhao (2014), which highlight such entrainment sensitivities within the Gregory and Rowntree (1990) and Bretherton et al. (2004) parameterizations respectively. Also under this category one might consider uncertainties in parameter values that are needed by a parameterization but which have to be determined externally. A good example is the uncertainty in surface parameters addressed by Tennant and Beare (2014). (4) Inherent process uncertainties. Given a particular state of the parent model as the available input, it may not be possible, even in principle, to determine properties of the subgrid scales sufficiently well for the feedback to the filtered scales to be unambiguous. An extreme example occurs for the situation of convective initiation in a model for which the convection is simulated explicitly. The initiation may be very marginal and sensitive to subtle, inhomogeneous details of boundary-layer structure (e.g., Hanley et al., 2011). More generally, various coarse-graining studies of cloud-resolving model (CRM) simulations (e.g., Shutts and Palmer, 2007) have demonstrated that a given large-scale state is consistent with many subgrid states. (5) Incomplete calculation of the process. Radiation calculations in three spatial dimensions are straightforward to formulate but too timeconsuming to compute for most practical simulations. A Monte Carlo sampling strategy has therefore been adopted by some modellers, which is itself a source of uncertainty and has been found to lead to some noise-induced drift (Hill et al., 2011; R¨ ais¨ anen et al., 2008). A note on the coarse-graining methodology may be in order here. The idea is to perform very high-resolution integrations that are regarded as a “truth” and aggregate the results in order to obtain fields representa-

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tive of some coarser scale. The averaged fields are then compared against low-resolution integrations and provide a target for developments of the lower-resolution modelling, in the sense that producing low-resolution simulations that behave more similarly to the dynamics of the aggregated highresolution simulations is often regarded as valuable progress. It is also worth remarking that, while useful, and while the above categories do accurately reflect the descriptions and thinking within the literature, in practice the distinctions between categories are not always clear. For example, the bulk entrainment rate (and the various dependencies associated with it in a particular scheme) might reasonably be regarded as a sub-model embedded within the convective parameterization, rather than simply as a number. As discussed in Vol. 1, Ch. 10, the entrainment rate encompasses many possible cloud–environment interactions, and in a bulk scheme it also implicitly encodes assumptions about the relative occurrence of different cloud types (Vol. 1, Ch. 9). Thus, although the entrainment rate has sometimes been treated as a parameter that can be freely varied with some plausible range, uncertainties from all five of the above categories might in fact reasonably be considered to also apply to treatments of entrainment. (For an alternative view on entrainment, see Ch. 24.)

3

Pragmatic treatments of parameterization uncertainty

In later sections of this chapter, the focus is mainly on the fourth category and exploration of the physical basis for the inclusion of stochastic effects within the design of a convection parameterization. First, however, some more pragmatic treatments of uncertainty are considered, which have been proposed and indeed used within operational models. Often, the physical source for the uncertainties is not directly considered in these methods. The usual practical rationale for an ensemble prediction system has been to increase the ensemble spread (standard deviation), so that it better corresponds to the model skill and thus provides a well-calibrated prediction system (e.g., Buizza et al., 2005).

3.1

Multi-model

One strategy is to sample different deterministic model configurations within the design of an ensemble. An extreme case is to use different models entirely. This provides a sample of structural model uncertainty

Stochastic aspects of convective parameterization

141

(although doubtless many possible and viable model structures exist for which no model has yet been constructed) and it is eminently practical. It may amount to simply collecting and combining data which would have existed in any event. There are more comprehensive studies, but a good, simple example of the approach is Mylne et al. (2002) who showed that a combination of the Met Office and ECMWF (European Centre for MediumRange Weather Forecasts) ensembles offered clear improvements over either ensemble when considered on its own. The approach can also be extended to incorporate multiple analysis methods for the initial state (e.g., Bowler et al., 2008a). 3.2

Multi-parameterization

Within the same NWP or GCM, there may be different options available for the various parameterization schemes. This is hardly an uncommon situation as it can be useful to have a back-up option, or a possible new scheme may be under construction or testing. Various studies have sought to exploit this overlap by constructing ensembles in which different members use different combinations of the parameterizations. Houtekamer et al. (1996) is an early example. Although such an approach is very practical, it is very easy to criticize on the basis that two (or perhaps three) deterministic formulations of a parameterized process are scarcely sufficient to span the full range of uncertainty. However, a more subtle but more serious issue is that not all combinations of the parameterizations may work well together and deliver physically reasonable results. Moreover, the approach may in fact prove impractical if considered for a full-scale operational ensemble prediction system as opposed to a one-off study. Maintaining a model code that allows all combinations of choices to be run becomes a major effort if at the same time one wishes to allow for research and testing of new model development projects. 3.3

Multi-parameters

A popular and very simple approach is to vary the values of some important parameters within the parameterization schemes, recognizing that these may not be well constrained. Typically the parameter range is established by so-called “expert elicitation”: essentially educated guesswork by experienced model developers. For the distribution of the parameter values

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within the acceptable range, typically no attempt is made at justification. Both pre-chosen and random sampling of the parameters have been used. In the simplest form of the approach, model parameters are determined at the start of each simulation and then fixed. Thus, it is supposed that each parameter has, in principle, a single correct value but that it is not known. Various attempts to quantify climate change uncertainties or to estimate GCM parameters in a probabilistic sense have followed this approach (e.g., Klocke et al., 2011; Knight et al., 2007; Murphy et al., 2004). Yang and Arritt (2002) is an example that focuses on convection parameters in a seasonal forecasting context, specifically the closure timescale and the depth of the initial test parcel, while Grell and D´ev´enyi (2002) considered many convective parameter variations within 12-hr NWP forecasts. A similar issue arises here as for the multi-parameterization, although arguably less acute in this case. The default model configuration uses a set of parameters that perform reasonably well as a set, but there may be compensating errors implicit within that set, and varying each parameter independently may lead to parameter sets that are unrealistic. It is possible, and probably desirable, to filter out some simulations from the ensemble that deliver obviously flawed results according to a few well-constrained metrics, but this serves only to ameliorate the problem rather than remove it. Moreover, it is known that some of the parameters within the models must be inter-related. For example, consider the rates of the microphysical processes under given parameters within a bulk microphysics scheme. As discussed in detail in Ch. 23, these may formally stand for various weighted averages of an underlying size distribution. By varying microphysical parameters independently, some choices might imply a size distribution that is not mathematically realizable (cf., Ch. 25). Thus, a more physically consistent approach would be to randomly vary the parameters characterizing the underlying distribution, and from that to calculate or estimate the corresponding effects on the various parameters of the bulk microphysics scheme. Unfortunately, no such method has been implemented as yet. 3.4

Randomly evolving parameters

A variant of the multi-parameter approach that has been used within an NWP ensemble context is to allow the parameters to vary randomly during a simulation. An early example is the random parameters scheme produced for the Met Office Global and Regional Ensemble Prediction System (MO-

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GREPS). Some results of early tests are described in Arribas (2004) and the operational implementation is presented in Bowler et al. (2008b, 2009). The parameter variations are made independently but are correlated in time by means of a first-order auto-regression model for the variations. There is no spatial component to the parameter variation, although this was essentially only for reasons of ease-of-implementation (Arribas, personal communication, 2008). Baker et al. (2014) have recently experimented with a similar method for the convective-scale MOGREPS ensemble. 3.5

Multiplicative noise

One approach to have enjoyed some practical successes (e.g., Buizza et al., 1999a, 2005; Charron et al., 2010; Teixeira and Reynolds, 2008) is to apply multiplicative noise to the tendencies of horizontal winds, temperature, and moisture that are predicted from the model’s standard deterministic parameterizations. For the scalar variables, the model equations therefore become: ∂T + u.∇T = ǫT PT (X, α) (3.1) ∂t ∂q + u.∇q = ǫq Pq (X, α), (3.2) ∂t where PT and Pq are the parameterized tendencies for T and q, temperature and moisture respectively, expressed with a dependence on the model state X and some parameters α. The prefactor ǫ is allowed to fluctuate randomly about unity. The method is sometimes known as the stochastic perturbation of parameterization tendencies (SPPT) and in the formulation above the tendencies from all of the parameterizations are combined before being multiplied. Note that a perturbation is not applied if it would generate supersaturation. The need for such a constraint is discussed by Buizza et al. (1999b), although it was somewhat relaxed later (Tompkins et al., 2007). The report of Buizza et al. (1999b) also describes some of the early testing that led to the implementation at the ECMWF, and focuses on the importance of choosing suitable correlation behaviour for the multiplier. The scheme adopted keeps ǫ constant in tiles of side 10◦ and chooses a new random value every 6 hr. The multiplier has a distribution that is uniform in the range [0.5, 1.5]. The current operational formulation is described by Palmer et al. (2009). A more complex spatial pattern generator has replaced the crude tiling and the same random variable ǫ is now chosen for all variables. There are sound

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reasons for this choice as demonstrated in Sec. 3.6. Further refinements are that the perturbations are tapered towards zero within the boundary layer (for numerical stability reasons) and are not applied within the stratosphere since the only important parameterization there is the radiation calculations which have relatively little associated uncertainty. Some interesting variations on the approach in respect of convection are those of Bouttier et al. (2012), who applied the method in a convectionpermitting ensemble and Teixeira and Reynolds (2008), who applied multiplicative noise in a medium-range ensemble only to the temperature tendency from the convection parameterization. Sanchez (2014) considered an implementation in which clear-sky radiation tendencies are not perturbed, which is more realistic but reduces the scheme’s impact. He also demonstrated that the SPPT approach can imbalance the moisture and energy budgets of a climate simulation, in the particular example with an excess of precipitation over evaporation by ∼ 0.17 mmday−1 . Whitall (2012) also discussed the moisture budget of some SPPT simulations, both in singlecolumn and aqua-planet configurations, noting that the budget is not closed when perturbations are applied to water vapour tendencies but not to the tendencies of layer-cloud condensate. The effects on the heat budget in this case were driven by the fact that the stochastic perturbations had the effect of somewhat shifting the partition between layer-cloud and convective cloud in the model. Although it is hardly difficult to make various criticisms of the multiplicative noise approach, its simplicity and its practical success in increasing ensemble spread should hardly be denied either. For example, in the application by Teixeira and Reynolds (2008), the stochastic variability becomes zero if the deterministic convection parameterization does not meet its triggering conditions, which is not realistic. Indeed, an explicit demonstration of the variability that remains in situations where a deterministic parameterization predicts no convective tendency can be seen in Shutts and Palmer (2007). A coarse-graining study by Hermanson (2006) argued that a stochastic term does arise that is proportional to the total parameterized tendency, but that an additive noise term may also be desirable, particularly in the vicinity of fronts (Hermanson et al., 2009). Recently, a coarse-graining analysis of the ECMWF model by Shutts and Pallare`es (2014) considered separately the mid-tropospheric temperature tendencies from each parameterization. The standard deviation at the coarse-grained scale for both the convective and layer-cloud parameterization tendencies was well described

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as being proportional to the square root of the mean tendency from a lowresolution model, rather than scaling linearly as currently assumed. Closer inspection of conditional frequency distributions provides further evidence of a Poisson-like scaling or a Craig and Cohen (2006) distribution rather than a Gaussian distribution. These results are consistent with the theory and practice of Sec. 8 and suggest that SPPT might be adapted to provide it with a firmer physical basis. 3.6

Tests of perturbation formulation

In this subsection there is a link made between the most common pragmatic approach, the SPPT, and more physically based approaches to be presented later. To do so, the use of SPPT and the sensitivity to its formulation is illustrated in order to highlight sensitivities to the character of stochastic perturbations. These results are based on the single-column model experimental set-up presented by Ball and Plant (2008) and Whitall (2012), in which the Met Office Unified Model (MetUM) is used for the TOGA-COARE test case (Petch et al., 2007). Ensembles of 39 singlecolumn simulations of the case are used, with small random perturbations to the initial state and different seeds for the stochastic schemes. Basing the stochastic term on the total parameterized tendency assumes a coherence between the parameterized processes. Alternatively, one might consider that the different processes described by different parameterizations have different associated variabilities. Figure 20.1a shows the application of SPPT to the tendencies from a single parameterization only, in terms of the resulting ensemble spread in temperature. Also plotted for reference are the results from the full SPPT scheme and the spread in a default ensemble with perturbations applied to the initial state only. Note first that in the stratosphere, the configurations with SPPT applied to a single scheme produce no more spread than the default MetUM, except for the perturbed radiation scheme. This confirms the remarks on spread in the stratosphere given the previous subsection, and explains why SPPT is no longer used at such levels by the ECMWF. In the troposphere, the ensemble spread profiles look remarkably similar, with the ensembles perturbing a single scheme tending to lie between the default MetUM and the full SPPT. Thus, wherever and however one introduces perturbations in the troposphere, the response of the parameterizations rather than the structure of the perturbations dominates the structure of the ensemble spread in this experiment. The analysis of Ball and Plant (2008) suggests that the

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4

x 10

 

  

pressure / Pa

2 4 6

  

8

10 0

1

0.5

1.5












 

    

ensem ble spread in T / K (a)



(b)

Fig. 20.1 Time-averaged vertical profiles of root-mean-square ensemble spread in temperature for days 21 to 23. The results in (a) are for the default MetUM (dashed), the SPPT scheme (black), and the SPPT scheme with perturbations applied only to the radiation scheme (red), the boundary-layer parameterization (green), the convection parameterization (magenta), and the large-scale cloud and microphysics parameterization (blue). The results in (b) are for the default MetUM (dashed), the SPPT scheme (black), the SPPT scheme with a different random number used to perturb each parameterization tendency (blue), and the SPPT scheme with different random numbers used to perturb the increments to T and q (red). í3

1.5

x 10

q increment / ggí1

1

0.5

0

í0.5

í1

í1.5 í3

í2

í1

0 1 T increment / K

2

3

Fig. 20.2 Scatter plot of the total parameterization increments in specific humidity and temperature, at all timesteps and in all ensemble members during the period from day 16 to 19. The SPPT scheme is used with ǫT = ǫq . Colours denote the altitude: green, below 850 hPa; red: 850–500 hPa; blue, 500–200 hPa; and, cyan, above 200 hPa. The blue diagonal indicates increments that conserve moist static energy.

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actual structure is governed by the convection parameterization. Another issue is how one chooses to correlate the perturbations made to the parameterization increments in different model variables. For example, Teixeira and Reynolds (2008) found that much greater ensemble spread can be produced by perturbing the temperature tendency directly and applying a moisture tendency perturbation in such a manner as to hold relative humidity fixed, as opposed to perturbing both the temperature and moisture increments with the same random multiplier. Note that this approach implies a positive correlation between the stochastic perturbations to temperature and water vapour. In order to understand the significance of inter-variable correlations in the applied perturbations, consider Fig. 20.1b which demonstrates that a dramatic increase in spread, relative to SPPT, occurs for uncorrelated temperature and moisture perturbations. Indeed, the spread produced by decorrelating the noise in the two tendencies is larger than that produced by quenched random noise (not shown; “quenched” means choosing a random number at the first timestep only, which is then left fixed for that ensemble member throughout the integration). Figure 20.2 is a scatter plot of the total parameterized tendencies in T and q. The stratospheric points (cyan) are barely visible on the figure as they are bunched tightly along the temperature axis. Points within the boundary layer (green) tend to lie near the moisture axis: their close coupling to a prescribed sea surface means that temperature changes are modest, but there is an interplay between gradual moistening by surface fluxes and less frequent but strong drying caused by deep convection. In the mid and upper troposphere (red and blue, respectively), points generally lie close to the line of constant moist static energy in the cooling/moistening direction; these are mainly associated with evaporation of layer cloud. There is a broader spread of points indicating warming and drying, associated with deep convection. Clearly there is a strong negative correlation between increments in temperature and moisture in the free troposphere. The plot is for the SPPT scheme with ǫT = ǫq , which scales the increments whilst preserving the correlation. However, if the perturbations are decorrelated, then they weaken the physical anti-correlation and damage features such as the moist-static-energy conserving behaviour when cloud water is evaporated. In this way, an increased ensemble spread can be produced by drawing independent random numbers, but only by disrupting the basic physical constraint Cp ΔT ≈ −Lv Δq.

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Filtering and scale dependencies

The filter operation of a model reduces the variables appearing in the original equation set to those which are used by the numerical model. In principle, any filtering operation might be performed on the raw variables but it is usually thought of as removing from explicit consideration the variations at small, sub-filter scales. In a limited-area model, another aspect of the filter will also be implied, in that scales above the domain size are excluded, or at least relegated to the status of prescribed boundary forcings. Although the meteorological literature does include examples of trying to parameterize interactions with larger unrepresented scales (e.g., Daleu et al., 2012; Romps, 2012; Sobel and Bretherton, 2000), parameterizations normally attempt to represent effects arising from filtering out the small spatial scales (Wyngaard, 2004). While NWP and GCMs rely on this general approach, an actual filtering operation is not normally explicitly stated. Herein lies the difficulty that plagues many discussions of deterministic and stochastic parameterizations. The use of a stochastic parameterization is sometimes seen as a controversial topic within the meteorological community, but, in the opinion of the authors of this chapter the controversy may be rather artificial. The essential issues are neither obscure nor debatable if one has decided upon some basic properties of the filter. Of course, one can (and should) debate at length the nature of the filter desired in order for a model to be most useful in addressing its stated purposes. However, having established the basic characteristics of a filter, following through the consequences of that choice to the design of appropriate parameterizations is, in principle, a matter only of self-consistency. There is an increasing recognition that a deterministic form of parameterization is not necessarily well founded (e.g., Palmer, 2001, 2012; Penland, 2003; Williams, 2005). Given some input state which represents the atmosphere at the filter scale of the model, the parameterization is required to produce tendencies in the filtered state that arise through the action of the sub-filter processes. In practice, the input to a parameterization is normally the grid-scale state rather than a filter-scale state, an issue discussed by Lander and Hoskins (1997). Neglecting this distinction for the moment, an important question is whether the required tendencies have a one-to-one functional relationship with the input state. If so, then the parameterization should be deterministic. If not, then the situation may be rather that we do have some information about the PDF of the tenden-

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cies, conditioned on the input state, but that the PDF has non-negligible width. In such circumstances, it would be natural to adopt a stochastic parameterization which randomly samples the conditional PDF. If we can assume a clean scale separation between the filtered and residual parts of the flow, then the latter could be considered to be in statistical equilibrium given the former. However, no such clean scale break exists in the atmosphere: aircraft and other observations indicate spectra that are well described by power laws (see Vol. 1, Ch. 3). In large-eddy simulations, the filter is placed within the inertial subrange, with its famous k −5/3 power-law dependence. Crucially, for a filter within the inertial subrange, we can assume the sub-filter turbulent motions to be homogeneous and isotropic, and that they contain relatively little energy compared to the turbulent eddies on the scales of primary interest. On the other hand, if the dominant energy-containing structures at sub-filter scales are instead coherent, organized features just below the filter scale then it follows that only a relatively small number of such features can be contained within any grid-box area. In other words, a local statistical equilibrium at the grid scale is no longer reliable. It may still be reasonable to consider a statistical equilibrium and associated deterministic relationships in an ensemble-averaged sense, or after averaging over multiple grid boxes, but there will be considerable fluctuations in a spatially averaged sense at the scale of a grid box. If the filtering operation used to define the model variables is a spatial average, then a strategy of sampling from the PDF of the sub-filter-scale flow will provide a more realistic description of the action of the smallerscale processes. Such a stochastic parameterization strategy provides possible realizations of the sub-filter scales conditioned on the filtered scale. Each model simulation will produce a structure close to the filter scale that is physically self-consistent but inherently unpredictable. Predictions that are valid close to the filter scale can only be generated from the statistics of an ensemble of such simulations. One might prefer to define a model in which the filter is constructed by taking an ensemble average of the possible flows at the sub-filter scales. A single model simulation is then sufficient in order to make predictions, and the results from the simulation will be ensemble-averaged (at least in respect of small scales) and therefore should be smooth close to the filter scale. Clearly this is a very much simpler strategy, albeit with limitations. It is designed to capture the ensemble average as efficiently as possible, but it does so at the expense of being able to make any statements about higher

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order statistics. In particular, because it deliberately excludes specific realizations in favour of the ensemble average, it excludes any considerations of spatial variability close to the filter scale and hence any consideration of extreme events at such scales (Wyngaard, 2004). Assuming, however, that we confine our attention to the ensemble mean, we have a choice of strategy: either ensemble average the sub-filter scales and make a single simulation of the filtered scales, or else consider specific realizations of sub-filter scales and ensemble average at the filtered scales. The optimum strategy must depend upon the character of the interactions between the filtered and the sub-filtered scales. The second (stochastic parameterization) strategy explicitly recognizes that such interactions may be important and attempts to capture them. In contrast, the great advantage of the first (deterministic parameterization) strategy is that it is very much more efficient if such interactions are not necessary in order to simulate the filtered scales reliably. It is far from obvious that this advantage holds for atmospheric flows, but of course practical parameterizations often involve making some uncomfortable compromises in order to realize large gains in efficiency. Such considerations should never be considered trivial because efficiency gains can easily be exploited for the benefit of other aspects of the modelling: to increase resolution, for example (cf., Berner et al., 2012; Mullen and Buizza, 2002). In summary, the existence of a PDF of states at the sub-filter scale, however broad it might be, does not necessarily demand that we sample it. We may instead construct suitable averages from the PDF to obtain a deterministic functional relationship. The stochastic strategy does become important, however, if the departures from the average are not merely manifest as noise at scales close to the grid scale and the timestep of the model. Local variability may cascade upscale in ways that influence systematically the quantitative, and possibly even qualitative, behaviour of the parent model. The chaotic nature of atmospheric flows means that the possibility cannot easily be excluded. While there are many physical systems, and no doubt meteorological situations, where small-scale fluctuations are irrelevant, it is also straightforward to find many systems for which the smallscale fluctuations are essential for a realistic physical description. Many systems close to a regime transition present good examples (e.g., Williams et al., 2004). To be specific, consider a simulation of a highly idealized situation in which a prescribed cooling is applied to destabilize a horizontally uniform atmosphere, ultimately leading to moist convection. The surface condi-

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tions and the imposed cooling are also horizontally uniform. A turbulent boundary layer will be present. Supposing that the model filter is an ensemble average of the turbulent scales, then the effects of the turbulence on the filtered state must also be horizontally uniform, and the atmosphere will continue to build up instability until moist convection develops, which will happen simultaneously everywhere across the domain. However, if the model filter is a finite spatial average, then some fluctuations of the turbulence will survive the averaging operation. Such fluctuations will depend on both the nature of the turbulence and the extent of the averaging. They will however provide some inhomogeneities to break the horizontal symmetry and cause moist convection to break out locally, and also earlier than for the simulation with ensemble-averaged turbulence. The above example is extremely idealized in order to give a specific illustration, and it may very reasonably be criticized as unrealistic since such a purely homogeneous situation will never arise within a climate model. Nonetheless, it serves to illustrate the distinction between deterministic and stochastic forms of parameterization, how the nature of the parameterization is dictated by the nature of the assumed filter, and how the stochastic form may prove important in some circumstances (the triggering event itself) and yet be entirely irrelevant in others. (As the instability gradually builds up, well before any moist convection occurs, the fact that individual turbulent eddies may be stronger or weaker locally is not our interest here.) What is truly missing from the current efforts is a formal procedure for deriving a practical stochastic parameterization from a filtering procedure (or alternatively, from a multi-scale asymptotic expansion). General procedures for deriving parameterizations were outlined in Vol. 1, Ch. 2 and the mass-flux convection parameterization was derived using SCA in Vol. 1, Ch. 7. Various applied mathematical methodologies (e.g., homogenization) may be useful to objectively decide whether a given subgrid-scale term is best represented in a stochastic or deterministic manner. A related difficulty is that an actual, specific filtering procedure that defines the filtered and residual parts of the flow in current NWP and climate models is never written down. Unfortunately, these issues are beyond the scope of this set.

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Scale dependence of parameterization

It is worth highlighting the simple point that the extent of the (explicit or implied) averaging must have implications for the strength of fluctuations. This is often not recognized by much of the literature that seeks to add some stochastic component to climate and NWP systems. That may be entirely reasonable if the purpose of the stochastic component is to consider uncertainty in a given parameter for example, perhaps probing for a potentially important sensitivity in the model formulation. In order to address many aspects of uncertainty, however, well-formulated stochastic methods should include a scaling of the applied fluctuations with the model grid length that can be justified from the physical motivation for the origin of the fluctuations. This is a particularly important consideration for the use of stochastic methods with models that are beginning to be developed with a varying grid length, such as ICON (Icosahedral Non-hydrostatic: Zangl et al., 2015). However, the point clearly also applies to some extent to any global model with a latitude–longitude grid. It seems fair to say that the resolution dependence of parameterizations, stochastic and deterministic, has not received sufficient attention to date. The purpose of a parameterization is to represent the effects of processes that take place below the parent model’s filter scale. It immediately follows that any change to the filter (e.g., the grid length) may require changes to the representation because of changes to what is sub-filter. Most parameterizations contain no explicit recognition of the length and timescales on which they operate. That may be an acceptable approximation over some range of the filter length and timescales, in terms of an ensemble mean of the unresolved processes. However, it is unlikely to be a good assumption to make within the convection-permitting regime (cf., Ch. 19) and will normally be implausible for the stochastic aspect of a parameterization. For a specific example of some issues of resolution dependence, see the discussion by Rauscher et al. (2013) of aqua-planet experiments across a range of resolutions and also on a variable-resolution grid, which together highlight difficulties with scale dependence. It is worth comparing that study to Zarzycki et al. (2014), which indicates that the issues may be strongly dependent on the convection parameterization in question. Sun et al. (2013) provided an illustration of how model convergence in simulations of a tropical cyclone depends on the convection parameterization used. The scale dependence of convection parameterizations is discussed fur-

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ther below, but Boutle et al. (2014) may also be noted here as a good example study in the context of microphysical schemes. It estimates from observations a dependence of the distributions of liquid water and rainwater on the grid-box size and demonstrates that it is useful to account for such scale dependence in microphysical calculations of autoconversion and accretion (cf., Ch. 23). 5.1

Further considerations close to the grid scale and timestep

Although stochastic terms are implied by a spatial filter, it is necessary to pay attention to their interactions with sources of model uncertainty arising from the need for numerical calculations of the dynamics of the filtered flow. Even supposing that parameterizations were able to deliver a perfect representation of the effects of sub-filter processes on the filtered flow, numerical issues would remain. One way to think of this is in terms of an effective model filter which depends upon the numerics. A given parameterization suite operating on a given grid may nonetheless produce important differences in the results if the advection scheme is changed (e.g., Brown et al., 2000). Idealistically, a parameterization designed to represent sub-filter scales based on a spatial filter would recognize and adapt to the effective filter and not merely to the nominal grid length. This may be a counsel of perfection, and there do not appear to be any convective or cloud parameterizations that explicitly account for the numerics of the parent model. However, it should be noted that some attempts have been made to separate boundary-layer flows into components above and below an effective filter scale (e.g., Beare, 2014). The actual behaviour of mass-flux convective parameterizations close to the grid scale and timestep is very interesting, in that the parameterizations are not generally formulated to be stochastic, and yet the tendencies produced may be sensitive to small perturbations in the input grid-scale state. The outcome, therefore, is often far from the smooth sequence or pattern that one would expect to obtain from an ensemble average applied to a smooth large-scale field. Strong on/off behaviour with strong timestep to timestep variability is often found. Examples can be seen in Plant and Craig (2008, their Fig. 4d) and Whitall (2012, their Fig. 2.15). This smallscale, high-frequency variability is well known by model developers but has not been well characterized or studied, and there is little in the journal literature that explicitly addresses the issue and its possible consequences.

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One exception is Stiller (2009) who highlighted problems that such noisy behaviour can produce for data assimilation. His Fig. 2 shows that autocorrelation of parameterized convective tendencies decays on a timescale that is effectively set by the model timestep rather than anything physical. There is almost no correlation in the tendencies at neighbouring timesteps. This example comes from some experiments with the Gregory and Rowntree (1990) scheme in the MetUM, but the more general issue is not specific to that scheme or model. Further description is provided by Willett and Milton (2006, their Sec. 5.2). Whenever the deep convective parameterization is activated it stabilizes the atmosphere by warming the column above the initiating layer and it is also likely to cool the column below via downdraughts and the evaporation of precipitation. Thus, the parameterization tends to stabilize the column against activation at the following timestep, particularly from levels close to the cloud base. A consistent response from the parameterization would require that the stabilization is closely and carefully balanced with the destabilization mechanism on each individual timestep. It is then easy to appreciate that any excess of parameterized activity may induce on/off activation behaviour. Given that an important consideration for the settings of an operational model is that it should not fail during its simulations, the fact that stabilization is sometimes liable to be a little excessive for a single column at a single timestep is hardly surprising. Artificial on/off noise is present in many models and may be having upscale effects which are not currently understood. The issue would seem to be worthy of some dedicated study, not least because artificial noise in the output from, say, the convection parameterization implies that noise will be present in the input to other parameterizations and to the dynamics. One look at the implications for physics–dynamics coupling is the study of Hodyss et al. (2013), which highlights that the impact of noisy physics will depend on the numerical scheme used for the dynamics. For example, noise of a diffusive character may lead to instability for an advection scheme that is off-centred in the direction towards implicitness. A further implication for stochastic parameterization methods for convection is the need for careful checks that the stochastic parameterization actually behaves as designed, and does not merely induce on/off noise with different characteristics. To achieve this, it may be necessary to take steps to remove the artificial noise before attempting to impose the physically based noise.

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155

Backscatter schemes

In general terms, we expect kinetic energy to cascade to progressively smaller scales, as motions on any given scale are dissipated by the growth of eddies on smaller scales. Kinetic energy in a turbulent fluid ultimately cascades to the very small scales at which the energy is lost through molecular viscosity. In numerical models the kinetic energy is lost much sooner, at the near-grid scales, either through numerical diffusion, or else by means of a diffusion that is deliberately introduced in order to suppress numerical non-linear instabilities associated with any build-up of energy towards the grid scale. Although the net energy cascade is downwards, there are also substantial energy transfers upscale. Some of the kinetic energy on scales near or below a model’s grid size would in reality backscatter upscale, and modify the motions at resolved scales. Explicit demonstrations of the mechanism can be found in the spectral transfer analysis of the barotropic vorticity equation by Thuburn et al. (2014) and from a coarse-graining of the vorticity equation in the ECMWF model by Shutts (2013). An upscale cascade process from unresolved scales cannot directly occur in a model, however. A consequence is that atmospheric models underestimate the kinetic energy in a broad part of their resolvable spectrum, to a worsening degree as the grid scale is approached (e.g., Shutts, 2005). With this motivation, methods have been developed to randomly backscatter kinetic energy onto near-grid scales. The approach was first developed for large-eddy simulations (Mason and Thomson, 1992) where it can be valuable in improving flow statistics close to boundaries. Here, the size of the main energy-containing eddies scales with distance from the boundary and so will approach the grid scale (e.g., Weinbrecht and Mason, 2008). Frederiksen and Davies (1997) first suggested the use of stochastic backscatter in GCMs and considerable efforts have been made since. Key issues that are important for the method are the estimation of how much energy has been unphysically diffused from model winds at poorly resolved scales and should be reinjected, and also the scale dependence of the pattern of winds to be used for the stochastic perturbations. Shutts (2005) developed a stochastic kinetic energy backscatter scheme (SKEB) for the ECMWF ensemble prediction system, in which a randomly generated streamfunction forcing field is added to the model winds. For further developments of SKEB which discuss these issues in some detail, see: Berner et al. (2008, 2009) for developments at the ECMWF;

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Bowler et al. (2009) and Tennant et al. (2011) for developments within MOGREPS; and Berner et al. (2011) for some experiments with the WRF (Weather Research and Forecasting) model. The method does benefit the kinetic energy spectrum and increases ensemble spread, and a lessening of a model bias through improved blocking statistics has also been highlighted (Berner et al., 2008; Tennant et al., 2011). A recent study (Sanchez, 2014; Sanchez et al., 2014) expresses some caution about the effects on blocking statistics, but does show that it produces some clear benefits for the tropical precipitation distribution and some further upscale effects (both positive and negative) on the spectrum of convectively coupled equatorial waves. Interestingly, the inclusion of noise from SKEB actually reduces the on/off behaviour and so also the high-frequency variability of the convection parameterization. 7

The physics of fluctuations

In the remainder of this chapter, some physically based (or at least physically motivated) approaches to stochastic effects in convective parameterizations will be discussed. The need for the characteristics of a noise term to respect the characteristics of the underlying physics leading to that noise is stressed in many studies, books and reviews. Good starting points are provided by Miguel and Toral (2000) and van Kampen (2007), with many useful references therein. Some examples are also given in Sec. 3.6, and so there will simply follow some brief additional remarks. In order to write down any specific form for the stochastic aspect of a parameterization, one is obliged to make some assumptions about relevant physics. There is no good reason why one should not consider such assumptions on the same footing as for any other assumptions made in parameterization design. Thus, the assumptions should be as carefully formulated, as fully justified, and as well tested as possible, and any compromises, simplifying assumptions, or ansatzes that may be required to make a practical implementation of an idea should be identified. As a specific and very simple example, consider the choice of variable which is to be subject to a source of noise. Suppose that we wished to introduce an additive noise contribution to the potential temperature equation, ∂θ + u.∇θ = Pθ (X, α) + ǫ. ∂t

(7.1)

Here, Pθ represents deterministic parameterizations that depend upon a set

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of parameters α and the model state X. ǫ is the noise term, the properties of which we do not need to specify for this discussion. We might then decide to reformulate our model in terms of the transformed variable η = eθ , which will evolve according to: ∂η + u.∇η = Pη (X, α) + ǫη, ∂t

(7.2)

so that the additive noise has become a multiplicative noise. Such terms merely describe the form of the equation being used and say nothing meaningful about the noise unless the variable in consideration is also specified. In other words, choosing an appropriate variable to subject to an appropriate source of noise requires some important assumptions about the physical origin of the noise, and it is impossible to be agnostic on the matter. One approach towards formulating the stochastic aspects of a process is to try to allow oneself to be guided by data as much as possible. Good examples are Lin and Neelin (2002), who attempted to parameterize convective heating statistics directly; Dorrestijn et al. (2013), who built a statistical emulator for BOMEX (Barbados Oceanographic and Meteorological Experiment) simulations of shallow convection; and, Gottwald et al. (2014), who used observational data to train random conditional sampling and Markov chain formulations for the convective-cloud fractional area. However, one must be very cautious about the range of validity of the purest forms of a data-driven analysis, and even the best representations of the data may not necessarily perform well when embedded for parameterization purposes. As concluded by Lin and Neelin (2002), it is “not very prudent to develop stochastic physics schemes outside an atmospheric model framework.” Or, as stressed by van Kampen (2007), “for internal noise one cannot just postulate a non-linear Langevin equation or a Fokker–Planck equation and hope to determine its coefficients from macroscopic data.” 8

Stochastic convection closure

The basis of the mass-flux formulation is that moist convection is characterized by deep cumulus clouds, or “hot towers”. Thus, the convective instability is released in a discrete fashion: within a given area, we find a finite number of deep convective clouds. Let us make a simple estimate of the number of such clouds that we might expect to find in a typical GCM grid box of size, say, (100 km)2 . In Vol. 1, Ch. 8, a typical cloud-base mass flux was stated to be MB = ρσw ∼ 1 × 10−2 × 1 = 10−2 kgm−2 s−1 .

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A cloud of radius 1 km ascending at 1 ms−1 is associated with a flux of ρπr2 w ∼ 3 × 106 kgs−1 . Combining these two estimates implies that there will be of order 30 clouds within the grid box, or a cloud every (17 km)2 on average. At higher resolutions, typical of NWP applications or of some of the newer climate simulations, the grid-box area approaches ∼ (17 km)2 . Here, the grid boxes might contain one cloud on average, but any particular grid-box area may very likely contain no clouds, one cloud, or several. This is a situation where the ensemble-averaged subgrid state and particular realizations of subgrid states may be very different (see Sec. 4). If the expected number of deep convective clouds is insufficient to produce a steady response for a given steady forcing, then a stochastic parameterization may be appropriate. The situation is straightforward to demonstrate explicitly by coarse-graining cloud-resolving model outputs (Plant and Craig, 2008; Shutts and Palmer, 2007). In order to compute the fluctuations arising from averaging a field of convective clouds over a finite region, a statistical mechanics approach can be used (Craig and Cohen, 2006). The assumptions used are entirely consistent with the standard assumptions of a mass-flux formulation, most notably that the deep convective clouds are independent, embedded within a uniform environment and that the full cloud ensemble at large scales is in a state of equilibrium with the large-scale forcing. The independence of clouds implies a Poisson distribution of cloud number within a given area. Furthermore, the distribution of mass flux amongst the clouds under these assumptions is exponential, as originally shown by Craig and Cohen (2006) and as also rederived in Vol. 1, Ch. 4, Sec. 6. The convolution of the cloud number distribution and the mass-flux distribution of individual clouds leads to the final result for the fluctuations in total mass flux:    2  M + M 

M  1 exp −

M M , (8.1) I1 p(M ) =

m M

m

m

where I1 is a modifed Bessel function, m is the mass flux of an individual cloud, and the angled brackets denote an ensemble average. The theoretical predictions are in excellent agreement with equilibrium CRM simulations that respect the theoretical assumptions about the forcing, irrespective of the forcing strength and of the height at which the statistics are computed (Cohen, 2001; Cohen and Craig, 2006; Plant and Craig, 2008). They also work surprisingly well in other conditions, including organized convection in the form of squall lines (Cohen and Craig, 2006), and for time-varying simulations with interactive radiation (Davoudi

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et al., 2010). In these conditions, the clouds are no longer independent and non-interacting, but the cloud number distribution nonetheless remains sufficently Poisson-like for Eq. 8.1 to continue to work well (cf., Vol. 1, Ch. 4, Sec. 6). The Craig and Cohen (2006) theory was developed into a full parameterization by Plant and Craig (2008). The scheme is structured as follows: (1) An average in the horizontal and over time is performed in order to determine the large-scale state. (2) Properties of the equilibrium statistics are determined dependent upon that large-scale state. The ensemble-mean cloud-base mass flux M  is determined from the scheme’s CAPE-based closure1 , and the mean mass flux of a single cloud m must also be determined. (3) Given the above quantities, the theoretical PDFs are then fully specified for the number and the properties of the clouds within the grid box. Those PDFs are sampled randomly to determine how many clouds are present and their associated mass fluxes. (4) The vertical profile for each sampled cloud must be determined through a plume model, an adpatation of the Kain and Fritsch (1990) approach being used in practice. (5) Output tendencies are computed for the sampled set of clouds. Two aspects of this parameterization raise general issues and are worth further comment. First, note that m is an important parameter in determining the extent to which a convective cloud ensemble has to be considered as being composed of discrete elements: it sets a scale for the fluctuations. In principle, it may be a function of the large-scale state or forcing, but available CRM data shows that at the cloud base it is a weak function, and so a constant value is used in the parameterization (Plant and Craig, 2008, their Sec. 3a). The main data source for this statement is the experiments of Cohen (2001) and Cohen and Craig (2006), which were conducted at a relatively coarse resolution (by modern standards) of 2 km. It is possible that this conclusion may have been influenced by the CRM grid length, since this may have exerted effects on typical updraught areas in the simulations (cf., Hanley et al., 2015). These experiments have recently been revisited by Scheufele and Craig (2014) using higher resolutions. They found some 1 More specifically, the closure computes a dilute CAPE for a spectrum of cloud profiles, and combines those into a weighted average using the exponential spectral distribution. The closure removes the weighted dilute CAPE over a specified timescale (cf., Vol. 1, Ch. 11). Other closures for M  could also be applied.

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increase of m with forcing strength, but reaffirmed that this is a much weaker effect than the changes in cloud number. Second, notice that the parameterization distinguishes between the gridscale state and the large-scale state, with the PDFs being dependent only on the latter. This is important both theoretically and practically. If noisy grid-scale input is used, then fluctuations in the input state can damage the closure calculations which should produce a smoothly varying M  that balances the large-scale forcing. Noisy input can instead produce an artificially fluctuating M , and this is the reason for the averaging of step (1). The grid-scale fluctuations here are entirely deliberate and need to be carefully controlled. This may be compared to the on/off triggering behaviour discussed in Sec. 5.1. Many of our deterministic schemes do not actually perform as ensemble-averaged methods with smooth responses to smoothly varying conditions, but rather more as though they were spatially averaged methods with uncontrolled, ad hoc noise. Keane and Plant (2012) demonstrated these points for radiativeconvective equilibrium experiments with prescribed radiation in a threedimensional domain with parameterized convection. This is a simple situation that corresponds to very basic thinking about tropical convection, so it is an important test of any parameterization that it should be able to describe this well, i.e., in agreement with basic characteristics of equivalent CRM experiments. However, it is not a trivial test and Keane and Plant (2012) described how some parameterizations generate spurious selforganization instead of random, scattered convection. The theoretically predicted distribution of mass flux in Eq. 8.1 can be successfully reproduced by the Plant and Craig (2008) scheme, regardless of the model resolution and of the area over which statistics are computed. An alternative, if indirect, approach could be to smooth the output from noisy closure calculations, although this appears to be much less effective than input averaging (Keane, 2011, personal communication). However, a similar effect might also be achieved as a by-product of a prognostic closure method (e.g., Pan and Randall, 1998; Wagner and Graf, 2010; Yano and Plant, 2012, cf., Vol. 1, Ch. 11, Sec. 10), which implies a memory of convective-ensemble evolution and hence a physically based autocorrelation between timesteps. Some other discussions of convective memory and its effects can also be found in Davies et al. (2013); Gerard et al. (2009); Jones and Randall (2011); Piriou et al. (2007); Scinocca and McFarlane (2004). Cellular automata might provide an alternative mechanism for the treatment of memory, and this is discussed further in Sec. 12.

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The Plant and Craig (2008) scheme accounts for the stochasticity due to finite cloud-number by sub-sampling the spectrum of cloud types at equilibrium. A natural question to ask is whether this might be extended to account for such variability in non-equilibrium systems. As shown by Plant (2012), this can be achieved by allowing the probability of producing an individual cloud to vary with time in a way that is modulated by the evolving cloud-work function, and which is fully consistent with the relevant prognostic differential equations in the limit of large system size. A simple numerical example is given in Fig. 20.3. 0.02

0.018

0.016

í1 í2

Mass flux (kgm s )

0.014

0.012

0.01

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0.006

0.004

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0

0

1

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3 Time (hours)

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Fig. 20.3 An example solution of the ordinary differential equations of Pan and Randall (1998) (dotted line) is shown alongside numerical results from a stochastic model that is equivalent in the limit of large system size. The actual system size used was such that there are an average of ten clouds present at equilibrium within the simulated domain. The solid line shows a single realization of the individual-level model and the dashed line the ensemble mean from 100 realizations.

The Plant and Craig (2008) scheme has been tested for ensemble forecasting over Europe by Groenemeijer and Craig (2012) for an implementation in the COSMO model at 7 km grid spacing. One hundred simulations were performed for each of seven cases, with ten realizations of the stochastic scheme for ten sets of initial and boundary conditions. The relative impact of the scheme was strongly dependent on the weather regime, with modest effects on cases with strong synoptic forcings for which large-scale uncertainty dominates, but dominating the grid-scale precipitation variability for cases that were weakly forced on the synoptic scales. The Plant

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and Craig (2008) scheme introduces noise at the grid-scale of the model and does not apply a pattern generator. However, the impact of the noise does propagate upscale, and was to shown to contribute substantially to variability on scales of 5 × 5 grid boxes. Kober et al. (2014) further assessed the skill of cases of strong and weak large-scale forcings from this set against radar observations. An important aspect of this analysis is the need to consider the dependence of skill on scale by using spatial verification techniques (see Ch. 21), such as the fractions skill score (Roberts and Lean, 2008, see also Ch. 21, Sec. 5.3), given that unpredictable noise is deliberately being applied at the grid scale. The stochastic aspect of Plant and Craig (2008) provides benefits for the weak large-scale forcing case but the precipitation is overly fragmented in the strong forcing case. Further analysis by Selz and Craig (2014) indicates that the processes of upscale error growth from convective uncertainties can be well reproduced by the Plant and Craig (2008) scheme, in good agreement with the behaviour of large-domain simulations in which the convection is simulated explicitly (Selz and Craig, 2015). The scale dependence of the approach was assessed by Keane et al. (2014) in aqua-planet simulations at three different resolutions. The PDF of precipitation obtained at the grid scale varies with resolution in a realistic manner, thereby showing an appropriate scale adaptivity. However, the PDFs from simulations at different resolutions collapse very effectively when computed for the same spatial scale, thereby showing good scale consistency. The use of a spatially averaged atmospheric state for input to the closure calculations is valuable for producing the correct scalings. The spatial averaging also produces more extreme, local precipitation values at the higher resolutions by allowing repeated precipitation at the same location and so avoiding artificial on/off behaviour due to too-strong local stabilization. A case study in the COSMO ensemble by R¨ ohner et al. (2013) has also shown benefits of this parameterization for predicting an extreme precipitation event. The Craig and Cohen (2006) and Plant and Craig (2008) approach has recently been extended to an analysis of the discrete character of shallow convection by Sakradzija et al. (2015). The shallow cumulus diagnosed in large-eddy simulations (LES) of the RICO (Rain in Cumulus over the Ocean) case (van Zanten et al., 2011) fall into two types: an active mode with a buoyant core, and a passive mode which dominates in terms of cloud number but not in terms of the vertical transports. An exponential distribution for mass flux would be a reasonable approximation for each of the

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modes (see Sakradzija et al., 2015, their Fig. 4a) but a linear superposition of Weibull distributions provides a better fit:    k  k  m k−1 m . (8.2) exp − fi p(m) = θi θi θi i Here, i labels the mode, fi is the fraction of the cloud ensemble, and ki and θi are the distribution parameters. The exponential shape is recovered by choosing k = 1 and the departure from that (to k = 0.7 in the simulated case) can be interpreted in terms of a memory carried by the cloud life cycles. Combining this distribution for mass flux with a Poisson distribution for cloud number, those authors were able to reproduce statistics of the variability of the cloud field and its scale dependence with a simple stochastic model. The analysis has yet to be developed into a full parameterization, but it strongly suggests that a stochastic parameterization for shallow convection might be formulated analogously to the Plant and Craig (2008) scheme with modest adaptations. Another approach to the stochastic aspects of closure is the cellular automata method explored by Bengtsson et al. (2013), which will be described further in Sec. 12. Two earlier studies on stochastic closure should also be mentioned here. Lin and Neelin (2000) added a stochastic term to the CAPE that was input to the convection parameterization in an intermediate complexity tropical atmosphere model. This led to improvements in slow modes of variability but was found to be highly sensitive to the autocorrelation timescale applied for the perturbations. A follow-up study, Lin and Neelin (2003), compared such stochastic perturbations to the closure with perturbations applied to the vertical heating profile. The latter approach had weaker effects on the model’s variability, but it is interesting to note that its effects were focused towards larger scales. Other perturbation strategies dealing with entrainment are discussed immediately below. 9

Stochastic entrainment

The stochastic nature of entrainment processes was discussed in Vol. 1, Ch. 10, Sec. 4.6, but it is useful to revisit here in the context of explicitly stochastic treatments within parameterizations. The so-called “stochastic mixing scheme” of the Emanuel (1991a) parameterization is based on the picture established observationally by Raymond and Blyth (1986) of entrainment as being episodic as opposed to a

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continuous mixing process. The formulation envisages contributions from many sub-plume elements, each of which experiences random entrainment events. However, assuming that there are many such elements which comprise the full plume, the actual implementation by Emanuel (1991a) was deterministic rather than stochastic. In other words, the distribution of all possible sub-plumes and their possible entrainment events is effectively summed over rather than sampled. The study of Byun and Hong (2007) makes a random selection of the convective cloud top within a simplifed Arakawa–Schubert scheme (based on Grell, 1993; Hong and Pan, 1998) that conceptually treats only the deepest possible cloud type. The entrainment parameter had to be reset for consistency with the random cloud top, so in effect the method amounts to a randomization of the entrainment. The method appeared to show some promise, both in a single-column model and seasonal forecasts, but those authors and their analysis were clearly little motivated by its stochastic aspect and more by the notion of producing a crude but cheap implementation of a spectral approach allowing different cloud types. It should therefore be considered as a Monte Carlo approach. Analyses of LES of shallow convection by Romps and Kuang (2010b) and Nie and Kuang (2012) focused on the variability of thermodynamic properties amongst cloudy udpraft parcels. A note of caution is that this variability is related to but clearly not the same as the variability in thermodynamic properties of convective plumes, nor indeed the same as the variability that might be associated with a bulk plume. Nonetheless, it is clear from these studies that neither the pointwise variability (Romps and Kuang, 2010b) nor the response of the cloud field to perturbations in the environmental state (Nie and Kuang, 2012) can be explained in terms of sub-plumes undergoing entrainment at a steady rate. Rather, an explicit recognition of episodic entrainment events is necessary. The key LES results could be reproduced with a model (Romps and Kuang, 2010a,b) in which parcels underwent entrainment episodes at random, with uniform probability per unit of distance travelled. The proportion of entrained air to parcel air in an entrainment episode was also taken as a random variable, somewhat arbitrarily assigned an exponential distribution. An explicitly stochastic treatment of episodic entrainment within a shallow-convection parameterization was proposed by Su˘selj et al. (2013) in the context of the eddy diffusion and mass flux (EDMF) approach (e.g., Soares et al., 2004; Su˘selj et al., 2012, see also Vol. 1, Ch. 7, Sec. 4.4.1). A dry plume is considered below the cloud base, and within-cloud this is for-

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mulated as ten sub-plumes which inherit somewhat different properties at cloud base and for which the entrainment is considered stochastically with the probability of an event scaled as dz/L0 with dz the distance travelled and L0 set at 10% of the cloudy part of the boundary layer. A difference from the parcel model developed by Romps and Kuang (2010b) is that each entrainment event involves a fixed proportion of environmental air, specifically 10% of that in the sub-plume. In contrast to fixing an equivalent mean entrainment, the sub-plumes in the Su˘selj et al. (2013) treatment are able to penetrate to a range of depths.

10

Stochastic aspects of triggering

The role of triggering as an on/off switch for the convection parameterization was discussed in Vol. 1, Ch. 11. As discussed in Sec. 5.1 of the present chapter, many models exhibit on/off behaviour as the instantaneous, grid-scale profile is allowed to fluctuate between states adjudged favourable or unfavourable for convection according to the triggering criteria in use. Given that these on/off fluctuations are essentially unphysical, and certainly uncontrolled, it is perhaps not unexpected that several studies have experimented with introducing a stochastic aspect to the trigger. Although these perturbations are often ad hoc, by altering the character of the on/off behaviour such studies are of interest in providing indications of how the on/off noise in deterministic parameterizations may be having some impact. Bright and Mullen (2002) applied stochastic perturbations to the vertical velocity value used in the trigger calculations of the Kain and Fritsch (1990) parameterization (see Vol. 1, Ch. 11, Sec. 13.3). Together with a similarly motivated perturbation to the critical Richardson number in the boundary-layer scheme, increases in spread and probabilistic skill were obtained for ensemble forecasts of the North American monsoon. Song et al. (2007) also implemented a stochastic variant of the Kain and Fritsch (1990) trigger, including several random parameters within the trigger formulation. Their treatment was rendered somewhat less ad hoc by using a Bayesian learning procedure with radar data in an attempt to estimate the distribution of these parameters. The outcome was that precipitation features could be generated with more spatial variability and thus with more fidelity to the radar patterns. A more systematic attempt to analyse stochastic aspects of convection

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initiation in LES has recently been pursued by Rochetin et al. (2014a,b). The simulation was for an AMMA (African Monsoon Multidisciplinary Analysis) case. The purpose of their analysis was to construct estimates characterizing the largest and most energeric boundary-layer thermals at the cloud base for use within the triggering conditions for the LMDZ5B (Hourdin et al., 2013) implementation of the Emanuel (1991a) parameterization. The LES analysis considers the population of thermals at the cloud base prior to the initiation of deep convection as being comprised of a two-mode exponential distribution,   fi S exp − , (10.1) p(S) = Si Si i

with i = 1 labelling a large population of small thermals, and i = 2 a much smaller population of stronger thermals (cf., Sakradzija et al., 2015). Here, S denotes their horizontal area, fi is the fractional number of thermals in each mode and, Si is the mean horizontal area for a thermal belonging to that mode. The parameter S2 was set to be a linear function of boundarylayer cloud depth and it was assumed that the initiation of deep convection is produced by the largest thermals in the ensemble. If there are N2 thermals of the strong mode in the area of interest, then the median value for the largest thermal size may be estimated as:  N2 Smax = S2 ln (10.2) ln 2 (Rochetin et al., 2014a, their Appendix B). The interest in Smax arises from s the idea that the vertical velocity distribution can be sampled on Smax /˘ independent occasions to obtain an estimate for the maximum vertical velocity within the largest thermal, and it is the median estimate for this quantity Wmax which is used in the triggering conditions. Here, s˘, the area over which samples of the vertical velocity are considered independent, was rather arbitrarily set to the LES grid length of 200 m, although fortunately the final results seem to be not overly sensitive to this choice. A key assumption made in the estimation of Wmax is that the vertical velocity distribution is uniform across and within the thermals, i.e., that both the mean velocity of a thermal and the intrathermal variability (assumed Gaussian) are independent of the size of the thermal at the cloud base. With the above assumptions in place, the result for Wmax is:

  1/2 2 2 Smax Smax Wmax = wp′ + wp′ ln − ln ln , (10.3) 2π(˘ s ln 2)2 2π(˘ s ln 2)2

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where wp′ is the mean in-thermal velocity. The final result for Wmax is used deterministically within the triggering 2 conditions, which require that (1/2)Wmax > |CIN| as a necessary condition to overcome the CIN barrier. The stochastic element of the formulation arises from the notion that there must be sufficient thermals present of a sufficient size. Defining a threshold size as Strig , the probability that any one thermal is smaller than the threshold size is 1 − exp(−Strig /S2 ). If there are N2 thermals present and if we define τ as a decorrelation time over which a new, independent set of thermals is generated, then the no-triggering probability that no thermal will exceed the size Strig within timestep Δt becomes:  N2τ∆t

 −Strig . (10.4) PΔt = 1 − exp S2 This aspect of the triggering is considered as a stochastic event, so that a random number from [0, 1] is chosen at each timestep and compared to PΔt . Note that N2 is proportional to the grid-box area and thus the method is scale aware, with a greater chance of triggering over a larger area. There are several strong assumptions within the derivation, not all of which were expounded upon above, and although they appeared to hold well (perhaps surprisingly well) for the LES data examined (Rochetin et al., 2014a), the generality of the results is not easy to judge. Careful analyses of other initiation episodes would also be useful (see further discussions in Vol. 1, Ch. 11, Sec. 11.5). Various issues with convection-permitting resolutions were discussed in Ch. 19. Another important aspect of such models is the initiation of explicitly simulated convective cells. In some regimes, a specific initiation mechanism such as a convergence line (Leoncini et al., 2013; Warren et al., 2014) or effects of orography (Soderholm et al., 2014) or upper-level cloud (Marsham et al., 2007) may be captured by the model, and so explicit convective cells can be be produced at an appropriate location. As alluded to in Sec. 4, however, there is also an important regime particularly associated with convective equilibrium, in which the actual initiation locations are not predictable, only the general area within which conditions are favourable for initiation. The importance of distinguishing these regimes for predictability, data assimilation, and ensemble analysis is a topic of increasing interest (e.g., Craig et al., 2012; Dey et al., 2014; Keil and Craig, 2011). In the equilibrium regime, small unresolved boundary-layer fluctuations can easily shift the locations of precipitating convective cells, and in an ensemble

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context it may be useful to introduce such fluctuations directly (Leoncini et al., 2010, 2013). 11

Stochastic convective vorticity

The stochastic convective vorticity (SCV) scheme of the UK Met Office is an unusual but interesting approach since it attempts to account for an effect of organized convective systems that is important but uncertain in a GCM. Mesoscale convective systems (MCSs) contain vertical gradients in the diabatic heating on relatively large horizontal scales, and these are associated with characteristic features in terms of potential vorticity: specifically a mid-level positive (cyclonic) anomaly and an upper-level negative (anti-cyclonic) anomaly. Such structures are typically not well represented in models (Gray, 2001) and the SCV scheme adds a resolved-scale vorticity dipole in the vicinty of convection (Bowler et al., 2008b; Gray and Shutts, 2002). Among the various parameters that describe the dipole, most are simply fixed but the dipole radius a is allowed to vary stochastically. It is given by: √ P CAPE , (11.1) a = αǫ |f | where P is the convective precipitation, ǫ is a random number uniformly distributed from [0, 1], and α is a tuning value. Here, the scaling of √ CAPE/|f | is taken from the adjustment arguments of Shutts (1987) whilst the proportionality to convective precipitation simply reflects an expectation that a more intense convective event is more likely to be larger in scale and more organized. It also provides a direct link to the convective parameterization. In the practical implementation, the Coriolis parameter f is assigned a minimum value of 2 × 10−5 s−1 . The approach was found by Gray and Shutts (2002) to have a neutral effect on model climatology, but it did induce effects on tropical cyclones and the southern hemisphere storm track (via sub-tropical convective disturbances that feed into it). 12

Lateral communication and cellular automata

There are long-standing issues in NWP and climate models regarding propagation and organization of convection. At least in part, this may be due to the fact that in an independent-column parameterization approach, there

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is no treatment of horizontal transports of heat, moisture, or momentum due to convection. In nature, dissipative systems involving microscopic irreversibility, or those open to interactions with their environment, may evolve from disordered to more ordered states. The states often exhibit complicated structures (Wolfram, 1983) such as the flow patterns in turbulent fluids and atmospheric convection interacting with the large-scale flow. Cellular automata have often been used as a simple mathematical model to describe, understand, and simulate such self-organizational behaviour. A famous study is that of Wolfram (1986), which considers a family of simple, one-dimensional cellular automaton rules. This work contributed to the study of cellular automata as important objects within statistical mechanics. The original concept, however, dates back to the 1940s and was introduced by Von Neumann and Ulam as a possible idealization for modelling biological self-reproduction (e.g., Von Neumann, 1951; Von Neumann and Burks, 1966). A cellular automaton describes the evolution of discrete states on a lattice grid, similar to that of a numerical weather or climate model. The states are updated according to a set of rules based on the states of neighbouring cells at the previous timestep. Typically, the update rule is deterministic, and is applied to the whole grid simultaneously, although stochastic cellular automata are also used, and sometimes known as locally interacting Markov chains. A stochastic cellular automaton simply means that new discrete states are chosen according to some probability distribution. Similarly to cellular automata, a Markov chain is a mathematical system that undergoes transitions from one state to another, among a finite or countable number of possible states. The changes of state are called “transitions”, and the probabilities associated with various state changes are called “transition probabilities”. As for cellular automata, the process is characterized by a state space, and an initial state across the state space. However, a Markov chain also has a transition matrix describing the probabilities of particular transitions. If this transition matrix takes into account the state of other Markov chains it can be considered a stochastic cellular automata. The given rules of the cellular automata generate self-organization of cells and complex patterns may emerge even from simple rules. With random initial configurations, the irreversible character of the evolution can lead to various self-organization phenomena (Wolfram, 1983). The famous cellular automaton called “life”, or “game of life” is a good example. It was

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proposed by John Conway in 1970, with the motivation of finding a simple rule leading to complex behaviours (Gardner, 1970). He proposed a twodimensional square lattice, in which each cell can be either alive (state one) or dead (state zero). The updating rules are that: a dead cell surrounded by exactly three living cells can be born (i.e., it changes from 0 to 1); and, a living cell surrounded by less than two or more than three neighbours dies of isolation or over-crowdedness. The game of life has unexpectedly rich behaviour and complex structures emerge out of a primitive initial state (Chopard and Droz, 2005). The irreversibility of life leads to configurations which cannot be reached by evolution from any other configuration (Wolfram, 1983). The idea of using cellular automata within NWP was first proposed by Palmer (2001). The cellular automata was proposed as a pattern generator to provide spatial structure to stochastic perturbations. The stochastic backscatter formulation of Shutts (2004, 2005) used a spatial pattern for vorticity perturbations (see Sec. 6) that was modulated with a cellular automaton pattern. The rules used were an extension to the cellular automaton family known as “Generations”, which in turn is based on the game of life but adds cell history to the rule set. Bengtsson et al. (2013) considered a two-way interaction between cellular automata and convection parameterization. The motivation was that the self-organizational characteristics of the cellular automata can allow for lateral communications between adjacent grid boxes and add additional memory (through a Generations form of cell history) to the convection. In this study, the cellular automata grid was in two horizontal dimensions, with finer grid spacing than the parent model. It was randomly seeded in regions where CAPE exceeded a threshold, and was coupled to the updraught area fraction, which was used as a prognostic variable within the deep convective closure. Both deterministic and probabilistic rules, coupled to the large-scale wind, were explored to evolve the cellular automata in time. Normally, horizontal communication only takes place via grid-scale circulations. The two-dimensional cellular automaton field in Bengtsson et al. (2011) acts on the subgrid of the numerical model, such that one cellular automaton cell aims to represent a horizontal scale characteristic of individual convective plumes; furthermore, it contains stochastic elements and has self-organizational properties. A somewhat similar idea – using a twodimensional field in order to represent convective organization – was also explored by Mapes and Neale (2011), where rain evaporation was used as

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a source for a convective-memory variable (see also Vol. 1, Ch. 10, Sec. 9). Another recent study to mention in this context is Grandpeix and Lafore (2010), who proposed a density current parameterization to describe cold pools. They also suggested that it might be extended to provide a mechanism for horizontal propagation. This explicit cold pool approach has been debated (Grandpeix and Lafore, 2012; Yano, 2012) and it is not stochastic, but it may be remarked that a stochastic aspect would appear natural in any such extension, given that cold pool propagation across a grid-box boundary must to some extent depend on the unknown and unknowable location within the grid box at which the downdraught source is assumed to occur.

13

Current status

The topic of stochastic parameterization is a recent but burgeoning one, meaning any remarks are liable to date very quickly, which makes the drawing of conclusions a dangerous business. At the time of writing, it is clear that there are strong practical benefits of introducing some stochasticity to ensemble weather forecasting. To give just one example, Berner et al. (2012) presented results demonstrating impacts from stochastic physics that are comparable to increasing the model resolution. The calls for the introduction of stochastic physics to climate models are growing stronger (e.g., Palmer, 2012) and indeed the current plans of the UK Met Office envisage the use of stochastic physics within their next submission to the CMIP (Coupled Model Intercomparison Project). At the same time, there is a widespread recognition that the existing methods used operationally are somewhat ad hoc, with various arbitrary settings and parameter choices. In the longer term, we may move towards parameterizations that are explicitly designed to be stochastic, providing physically based one-to-many feedbacks which are conditioned on the gridscale input state but which are not a one-to-one function of the input. There are various candidates emerging, as described above, as well as other approaches a little outside the scope of this set and yet more under active development. On the other hand, we may retain simple approaches such as SPPT or random parameters, but seek to refine them with physical constraints more firmly in mind. Sections 3.5 and 3.6 offered some simple examples. There may also be a middle approach in which some of the most important elements of more physically based methods can be brought to

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bear in developing pragmatic operational methods. For example, Shutts and Pallare`es (2014) might be seen as being somewhat intermediate between Plant and Craig (2008) and SPPT. Any final answer, and the ultimate balance between pragmatism and physical realism, will require better understandings of how and why small-scale, high-frequency variability interacts with large-scale, low-frequency variability in the atmosphere.

Chapter 21

Verification of high-resolution precipitation forecast with radar-based data ˇ aˇcov´ D. Rez´ a, B. Szintai, B. Jakubiak, J.-I. Yano, and S. Turner Editors’ introduction: The forecasting of precipitation by numerical weather prediction models is one of their most important and high-profile functions. In order to assess the quality of the forecasts, comparison against observations is required. Radar estimates of precipitation, with their extensive spatial coverage, provide the most natural data source for this purpose. However, the method of making comparisons is an important topic in its own right. Recent developments in the field have been motivated by the increasing use of convection-permitting models. Although these higher resolution models are subjectively highly valued by operational forecasters, a naive application of traditional point-to-point verification scores can produce worse scores than for relatively low resolution models. One possible issue is the so-called “double-penalty” problem: a localized storm that is slightly displaced spatially but otherwise well predicted would be considered as a large error at the forecasted location and as a second large error at the actual location. It may be worth stressing that such behaviour is characteristic of any point-to-point comparison measure, though of course the issues become increasingly acute when verifying higher-resolution forecasts on their own native grids. This chapter describes various modern approaches towards verification measures that are more appropriate for assessing the fidelity of convective storms, accounting for their localized and somewhat spatially unpredictable character.

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Introduction

Current operational forecast centres, especially in Europe, are pushing hard to increase the horizontal resolutions of their numerical forecast models. For example, the Consortium for Small-scale Modelling (COSMO) currently plans to reach a horizontal resolution of 500 m for their regional forecast models by 2020. As noted in Ch. 17, the ECMWF plans to increase the horizontal resolution of their global forecast model to 10 km by 2015, and to 5 km around 2020. Thus, forecast models both regional and global are now ready to resolve individual elements of convective storms. We may call this regime “convection permitting”. Under this convection-permitting regime, we face new challenges. First of all, convection parameterization must be modified, as discussed in Ch. 19. The verification methods of the forecasts must also be modified. Individual convective elements are unlikely to be predicted at exactly the right locations, but we might for example wish to assess whether the convective elements are predicted with the correct intensity within some suitably sized area. The spatial displacement of the storms might also be assessed. Within such a forecast environment, the model performance must be evaluated differently from simple mean-square-error measures. The question of how convection-permitting forecasts can be objectively verified is the theme of the present chapter. Precipitation is probably the most important information required by users of numerical weather forecasts, usually expressed as the expected amount of precipitation accumulated over a specified time period and area. This is the issue of the quantitative precipitation forecast (QPF). Although QPF has improved significantly over recent decades, it still remains the most difficult problem in operational forecasts, with large uncertainties. This is particularly the case for convective precipitation because of its rapid evolution and strong spatial variability. The importance of a proper understanding of convective dynamics and improvements to associated model physics can hardly be overemphasized. Thus, model verification techniques should be developed in such a manner that their results are helpful both for the dynamical interpretation of forecasts as well as further developments and improvements of model physics. Current operational high-resolution numerical weather prediction (NWP) models can produce results for rainfall with a horizontal resolution of the order of 1 km. This scale happens to correspond to a typical horizontal resolution of operational radar data. For this reason, radar-based

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rainfall values become extremely relevant for verifications of high-resolution precipitation forecasts. This chapter summarizes the state of the art of QPF verification, in which radar data is included as a part of the verification data sets. Verification needs to meet the demands of many diverse groups, including the model developers, the forecasters, and the end users. For example, developers may not wish to focus only on user-oriented scores. More detailed aspects of precipitation forecasts such as timing, localization, and the structure of precipitation fields would be more important for modellers. Convective rainfall with short duration, and heavy local rainfall in areas with rapid hydrological responses are difficult to forecast quantitatively. The forecast performance strongly depends on the convection parameterization adopted in a high-resolution NWP model. Thus, QPF verification must be designed to provide baseline information for developing and improving convection parameterizations. In particular, it must provide an objective tool for better quantifying the sensitivities of forecasts to choices within convection parameterizations. The next section reviews some principles of radar measurements and considers radar-based products as verification data sets. The following sections examine both traditional and spatial verification techniques and their application in verifying high-resolution QPF. Section 3 summarizes traditional techniques, which are still common in the verification of highresolution precipitation. Spatial verification, which relaxes the condition of an exact match to the observation at fine scales, is the topic of Sec. 4, and three examples of spatial verifications are more specifically considered in Sec. 5. Section 6 reviews the use of polarimetric measurement for verifications of microphysics. The concluding section summarizes fundamental aspects and formulates an outlook on high-resolution verification.

2

Verification data

Two basic data sources are used for the verification of precipitation forecasts: point-wise ground measurements by gauges, and volume-based radar reflectivity data. Gauge measurements provide direct information about point-wise precipitation. However, they have major limitations due to the limited gauge station distribution and the issue of the representativeness of single-point measurements. In order to use radar measurements to estimate rainfall data, the measured radar reflectivity Z must be transformed into

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a rainfall rate R by using a suitable form of Z–R relationship. Errors and imperfections in radar precipitation values must also be eliminated, based on the principles of radar-volume measurements and radar-scan strategy. A merger of both ground-based gauges and radar is also recognized as a good way of compiling verification data. 2.1

Ground precipitation measurements

Gauge measurements give direct information about point-wise values of rain intensity and rain amount for various accumulation periods. In considering gauge data, we often meet the term “ground truth”. Comprehensive monographs cover the state-of-the-art gauge-based precipitation measurements (e.g., Sevruk, 2004; Strangeways, 2007). There are many problems including a limited collector size, evaporative loss, out-splashes, and effects of wind. A basic problem in using gauge data for QPF verification is related to the estimates of area distribution of rainfall values. The spatial density of gauges can easily be too low for capturing convective rain distribution, because the resulting single-day rainfall can differ significantly even over just several km. We typically find only one operational gauge over 50–100 km2 in central Europe, and the gauge density is much less over the continents of the southern hemisphere. It is difficult to define universal gauge representativeness because it depends on gauge type, precipitation type, orography, and other factors. Nevertheless, it is commonly assumed that a gauge measurement represents a true value in the radar pixel that covers the gauge position. Pairing of radar pixel values with gauge data is a basis for statistical correction of the radar-based rainfall estimates. Several experiments have been performed using local gauge networks of a high density. For example, Wood et al. (2000) studied the differences in 15-min rainfalls among eight gauges located over an area of (2 km)2 . The standard deviation against the mean of the eight measurements increased with rainfall value and reached about 4 mm for 10 mm rainfall over one rain gauge. Quantification of spatial variability of precipitation is a topic that requires more intensive study, especially for verifying convection representation in models more objectively.

Verification of high-resolution precipitation forecast with radar-based data

2.2

177

Determination of radar-based rainfall from radar reflectivity measurements

Weather radar is commonly considered capable of capturing the spatial distribution of precipitation well, but in a relative sense. A radar-scanned area provides a reflectivity field (Plan Position Indicator: PPI) which is typically transformed into a horizontal distribution for several elevation levels (Constant Altitude Plan Position Indicator: CAPPI). The horizontal resolution of radar reflectivity data is, typically, of the order of 1 km, and typical radar pixel areas are from (1 km)2 to (5 km)2 . Weather radar operates by emitting pulses of microwave radiation and sampling the backscattered power. We can use a general radar equation (e.g., Doviak and Zrnic, 1984; Meischner, 2004) to deduce the radar reflectivity from the average received power. The radar reflectivity, which is the sum of all backscattering cross-sections in a unit volume, can be related to the radar reflectivity factor Z, which is a sum of the cross-sections of the individual drops within a unit volume. The radar reflectivity factor is meteorologically a more meaningful way of expressing the radar reflectivity and is often just referred to as radar reflectivity (Meischner, 2004). For a given drop-size distribution N (D), the radar reflectivity factor Z is defined by:  ∞  N (D)D6 dD, (2.1) D6 = Z= V

0

where V is a unit volume, D is the diameter of a spherical particle and N (D) is the size distribution. Unfortunately, the assumption of Rayleigh scattering is a coarse approximation for hydrometeors, and it is generally not valid in the atmosphere. For this reason, the convention is that Eq. 2.1 is used to compare the measured return power against the equivalent radar reflectivity factor Ze , which would be equal to the radar reflectivity factor for a population of liquid, spherical particles satisfying the Rayleigh approximation. A conventional unit for Z and Ze is mm6 m−3 . However, it is often expressed on a logarithmic scale (10 log Z) with the unit of dBZ. The reflectivity factor (Z or Ze ) can be converted to a radar-based rainfall rate estimate (R) by using an empirical Z–R relationship. The most common form of Z–R relationship is: Z = aRb , 6

−3

(2.2)

where Z is in mm m , R in mmhr−1 , and a and b are empirical constants. The parameters a and b primarily depend upon the type of drop-size distribution (DSD). The power-law form of the Z–R relation follows from a

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negative exponential rain DSD of the Marshall–Palmer type or a more general Gamma distribution (cf., Ch. 23). Based on a historical Z–R relation suggested by Marshall and Palmer (1948), the parameter values of a = 200 and b = 1.6 are obtained. These are often regarded as operationally acceptable values. However, there are many other forms of Z–R relation in use, and their parameters vary with regions, storm structures, and cloud microphysical properties. In order to obtain the R value representative of surface rain intensity, the Z value for low horizontal levels is used, typically CAPPI data at 1– 1.5 km height. Rainfall for a given accumulation period can be estimated by time integrating over all radar pixels inside a verification domain. However, this initial estimate must be corrected against the errors arising from the radar measurement and other factors. Villarini and Krajewski (2010) provide an extensive literature survey on the principal sources of errors affecting single-polarization radar-based rainfall estimates. These include radar miscalibration, attenuation, ground clutter, anomalous propagation, beam blockage, variability of the Z–R relation, range degradation, vertical variability of the precipitation system, vertical air motion, precipitation drift, and temporal sampling errors. Analyses of meteorological and non-meteorological error sources in radar measurements are also found in Collier (1996) and Meischner (2004), as well as in many of the conference proceedings of ERAD (European Conference on Radar in Meteorology and Hydrology). Meteorological services, based on their operating meteorological radars, provide products in which the basic technical errors are eliminated as optimally as possible over the operated territory (clutter, radar calibration, etc.). 2.3

Polarimetric radar measurement

Hydrometeors in the form of raindrops and ice particles are characterized by different shapes, different orientations during fall, and different dielectric constants. Thus, they backscatter differently the signals of different polarization. Polarimetric (or dual-polarization) radar is able to control the polarization of the transmitted signal and to detect that of the returned signal. Most polarimetric radars use horizontal and vertical polarizations for transmission and reception. Polarimetric weather radars have a significant advantage over single-polarization systems because they allow multi-parameter measurements useful for estimating DSD and rainfall rate. Thus, they can lead to overall improvements of quantitative precipitation

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179

estimation (QPE). Comprehensive descriptions of polarimetric radar measurements are found in Bringi and Chandrasekar (2001); Doviak and Zrnic (1984); Giangrande (2007); Meischner (2004), for example. The most common additional parameters from polarimetric radar measurement are: • The differential reflectivity ZDR= 10 log(ZHH /ZV V ), where ZHH (ZV V ) is the reflectivity of a horizontal (vertical) polarized pulse. ZDR depends on the asymmetry of particles. It is positive for oblate raindrops, and zero or slightly negative for hail and graupel. • The linear depolarization ratio LDR= 10 log(ZV H /ZV V ), with ZV H (ZHH ) the vertically (horizontally) polarized return from a transmission with horizontal polarization. Depolarization of the horizontally polarized pulse is normally small for rain but high for melting snow and water-coated hail and graupel. • The specific differential phase shift KDP, which is a difference of phase shifts between horizontally and vertically polarized radiation measured in degrees per kilometre. It arises as a result of different propagation characteristics for different polarizations. • The co-polar correlation coefficient ρHV , evaluated from timeseries of ZH and ZV , indicates variability of the scattering particles in terms of their shape, size, and thermodynamic phase. Extended discussions of parameters derived from polarimetric measurements were compiled by Illingworth (2004). Dual polarimetric measurements can improve rainfall rate estimations by considering the relationship between polarimetric variables and the parameters of the drop-size spectrum. The raindrop-size distribution model presented in many polarimetric radar rainfall studies (e.g., Bringi and Chandrasekar, 2001; Illingworth and Blackman, 2002; Testud et al., 2000) is the normalized Gamma distribution: 

μ  D (3.67 + μ)D exp − , (2.3) N (D) = NW f (μ) D0 D0 where N (D) in m−3 mm−1 is the volume density, D0 in mm is the median volume drop diameter, NW in m−3 mm−1 is the intercept parameter, and μ (no units) is the parameter of DSD shape. When μ = 0, Eq. 2.3 reduces to a simple exponential Marshall–Palmer DSD with concentration parameter N0 = NW . From polarimetric parameters, one can derive the parameters D0 , NW , and μ, and hence a rainfall rate R.

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Various relationships between polarimetric parameters and rainfall rate have been derived (e.g., Bringi and Chandrasekar, 2001; Giangrande, 2007; Illingworth, 2004). As an example used by several authors, Illingworth (2004) presents the equation: R(KDP, ZDR) = cKDPa ZDRb ,

(2.4)

with the values a, b, and c obtained by a regression analysis of values obtained by scanning over the N0 , D0 , and μ range given by Ulbrich (1983). A table of the coefficients a, b, and c for use in Eq. 2.4 was provided by Bringi and Chandrasekar (2001). The use of dual-polarization radar for QPE has been demonstrated in a number of recent studies, which are designed to test and compare various algorithms for rainfall rate estimation from polarimetric measurements. For example, Anagnostou et al. (2013) evaluated a new algorithm (SCOPME: Self-Consistent with Optimal Parameterization attenuation correction and rain Microphysics Estimation algorithm) by using long-term X-band dual-polarization measurements and disdrometer DSD data, acquired in Athens, Greece. The retrieval of median volume diameter D0 and intercept parameter NW were compared with two existing rain microphysical estimation algorithms and the retrievals of rainfall rate with three available radar rainfall estimation algorithms. Relations for Z–R, R–KDP, R–ZH , and ZDR–KDP were used here with coefficients obtained from multiple regression. Error statistics for rainfall rate estimates, in terms of relative mean and root-mean-square error, showed that the SCOP-ME has a low relative error compared to the existing algorithms, which systematically underestimate rainfall. Rainfall rate estimates with SCOP-ME mostly depend on D0 , which is estimated much more efficiently than the intercept parameter NW . There are a large number of studies documenting methods for retrieving meteorological information from polarimetric radar measurement. An important issue in polarimetric QPE is in determining a method to employ for a given set of observed polarimetric parameters. At Colorado State University (CSU), an optimization algorithm has been developed and used for a number of years to estimate rainfall based on thresholds of ZH , ZDR, and KDP. In the study by Cifelli et al. (2011) a new rainfall algorithm using hydrometeor identification (HID) was presented to guide the choice of the particular rainfall estimation algorithm. Both the data collected from the S-band radar and a network of rain gauges were used to evaluate the performance of the new algorithm in mixed rain and hail in Colorado. Results

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181

showed that the new HID-based algorithm provided good performance for the Colorado case studies. Dual-polarization radars have also been increasingly used for investigations of cloud microphysics. According to Straka et al. (2000), who provide a basic review of microphysical interpretations of polarimetric data, the identification of hydrometeor types by polarimetric radar is accomplished by associating different bulk hydrometeor characteristics with the unions of subsets of values of the various polarimetric variables. Consequently, the polarimetric radar-based microphysical categorizations can be applied as verification data for prognostic microphysics schemes. The retrieval of cloud microphysical structure from polarimetric characteristics has been a topic of many studies for several decades and the interest is increasing as dual-polarization radars become more widely operational. The techniques of extracting microphysical categories and their use in NWP models are further discussed in Sec. 6. 2.4

Quantitative precipitation estimates by merging rain-gauge and radar measurements

Many different approaches have been used to improve the accuracy of radarbased precipitation estimates. A common strategy is the merger of radarand gauge-based data, combining their different strengths so that systematic errors can be substantially reduced. The main attempts are based on reducing radar biases using additional rain gauge data. However, particularly for highly variable convective precipitation, the use of a complex gauge adjustment can be detrimental, depending on the density of the rain-gauge network (e.g., Collier et al., 2010). For this reason, many procedures are performed simply by employing correction factors based on the radar/rain-gauge ratio of rainfall, in which area-averaged radar rainfall rates are compared with the average of a number of gauges over the given area. A key role of gauge adjustment is to make sure that the radar-based precipitation estimates are unbiased against gauge measurements on a long-term basis. At a more general level, the issue of merging rain-gauge and radar data can be considered as an issue of data assimilation. Along these lines, various more complex techniques can be developed based on methods such as kriging and other types of weighted interpolation. Some such methods are used operationally, with adjustment functions derived from historical as well as real time data. Alternatively, data representing a certain time

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and area window may be used (see Gjertsen et al., 2004). Regression-based techniques are discussed by Gabella et al. (2001); Kracmar et al. (1999); Morin and Gabella (2007); Sokol (2003), for example. A technique based on weighting the radar-based and gauge-measured rainfall rates, where the weights depend on the distance of the gauges from a radar pixel (Seo and Breidenbach, 2001), was applied operationally in the Czech Republic (Salek et al., 2004). It was also combined with a krigingbased technique (Salek, 2010). The Probability Matching Method, proposed by Rosenfeld et al. (1994), derives an adjustment function from an analysis of historic radar and gauge data. It can provide a statistically stable relationship, and has formed the basis for other products and procedures (Collier et al., 2010). Several operational products provide radar-based QPE that can be used for verification of high-resolution forecasts. The RANIE (RadarNiederschlagsmessungen) product at Deutscher Wetterdienst (DWD), Germany (Pfeifer et al., 2008, 2010), and the MERGE (Merging radar and gauge data) product at Czech Hydro-Meteorological Institute (CHMI), Czech Republic (Salek, 2010) are examples.

2.5

Use of direct radar data for verification by comparison with forward-modelled results

A difficulty in evaluating NWP model forecasts against observations is that the latter may not be directly linked to model parameters. Two major approaches may be used. The first is the “observation-to-model approach”, which converts the observations into model variables. The second is the “model-to-observation approach”, which simulates observation variables from model outputs. In the second approach, comparisons are performed in terms of observation characteristics. Several operational NWP models include radar-forward operators, which enable the calculation of radar reflectivity from model outputs. Simulated radar data is then compared with the radar observations for evaluation. This approach was applied to single-polarization radar by Haase and Crewell (2000), for example, and to polarimetric radar by Pfeifer et al. (2008). The present chapter mostly focuses on the first approach, although the use of a polarimetric radar-forward operator by Pfeifer et al. (2010) is considered further in Sec. 6.

Verification of high-resolution precipitation forecast with radar-based data

3

183

Traditional verification techniques and their limitations

Forecast verification is indispensable in meteorological research as well as for operational applications. A well-designed verification method helps to identify model shortcomings and systematic errors. It also provides quantitative assessments of the improvement of the forecasts over time. Various international research projects have provided good frameworks to develop and test new verification strategies. Examples include the Sydney 2000 and Beijing 2008 Olympic Forecast and Research Demonstration Projects, Mesoscale Alpine Programme (MAP) studies, and the Forecast Verification Method Inter-comparison Project. A Joint Working Group on Verification (JWGV) under the WMO/WWRP (World Meteorological Organization/World Weather Research Project) was established in January 2003 in order to promote verification practices and research. International conferences and workshops on verification are organized by the WWRP/WGNE (Working Group for Numerical Experiments) Joint Working Group on Forecast Verification Research. This group initiated special issues of the journal Meteorological Applications on forecast verification, which provide a good overview of the topic (Casati et al., 2008; Ebert et al., 2013). Although QPF verification is of particular importance, verification strategies are also under development focusing on other issues, notably the verification of extreme events. The WWRP/WGNE group have recently produced a set of recommendations (Brown et al., 2009), including the recommendation that “where possible, combined radar–gauge rainfall analyses be used to verify model QPFs and PQPFs [Probabilistic QPFs] at high spatial and temporal resolution.”

3.1

Traditional skill scores

Deterministic precipitation forecasts can be verified with two distinct approaches: (1) categorical (dichotomous, binary, Yes/No); and, (2) continuous. Various verification measures (scores) can be adopted for both approaches. As an example of a categorical score, QPF is often judged by whether the rainfall exceeds a threshold. On the other hand, continuous variables, such as the rainfall amount, can be more directly adopted as a score measure. Quality measures such as the mean-square-error and correlations are also defined in continuous terms. Note, however, that:

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Part III: Operational Issues

“. . . because rainfall amount is not normally distributed and can have very large values, the continuous verification scores (especially those involving squared errors) which are sensitive to large errors may give less meaningful information for precipitation verification than categorical verification scores” (Brown et al., 2009).

This points to the importance of careful statistical quantifications of rainfall variability both in time and space, as already emphasized in Secs. 2.1 and 2.5. Brown et al. (2009) lists the recommended verification measures as: (i) forecasts of rain occurrence meeting or exceeding specific thresholds; (ii) forecasts of rain amount; (iii) probability forecasts of rain meeting or exceeding specific thresholds; and, (iv) verification of ensemble probability distribution. As any verification score must be regarded as a sample estimate of the true value, it is further recommended to estimate confidence intervals to produce bounds on the expected value of the score. In the following, the recommended measures for the types (i) and (ii) are summarized, including more comprehensive information about their applications. A traditional categorical verification of grid-point precipitation defines an event as the accumulated grid-point precipitation greater than or equal to a threshold. An alternative approach is to consider a spatial rainfall pattern over an elementary area covered by a finite number of grid points. A radar either observes an event (o = 1) or not (o = 0). A model forecast either predicts the event (f = 1) or not (f = 0). The contingency table (Table 21.1) counts the number of grid points (elementary areas) with hits (o = 1, f = 1), false alarms (o = 0, f = 1), misses (o = 1, f = 0), and correct rejections (o = 0, f = 0). There are number of scores based on a contingency table (e.g., Jolliffe and Stephenson, 2003; Wilks, 2006). Examples for this category of QPF verification are found in Damrath et al. (2000) for Germany and Ebert et al. (2003) for the United States, Australia, and Germany. Categorical scores recommended by Brown et al. (2009) are summarized in Table 21.2. During COST Action 717, a survey of traditional verification measures was compiled by C. Wilson1 . Table 21.3 lists those measures not included in Brown et al. (2009). In verifying a forecast of a continuous variable (rainfall amount), we have to take into account the sensitivity of continuous verification scores to outliers. According to Brown et al. (2009), the sensitivity can be reduced 1 Available

at http://www.smhi.se/hfa coord/cost717/doc/WDF 02 200109 1.pdf

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185

Table 21.1 Categorical contingency table: A is the number of hits (correctly forecast events), B the number of false alarms (incorrectly forecast non-events), C the number of misses (incorrectly forecast events), and D the number of correct rejections (correctly forecast non-events). The observed frequencies of events and non-events are given by (A + C)/N and (B + D)/N , respectively. Similarly, the forecast frequencies of events and non-events are (A + B)/N and (C + D)/N . observation YES, o=1 NO, o=0 forecast

YES, f=1 NO, f=0

observational total

forecast total

A C

B D

A+B C+D

A+C

B+D

N=A+B+C+D

if we normalize the rainfall amount values using a square-root transformation (Stephenson et al., 1999). If necessary, an inverse transformation by squaring is applied to return to physical units. An overview of continuous scores suggested by Brown et al. (2009) is reproduced as Table 21.4. Verifications adopting categorical scores involve a multi-parametric task. Graphical representations of verification results are useful for analysing various aspects of the QPF quality. Various choices are possible for graphical representations of multiple measures of traditional yes–no forecast quality, as summarized by Roebber (2009). An example for a high-resolution forecast of heavy, local, convective rainfall is given in Fig. 21.1. 3.2

The double-penalty problem

Grid point-related error measures are problematic for phenomena such as convective precipitation, which are characterized by complex structures on scales of less than 100 km. A classic example illustrating the limits of grid point-based scores is the well-known double-penalty problem. Prediction of a precipitation event with the correct size and structure might yield very poor verification scores if, for example, a feature is displaced slightly in space, because categorical error scores penalize such a situation heavily. From the point of view of traditional verification, a displacement leads to a false alarm at the displaced location and to a miss at the location of observed rainfall (Davis et al., 2006). However, the very fact that an event is predicted nearby should be positively evaluated, because from a physical point of view, it is clearly a better forecast than completely missing an

Name

Definition

Range

Best

Recom.

BIAS

A+B A+C

0, ∞

1

***

PC

A+D N

0, 100

100

***

POD, HR FAR

A A+C

0, 1

1

***

0, 1

0

***

POFD

B B+D

0, 1

0

**

TS, CSI ETS, CSISC

A A+B+C

0,1

1

**

A−E A+B+C−E

1/3, 1

1

***

HK,TSS

B A − B+D A+C POD−POFD

−1, 1

1

**

HSS

A+D−E ∗ N−E ∗

−∞, 1

1

**

OR

AD BC

ORSS

AD−BC AD+BC

B A+B

× 100

= 1−FOH

=

**

−1, 1

**

Part III: Operational Issues

Bias, Frequency Bias: ratio of the forecast rain frequency to the observed rain frequency Proportion (Percentage) Correct: fraction of all correct forecasts Probability of Detection (Hit Rate): fraction of observed events that were correctly forecast False Alarm Ratio (Rate): fraction of forecast events that were observed to be non-events Probability of False Detection (False Alarm Rate): fraction of observed non-events that were forecast to be events Threat Score (Critical Success Index): fraction of all events forecast and/or observed that were correctly forecast Equitable Threat Score (Skill-Corrected CSI, Gilbert Skill Score): fraction of all events forecast and/or observed that were correctly forecast accounting for the hits E that would occur due to random chance: E = (A + C)(A + B)/N Hanssen and Kuipers score (True Skill Statistic, Pierce Skill Score): measures the ability to separate the observed YES cases from the NO cases Heidtke (Total) Skill Score: measures the increase in proportion correct for the forecast system, relative to that of random chance; E ∗ = E + [(B + D)(C + D)/N ] Odds Ratio: ratio of the odds (see Table 21.3) of making a hit to the odds of making a false alarm, taking prior probability into account Odds Ratio Skill Score: transformation of odds ratio to the range [−1, 1]

Abbr.

186

Table 21.2 A summary of categorical scores (Brown et al., 2009). The last column indicates recommendations by the WWRP/WGNE Joint Working Group on Verification: highly recommended (***), recommended (**), or worth a try (*, not listed here).

187

Verification of high-resolution precipitation forecast with radar-based data

Table 21.3 A summary of further categorical scores according to Wilson (2001). Also shown are confidence limits for the HK and OR scores. See Table 21.1 for the meanings of A, B, C, and D. Name

Abbr.

Definition

Range

Best

0, 1 0, 1

0 1

Stratification by observation: Frequency of misses Probability of Null Event

C A+C D B+D

FOM PON

= 1−POD = 1−POFD

Stratification by forecast: Frequency of Hits, Success Ratio, Post Agreement Detection Failure Ratio Frequency of Correct Null Forecasts Odds scores have the form Odds (hit)

FOH, SR

A A+B

0,1

1

DFR

C C+D

0, 1

0

FOCN

D C+D

0, 1

1

= 1−DFR

p(x) p(xmean )

Odds (false alarm)

=

p 1−p

A C

=

POD 1−POD

B D

=

POFD 1−POFD

0, ∞

Confidence limits: Hanssen and Kuipers score Odds Ratio

s2 (HK)

N 2 −4(A+B)(C+D)(HK)2 4N(A+B)(CD)

s2 (log OR)

A−1 + B −1 + C −1 + D −1

event. This issue becomes increasingly acute as model resolutions increase (towards the horizontal grid spacing of 1–4 km). Current high-resolution models can produce precipitation fields that are comparable in resolution to radar information. The complexity and variety of structures generated by a high-resolution model must be objectively scrutinized. High-resolution precipitation forecasts from numerical models may look quite realistic visually, and may provide useful guidance for forecasters. However, the usefulness of such high-resolution forecasts cannot be objectively quantified with traditional verification scores. Spatial verification techniques have been developed in recent years in order to overcome limitations associated with the traditional methods. These are discussed next.

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Part III: Operational Issues

Table 21.4 A summary of continuous scores for forecasting rain (Brown et al., 2009). The last column indicates recommendations by the WWRP/WGNE Joint Working Group on Verification: highly recommended (***), recommended (**), or worth a try (*, not listed here). Any of the accuracy measures can be used to construct a skill score that measures the fractional improvement of the forecast system over a reference forecast (Brown et al., 2009; Wilks, 2006). The most frequently used scores are the MAE and the MSE. The reference estimate could be either climatology or persistence for 24-hr accumulations, but persistence is suggested as a standard for short-range forecasts and for shorter accumulation periods. Name

Abbr.

Mean observed Sample standard deviation

o¯ s

Description

Interquartile range

IQR

Mean error

ME

Mean absolute error Mean square error

MAE MSE

Root-mean-square error

RMSE

Root-mean-square factor

RMSF

(Product moment) correlation coefficient

r

Spearman rank correlation coefficient

rs

MAE skill score

MAE SS

The square root of the sample variance, which provides a variability measure in the same units as the quantity being characterized. The “typical” rain amount. Since the most common rain amount will normally be zero, the conditional median should be drawn from the wet samples in the distribution. It is more resistant to outliers than the mean. The 75th percentile minus the 25th percentile of the distribution of rain amounts. It reflects the sample variability and is more resistant to outliers than the standard deviation. Like the conditional median, the IQR should be drawn from the wet samples. The average difference between the forecast and observed values. The average magnitude of the error. The average squared error magnitude, and often used in the construction of skill scores. Larger errors carry more weight. Measures the average error magnitude but gives greater weight to the larger errors. It is useful to decompose the RMSE into components representing differences in the mean and differences in the pattern or variability The exponent of the root-mean-square error of the logarithm of the data; gives a scale to the multiplicative error. Measures the degree of linear association between the forecast and observed values, independent of absolute or conditional bias. Highly sensitive to large errors; it benefits from the square-root transformation of rain amounts. Measures the linear monotonic association between the forecast and observations, based on their ranks: i.e., the position of the values when arranged in ascending order. rs is more resistant to outliers than r. MAEforecast 1 − MAE

MSE skill score

MSE SS

1−

Conditional median

reference

MSEforecast MSEreference

Recom. *** ***

***

**

*** ** **

***

**

***

** **

Verification of high-resolution precipitation forecast with radar-based data

189

Fig. 21.1 Verification of a rainfall forecast using POD, FAR, BIAS, and CSI. The rainfall was accumulated over the time period 1600 to 2200 UTC (10-16-hr forecast lead time) for five convective events with heavy local rainfall. The dates of the events are given in the legend. (a) Grid-point precipitation that exceeds threshold values, the values used being indicated on the curves for 30 May 2005. (b) Dependency on an elementary square area for the analysis, with the length of the square considered (as a number of grid points) being indicated on the curves for 15 May 2002. The threshold used in panel (b) is 1 mm(6 hr)−1 . The forecasts are from the COSMO model with a horizontal grid length of 2.8 km, adapted to cover the territory of the Czech Republic. (Reprinted from ˇ aˇ Fig. 6 of Rez´ cov´ a et al. (2009), with permission from Elsevier.)

4

Spatial verification techniques

The basic principle of spatial verification is to relax the requirement for an exact match to the observation at fine scales. Spatially based techniques stress the usefulness of the forecasts by analogy with visual verification (verification by eye). According to Ebert (2008), a useful forecast predicts an event somewhere near the observation, over a similar area, and with a similar distribution of intensities as observed. The spatially based methods focus on the verification of gridded forecast data with an observation field that is on the same grid, as may be formed, for example, by radar-based QPE. Such gridded observations have greater uncertainty at high resolution than at lower resolutions. Pointwise precipitation measurements, which are another source of verification data, may suffer from an issue of representativeness, particularly for highly variable fields such as convective precipitation. As a result, an estimated forecast error is biased by observational uncertainty. In other words, a formally obtained forecast error should not necessarily be considered prac-

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Part III: Operational Issues

tically meaningful. Several spatially based methods are built upon the idea of identifying weather events as “objects” or “features”. Using this perspective, the forecast (F) and observed (O) fields of rainfall values are not compared directly at the same locations (i.e., on identical or nearby grids) but instead the objects of interest are extracted from the F/O data and then compared together to obtain the verification statistics. For this reason, the displacement method subclass (b1: to be discussed in Sec. 4.1) is called “object-oriented” or “feature-based”. A large number of spatial verification methods are proposed in the literature. Comparative review studies are found, for example, in Ahijevych et al. (2009); Ebert (2008); Gilleland et al. (2009). The Spatial Forecast Verification Methods Inter-comparison Project (ICP), which stemmed from an international verification workshop held in Boulder, Colorado in 2007, is an effort to analyse and compare the newly proposed methods for verifying high-resolution forecasts. A division of the new methods into several categories is one of the major ICP results, and is presented next. 4.1

Categories of spatial methods

According to Gilleland et al. (2009), many spatial methods can be classified into two basic categories: (a) filtering methods; and, (b) displacement methods. The filtering methods apply a spatial filter to one or both data fields (or sometimes to the difference field), and then calculate verification statistics on the filtered fields. A filter is usually applied at progressively coarser scales to provide information about the scales at which a forecast has skill. The displacement methods seek a best fit of a forecast with observations by adjusting their mutual positions. The “fitting” procedure quantifies the extent to which a forecast field needs to be manipulated spatially (displacement, rotations, scaling, etc.) as well as the associated residual errors. The filtering methods (a) can be further classified into (a1) neighbourhood (or fuzzy) and (a2) scale-separation (or scale-decomposition) methods. The neighbourhood methods apply a smoothing filter, whilst the scale-separation techniques apply several single-band spatial filters (Fourier, wavelets, etc.) so that performance at separate scales can be evaluated separately. The displacement methods (b) can be further classified into (b1) featurebased (or object-oriented) and (b2) field-deformation methods. The pri-

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mary difference between the two is that the feature-based methods first identify features (or objects) of interest (e.g., storm cells), and analyse each feature separately, whereas the field-deformation approaches analyse the entire field or a subset thereof. Gilleland et al. (2009) presented examples of verification techniques for each category with references. Their list of methods (their Table 1) includes two traditional and sixteen spatial techniques. Not all of the methods are unambiguously classified into one of the four categories above, with some being classified into more than one category. Ahijevych et al. (2009) applied various spatial techniques to a set of artificial and perturbed forecasts with prescribed errors, and to a set of real forecasts of convective precipitation on a 4 km grid. They summarize that: “. . . each method provided different aspects of forecast quality. Compared to the subjective scores, the traditional approaches were particularly insensitive to changes in perceived forecast quality at high-precipitation thresholds. In these cases, the newer features-based, scale-separation, neighbourhood, and field-deformation methods have the ability to give a credit to close forecasts of precipitation features or resemblance of overall texture to observation. In comparing model forecasts with real cases, the traditional verification scores did not agree with the subjective assessment of the forecasts.”

Basic aspects of spatial verification categories are summarized in Table 21.5 as compiled by Ebert (2011). Table 21.5 Some basic aspects of spatial categories for new precipitation verification methods. Adapted from Ebert (2011). Attribute

Traditional methods

Featurebased

Neighbourhood Scale

Field deformation

Performance at different scales Location errors Intensity errors Structure errors Hits etc.

indirectly

indirectly

Yes

Yes

No

No Yes No Yes

Yes Yes Yes Yes

indirectly Yes No Yes

indirectly Yes No indirectly

Yes Yes Yes Yes

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The neighbourhood techniques

The neighbourhood techniques are probably the most elaborated approach thanks to studies by Ebert (2008, 2009). Compared to the other categories, the idea, the procedures, and the applications are easier and more intuitive. Neighbourhood (also called “fuzzy”) approaches compare values of a forecast in space–time neighbourhoods relative to a point in an observation field. In most applications, spatial windows (also called “elementary areas”) are employed for this purpose. It is also straightforward to extend the technique to include neighbourhoods in time (Ebert, 2008). A suitable window size should depend on the grid spacing, the timestep of the available data, and the meteorological situation. Thus, a single choice of a window may not work well for all forecasts and domain sizes. Fuzzy verification techniques address this question by allowing the neighbourhood size to vary. The scale that attains a desired level of forecast skill is determined by performing a comparison over incrementally larger neighbourhoods. There are two main neighbourhood strategies (Ebert, 2008). The “single-observation neighbourhood-forecast” strategy matches a grid-box observation to a corresponding neighbourhood of grid boxes in the forecast. The other strategy is called “neighbourhood-observation neighbourhoodforecast” and also takes into account the neighbourhood surrounding the observations. This second strategy takes a model-oriented viewpoint, in which observations must be upscaled in one way or another, and then treated as representing those scales resolved by a model, usually of several grid lengths. The single-observation neighbourhood-forecast strategy represents a user-oriented viewpoint, in which it is important to verify a predicted value for a particular location of interest. The earliest and perhaps simplest of these methods is more specifically called upscaling, in which both the forecasts and observations are averaged consecutively to coarser scales and then compared by traditional scores (e.g., Yates et al., 2006; Zepeda-Arce et al., 2000). A disadvantage of upscaling, however, is a loss of small-scale variability that is crucial for depicting high impact events such as heavy local convective precipitation. Such fine-scale variability is certainly captured by a high-resolution model in a gross sense, although details such as the precipitation locations may be displaced from those observed. Theis et al. (2005) was probably the first to clearly express the idea of a fuzzy approach, and they compared the forecast fractional coverage around a neighbourhood to the observed occurrence of an event. Marsigli et al. (2008) took a more general approach

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by comparing the statistical moments of the distribution of observations in the neighbourhoods. Several independently developed fuzzy techniques were reviewed by Ebert (2008), who compared them by using radar-based data for an intense storm event over the United Kingdom in May 1999. This case is also examined in detail by Casati et al. (2004). The verification referred to a 3-hr forecast of rainfall rate (mmhr−1 ) at 5 km spatial resolution. A sub-domain of 256 × 256 grid boxes centred on the rain system was considered and the Nimrod quality-controlled radar rainfall analysis (Golding, 2000) provided the verification data. These case studies demonstrated the strengths and weaknesses of several fuzzy verification methods, giving the impression that other cases may show different responses at different scales. These studies suggest that it makes sense to first consider carefully the requirements for a good forecast and then to focus on those fuzzy methods that address those requirements through their decision models.

5

Examples of spatial verification

A large number of techniques apply the principle of spatial verification (see Ahijevych et al., 2009; Ebert, 2008; Gilleland et al., 2009). Some of them are used solely by proposers of the methods, but others find more general use. Among those, this section will focus on the CRA (contiguous rain area: Ebert and McBride, 2000), SAL (structure-amplitude-location: Wernli et al., 2008, 2009), and FSS (the Fractions Skill Score: Roberts and Lean, 2008) techniques. All results in this section were obtained by using single-polarization radar measurements as verification data. In Sec. 5.1, CRA as developed at the Interdisciplinary Centre for Mathematical and Computational Modelling (ICM), Warsaw University (Poland) is discussed. The second example is from the Hungarian Meteorological Service (HMS), applying SAL (Sec. 5.2). This analysis specifically seeks to quantify the effects of the boundary-layer parameterization and its modification at HMS. The third example is an application of FSS (Sec. 5.3) in comparing the performance of NWP models and their options (Zacharov et al., 2013). These results were obtained at the Institute of Atmospheric Physics (IAP), Czech Academy of Sciences, in collaboration with the Czech Hydrometeorological Institute (Czech Weather Service).

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Part III: Operational Issues

CRA technique and its modification

The examples in this subsection come from the mesoscale numerical prediction system that is being explored operationally at Warsaw University. The model precipitation forecasts are compared with radar observations, consisting of 15-min reflectivity data on the 500 m CAPPI level and collected from all radars operated in the area of Baltic Sea catchments. After some basic corrections, these data were integrated into 1-hr precipitation accumulations using a standard Z–R relationship. To facilitate comparisons, estimated precipitation observations were converted to the projection and the resolution of the model. The mesoscale model used here is the Met Office Unified Model with a spatial resolution of 4 km. The convection parameterization was switched on for these runs but its action is heavily restricted by the Roberts (2003) modification. The CAPE closure timescale is substantially increased for larger values of CAPE, so that the cloud-base mass flux that can be produced by the parameterization is limited. Thus, although some of the convection is parameterized, most is represented explicitly by the model dynamics. The present subsection considers CRA, as introduced by Ebert and McBride (2000), which is based on areas of contiguous observed and/or forecast rainfall enclosed within a specified isohyet. This approach consists of four major steps: (1) (2) (3) (4)

identifying separate objects; describing characteristics of interest; finding matching objects in both fields; and, calculating verification statistics.

Although there are some differences in the definitions and techniques presented herein, the philosophy behind the algorithm is that of Ebert and McBride (2000). These four steps are discussed below. (1) Object definition: As a working hypothesis, a definition for the object proposed by Ebert and McBride (2000) is adopted here. The contiguous rain area (CRA) is defined as the area of contiguous observed or forecasted rainfall enclosed within a specific isohyet. Figure 21.2 shows an example of observed and forecasted precipitation fields projected onto the same spatial grid. A general view of this picture gives an impression that the fields are broadly similar, but with differences that should be evaluated in order to determine the quality of the forecast. The first step is to look for distinct

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entities that can be associated with merged observation and forecast fields. However, as a difference from the original algorithm, we treat the forecast and observed fields separately so that the CRAs contain only observed or forecast rain. We apply a minimal threshold for the precipitation to be considered, and change sub-threshold values to no-rain values. (2) Describing characteristics of interest: Let us apply an algorithm that distinguishes CRAs based on the spatial coordinates of the objects. Once these spatial coordinates are defined, various statistics such as the maximum precipitation area, average precipitation, and total rainfall can be evaluated for every identified object over a forecast period. (3) Matching the forecast and observed objects: The most obvious measure of similarity between objects in the F/O fields is a mutual distance. Figure 21.3 shows examples of observed and forecast objects and the method for estimating the distance between the objects. The centre of mass of an object area can be calculated, and the obtained position is used as a reference for calculating distance. A drawback of this approach is that it may lead to awkward results for elongated, curved shapes, in that the centre may lie outside of the object. Those problems are circumvented by using a variation of the Hausdorff distance (Venugopal et al., 2005), which measures the maximum distance from a point in one object to the nearest point in another object. In general, a distance threshold has to be set in such a manner that the maximum distance between the objects under consideration remains close enough (compared to other forecast objects within a range) for further inspection. The choice of a suitable threshold depends on the type of data and on the judgement of the analyst. For synopticscale data sets, 400 km may be a reasonable separation, whilst for local convective events 50 km may be too much. If the two objects overlap, the separation distance is zero under this definition. In some cases, a forecast CRA may be found within the range of more than one observed object. In order to make the matching unique, the approach presented by Ebert and McBride (2000) is slightly modified here. Once a pair of observed and forecast objects is selected, we calculate one of the classical statistics such as root-mean-square (RMS) error, or a correlation coefficient. Next, we shift the forecast object over the observed object in order to minimize the error (or maximize a correlation). We choose the forecast pair with the best score by examining all the statistics obtained in respect of all observed objects within a range. (4) Verification statistics: Once the matched pairs are identified, we can compute a set of errors that measures the model performance. The set leads

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Fig. 21.2 An example of observed (left panel) and forecasted (right panel) precipitation fields from 20 July 2009. The precipitation rate is measured by a 15-min accumulation (mm).

Fig. 21.3 An illustration of the technique for matching observed (left panel) and forecasted (right panel) objects. The objects are identified from the precipitation measured by 15-min accumulations (mm). In this case, the model forecasts a precipitating object too far to the lower left of the area shown. Thus, the forecast object must be displaced to the upper right as indicated by an arrow. The displaced object, also shown in the illustration, then provides a better fit with the observation.

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to a categorical contingency table. Typically, not all observed objects find a corresponding pair. Those which are left out are called “missed events”. It may also happen that not all forecast events are matched, and these are called “false alarms”. The matched objects are called “hits”. Note that a category of correct rejections does not make sense in this setting. In Fig. 21.4 statistics computed from all forecasts produced by the Unified Model in July 2009 over Poland are presented. The RMS error is calculated before and after the shift of the forecast objects. The error remaining after the shift is referred to as a “shift error” while the difference between the two calculations produces the so-called “displacement error”. The left panel of Fig. 21.4 shows these two contributions to the overall forecast error, as a function of the lead time. The total RMS error, shown on the right, has a diurnal cycle in the first and the second 24 hours of the forecast. Some evidence for larger high-frequency fluctuations during convection development on the second day of the forecasts may be observed.

Fig. 21.4 Shift and displacement forecast errors (left panel: see text for the definition) and the total error (right panel: sum of the two in the left).

5.1.1

Further modifications and possible extensions

In this subsection some possible extensions of the CRA approach are outlined that may be useful for high-resolution verification. Ensemble forecasts: An ensemble forecast is generated by means of perturbations to the initial-condition data and/or to the model parameters. Due

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to uncertainty in the initial data as well as inherent model imperfections, all ensemble members may be considered equally probable. In the simplest treatment of verification, the ensemble forecast field is defined by a mean over the ensemble members. Suppose that an ensemble has N members and let Fn , n = 1, 2, . . . , N denote the individual forecast fields, then an ensemble-averaged forecast is given by: E=

N 1  Fn . N n=1

(5.1)

A rain/no-rain threshold R may be applied to the fields Fn and we may define rain occurrence by E ≥ R for each spatial point. This condition may be satisfied even if a majority of members show no rain, but the values of some members are high enough. Clearly, such a method does not fully reflect the probabilistic aspect of ensemble prediction. An alternative approach would be to use the existence of CRAs as a criterion for the occurrence of an event. The identified CRA objects may be considered as robust against perturbations of the initial conditions, if they are present in most of the ensemble members. The objects would be expected to lie in the vicinity of an object identified from the ensemble mean field. Thus, objects from different ensemble members could be identified as being associated with the same ensemble-mean object under a perturbation. Once objects are grouped together, the probability of an event can be assigned from the fraction of members that predicted the given object. In this way we can specify likelihoods of specific events based on an ensemble forecast. The method may be particularly useful for the verification of rare but intense events, with a given measure of reliability. Clustering: The CRA algorithm enables us to isolate and describe individual objects in the F/O fields. At the same time, many of the objects can be a part of larger-scale structures such as mesoscale organizations or synoptic-scale fronts, which are of interest in their own right. Furthermore, large-scale dynamics may provide a pre-condition for defining a drift bias of the CRA objects. In order to reunite objects isolated from the F/O fields, we may introduce a clustering algorithm which collects together those objects which satisfy proximity criteria. In this way, we can create sets of non-overlapping clusters of CRAs, which may then be useful for comparison of their total precipitation, area, and maximum precipitation, for example. Time evolution of CRAs: An interesting question is that concerning a model’s performance in respect of forecast lead time. Clearly, the reliability

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of a forecast would be expected to decrease with time. However, the peak of performance is not at the beginning of the forecast period, due to spinup of the model. In order to identify an optimal time lag and determine a reliability profile, we need to compare verification results from various times in a consistent manner. A direct solution to the problem would be to determine statistics independently for the different lead times. A more sophisticated method might require, for example, tracking of objects in time. When the objects in forecast and observed fields are examined in time, continuity in the evolution of objects would be seen as an object appears, develops, and disappears. This means that when two objects are matched together at a certain time, their immediate future states (for the next few snapshots) should be comparable as well. The CRA algorithm, however, tries to minimize errors for every snapshot separately. A procedure is still to be developed which would assess model performance in respect of the evolution of CRAs or clusters in time. Another important aspect is a possibility for extending the Ebert– McBride error minimization procedure to the space–time domain. The current algorithm employs a shift in space in order to find the minimal error. However, by extending the verification domain to both space and time, a more flexible procedure would become possible so that an object can be shifted also in time. Correction of drift: Among the outputs of the CRA algorithm is a vector field showing mutual dislocations of respective objects. Since a pairing algorithm does not take into account any additional conditions concerning the general behaviour of the forecast, this dislocation field should not display any particular tendency. The atmospheric circulation, however, tends to display persistence: a forecast failure of a synoptic-scale circulation results in a tendency for dislocation of CRAs. A way of introducing a tendency to the pairing phase would be to impose a constraint on the search region so that an observed object can only look for a matching forecast object in a preferred sector. We might consider two methods for obtaining a bias for use in such an algorithm. The first method uses tracking, as mentioned above. Let F and O be matched objects at time t. A dislocation vector is a minimization of a function Error(F + v, O), where Error is a discrepancy measure of choice (e.g., MSE or a correlation coefficient). A tracking algorithm can be used to find O′ , the associated observed object at the next time point. A dislocation vector is then calculated for O and O′ , i.e., minimization of Error(O, O′ ) yielding a local tendency in the observed field. Likewise, we

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can calculate a local tendency of the forecast field. With tendency vectors calculated for all of the objects admitting at least one future state, then we can compare the evolution of the resulting vector fields. 5.2

The SAL technique

The SAL technique (Wernli et al., 2008, 2009) provides a three-component feature-based quality measure for QPF, in which the three components quantify a forecast in terms of structure S, amplitude A, and location L. The method is object-based in the sense that precipitation objects are identified within a verification domain in order to determine S and L for contiguous precipitation areas exceeding a fixed or statistically defined precipitation threshold. However, SAL does not require one-to-one matching between the objects, which are identified separately for the observed and forecast fields. The A component represents a normalized difference between the domain-averaged QPF and the domain-averaged QPE. A positive value of A indicates an overestimation of predicted total precipitation; a negative value indicates an underestimation. The value of A is in the range [−2, 2] and 0 corresponds to a perfect forecast for system-averaged precipitation intensity. The L component combines information about the predicted precipitation centre of mass and the error in a weighted-average distance between the precipitation objects’ centres of mass. It consists of two parts: L = L1+L2. L1 measures the normalized distance between the centres of mass of the modelled and observed precipitation fields; the values range from [0, 1]. A value of L1 = 0 indicates that the centres of mass of the predicted and observed precipitation fields are identical. L2 is based on an averaged distance between the centre of mass of the total precipitation field and those of the individual precipitation objects. The value of L2 is in the range [0, 1]. As a whole then, L ranges from [0, 2]. The S component compares the volumes of the normalized precipitation objects, providing information about their size and shape. The range of S is [−2, 2]. A positive value occurs when the precipitation objects are too large or too flat, and a negative value when the objects are too small or too strongly localized. A perfect QPF is characterized by zero values for the all three SAL components. SAL can also be used to provide information about the systematic differences in the performance of two NWP models. It has been

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applied in numerous verification studies, as cited in Wernli et al. (2009), for example. 5.2.1

Application of SAL at the Hungarian Meteorological Service

At the Hungarian Meteorological Service (HMS), the SAL technique has been applied since the end of 2010. It is used both for the evaluation of model development and for the routine verification of operational NWP models. For the SAL verification discussed here, an operational radar product of HMS for accumulated rainfall was used. This product is a composite radar image produced from the reflectivity fields of three radars operated by HMS. First, column-maximum reflectivity fields (from ten elevation angles) are produced for each of the three radars, and then the composite image is calculated by taking the maximum value out of the three radar column-maximum fields for each pixel. The domain of the composite radar image covers a slightly larger area than Hungary with a spatial resolution of 2 km. A composite column-maximum reflectivity image is produced every 5 min. In making the accumulated rainfall product, a spatio-temporal interpolation method (Li et al., 1995) is applied to the composite images to produce interpolated images at 1-min intervals. With the use of such images, unrealistic rainfall amounts from rapidly propagating convective cells can be avoided. From the interpolated column-maximum composite-reflectivity fields the precipitation intensity is calculated with the Marshall–Palmer relationship. The radar rainfall product is corrected with surface rain gauge measurements only for the 12-hourly and 24-hourly accumulation periods. When evaluating NWP forecasts with the SAL technique, a three-hourly accumulation period was chosen. This radar product was not corrected, however, because the correction of three-hourly radar amounts was still under development. The following example shows the usefulness of SAL for the development of the AROME (Applications de la Recherche l’Op´erationnel a´ M´esoEchelle) non-hydrostatic model, specifically here for the evaluation of different boundary-layer parameterizations on forecasts of resolved deep convection. Two different model configurations are compared. For the first configuration, boundary-layer turbulence and shallow convection were parameterized separately; the CBR (Cuxart–Bougeault–Redelsperger: Cuxart et al.,

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2000) scheme was adopted for turbulence parameterization, whilst the Kain–Fritsch scheme (Kain and Fritsch, 1990) was used for shallow convection. For the second configuration, an eddy-diffusivity mass-flux approach, as originally proposed by Soares et al. (2004) and modified by Pergaud et al. (2009), was used, which describes boundary-layer turbulence and shallow convection together (cf., Vol. 1, Ch. 7, Sec. 4.4.1). Pergaud et al. (2009) took the Kain–Fritsch scheme (Kain and Fritsch, 1990) for the mass flux component of the EDMF, and thus this approach is labelled as EDKF here, as distinct from the case with separate parameterizations, labelled no EDKF. For the evaluation of the two AROME versions, a one-month period from 17 July 2010 to 17 August 2010 was taken. Figure 21.5 shows example precipitation forecasts with the two versions of the AROME model, as well as the radar observations, together with the defined SAL objects. The objects were defined using a dynamic threshold equal to 1/15th of the precipitation maximum over the domain. The no EDKF run overestimates the number of convective cells (objects), whilst the cell number for the EDKF run is much closer to reality. For both model runs, the size of the cells was smaller than observed and the maximum intensity within a cell was overestimated (i.e., objects were too strongly peaked in the model). The reason for the difference between the two AROME versions was traced to the fact that the EDKF enhanced mixing in the boundary layer during unstable conditions compared to the no EDKF version due to the non-local (mass-flux) transport added in the EDMF approach. This enhanced mixing prevented the build-up of warm air close to the surface and hampered the generation of small convective cells in the early afternoon hours. The three components of SAL verification can be visualized by the socalled “SAL-plot”. Figure 21.6 shows that the SAL-plots for the two different AROME runs are very similar. The domain-averaged precipitation (A component) is very well forecast whilst the size of the precipitation objects (S component) is underestimated (as expected from Fig. 21.5). It is interesting to compare the SAL results of AROME to hydrostatic models as well. From the SAL-plots of the IFS model at the ECMWF and of the operational ALADIN (Aire Limit´ee Adaptation dynamique D´eveloppement International) model run at HMS (Fig. 21.6) it is seen that the hydrostatic models with coarser resolution than AROME tend to overestimate the size of precipitation objects (positive S values), while the simulated domain-averaged precipitation was similar to the AROME model. It may be worthwhile to note that A and S tend to be correlated with each other

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Fig. 21.5 Top row: Three hourly accumulated precipitation from the AROME no EDKF run (left), the AROME EDKF run (middle), and radar (right) between 0900 and 1200 UTC on 22 July 2010 (9–12 hr forecasts). Bottom row: the defined SAL objects.

within a random-looking scatter. However, importantly, the precipitation pattern is quite different between the two AROME runs, although that is not strongly reflected in the SAL plots. The reason for this is that the two versions mainly differ in the number of precipitation objects (Fig. 21.7), which is not directly captured by the three SAL components. The diurnal cycle of convection and precipitation is a key operational issue. The domain-averaged diurnal cycle of precipitation is compared between different models and the radar measurements in Fig. 21.8. As shown in previous studies (e.g., Bechtold et al., 2004; Brockhaus et al., 2008), hydrostatic models with parameterized deep convection (ECMWF/IFS, ALADIN, ALARO in Fig. 21.8) tend to initiate convection too early, whilst non-hydrostatic models (both versions of AROME in Fig. 21.8) are quite accurate with the timing of convective precipitation. Unfortunately, the classical SAL method is not suitable for the verification of extreme values in simulated precipitation such as the maximum precipitation intensity within an object. New verification scores were introduced at HMS in order to enhance the SAL information in this respect. Specifically, let us consider here the averaged intensity of the three strongest objects, the averaged maximum intensity of all objects, and the averaged maximum intensity of the three strongest objects. These statistics are smoothed using a moving average over time. As an example, they are plotted as a function of lead time in Fig. 21.9 for two experiments with AROME. This comparison tests the sensitivity of the AROME forecast to the initial surface state of the model over a one-month summer period. In

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Fig. 21.6 SAL plots for ALADIN with 8 km grid length (upper left), for ECMWF/IFS at 16 km grid length (upper right), for the no EDKF AROME run (lower left), and for the EDKF AROME run (lower right) for a one-month period. Single forecasts are represented by marks, whose colours show the magnitude of the L component (the colour scale in each panel). The grey areas indicate the 25–75% percentiles and the dashed lines depict the median values of the S and A components. The contingency table in the bottom-right corner of each panel provides the number of cases (3-hr intervals) for which the threshold of 0.1 mm(3 hr)−1 is exceeded at least at one grid point in the model (MY) and the observations (OY) respectively, or not (MO, ON). Only (OY, MY) pairs are shown on the SAL plot.

the first experiment, the initial surface data was interpolated from the ALADIN model, whilst in the second, a surface-data assimilation cycle was run for AROME. For both experiments, the same atmospheric analysis (from a 3DVAR system) was used. Figure 21.9 shows that the forecast with AROME surface assimilation predicts the averaged maximum inten-

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Fig. 21.7 The number of precipitation objects (vertical axis) identified in the no EDKF AROME run (left) and the EDKF AROME run (right) against the number identified by radar measurements (horizontal axis). A one-month period is investigated (as in Fig. 21.6) using three-hourly precipitation accumulations.

sity of all convective cells well, although the intensity of the strongest cells is overestimated. In this respect, the experiment using ALADIN surface data performs better. This example highlights a general point that different scores (averaged and maximum cell intensity in this case) could favour different modelling configurations, making it difficult to single out the best prediction. A suite of scores helps to identify certain strengths and weaknesses of each forecast system. The importance of each score depends on the purpose of the forecast. For instance, forecasts of cell-averaged precipitation would be more important for hydrological applications (e.g., when coupling the NWP model to a hydrological model), while performance of the cell-maximum prediction would be more relevant for aviation meteorology. 5.3

Fractions skill score: QPF from different models

Use of the fractions skill score (FSS) belongs to the (a1) category of neighbourhood (or fuzzy) verification techniques and applies the strategy of neighbourhood-observation–neighbourhood-forecast. The FSS compares the fractional coverage of events (occurrences of rainfall values exceeding a certain threshold) over a window surrounding the observation and the forecast (Mittermaier and Roberts, 2010; Roberts, 2008; Roberts and Lean, 2008). When only a spatial window is considered, we may call the window an elementary area (EA). According to Ebert (2008), the FSS is defined

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Fig. 21.8 Composite of domain-averaged precipitation through the diurnal cycle for different models and radar for the one-month period. All model runs started at 0000 UTC, so that the lead time corresponds to an equivalent time of day. Black: radar measurements; red: AROME EDKF version; green: AROME no EDKF version; light blue: ECMWF/IFS; purple: ALADIN (operational at HMS with 8 km grid length); grey: ALARO (ALADIN model with different physics, run experimentally at HMS).

Fig. 21.9 Extreme precipitation statistics based on the SAL object definition. Both model runs started at 0000 UTC, so that the lead time corresponds to an equivalent time of day. Results for one month of simulations are shown, with two different versions of the AROME model. Red: with a surface data assimilation cycle for AROME; green: with initial surface data interpolated from the ALADIN model. Three-hourly radar observations are marked with black. Left: averaged maximum intensity of all objects. Right: averaged intensity of the three strongest objects.

Verification of high-resolution precipitation forecast with radar-based data

by: F SS = 1 −

N



2 0) N (PF − P 2 −1 N PF + N N

N −1  −1

P02

,

207

(5.2)

where PX (for X = F, O) is the fraction of the EA that is covered by rainfall that exceeds the given threshold, and N is the number of grid points in the EA. The expression in the numerator is a version of the fraction Brier score (FBS), in which fractions are compared. The denominator gives the worst possible FBS, in which there is no overlap of non-zero fractions. The FSS spans [0, 1] with 0 for a complete forecast mismatch and 1 for a perfect forecast. The FSS is zero if the forecast does not exceed the threshold but the observation does, or if threshold-exceeding values are forecast but not observed. The score is sensitive to rare events or small rainfall areas. The FSS depends on EA size and a precipitation threshold. Roberts and Lean (2008) show that when the EA size is increased, the FSS increases until it reaches an asymptotic value of fractions skill score (AFSS). If there is no bias, it asymptotes to AFSS = 1. If there is a bias, then AFSS is linked to the conventional frequency bias f0 /fM , where f0 is the fraction of observed points exceeding the threshold over the domain, and fM is the corresponding forecast frequency. Two reference FSS values are considered by Roberts and Lean (2008). The first is FSSrandom= f0 , which is obtained from a random forecast with a mean fractional coverage f0 . The second is FSSuniform= 0.5+f0 /2, which is obtained from a forecast with a fraction equal to f0 at every point. While the random forecast has low skill unless f0 is large, the uniform forecast is always reasonably skilful. The smallest EA size corresponding to FSSuniform represents the smallest scale over which the forecast output contains useful information. The actual scale over which output should be presented becomes a compromise between user requirements, cost effectiveness, and the forecast skill. In this manner, it is possible to compare the performance of different models using the scales for which the FSS exceeds FSSuniform. An example is shown in Fig. 21.10, which presents an FSS verification of forecasts of heavy local convective rainfall that caused flash floods over the Czech Republic in 2009 (Zacharov et al., 2013). In this study, a traditional verification, the SAL technique, and the FSS verification were all considered. Events occurring between 22 June and 5 July 2009 were analysed and simulated by COSMO (as adapted at IAP for the Czech Republic) and

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ALADIN. The non-hydrostatic model COSMO was run without a cumulus parameterization at a horizontal grid length of 2.8 km (C3 in Fig. 21.10) and with a cumulus parameterization at a horizontal grid length of 7 km (C7 in Fig. 21.10). The operational hydrostatic model ALADIN in its 2009 configuration was run with 9 km grid length (A9 in Fig. 21.10). At present, a 4.7 km grid length is used and results with this model configuration were obtained by rerunning for the 2009 flash flood period at the Czech HydroMeteorological Institute (CHMI) (A5 in Fig. 21.10). The forecasts of three hours of rainfall were verified against gauge-adjusted radar data from the Czech radar network CZRAD (Novak, 2007). The operational CHMI product MERGE (Salek, 2010) was used, which provides gauge-adjusted 3-hr rainfall in radar pixels covering the Czech territory with a resolution of (1 km)2 . Figure 21.10 plots the fraction of the 56 forecasts, as a function of the EA, which give FSS values larger that FSSuniform. In other words, it gives the relative number of forecasts leading to useful forecasts (FSS > FSSuniform) at each spatial scale. The plots show a change in forecast performance with the precipitation threshold. The C3 results are the least accurate for the threshold of 1 mm(3 hr)−1 , and the remaining models yield nearly the same forecast performance. The C3 behaviour for a wider EA corresponds to an under-prediction of lower precipitation rates. With an increasing threshold, the C3 model begins to improve its relative performance, and with the threshold of 10 mm(3 hr)−1 , the C3 achieves the best performance in terms of FSS. The analysed episode with heavy convective precipitation resulted in flash floods in several parts of the Czech Republic. The beginning of the period was marked by a front crossing the Czech Republic. The largescale precipitation associated with this front was generally forecasted well. However, the precipitation was difficult to forecast over the latter part of the period when convection dominated. At the time of writing, a second verification of the ALADIN forecasts is under way, which tests the QPFs from the 2009 period obtained after a modification of the operational model. Preliminary results indicate a significant improvement, especially on days with small-scale convection without any synoptic forcing. The simultaneous use of a traditional verification, the SAL technique, and the FSS verification by Zacharov et al. (2013) showed that the combination of several verification techniques was useful. Each technique evaluated the precipitation forecast in a different way and provided complementary information about the forecasts. This was especially true for the high-

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Fig. 21.10 The cumulative fractions of 56 forecasts with FSS>FSSuniform plotted as a function of EA scale (km) for precipitation thresholds of 1 mm(3 hr)−1 (upper plot), 5 mm(3 hr)−1 (middle plot), and 10 mm(3 hr)−1 (lower plot). The results are shown with four NWP models: COSMO-CZ (2.8 km grid length, C3: red), COSMO-CZ (7 km grid length, C7: black), ALADIN (4.7 km grid length, A5: green) and ALADIN (9 km grid length, A9: blue). (Reprinted from Fig. 6 of Zacharov et al. (2013), with permission from Elsevier.)

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resolution QPF, where the double-penalty problem occurred extensively.

6

Use of dual-polarization radar for microphysics verification

There are several techniques available for retrieval of hydrometeor categories from polarimetric radar (PR) measurements. Tree methods (classification techniques) use threshold values in a phase space of several PR characteristics. For instance, one of the first studies based on threshold values examined the microphysical structure of a supercell storm that occurred near Munich on 30 June 1990 (H¨ oller et al., 1994). The PR variables, ZDR and LDR, together with the height of the melting layer were used for an empirical interpretation for nine microphysical categories. Verification against observations focused mainly on hail occurrence and the results showed a consistency with the storm dynamics that proved to be useful for hail detection. Straka et al. (2000) provided an extended review of a range of PR variables which can be useful for the identification of several microphysical categories (hail, graupel, rain, wet hail mixture, snow crystals, and aggregates). Temperature is also included in the classification in order to avoid some obvious unphysical situations that cannot be avoided otherwise. For example, ice crystals would not be expected above 15◦ C, and rain would not be expected below −30◦ C. Vivekanandan et al. (1999a) have reported on many classification studies based on thresholds, or hard boundaries of polarimetric parameters. They proposed a fuzzy approach as being more suitable for the retrieval of information about hydrometeor categories. It is known that the use of hard boundaries can lead to wrong classifications due to an overlap between PR variable ranges for various hydrometeor types. Fuzzy boundaries between polarimetric observables are best considered under a fuzzy logic (Mendel, 1995) that enables a smooth transition in polarimetric observable boundaries among precipitation types. The use of neural networks (NN: e.g., Haykin, 1994) has also been considered. Although powerful, the NN approach needs a training set of considerable size, which is difficult to obtain (e.g., Straka et al., 2000). For example, an NN-based method was applied to large data sets of radar and surface observations by Vulpiani et al. (2009). Point-wise estimates

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of hourly rainfall accumulations and instantaneous rainfall rates by NN using parametric polarimetric rainfall relations were compared with dense surface-gauge observations. Liu and Chandrasekar (2000) proposed an NN system in combination with a fuzzy logic classification. The performance of a fuzzy classification depends critically on the shape of the so-called “membership functions”, which enter a “fuzzification” component and convert measured values to fuzzy sets with a different membership degree. In their study, a hybrid neuro-fuzzy system was proposed, in which a training algorithm for the NN was also used to determine the parameters for a fuzzy logic. The state of the art of using PR measurements for retrieving microphysical structures is reviewed by Chandrasekar et al. (2013). The review declared that there had been much progress in hydrometeor classification over the last decade due to the introduction of fuzzy logics to classification systems. The implementation procedures have expanded from point-wise classifications to areal analyses by using texture information. At present, integrated weather-radar classification systems include three aspects: data quality, echo classification, and hydrometeor identification. The review concluded that “the hydrometeor classification topic presents exciting future research opportunities, and is likely to remain active for a long time.” Straka et al. (2000) summarized scientific and operational reasons for deducing hydrometeor types from PR data. The verification of microphysical parameterizations in NWP models, and QPF verification were included in the set of reasons. At present, many studies focus on verification of the polarimetric identification of microphysical categories by various direct measurements. Use of PR variables for quantitative verification of highresolution NWP models is still limited, with the majority of verification studies being limited to qualitative comparison of retrieved and modelled microphysics. Several studies have used data collected during Intensive Observing Periods (IOPs) of the MAP project (cf., Sec. 3) to evaluate cloud microphysical structures during heavy precipitation events. The non-hydrostatic Meso-NH model was run for several MAP events, and the simulated microphysical structure was compared with cloud microphysical retrievals from National Center for Atmospheric Research (NCAR) S-band dual-polarized (S-Pol) radar data (Lascaux et al., 2006; Pujol et al., 2005, 2011). Simulations made use of a bulk microphysics scheme to predict the evolution of six water species (vapour, cloud droplets, raindrops, pristine ice, snow/aggregates, and frozen drops/graupel) included in the Meso-NH ICE3

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scheme. The scheme was later extended to account for hail (Meso-NH ICE4 scheme). Identification of hydrometeor classes from polarimetric variables used the NCAR algorithm (Vivekanandan et al., 1999b) together with the determination of the 0◦ C level from radiosondes. Detailed microphysical analysis was focused on microphysical processes leading to the development of heavy precipitation in the Lago Maggiore region of Italy. In the study by Jung et al. (2012), observed polarimetric variables were compared with polarimetric signatures simulated by the NWP model ARPS in order to assess the ability of single- and double-moment microphysics parameterizations to reproduce observed polarimetric signatures. The study analysed a tornadic thunderstorm that occurred in May 2004 in central Oklahoma. An ensemble Kalman filter technique was used to assimilate data from one single-polarized radar, whilst observations from another singlepolarized radar and a dual-polarized radar were used for verification. Both microphysics parameterizations were able to capture well the observed reflectivity fields, but a comparison of simulated and observed polarimetric signatures showed better agreement when using the double-moment microphysics parameterization. A comprehensive verification study using satellite and radar data was performed by Pfeifer et al. (2010). Quantitative verification was applied to precipitation forecasts produced by the COSMO-DE model of the German weather service, with a horizontal grid length of 2.8 km. Two cases with heavy precipitation were analysed and compared with observed data under a model-to-observation approach using forward operators (cf., Sec. 2.5), including a polarimetric radar-forward operator (SynPolRad: Pfeifer et al., 2008). The verification applied traditional skill scores such as the rootmean-square error, probability of detection, frequency bias, false alarm ratio, and the Heidtke skill score, as well as a spatial technique, the fractions skill score.

7

Summary and conclusions

Radar-based verification data and techniques for QPF verification have been critically reviewed in this chapter. In order to verify high-resolution QPF performance, it is necessary to identify a suitable verification technique together with suitable verification data, which correspond to forecast variables and the forecast horizontal resolution. The horizontal resolution of radar data, which is of the order of 1 km, is well matched to current

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operational NWP and so radar-based QPE has become an important data source for QPF verification. Single-polarization radars provide QPE values based on a Z–R relation. A suitable merging of radar rainfall estimates with ground precipitation measurements offers a verification data set with sufficient spatial and temporal resolution. However, radar-based QPE can be difficult to obtain over mountainous terrain and where gauge density is low. Over the last decade, dual-polarization radar has grown into an operational technology in the meteorological services of many countries. Polarimetric characteristics provide multi-dimensional information. When properly evaluated, polarimetric variables can not only improve the rain rate estimate but can also provide information about the structure of cloud microphysical categories in time and space. Different verification techniques evaluate QPF in different ways, providing different information about the forecast performance. It is important to examine all aspects of a forecast system in order to reveal its strengths and weaknesses. Traditional verification is based on grid-point information. However, the use of spatial methods is particularly worthwhile for high-resolution forecasts that may have difficulty in predicting the localization of high-precipitation areas. Simultaneous use of multiple verification techniques is therefore recommended for modellers. In summary, the authors expect the following in the coming years: • More studies comparing polarimetric and ground-based rainfall values and using polarimetric QPE in high-resolution QPF verification. • Increasing applications of spatial techniques in modeller-oriented verifications. The spatial techniques are able to take into account QPF uncertainty and to reflect various aspects of forecast performance. • Use of spatial verifications in regional ensemble forecasts, especially for quantitative diagnosis of forecast uncertainties. • Increasing applications of quantitative verification for microphysical studies. For this purpose, polarimetric parameters provide the primary verification data, which need to be converted into information on cloud microphysical structures. An opposite approach of comparing model-generated polarimetric information with measurements is also plausible. The importance of understanding physical uncertainties in a model must be emphasized. In the present chapter, for example, a goal was to show how the impact of different treatments for boundary-layer turbulence can be assessed from a SAL-based analysis. There is a need to develop more phys-

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ically based verification methods to make such assessments. Such developments will be especially crucial in order to fully exploit multi-dimensional information obtained from polarimetric radar in the future. In the longer term, the need for probabilistic quantifications of the forecast should be emphasized, as already suggested at several places in the chapter (cf., Secs. 3 and 5). We especially refer to Jaynes (2003) for the basics of probability as an objective measure of uncertainties. From the point of view of fundamental probability theory, the goal of model verification would be to reduce model uncertainties by objectively examining the model errors. In order to make such a procedure useful and effective, forecast errors and model uncertainties must be linked together in a direct and quantitative manner. Unfortunately, many of the statistical methods found in the general literature are not satisfactory for this purpose. It may be worthwhile to see how Jaynes (2003) in his Ch. 16 reviews traditional statistics: he summarizes it as a “cookbook” without clear guiding principles. The current efforts of object-oriented verifications also appear to be suffering from a similar problem. Although each proposed method provides a certain quantification of required forecast error measures such as position, size, and amplitude of a storm, we lack any principles to judge the usefulness of these quantifications for forecast verification. We rather tend to be slaved by these quantified measures for judging the forecasts without appreciating their usefulness. The Bayesian principle would, on the other hand, provide such a direct link that from a given forecast error, an uncertainty associated with a particular parameter in a parameterization, for example, could be objectively and quantitatively estimated by invoking the Bayes theorem. In this respect, the forecast verification process may be considered a type of inverse data assimilation, in which the model outputs are reassimilated into observations and compared with the latter in an objective manner. In the same sense as the standard variational data assimilation works, such an inverse assimilation procedure must also be based on model physics in such a manner that any weaknesses of model physics can be assessed objectively. The Bayesian principle also tells us that ensemble, sample space, randomization, etc. as typically invoked in statistical methods are not indispensable ingredients for uncertainty estimates, although they may be useful. Such a perspective could promote approaches for evaluating model uncertainties without relying on computationally intensive ensemble runs.

PART IV

Unification and consistency

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Introduction to Part IV

The convection parameterization problem was considered within wider operational contexts in Part III. That wider view is retained in Part IV with the representation of convection considered still as one aspect within the full modelling system, albeit of course the focal point. However, we now consider convection from wider theoretical perspectives, some of which are borrowed and developed from approaches that have been pursued for the parameterization of other physical processes. Part IV is titled “Unification and Consistency” because these perspectives naturally lead to such questions and goals. In order to set the scene, Ch. 22 is devoted to a very basic issue: the exact formulation of moist thermodynamics for atmospheric modelling. It is common practice today that formulations of moist thermodynamics are presented in a grossly approximated manner. This chapter shows how a formulation without unnecessary approximations is possible, and also suggests what kind of impacts and implications can be expected for parameterization. In Ch. 23 attention returns to key issues of cloud microphysics. Currently there are two major approaches for numerically modelling cloud microphysics: bulk and spectral formulations. The problem may be considered by analogy with a similar choice in the context of mass-flux parameterization (cf., Vol. 1, Ch. 9). In critically discussing these two major possibilities, the chapter also serves as an introduction to cloud microphysics modelling, and to suggest the extent to which the complexities of microphysics can and should be incorporated into convection parameterization and explicit convection modelling. Thus, the chapter is complementary with Ch. 18. The discussions of this set so far have been focused on a standard ap217

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proach for deep-convection parameterization based on mass flux. This focus arises partly because this is one of the first systematic formulations for deep-convection parameterization, as proposed by Arakawa and Schubert (1974), and partly because this is the formulation that has been adopted (and adapted) in actual practice by the majority of current operational models (global and regional weather forecast models) as well as global climate projection models. However, this close focus should not be interpreted as meaning that the mass-flux approach is necessarily superior to all other approaches. It has various known problems that we have discussed at length, some of which may be avoided in alternative approaches, although of course these may introduce difficulties of their own. Part IV is, thus, partially devoted to discussions of alternative, equally legitimate possibilities. A particular interest is whether alternative approaches might prove valuable for improving the interactions between different parameterizations or for use at higher resolutions. Chapter 24 is devoted to examining the possibility of treating moist convection as a type of turbulent flow. From this perspective, analysis of the flow proceeds by searching for key physical combinations that allow one to scale the terms in the turbulence kinetic energy equation. Once suitably scaled, the turbulent fluxes required for parameterization can then be expressed as functions of dimensionless quantities following the prescription of similarity theory. The similarity approach is an attractive alternative framework in its own right, but is also of interest in providing intriguing new perspectives on the mass flux-based approach. For example, a dimensionless quantity analogous to the fractional area of convection is shown to play an important role. Moreover, the closure problem can be interpreted as arising from a matching/consistency condition at the cloud base, connecting the turbulent scalings that are appropriate for the cloud layer and the sub-cloud layer. Chapter 25 introduces the Liouville equation as a formal approach for describing the subgrid-scale distribution of physical variables in terms of a distribution density function (DDF). In the literature, unfortunately, it is often misleadingly called a “probability density function (PDF)”. When stochasticity is further introduced, the DDF can then be legitimately reinterpreted as a PDF: the description of a system becomes inevitably probabilistic due to the existence of stochasticity (randomness) as separately discussed in Ch. 20. Various current cloud parameterizations can be understood as arising from various simplifications and assumptions applied to the Liouville formulation.

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Chapter 26 is devoted to the issue of unification of subgrid-scale parameterizations. In current practice, various parameterizations for individual subgrid-scale physical processes are developed and implemented into models relatively independently with often rather modest attention to the other physical processes and schemes. Thus, in an operational context various issues of consistency and unification arise. This chapter discusses such issues for an example operational system. It also steps back from the immediate operational context, and asks in what sense and to what extent do we really need consistency and unification of parameterizations at a more fundamental level. Consistency of subgrid-scale parameterizations is not an issue as long as one begins with the first principles of the physics, but it is not necessarily satisfied at each point where an empirical formula, hypothesis, or approximation is introduced in the course of a deduction towards a specific parameterization. The chapter further explains how a consistent approach may be possible.

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Chapter 22

Formulations of moist thermodynamics for atmospheric modelling P. Marquet and J.-F. Geleyn Editors’ introduction: The meteorological literature contains a plethora of temperatures and potential temperatures which have been introduced in order to study various aspects of the thermodynamics of the moist atmosphere. Many of these are based upon approximations that may be very well suited to particular applications, but which are not necessarily valid more generally. Given the theme of consistency in Part IV, it is therefore appropriate to begin with this study of moist thermodynamics, which focuses on the construction of very general representations of enthalpy and entropy. If we consider two separate parameterizations within our model codes (or perhaps even two separate calculations within a single parameterization) then the use of subtly different thermodynamic variables that may be separately appropriate for the particular calculations could nonetheless produce inconsistencies in the overall energy, enthalpy, or entropy budgets. More generally applicable formulations are developed here by seeking suitable standard values on the basis of the third law of thermodynamics.

1

Introduction

Internal energy, enthalpy, and entropy are the key quantities to study in an examination of the thermodynamic properties of the moist atmosphere, because they correspond to the first (internal energy and enthalpy) and second (entropy) laws of thermodynamics. The aim of this chapter is to search for analytical formulae for the specific values of enthalpy and entropy and for the moist-air mixture composing the atmosphere. It may seem a little bit surprising to initiate this kind of research, be221

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cause one may consider that all the basic moist-air properties have already been described in the available textbooks on atmospheric thermodynamics (Ambaum, 2010; Bejan, 1988; Bohren and Albrecht, 1998; de Groot and Mazur, 1962; Dufour and Van Mieghem, 1975; Emanuel, 1994; Glansdorff and Prigogine, 1971; Iribarne and Godson, 1973; Zdunkowski and Bott, 2004). It is, however, explained in this chapter that there are some aspects which are not easily understood and some problems which are not easy to solve cleanly in order to compute locally the moist-air enthalpy and entropy. It is also shown that the computations of these local state functions requires the knowledge of reference values of enthalpies and entropies for all species forming the moist air (N2 , O2 , and H2 O), and that the third law of thermodynamics can be used to improve on the present formulae for moist-air enthalpy and entropy. The third law consists of specifying zero entropy at 0 K for the more stable crystalline states of atmospheric species (solid-α O2 , solid-α N2 , and Ice-Ih without proton disorder). Differently, it is common in atmospheric science to specify zero entropies at higher temperatures (typically at 0◦ C), for the dry air and for one of the water species (typically either for the liquid or the vapour state, depending on the authors). It is explained in the next sections that it is possible to avoid these arbitrary assumptions and simply rely on the third law, leading to interesting new results concerning the study of local values of moist-air entropy. The same programme can be applied to the computation of specific values of moist-air enthalpy, with special attention to be paid to determining the kind of enthalpy that must be considered and for which zero values can be set at 0 K. The theoretical aspects of the new third law-based formulations for specific values of enthalpy and entropy, with possible applications of them, have been published in a series of four papers (Marquet, 2011, 2014, 2015; Marquet and Geleyn, 2013). This chapter describes and summarizes the content of these four papers. The reason internal energy is not considered in the previous and next paragraphs is that enthalpy is more suitable than internal energy when studying flowing fluids like the moist atmosphere. Following the notations collected in Sec. 10, the use of enthalpy h simplifies the description of the pressure–volume work in open systems, whereas internal energy ei is of more common use at the laboratory scale and for portions of fluids at rest. Indeed, the first law can be expressed in terms of the specific enthalpy defined by h = ei + RT = ei + p/ρ, due to the perfect gas and Mayer’s laws (p = ρRT and Cp = Cv + R) valid for the moist-air definitions of Cp ,

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Cv , and R. Therefore, internal energy will not be studied in the following because it can be easily computed from the enthalpy, since the reverse formulae ei = h − RT = h − p/ρ are valid for the moist-air definitions of the variables. Computing local values for enthalpy and entropy is a new feature here. This feature should be clearly distinguished from the usual need for determining the equations for observed variables, like the temperature. It is explained in the following sections that the temperature equation can indeed be expressed in terms of Cp dT /dt or d(Cp T )/dt, with no need to apply the third law for determining the reference values of enthalpies. Similarly, the present moist air-conserved variables are built by assuming adiabatic evolution of closed systems (i.e., isentropic processes with no change in total water content qt ). These assumptions avoid the need to determine reference values of enthalpies or entropies. However, the knowledge of third law-based specific values for h and s can allow original, direct computations of h, s, dh/dt, and ds/dt, without making the hypothesis of constant values of qt and with the possibility of analysing isenthalpic or isentropic properties for open-system evolution of moist air, namely for qt varying in time and in space. The chapter is organized as follows. An overview of moist-air atmospheric thermodynamics is presented in Sec. 2, where the role of reference values is described in some detail. The moist-air specific entropy and enthalpy are then defined in Secs. 3 and 4 respectively, starting from the standard values computed by applying the third law. Some physical properties of third law moist-air entropy are presented in Sec. 5, including comparisons with well-known alternative potential temperatures. Several applications are shown in Sec. 6: isentropic features, transition from stratocumulus to cumulus, top-PBL entrainment and CTEI criterion, turbulence parameterization, squared Brunt–V¨ ais¨ al¨ a frequency, and potential vorticity. Section 7 focuses on conservation properties associated with third law moist-air entropy. Some physical properties of moist-air enthalpy are described in Sec. 8. Comparisons are made with the well-known concept of moist static energy and possible applications to turbulent surface fluxes are presented. Final comments are made in the concluding Sec. 9 about possible new applications of moist-air entropy and enthalpy.

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Part IV: Unification and consistency

Overview of atmospheric thermodynamics Moist-air thermodynamics

The results derived in this section correspond to the usual hypotheses made in most textbooks on moist-air thermodynamics. Moist air is made up of dry air, water vapour, liquid water, and ice species, denoted by the indices d, v, l, and i, respectively. Cloud liquid or solid species, together with liquid or solid precipitations, are all considered as being at the same temperature as the rest of the parcel (i.e., the one for dry air and water vapour). It is assumed that the dry air and water vapour species are perfect gases. The same hypothesis of perfect gas is extended to moist air containing cloud droplets and crystals or precipitations, with the condensed phases having zero volume, though impacting on moist values of Cp and R. Due to Dalton’s law, the moist-air specific enthalpy and entropy are assumed to be additive functions, leading to weighted sums equal to: h = qd hd + qv hv + ql hl + qi hi ,

(2.1)

s = qd sd + qv sv + ql sl + qi si .

(2.2)

The parcels of fluid are assumed to be small enough to be homogeneous, but large enough to allow the definition of temperature and pressure variables. The total mass and density of a parcel are equal to m (in kg) and ρ. The mass and density of the species are denoted by mk and ρk (values of k are d for dry air, v for water vapour, l for liquid water, and i for ice). The specific contents and the mixing ratios are equal to qk = mk /m = ρk /ρ and rk = qk /qd = ρk /ρd .1 The total-water specific contents and mixing ratios are denoted by qt = qv + ql + qi and rt = qt /qd . The sum of the specific contents is equal to qd + qt = 1. Three equations of state exist for the moist air, dry air, and water vapour perfect gases. They can be written p = ρRT , pd = ρqd Rd T and e = ρqv Rv T . The property p = pd + e corresponds to R = qd Rd + qv Rv . It is assumed that the specific heat at constant pressure of vapour species (Cpd and Cpv ), liquid water (Cl ), and ice (Ci ) are constant terms in the atmospheric range of temperatures (from 150 to 350 K). The specific heat of moist air is equal to the weighted sum Cp = qd Cpd + qv Cpv + ql Cl + qi Ci . The latent heats of fusion, sublimation, and vaporization are created with changes in enthalpy associated with changes of phases. For water 1 Unlike elsewhere in this set where q designates a mixing ratio, in this chapter q designates a specific content, while the mixing ratio is designated by r.

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species Lv = hv − hl > 0, where hv and hl are the specific values computed per unit mass of moist air. Similarly, Ls = hv −hi > 0 and Lf = hl −hi > 0. The Gibbs function is equal to μ = h − T s, and depends on enthalpy, entropy, and absolute temperature. The important property is that Gibbs functions are the same for two phases in stable equilibrium states (for instance, for liquid and vapour states at the saturation equilibrium curve). This excludes the metastable states (for instance, if supercooled water is considered). The consequence is that the change in entropy created by a change of phase is equal to the latent heat divided by the equilibrium temperature (Lv /T > 0 for vaporization processes). Kirchhoff’s laws are derived with the assumption of constant values for the specific heat at constant pressure, leading, for instance, to dLv /dT = d(hv − hl )/dT = Cpv − Cl for the latent heat of vaporization. The same relationships are valid for the fusion and sublimation processes (dLs /dT = Cpv − Ci and dLf /dT = Cl − Ci ). If standard values of the latent heats are known at a temperature T0 , the latent heat at any temperature T can be determined from Lv (T ) = Lv (T0 ) + (Cpv − Cl )(T − T0 ), with the equivalent relationships for Ls (T ) and Lf (T ). The Clausius–Clapeyron equation describes the equilibrium curves between liquid water (or ice) and water vapour. The water-saturation equilibrium corresponds to Rv T 2 desw /esw = Lv dT while that for ice saturation corresponds to Rv T 2 desi /esi = Ls dT .

2.2

Special features of atmospheric thermodynamics

Many specialized quantities associated with the first and second laws of thermodynamics have been derived and are commonly used in the context of atmospheric science, in particular for applications to turbulent and convective processes. Examples of those atmospheric quantities are the static energies and the potential temperatures. The second law is mainly expressed in meteorology in terms of the potential temperature θ via the generic relationship s = Cp ln(θ) + const, with, for instance, θ = T (p0 /pd )κ for the dry air limit. Moist-air gener′ alizations of θ exist: they are denoted by θl , θe , θe′ , and θw , and are all associated with the moist-air entropy and with adiabatic (closed system) or pseudo-adiabatic (saturated open system) assumptions. The liquid-water and equivalent potential temperatures θl and θe defined in Betts (1973) and Emanuel (1994) are conserved quantities for adi-

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abatic motion of a closed parcel of atmosphere, and therefore for isentropic processes of closed parcels of moist air. The pseudo-adiabatic wet-bulb ver′ computed in Saunders (1957) is a conserved quantity for motions of sion θw saturated moist air, provided that all the condensed water is withdrawn by precipitation. Even if entropy is conserved by adiabatic upward motions, entropy is removed by precipitation, leading to Saunders’s (1957) formulae, which are associated with that increase in specific moist-air entropy which balances the loss by precipitation. ′ . It is Another equivalent potential temperature is deduced from θw equal to the dry-air value of θ attained by a saturated parcel at very low ′ pressure, following a path at constant θw until all the water vapour is withdrawn by precipitation. This pseudo-adiabatic potential temperature will be denoted by θe′ . The numerical values of θe and θe′ are close to each other. The links between the moist-air entropy and the conservative features of these potential temperatures are thus subject to hypotheses of closed and adiabatic systems or pseudo-adiabatic processes. They are not valid for the more general atmospheric states of arbitrarily varying water contents. This is the motivation for the search for a moist-air potential temperature θs , which would be valid for open systems and would be equivalent to the moist-air entropy via s = Cpd ln(θs ) + const. The possibility of computing θs (and thus s) locally would be interesting as it would permit new studies of atmospheric energetics (on the basis of only those general assumptions recalled in Sec. 2.1). There is another moist-air potential temperature denoted by θv , which is not directly associated with moist-air entropy. It is involved in the computation of the buoyancy force and represents the leading order approximation of the impact of fluctuations of density. It is only for the dry-air case that ′ coincide with θ, all the moist-air potential temperatures θl , θe , and θw including the virtual potential temperature θv . The computations of moist-air enthalpy are usually conducted with the same hypotheses of closed systems as for the derivation of θl or θe . The moist-air enthalpy is often expressed by Cpd T + Lv qv or Cpd T − Lv ql , up to some constant values. This demonstrates a close link with what are called “moist static energies” (MSE). The MSEs are associated with the moist-air enthalpy and are defined by quantities like Cpd T + Lv qv + φ or Cpd T −Lv ql +φ. The potential energy gz is added to the moist-air enthalpy to form the quantity called “generalized enthalpy”, h + φ (Ambaum, 2010). It is explained in the following sections that these MSE formulae are

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derived by making additional hypotheses concerning the zero enthalpy of dry air and water atmospheric species. This is the reason the search for a moist-air enthalpy temperature Th (Derbyshire et al., 2004), which would be equivalent to the moist-air enthalpy via h = Cpd Th + const, could be interesting in allowing new studies of atmospheric energetics (again with only those general assumptions recalled in Sec. 2.1). 2.3

The role and the calculation of reference values

The aim of this subsection is to explain why it is difficult to compute the moist-air enthalpy and entropy, and why this is due to the necessity to determine the reference (or standard) values of enthalpies and entropies. It turns out that it is difficult to compute precisely the specific values of enthalpy and entropy for the moist air understood as an open system made of dry air (O2 , N2 ), water vapour, and condensed water species (liquid and ice droplets, and precipitations). The open feature corresponds to the fact that different phases of water can be evaporated from (or precipitated to) the surface, mixed by convective or turbulent processes, or entrained (or detrained) at the edges of clouds. For these reasons, except in rare conditions, qt is a constant neither in time nor in space. Let us examine the consequences of varying qt on the reference values by analysing the usual methods employed in atmospheric science to compute the specific enthalpy (Ambaum, 2010; Emanuel, 1994; Iribarne and Godson, 1973). For the sake of simplicity, the ice content is assumed to be zero. The definition of Eq. 2.1 for h can be rewritten with ql = qt − qv or qv = qt − ql as any of: h = qd hd + qt hv + ql (hl − hv ) ,

h = qd hd + qt hl + qv (hv − hl ) .

(2.3) (2.4)

The next step is to use the properties qd = 1 − qt and Lv = hv − hl , leading to: h = hd + qt (hv − hd ) − ql Lv ,

h = hd + qt (hl − hd ) + qv Lv .

(2.5) (2.6)

Since the specific heats at constant pressure are assumed to be independent of T , the enthalpies at the temperature T can be expressed from the enthalpies at the reference temperature Tr . The result is hd = (hd )r + Cpd (T − Tr ) for the dry air, with equivalent relationships for water vapour (index v) and liquid water (index l). Equations 2.5 and 2.6

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Part IV: Unification and consistency

transform into: h = [Cpd + (Cpv − Cpd )qt ] T − Lv ql

+ [(hd )r − Cpd Tr ] + qt [(hv )r − (hd )r ] − qt (Cpv − Cpd )Tr , (2.7)

h = [Cpd + (Cl − Cpd )qt ] T + Lv qv

+ [(hd )r − Cpd Tr ] + qt [(hl )r − (hd )r ] − qt (Cl − Cpd )Tr .

(2.8)

The second lines of Eqs. 2.7 and 2.8 are usually discarded. Since the term (hd )r −Cpd Tr is a true constant, it can be removed in both Eqs. 2.7 and 2.8 because it represents a global offset having no physical meaning. The other terms depending on qt can also be neglected if qt is a constant (i.e., for motions of closed parcels of moist air). However, if qt is not a constant, the last quantities depending on Cl − Cpd and Cpv − Cpd must be taken into account as correction terms for the first lines. These last terms could be easily managed, since the numerical values of specific heats are known. However, the other terms (hl )r − (hd )r and (hv )r − (hd )r depending on the reference values are more problematic for varying qt , since these reference values are not known. The only possibility for discarding these terms is to set arbitrarily (hv )r = (hd )r in Eq. 2.7 or (hl )r = (hd )r in Eq. 2.8. But these arbitrary choices have never been justified in general or atmospheric thermodynamics. Alternative formulations are usually derived in terms of the enthalpy expressed per unit mass of dry air. The quantities kw and k are defined by Eqs. 4.5.5 and 4.5.4 in Emanuel (1994) and are called “liquid-water enthalpy” and “moist enthalpy”, respectively. They correspond to h/qd with h given respectively by Eqs. 2.3 and 2.4. They may be written in terms of the mixing ratios, leading to kw = hd + rt hv − Lv rl and k = hd + rt hl + Lv rv . Equations 2.7 and 2.8 are then replaced by: kw = (Cpd + Cpv rt )T − Lv rl

+ [(hd )r − Cpd Tr ] + rt [(hv )r − Cpv Tr ] ,

(2.9)

k = (Cpd + Cl rt )T + Lv rv

+ [(hd )r − Cpd Tr ] + rt [(hl )r − Cl Tr ] .

(2.10)

The last lines of Eqs. 2.9 and 2.10 are discarded in Emanuel (1994). This is indeed valid if rt is a constant term. But, this is not valid for varying rt , except if the arbitrary hypotheses (hv )r = Cpv Tr or (hl )r = Cl Tr are made. The conclusion of this subsection is that if qt is not a constant, it is necessary to determine reference values of (hd )r for dry air, and (hl )r or

Formulations of moist thermodynamics for atmospheric modelling

229

(hv )r for water species, in order to compute the moist-air enthalpy. Even if there is a link between the water-vapour and the liquid-water enthalpies via (hv )r = (hl )r + Lv (Tr ), no such link exists between the dry-air and any of the water reference values. The same problem is observed in attempting to compute specific values of the moist-air entropy, with the need to know the reference entropies and with additional problems created by the need to manage the reference partial pressures for the dry air and the water vapour.

2.4

New proposals based on the third law

It is possible to compute the first and second laws’ moist-air specific quantities h and s without making any a priori assumptions of adiabatic, pseudoadiabatic, or closed parcel of fluid, and without making the arbitrary hypotheses described in the previous subsection concerning the reference values. The theoretical aspects of the new third law moist-air formulations for h and s, with possible applications of them, are published in Marquet (2011, 2014, 2015) and Marquet and Geleyn (2013). Following the advice given in Richardson (1922) and continuing the method described in Hauf and H¨ oller (1987), it is shown in Marquet (2011) that it is possible to use the third law in order to define a moist-air potential temperature denoted by θs , with the moist-air entropy written as sref + Cpd ln(θs ), where sref and Cpd are two constants. This new moist-air potential temperature θs generalizes θl and θe to open systems and becomes an exact measurement of the moist-air entropy, whatever the changes in water contents qv , ql , or qi may be. The same method used for defining the moist-air entropy based on the third law is used in Marquet (2015) to express the moist-air enthalpy without making the assumptions of pseudo-adiabatic, adiabatic, or zeroenthalpy at temperature different from 0 K. Applications of open-system moist-air features described in the four articles are to stratocumulus (see Secs. 6.1 and 6.2), to moist-air turbulence via calculations of the Brunt–V¨ ais¨ al¨ a frequency (see Secs. 6.3 and 6.4), to the definition of a moist-air potential vorticity (see Sec. 6.5), and to comparisons with moist static energies (see Sec. 8.3).

230

2.5

Part IV: Unification and consistency

Computation of standard values for entropies and enthalpies

The aim of this subsection is to compute the standard values of enthalpy and entropy. They are denoted by the superscript 0, leading to h0 and s0 , respectively. They must be computed for the three main species which compose the moist air: nitrogen (N2 ), oxygen (O2 ), and water (H2 O). It is worthwhile explaining more precisely how the reference values are different from the standard values. • The standard temperature is commonly set to any of 0◦ C, 15◦ C, 20◦ C, or 25◦ C, whereas the standard pressure p0 is set to either 1000 hPa or 1013.25 hPa. The standard values are set to T0 = 0◦ C and p0 = 1000 hPa for all species in the following sections of this chapter. • The reference temperature corresponds to the same value Tr for all species. Unlike the standard definitions (pd )0 = e0 = 1000 hPa, the reference partial pressures (pd )r for dry air and er for water vapour are different. The reference total pressure is equal to pr = (pd )r + er and is set to pr = p0 = 1000 hPa hereafter. The water-vapour partial pressure er is equal to the saturation pressure value at Tr . It is thus close to 6.11 hPa for Tr = 0◦ C. This value is much smaller that p0 and this explains why the reference values of water-vapour entropy will be different from the standard ones. The dry-air partial pressure is equal to (pd )r = pr − er ≈ 993.89 hPa. It is not possible to compute enthalpy and entropy from the ideal-gas properties only, because the ideal-gas assumption ceases to be valid at very low temperatures. Moreover, it is not possible to use the Sakur–Tetrode equation for computing the expressions for the entropies of N2 , O2 , or H2 O, since they are not mono-atomic ideal gases. It is assumed that the three main gases N2 , O2 , and H2 O are free of chemical reactions, with the consequence that the specific enthalpies cannot correspond to the concept of “standard enthalpies of formation or reaction” (denoted by ΔHf0 and ΔHr0 in chemical tables for most species). The general method used in Marquet (2015) is to take advantage of the fundamental property of h and s of being state functions, and to imagine at least one reversible path which connects the dead state at T = 0 K and p0 = 1000 hPa, and the standard state at T0 and p0 . The standard entropy s0 of a substance at T0 and p0 is obtained by:

Formulations of moist thermodynamics for atmospheric modelling

231

(1) setting to zero the value at 0 K by virtue of the third law for the most stable solid state at 0 K; (2) computing the integral of Cp (T )/T following the reversible path from 0 K to T0 , for the solid(s), the liquid, and the vapour; and, (3) adding the contributions due to the changes of state (latent heats) for solid(s)-to-solid(s), solid-to-liquid, and liquid-to-vapour transitions. The second and third steps must be understood for a series of solid states between 0 K and up to the fusion point, possibly different from the most stable one at 0 K.2 Similarly, the thermal part of enthalpy is a thermodynamic state function which is fully determined by the integral of Cp (T ) following the same reversible path as the one described for the computations of the entropy. The thermal enthalpies for N2 , O2 , and H2 O substances are generated by the variations of Cp (T ) with T corresponding to progressive excitations of the translation, rotation, and vibrational states of the molecules, and by the possible changes of phase represented by the latent heats (but with no or very small impact of changes of pressure). The aim is thus to compute integrals of Cp (T ) and Cp (T )/T , plus the impacts of latent heats existing for N2 , O2 , and H2 O. Integrals extend from 0 K to the standard temperature T0 by following a reversible path involving the most stable forms of the substance at each temperature. These processes are represented by the different terms in the formulae:  T0  h0 = h(T0 ) = h(0) + Lj , (2.11) Cp (T )dT + 0

j

and s0 = s(T0 , p0 ) = s(0, p0 ) +



0

T0

 Lj Cp (T ) dT + . T Tj j

(2.12)

The latent heats represented by the Lj terms are due to changes of phase (solid-to-solid, melting, and vaporization). Another contribution must be added to Eq. 2.12 if the standard entropy is computed at a pressure different from 1000 hPa (for instance, at the saturation pressure over liquid water or ice). 2 Almost all atmospheric species (N , O , Ar, CO ) are gaseous at T = 273.15 K= 0◦ C. 2 2 2 0 Only water is different, due to the unusual high fusion point temperature. More precisely, there are three solid states for O2 (α, β, and γ), two solid states for N2 (α and β), and only one for Ice-Ih (hexagonal ice).

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Part IV: Unification and consistency

The standard values h0 and s0 are computed from Eqs. 2.11 and 2.12 and using the cryogenic datasets described in Marquet (2015). The specific heats and the latent heats for O2 , N2 , and H2 O (ice-Ih) are depicted in Figs. 22.1 and 22.2. They are based on the books and papers of Fagerstroem and Hallet (1969), Feistel and Wagner (2006), Jacobsen et al. (1997), Kudryavtsev and Nemchenko (2001), Lipi´ nski et al. (2007), and, Manzhelii and Freiman (1997).

(a)

(b)

Fig. 22.1 The specific heat Cp (T ) at constant pressure (1000 hPa) for (a) O2 and (b) N2 . Units are JK−1 kg−1 . There are three solid phases (α, β, and γ) for O2 and two solid phases (α and β) for N2 . Latent heats are expressed in kJkg−1 at each change of phase (solid-to-solid, solid-to-liquid, liquid-to-vapour), together with the corresponding c Royal Meteorological temperature. (From Figs. 4 and 5 of Marquet (2015), which is  Society, 2015.)

Fig. 22.2 The specific heat Cp (T ) at constant pressure (1000 hPa) for H2 O (ice-Ih). Units are JK−1 kg−1 . The value 2106 JK−1 kg−1 is the commonly accepted one in c Royal Meteorological atmospheric science. (From Fig. 6 of Marquet (2015), which is  Society, 2015.)

The third law means that the entropy of a perfect crystal approaches zero as temperature approaches absolute zero, leading to s(0, p0 ) = 0.

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233

The specific entropies at the pressure 1000 hPa are shown in Fig. 22.3 for temperatures between 0 and 340 K. The standard entropies computed at T0 = 273.15 K= 0◦ C and p0 = 1000 hPa are given in Eqs. 2.13 to 2.16: s0d ≈ 6846[6775] JK−1 kg−1 ,

s0v s0l s0i

−1

≈ 10327[10320] JK

−1

≈ 3520[3517] JK

−1

≈ 2298[2296] JK

kg

−1

(2.13) ,

(2.14)

kg

−1

,

(2.15)

kg

−1

.

(2.16)

These values of standard entropies are close to those published in Thermodynamic Tables (recalled in bracketed terms in Eqs. 2.13–2.16) and retained in Hauf and H¨ oller (1987) and Marquet (2011). It is worthwhile noting that a residual entropy of about 189 JK−1 kg−1 must be taken into account for H2 O at 0 K (see the vertical shift for H2 O (ice-Ih) in Fig. 22.3), due to proton disorder and remaining randomness of hydrogen bonds at 0 K (Pauling, 1935; Nagle, 1966).

Fig. 22.3 The entropy diagram at 1000 hPa for O2 , N2 , dry air, and the three phases of H2 O (ice, liquid, vapour). Units are kJK−1 kg−1 .

The ice, liquid water, and water vapour values are 2, 3, and 7 units higher than those retained in Hauf and H¨ oller (1987). This is in good agreement with the accepted values, with an accuracy better than 0.1%, and provides a validation of the cryogenic datasets described in Marquet (2015) and depicted in Fig. 22.2. The dry-air value of standard specific entropies is 71 units higher than the one retained in Hauf and H¨oller (1987). The accuracy is thus coarser for the dry air (1%), than for the water species (0.1%), and the higher uncertainty for the dry air value is mainly due to the dataset for oxygen (0.5% for N2 and 1.3% for O2 ). This may be explained by the solid α-to-β transition of O2 occurring at 23.85 K. It is a second order transition, with

234

Part IV: Unification and consistency

no latent heat associated to it, but with infinite values of Cp leading to a kind of Dirac function for Cp (T ), partly visible on Fig. 22.1(a) (up to 1800 JK−1 kg−1 only). This second order transition is not always taken into account in published values of standard entropy for O2 . If a smoother transition were introduced in the cryogenic datasets, the computed value for s0d would be 3% lower, making the value retained in Hauf and H¨oller (1987) compatible with the present one. The same cryogenic datasets are used to compute dry air and water species standard values of specific thermal enthalpies. It is assumed that a kind of third law may be used to set the thermal part of the enthalpy to zero at 0 K. The specific thermal enthalpies are shown in Fig. 22.4 for temperatures from 0 to 360 K. The (cst ) curves for water vapour, liquid water, and ice are plotted in Fig. 22.4b from the values of enthalpy at 0◦ C and by assuming that the specific heat at constant pressure is a constant (this explains the difference from the true enthalpy for ice below 200 K).

(a)

(b)

Fig. 22.4 The enthalpy diagram at 1000 hPa for (a) O2 , N2 , and dry air and (b) dry air and the three phases of H2 O (ice, liquid, vapour). Units are kJkg−1 . The enthalpies for N2 and O2 are equal for T ≈ 220 K in (a) and those for dry air and water vapour are equal for T ≈ 241.4 K in (b). Grey spots label these locations. The circle–cross in (a) and (b) represents the dry air value at 0◦ C (530 kJkg−1 ). (Adapted from Figs. 7 and 8 c Royal Meteorological Society, 2015.) of Marquet (2015), which is 

The standard values computed at T0 = 273.15 K= 0◦ C are equal to: h0d ≈ 530 kJkg−1 , (2.17) h0v ≈ 3133 kJkg−1 , h0l h0i

−1

≈ 632 kJkg

−1

,

(2.18)

(2.19)

≈ 298 kJkg . (2.20) −1 The value of is in close agreement with the value 298.35 kJkg published in Feistel and Wagner (2006). The standard values given in Eqs. 2.17–2.20 h0i

Formulations of moist thermodynamics for atmospheric modelling

235

are computed from cryogenic datasets and are different from the results published in Bannon (2005) and Chase (1998), because the impacts of the solid phases and of the latent heats were not taken into account in these previous studies, where only the vapour state (perfect gases) of the species was considered.

3

Moist-air specific entropy

The computations of the specific moist entropy are presented in detail in this section, according to the method published in Marquet (2011). Moist air is assumed to be a mixture of four ideal gases at the same temperature3 T : dry air, water vapour, liquid-water droplets, and ice crystals. The condensed phases cannot coexist4 , except at the triple point (273.16 K). There is no supersaturation and no metastable phase of water5 (thus, supercooled water is excluded). If a condensed phase exists, the partial pressure of water vapour is equal to the saturation pressure over liquid water (esw ) if T > 0◦ C or ice (esi ) if T < 0◦ C. The specific entropy is written as the weighted sum of the specific partial entropies, and can be expressed by Eq. 2.2. Following Hauf and H¨ oller (1987), the total-water content qt = 1−qd = qv +ql +qi is used to transform Eq. 2.2 into: s = qd sd + qt sv + ql (sl − sv ) + qi (si − sv ) .

(3.1)

The differences in partial entropies can be expressed in terms of the differences in specific enthalpy and Gibbs functions, leading to: T (sl − sv ) = −(hv − hl ) − (μl − μv ) ,

T (si − sv ) = −(hv − hi ) − (μi − μv ) .

(3.2) (3.3)

When the properties Lv = hv − hl , Ls = hv − hi , μl − μv = Rv T ln(esw /e), and μi − μv = Rv T ln(esi /e) are inserted into Eqs. 3.1–3.3, the specific 3 This hypothesis is not strictly valid in regions where precipitating species are observed ′ . at about Tw 4 They coexist in the real atmosphere and in several climate, numerical weather prediction, and other models. However, the mixed phase is not taken into account in this study. 5 This hypothesis may be easily overcome by using the affinities defined in Hauf and H¨ oller (1987), where the relative humidities e/esw (T ) and e/esi (T ) may be greater than 1 in clouds.

236

Part IV: Unification and consistency

moist-air entropy may be written as:  Lv ql + Ls qi s = qd sd + qt sv − T   e  e  si sw + qi ln . (3.4) − Rv ql ln e e The final term of Eq. 3.4 in square brackets cancels out for clear-air regions (where ql = qi = 0, but with qv different from zero). It is also equal to zero for totally cloudy air (where ql and qi are different from zero and qv is equal to the saturation value), because for liquid water ql = 0 with e = esw and ln(esw /e) = 0, and for ice qi = 0 with e = esi and ln(esi /e) = 0. The terms on the first line of Eq. 3.4 are then computed with the dry-air and water-vapour entropies expressed as:   pd T sd = (sd )r + Cpd ln − Rd ln , (3.5) Tr (pd )r   T e sv = (sv )r + Cpv ln − Rv ln . (3.6) Tr er It is assumed that the specific heats and the gas constants are indeed constant within the atmospheric range of temperature and pressure6 , with the reference values (sd )r and (sv )r associated with the reference conditions Tr and pr = (pd )r + er , with er (Tr ) equal to the saturation pressure at the temperature Tr . When Eqs. 3.5 and 3.6 are inserted into the first line of Eq. 3.4, and after several rearrangements of the terms which are described in Appendix B of Marquet (2011), the moist-air specific entropy can be written as:  θs = sref + Cpd ln (θs ) . (3.7) s = sr + Cpd ln θsr The reference potential temperature θsr is equal to:  κ p0 exp (Λr qr ) (1 + ηrr )κ θsr = Tr pr

(3.8)

and the associated reference entropy sref is equal to sr − Cpd ln(θsr ), with sr = (1 − qr )(sd )r + qr (sv )r . The reference-specific vapour content and mixing ratio are equal to qr = rr /(1 + rr ) and rr = ε˜er (Tr )/[pr − er (Tr )], where er (Tr ) is the saturation pressure at Tr . Both qr and rr are thus completely determined by the couple (Tr , pr ). 6 This is valid with high accuracy for dry air, water vapour, and liquid water, but is not true for ice. It is, however, possible to derive a version of moist-air entropy and θs where Ci (T ) depends on temperature.

Formulations of moist thermodynamics for atmospheric modelling

237

The moist-air potential temperature θs appearing in the logarithm of Eq. 3.7 is equal to:  Lv ql + Ls qi θs = θ exp − exp (Λr qt ) Cpd T γqt  λqt  −κδq ˆ t ˆ p rr (1 + ηrv )κ(1+δqt ) T . (3.9) × ˆ t Tr pr rv (1 + ηrr )κδq It is defined in terms of the leading-order7 quantity (θs )1 given by the first line of Eq. 3.9 and equal to: (θs )1 = θl exp (Λr qt ) ,

(3.10)

where Λr =

(sv )r − (sd )r Cpd

(3.11)

is a non-dimensional value depending on the reference entropies of water vapour (sv )r and dry air (sd )r . The liquid-ice potential temperature θl appearing in Eq. 3.10 is equal to:  Lv ql + Ls qi θl = θ exp − . (3.12) Cpd T The quantity θs , determined by Eqs. 3.9–3.12, has a rather complex expression, although it is roughly similar to the ones published in Emanuel (1994), Hauf and H¨ oller (1987), and Marquet (1993). The reference value sref = sr − Cpd ln(θsr ) can be computed with θsr given by Eq. 3.8 and with the reference values Λr evaluated with (sv )r and (sd )r expressed in terms of the standard values s0v and s0d computed at T0 and p0 , leading to:   Tr (pd )r (sd )r = s0d + Cpd ln − Rd ln = 6777 JK−1 kg−1 , (3.13) T0 p0   Tr er (sv )r = s0v + Cpv ln (3.14) − Rv ln = 12673 JK−1 kg−1 . T0 p0 The important and expected result is that most of the terms cancel out and that sref = s0d − Cpd ln(T0 ). It can be computed from the third law value s0d given by Eq. 2.13, leading to sref ≈ 1138.6 JK−1 kg−1 . It is thus a constant term and is independent of the choice of the reference values Tr 7 These first-order (θ ) and higher-order approximations of θ can be derived mathes 1 s matically as Taylor expansions of Eq. 3.9 expressed in terms of the small water contents rv ≈ qt ≪ 1. This is described at the end of Sec. 5.2.

238

Part IV: Unification and consistency

and pr , provided that rr = ε˜er (Tr )/[pr − er (Tr )] is equal to the saturation value at Tr . Since the specific heat for dry air Cpd is also a constant term, the moistair entropy potential temperature θs given by Eq. 3.7 becomes a true equivalent of the moist-air entropy s, whatever the changes in dry air and water species contents might be. This important property does not hold for the potential temperatures θS , θ⋆ , θl , or θe previously defined in Emanuel (1994), Hauf and H¨ oller (1987), and Marquet (1993).8 It is possible to write θs differently, in order to show more clearly that the entropic potential temperature is independent of the reference values. The aim is to put together all the terms containing Tr , pr , or rr to form the last line of  Lv ql + Ls qi ˆ t ˆ t (λ−δ)q 1+δq exp − T θs = θ Cpd T ˆ

×

(1 + ηrv )κ(1+δqt ) × exp {[Λ0 − λ ln(T0 )] qt } . (ηrv )γqt

(3.15)

The last exponential factor in Eq. 3.15 is computed with (sd )r and (sv )r evaluated as for Eq. 3.9 in terms of the standard values s0d and s0v , and computed via Eqs. 3.13 and 3.14 at T0 and p0 , leading to:   Λ0 = s0v − s0d /Cpd ≈ 3.53 and Λ0 − λ ln(T0 ) ≈ −1.17, (3.16)

which are clearly independent of Tr , pr , or rr . The standard value Λ0 corresponds to the definition retained in Hauf and H¨ oller (1987) and is different from the reference Λr in Eq. 3.11, due to the change of wateroller (1987) to vapour partial pressure, from p0 = 1000 hPa in Hauf and H¨ er = 6.11 hPa in Λr . The advantage of Eq. 3.15 over Eq. 3.9 is that all the reference values disappear in Eq. 3.15, since they can be replaced by the standard values. The associated drawback is that all the other terms must be retained in Eq. 3.15 for computing accurately the numerical value of θs . The main difference between the formulation s = sref + Cpd ln(θs ) and the other formulations of the moist-air specific entropy is the term Λr defined by Eq. 3.11. It is a key quantity in the formulations for θs . It is explained in Marquet (2011) that the advantage of Eq. 3.9 lies in that it is possible to retain as a relevant approximation of θs the quantity (θs )1 defined by Eq. 3.10. The interesting property is that (θs )1 is equal to 8 The notations θ and θ ⋆ were used in Emanuel (1994), Hauf and H¨ oller (1987), and S Marquet (1993) to denote approximate moist-air entropy potential temperatures.

Formulations of moist thermodynamics for atmospheric modelling

239

θl exp(Λr qt ) if qi = 0, i.e., a formulation formed by a simple combination of Betts’s variables θl and qt . Moreover, if qt is a constant, θs varies like Betts’s potential temperature θl up to constant multiplying factors which have no physical meaning when ln(θs ) is computed in Eq. 3.7. However, if all the terms in Eq. 3.7 are considered, if qt is not a constant, and if qi is different from zero, it is expected that the potential temperature θl , depending on qi , and the new factor exp(Λr qt ), depending on the reference entropies, may lead to new physical properties. Differently from s and θs , which are independent of the choice of the reference values, both (θs )1 and Λr depend on the choice of the couple (Tr , pr ). There is no paradox associated with this result. The fact that numerical values of (θs )1 vary with (Tr , pr ) is compatible with the fact that θs does not depend on them. The explanation for this result is that the extra terms in the factor of (θs )1 in Eq. 3.9 also depend on (Tr , pr ) and that the changes in these terms balance each other in order to give values of θs which are independent of (Tr , pr ). The numerical tests shown in Table II of Marquet (2011) validate these results.

4

Moist-air specific thermal enthalpy

The method used to compute the moist-air specific thermal enthalpy h is described in Marquet (2015). It is similar to the one used in Sec. 3 to derive the moist-air specific entropy. The first step is to write h as the weighted sum of Eq. 2.1. The properties Lv = hv −hl and Ls = hv −hi , together with qd = 1−qt and qt = qv +ql +qi , are then used to transform h into the sum hd + qt (hv − hd ) − (ql Lv + qi Ls ). The next step is to write hd = (hd )r + Cpd (T − Tr ) and hv = (hv )r + Cpv (T − Tr ) in terms of the reference values (hd )r and (hv )r expressed at temperature Tr , with (hd )r and (hv )r to be determined from the standard values h0d and h0v given by Eqs. 2.17 and 2.18. After rearrangement of the terms the moist enthalpy can be written as: h = href + Cpd T − Lv ql − Ls qi + Cpd (λT + TΥ ) qt .

(4.1)

The term TΥ = Tr [Υ(Tr ) − λ] depends on Υ(Tr ) = [(hv )r − (hd )r ]/(Cpd Tr ), which is the equivalent of Λr in the definition of the moist-air entropy potential temperature θs . It is possible to define an enthalpy temperature

240

Part IV: Unification and consistency

Th by setting h = href + Cpd Th , where Cpd Th corresponds to:  Lv ql + Ls qi Th = T − (4.2) + (λT + TΥ ) qt . Cpd It is shown in Marquet (2015) that TΥ is independent of Tr , even if Υr depends on Tr . The quantity href is also independent of the reference value Tr . Both href and TΥ can be expressed in terms of the standard values h0d and h0v given by Eqs. 2.17 and 2.18, leading to the numerical values: and

5 5.1

TΥ = T0 [Υ(T0 ) − λ] = 2362 K ,

(4.3)

href = h0d − Cpd T0 = 256 kJkg−1 .

(4.4)

Physical properties of the moist-air entropy and θs Comparisons between θl , θs , (θs )1 , and θe

Let us assume for the moment that θs can be approximated by (θs )1 . Comparisons of (θs )1 with the well-known potential temperatures θl , θv , and θe are facilitated by the study of the approximate formulations:  Lv ql (5.1) θl = θ exp − ≈ θ (1 + 0qv − 9ql ) , Cpd T   ˆ v − ql ≈ θ (1 + 0.6 qv − ql ) , θv = θ 1 + δq (5.2)  Lv ql + Λr qt ≈ θ (1 + 6qv − 3ql ) , (5.3) θs ≈ (θs )1 = θ exp − Cpd T  Lv qv θe ≈ θ exp (5.4) ≈ θ (1 + 9qv ) . Cpd T Only liquid water is considered in this section, but qi and Ls terms may be added in the first three equations if needed. The approximations are obtained with the properties Λr ≈ 6 (see the next subsection), exp(x) ≈ 1 + x valid for |x| ≪ 1 and Lv /(Cpd T ) ≈ 9 valid for atmospheric conditions. The impact of water vapour (qv ) is zero for θl and very small for θv (factor 0.6). The factor 9 is much larger for θe , and it is 2/3 of it for (θs )1 . This means that θl and θv must remain close to θ, whereas θs and θe must differ from θ by more from the other potential temperatures. Moreover, if liquid-water content is discarded (ql is small in stratocumulus), (θs )1 must be in about a 6/9 = 2/3 proportion between θl and θe . These consequences are confirmed for the observed dataset depicted in Fig. 22.5a for the gridaverage vertical profiles of the FIRE-I (RF03B) case.

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(a)

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(b)

Fig. 22.5 (a) Vertical profiles of several potential temperatures plotted from the FIRE-I (RF03B flight) datasets. (Adapted from Fig. 2 of Marquet and Geleyn (2013), which is c Royal Meteorological Society, 2013.) Different values of θe are plotted by using four  different definitions suggested in older studies (Betts, 1973; Bolton, 1980; Emanuel, 1994) and a recent one used in the ARPEGE-IFS model. (b) Conservative variable diagram at 900 hPa, with qt plotted in ordinates and moist-air entropy computed either with θs or (θs )1 plotted in abscissae. Isotherms are plotted every 10 K, with solid lines for θs and dashed lines for (θs )1 . Contours of relative error in moist-air entropy (i.e., if θs were replaced by (θs )1 ) are plotted as thin solid lines.

5.2

Use of (θs )1 as an approximation to θs

The full expression of θs given in Eq. 3.9 can be used as such in order to compute θs numerically, for either undersaturated or saturated moist air. The interest of searching for a simpler expression for θs is to better understand the physical meaning of θs and to facilitate the analytical comparisons with θl and θe . The result shown in Marquet (2011) is that (θs )1 seems to be a relevant approximation to θs for Tr = T0 and pr = p0 , with er = esw (T0 ) ≈ 6.11 hPa and (pd )r = p0 − esw (T0 ), differently from the standard conditions where (pd )r = er = p0 . For these conditions, the numerical value of Λr is close to 5.87. It is possible to appreciate on Fig. 22.5a how far this expression of (θs )1 (computed with this value 5.87 for Λr ) is a relevant approximation of θs . This is true from the surface layer up to the level 1,500 m in Fig. 22.5a

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and the accuracy of the approximation θs ≈ (θs )1 even improves above 1, 500 m, because the total-water content qt is smaller. Other comparisons of θs and (θs )1 are available in Marquet (2011, 2014); Marquet and Geleyn (2013). Differences between (θs )1 and θs remain close to 0.4 to 0.6 K at all levels for FIRE-I vertical profiles. The fact that the difference θs − (θs )1 is almost constant with height is an important result if gradients of θs are to be considered (see Secs. 6.4 and 6.5 dealing with the moist-air Brunt–V¨ais¨ al¨ a frequency and the potential vorticity, respectively). Other comparisons between θs and (θs )1 (with Λr = 5.87) are shown in the entropy diagram plotted in Fig. 22.5b, where a wide range of saturated and non-saturated conditions are explored at p = 900 hPa. The same differences of the order of ±0.2 % or ±0.6 K are observed for most parts of the diagram. The same entropy diagram is plotted in Marquet and Geleyn (2013), but with the set of isotherms computed with the definition of entropy of Pauluis and Schumacher (2010), based on the work of Emanuel (1994) in terms of θe . It is shown that the definition of moist-air entropy in terms of θs or θe leads to incompatible features in the saturated regions, where isotherms correspond to almost constant values of θe and to clearly decreasing values of θs . These isotherms are also different in the non-saturated region, although they correspond to increasing values of both θe and θs . This demonstrates that the way in which the moist-air entropy is defined may generate important differences in physical interpretation, with θs being the only general formula valid whatever qt may be. First- and second-order approximations of θs can be derived by using the Taylor expansion of several factors in the second line of Eq. 3.9. The ˆ first-order expansion of (1 + ηrv )[κ(1+δqt )] for small rv ≈ qt is equal to (γqt ) exp(γqt ). The factor (rr /rv ) is exactly equal to exp[−γqt ln(rv /rr )]. Other factors depending on temperature and pressure are exactly equal to ˆ t) ˆ t ln(p/pr )]. The final factor (1 + ηrr )(κδq exp[λqt ln(T /Tr )] and exp[−κδq ˆ t rr ] ≈ exp[O(q 2 )], since rr ≈ qt ≪ 1. leads to the higher-order term exp[γ δq t The result is that the Taylor expansion of θs can be written as:  Lv ql + Ls qi θs ≈ θ exp − exp (Λ∗ qt ) Cpd T 

    p T × exp qt λ ln (5.5) − κδˆ ln + O(qt2 ) , Tr pr where Λ∗ = Λr − γ ln(rv /r∗ ) and r∗ = rr × exp(1) ≈ 10.4 gkg−1 .

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It is thus possible to interpret (θs )1 by the value of θs obtained with the approximation Λ∗ ≈ Λr and with the second line of Eq. 5.5 neglected. An improved second-order approximation is obtained by taking into account the small term −γ ln(rv /r∗ ) and so using Λ∗ instead of Λr . The impact of the second line, containing terms depending on temperature and pressure, leads to a third-order correction of r∗ , which should be pragmatically increased up to about 12.4 gkg−1 for usual atmospheric conditions (unpublished results).

5.3

Comparisons with alternative moist-air entropy formulae

The issues of whether the third law can be applied to atmospheric studies or not, and if so, how this can be managed practically, are old questions. Richardson (1922, pp. 159–160) wondered if it could be possible to ascribe a value for energy and entropy for a unit mass of (water) substance. He proposed to take the absolute zero temperature as the zero origin of entropies. He recognized that the entropy varies as Cp dT /T and that the integrand has an infinity where T = 0 K. But he mentioned that Nernst had shown that the specific heats tend to zero at T = 0 K in such a way that the entropy remains finite there. This is due to Debye’s law, which is valid for all solids and for which Cp (T ) is proportional to T 3 (the entropy is defined by ds = Cp dT /T and is also proportional to T 3 , a formulation which is not singular for T approaching 0 K). However, probably due to the lack of available standard values of entropies and energies in the early 20th century, Richardson did not really use the third law. He suggested considering the lowest temperature occurring in the atmosphere (180 K) as the more practical value for the zero origin of entropies. This is in contradiction with the above computations based on the third law, with the definition depending on θs given by Eq. 3.9 that can be considered as being the moist-air specific entropy imagined by Richardson. A synthetic view of existing formulations of moist-air entropy derived in Betts (1973), Emanuel (1994), and Iribarne and Godson (1973) in terms of θl or θe is suggested in Pauluis et al. (2010). Two moist-air entropies are defined. The first was called “moist entropy” and was denoted by Sm . It is rather written as Se in Eq. 5.6, since it is associated with θe . It can be

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written as:

Part IV: Unification and consistency



T Tr



Lv + qv Se = [Cpd + qt (Cl − Cpd )] ln T   p−e e − qd Rd ln − qv Rv ln . pr − e r esw

(5.6)

The second is given by Eq. 5.7. It was called “dry entropy” and was denoted by Sl in Pauluis et al. (2010).9 It is associated with the potential temperature θl and is written as:  T Lv Sl = [Cpd + qt (Cpv − Cpd )] ln − ql Tr T   e p−e − qd Rd ln − qt Rv ln . (5.7) pr − e r er

It is possible to show that the difference between Eqs. 5.6 and 5.7 is equal to Se − Sl = qt Lv (Tr )/Tr .10 The important feature is that Se − Sl depends on the product of qt with a constant factor Lv (Tr )/Tr . Therefore, since there is only one physical definition for the moist-air entropy, and since qt is not a constant in the real atmosphere, Se and Sl cannot represent at the same time the general form of moist-air entropy. More precisely, the comparisons between the third law-based formulation s(θs ) given by Eqs. 3.7 and 3.9 and the two formulations Se (θe ) or Sl (θl ) can be written as: s(θs ) = Se (θe ) + qt [(sl )r − (sd )r ] + (sd )r , s(θs ) = Sl (θl ) + qt [(sv )r − (sd )r ] + (sd )r .

(5.8) (5.9)

The first result is that, if qt is a constant, then Se and Sl can become specialized versions of the moist-air entropy associated with the use of the conservative variables θe and θl , respectively. However, even if Se and Sl are equal to s(θs ) up to true constant terms, the constant terms are not equal to zero and they are not the same for Se and Sl . Moreover, they depend on the value of qt . Therefore, even if qt is a constant (for instance for a vertical ascent of a closed parcel of moist air), it is not possible to compare values of Se or Sl with those for other columns, since the values of qt and hence the constant terms in Eqs. 5.8 and 5.9 are different from one column to another. This means that it is not possible to compute relevant 9 This is slightly different from the value given by Pauluis et al. (2010), because the last term e/esw (T ) in Eq. 5.6 was written as e/esw (Tr ) in Eq. A3 of Pauluis et al. (2010). 10 This difference is different from Eq. A5 given in Pauluis et al. (2010), because of the change of esw (T ) into er = esw (Tr ) in their Eq. A3.

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spatial averages or horizontal gradients of Se (θe ) or Sl (θl ), because the link between θe or θl with the moist-air entropy s(θs ) must change in space. It is considered in this study that only the third law-based formulation s(θs ) is general enough to allow relevant computations of spatial averages, vertical fluxes or horizontal gradients of moist-air entropy. Indeed, if qt is not a constant, then s = Se + (sd )r only if (sl )r = (sd )r . Similarly, s = Sl + (sd )r only if (sv )r = (sd )r . Therefore, moist-air entropy for an open system with varying qt cannot be represented by Se (θe ) or Sl (θl ), because the arbitrary choices for the reference entropies (sl )r = (sd )r or (sv )r = (sd )r are not compatible with the third law of thermodynamics. It is suggested in Pauluis et al. (2010, their Appendix C) that the weighted sum Sa = (1 − a)Se + aSl is a valid definition of the entropy of moist air, where a is an arbitrary constant going from 0 to 1. The adiabatic formulation Sl corresponds to a = 1 and θl , whereas the pseudo-adiabatic formulation Se corresponds to a = 0 and θe . The weighted sum Sa applied to Eqs. 5.8 and 5.9 leads to:

 Lv (Tr ) s(θs ) = Sa + (sd )r + qt (sl )r − (sd )r + a . (5.10) Tr Equation 5.10 shows that if the factor multiplying the qt term is not equal to zero, then Sa will be different from the third law formulation depending on θs . If the value Λr ≈ 5.87 derived in Marquet (2011) is retained, this factor is equal to zero for a = [(sd )r − (sl )r ]Tr /Lv (Tr ) ≈ 0.356. This special value represents (and allows the measurement of) the specific entropy of moist air in all circumstances. No other hypotheses are to be made on the values of the reference entropies, on adiabatic or pseudo-adiabatic properties, or on constant values for qt . This provides another explanation for the results observed in Secs. 5.1 and 5.2: the third law potential temperature θs is almost in a (1 − a) ≈ 2/3 versus a ≈ 1/3 position between θl and θe . The third law is not used in Pauluis et al. (2010), where it is mentioned in Appendix A that “. . . the entropy used in atmospheric sciences . . . does not, however, correspond to the absolute entropy based on Nernst’s theorem”. It is explained in their Appendix A that the entropy of an ideal gas is fundamentally incompatible with Nernst’s theorem as it is singular for T approaching 0 K, due to the term ln(T ). As explained in Richardson (1922), this statement may be true for ideal gases, but it does not invalidate the application of the third law in atmospheric science, since only the more stable solid states and Debye’s law must be considered to apply Nernst’s theorem, with finite results at the limit T = 0 K.

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It is mentioned in Pauluis et al. (2010, their Appendix A) that “A common practice is to . . . set the reference value for the specific entropies of dry air and of either liquid water or water vapour to 0.” This is in contradiction with the third law and the absolute entropies derived in Sec. 2.5. Really, the third law cannot be bypassed when evaluating the general formula of moist-air entropy. Reference values must be set to the standard ones obtained with zero entropy for the most stable crystalline form at T = 0 K. If qt is not a constant, the formulations for s(θs ), Se , Sl , and Sa are thus different and it is claimed that the third law formulation s(θs ) is the more general one: it is the only formula which can be applied in all circumstances. 6

6.1

Applications of the moist-entropy potential temperatures θs and (θs )1 Isentropic features: Transition from stratocumulus to cumulus

It is shown in Fig. 22.5a that (θs )1 is in about a 2/3 proportion between θl and θe . This confirms the evaluation made in Sec. 5.1. The other important result shown in Fig. 22.5a is that θs is almost a constant from the surface to the very top (1,000 m) of the planetary boundary layer (PBL), including the entrainment region (850–1,000 m) where large jumps in qt and θl are observed. This means that the whole PBL of this FIRE-I (RF03B) marine stratocumulus exhibits a surprising moist-air isentropic state. Constant values of the moist-air quantities s and θs are obtained in spite of observed vertical gradients (and PBL-top jumps) in qt and θl , and also in θe . This means that the gradients of the two terms θl and exp(Λr qt ) combine in such a way that their product remains constant. This is only observed for that value Λ ≈ 5.87 determined from the third law reference entropies of dry air and water vapour. It is not valid for the value Λr = 0 (leading to θl ) or the value Λr = Lv /(Cpd T ) ≈ 9 (leading to θe ). It is shown in Marquet (2011) that the moist isentropic feature (constant values of θs ) is also observed for other FIRE-I profiles (radial flights 02B, 04B, and 08B) and for the vertical profiles of other marine stratocumulus (ASTEX, EPIC, DYCOMS). This is the confirmation that the third lawbased moist-air entropy defined with θs can reveal unexpected new and important physical properties which have not been revealed by the previous studies based on θl or θe . Since the moist air isentropic feature is observed in marine stratocumulus, there is a hope that this very special property

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might be taken into account for building (or improving) moist turbulent parameterizations, and possibly shallow-convection ones (both acting in the cloudy parts of these marine stratocumulus profiles), including the PBL-top entrainment processes. In order to determine if these isentropic patterns are also observed during the transition from stratocumulus to cumulus (or how they are modified), several vertical profiles of θl , θs , and θe are plotted in Fig. 22.6 for the aircraft measurements collected during the first ASTEX Lagrangian campaign and used for the EUCLIPSE model intercomparison study (de Roode and Duynkerke, 1997; de Roode and van der Dussen, 2010). The moist-air isentropic feature (constant values of θs ) is indeed observed for Flight 4, with an increase of moist-air entropy at the PBL-top of Flights 2 and 3, and a decrease in θs for Flight 5 (cumulus). The patterns of θl are almost the same for all profiles, with positive PBL-top jumps that cannot be differentiated at first sight. The patterns of θe are more reactive than those of θs , with a neutral feature in between Flights 3 and 4. For this reason, θs also seems to be in an intermediate position between θl and θe . The sketch of the cumulus profile depicted in Flight 5 is compatible with the other cumulus profiles of θl , θs , and θe (unpublished results for ATEX, GATE, BOMEX, and SCMS). It is explained in the next section that a neutral vertical gradient of θs seems to correspond to the Δ2 CTEI criterion line. This may demonstrate a new possibility for analysing the strength of a PBL-top inversion: it is respectively stable/neutral/unstable for a positive/zero/negative PBL-top jump in θs . This is an interesting new property for moist-air entropy expressed in terms of θs , since it allows a rapid and graphical analysis of entrainment instabilities, and thus an estimation of the evolution of clouds. 6.2

PBL-top entrainment and CTEI criterion

The concept of cloud-top entrainment instability (CTEI) was defined in Deardorff (1980), Lilly (1968), and Randall (1980) in order to find an explanation for the stratocumulus cloud break-up. It was suggested that critical values exist for the PBL-top jumps in θe or θl (hereafter denoted by (Δθe )crit and (Δθl )crit ) above which a mixed-air parcel will be positively buoyant in respect of cloud-top entrainment processes. Instabilities are thus expected if Δθe is smaller than the value (Δθe )crit = kRD Lv Δqt /Cpd < 0, with Δqt equal to the PBL-top jump in qt . The Randall–Deardorff parameter kRD has been evaluated as 0.23

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Fig. 22.6 Vertical profiles from the first ASTEX Lagrangian campaign studied during the EUCLIPSE model intercomparison. Top panel: schematic of the observed five aircraft flights (from de Roode and Duynkerke, 1997; de Roode and van der Dussen, 2010). Bottom panel: the vertical profiles of θl , θs , and θe computed for Flights 2 to 5 (Flight 1 is similar to Flight 2).

in the first studies and it varies between 0.18 and 0.7 in the literature. It is possible to express the CTEI criterion in the conservative diagram (θl , qt ), leading to a buoyancy reversal criterion line usually denoted by Δ2 (de Roode and Wang, 2007). The corresponding threshold is equal to (Δθl )crit = −(1/kL )Lv Δqt /Cpd > 0, where kL ≈ 1/(1 − kRD ) is the Lilly parameter (Lilly, 1968). The fact that moist-air entropy and θs ≈ (θs )1 are almost conserved within the whole PBL of marine stratocumulus may help to give a new insight for the CTEI criterion. Indeed, the property Δ(θs )1 = 0 corresponds

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to the value kRD = 1 − Cpd Λr θ/Lv ≈ 0.29 for Λr = 5.87 (Marquet, 2011). It seems that higher values up to kRD = 0.35 are associated with the use of the improved formulation of Eq. 5.5 to compute Δ(θs ) ≈ 0 (unpublished results). These third law-based values of kRD varying between 0.29 and 0.35 are close to those derived by Randall (1980) and Deardorff (1980) (0.23) and are located within the range of the published ones (0.18 to 0.7). It can be inferred that the CTEI mechanism seems to be controlled by a neutral (zero) PBL-top jump in moist-air entropy (in fact by an isentropic state of the whole PBL) without additional assumptions needed to derive this value of the Randall–Deardorff parameter kRD . The potential temperatures θl or θe cannot represent changes in moist-air entropy or isentropic processes for marine stratocumulus, which are open systems where qt is rapidly varying with height. Only θs can reveal isentropic patterns for such moist-air open systems. 6.3

Moist-air turbulence parameterization

The moist PBL of marine stratocumulus is considered a paradigm of moist turbulence. The property that moist-air entropy (and θs ) is almost uniform in the whole PBL of marine stratocumulus, including the PBL-top region, is thus an important new result. The first possible consequence is that it could be meaningful to replace Betts’s variables (θl , qt ) with the pair (θs , qt ). If θs is approximated by (θs )1 = θl exp(Λr qt ), the vertical turbulent fluxes of (θs )1 , θl , and qt are then approximated by w′ θs′ ≈ w′ (θs′ )1 = exp(Λr qt )w′ θl′ + θs Λr w′ qt′ . If linear flux–gradient relationships w′ ψ ′ = −Kψ ∂ψ/∂z are assumed to be valid for ψ = (θs )1 , θl or qt , they define corresponding exchange coefficients Ks , Kh , and Kq . It is then worthwhile to compare the two methods: either computing w′ (θs′ )1 by using −Ks ∂(θs )1 /∂z, or by using the weighted sum of the vertical fluxes w′ θl′ and w′ qt′ described above with these fluxes expressed with the corresponding flux–gradient relationships. The two methods are only equivalent if Ks = Kh and Ks = Kq , this implying Kh = Kq . This equality is often assumed to be valid in numerical modelling, with the Lewis number Le = Kh /Kq ≈ 1. These results are, however, based on old papers (Dyer and Hicks, 1970; Oke, 1970; Swinbank and Dyer, 1967; Webb, 1970) and are sometimes invalidated in other old articles (Blad and Rosenberg, 1974; Brost, 1979; Verma et al., 1978; Warhaft, 1976). More recent studies based on LES outputs suggest that Kh could be different from Kq (Siebesma et al., 2003; Stevens et al., 2001). Moreover,

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neutral bulk coefficients are different for heat and water in some surface flux parameterizations over oceans, like in ECUME/SURFEX (Belamari, 2005; Belamari and Pirani, 2007; Le Moigne et al., 2013; Masson et al., 2013). The two methods are different if either Ks = Kh or Ks = Kq , in which case it is not equivalent to use Betts’s variables or the pair (θs , qt ). This could provide a possible way of expressing the turbulent scheme with the moist-air entropy, since θl is only an approximation of (θs )1 for the case of constant values of qt . Other possible applications for turbulent parameterizations and shallow convection may be based on the use of moist-air Richardson numbers, equal to the moist-air square of the Brunt–V¨ais¨ al¨ a frequency divided by the squared wind shear, Ri = N 2 /S 2 . It is in this context that the results for the moist-air definition of N 2 (C) in Sec. 6.4 may provide new insights for computations of Ri (C). The transition parameter C might be somehow generalized to the grid-cell concept and to sub-grid variability, with C considered as the proportion of an air parcel being in saturated conditions. This could lead to extended grid-cell interpretations for the local formulation of N 2 (C) given by Eq. 6.6. One could also imagine possible bridges with the programme followed by Lewellen and Lewellen (2004), where a ˆ is introduced to mix the dry and wet limits in the context of parameter R a computation of moist-air buoyancy fluxes. 6.4

The moist-air squared Brunt–V¨ ais¨ al¨ a frequency

One of the applications of the moist-air entropy defined in terms of θs given by Eq. 3.9 is to compute the moist-air adiabatic lapse rate and squared Brunt–V¨ais¨ al¨ a frequency (BVF). Indeed, they both depend on the definition of reversible and adiabatic displacements of a parcel of air; that is to say at a constant value of entropy. The squared BVF can be defined for dry air by N 2 = (g/θ)∂θ/∂z, or equivalently in terms of the dry-air entropy by N 2 = (g/Cpd )∂s/∂z, since s = Cpd ln(θ) up to a constant term. Therefore, the way the entropy is defined may modify the formulation of both the adiabatic lapse rate and the squared BVF. This is the initial motivation for using the formulation θs for searching for a moist-air generalization of the dry-air expression. Moist-air versions have been derived by using different methods (Durran and Klemp, 1982; Emanuel, 1994; Lalas and Einaudi, 1974). Unsaturated and saturated moist-air versions of N 2 have been recently revisited in Mar-

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quet and Geleyn (2013), where the moist-air entropy is defined in terms of the more general formulation for θs given by Eq. 3.9. Additional results are described in Marquet and Geleyn (2013) concerning the computation of the adiabatic lapse rates. A synthetic view of these results is presented in this subsection. The adiabatic lapse rates are defined in Marquet and Geleyn (2013) in terms of a coefficient C which labels the unsaturated case by C = 0 and the saturated case by C = 1. This transition parameter allows compact expressions to be written that are valid for both unsaturated and saturated moist air. It is worthwhile remembering that unsaturated moist air is different from dry air because it includes water vapour (with relative humidity lower than 100%). The adiabatic lapse rate computed with θs can be written as: g × M (C) , (6.1) Γ(C) = Cp where 1 + DC Lv (T )rsw M (C) = , (6.2) , DC = 1 + F (C)DC Rd T

 Lv (T ) R F (C) = 1 + C −1 . (6.3) Cp T Rv Since F (0) = 1 and M (0) = 1, the unsaturated adiabatic lapse rate is exactly equal to Γ(0) = g/Cp , where Cp is the moist value depending on the dry-air and water-vapour contents. The saturated adiabatic lapse rate is slightly different from those derived in Durran and Klemp (1982) and Emanuel (1994). It is equal to Γ(1), leading to: −1



g Lv (T )rsw RL2v (T )rsw × 1+ . (6.4) 1+ Γsw = Cp Rd T Rd Cp Rv T 2 This formulation enables comparison with previous formulations derived in Durran and Klemp (1982) and Emanuel (1994). The numerator 1 + (Lv rsw )/(Rd T ) is multiplied by a term (1 + rt ) in previous studies, where the denominators are also different. The term g/Cp is replaced by g/Cpd in Durran and Klemp (1982) and by g/c∗p in Emanuel (1994), where c∗p = Cpd + Cpv rsw . The squared BVF is defined in Marquet and Geleyn (2013) by the general formula       ∂q ∂s ∂ρ g ∂ρ t   + N2 = − , (6.5) ρ ∂s p,qt ∂z ∂qt p,s ∂z

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in terms of the vertical gradients of moist-air entropy and total-water content. The gradients are obtained only after long computations, because s is defined in terms of θs , as given by Eq. 3.9. The result can be written using the same definitions for Γ(C), M (C), DC , and F (C) described in Eqs. 6.1–6.3. The same transition parameter C is used to express with a single formula the unsaturated (C = 0) and the saturated (C = 1) cases: ∂ ln(qd ) ∂s +g ∂z

∂z  ∂qt Rv + Γ(C) Cp (1 + rv ) F (C) − Cpd (Λr + Λv ) , R ∂z

N 2 (C) = Γ(C)

(6.6)

where Λv is an additional term defined in Marquet and Geleyn (2013). The impact of F (C) in the term Cp (1 + rv )(Rv /R)F (C) in the second line is to replace the unsaturated value (1 + rv )Cp Rv /R if C = 0 by the saturated version (1 + rsw )Lv /T for C = 1. The unsaturated version (C = 0) leads to the result: 

g ∂θv g θ 2 ˆ Nns = − (λ − δ)qv . (6.7) θv ∂z Cp T This expression is close to the expected result (g/θv )(∂θv /∂z), because the countergradient term is much smaller (typically one hundredth of the other). The saturated version is more complicated, but it is compared in Marquet and Geleyn (2013) with the older results derived in Durran and Klemp (1982) and Emanuel (1994), where N 2 was based on a moist-air entropy mainly expressed in terms of θe , instead of θs . The important feature is not the numerical value of N 2 , but rather the way in which N 2 is separated in Eq. 6.6 into three terms depending on the vertical gradients of s, qd , and qt . This separation depends on the way that the moist-air entropy is defined. Following Pauluis and Held (2002a,b), the first term depending on s can be interpreted as a conversion term between the turbulent kinetic energy and the available potential energy. The second term depending on qd is well known and was already included in Lalas and Einaudi (1974) as well as in other studies. It corresponds to the total-water lifting effect (conversion between the turbulent kinetic energy and the potential energy). The third term depending on qt is different from those derived in previous studies, simply because the moist-air entropy is defined differently. It corresponds to a new Λr scaled differential expansion and latent heat effects, associated with the use of θs instead of θe or θl .

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6.5

253

The moist-air potential vorticity PVs

Another logical application of the moist-air entropy, defined in terms of θs given by Eq. 3.9, is the computation of the moist-air potential vorticity (PV). The choice of the conservative variable to enter the definition of PV was open in the seminal papers of Ertel (1942)11 and Hoskins et al. (1985). It was noted that the modern choice of one of the potential temperatures θ (Ertel, 1942; Hoskins et al., 1985), θw (Bennetts and Hoskins, 1979), θv (Schubert et al., 2001), θes , or θe (Emanuel, 1994) might be replaced by an entropy variable like the dry-air value Cpd ln(θ) + const. It was thus tempting to study the properties of the moist-air PV computed starting with Cpd ln(θs ) + const. The results are published in Marquet (2014). The description of the meteorological properties of PV and the possibilities of an associated invertibility principle are beyond the scope of this set, which is focused on convective processes. However, those regions where negative values of moist PV are associated with positive gradients of the associated potential temperature correspond to criteria for possible slantwise convection instabilities (Bennetts and Hoskins, 1979). It is shown in Marquet (2014) that these criteria are more clearly observed with θs than with θes or θe . This result may encourage more systematic studies of slantwise convection with the help of the moist-air potential vorticity denoted by PVs and based on PV(s).

7 7.1

Conservation properties of θs Conservative properties of θs for pseudo-adiabatic processes

It is easy to use θs to determine whether or not a process is isentropic, i.e., by analysing whether or not θs = const. It is however important to explain why the moist-air entropy increases if pseudo-adiabatic processes are involved, and to compute precisely the change in θs associated with either reversible and adiabatic, or irreversible and pseudo-adiabatic, processes. Let us first consider pseudo-adiabatic processes for open systems associated with an ascending parcel of just-saturated moist air of mass m = md +mv , with no liquid water retained within the parcel. The pseudoadiabatic processes can be understood as a series of three elementary steps. 11 In

English in Schubert et al. (2004).

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Part IV: Unification and consistency

The first step is an infinitesimal and isentropic rising of this parcel, with the appearance of liquid water by condensation processes. The new mass is equal to m = md + (mv + dmv ) + dml , with dmv = −dml by virtue of the conservation of the mass. During the first step the moistair entropy is assumed to be a constant. The initial entropy is equal to Si = md sd + mv sv + 0 × sl , whereas the final entropy is equal to Sf = (md + dmd )(sd + dsd ) + (mv + dmv )(sv + dsv ) + dml (sl + dsl ). If the secondorder terms are discarded, the change in entropy divided by md is then equal to: (Sf − Si )/md = 0 = dsd + rsw dsv + (sv − sl )drsw .

(7.1)

The water-vapour mixing ratio is equal to its saturation value rsw = mv /md in pseudo-adiabatic processes. The water-vapour and dry-air specific entropies sv and sd must be evaluated at the temperature T and at the saturated partial pressures esw (T ) and p − esw (T ), respectively. The second step is the removal of the condensed liquid water dml = −dmv by falling rain, with the total mass being reduced to md + mv − dml and with an amount of entropy dml sl withdrawn by the precipitation. The consequence of this second step is that rt is equal to rsw during the whole ascent. The paradox is that this step does not mean that the specific entropy will decrease. It is shown hereafter that s increases during pseudoadiabatic ascent. The third step is to consider a resizing of the parcel of mass md + mv − dmv into a bigger volume of mass equal to its initial value m = md + mv . The aim of this step is to be able to compute specific quantities for a constant unit mass of moist air (not of dry air). The change in specific entropy (s) associated with the pseudo-adiabatic processes can be computed by rewriting Eq. 7.1 as 0 = d(sd + rsw sv ) − sl drsw . The term in parenthesis is equal to the just-saturated moist-air entropy expressed per unit of dry air (s/qd ). The result is that pseudoadiabatic processes correspond to d(s/qd ) = sl drsw . Since the property 1/qd = 1/(1 − qsw ) = 1 + rsw is valid for just-saturated moist air, it can be used to derive the differential equation:

 −drsw (T, p) dθs ds = Cpd = (s − sl ) . (7.2) θs 1 + rsw (T, p) It can be deduced from Eq. 7.2 that the change in specific moist-air entropy is positive for upward pseudo-adiabatic displacements, since they are associated with the properties drsw (T, p) < 0 and s > sl . The first property occurs because T decreases with height within the updraught. The second

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255

property occurs because qd ≫ qsw and thus s = qd sd + qsw sv is close to the dry-air entropy sd . Moreover, according to the curves depicted in Fig. 22.3, sd is always greater than sl for the atmospheric range of temperatures. It is worth noting that the impact of a change in pressure from p0 to pd in sd does not much modify the values of sd (T, pd ), leading to values of sl (T ) which are always smaller than those of s. The physical meaning of Eq. 7.2 is that the impact of the precipitation is to remove liquid-water entropy during pseudo-adiabatic processes, to be replaced by the local value of the moist-air entropy. This last action can be interpreted as a detrainment process. It corresponds to the third step described above, leading to a resizing of the moist-air parcel in order to be able to compute specific quantities expressed “per unit of moist air”, with the need to replace the lost mass of liquid water by an equal mass of moist air. For these reasons, moist-air entropy and θs must increase with height in regions where pseudo-adiabatic conditions prevail. It can be shown that Eq. 7.1 exactly corresponds to the differential equation:

(Cpd + rsw Cl )

dpd dT − Rd +d T pd



rsw Lv T



=0.

(7.3)

This differential equation is the same as those derived in Saunders (1957) or in the Smithsonian (1966) Tables to define the water-saturation pseudoadiabats. Equation 7.3 is derived from Eq. 7.1 with the use of the Kirchhoff and Clausius–Clapeyron equations recalled in Sec. 2.1, together with the equalities dsd = Cpd dT /T − Rd dpd /pd , dsv = Cpv dT /T − Rv desw /esw and sv −sl = Lv /T . The integration of Eq. 7.3 leads to the definition of the wet′ and the equivalent version bulb pseudo-adiabatic potential temperature θw ′ θe . This is a confirmation that the third law definition of θs does not modify ′ (nor θe′ ), since Eq. 7.3 can be derived from Eq. 7.1 withthe definition of θw out the use of Eqs. 3.7–3.12 which define s(θs ). The two kinds of potential temperatures correspond to different properties: θs always represents the moist-air entropy (for open or closed systems, for adiabatic or diabatic pro′ and θe′ are conservative properties for pseudo-adiabatic cesses), whereas θw processes only.

256

7.2

Part IV: Unification and consistency

Conservative properties of θs for adiabatic or isentropic processes

Let us consider the adiabatic conservation laws. The equation defining the water-saturation reversible adiabats is written in Saunders (1957) and Betts (1973) as:   rsw Lv dT dpd s +d − Rd = (Cpd + rt Cl ) =0. (7.4) d qd T pd T In comparison with Eq. 7.3, Cpd +rsw Cl is replaced by Cpd +rt Cl in Eq. 7.4. It is assumed in Betts (1973) that Eq. 7.4 corresponds to the conservation of entropy per unit of dry air, namely d(s/qd ) = 0, with the hypotheses that qd and qt = 1 − qd are constant and that qi = 0. The important feature is that the same equation, Eq. 7.4, can be obtained in a more straightforward way, by differentiating s/qd and with s given by Eq. 3.1, providing that the same assumptions of constant values for qd and qt = 1 − qd and of qi = 0 prevail. Therefore, the water-saturation reversible adiabats can be defined by constant values of the third law-based specific entropy s(θs ), or equivalently by θs = const. However, the moist-air isentropic equation ds = Cpd dθs /θs = 0 is more general than Eq. 7.4, since it can be derived analytically, without the assumption of constant qt and qd = 1 − qt and is thus valid for open systems and for varying values of qt . Therefore, the third law potential temperature θs and the corresponding moist-air entropy s defined by Eqs. 3.9 and 3.7 can be interpreted as the general integral of ds = 0. It is the quantity qd times Eq. 7.4 that is integrated via some approximations in Betts (1973) and Betts and Dugan (1973), to arrive at the definitions of the well-known liquid-water and saturation equivalent potential temperatures θl and θes , respectively. The water-saturation reversible adiabatic conservative properties verified by θs , θl , and θes correspond to the formal equations: ds = 0 = Cpd d [log(θs )] , s = 0 = Cp d [log(θes )] , qd d qd  Lv (T ) s − qd d dqt = 0 = Cp d [log(θl )] . qd T 

(7.5) (7.6) (7.7)

The specific heat Cp = qd Cpd + qsw Cpv is a moist value in Eqs. 7.6 and 7.7, with a corresponding moist value of the gas constant R = qd Rd + qsw Rv being involved in the terms qd d(s/qd ). Equation 7.5 can be solved exactly,

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257

and the result is θs as given by Eq. 3.9. The two other equations are solved with the approximations Cp ≈ Cpd and R ≈ Rd in Betts (1973) and Betts and Dugan (1973). The term −(Lv /T )dqt is equal to zero in Eq. 7.7, since qt is assumed to be a constant. This term is further approximated by −d[(Lv qt )/T ] and it corresponds to a differential of a certain constant term depending on qt . Therefore, this constant term must added to Cp ln(θes ) in order to define Cp ln(θl ), and thus the symmetrical pair of potential temperatures θes and θl . This precisely corresponds to the exponential term exp[−(Lv (T )qt )/(Cpd T )] to be included as a factor of θes to obtain θl . It is worthwhile noticing that the consequence of the approximations is that this exponential term is not a constant, since it depends on both qt and T , with the absolute temperature being a non-conservative variable. Equations 7.6 and 7.7 do not correspond to changes in specific moistair entropy ds, nor to the associated potential temperature θs . The equivalent and liquid-water potential temperatures are defined by θes ≈ θ exp[(Lv qsw )/(Cpd T )] and θl ≈ θ exp[−(Lv ql )/(Cpd T )], respectively. They are approximately constant during pseudo-adiabatic processes undergone by a closed parcel. The approximate features are due to Cp ≈ Cpd , R ≈ Rd plus the hypotheses (Lv /T )dqsw ≈ d(Lv qsw /T ) and (Lv /T )dql ≈ d(Lv ql /T ) recalled in Deardorff (1980). These approximations, together with the fact that qt is assumed to be a constant, explain the differences between θs and θes or θl : only θs can be used as a conservative (isentropic) variable for open systems, because the additional terms depending on qt in θs given by Eq. 3.9 could not be derived from Eqs. 7.6 and 7.7, due to the arbitrary terms corresponding to the hypothesis dqt = 0 and due to the aforementioned approximations needed to define θes or θl . There is no need to make these approximations for deriving the third law-based moist-air entropy and θs . Similar “conserved” quantities depending on qt are included by hand in Eq. 4.5.15 of Emanuel (1994), to form another liquid-water potential temperature also denoted by θl . The “conserved” arbitrary quantity is equal to (Rd + rt Rv ) ln(p0 ) − (Rd + rt Rv ) ln(1 + rt /ǫ) + rt Rv ln(rr /ǫ). It is indeed a complex term that can only be justified by the desire to arrive at a result prescribed a priori. Any other terms depending on qt and rt would lead to other possible conservative quantities different from θl , making these manipulations unclear and questionable, in particular if qt is not a constant. The important result obtained with the specific moist-air entropy expressed by Eq. 3.7 is that Cpd ln(θs ) is an exact integral of Eq. 7.4 expressed

258

Part IV: Unification and consistency

as ds = 0, which corresponds to the second law property. This result is obtained without manipulation of terms depending on qt . Therefore, θs is a true measurement of the moist-air entropy and an exact generalization of θl or θe . New isentropic conservative properties can thus be observed for θs for open systems, with changes in qt that must be balanced by changes in other variables, in order to keep θs unchanged. 7.3

The conservative properties of θs : A synthesis

A synthetic view is given in Table 22.1, where the moist-air conserved feature is tested for six potential temperatures (listed on the first line) and four moist-air conservative properties (described on the first column). Table 22.1 Analyses of several moist-air conservation properties for several moist-air potential temperatures. A=approximate; I=irrelevant; N=no; Y=yes; Y/N=yes or no. List of moist–air properties

|

θv

θl

θe

θe′

′ θw

θs

unsat. adiab./closed parcel/constant qv sat. adiabatic/closed parcel/constant qt pseudo-adiab./open parcel/qt = q ∗ (T, p) isentropic/open parcel/varying qt , ql , or qi

| | | |

Y/N Y/N Y/N Y/N

A A I I

A A I I

A N Y I

A N Y I

Y Y N Y

• The virtual potential temperature θv = Tv (p/p0 )κ is conserved for neutral buoyancy conditions. Adiabatic or pseudo-adiabatic motions of closed or open parcels of moist air do not automatically imply the conservation of θv . For these reasons, θv may be conserved, or not, independently of the adiabatic, pseudo-adiabatic, or isentropic properties. • Betts’s liquid-water and equivalent potential temperatures (θl and θe ) are both approximately conserved for adiabatic motions of closed parcels of moist air, where qt is a constant. Since it is an equivalent of the moist-air entropy, only θs is a true conserved adiabatic quantity and the approximate feature for both θl and θe is due to the two terms in Eq. 3.9 which depend on T and p and which are not included in (θs )1 given by Eq. 3.10 (which behaves like θl if qt is a constant). Both θl and θe are irrelevant for studies of open systems (such as motions of real parcels of moist air with varying qt ). This is due to the method described in Sec. 7.2, where θl and θe are derived from two (approximate) differential equations, with the assumption of constant qt .

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259

′ • The two potential temperatures θe′ and θw are defined in order to be conserved for pseudo-adiabatic processes only (see Sec. 7.2). Since pseudo-adiabatic processes are incompatible with existing condensed water species, neither θl nor θe are conserved for adiabatic motions of closed parcels, and they are irrelevant for describing isentropic feature for open parcels (where contents in all water species may vary in time and in space). Only approximate conservative features are observed for θe′ and ′ for adiabatic motions of unsaturated parcels of moist air, where it θw ′ by an adiabatic saturation process by is the usual practice to define θw imposing a constant value for the dry-air potential temperature θ. This can be explained because only θs is conserved during these moist-air adiabatic motions, and even if qt = qv is a constant, the conservation of θs given by Eq. 3.9 does not mean that θ is conserved, due to the two terms in Eq. 3.9 which depend on T and p and which are not included in (θs )1 as given by Eq. 3.10 (which behaves like θ if ql = qi = 0 and qt = qv is a constant). • The moist-air entropy potential temperature θs is not conserved by pseudo-adiabatic processes, since, as is explained in Sec. 7.1 the increase in θs is given by Eq. 7.2. θs is, however, interesting in that it is conserved for all other moist-air adiabatic processes (last column).

The conclusion of this section is that pseudo-adiabatic processes must ′ , and that other adiabatic processes should be be studied by using θe′ or θw analysed with the use of θs , in order to avoid unnecessary approximations. The well-known potential temperatures θl and θe are, however, relevant approximations of θs for studying conservative properties of closed parcels of moist air. Studies of isentropic features for open systems (last line in Table 22.1) require the use of θs and they exclude the use of other moistair potential temperatures. As for the virtual potential temperature θv , it is the specialized quantity suitable for studying the buoyancy force and it cannot be used for analysing isentropic features.

8 8.1

Physical properties of the moist-air enthalpy and Th Applications for the moist-air enthalpy: Turbulent surface fluxes

The problem of the need to manage relevant values for the reference enthalpies in atmospheric science is explicitly addressed in Businger (1982).

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Part IV: Unification and consistency

The specific enthalpies are written as hk − (hk )r = Cpk (T − Tr ), where the index k goes for dry air (d), water vapour (v), liquid water (l), and ice (i). They can also be written as hk = Cpk T + bk , where bk = (hk )r − Cpk Tr . The question asked in Businger (1982) is whether or not it is important to determine the numerical values of bk (and thus the reference values (hk )r ) in order to compute the vertical turbulent fluxes of the moist-air enthalpy h. It is common practice in atmospheric science to set reference enthalpies of dry air equal to zero for a given reference temperature, typically at the high temperature Tr = 273.15 K = 0◦ C. The choices made for the zeroenthalpies for water species are more variable. They may be set to zero either for the water-vapour or liquid-water enthalpies, for the same reference temperature 0◦ C, and with the latent heats obviously connecting the other water enthalpies by Lv (Tr ) = (hv )r − (hl )r or Lf (Tr ) = (hl )r − (hi )r . It was shown by Businger (1982) that it is the choice of zero-enthalpies for both dry air and liquid water at Tr = 0◦ C that is in agreement with well-established procedures for computing surface turbulent fluxes (the ones still retained nowadays). Otherwise, additional fluxes of qt should be added to the moist-air enthalpy flux, leading to other definitions of this turbulent flux. The same hypothesis (hd )r = (hl )r = 0 is retained in the review of Fuehrer and Friehe (2002) and for Tr = 0◦ C. It is, however, unlikely that such arbitrary fluxes of qt may be added to or cancelled from the enthalpy flux, leading to arbitrary closure for the computation of turbulent fluxes of moist air. The same is true for the vertical integral and the horizontal or vertical gradients of h, for which terms depending on qt could be of real importance if qt is not a constant. The accuracy and the relevancy of the hypotheses (hd )r = (hl )r (= 0?), (hd )r = (hv )r (= 0?), bv = (hv )r − Cpv Tr = 0, or bl = (hl )r − Cl Tr = 0 are analysed in the next subsection. 8.2

Impacts of Trouton’s rule and of the coincidence (hl )r ≈ (hd )r

The analyses of the constant heat capacity curves (cst straight lines) of enthalpies depicted in Fig. 22.4b can be used to test the three kinds of assumptions listed in Businger (1982): (1) hl (Tr ) = hd (Tr ) for some reference temperature Tr > 150 K; (2) hl (Tr ) = 0 and hd (Tr ) = 0 for some reference temperature Tr > 150 K;

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261

and, (3) hl (Tr ) = 0 and hd (Tr ) = 0 for the extrapolated limit Tr = 0 K, with the consequence of linear laws hv (T ) = Cpv T and hl (T ) = Cl T valid for temperature T > 150 K. The reference temperature for which hl (Tr ) = hd (Tr ) can be computed with h0l and h0d as given by Eqs. 2.19 and 2.17 respectively, and with the properties hl (Tr ) = hl (T0 )+Cl (Tr −T0 ) and hd (Tr ) = hd (T0 )+Cpd (Tr −T0 ). It is equal to T ≈ 241.4 K= −31.75◦C (the grey spot in Fig. 22.4b). It is clear from Fig. 22.4b that the differences between hl (T ) and hd (T ) increase with increasing temperatures. For example, at 30◦ C, hl ≈ 760 kJkg−1 and is 36% larger than hd ≈ 560 kJkg−1. It is therefore only a coincidence and a crude approximate property that the dry-air and liquid-water thermal enthalpies are close to each other for the atmospheric range of temperature. Except for the too-low reference temperature Tr = −31.75◦C, this result invalidates the hypothesis recalled in Sec. 2.3, where the property (hl )r = (hd )r is often assumed in order to cancel the second term in the second line of Eq. 2.8. It also demonstrates that additional terms should be added to the usual turbulent fluxes of enthalpy, consistent with the conclusions of Businger (1982). Values of h0i are about half those of h0l , and due to Trouton’s rule, values of h0v are much higher than those of h0i and h0l . Trouton’s rule (see the review of Wisniak, 2001) states that the general property Lv /T ≈ 88 Jmol−1 K−1 holds true for almost all substances at their boiling temperature. Since boiling temperature for H2 O (373.15 K) is about 4.4 times greater than those for N2 (77.4 K) and O2 (90 K), the latent heats of sublimation and vaporization of water are thus logically the dominant features in Fig. 22.4. This invalidates the possibility that (hv )r = (hd )r for which the second term in the second line of Eq. 2.7 could be discarded. Even if (hv )r − (hd )r can be written as (hl )r − (hd )r + Lv (Tr ), with the dominant term Lv (Tr ) leading to possible correction values in Eq. 2.7, the remaining term (hl )r − (hd )r is different from zero in Eq. 2.8 and must be taken into account. Moreover, since the lines denoting liquid water and water vapour on Fig. 22.4b above 150 K do not intersect the origin (if these straight lines were continued towards T = 0 K), this does not confirm the properties bv = 0 or bl = 0, namely the linear laws (hv )r = Cpv Tr or (hl )r = Cl Tr , which could allow the cancellation of the second lines of Eqs. 2.9 and 2.10 whatever the values of qt may be.

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8.3

Part IV: Unification and consistency

Comparisons of moist-air enthalpy with various moist static energy formulae

It is recalled in this subsection that the concept of moist static energy is intimately linked with the conservation of the moist-air entropy (s), and with the computation of the moist-air enthalpy (h). The link between the moist-air enthalpy and entropy is due to the Gibbs equation: T

dh 1 dp  dqk ds μk = − − , dt dt ρ dt dt

(8.1)

k

where the local Gibbs function is defined by μk = hk − T sk and where the sum over the index k goes for dry air (d), water vapour (v), liquid water (l), and ice (i). The reason it is possible to associate the moist-air enthalpy h with s is that, for stationary pure vertical motions, the material derivative is equal to d/dt = w∂/∂z, with the vertical velocity w factorizing all of the terms in Eq. 8.1 and which can therefore be omitted. For vertical hydrostatic motions −ρ−1 ∂p/∂z = ∂φ/∂z. If the parcel is closed (constant qt ) and undergoes reversible adiabatic processes (no supercooled water, but with possible reversible condensation, evaporation, and fusion processes), then  ∂s/∂z = 0 and k μk ∂qk /∂z = 0. The Gibbs equation can then be written as: T

∂(h + φ) ∂s =0= . ∂z ∂z

(8.2)

The quantity h + φ is thus a conserved quantity for vertical motions provided that all the previous assumptions are valid (adiabatic and hydrostatic vertical motion of a closed parcel). This sum h+φ is called the “generalized enthalpy” in Ambaum (2010). The consequence of the adiabatic conservative property of Eq. 8.2 is that the generalized enthalpy can be used to represent convective saturated updraughts. There is thus a need to compute h + φ at the bottom of an updraught, and therefore to compute the moist-air enthalpy h itself, before using the property h + φ = const to determine the local properties of the updraught (T , qv , ql , qi ) at all levels above the cloud base. It is expected that the moist-air specific thermal enthalpy h given by Eq. 4.1 is the quantity that enters the generalized enthalpy h + φ of Eq. 8.2. It is, however, a common practice in studies of convection to replace h + φ

Formulations of moist thermodynamics for atmospheric modelling

263

with one of the MSE quantities defined by: MSEd = Cpd T + Lv qv + φ ,

(8.3)

MSEl = Cpd T − Lv ql + φ ,

(8.4)

LIMSE = Cpd T − Lv ql − Ls qi + φ .

(8.5)

The formulation MSEd is the most popular. It is considered in Arakawa and Schubert (1974) as “approximately conserved by individual air parcels during moist adiabatic processes”. It is mentioned in Betts (1975) that it is approximately an analogue of the equivalent potential temperature θe . It is used as a conserved variable for defining saturated updraughts in some deep-convection schemes (Bougeault, 1985). A liquid-water version, MSEl , is defined in Betts (1975) by removing the quantity Lv qt (assumed to be a constant) from Eq. 8.3, leading to Eq. 8.4. It is considered in Betts (1975) that MSEl is an analogue of the liquidwater potential temperature θl . The formulation MSEl is generalized in Bretherton et al. (2005) and Khairoutdinov and Randall (2003) by further subtracting a term Ls qi from Eq. 8.4, for the sake of symmetry, leading to the liquid-ice static energy LIMSE given by Eq. 8.5, which is used as a conserved variable for defining saturated updraughts in some shallowconvection schemes (e.g., Bechtold et al., 2001). It is shown in Marquet (2015) that the factor multiplying qt in Eq. 4.1 can be roughly approximated by the dominant term Lv (T ). This corresponds to the coincidence h0d ≈ h0l and to Trouton’s rule described in Sec. 8.2. Accordingly, the generalized moist-air thermal enthalpy can be approximated by h + φ ≈ href +FMSE, where FMSE = Cpd T + Lv qv − Lf qi + φ .

(8.6)

This formulation corresponds to the frozen moist static energy as defined by Vol. 1, Ch. 10, Eq. 4.5, and as studied in both Vol. 1, Ch. 10 and Bretherton et al. (2005). It is equal to the sum of LIMSE plus Ls qt and it is equal to MSEd for positive Celsius temperatures (qi = 0). The drawback of this approximate formula is that it does not possess the same symmetry as in Eq. 4.1, where Lv is logically replaced by Ls if ql is replaced by qi . This means that the approximation used to establish Eq. 8.6 is probably not accurate enough. MSE is sometimes defined with Cpd replaced in Eq. 8.3 by the moist value Cp , leading to: MSEm = Cp T + Lv qv + φ .

(8.7)

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Part IV: Unification and consistency

This version is used in some convective schemes (e.g., Gerard et al., 2009). Other MSE quantities are expressed per unit mass of dry air and with the assumption that h0d = h0l = 0, such as the moist enthalpy k in Emanuel (1994), which is equivalent to the MSE defined by h⋆v in Ambaum (2010): k + φ/qd =

h⋆v = [Cpd + qt (Cl − Cpd )]T + Lv qv + φ ,

h⋆v /qd

= (Cpd + rt Cl )T + Lv rv + (1 + rt )φ .

(8.8) (8.9)

The aim of this subsection is thus to compare the generalized enthalpy h + φ, with h given by Eq. 4.1, to some of the other MSE quantities given by Eqs. 8.3–8.9. The properties of surface deficit charts are commonly used in studies of convective processes and two of them are depicted in Fig. 22.7, for the FIRE-I (RF03B) and BOMEX datasets.

Fig. 22.7 Surface deficit in generalized enthalpy h + φ and in MSE quantities for the FIRE-I (RF03b) stratocumulus and the BOMEX shallow cumulus datasets. The units c Royal Meteorological Society, are kJkg−1 . (From Fig. 11 of Marquet (2015), which is  2015.)

It is shown that the two formulations MSEm and MSEd (=FMSE) are close to h + φ at all levels for both cases, with the generalized enthalpy located in between the others and with MSEm being a better approximation for h + φ. The generalized enthalpy departs more strongly from the other quantities, LIMSE and h⋆v (with k + φ/qd very close to h⋆v , not shown). Thus, LIMSE and h⋆v cannot represent accurately the generalized enthalpy h + φ. Large jumps in all variables are observed close to the surface for the BOMEX case, except for LIMSE, due to the impact of large values of specific humidity that are not taken into account in LIMSE. Such impacts of surface values are not observed for FIRE-I, since the air-flight measurements were taken above sea level.

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265

The conclusion of this subsection is that MSEm is probably the best candidate for approximating h + φ. However, the fact that systematic differences between h + φ and MSEm exist in the moist lower PBL only, and not in the dry air above, may have significant physical implications if the purpose is to analyse moist-enthalpy budgets accurately, or differential budgets, or to understand convective processes (entrainment and detrainment), or to validate long-term budgets for NWP models and GCMs by comparing them with climatology or reanalyses. For these reasons, it is likely important to define the moist-air enthalpy h by Eq. 4.1 if the aim is to use conservative MSE quantities based on the generalized enthalpy.

9

Summary and discussion of possible new applications

It has been shown in this chapter that it is possible to compute directly the moist-air entropy expressed by s = sref + Cpd ln(θs ), in terms of the potential temperature θs given by Eq. 3.9. The moist-air thermal enthalpy can be similarly expressed by h = href + Cpd Th , in terms of the enthalpy temperature Th given by Eq. 4.2. These new formulae are associated with the first, second, and third laws of thermodynamics and are derived with a minimum of hypotheses. The important result is that the formulae offer the opportunity of analysing new atmospheric processes, since it is possible to evaluate whether isentropic conditions prevail within those regions where moist air must be considered as an open system, namely where qt is varying in time and in space. In particular, computations of changes in moist-air entropy are suitable and relevant for convective processes, simply by examining the distribution of θs in those regions where entrainment, detrainment, precipitation, or evaporation take place. These studies cannot be realized by using other meteorological potential temperatures. A first example of a possible application of moist-air entropy to convective processes concerns the PBL-top entrainment in marine stratocumulus, where the PBL-top jump in moist-air entropy seems to control the turbulent and convective processes, by maintaining or evaporating the cloud depending on the sign of the jump in θs . This result is not observed for the sign of PBL-top jump in θl or θe : these jumps cannot directly explain the instability of marine stratocumulus simply because θl and θe cannot represent the moist-air entropy if qt is not a constant. A second example is given by theories of conditional symmetric instabil-

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ity and slantwise convection. These theories are expressed in Bennetts and Hoskins (1979) in terms of the moist-air Brunt–V¨ ais¨ al¨ a frequency and the ′ ′ . The wet-bulb potential temperature θw is replaced vertical gradient of θw by θe in subsequent studies (Emanuel, 1983a,b, 1994; Emanuel et al., 1987; ′ Thorpe and Emanuel, 1985a,b). Both θw and θe are used as if equivalent to the moist-air entropy, with dθe /dt = 0 only for diabatic processes other than latent heat release. It is, however, irrelevant to use θe in those regions where qt is not a constant. This is a clear motivation for replacing the use of θe with the use of θs for studying slantwise convection. A third example concerns the change in moist-air entropy, which is a key quantity entering Emanuel’s vision of a tropical cyclone as a Carnot heat engine (Bryan and Rotunno, 2009; Camp and Montgomery, 2001; Emanuel, 1986, 1991b; Holand, 1997; Tang and Emanuel, 2012). It is mostly the quantity θe which is used to compute the difference between the entropies (Δs) of moist air in the ambient environment and that near the storm centre. However, qt varies rapidly between these regions. This is a motivation for replacing the use of θe with the use of θs when studying hurricanes, typhoons, or tropical cyclones. Even if it is valuable to continue to analyse pseudo-adiabatic processes ′ (see Table 22.1), the three with the help of constant values of θe′ ≈ θe or θw aforementioned examples are typical applications where θl or θe must be replaced by θs in order to compute and compare values of moist-air entropy in regions with varying qt . The first example demonstrated in Marquet (2011) and recalled in Sec. 6.1 concerns the constant feature of θs in the moist-air PBL of marine stratocumulus. The second example dealing with the moist-air potential vorticity and the slantwise convection criteria is documented in Marquet (2014) and in Sec. 6.5. The third example has been documented by analysing drop-sounding and LAM outputs for several tropical cyclones (unpublished results). Three classes of convective phenomena are depicted in Fig. 22.8 along with variables that are well suited to their study. From left to right are shown: stratiform clouds (stratocumulus), shallow convection (nonprecipitating cumulus), and deep convection (precipitating cumulus or cumulonimbus). The goal of this figure is to recall that the study of θv is relevant for all clouds, since it is the relevant parameter for the buoyancy force. Therefore, the conservative properties observed for θv are not related to the second law and they do not correspond to the moist-air entropy. ′ It is recalled that θe (in fact θw or θe′ ) is conserved during pseudo-

Formulations of moist thermodynamics for atmospheric modelling

Fig. 22.8

267

A schematic showing relevant variables for different regimes of convection.

adiabatic processes and is thus relevant for describing the precipitating saturated updraughts in deep convective clouds (provided that any preexisting non-precipitating cloud condensed water species are discarded). However, θe is not suitable for describing the non-precipitating parts of deep convective clouds (since closed parcels imply constant qt and are not compatible with pseudo-adiabatic processes), nor the impact on moist-air entropy of the flux of matter occurring at the edge of clouds (since entrainment and detrainment are not pseudo-adiabatic processes and since they act on open systems, where changes in qt are large and rapid). It is also recalled that a stratocumulus system exhibits large PBL-top jumps in θ, θl , θe , and qt . Only θs remains close to neutrality (both within the PBL and the entrainment region, with a continuous variation within the free air located above). For these clouds, the entropy potential temperature θs should replace θl and θe , since qt is not a constant and since that θl and θe cannot represent the moist-air entropy of such open systems. An important problem concerns shallow convection, where no conserved variable really exists. Since the condensed water is not withdrawn by precipitation, pseudo-adiabatic variables cannot represent the updraughts of non-precipitating cumulus. If adiabatic motions of closed parcels are considered, namely if isentropic processes are imagined, then the use of θs should replace that of θl and θe , according to the conclusion of Table 22.1. Moreover, only θs remains relevant for studying the impact of lateral entrainment and detrainment processes, due to the edges of clouds which can be considered as open systems. However, the notable limitation of the possibility that θs may be a

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conserved quantity for shallow convection is that the diabatic impact of radiation is important for these clouds. It could be useful to interpret the problem differently: namely to analyse the non-equilibrium state of the cloud in terms of the diabatic impacts on moist entropy and θs , rather than to search for an adiabatic magic quantity which may not exist. The common feature of the three types of clouds is that it is always possible to compute the moist-air entropy in terms of θs , whatever diabatic or adiabatic processes, and closed or open parcels, are considered. Except for very specialized cases (pseudo-adiabatic or closed adiabatic parcels), it should be worthwhile to analyse the conservative feature of s = sref + Cpd ln(θs ) within clouds, at the edges of clouds, and outside the clouds, automatically leading to the isentropic state identified as constant values of s (whatever the changes in qt may be), or to diabatic heating production of moist-air entropy given by Eq. 7.2 for varying values of s. It is explained in Sec. 8.3 that there is an alternative description of conservative properties of clouds in terms of the moist-air generalized enthalpy h + φ defined by Eq. 8.2, with the thermal enthalpy h given by Eq. 4.1. The validity of the equation ∂s/∂z = ∂(h + φ)/∂z = 0 relies on several assumptions: vertical, adiabatic, and stationary motion of a closed parcel. Since the second law is invoked via the use of ∂s/∂z = (Cpd /θs )∂θs /∂z = 0, the conservation of h + φ is equivalent to the conservation of θs , and approximately of θl and θe , since qt is assumed to be a constant. This a priori excludes the use of h + φ in pseudo-adiabatic regions and suggests the use of θs instead of θe or θe′ . It is further explained in Sec. 8.3 that h + φ can be approximated with a good accuracy by MSEm , MSEd , or FMSE, but not by any of MSEl , LIMSE, k + φ/qd or h⋆v . Moreover, it is likely that systematic differences between the generalized enthalpy and MSE quantities are important enough to justify the use of the more general formula h + φ valid for the moist-air thermal enthalpy given by Eq. 4.1. The fact that it is possible to compute the entropy of matter in the atmosphere by the third law-based formula may be used for addressing in a new way both computations of the budget of moist-air entropy and the assessment of the maximum entropy production (or MEP12 ) principle, for 12 It is quoted in Ozawa et al. (2003) that “it was Paltridge (1975, 1978) who first suggested that the global state of the present climate is reproducible, as a long-term mean, by a state of Maximum Entropy Production”. The principle of least action is applied using variational methods in order to implement the MEP principle. The relevance of the MEP principle for the atmosphere is described, analysed and discussed by many authors (Dewar, 2003; Ebeling and Feistel, 2011; Goody, 2007; Grassl, 1981; Jaynes,

Formulations of moist thermodynamics for atmospheric modelling

269

either the global atmosphere or for local turbulent or convective processes. Indeed, the change in moist-air entropy ds/dt = −∇ · Js + σs is usually computed via the flux (Js ) and a positive production rate (σs ) of moistair entropy (Pascale et al., 2011; Pauluis and Held, 2002a; Peixoto et al., 1991; Stephens and O’Brien, 1993). The aim of such studies is to analyse whether the production of entropy by the moist-air atmosphere, land, and oceans can balance the average sink of about 1 WK−1 m−2 imposed by the imbalance of incoming low-value solar and outgoing high-value infrared radiation entropies, including impact studies of local turbulent or convective processes. The novelty offered by the third law-based entropy is that it is possible to compute directly the left-hand side of the moist-air entropy equation ds/dt by using s = sref + Cpd ln(θs ) and a finite difference formulation [s(t + dt) − s(t)]/dt, leading to new comparisons of NWP models or GCMs with analysed or observed values. The third law-based entropy can also be used to study the MEP principle in a new way. It is explained in Secs. 5.1 and 6.1 that θs is about in a 2/3 position between θ ≈ θl and θe ≈ θe′ . This means that the moist-air entropies computed in Emanuel (1994), Pauluis et al. (2010), and Pauluis (2011) are underestimated or overestimated when they are computed with θl or θe , respectively. Therefore, the moist-air entropy productions based on θl or θe must be different from those based on θs , especially in regions with high values of qt like convective regions, storms, or cyclones. For these reasons, an MEP principle based on the moist-air entropy expressed in terms of θs could be more relevant than those based on the dry-air entropy (namely θ or Cpd T + φ). From Λr = [(sv )r − (sd )r ]/Cpd , the moist-air entropy s(θs ) depends on reference entropies determined from the third law, and thus on experimental values. Therefore, s(θs ) can be obtained  neither from the Shannon information entropy − i pi log pi nor from the  relative (Kullback and Leibler) information entropy i pi log(pi /qi ), two theoretical quantities which can only deal with one-component systems like dry air, but not with moist air.

1980; Kleidon, 2009; Kleidon et al., 2006; Martyushev, 2013; Martyushev and Seleznev, 2014; Nicolis, 1999, 2003; Nicolis and Nicolis, 1980, 2010; Paltridge, 1981, 2001; Paltridge et al., 2007; Pascale et al., 2012; Seleznev and Martyushev, 2014; Stephens and O’Brien, 1993).

270

10 a bk C Cpd Cpv Cl Ci Cp

Part IV: Unification and consistency

List of symbols and acronyms

a weighting factor in Pauluis et al. (2010) (hk )r − Cpk Tr for species k in Businger (1982) a transition parameter (Marquet and Geleyn, 2013) specific heat for dry air (1004.7 JK−1 kg−1 ) specific heat for water vapour (1846.1 JK−1 kg−1 ) specific heat for liquid water (4218 JK−1 kg−1 ) specific heat for ice (2106 JK−1 kg−1 ) specific heat at constant pressure for moist air, = qd Cpd + qv Cpv + ql Cl + qi Ci Cp⋆ specific heat at constant pressure depending on rt Cv specific heat at constant volume for moist air (d, v, l, i) subscripts for dry air, water vapour, liquid water, and ice a moist-air parameter in Marquet and Geleyn (2013) DC ΔHf0 the standard enthalpies of formation the standard enthalpies of reaction ΔHr0 the CTEI line in the (qt , θl ) diagram Δ2 PBL-top jump in qt for stratocumulus (and Δ(θl ), Δ(θs ), ...) Δ(qt ) δˆ = Rv /Rd − 1 ≈ 0.608 η = 1 + δˆ = Rv /Rd ≈ 1.608 ε˜ = 1/η = Rd /Rv ≈ 0.622 κ = Rd /Cpd ≈ 0.2857 γ = ηκ = Rv /Cpd ≈ 0.46 λ = Cpv /Cpd − 1 ≈ 0.8375 e the water-vapour partial pressure esw (T ) partial saturation pressure over liquid water esi (T ) partial saturation pressure over ice the water-vapour reference partial pressure, er with er = esw (Tr = T0 ) ≈ 6.11 hPa the water-vapour standard partial pressure (1000 hPa) e0 the internal energy (= h − p/ρ = h − RT ) ei φ the gravitational potential energy (= gz + φ0 ) Γ the moist-air adiabatic lapse-rate the liquid water-saturated moist-air adiabatic lapse-rate Γsw F (C) a parameter in Marquet and Geleyn (2013) g 9.8065 ms−2 , magnitude of gravity h the moist-air specific enthalpy enthalpy of dry air hd

Formulations of moist thermodynamics for atmospheric modelling

hv hl hi href (hd )r (hv )r (hl )r (hi )r h0d h0v h0l h0i h⋆v k kw , k kRD kL Kψ Kh Kq Ks Λr Λ0 Lv L0v Lf L0f Ls L0s M (C) MSE m μ μd μv μl μi N2

271

enthalpy of water vapour enthalpy of liquid water enthalpy of ice a reference value for enthalpy reference enthalpy of dry air at Tr reference enthalpy of water vapour at Tr reference enthalpy of liquid water at Tr reference enthalpy of ice at Tr standard specific enthalpy of the dry air (530 kJkg−1 ) standard specific enthalpy of the water vapour (3133 kJkg−1 ) standard specific enthalpy of the liquid water (632 kJkg−1 ) standard specific enthalpy of the ice water (298 kJkg−1) a specific moist static energy (Ambaum, 2010) a dummy subscript for (d, v, l, i) (for instance in bk ) liquid-water and moist enthalpy in Emanuel (1994) the Randall–Deardorff CTEI parameter in the (qt , θe ) diagram the Lilly CTEI parameter in the (qt , θl ) diagram a general exchange coefficient for any ψ the exchange coefficient for heat the exchange coefficient for water content the exchange coefficient for entropy = [(sv )r − (sd )r ]/Cpd ≈ 5.87 = [s0v − s0d ]/Cpd ≈ 3.53 = hv − hl : latent heat of vaporization = 2.501 × 106 Jkg−1 at T0 = hl − hi : latent heat of fusion = 0.334 × 106 Jkg−1 at T0 = hv − hi : latent heat of sublimation = 2.835 × 106 Jkg−1 at T0 a parameter in Marquet and Geleyn (2013) moist static energy (MSEd , MSEm , MSEl , LIMSE, FMSE) a mass of moist air (and md , mv , ml , mi ) moist-air Gibbs function h − T s (with moist-air h and s) dry-air Gibbs function (hd − T sd ) water-vapour Gibbs function (hv − T sv ) liquid-water Gibbs function (hl − T sl ) ice Gibbs function (hi − T si ) the squared Brunt–V¨ ais¨ al¨ a frequency

272 2 Nsw PV PVs p pr pd (pd )r (pd )0 p0 qd qv ql qi qt rv rl ri rr

rsw rt ρd ρv ρl ρi ρ Rd Rv R Ri ˆ R s sref sd sv sl si

Part IV: Unification and consistency

the liquid water-saturated version of N 2 the potential vorticity the moist-air entropy potential vorticity (= pd + e) the local value for the pressure (= (pd )r + er ) the reference pressure (pr = p0 = 1000 hPa) local dry-air partial pressure reference dry-air partial pressure (≡ pr − er = 993.89 hPa) standard dry-air partial pressure (1000 hPa) a conventional pressure (1000 hPa) = ρd /ρ: specific content for dry air = ρv /ρ: specific content for water vapour = ρl /ρ: specific content for liquid water = ρi /ρ: specific content for ice = qv + ql + qi : total specific content of water = qv /qd : mixing ratio for water vapour = ql /qd : mixing ratio for liquid water = qi /qd : mixing ratio for ice reference mixing ratio for water species, with ηrr ≡ er /(pd )r , leading to rr ≈ 3.82 gkg−1 saturation water vapour mixing ratio over liquid water = qt /qd : mixing ratio for total water specific mass for dry air specific mass for water vapour specific mass for liquid water specific mass for ice specific mass for moist air = ρd + ρv + ρl + ρi dry-air gas constant (287.06 JK−1 kg−1 ) water-vapour gas constant (461.53 JK−1 kg−1 ) = qd Rd + qv Rv : gas constant for moist air Richardson number (N 2 /S 2 ) a parameter similar to C to mix the dry and wet limits for turbulent values (Lewellen and Lewellen, 2004) moist-air specific entropy and sr : reference values for entropy entropy of dry air entropy of water vapour entropy of liquid water entropy of ice

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273

(sd )r reference value for dry-air entropy (6777 JK−1 kg−1 ) (sv )r reference value for water-vapour entropy (12673 JK−1 kg−1 ) s0d standard specific entropy for dry air (6775 JK−1 kg−1 ) 0 sv standard specific entropy for water vapour (10320 JK−1 kg−1 ) 0 standard specific entropy for liquid water (3517 JK−1 kg−1 ) sl 0 si standard specific entropy for ice (2296 JK−1 kg−1 ) two moist-air entropies in Pauluis et al. (2010) Se , Sl a weighted sum of Se and Sl (Pauluis et al., 2010) Sa S2 the squared of the wind shear (in Ri ) t, dt time and timestep T local temperature moist-air enthalpy temperature Th the reference temperature (Tr ≡ T0 ) Tr the zero Celsius temperature (273.15 K) T0 a constant temperature (2362 K) TΥ θ = T (p0 /p)κ : the dry-air potential temperature θl the liquid-water potential temperature (Betts, 1973) the virtual potential temperature θv the equivalent potential temperature (companion to θl ) θe the saturation value of θe θes ′ the equivalent potential temperature deduced from θw θe′ ′ θw the pseudo-adiabatic wet-bulb potential temperature the moist-air entropy potential temperature (Marquet, 2011) θs the reference value for θs θsr (θs )1 an approximate version for θs θS a moist-air potential temperature (Hauf and H¨ oller, 1987) a moist-air potential temperature (Marquet, 1993) θ⋆ Υ(T ) = [hv (T ) − hd (T )]/(Cpd T ) = [(hv )r − (hd )r ]/(Cpd Tr ), Υ(T0 ) ≈ 9.5 Υ(Tr ) w the vertical component of velocity z the vertical height ASTEX Atlantic Stratocumulus Transition Experiment BOMEX Barbados Oceanographic and Meteorological Experiment CTEI Cloud Top Entrainment Instability DYCOMS DYnamics and Chemistry Of Marine Stratocumulus FIRE First ISCCP Regional Experiment EPIC East Pacific Investigation of Climate EUCLIPSE European Union CLoud Intercomparison,

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GCM ISCCP NWP PBL

Part IV: Unification and consistency

Process Study and Evaluation project General Circulation Model International Satellite Cloud Climatology Project numerical weather prediction planetary boundary layer

Chapter 23

Representation of microphysical processes in cloud-resolving models

A.P. Khain Editors’ introduction: The use of cloud-resolving or large-eddy models is widespread in order to study convection and in order to inform or test ideas for parameterization development. There are many examples throughout this book (e.g., in Vol. 1, Ch. 10 and Chs. 17, 18, 19, 24). The results from such simulations are often treated as truth in comparisons with large-scale models. For that approach to be valid, it is a necessary condition that the simulations be run with small enough grid lengths and timesteps in order to provide good numerical resolution of the phenomena of interest. While necessary, however, that is not in itself sufficient. An adequate treatment of microphysical processes is also required for fully realistic results. This chapter may be considered a discussion of the possibilites for microphysical representations and an exploration of the meaning of “adequate” in the previous sentence. The chapter may also be considered as a useful companion piece and as a stark contrast to Ch. 18, which considered the possibilities for “adequate” representations of microphysics within mass-flux convection parameterizations.

1

Introduction: Two methods for representing cloud microphysics

Limited area models (LAMs) and general circulation models (GCMs) do not resolve cloud scales. To describe convective heating/cooling, the traditional methods of convection parameterization are usually used (see Vol. 1, Part II). The goal of these parameterizations is to describe the overall effect of sub-grid cumulus convection on the large spatial scales explicitly repre275

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sented in the models. Convection parameterizations use simple representations of clouds in the form of plumes and typically consider microphysics only in a very crude manner for practical reasons (cf., Ch. 18). Within LAMs and GCMs, convectively induced heating/cooling and cloud-induced radiative effects are treated by two separate parameterization schemes, despite the fact that these effects are caused by the same clouds. The convection parameterizations do not allow for the determination of cloud coverage. The cloud coverage may instead be empirically related to the horizontally averaged relative humidity (cf., Ch. 25). The rapid and continuous increase in computing capabilities has made it feasible to use cloud-resolving grid spacing over continental scale model domains. As a result of the increases in model resolution, not only the large-scale processes but also the cloud ensembles and single clouds are beginning to be explicitly resolved. Explicit simulation of clouds allows us to handle consistently all aspects of processes associated with clouds, including thermodynamics, radiation, and microphysical processes leading to precipitation formation. Explicit simulation of clouds (with appropriate microphysics) in large-scale models would be likely to lead to dramatic improvements in large-scale and mesoscale atmospheric modelling. In cloud-resolving models, microphysics contributes to cloud evolution. From the very beginning, the development of microphysical schemes has been pursued in two separate directions, known as the “bulk microphysics” (or “bulk schemes”) and “spectral-bin microphysics” (SBM). Despite the fact that in both approaches the same microphysical processes are described and they both have similar outputs (cloud and rain water contents, cloud ice content, precipitation rates, etc.), the two approaches are quite different. The bulk schemes aim at representing the most general microphysical cloud properties using a semi-empirical description of particle size distributions. Accordingly, this approach is supposed to be computationally efficient. Historically, the first bulk scheme allowing reproduction of cloud microphysics and cloud evolution in numerical models was developed by Kessler (1969). The advantage of the computational efficiency of bulk schemes arises from the consideration of microphysical equations not for the particle size distributions (PSD) of different hydrometeor types (e.g., cloud droplets, raindrops, ice crystals, aggregates, graupel, hail) themselves, but for a number of moments of the PSDs, n(m), with m being the particle mass. The

Representation of microphysical processes in cloud-resolving models

k-th moment of the PSD is defined as:  ∞ mk n(m)dm, M (k) =

277

(1.1)

0

with k normally taken as an integer. The schemes that use only one moment (typically the mass densities of hydrometeors k = 1) are known as “one-moment” or “single-moment schemes”, while the schemes using two moments (typically number concentrations k = 0 and mass contents k = 1) are known as “two-moment schemes”. Less frequently, three-moment bulk schemes are used, in which the variables are number densities (concentrations), mass densities, and radar reflectivity (k = 2), assuming Rayleigh backscattering. The system of equations for the moments of the PSD is not closed, since the equations for k-th moment M (k) include terms with a higher-order moment M (k+1) (Seifert and Beheng, 2001). The closure problem is circumvented by representing the PSD n(m) in the form of specific mathematical functions which are completely determined by a few parameters only. A four-parameter gamma distribution is typically used as the master function. The first bulk scheme by Kessler (1969) describes only warm (no ice) microphysical processes. Since the schemes by Lin et al. (1983) and Rutledge and Hobbs (1984), all bulk formulations have begun to describe both warm and ice processes. A great number of bulk schemes have been used in different mesoscale models with spatial resolutions of several kilometres. Bulk schemes are also implemented in climate models in simplified forms (e.g., Boucher and Lohmann, 1995; Ghan et al., 2001; Lohmann and Feichter, 1997). Despite the significant variety of different components of bulk schemes, the basic assumption about the shape of the PSD remains unchanged: in any bulk scheme, even in those containing more than ten hydrometeor types, the PSDs of particles belonging to each hydrometeor type are approximated by exponential or gamma distributions, and much more rarely by lognormal functions. The spectral-bin microphysics (SBM) approach is also referred to as explicit microphysics, bin microphysics, bin-resolving, or size-resolving microphysics. The major goal of the SBM approach is to reproduce the cloud microphysical and precipitation processes as accurately as possible. The main concept of the bin microphysics approach is the calculation of the PSD by solving explicit microphysical equations. Therefore, no a priori information about the form of PSDs is required or assumed. Instead, the PSDs are defined and calculated, for example, on a finite difference mass

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grid containing from several tens to several hundred mass bins. It should be stressed that contemporary bin microphysics models substantially differ from the first-generation approaches (Clark, 1973; Kogan et al., 1984; Tzivion et al., 1987). The difference between the earlier and the later schemes is apparent by the degree of detail in the description of the microphysical processes. The increase in computing power makes it possible to describe explicitly the cloud microstructure using the accumulated knowledge of cloud physics processes. In parallel to the improvement of the representation of separate microphysical processes, a number of bin microphysics schemes were implemented and successfully used in cloud models and cloud-resolving mesoscale models for the simulation of a wide range of meteorological phenomena. Since the equations used in bin microphysics do not depend on the particular meteorological situations, the great advantage of bin microphysics is its universality, i.e., the same scheme can be used without any modifications for simulations of different atmospheric phenomena from Arctic stratiform clouds to tropical cyclones. This may not always be true for bulk schemes in which microphysical parameters may need to be chosen based on the cloud system being simulated. A third, modern approach for representing cloud microphysical processes are hybrid schemes that combine SBM and bulk representations. These approaches developed from the need to incorporate the more accurate representation of microphysical processes by bin schemes while retaining some of the computational efficiency of bulk formulations. Taking into account the necessity for very accurate descriptions of drop formation and growth and comparatively large uncertainties in processes related to ice formation, Onishi and Takahashi (2012) developed a scheme in which warm microphysical processes are described using the SBM approach, while processes related to ice formation and evolution are described using a bulk formulation. Another form of bulk formulation is referred to as the “bin-emulating” approach, and is another example of a hybrid scheme. This approach is used in the Colorado State University Regional Atmospheric Modeling System, RAMS (e.g., Carri´ o et al., 2007; Cotton et al., 2003; Igel et al., 2013; Meyers et al., 1997; Saleeby and Cotton, 2004, 2008; Seigel and van den Heever, 2012; Storer and van den Heever, 2013; van den Heever et al., 2006; van den Heever and Cotton, 2004, 2007; van den Heever et al., 2011). In this approach the rates of various microphysical processes for a wide range of atmospheric conditions are calculated offline using SBM within a Lagrangian

Representation of microphysical processes in cloud-resolving models

279

parcel model (Feingold and Heymsfield, 1992; Heymsfield and Sabin, 1989). The results of the SBM parcel model calculations for each of these processes are then included in lookup tables incorporated within the bulk microphysical module of the RAMS model, in which gamma distributions are used as the master functions. The bin results are therefore fully accessible to the model. In this way, the sophistication of bin schemes are exploited while at the same time ensuring relatively high computational efficiency. Researchers have high hopes that cloud-resolving models with advanced descriptions of microphysical processes can help with problems of climate change as well as local and global change of precipitation regimes. One of the main problems is the effects of anthropogenic emission of large masses of aerosols into the atmosphere. During the past two decades a great number of observational and numerical studies have been dedicated to quantifying the effects of aerosols on radiation, precipitation, and other atmospheric features on a wide range of scales from that of a single cloud to global scales (e.g., Fan et al., 2013; Khain, 2009; Levin and Cotton, 2008; Rosenfeld et al., 2008; Tao et al., 2007, 2012; van den Heever et al., 2011). Aerosols affect cloud microphysics through the influence on cloud particle number concentration and their sizes. Two-moment bulk schemes are able to take into account the effect of a decrease in droplet size with an increase in aerosol concentration. Many studies report advantages of two-moment bulk schemes as compared to one-moment schemes, as well as advantages of three-moment schemes as compared to two-moment schemes. One of the first applications of SBM models was the investigation of possible precipitation enhancement and hail suppression by the seeding of clouds with hygroscopic or ice-forming aerosols (e.g., Khvorostyanov et al., 1989; Reisin et al., 1996c; Yin et al., 2000). Thus, accounting for aerosol effects on cloud microphysics is a characteristic feature of bin microphysics, beginning with the design of the first bin schemes. The choice of which microphysical approach to use in a particular case is not easy. Advantages and disadvantages of these two microphysical approaches must first be carefully evaluated both for short- and long-term simulations. It is always important to identify possible shortcomings of each microphysical process for any given simulation, and seek possible improvements of a given algorithm. In this chapter, the following questions are addressed: • What is the shape of PSDs in real clouds and how are they approximated in the bulk schemes? (Sec. 2)

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• What are the main concepts and basic equations of SBM and bulk schemes? (Sec. 3) • How are the basic microphysical processes in SBM and bulk schemes represented in cloud-resolving models? (Sec. 4) • What conclusions can be derived from the comparison of the results obtained using SBM and bulk formulations? (Sec. 6) The chapter ends by discussing perspectives for the ongoing development of microphysical schemes.

2

2.1

Particle size distributions and their approximation using master functions PSDs in bulk schemes

The rates of all microphysical processes depend on particle size distributions (PSDs). Any microphysical scheme either calculates PSDs, or parameters of the PSDs, the shape of which is assumed a priori. Most bulk schemes use the gamma distribution as a master function for the approximation of PSDs of different hydrometeors in clouds. This function has the form (e.g., Seifert and Beheng, 2001): n(m) = N0 mν exp(−λmμ ),

(2.1)

where N0 is the intercept, ν is the shape parameter, λ is the slope or scale parameter, and μ is the dispersion parameter. The parameters ν and μ determine the shape of n(m) for very small m and for very large m respectively. Note that sometimes radius r or effective diameter D is used as the independent variable instead of mass m. In this case, the PSD can be written as (e.g., Ferrier, 1994; Milbrandt and McTaggart-Cowan, 2010): ′

′

n(D) = N0′ Dν exp(−λ′ Dμ ).

(2.2)

Note that throughout the present chapter, n is used for designating the particle (or droplet) number density, or PSD. It important to keep in mind that the actual functional forms for n are different depending on which dependent variables to take. In the case of a one-to-one relation between radius and mass, as for spherical particles, one can easily convert the respective number densities (PSDs) by invoking the relation: n(m)dm = n(D)dD.

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281

Further noting the relation m = πρD3 /6 (ρ is the bulk density of the hydrometeor), one obtains the following relations between parameters in Eqs. 2.1 and 2.2:  πρ ν+1  πρ μ N0′ = 3N0 , ν ′ = 3ν + 2, λ′ = λ , μ′ = 3μ. (2.3) 6 6 Other expressions for the PSD n(D) are also used, for instance (Saleeby and Cotton, 2004): n(D) =

Γ(ν ′′

′′ N0′′ Dν exp(−D/β). + 1)β ν ′′ +1

(2.4)

In the case of spherical particles one can easily convert the respective PSDs. However, in the case of non-spherical particles, such as pristine ice crystals, the use of a PSD as a function of mass is more convenient. The gamma function is equal to zero at m = 0 when ν > 0, and it has one maximum and tends to zero when m → ∞. Since observed drop-size distributions have as a rule two modes (the first corresponding to cloud droplets with radii below 20–25 μm and centred at r ≈ 10–15 μm, and the second to raindrops with radii sometimes as large as 3–4 mm: see Pruppacher and Klett (1997); see also the schematic in Fig. 23.8), size distributions containing both small cloud droplets and raindrops obviously do not obey a single gamma distribution. To avoid this problem, all bulk schemes without any exception distinguish between small cloud droplets with a distribution described by Eq. 2.1 and raindrops with a distribution approximated usually, although not always, by exponential functions. Thus, liquid drops are represented by two types of hydrometeors: cloud droplets and raindrops. Such a separation is physically grounded owing to the fact that the collision process leads to a cloud droplet mode that is separated from the raindrop mode with a well-pronounced minimum located between the modes within the radii range 25–60 μm. The drop radius separating cloud droplets and raindrops is typically accepted to be 30 μm. Figure 23.1 shows examples of general gamma distributions used for approximation of wide and narrow cloud droplet size distributions (DSDs) in bulk schemes. Morrison and Gettelman (2008) said that: “A major advantage of using gamma functions to represent subgrid variability of cloud water is that the grid-average microphysical process rates can be derived in a straightforward manner.” Following this statement, a major question that should be asked is whether gamma functions approximate observed PSDs well enough. For liquid drops, this question should be addressed to PSDs of cloud droplets (i.e., DSDs) and raindrops (RSD) separately.

282

Part IV: Unification and consistency

Fig. 23.1 Examples of gamma distributions chosen for approximation of (left) broad and (right) narrow DSDs. The values of the intercept parameter were chosen in such a way as to reproduce the concentrations of droplets indicated in the boxes. Solid, dashdotted, and dotted lines show DSDs typical of maritime, intermediate, and continental c American Geophysical Union 2010, from Fig. 11 of Noppel conditions, respectively.  et al. (2010b).

2.2

Observed PSDs

Mazin et al. (1989) demonstrated that DSDs averaged over large distances and over many clouds can be reasonably approximated by gamma distributions written in the form of Eq. 2.4, with droplet diameter serving as the independent variable. ν ′′ = 2 is known as the Khrigian–Mazin distribution. It is common practice to describe PSDs of precipitating particles assuming ν ′ = 0 in Eq. 2.2. The obtained PSD is known as the Marshall–Palmer distribution. Such distributions are also obtained by averaging over large distances of hundreds of kilometres. Note that in real clouds the rates of different microphysical processes depend not on the DSDs averaged over large distances and over many clouds, but on the local DSDs. Figure 23.2 shows examples of DSDs measured in situ in deep convective clouds. One can see that the DSDs averaged over long traverses resemble the gamma distributions illustrated in Fig. 23.1. At the same time, there are obvious differences between the shapes of observed local DSDs and of the gamma functions. Local DSDs vary significantly along the flight traverses; many DSDs are bi-modal. The existence of bimodal DSDs is regularly reported in observations (Korolev, 1994; Prabha et al., 2011; Warner, 1969). The DSDs averaged over a flight traverse are often substantially broader than local DSDs and uni-modal, as illustrated in Fig. 23.2a. In some cases the averaged DSDs are also bi-modal (Fig. 23.2b). Thus, the shape of observed DSDs depends on the spatial and time averaging.

Representation of microphysical processes in cloud-resolving models

283

Fig. 23.2 Plots of 1 Hz droplet size distributions measured along a flight pass length during the Cloud Aerosol Interaction and Precipitation Enhancement Experiment (CAIPEEX). (a) DSDs measured on 21 June 2009 at an altitude of 6.7 km; and, (b) DSDs measured on 22 June 2009 at an altitude of 7 km. Averaged DSDs are denoted by solid lines. The first five numbers denote time in seconds, and the last four numbers c American Meteorological Society (AMS) 2011, from Fig. 14 denote the altitude in m.  of Prabha et al. (2011). Reproduced by permission of the AMS.

Ice particle size distributions above about 50–100 μm as measured by two-dimensional imaging probes have been represented by exponential (Sekhon and Srivastava, 1970), bi-modal (Lawson et al., 2006; Mitchell et al., 1996; Yuter et al., 2006), normal (Delano¨e et al., 2005), by the sum of an exponential and gamma (Field et al., 2005), lognormal (Tian et al., 2010), and by gamma (Heymsfield et al., 2013) size distributions. In most studies, the PSDs were obtained by averaging of measurements along long traverses of several hundred kilometres and over different clouds. As follows from numerical simulations using SBM models, the local PSDs (at spatial scales of a few hundred metres) of aggregates, graupel, and hail can be gamma (exponential) type, bi-modal, or of a more complicated shape (e.g., Khain et al., 2011; Phillips et al., 2015). Ovchinnikov et al. (2014) highlighted the very important role of accurate representation of ice PSD in determining the partitioning of liquid and ice and the longevity of mixedphase clouds. 2.3

Fit of observations into a gamma distribution

To determine four parameters in Eqs. 2.1–2.4, four equations are to be solved. So, in principle, four-moment bulk schemes should be used. How-

284

Part IV: Unification and consistency

ever, currently only one- to three-moment schemes are used. The remaining parameters in Eqs. 2.1–2.4 are either fixed a priori or are determined using additional semi-empirical relationships (cf., Sec. 2.4). In several studies attempts were made to approximate measured DSDs by gamma, lognormal, and exponential master functions (e.g., Costa et al., 2000). Significant differences regarding droplet concentration and spectrum shape were observed among four different cloud types: maritime, coastal, continental, and urban types. The exponential distribution was shown to be unsuitable for most of the observed DSDs. An exponential distribution in m (ν = 0, μ = 1), or a size distribution with ν ′ = 2 and μ′ = 3 provided the best fit of observed DSDs, better than the fits using gamma and lognormal distributions (Costa et al., 2000). Liu et al. (1995) showed that the maximum entropy principle introduced in Vol. 1, Ch. 4, Sec. 6.1 leads to this distribution when the DSD is constrained by the total mass and the total particle number. Recall that this principle assumes that a given distribution is defined such that the maximum number of combinations (or partitioning) is retained among all the possible distributions. However, large differences in the width and shape parameters were found between maritime, coastal, continental, and urban clouds. Significant variability was also found in the values of the parameters from cloud to cloud. Figure 23.3 shows histograms characterizing the occurrence of the shape and scale parameters in all gamma fits. It was also found that for a relatively large number of cases (> 10%) no fit was possible. Here, no fit is attempted if a standard deviation error of less than 2% could not be achieved relative to the total number concentration. Costa et al. (2000) concluded that such variability imposes important limitations on bulk microphysical modelling. Analyses of observed data by Tampieri and Tomasi (1976) and later by Dooley (2008) showed that the parameters of gamma distributions used for approximation of DSDs cannot be considered as independent. This conclusion is illustrated in Fig. 23.4. Geoffroy et al. (2012) tried to relate the values of parameters of lognormal and gamma distributions with the values of DSD moments using DSDs measured in stratocumulus and small cumulus clouds. The possibility of using a third parameter as a prognostic variable in a bulk scheme has also been explored. However, dispersion in the values of parameters turned out to be very large as each microphysical process affects the values of the parameters differently. Geoffroy et al. (2012) were not able to isolate one process that could be considered as dominant. In conclusion, Geoffroy

Representation of microphysical processes in cloud-resolving models

285

Fig. 23.3 Occurrences of the (upper panel) shape parameter and (lower panel) scale diameter in all gamma fits. DSDs were measured in shallow cumuli over Northeast Brazil. (Reprinted from Fig. 5 of Costa et al. (2000), with permission from Elsevier.)

et al. (2012) wrote that: Considering the limitations inherent to the bulk approach, one might also conclude that the accuracy of the parameterizations [representations] proposed here is sufficient for most of the topics that can be addressed with a bulk scheme.

However, the question as to what topics can be addressed with such schemes remains unanswered. In other words, the authors appreciate that bulk schemes have limitations because of the extremely high variability of observed DSD shapes and the corresponding problem of the choice of the master functions and their parameters. There are several studies in which parameters of gamma distributions were determined for constructing Z–R relationships, allowing estimation of rainfall rates using the radar reflectivity or radar polarimetric parameters (e.g., Handwerker and Straub, 2011; Illingworth and Blackman, 2002). In these studies, a high variability of the parameters was mentioned as well.

286

Part IV: Unification and consistency

Fig. 23.4 Scatter plots of parameters of gamma distributions used to fit observed DSDs. N0 and λ acquired by 10 s gamma fits to observed DSDs. The solid red line represents the best fit line for the variation of N0 in respect of λ obtained by McFarquhar et al. (2007) (from Dooley, 2008).

Fig. 23.5 Dependence of graupel intercept parameter on supercooled liquid-water content and graupel content in the Thompson bulk scheme (Thompson, 2014, personal communication).

Representation of microphysical processes in cloud-resolving models

287

The relationship between these parameters turned out to be dependent on whether standard or generalized gamma functions were used. 2.4

Approaches to improving the PSD approximation by gamma functions

Three main approaches have been suggested to improve the PSD approximation by gamma functions in bulk schemes. The first approach widely applied in two-moment bulk schemes consists of the implementation of dependencies between parameters of gamma functions similar to those shown in Fig. 23.4. Such relations between the size distribution parameters were used in a bulk microphysics scheme by Thompson et al. (2004) who derived them from bin-microphysics simulations. For graupel with a gamma size distribution, the scheme by Thompson et al. (2004) provides a relation for a one-moment bulk scheme between the intercept parameter N0 and the slope parameter λ. They fix the shape parameter with a constant value. For rain in the same scheme, Thompson et al. (2004) used a relation between N0 and rain mass mixing ratio. In the recent version of the Thompson et al. (2004) bulk formulation, the graupel intercept parameter depends on supercooled liquid-water content and graupel content as shown in Fig. 23.5. For snow, Thompson et al. (2004) used a relation between N0 and temperature. Formenton et al. (2013b,a) applied a similar idea by deriving an iterative numerical solution for shape, slope, and intercept parameters of a gamma size distribution in a one-moment treatment of snow. They are all diagnosed from a given snow-mass mixing ratio at every grid point for each timestep. Thompson et al. (2004) used an observed relation between the parameters of the gamma distributions, obtained by Heymsfield et al.’s (2002) analysis of aircraft data from various field campaigns. This method was also applied by Kudzotsa (2014). The use of relationships between parameters of the gamma distribution still keeps the PSDs uni-modal. Note that the shapes of DSDs contain very important information concerning microphysical processes. For instance, the bi-modality and multi-modality of DSDs may reflect the existence of in-cloud nucleation, when new small droplets arise within a wide range of heights above the cloud base of convective clouds (Khain et al., 2000; Pinsky and Khain, 2002; Segal et al., 2003). Some observations (Khain et al., 2012; Prabha et al., 2011, 2012) indicate that such in-cloud nucleation may take place continuously over a significant range of altitudes. These small

288

Part IV: Unification and consistency

droplets can be a source of a large number of ice crystals in thunderstorm anvils around the level of homogeneous freezing, can intensify hail and graupel riming, and foster lightning in clouds (Heymsfield, 2009; Khain et al., 2012; Rosenfeld and Woodley, 2000). Brenguier and Grabowski (1993) reported the bi-modality of DSDs caused by droplet activation at cloud edges of shallow clouds through entrainment. Khain et al. (2013) found cloud condensation nuclei (CCN) nucleation at the edges of deep convective clouds. Bi-modal DSDs in warm stratocumulus clouds were also observed by Korolev (1994, 1995) and have been simulated (e.g., Magaritz et al., 2009, 2010). So, representations of DSDs in the form of a single-mode gamma distribution may substantially distort the cloud microphysical structure and the associated microphysical processes. The second approach to improve the representation of the PSD consists of implementing additional modes into the master functions. This approach, still using gamma functions, is applied in RAMS where a third liquid water, so-called “drizzle”, mode has been implemented (Saleeby and Cotton, 2004). The drizzle mode, which includes droplets with diameters of 50–100 μm, is located between the cloud (diameters of 2–50 μm) and raindrop (diameters greater than 100 μm) modes. The implementation of the drizzle mode has been found to slow down the process of rain formation, and has led to significant improvements in simulated precipitation rates when compared with observations. The third (drizzle) mode was also introduced by Sant et al. (2013) for the simulation of drizzle formation in a warm stratocumulus cloud. Thompson et al. (2008) approximated the PSD of snow by a sum of two gamma functions. This approach provides more degrees of freedom for the construction of the PSD. The third approach consists of three-moment bulk schemes, thereby allowing for the calculation of three parameters in Eqs. 2.1–2.4. This approach allowed Loftus et al. (2014a,b,c) and Milbrandt and Yau (2005a) to simulate successfully big hail with a broad size distribution, which was not achievable in simulations when using two-moment schemes. Summarizing this section, it can be concluded that the use of gamma distributions to represent PSDs appears too simple to adequately describe the variability and some specific features of local PSDs in clouds. At the same time, answering the question of to what extent the errors in the reproduction of PSD shape affect the model results for say, precipitation, temperature, radiation, etc. is not straightforward, because the predicted

Representation of microphysical processes in cloud-resolving models

289

variables are typically affected by many factors, with the microphysics being only one of them. Moreover, some of the integral quantities (e.g., accumulated precipitation) may be largely sensitive to the mass contents and concentration of hydrometeors, while the exact shape of the PSD may be of secondary importance. Some evaluations of these issues will be presented in Sec. 6.

3 3.1

Basic equations in SBM and bulk schemes SBM: Two approaches to represent size distributions

The kinetic equations for the PSD of the k-th hydrometeor type nk (m) that are used in SBM models can be written in the form:  ∂ ∂ ∂ ∂ ∂ ρnk + ρunk + ρvnk + ρ(w − vt (m))nk = nk ∂t ∂x ∂y ∂z ∂t nucl +



∂ nk ∂t



c/e

+



∂ nk ∂t



+

d/s

+



∂ nk ∂t



f/m

∂ ∂ Kθ ρnk , ∂xj ∂xj

+



∂ nk ∂t



col

+ ···

(3.1)

where u, v, and w are components of wind speed, vt is the fall velocity that depends on the mass and type of the hydrometeor (including the characteristics of particle shape), and ρ is the air density. Terms on the right-hand side of Eq. 3.1 determine the rates of different microphysical processes such as nucleation (nucl), condensation/evaporation (c/e), deposition/sublimation (d/s), freezing/melting (f/m), collisions (col), etc. The last term determines the change of the PSD due to turbulent mixing. The descriptions of the rates of microphysical processes are discussed below. The number of equations for PSDs is equal to the number of the bins multiplied by the number of hydrometeor types. There are two main schemes used to compute the time evolution of the PSD by Eq. 3.1. In the following discussions they will be referred to as “bin microphysics” (BM) and the microphysical method of moments (MMM), respectively. In many publications, BM is often referred to as SBM. However, the present chapter makes the distinction between BM and SBM clear whenever relevant.

290

Part IV: Unification and consistency

3.1.1

Bin microphysics (BM)

The development of this method goes back to the classical studies by Berry and Reinhardt (1974a,b,c). The basic ideas of this method were further developed and used in the cloud models designed by Benmoshe et al. (2012); Hall (1980); Khain and Sednev (1996); Khain et al. (2000, 2004b, 2005, 2008a, 2012, 2013); Khairoutdinov and Kogan (2000); Kogan (1991); Ovtchinnikov and Kogan (2000). The PSD representation proposed by Berry and Reinhardt (1974a,b,c) is used in different versions in the Hebrew University Cloud Model (HUCM), as well as in mesoscale cloud-resolving models such as WRF, SAM, RAMS (e.g., Fan et al., 2009, 2012a,b, 2013; Iguchi et al., 2008, 2012b,a, 2014; Khain and Lynn, 2009; Khain et al., 2009, 2010; Ovchinnikov et al., 2011, 2014). In this method, the PSD n(m) is defined on a logarithmic equidistance mass grid containing several tens of bins (Fig. 23.6).

Fig. 23.6 A mass grid used for the representation of PSDs in the bin microphysical approach.

The important parameter of the grid is the ratio mi+1 /mi = α = constant. In most cloud and cloud-resolving models α = 2, but other values are also used depending on the problem to be tackled and the number of bins being used. The advantage of such a grid is that the resolution is highest for small particles and gradually decreases with the increase in particle mass over a wide range of particle masses. Particles belonging to a particular bin are characterized by the mass and type of hydrometeor. All other parameters such as bulk density, equivalent radius (or diameter), shape parameters, fall velocity, etc. are expressed via their mass using empirical relationships (e.g., Pruppacher and Klett, 1997). In the most advanced schemes, ice particles are also characterized by rimed- or liquid-water fraction (e.g., Benmoshe et al., 2012; Phillips et al., 2014, 2015). The concentration of particles N and particle mass content M can be calculated by appropriate integration of the PSD:  mmax  mmax mn(m)dm. (3.2) n(m)dm ; M = N= mmin

mmin

Following Berry and Reinhardt (1974a), it is convenient to define an-

Representation of microphysical processes in cloud-resolving models

other PSD g˜(ln r), obeying the normalization condition:  ln rmax g˜(ln r)d ln r. M=

291

(3.3)

ln rmin

Here, r is the radius of the “equivalent” sphere with mass m. The functions n(m) and g˜(ln r) are related by g˜(ln r) = 3m2 n(m). The finite-difference equivalent of the PSD is expressed as (Berry and Reinhardt, 1974a): (3.4) Gi ≡ g˜(ln ri ) = 3m2i f (mi ), where i is the index for a bin. The finite-difference representation of Eq. 3.3 on the logarithmically equidistant mass grid means that Δ ln ri ≡ ln ri+1 − ln ri = ln ri+1 /ri = ln α1/3 = (1/3) ln α = constant. So, particle concentrations and mass contents in the i-th bin can be written in a simple way as Ni = (1/3) ln α(Gi /mi ) and Mi = (1/3) ln αGi . The total concentration and total mass content are determined as a sum of the corresponding values over all bins:  Gi  1 1 Gi . (3.5a,b) ; M = ln α N = ln α 3 mi 3 i i Such a representation of the PSD on the discrete mass grid is straightforward for numerical simulations. Equations 3.5a,b uniquely define the number concentration and the mass in the bins. Note that size distributions for aerosol particles are also mostly represented on a logarithmically equidistant mass (radius) grid covering a wide range of aerosol sizes (or masses). Using Eq. 3.1, the kinetic equation for the PSD determined on the mass grid can be represented as a set of equations for the values of the PSD in the i-th bin:  ∂ ∂ ∂ ∂ ∂ ρni,k + ρuni,k + ρvni,k + ρ(w − vt (m))ni,k = ni,k ∂t ∂x ∂y ∂z ∂t nucl     ∂ ∂ ∂ ∂ ni,k ni,k ni,k ni,k + + + + + ··· ∂t ∂t ∂t ∂t c/e d/s f/m col ∂ ∂ + Kθ ρni,k . (3.6) ∂xj ∂xj Some research problems of atmospheric chemistry and atmospheric electricity require a more comprehensive representation of the PSD according to which particles having the same size (i.e., belonging to the same mass bin) are further categorized by, for example, solute concentration or charge (Beheng and Herbert, 1986; Ramkrishna, 2000). For this purpose, higher dimensional PSDs are used, so that the particles having the same mass are distributed according to the secondary parameters of interest (e.g., Bott, 2000; Khain et al., 2004a).

292

3.1.2

Part IV: Unification and consistency

The microphysical method of moments (MMM)

The development of the MMM is rooted in works by Enukashvily (1980); Young (1975). In the studies by Feingold et al. (1988); Reisin et al. (1996a,b,c); Tzivion et al. (1987, 1989) the method was further developed and both axisymmetric and slab-symmetric models using the MMM were created. In this method, the axis of mass is separated into categories (Fig. 23.7). The boundaries of the categories form a logarithmic-equidistant grid. In each category, the PSD is characterized by several moments. In the studies by Tzivion et al. (1987, 1989) two moments are used: the mass content and the number concentration. Thus, the equations for each category are formulated not for the PSD, as in the point BM, but for its moments, which are defined for each category as:  mi+1 (v) mv n(m)dm. (3.7) Ri = mi

(0)

Note that the zero-order moment Ri is particle concentration in the i-th (1) category, the first-order moment Ri is mass content in the i-th category (2) and the second-order moment Ri is radar reflectivity formed by particles belonging to the i-th category.

Fig. 23.7

Separation of the mass axis into categories in the MMM.

To calculate these integrals, two steps are required. First, it is necessary to make some assumptions about the behaviour of the size distribution within each category. In the MMM, the size distributions are assumed to be linear functions. Second, to write equations for moments in each category it is necessary to perform averaging of the basic equation of Eq. 3.1 over each category. This averaging creates some mathematical problems, such as the problem of the equation closure. As a closure condition, Tzivion et al. (1987) used a certain relationship between three consecutive moments. To calculate collision integrals, the collision kernel in the form of Long (1974) is used. The number of equations in MMM is equal to the product of the number of categories and the number of moments used for each category. Tzivion et al. (1987) and many successive studies that used the MMM (e.g., Saleeby

293

Representation of microphysical processes in cloud-resolving models

and Cotton, 2004; Teller and Levin, 2008; Teller et al., 2012; Xue et al., 2010, 2012; Yin et al., 2000) solve a system of equations for the zeroth and the first moments in each category. The form of the equations is similar to that of Eqs. 3.5a,b. Tzivion et al. (1987) showed that using 36 categories gives accurate results, comparable to those obtained by using 72 or even 144 categories. Furthermore, Tzivion et al. (2001) applied a new approximation of the distribution function used in the MMM and implemented it into the numerical algorithm used to solve the stochastic kinetic collection equation. This method provides an accurate and efficient numerical solution of the kinetic collection equation, appropriate for use in dynamical cloud models. The results show a significant improvement in the accuracy of the calculations with the increase in the number of categories. Compared to the original spectral multi-moment method (SMMM) the computation time using this method can be reduced by more than an order of magnitude while maintaining similar accuracy. The MMM scheme originally developed in Tel Aviv University is used in one form or another at NCAR (WRF), in the RAMS model of Colorado State University, at NOAA and at other universities in the USA, China, Vietnam, Brazil, and elsewhere (e.g., Levin et al., 2003; Muhlbauer et al., 2010; Teller and Levin, 2008; Teller et al., 2012; Xue et al., 2010, 2012). The scheme has been used to study squall lines, orographic clouds, deep tropical convection, Mediterranean convective clouds, marine stratocumulus clouds, and warm cumulus using large-eddy simulation (LES). In its current state, the scheme is highly versatile and can be used for almost any WRF application from simple two-dimensional idealized cases to complicated three-dimensional real cases. 3.2

Equation system for PSD moments in bulk schemes

Multiplying Eq. 3.1 by mk and integrating over all particle masses one can obtain, after some simplifications, the equations  ∞ for the k-th PSD moment (k) of the hydrometeor of the i-th type Mi = 0 mk ni (m)dm, which are:  ∂ (k) ∂ ∂ ∂ ∂ (k) (k) (k) (k) (k) ρMi + ρuMi + ρvMi + ρ(w − v t,i )Mi = M ∂t ∂x ∂y ∂z ∂t i nucl +



∂ (k) M ∂t i



c/e

+



∂ (k) M ∂t i



d/s

+



∂ (k) M ∂t i



f/m

+



∂ (k) M ∂t i



col

+ ···

294

Part IV: Unification and consistency

+

∂ ∂ (k) Kθ M . ∂xj ∂xj i

Here, (k) v t,i

1 = (k) M



∞

mk n(m)vt,i (m)dm

(3.8)

(3.9)

0

is the averaged fall velocity of the k-th moment of the PSD. The physical meanings of the terms in Eq. 3.8 are similar to the corresponding terms in Eq. 3.1, but all microphysical process rates are written for the moments of the PSD. As mentioned above, the number of equations of the form of Eq. 3.8 is equal to the number of moments used in a particular bulk scheme multiplied by the number of hydrometeor types. Thus, the number of equations in bulk schemes is typically an order of magnitude lower than in the SBM method. Note that current bulk schemes are much more sophisticated than those developed a decade ago. New schemes contain more types of hydrometeor, and more moments are used to describe the PSDs of the hydrometeors. See Tables 3.2–3.2 for an overview. 4

Representation of microphysical processes in SBM and bulk schemes

In this section, the representation of the main microphysical processes in SBM and in bulk schemes will be described. Note that the rates of all microphysical processes strongly depend on particle size. The use of PSD moments (i.e., quantities integrated over the entire size range) leads to some loss of the sensitivity of the rates to the particle size. It will be seen that substantial simplifications related to the replacement of equations for size-dependent PSDs by the equations for one or two moments of the PSD leads to significant problems in the representation of the rates of the main microphysical processes. 4.1

Droplet nucleation

It is known that cloud condensational nuclei are activated and turn into cloud droplets if their sizes exceed a critical value which is a function of the supersaturation in respect of water. With higher supersaturations, smaller CCNs can be activated. The process of drop nucleation typically occurs in the vicinity of the cloud base where supersaturation reaches its local

Table 23.1

The main characteristics of widely used bulk schemes.

First warm rain bulk formulation. One-moment (1M) scheme. Hail is treated as a high density hydrometeor. 1M scheme. First bin formulation used in RAMS. Processes of ice-multiplication, melting, and shedding are included. 1M NASA Goddard microphysics scheme, in many details similar to that of Lin et al. (1983). A new procedure of ice-water saturation adjustment is included. 1M, snow includes separate snow crystals and aggregates; many algorithms are taken similar to those in Lin et al. (1983); Cotton et al. (1986). Improved approach for calculation of relative fall velocity between different hydrometeors. All particles are spherical. RAMS, as Cotton et al. (1986), 1M. Deriving analytical expressions for integrals of collisions, development of lookup tables. Two-moment (2M) scheme. Number concentrations are predicted for ice/precipitation species. RAMS, 1M. Ice crystals are separated into two hydrometeor types: small pristine. crystals and large pristine crystals which are attributed to snow. Hail and graupel are of different density. RAMS, 2M. Number concentrations are predicted for ice/precipitation species. NCAR/Penn State Mesoscale Model Version 5 (MM5), 2M. 2M, warm microphysics. Implementation of some analytical expressions for rates. 2M scheme of warm processes, analytical development of formulae for autoconversion, accretion, and self-collection. RAMS 2001; 2M. Development of studies by Verlinde et al. (1990) and Walko et al. (1995). Wide use of lookup tables; development of bin-emulating bulk scheme; improvement of process of ice nucleation. Ice is categorized into pristine ice, large pristine ice (snow) that can be rimed, aggregates, graupel, and high-density frozen drops/hail. WRF, single-moment six-class scheme WSM6. The formulation is based on Lin et al. (1983) and Rutledge and Hobbs (1984). Improvement of description of microphysical processes, including ice nucleation as a function of temperature. 3M scheme; radar reflectivity is the third moment. Implementation of dependencies between parameters of gamma distributions, simulation of hail. WRF, a 1M scheme with empirical relation between parameters of gamma distributions, as well between these parameters and environmental conditions. Further development of the Thompson et al. (2008) scheme, with incorporation of aerosols explicitly in a simple and cost-effective manner. The scheme nucleates water and ice from their dominant respective nuclei and tracks and predicts the number of available aerosols.

Tao et al. (1989) Murakami (1990)

Verlinde et al. (1990) Ferrier (1994) Walko et al. (1995)

Meyers et al. (1997) Reisner et al. (1998) Cohard and Pinty (2000) Seifert and Beheng (2001) Cotton et al. (2003)

Hong and Lim (2006)

Milbrandt and Yau (2005; 2006) Thompson et al. (2008) Thompson and Eidhammer (2014)

295

Comments

Kessler (1969) Lin et al. (1983) Rutledge and Hobbs (1984) Cotton et al. (1986)

Representation of microphysical processes in cloud-resolving models

Authors

296

Table 23.2

The main characteristics of widely used bulk schemes.

Comments

Morrison et al. (2005a, 2009b)

2M scheme predicting the mixing ratios and number concentrations of droplets, cloud ice, rain, and snow. Improvement of calculation of supersaturation. Concentration of CCN is given. WRF double-moment six-class scheme, WDM6. This scheme is double-moment for droplets and rain. Concentrations of graupel and snow are diagnosed in the WDM6 as in 1M. A prognostic treatment of CCN is introduced. Stony Brook University Lin scheme. This is a five-class scheme with riming intensity predicted to account for mixed-phase processes (in a single snow array). RAMS 2M, bin-emulating scheme. A number of microphysical processes are determined from a bin scheme that is run previously offline and made accessible to the bulk scheme through the use of lookup tables. A new hydrometeor, drizzle, has been included to describe DSD better. Fully interactive with prognostic CCN and IN aerosol schemes. 2M scheme. The scheme is based on calibration of rates of collisions against a bin scheme. Time dependence of collision rate is taken into account. Same as Saleeby and Cotton (2004) except that significant modifications were made to the prognostic aerosol schemes, including the incorporation of the DeMott et al. (2010) IN scheme, multiple aerosol types, and a variety of aerosol processes. RAMS, 3MHAIL. Same as Saleeby and van den Heever (2013) except that a triplemoment scheme has been included for hail. Bin-emulating bulk scheme (not publically available). Emulated bin microphysics for coagulation for snow, rain, and graupel/hail; predicted in-cloud supersaturation and bulk treatment of six aerosol species

Lim and Hong (2010)

Lin and Colle (2011) Saleeby and Cotton (2004)

Seifert and Beheng (2006a,b) Saleeby and van den Heever (2013) Loftus et al. (2014a) Phillips et al. (2007, 2008, 2009, 2013) Formenton et al. (2013), Kudzotsa (2014)

Part IV: Unification and consistency

Authors

Table 23.3 Hydrometeor types included in the widely used bulk schemes of Tables 23.1 and 23.2. An X is included for each moment of each type. “aggr” indicates aggregates and “graup” indicates graupel. Type of hydrometeors cloud Kessler (1969) Lin et al. (1983) Rutledge and Hobbs (1984) Cotton et al. (1986) Tao et al. (1989) Murakami (1990) Verlinde et al. (1990) Ferrier (1994) Walko et al. (1995) Meyers et al. (1997) Reisner et al. (1998) Cohard and Pinty (2000) Seifert and Beheng (2001) Cotton et al. (2003) Hong and Lim (2006) Milbrandt and Yau (2005b, 2006) Thompson et al. (2008) and Thompson and Eidhammer (2014) Morrison et al. (2005a, 2009b) Lim and Hong (2010) Lin and Colle (2011) Saleeby and Cotton (2004) Seifert and Beheng (2006a,b) Saleeby and van den Heever (2013) Loftus et al. (2014a) Phillips et al. (2007b, 2008, 2009, 2013) and Formenton et al. (2013b,a), Kudzotsa (2014) ice

rain

ice

X X X X X X X X X XX XX XX XX XX X XXX

XX X XXX

X XX XX X XX XX XX XX

X XX XX X XX XX XX XX

X XX X X XX XX XX XX

X

XX

XX

XX XX XX

X X X X X X XX X XX XX

aggr.

snow X X

X X X

XX

XX X Xa XX XX XX XX

hail X

X X

XX X XX XX

X X XX X XX XX

XX X XXX

XX X XXX

X XX X

X

XX XX XX XX

XX XX XX XX

X

X

X X XX

graup

X X X XX X XX

XX XXX

XX XX XXX

297

a Precipitating

drizzle

X X X X X X X X X X X XX XX X X XXX

Representation of microphysical processes in cloud-resolving models

Authors

298

Part IV: Unification and consistency

maximum. The concentration of nucleated droplets depends on the vertical velocity, the size distribution of CCN particles and, to a lesser extent, on the chemical composition of CCN (Dusek et al., 2006; Khain et al., 2000). Since the supersaturation maximum near the cloud base is not resolved in many, although not all, cloud and cloud-resolving models, it must be parameterized. Various formulae and lookup tables for the supersaturation maximum value and for the concentration of nucleated droplets have been proposed (Ghan, 2011; Pinsky et al., 2012, 2013; Segal and Khain, 2006). In order to apply these formulations, CCN size distributions (e.g., maritime, continental, background, urban) must be specified first (Ghan, 2011). All in situ observations indicate the existence of the smallest droplets with diameters smaller than 10 μm within clouds at any height above the cloud base. These droplets form a new, second mode of DSDs due to incloud nucleation. The second mode is sometimes very pronounced, but sometimes is smoothed (cf., Sec. 2). According to Khain et al. (2012); Pinsky and Khain (2002); Prabha et al. (2011), the in-cloud nucleation and formation of the second mode is caused by a significant increase in the supersaturation in zones where the vertical velocity rapidly increases with height. The supersaturation value also increases in areas of effective collection of cloud droplets by raindrops and with sequential fallout of raindrops. Unloading in maritime clouds is one of the reasons for an increase in the vertical velocity in these clouds. Far above cloud base, in the interior of deep clouds, as soon as the supersaturation in the updraughts exceeds the supersaturation maximum near cloud base, nucleation of small CCN takes place. The existence of such small CCN in the atmosphere and their role is discussed in detail in by Khain et al. (2012). 4.1.1

Spectral-bin microphysics

A description of cloud droplet nucleation depends on the particular bin microphysics scheme. In the case of a large number of mass bins (e.g., several hundred to two thousand; Pinsky et al., 2008b), the exact equation for diffusion growth is solved for each bin, and wet aerosol particles that exceed critical values begin growing as cloud droplets automatically without applying any specific procedure of droplet nucleation. Smaller wet aerosols remain as haze particles, the size of which is determined by environmental humidity. In order to simulate the growth of wet aerosol particles, diffusion growth is calculated using timesteps as small as 0.01 s. In schemes where the DSD and size distribution of aerosol particles are

Representation of microphysical processes in cloud-resolving models

299

determined on a grid containing several tens of bins or categories, and the timestep is several seconds, the growth of haze particles cannot be explicitly calculated. Instead, special procedures for droplet nucleation are applied. In the main, these procedures are similar in BM schemes (e.g., Khain et al., 2004b; Kogan, 1991), and in MMM (e.g., Yin et al., 2000). The procedures require knowledge of the size distribution of dry CCN, which is treated as a separate hydrometeor type, and the supersaturation. Using the Kohler theory, the critical CCN radius is calculated for a given chemical composition of CCN. CCN with radii exceeding the critical value are activated and converted to small cloud droplets. Corresponding bins in the CCN size distribution then become empty or partly filled. The size of nucleated droplets is determined by either Kohler theory (for the smallest CCN) or by using the results of simulations from a parcel model, which are typically considered as the benchmark of accuracy. The modern version of HUCM uses an analytical approach to calculate the superaturation maximum near cloud base (Pinsky et al., 2012, 2013). Since the SBM schemes calculate the CCN size distributions and supersaturation at each grid point, they describe processes of in-cloud nucleation leading to the formation of small droplets at significant heights above the cloud base (Khain et al., 2012; Pinsky and Khain, 2002). These small droplets form new modes in the DSD leading to the formation of bi-modal and, sometimes, multi-modal DSDs similar to those presented in Fig. 23.2. These smallest droplets have relatively little effect on the formation of the first raindrops, but may affect ice microphysics, cloud dynamics, lightning, etc. 4.1.2

Bulk schemes

In most of the two- and three-moment bulk schemes, the concentration of nucleated droplets is calculated using different expressions for CCN activity spectra which represent different modifications of the Twomey formula (Ghan, 2011). These formulae describe a dependence of the number concentration of activated CCN on supersaturation. The supersaturation near the cloud base is calculated either directly at the model grid points or using expressions for the supersaturation maxima (Ghan, 2011). Most bulk schemes do not include the CCN budget: i.e., the concentration of CCN does not change despite the CCN nucleation. The lack of a CCN budget may lead to significant errors in the representation of cloud microphysics and precipitation (cf., Sec. 6).

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Part IV: Unification and consistency

In bulk schemes, the process of in-cloud nucleation is typically not considered in detail. In those bulk schemes that do take in-cloud nucleation explicitly into account, the new droplets nucleated at a certain altitude above the cloud base must obey the same gamma distribution as other cloud droplets. This means that the new droplets are assumed to be immediately distributed over a wide range of sizes. Such a procedure may lead to the erroneous appearance or disappearance of large cloud droplets in the DSD. The droplets of this size are able to absorb a substantial mass of water vapour thereby decreasing supersaturation and hindering the growth of the effective radius, and so preventing the formation of raindrops (Slawinska et al., 2012). In real clouds, the effective radius typically grows with height, which allows one to relate raindrop formation to the critical value of effective radius (e.g., Freud et al., 2008; Freud and Rosenfeld, 2012). In the bin-emulating double-moment bulk scheme used in RAMS, the treatment and activation of CCN are very similar to those described in the previous subsection on bin droplet nucleation. As described above, a Lagrangian parcel model is initially used to perform offline bin calculations, and the critical CCN radius and the size of the nucleated drops are determined using Kohler theory (Saleeby and Cotton, 2004). Activation of CCN within RAMS then occurs as a function of the model-predicted vertical velocity, temperature, aerosol number concentration, aerosol median radius, and aerosol solubility fraction via the use of lookup tables (Saleeby and van den Heever, 2013). Aerosol sources (such as aerosol regeneration following hydrometeor evaporation) and sinks (such as that following nucleation) are represented, and CCN budgets are carefully maintained. Representation of droplet nucleation in this manner allows for a better representation of secondary nucleation throughout the vertical depth of the cloud, and tends to offset some of the problems just described for bulk schemes. 4.2

Droplet growth by diffusion

After nucleation the newly formed droplets continue growing through vapour diffusion. This is one of the key microphysical processes. The diffusion growth of droplets is a comparatively slow process compared with the process of collisions. Latent heat released during diffusional growth is an important energy source for cloud updraughts. The rate of particle growth is determined by the supersaturation, which in turn depends on the PSDs of all hydrometeors and on the vertical velocity. The rate of diffusional droplet growth determines the altitude and time of raindrop formation and

Representation of microphysical processes in cloud-resolving models

301

the comparative contribution of warm rain and ice processes during the cloud lifetime. 4.2.1

Spectral-bin microphysics

Droplet growth by diffusion as well as deposition growth of ice particles is calculated by solving the classical equations of diffusional growth (Pruppacher and Klett, 1997, see also Vol. 1, Ch. 13, Sec. 10). In advanced SBM models the equation for diffusional growth is solved together with the equations for supersaturation, so that the value of supersaturation changes during diffusional growth. Some of the SBM-based models include the representation of aerosol recycling in the case of droplet evaporation (e.g., Lebo and Seinfeld, 2011; Magaritz et al., 2009, 2010; Teller et al., 2012; Xue et al., 2010, 2012), as does the bin-emulating bulk scheme of RAMS (Saleeby and van den Heever, 2013). It is highly complicated to account for the aerosol mass returned from an evaporated droplet, especially due to the modification of the aerosol chemistry within the droplets arising from chemical reactions (Wurzler et al., 1995). Formulations were therefore proposed to connect the remaining aerosol mass to the size of evaporated droplets. The most serious numerical problem arising in the SBM models containing a comparatively low number of bins is the problem of artificial numerical DSD broadening while solving the equation of diffusional drop growth on a regular, unmovable mass grid. This problem resembles the spatial dispersion of an advection solution by diffusive finite-difference schemes. At each diffusional-growth substep, it is necessary to interpolate the DSD to the regular, fixed mass grid. During this interpolation (remapping), some fraction of drops are transferred to bins corresponding to masses larger than would follow from the diffusion growth equation, which may eventually artificially accelerate raindrop formation. Schemes with high DSD broadening (e.g., Kovetz and Olund, 1969) lead to rapid rain formation at low levels, even in clouds developing under high aerosol concentration conditions. There were several attempts to decrease such broadening in SBM schemes. For instance, Khain et al. (2008a) proposed a method of remapping in which three DSD moments are conserved. This method substantially decreases the artificial broadening and allows for the reproduction of the height dependence of the DSD width. Another efficient method to avoid artificial DSD broadening is to use a movable mass grid on which to define the DSD. In this method, the masses

302

Part IV: Unification and consistency

corresponding to the centres of the bins increase (in the case of condensation) or decrease (in the case of evaporation) according to the equation of diffusional growth. Such a method does not require remapping which makes the description of the condensation process very accurate. Movable mass grids are used largely in Lagrangian parcel models or trajectory ensemble models (e.g., Magaritz et al., 2010; Pinsky and Khain, 2002; Pinsky et al., 2008b). 4.2.2

Bulk formulation

In most bulk schemes the equation for diffusional growth/evaporation of drops is not solved explicitly. Instead, the hypothesis of saturation adjustment is usually applied according to which supersaturation over water is forced to zero at the end of each timestep (Straka, 2009; Tao et al., 1989). Knowing the decrease in supersaturation one can calculate the watervapour mass condensed during each timestep (i.e., the increase in droplet mass). New parameters of the gamma distributions are then calculated using the droplet concentration obtained as a result of nucleation and the new droplet mass content. Note that the saturation adjustment assumption is valid under some conditions, such as when the vertical velocity is low, or when the diffusional growth of droplets rapidly decreases the supersaturation to very small values. However, in convective updraughts, supersaturation never falls to zero, and in contrast, may reach several percent as a result of an increase in the vertical velocity and a decrease in drop concentration by collisions (Khain et al., 2012; Prabha et al., 2011). Application of saturation adjustment under these conditions may lead to an overestimation of the condensate mass and latent heat release at each timestep, and hence to the overestimation of vertical velocity and the rate of convective precipitation. One of the ways to improve the bulk schemes is to avoid the procedure of the saturation adjustment. Some steps in this direction were described by Straka (2009). There are some exceptions to this treatment of supersaturation in bulk schemes. In the RAMS model for example, the diffusional growth equation is solved, and hence supersaturation is not forced to zero at the end of each timestep, thereby leading to better representations of condensate production and latent heating, particularly within deep convection. Phillips et al. (2007b) predicted the in-cloud supersaturation and diffusional growth of all hydrometeors in a two-moment bulk scheme, treating in-cloud droplet activation. The procedure involves a linearized scheme

Representation of microphysical processes in cloud-resolving models

303

and some sub-cycling of the time integration for diffusional growth. 4.3 4.3.1

Collisions between cloud particles Drop collisions

Drop collisions are one of the most important processes in raindrop formation. 4.3.1.1 Spectral-bin microphysics The evolution of the DSD n(m), caused by collisions of liquid drops while neglecting drop break-up, is described by the stochastic collection equation (SCE: Pruppacher and Klett, 1997; Ramkrishna, 2000):  m/2 dn(m, t) = n(m′ )n(m − m′ )K(m − m′ , m′ )dm′ dt 0    −

 

0

gain

∞

n(m)n(m′ )K(m, m′ )dm′ .  

(4.1)

loss

The first integral on the right-hand side of Eq. 4.1 is known as the gain integral, and describes the rate of generation of drops with mass m by coalescence of drops with masses m′ and m − m′ . The second integral is the loss integral, which describes the decrease in the concentration of drops with mass m. The gravitational collection kernel K(m, m′ ) is: Kg (m1 , m2 ) =

π (D1 + D2 )2 E(m1 , m2 )|Vt1 − Vt2 |, 4

(4.2)

where E(m1 , m2 ) is the collection efficiency between drops with masses m1 and m2 . In all BM schemes, collisions are calculated by solving Eq. 4.1. The collection kernel K(m, m′ ) depends on the mass of the collecting drops (which automatically takes into account the effects of drop shape). The collision kernel in the case of gravitational collisions increases with height. The increase in the kernel is caused by the increase in the difference in the terminal fall velocities with the decrease in the air density. Pinsky and Khain (2001) showed that the increase in the difference in the fall velocities also leads to an increase in collision efficiency. As a result, for some droplet pairs, the collision kernel at an altitude of 5–6 km is twice as large as that at the surface.

304

Part IV: Unification and consistency

It may be worthwhile to note that despite its name, the SCE does not involve any stochasticity in its formulation. It is rather a deterministic representation of the collection process, which may behave in a stochastic manner. Clouds are zones of enhanced turbulence (cf., Ch. 18, Sec. 3.3). In turbulent flows the collision kernel increases depending on the intensity of turbulence and the drop (or ice particle) inertia (i.e., the mass). In deep convective clouds the collision kernel for some droplet pairs can increase 5–10 times (e.g., Ayala et al., 2008; Devenish et al., 2012; Pinsky et al., 2008a). Benmoshe et al. (2012) simulated the evolution of convective clouds with collisions being calculated using a collision kernel parameterized as a function of four parameters: Kturb (m1 , m2 , ǫ, Reλ ) = Kg (m1 , m2 )Pkern (m1 , m2 , ǫ, Reλ )Pclust (m1 , m2 , ǫ, Reλ ), where ǫ and Reλ are the dissipation rate and the Taylor microscale Reynolds number, respectively. Pkern and Pclust are collision enhancement factors related to the effects of turbulence on the hydrodynamic interaction between droplets and on the droplet clustering, respectively. The values of ǫ and Reλ are calculated at each timestep and at each point of the spatial grid. Then the values of Pkern and Pclust are calculated using lookup tables presented by Pinsky et al. (2008a). There are several methods of solving the SCE (Khain et al., 2000; Straka, 2009). In the current version of HUCM, an efficient scheme of Bott (1998) is used. In the MMM, Eq. 4.1 is integrated within each category to produce equations that are similar to Eq. 4.1, but for the PSD moments within the categories. The method of Bott (1998) as well as the methods applied in the MMM (Tzivion et al., 1987; Wang et al., 2007) are accurate and show good agreement with known analytical solutions. Since Eq. 4.1 is solved explicitly for entire drop spectra without a separation into cloud droplets and raindrops, the problems of treating autoconversion, accretion, etc. do not arise. Some properties of Eq. 4.1 are discussed in Sec. 5. 4.3.1.2 Bulk formulation It is necessary to distinguish the following types of collision in the mixture of cloud droplets and raindrops (Fig. 23.8): self-collection (sc) is the collision of drops belonging to the same type of hydrometeor; autoconversion (au) is the process of collision of two cloud droplets resulting in raindrop formation; and, accretion (ac) is the process of collision between raindrops and cloud droplets leading to the growth of the raindrops.

Representation of microphysical processes in cloud-resolving models

305

Fig. 23.8 Schematic separation of drop spectrum into cloud droplets and raindrops, and definition of collision processes. “sc” denotes self-collection, “au” denotes autoconversion and “ac” denotes accretion. m∗ is the mass separating cloud droplets and raindrops (From Beheng, 2012, personal communication.)

Using Eq. 4.1 one can derive expressions for the PSD moments describing the rates of the corresponding microphysical processes (Beheng, 2010):   m∗  m∗ ∂M (k) =− f (m′ )f (m′′ )K(m′ , m′′ )(m′ )k dm′′ dm′ ∂t m′ =0 m′′ =m∗−m′ au (4.3)   m∗  ∞ (k) ∂M f (m′ )f (m′′ )K(m′ , m′′ )(m′ )k dm′′ dm′ =− ∂t m′ =0 m′′ =m∗ ac (4.4)   m∗  m∗−m′ (k) 1 ∂M = f (m′ )f (m′′ )K(m′ , m′′ ) ∂t 2 ′ =0 ′′ =0 m m sc ! (m′ + m′′ )k − 2(m′ )k dm′′ dm′ . (4.5)

Equations 4.3 and 4.4 represent the loss integrals in the stochastic collision equations written for the corresponding moments of cloud droplets and raindrops, while Eq. 4.5 represents the gain integral showing the rate of raindrop production. A major difficulty is found in writing down the autoconversion term in a closed analytical form under a bulk formulation (cf., Yano and Bouniol, 2010). Thus, the following discussion is focused on this process. More difficulties are encountered when a mixed phase is considered as further discussed in Sec. 4.3.2. On the other hand, it is relatively straightforward to write down the accretion term in a closed analytical form, and the selfcollection term is not usually considered in bulk approaches.

306

Part IV: Unification and consistency

Historically, the first formulation of autoconversion was proposed by Kessler (1969) and has been used in a great number of bulk schemes. According to Kessler (1969), the rate of rain mass production due to droplet collisions is proportional to the cloud-droplet mixing ratio (or cloud-water content, CWC):

 k(qc − qcr ) if qc > qcr ∂qr ∂M (1) (4.6) = = ∂t ∂t 0 otherwise. au In Eq. 4.6, qc and qr are cloud and rain water contents, while k and qcr are parameters to be tuned to simulate rain formation in different clouds. The threshold value qcr is typically chosen equal to values of 0.5–1 gcm−3 , and k = 10−3 s−1 (Straka, 2009). This formula does not take into account the shape of the DSD and is usually used in single-moment bulk schemes (e.g., Lin et al., 1983; Reisner et al., 1998). Equation 4.6 predicts identical raindrop production rates when the CWCs are the same, even with different DSDs. At the same time, as has been shown in many studies, rain production dramatically depends on DSD parameters, even under the same CWC. For instance, in the case of high aerosol concentration, cloud droplets are small, but the CWC is high. In this case the rain production should be low despite the high CWCs, because collisions of small droplets are inefficient. To parameterize effects of aerosols when using Eq. 4.6 it is necessary to apply high values of qcr for polluted clouds and low values of qcr for clouds developing in a clean atmosphere. The Kessler formula is simple and user-friendly for this purpose. At the same time, the linear expression of Eq. 4.6 is fully empirical and has no link to any solution of the non-linear SCE, Eq. 4.1. In particular, this formula relies on the incorrect assumption of fixed collection efficiency independent of droplet size. Several Kessler-type schemes have been developed to improve the original Kessler formulation (e.g., Liu et al., 2006). In most two-moment bulk schemes the rate of autoconversion is calculated using expressions based on the studies by Berry and Reinhardt (1974b). Berry and Reinhardt (1974b) derived their formulation by analysing the results of a limited number of numerical solutions of Eq. 4.1, using collision efficiencies presented by Hocking and Jonas (1970), which cannot be considered as very accurate from a modern point of view. The values of collision efficiencies calculated in that study differ significantly from those reported by Pinsky et al. (2001) and Wang et al. (2005). Berry and Reinhardt (1974b) took the initial DSD in the form of a gamma distribution; the initial mean-volume radius varied from 10 μm to 18 μm,

Representation of microphysical processes in cloud-resolving models

307

and the liquid-water content (LWC) was taken equal to 1 gm−3 . During the simulations, the time Δtau during which the mean drop radius reached 50 μm was determined. The rate of autoconversion was then determined as a ratio of the rainwater mass to Δtau . As emphasized by Beheng and Doms (1986), the Berry and Reinhardt (1974b) autoconversion relation could not discriminate between autoconversion and accretion, such that their relation included contributions from both. Since the formulation is based on a comparatively low number of simulations performed using only one value of LWC, Berry and Reinhardt (1974b) stressed that extrapolation of their results to other cases should be made with care. This gave rise to the development of more than ten similar formulations (Gilmore and Straka, 2008). These formulations differ in their choice of initial DSD shapes and by different definitions of the characteristic timescale of the formation of the first raindrops (Cohard and Pinty, 2000; Milbrandt and Yau, 2005a,b). Despite the fact that most formulae for autoconversion are applicable to the initial stage of the first raindrop formation only, the rates predicted by these formulae differ by orders of magnitude (Gilmore and Straka, 2008). These large differences between the predictions can be attributed to the highly non-linear nature of the SCE, where rates depend on the DSD shape, its change with time, and on the mass content. The only current scheme that treats raindrop production at later times, when raindrop diameters exceed ∼ 80 μm, appears to be that of Seifert and Beheng (2001). To take into account the time evolution of the autoconversion process during a typical rain event, as well as to use a more realistic collision kernel, Seifert and Beheng (2001) introduced “universal functions” that depend on the fraction of raindrop mass within the total LWC. The universal functions were derived by comparison with the exact solution of the SCE when using a specific type of collision kernel, known as the “Long kernel” (Long, 1974). If other kernels were to be considered then the expressions for the universal functions should be reconsidered. The RAMS model solves the full stochastic collection equations, rather than making use of continuous growth assumptions or some of the other approaches just described. The evolution of the representation of droplet collisions within the bin-emulating bulk scheme of the RAMS model is rather interesting, and is described in detail by Saleeby and Cotton (2004). Verlinde et al. (1990) showed that analytical solutions to the collection equation were possible for the predictions of hydrometeor mixing ratios and number concentrations if the collection efficiencies are held constant, an approxi-

308

Part IV: Unification and consistency

mation that is also made in those bulk schemes based on Kessler (1969). Walko et al. (1995) and Meyers et al. (1997) implemented this approach in earlier single- and double-moment versions of RAMS, respectively. The computational efficiency afforded by the use of lookup tables meant that it was no longer necessary to assume constant or average collection efficiencies, and hence the full stochastic collection equations could be solved using the bin scheme within a Lagrangian parcel model, as described above. The lookup tables were calculated using the collection kernels of Hall (1980) and Long (1974). Feingold et al. (1999) demonstrated that the closest agreement between the full bin representation of the microphysics and the bulk scheme occurred in simulations of marine stratocumulus when collection and sedimentation were represented using this bin-emulating approach. As mentioned above, the collision kernels depend on height and the intensity of turbulence. Hence, they vary with time and in space. The effects of turbulence on the collision kernels in a bulk scheme were taken into account by Seifert et al. (2010). A long set of simulations with a bin microphysics model was carried out in which the SCE was solved using the Ayala et al. (2008) kernel under different values of dissipation rate Reλ , the cloud-water content, the mean radius of cloud droplets, and the shape parameter ν from Eq. 2.1. Overall, more than 10,000 simulations were performed. The rates of autoconversion were calculated for each simulation. As a result of the comparison of the autoconversion rates calculated in turbulent and gravitational cases, a complicated expression for the autoconversion rate enhancement factor was derived. Thus, to change the collision kernel in the bulk schemes it was necessary to perform a derivation of a new formulation of autoconversion using numerous simulations with a bin microphysics model. 4.3.2

Collisions in mixed-phase and ice clouds

Collisions in mixed-phase clouds determine the formation and growth of aggregates (snow), graupel, and hail. These collisions are therefore responsible for precipitation in mixed-phase clouds. 4.3.2.1 Spectral-bin microphysics Collisions in mixed-phase clouds in the BM and MMM models are based on Eq. 4.1 (or its analogues written for the PSD moments in each category), extended to collisions between hydrometeors of different types. The domi-

Representation of microphysical processes in cloud-resolving models

309

nant process in mixed-phase convective clouds leading to the formation of graupel and hail is the process of droplet–ice collisions. In many studies using BM (e.g., Fan et al., 2009, 2012a,b; Iguchi et al., 2008, 2014; Khain and Sednev, 1996; Khain et al., 2004b, 2013), the collision kernels between drops and ice particles of different densities are assumed to be equal to those between two spheres with the corresponding densities. These kernels were calculated by solving the problem of the hydrodynamic interaction between particles within a wide range of Reynolds numbers (Khain et al., 2001). The collection kernels between ice crystals and water drops are taken from Pruppacher and Klett (1997). The collision kernels between ice crystals in these studies are calculated under the simplifying assumption that particles fall with their maximum cross-section oriented perpendicular to the direction of their fall. In recent studies (e.g., Benmoshe et al., 2012; Phillips et al., 2014, 2015) collision kernels are calculated taking into account the dependence of ice-particle fall velocities on the amount of rimed or liquid fractions as well as on the roughness of the particle surface. Equation 4.1 is strictly correct only when the concentration of particles in each bin is large enough. However, the concentration of large hail can be quite small. In this case the procedure for solving Eq. 4.1 proposed by Bott (1998) is reduced to solving the equation of continuous growth that is suitable under low concentrations of large particles (cf., Vol. 1, Ch. 4, Sec. 5.2). 4.3.2.2 Bulk formulations Collisions in mixed-phase clouds pose especially complicated problems for bulk schemes. In mixed-phase clouds, collision of particles belonging to hydrometeors of type X and Y can lead either to formation of particles of type Z (X + Y → Z) or of type X (X + Y → X). An example of the collisions X + Y → Z are those between snow and raindrops resulting in graupel formation. An example of collisions X + Y → X are collisions between graupel and drops that lead to graupel growth (riming). Equations for the time evolution of the moments of PSDs of collecting (k) (k) particles Mx , collected particles My , and the resulting hydrometeors

310

Part IV: Unification and consistency

(k)

Mz can be written as (Seifert and Beheng, 2006a):    ∞ ∞ (k) ∂MX =− KXY nX (mX )nY (mY )mkX dmX dmY ∂t 0 0 X+Y →Z

 

(k)

∂MY ∂t

(k) ∂MZ

∂t

 

(4.7)

X+Y →Z

=−



0

∞

0

∞

KXY nX (mX )nY (mY )mkY dmX dmY (4.8)

= X+Y →Z



0

∞



∞

KXY nX (mX )nY (mY )(mX + mY )k dmX dmY ,

0

(4.9) where the collision kernel is given by Eq. 4.2. The integrals in Eqs. 4.7 and 4.8 describe a decrease in concentration (k = 0) and mass content (k = 1) of the hydrometeors of types X and Y . Equation 4.9 represents the rate of the increase in the concentration and in mass content of hydrometeors of type Z. The integrals in Eqs. 4.7–4.9 are analogues of Eqs. 4.3–4.5 written for drop collisions. In different bulk schemes the integrals in Eqs. 4.7–4.9 (as well as Eqs. 4.3–4.5) are treated differently. Since these integrals cannot be solved analytically, several simplifications or approximations are used. The first simplification is that the collection efficiencies EXY are replaced by a mean value EXY (i.e., an averaged or effective collection efficiency between hydrometeors of type X and Y ). This value can then be moved in front of the integrals. The second simplification is that the difference in fall velocities needed for calculation of the collision kernels is also replaced by a mean value |ΔVXY |, which can then be taken out of the integral as well. As a result, Eqs. 4.7–4.9 are simplified. For example, Eq. 4.9 can be written as:    ∞ ∞ (k) ∂MZ π = EXY |ΔVXY | nX (mX )nY (mY ) ∂t 4 0 0 X+Y →Z

2 + DY2 )dmX dmY . (4.10) (mX + mY )k (DX

The particle diameters DX and DY in Eq. 4.10 have to be expressed as functions of mass. It is clear that the rates of conversions in Eqs. 4.7– 4.9 depend on the values of |ΔVXY |. Different bulk schemes use different formulae to calculate |ΔVXY | (see Table 23.4). To ease these calculations, it is often assumed that the fall velocity of the collector particle is much

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311

higher than that of the collected particles, so that the velocity of collected particles can be neglected. For instance, the fall velocity of the collected drops during riming is often neglected as compared to the fall velocity of ice particles (e.g., Cotton et al., 1986). In this case VY ≪ VX and one can use the approximated formula F1 in Table 23.4. Such a simplification is typically used in one-moment bulk schemes, where VX and VY are massweighted fall velocities. Table 23.4

Expressions for the calculation of |ΔVXY | used by different authors.

DX and DY in

F5

are particle diameters and the constants Cn , C1 , C2 , and C3

and

F6

depend

References

on

parameters

PSDs

(Straka,

2009).

Expressions for |ΔVXY |

F1 Cotton et al. (1986)

VX

F2 Wisner et al. (1972); Cotton et al. (1986)

|VX − VY |

F3 Murakami (1990); Milbrandt and Yau (2005a,b); Morrison et al. (2005a,b)

of



F4 Mizuno (1990)



F5 Murakami (1990)



(VX − VY )2 + 0.04VX VY

(1.2VX − 0.95VY )2 + 0.08VX VY   2 D 2 f (m ) (1/Fn ) 0∞ 0∞ (VX (mX ) − VY (mY ))2 DX X Y X

Mizuno (1990); Seifert

ny (mY )mY dmX dmY ]1/2 , where

and Beheng (2006a,b)

2 (m )D 2 (m )m Fn = Cn NX NY DX X Y X Y   2 2 1/2 (1/Fn )NX NY mX (C1 VX − 2C2 VX VY + C3 VY )   where Fn = 0∞ 0∞ (DX (mX ) + DY (mY ))2 nX (mX )

F6 Flatau et al. (1989); Ferrier (1994)

nY (mY )mX dmX dmY

In some schemes (e.g., Cotton et al., 1986; Wisner et al., 1972) it is assumed that |ΔVXY | is equal to the absolute value of the difference between the mass-weighted or concentration-weighted fall velocities (F2 in Table 23.4). The expression F2, used in many bulk schemes, is not only a mathematically crude approximation, but it also leads to the physical paradox of vanishing in the case of self-collisions. According to this formula there are no collisions between hydrometeors belonging to the same type. Since large snowflakes have an average fall velocity close to the average fall velocity of small raindrops, use of this formula also may erroneously discard snow–raindrop collisions, in spite of the fact that for particular

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snow–raindrop pairs the fall speeds of colliding particles are different and collisions can take place. To avoid this problem, many bulk schemes (e.g., Milbrandt and Yau, 2005a; Morrison et al., 2005a,b) use the formula proposed by Murakami (1990) and shown as F3 in Table 23.4. The term with a coefficient of 0.04 in formula F3 is introduced to avoid the physical paradox of vanishing of collisions in the case of VX = VY . This term characterizes the dispersion of fall velocities within the particles belonging to hydrometeors of the same type. Mizuno (1990) proposed a similar approximation of |ΔVXY | as F4 in Table 23.4. To get a better approximation of the collision rate Seifert and Beheng (2006a) use root-mean-square values instead of the mean absolute values, as in F5 of Table 23.4. Ferrier (1994) and Flatau et al. (1989) calculated |ΔVXY | using F6, which has the same meaning as F5. The advantage of these last two formulations is that they take into account specific forms of PSDs, obtained in current model calculations. The substitution of the expressions for PSDs with exponential or gamma-type relations into collision integrals like Eq. 4.10 leads to quite complicated expressions for the collision rates (e.g., Seifert and Beheng, 2006a,b). Significant problems arise in the application of the bulk approach in the case of self-collection X + X → X (e.g., snowflakes), because in this case there is no difference between the averaged fall velocities. One can expect the application of different representations of |ΔVXX | to lead to a high variability in results of the bulk schemes. In bin-emulating bulk schemes (such as in RAMS) the collision integrals of Eqs. 4.7–4.9 are calculated numerically. To reduce the computational cost when first using a single-moment bulk scheme, Walko et al. (1995) assumed collection efficiencies to be constant and calculated a large number of solutions for the integrals and compiled three-dimensional lookup tables. Two of the table dimensions are the characteristic diameters of the colliding particles. The third dimension is the value of the integral for particle pairs which result from collisions between the seven types of liquid and ice particles used in RAMS. As soon as the calculations are performed, the parameters of the underlying gamma distributions are calculated using the number concentration and mass content at each model grid point. In more recent versions of RAMS, the full stochastic collection equations are solved for all of the interactions amongst all liquid and ice species without assuming constant collection efficiencies, as described in the previous subsection. According to measurements, many aggregates consist of different comparatively large parts connected by thin necks. Such a structure supposedly

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313

indicates the presence of electrostatic forces. The measurements of charges of ice crystals are needed to perform quantitative evaluations. 4.4

Sedimentation

Sedimentation determines to a large extent the microphysical structure of clouds, as it determines the redistribution in the vertical of hydrometeor particles in respect of their size (so-called “size sorting”), and affects other microphysical processes, such as collisions, growth/evaporation, etc. Sedimentation also plays a strong role in determining surface precipitation. 4.4.1

Spectral-bin microphysics

Sedimentation of all particles in SBM is calculated by solving the following equation for the PSD: ∂Vt:i,k (z)ni,k ∂ni,k = , (4.11) ∂t ∂z where ni,k is the PSD of k-th hydrometeor belonging to i-th mass bin. Typically in BM and MMM models, the fall velocity of particles Vt:i,k (z) depends on the hydrometeor type and on the particle mass. The dependence of the fall velocity on air density (height) is also often taken into account. In some cases (e.g., Benmoshe et al., 2012; Phillips et al., 2007b, 2014, 2015), the fall velocity also depends on particle composition (rimed fraction, or liquid-water fraction in ice particles) and the roughness of the particle surface (dry or wet). Size sorting is simulated automatically since particles of different masses and of different types fall with different velocities. 4.4.2

Bulk formulation

Sedimentation in bulk schemes is performed not for single particles, but for the PSD moments. This process is described by the equation (k)

∂Vt M (k) ∂M (k) = , (4.12) ∂t ∂z in which k = 0 corresponds to the drop concentration, k = 1 corresponds to the mass content, and  1 (k) Vt = (k) mk n(m)Vt (m)dm (4.13) M is the averaged fall velocity. For the moments of different order, the averaged fall velocities are different. In the case of one-moment bulk schemes,

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sedimentation is performed for the first moment, so that only the massweighted fall velocity is used. For instance, all raindrops with fall velocities varying from, say, 0.5 ms−1 to 10 ms−1 fall with the same velocity equal to that of some mean mass raindrop, say, 3 ms−1 . It is clear that such an approach cannot reproduce the size sorting when larger drops fall faster than smaller ones. Milbrandt and Yau (2005a,b) showed that one-moment schemes are not able to reproduce size sorting and that the errors in reproducing particle sedimentation are very large (see also Straka, 2009). These errors can affect the rate and time duration of precipitation (Milbrandt and McTaggart-Cowan, 2010; Milbrandt and Yau, 2005a,b). Better results can be achieved using two-moment bulk schemes. However, the description of sedimentation is far from being perfect (as compared to the exact bin schemes). In two-moment bulk schemes, particle sedimentation is described using two equations of the form of Eq. 4.12; one for the drop concentration and one for the drop mass. In the case of constant fall velocity, Eq. 4.12 represents the classical equation of advection ∂M/∂t = −Vt ∂M/∂z (here, the z axis is directed downwards), which has an analytical solution M (z, t) = M (z − Vt t, 0). The initial vertical profile of the moment is translated downwards, without changing its shape. Note that the mass-averaged fall velocity should be larger than the concentration(1)

(0)

averaged one Vt > Vt , to reflect the fact that smaller particles fall more slowly than larger ones. The difference between these velocities leads to a problem when mass content sediments faster than number concentration. As a result, it may occur that in one area there is mass but with a negligible concentration, whilst in another there is concentration but with a negligible mass. This implies mean radii of particles that may be extremely large or extremely small. In such cases the value of the radius is artificially changed to shift it into a reasonable range. To decrease the number of such undesirable situations, Eq. 4.12 is typically solved using an upstream finite-difference scheme with significant computational diffusivity. The use of diffusive numerical schemes increases the overlap of the vertical profiles of the moments and leads to more reasonable values of particle diameter. At the same time, the mean volume diameter turns out to be dependent on the diffusivity of the numerical scheme, which is determined by the timestep and the vertical resolution as well as the fall velocity, which is an obvious disadvantage of such schemes. Aiming to improve sedimentation representations in bulk schemes, Mil-

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315

brandt and McTaggart-Cowan (2010); Milbrandt and Yau (2005a,b) compared the results of sedimentation simulated by a one-moment, a twomoment, and a three-moment bulk scheme with the solution obtained using a bin method, results from which were considered as exact. In the threemoment scheme, the additional prognostic equation for the k = 2 moment (radar reflectivity) was also solved. At the initial time, the hydrometeors (e.g., rain or hail particles) were located at heights of 8–10 km. It was shown that the vertical profiles of the concentration calculated using the one-moment scheme dramatically differed from the exact solution. The mean-mass particle diameters were unrealistically small and practically did not depend on height. Despite the fact that the profiles produced by the two-moment scheme were more realistic than those of the one-moment scheme, the size sorting was significantly stronger than that in the exact method. The results of the three-moment scheme turned out to be considerably better, so the application of three-moment bulk schemes to describe particle sedimentation was recommended. The same conclusion was reached by Loftus et al. (2014a). Milbrandt and McTaggart-Cowan (2010) also proposed an approach to improve the reproduction of sedimentation in two-moment bulk schemes. In this approach, the slope parameter was parameterized as a function of the ratio of the two moments calculated when solving Eq. 4.12. The values of the shape parameter recommended for the calculation of the fall velocity differ from those used in the rest of the bulk scheme. Note that all simulations were performed using a diffusive numerical scheme. In the bin-emulating method of RAMS, the PSD given in the form of a gamma function is discretized into several tens of bins before the sedimentation time sub-step (Cotton et al., 2003; Feingold et al., 1998). Then, the sedimentation algorithm is applied to each bin. After sedimentation, the values of the mass and the concentration calculated at different grid points are used to restore the parameters of the gamma distributions. The bin-emulating schemes have a greater computational cost than more standard bulk schemes. To decrease the computational time of bin-emulating schemes, lookup tables are used to calculate fall distances of particles belonging to a particular bin. It is widely accepted that such an approach produces results closer to the exact solution as compared to those based on the sedimentation of moments according to Eq. 4.12. However, Morrison (2012) showed that results of the bin-emulating approach give a similar vertical distribution of the moments as the standard bulk schemes. Morrison (2012) concluded that improvement in the repre-

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Part IV: Unification and consistency

sentation of sedimentation in bulk schemes can be achieved only in twoor three-moment schemes in which the effects of the change of PSD shape (e.g., the shape parameters of the PSD) during sedimentation are taken into account, as in the studies by Milbrandt and McTaggart-Cowan (2010); Milbrandt and Yau (2005b); Wacker and L¨ upkes (2009). These effects should be taken into account regardless of whether bin-emulating or standard bulk schemes are employed. Note that in the SBM approach such changes in the PSD shape during sedimentation are taken into account automatically.

4.5

Other microphysical processes

Cloud microphysics includes many other microphysical processes, such as freezing/melting, ice multiplication, and raindrop break-up. The conversion of particles of one hydrometeor type into particles belonging to another type is a very important problem to resolve. Some practical problems require the calculation of particle salinity, charge, etc. The scope of the present survey does not allow discussion of all of them, so the focus is only on some. 4.5.1

Spectral-bin microphysics

There is a wide range of approaches that may be used to represent different microphysical processes such as melting, freezing, etc. in SBM models. For instance, in some schemes immediate melting at the melting level is assumed (e.g., Khain and Sednev, 1996; Reisin et al., 1996a,b), while in other models melting is calculated in a simplified way by prescribing a characteristic melting time for ice particles of different sizes (e.g., Fan et al., 2012a). There are BM schemes where detailed melting is included through tracing the history of liquid water within melting particles and the advection of liquid-water mass within ice particles (Phillips et al., 2007b). In HUCM, liquid-water fractions of snowflakes, graupel, and hail are calculated. This means that three additional size distributions are treated in the scheme. Substantial differences exist also in the description of the process of heterogeneous freezing of drops. In most schemes, semi-empirical formulae determining the probability of freezing are used (e.g., Khain and Sednev, 1996; Reisin et al., 1996a). In more comprehensive schemes, two-step freezing is considered when the very short adiabatic stage is followed by timedependent freezing based on the calculation of the heat balance within the freezing drops. Freezing drops have been introduced as a new hydrometeor type (Kumjian et al., 2014; Phillips et al., 2014, 2015) and are converted to

Representation of microphysical processes in cloud-resolving models

317

hail when the liquid-water fraction reaches zero. Time-dependent freezing of accreted water was also implemented when riming of graupel and hail was considered. In these studies, as well as in the study by Phillips et al. (2007b), shedding of liquid water is taken into account. The process of shedding is described using the expressions for maximum thickness of liquid film obtained in laboratory experiments. The drops formed by shedding are added to the distribution of raindrops. In a set of studies by Flossmann et al. (1985); Flossmann and Pruppacher (1988); Respondek et al. (1995), a BM model was developed for the investigation of the formation of acid rain in cumulus clouds. The aerosol particle mass within water droplets is traced and is treated as a separate mass distribution function. Accordingly, processes of aerosol mass growth within droplets and aerosol scavenging can be calculated. Similarly, effects of microphysical processes (e.g., collisions) on the salinity of drops, the rate of their growth and on aerosol size were investigated in Lagrangian trajectory ensemble BM models (Magaritz et al., 2009, 2010; Shpund et al., 2011). 4.5.2

Bulk formulations

Analogously to collisions, the rates of melting in bulk schemes are calculated by averaging of the basic thermodynamic equations for melting over particle size spectra (e.g., Milbrandt and Yau, 2005a,b; Seifert and Beheng, 2006a,b). Typically, these equations are treated in a simplified form, say, in neglecting collisions and in the assumption of immediate shedding of all melted water. The approach often reduces to the evaluation of characteristic timescales of melting for ice hydrometeors of different types. Serious simplifications are made during the averaging of these equations. The process of drop freezing is typically calculated using spectrallyaveraged semi-empirical formulae. According to many such formulae the rate of freezing is proportional to drop mass, i.e., it is highly size dependent. Note that most particular microphysical processes such as diffusional growth, sedimentation, melting, and freezing should lead to deviation of the PSD from the gamma distribution, even if the initial distribution was gamma in shape. For instance, condensation leads to the appearance of a minimum size that differs from zero, and evaporation may lead to the appearance of a maximum concentration in the vicinity of zero size. Being proportional to the drop mass, drop freezing should lead to the disappearance of the tail of largest drops in the raindrop distribution. In this sense,

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Part IV: Unification and consistency

the assumption that after each microphysical process the PSD remains a gamma distribution (even with other parameters) may lead to errors in the representation of the microphysical structure of clouds. In summary, the use of a few moments of the PSD leads to uncertainties in treating different microphysical processes because of crucial dependencies of the rates of these processes on particle size. Different approximations of these processes in different bulk schemes may lead to a high variability of the results of simulations of the same atmospheric phenomena (cf., Sec. 6).

5

Examples of accurate description of microphysical cloud structure

Comparisons of in situ measurements of the microstructure of liquid and ice clouds suggest that the existing analytical equations provide a good basis for a reasonably accurate representation of the cloud microstructure. The deviation of numerical results from observations can be attributed to errors in the numerical methods applied to solve particular equations rather than to errors in the form of the equations themselves. For instance, the correct representation of the droplet concentration in clouds as a result of diffusional growth of aerosol and droplets can be used to justify the corresponding growth equations. It is noteworthy that the more accurate the method used for solving the diffusional growth equation, the better the agreement with the observed data. This agreement is especially good in parcel and Lagrangian models, where the equation of diffusional growth is solved using movable mass grids and the process of aerosol growth is described explicitly (e.g., Magaritz et al., 2009; Pinsky and Khain, 2002; Segal et al., 2004; Segal and Khain, 2006). The errors in the reproduction of the observed droplet concentration and the observed liquid-water content in maritime stratocumulus clouds from such models do not exceed 5–10% (e.g., Magaritz et al., 2009, 2010; Magaritz-Ronen et al., 2014). The same accuracy is reached in the reproduction of the DSD, effective and mean volume radii, and Z-LWC relationships in these clouds. An analysis based on observations with a helicopter-borne measurement platform and an SBM parcel model allowed Ditas et al. (2012) to reproduce the values of in situ measured droplet concentration in stratocumulus clouds to within an accuracy of 5%. Numerous simulations show that accurate solutions to the equations of diffusional growth and SCE allow for accurate simulations of the DSD in

Representation of microphysical processes in cloud-resolving models

319

convective clouds. For example, using the Tel Aviv University rain shaft model with the MMM, including drop break-up and evaporation, Levin et al. (1991) showed good agreement between the calculated raindrop-size distribution and that measured at the top and bottom of Mt Rigi in Switzerland. Similarly, the HUCM SBM model has accurately simulated DSDs and their changes with height under different aerosol loadings. Figure 23.9 compares simulated DSDs with in situ measured DSDs during the LBASMOCC 2002 field experiment in the Amazon region. Clouds developing under different aerosol concentrations were distinguished as Blue Ocean (BO) clouds (clean atmosphere), Green Ocean (GO) clouds developing under intermediate aerosol concentrations, and Smoky clouds (S) developing in a very polluted atmosphere with CCN concentrations of about 104 cm−3 (Rissler et al., 2006). Sounding data from Andreae et al. (2004) were used for verification purposes. Figure 23.9 shows DSDs at different heights in GO and S clouds during the earlier stages of cloud development, before the formation of the first raindrops. These DSDs were obtained by averaging the observed DSDs over the whole cloud traverses. In the simulations, the effects of turbulence on cloud-droplet collisions were taken into account (Benmoshe et al., 2012). Figure 23.9 shows that the model reproduces the evolution of DSD in clouds quite well. The DSDs in GO clouds are wider than in S clouds and centred at 12–14 μm, as compared with 10 μm in S clouds. One of the important microphysical characteristics of clouds indicating their ability to produce raindrops is the effective radius reff . Raindrop formation takes place if reff exceeds 14 μm in cumulus clouds developing in comparatively clean air, and 10–11 μm in clouds developing in a polluted atmosphere (Freud et al., 2008; Freud and Rosenfeld, 2012). Figure 23.10 shows vertical profiles of the effective radius reff calculated for BO, GO, and S clouds. The figures were plotted for the period of cloud development when the cloud-top heights grew from 2 to 6 km in the BO and GO clouds, and to 6.5 km in the S clouds. The vertical profiles of reff calculated using in situ observations in many clouds during LBA-SMOCC are presented as well. The vertical arrows show the values of reff corresponding to the formation of the first raindrops. One can see good agreement of the model results with observations: the first raindrops form at reff = 14 μm in BO, at 12 μm in the GO, and at 10.5–11 μm in highly polluted S clouds. The heights of the first raindrop formation marked in Fig. 23.10 are also in good

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Part IV: Unification and consistency

Fig. 23.9 The mass distributions measured in situ on 5 October 2002 up to a height of 4.2 km (dashed lines; from Andreae et al. (2004)) and calculated using HUCM (solid lines). Left: developing S-clouds; right: developing GO clouds. The CCN concentration at 1% of supersaturation is 5,000 cm−3 in S and 1,000 cm−3 in GO cases. The values of ǫ and Reλ were calculated using the equation for turbulent kinetic energy at each timestep c American Geophysical Union 2012, from Figs. 9 and and at each model grid point.  10 of Benmoshe et al. (2012).

Fig. 23.10 Height dependence of the effective radius in S (top), GO (middle), and BO (bottom) clouds. Simulations with gravitation collision kernel are marked by blue circles; simulations with turbulent collision kernels are marked by red asterisks. The solid lines show the observed data. The levels of the first rain formation found in the simulations c American Geophysical Union 2012, from Fig. 8 of are plotted by the vertical arrows.  Benmoshe et al. (2012).

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agreement with the observations to an accuracy of about 100–200 m. Very accurate reproductions of the observed vertical profile of reff (z) with low dispersion were reported in the simulation of deep convective clouds using the System for Atmospheric Modeling (SAM) model with the SBM (Khain et al., 2013). It was found in that study that in nonprecipitating, developing convective clouds reff = 1.08rv , where rv is the mean volume radius. The same relationship was found by Freud and Rosenfeld (2012) in numerous flights in different regions, as well as by Prabha in the field experiment CAIPEX in India (Khain et al., 2013). Very accurate representations of radar reflectivity, as well as signatures of dual-polarimetric radars in hailstorms using an SBM model were reported recently by Kumjian et al. (2014). These examples show that SBM models are able to reproduce fine features of the PSD determined by high-order PSD moments.

6

6.1

Comparison of results obtained using SBM and bulk schemes Criteria of comparison

Currently the following four main approaches are followed for intercomparisons. In the first approach the rates of separate microphysical processes (e.g., various autoconversion and sedimentation schemes) are compared (e.g., Beheng, 1994; Kumjian et al., 2012; Kumjian and Ryzhkov, 2012; Milbrandt and Yau, 2005a,b; Seifert and Beheng, 2001; Suzuki et al., 2011; Wood, 2005). Schemes describing particular microphysical processes are considered as elementary units of the entire microphysical schemes. In these studies the bin approach is typically considered as exact or the benchmark scheme. This approach aims at testing and tuning the particular elementary microphysical units against solutions obtained using basic microphysical equations. Some examples of this approach were discussed in previous sections. Such an approach has been widely used for the modification of existing bulk schemes and for the development of new bulk schemes. The second approach is to use kinematic models with prescribed velocity fields, thereby allowing for the advective and gravitational transport of hydrometeors (e.g., Ackerman et al., 2009; Fridlind et al., 2012; Morrison and Grabowski, 2007; Ovchinnikov et al., 2014; Shipway and Hill, 2012;

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Part IV: Unification and consistency

Stevens et al., 2005; Szumowski et al., 1998). The prescribed kinematic framework allows for the comparison of microphysical parameters simulated by different schemes under the same dynamical conditions. This approach also allows for the identification of a better combination of the elementary units to design a better bulk scheme. Note that a good agreement between bulk and bin schemes in the case of prescribed dynamics does not guarantee agreement when the mutual interactions of microphysics and dynamics are taken into account. The third approach is to compare bulk and bin schemes within the same dynamical framework, but in the case when all mutual influences of model microphysics and dynamics are taken into account (e.g., Adams-Selin et al., 2013; Fan et al., 2012a; Lebo and Seinfeld, 2011; Li et al., 2009a,b; Seifert et al., 2006). In this approach it is more difficult to get agreement with observations as well as between results of bulk and bin schemes than in the first two approaches. Finally, the fourth approach is to compare different microphysical schemes using different models, but simulating the same well-documented case study. Several examples of such comparisons of bulk and SBM schemes using different approaches are presented below beginning with simulations of particular microphysical processes and ending with tropical cyclones. In some examples, the responses of different microphysical schemes are compared to variations in the aerosol concentration. The ability to represent aerosol effects on clouds, precipitation, heat, and radiation balance is now considered as a measure of suitability of models to investigate local and global climatic changes. Reviews of a great number of studies and concepts of aerosol effects on clouds and precipitation can be found in Khain (2009); Levin and Cotton (2008); Rosenfeld et al. (2008); Tao et al. (2012). The interest in aerosol effects on cloudiness and precipitation has produced a great number of investigations of sources of aerosol, size distribution of aerosols, their solubility and chemical composition, etc. Results of a great number of measurements were classified to derive typical aerosol size distributions for different climatic zones (maritime, continental, urban, etc.) (Ghan, 2011). While soluble aerosols act as CCN, insoluble aerosols may act as ice nuclei (IN). Often, atmospheric aerosols contain both soluble and insoluble fractions. Numerical models with SBM make it possible to reproduce the observed drop-aerosol concentration dependencies (Ghan, 2011; Pinsky et al., 2012; Segal and Khain, 2006). Aerosols affect DSDs, and through this, all microphysical cloud properties. Two-moment bulk schemes allow

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323

for the calculation of mass contents and number concentrations: i.e., they allow an evaluation of the mean volume radius of droplets, which decreases with an increase in CCN concentration. The similarity in the response of bulk models and of SBM to changes in the CCN concentration would provide a solid basis to account for these effects in large-scale models. Several examples are presented below aimed at giving the reader a sense of the abilities of bulk formulations and SBM to simulate different atmospheric phenomena of different space and timescales. The following comparison studies must be viewed with some caution. In many studies it is found that the effects of microphysics on clouds and precipitation are not isolated, and should be considered within the framework of the entire system describing cloud–radiation–aerosol interactions that also includes the interactions of the atmosphere with the surface, and specifically with the ocean surface. Such complicated systems always contain various negative feedbacks, which makes the entire system more stable than might be expected in comparison to cases when only particular microphysical forcing is considered (Morrison et al., 2011; Stevens and Feingold, 2009). The existence of synergistic mechanisms is also possible. 6.2

Effects of size sorting on polarimetric signatures

As an example showing the importance of accurate reproduction of microphysical processes for calculations of polarimetric radar returns, results from Kumjian and Ryzhkov (2012) are discussed. In this study, polarimetric radar signatures produced by raindrops falling from 3 km altitude within an atmospheric layer with a horizontal wind shear of 6 ms−1 km−1 were calculated using the BM approach as well as single-, two-, and threemoment bulk schemes. Raindrop-size distributions were assumed to have the Marshall–Palmer form at 3 km. The results are illustrated in Fig. 23.11 showing the fields of radar reflectivity ZH and the differential radar reflectivity Zdr. Substantial differences can be seen in the ZH and Zdr obtained in the bin approach (benchmark) and a single-moment scheme. While in the bin approach large values of ZH and Zdr extend to the surface, a single-moment scheme dramatically underestimates these values below the 1.5 km level. So, in agreement with the conclusions of Sec. 4, these results led Kumjian and Ryzhkov (2012) to the conclusion that “one-moment parameterizations are incapable simulating size sorting”. This inability to reproduce size sorting results in large errors in Zdr computed from the resulting DSDs. Despite the

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Fig. 23.11 Results from the two-dimensional wind shear model obtained using different sedimentation schemes. Panels show the two-dimensional fields of (a) ZH, (b) Zdr. Overlaid on (a) are Zdr contours (0.5–2.5 dB in 0.5 dB increments) and on (b) are ZH contours (10–40 dBZ, in 10 dBZ increments). For ZH of 0 dBZ, all fields are set to zero. Note that the colour scales of ZH and Zdr for the two-moment scheme differ from those c American in other panels. (Figure reproduced from Kumjian and Ryzhkov (2012).  Meteorological Society. Used with permission.)

fact that errors introduced by the two-moment and three-moment schemes are much smaller, the two-moment schemes with fixed shape parameters suffered from excessive size sorting. This leads to a dramatic overestimation of Zdr (by several dB), ZH, and other parameters in large zones of a simulated rain shaft encountering wind shear, as well as beneath newly precipitating clouds. The use of a diagnosed shape parameter in a two-moment

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325

scheme or of a calculated shape parameter in a three-moment scheme largely mitigates the errors associated with size sorting. Figure 23.11 shows that the three-moment bulk scheme provides the best agreement with the results of the bin approach. At the same time, even the three-moment scheme did not manage to simulate high Zdr near the surface in the zone containing the largest falling raindrops (at x-values of 2–4 km in Fig. 23.11). Shipway and Hill (2012) compared the rates of the processes of diffusional growth, collisions, and sedimentation, as well as the surface precipitation in bulk microphysical schemes and in the MMM method. Among the bulk schemes were three one-moment and three two-moment schemes, as well as a three-moment scheme. The warm microphysical processes were calculated using a simple one-dimensional kinematical model. The results obtained using the MMM method (with 34 mass categories for drops) were considered as a benchmark. The schemes tested produced very different surface precipitation rates and onset timing, as well as accumulated surface precipitation. The early onset of precipitation was shown to be a persistent feature of single-moment schemes. Two-moment schemes offered a better comparison with the MMM results. However, the peak of precipitation rate was larger than that in the MMM. The best agreement with the MMM results was reached using the three-moment scheme. 6.3

Simulation of a mesoscale rain event

The studies by Lynn et al. (2005a,b) were, supposedly, the first in which an SBM scheme had been used in a mesoscale model (specifically the fifthgeneration Pennsylvania State University–NCAR Mesoscale Model, MM5: Grell et al., 1995) to simulate the evolution of a large convective system. A rain event over Florida on 27 July 1991, during the Convection and Precipitation Electrification Experiment (CaPE), was chosen. A 3 km grid length was used to simulate the mesoscale event for a comparatively long period of time and over a significant area. The experiments were performed for both low and high CCN concentrations. SBM results were compared to those obtained using different one-moment bulk schemes. All simulations produced more rain at certain locations than was found in observations. While the SBM simulations reproduced observed large areas of stratiform rain, the bulk schemes tended to underestimate the area of weak, stratiform rain (Fig. 23.12). Figure 23.13 shows time dependencies of the area-average and maximum rain amounts. The SBM produces results closer to observations. The

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Fig. 23.12 Three-dimensional structure of rainwater content in the experiments SBM FastM and in Reisner2 at 2300 UTC. The solid line denotes the land–sea boundary. c American Meteorological Society (AMS) 2005, from Fig. 8 of Lynn et al. (2005b).  Reproduced by permission of the AMS.

one-moment bulk models produced unrealistically large rain rates within a comparatively narrow line of cumulus clouds. One of the possible reasons for such an effect is incorrect representation of the process of precipitation sedimentation. Thus, this example illustrates the substantial limitations of one-moment bulk schemes even in the simulation of general characteristics such as the structure of a cloud ensemble and precipitation rates. Lynn and Khain (2007) simulated the same storm using a larger computational area. They compared the results of SBM simulations with those obtained using other bulk schemes than those tested by Lynn et al. (2005a,b). In particular, they considered the Thompson scheme (Thompson et al., 2006), the Reisner–Thompson scheme which is a modified form of Reisner2, and the Fully Two-Moment Microphysical Scheme of Seifert et al. (2006). The results obtained were quite similar to those shown in Fig. 23.13: all bulk schemes substantially overestimated averaged and maximum rain rates, by a factor of 2 to 3 compared to the observations. The two-moment scheme predicted rain rates better than the one-moment schemes. The SBM scheme overestimated the maximum rain rate by about 20%. 6.4

Cumulus and stratocumulus clouds

Morrison and Grabowski (2007) used a two-dimensional kinematic model with prescribed dynamics to compare the microphysical structure of warm

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Fig. 23.13 Average (A) and maximum (B) rainfall rates obtained from observations and model simulations. The labels are Reisner2 (Reisner et al., 1998), GSFC (Goddard Space c American Meteorological Flight Center) (Tao et al., 2003) and Schultz (Schultz, 1995).  Society (AMS) 2005, from Fig. 6 of Lynn et al. (2005b). Reproduced by permission of the AMS.

stratocumulus clouds and warm cumulus clouds simulated using the Morrison et al. (2005a) two-moment bulk scheme against a bin scheme for different aerosol loadings. Three different formulations for the coalescence process were tested: those of Beheng (1994), Khairoutdinov and Kogan (2000), and Seifert and Beheng (2001). Simulations showed that the different schemes led to differences in horizontally averaged values of rain water content (RWC), as well as mean raindrop diameters by several times. It is interesting that the difference in the accumulated rain produced by the different schemes was comparatively low (up to 10–20%). This

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result suggests that accumulated rain is determined to a large extent by general properties of the atmosphere, such as environmental humidity and the dependence of saturation water vapour pressure on temperature, which is similar in all schemes. However, the precipitation rates can be quite different in different schemes. Morrison and Grabowski (2007) showed that results of the one-moment scheme were much worse than those of two-moment schemes. They found that in order to adjust the results of the one-moment schemes to those in two-moment schemes, the intercept parameter N0 had to be changed to vary with height by five orders of magnitude. Taking into account this high sensitivity of the results to the value of the intercept parameter, Morrison and Grabowski (2007) concluded that one-moment schemes are not suitable for use in regional or global climate simulations using cloud-resolving models. Detailed comparison of bulk and bin approaches in simulations of the structure and precipitation of continental (Texas) clouds and tropical maritime (GATE) clouds was performed by Seifert et al. (2006). In order to do so, the two-moment scheme of Seifert and Beheng (2006a) was implemented into the HUCM. Simulations were performed for two aerosol loadings: a high CCN concentration of 1,260 cm−3 and a low CCN concentration of 100 cm−3 . It was found that the most important ingredient to achieve a reasonable agreement between concentrations and mass contents in the bulk formulation and the SBM is an accurate representation of the warm-phase autoconversion process, parameterized in that study using the scheme of Seifert and Beheng (2001). Some modifications of certain parameters of the two-moment bulk scheme were necessary to obtain comparable results. This was mainly achieved by adjusting the parameters of the various PSDs. In spite of differences between the PSDs of the bin and the bulk simulations, the precipitation rates and accumulated precipitations were quite close, especially for the case of high CCN concentration (Fig. 23.14). The substantial difference in the accumulated rain amounts from maritime clouds at low CCN concentration seen in Fig. 23.14 (right panel) is related to the fact that the bulk scheme did not take into account a decrease in the aerosol concentration due to nucleation scavenging. Neglecting the aerosol budget in the BM scheme led to a substantial decrease in the difference between accumulated rain amounts in BM and in the bulk scheme (see the line “spectral bin-maritime bulk CCN” in Fig. 23.14). Neglecting such cloud–aerosol feedback led to differences in the time dependencies of precipitation rates, as shown in Fig. 23.15. In particular, one can see for-

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Fig. 23.14 Time series of grid-averaged accumulated surface precipitation from (left) the four Texas-case simulations, and (right) the five GATE-case simulations, comparing bin and two-moment bulk microphysics and assuming continental or maritime CCN as indicated on the figure. (Reprinted from Figs. 11 and 14 of Seifert et al. (2006), with permission from Elsevier.)

mation of secondary convective clouds in the bulk-scheme simulation with low CCN (Fig. 23.15d), while no such secondary convective clouds occurred in the SBM simulation, in which most CCN were scavenged. The same conclusion regarding the importance of cloud–aerosol feedback was reached by Fan et al. (2012a) who compared the results of the two-moment Morrison and Grabowski (2007) bulk scheme, as modified by Solomon et al. (2009), with the results of the SBM also implemented into the WRF (Fan et al., 2012a,b). Two case studies were simulated: firstly, stratocumulus and convective clouds observed during the Atmospheric Radiation Measurement Mobile Facility field campaign in China (AMF-China, 2008), and secondly, marine stratocumulus clouds observed during the VAMOS Ocean–Cloud–Atmosphere–Land Study (VOCALS) field experiment. In both case studies, WRF/SBM reproduced well the observed geometrical cloud structure, microphysics, and surface precipitation. Fan et al. (2012a) and Wang et al. (2013) obtained substantial improvement in the reproduction of microphysical parameters such as CWC, drop concentrations, mean radius, etc. through the implementation of nucleation scavenging due to the CCN activation. The conclusion is that to improve the representation of cloud–aerosol interactions in atmospheric models using bulk schemes, the schemes should include an aerosol budget, aerosol transport and the process of nucleation scavenging. Saleeby and van den Heever (2013) also showed that for bulk schemes that represent aerosol budgeting, transport, and nucleation scavenging, the reproduction of microphysical parameters appeared to be significantly im-

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Fig. 23.15 Hovm¨ oller diagrams showing the surface precipitation in mmhr−1 from four GATE-case simulations applying spectral-bin (a,c) and two-moment bulk (b,d) microphysics and assuming continental (a,b) and maritime CCN (c,d). (Reprinted from Fig. 12 of Seifert et al. (2006), with permission from Elsevier.)

proved. Sensitivity experiments performed using four different types of autoconversion schemes reveal that the saturation adjustment employed in calculating condensation/evaporation in the bulk scheme is the main factor responsible for the large discrepancies in predicting cloud water content. Thus, an explicit calculation of diffusional growth with predicted supersaturation (such as is included in the bin-emulating bulk scheme of the RAMS model) is a promising way to improve bulk microphysical schemes. In addition, it was also found that rain evaporation in the bulk schemes was too fast compared to the SBM. In summary, comparison of the results of bulk schemes and SBM have allowed efficient ways to be found for improving bulk schemes. These results again indicate that two-moment schemes have significant advantages over one-moment schemes. It is worth noting that the implementation of nucleation scavenging by

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Wang et al. (2013) doubled the computer time required. As a result, the computer time cost ratio of SBM/bulk decreased from ∼ 12 to about 5. Implementation of the equation for diffusional growth of droplets would require substantial changes in bulk schemes and further lead to an increase in the computational time. 6.5

Squall lines

Squall lines are a vivid example of mesoscale systems in which convective clouds form a line of several hundred kilometres length. The front of the squall line represents an elongated thunderstorm characterized by heavy rain, hail, and lightning. The width of the leading updraught core ranges typically from ten to a few tens of kilometres. A wide zone of light precipitation forms behind this convective zone. The light-precipitation zone is caused largely by aggregates and has a width of a few hundred kilometres. Squall lines represent a typical type of severe weather phenomena both over land and sea. However, over the sea squall lines represent the dominant type of mesoscale convective system. The lifetime of squall lines can be as long as 10 h, as compared to about 1 h for isolated storms. The mechanisms of the formation of squall lines and the reasons for their long lifetimes have been investigated in many studies (e.g., Heymsfield and Schotz, 1985; Rotunno et al., 1988; Tao et al., 1993). Li et al. (2009a,b) simulated a squall line typical of continental conditions observed during the Preliminary Regional Experiment for Stormscale Operational and Research Meteorology 1985 (PRE-STORM: Braun and Houze Jr, 1997; Zhang et al., 1989). Simulations were performed using the two-dimensional anelastic version of the Goddard Cumulus Ensemble (GCE) model with two types of microphysical schemes: specifically, a one-moment bulk microphysical scheme based on Lin et al. (1983) with prognostic equations for mixing ratios of cloud water, rain, ice, snow, and graupel/hail, and SBM (Khain et al., 2004b; Phillips et al., 2007b). The model resolution in the zone of interest was 1 km, and the timestep was 6 s. Figure 23.16 shows the surface rainfall time-domain plots obtained in these simulations. During the developing stage (the first 6 h), new convection is generated at the leading edge of the squall line. The trailing stratiform-rain area expands gradually until 6 h and then settles into a quasi-steady state. The development of the simulated systems agrees qualitatively with the surface radar observations of Rutledge et al. (1988), which show the sur-

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Fig. 23.16 Time-domain diagram of surface rainfall for the (left) bulk and (right) bin c American Meteorological Society (AMS) 2005, from Fig. 1 of Li scheme simulations.  et al. (2009a). Reproduced by permission of the AMS.

face precipitation area expanding to more than 180 km with an extensive stratiform-rain area within about 2 h. Figure 23.16 shows that the SBM has a much larger trailing stratiform area than the bulk scheme. These results agree well with those shown in Fig. 23.12. Consequently, the SBM indicates a much larger contribution of light rain to the total surface rainfall. The SBM did not generate rainfall rates higher than 90 mmhr−1 . During the mature stage, about 20% of the total surface rainfall was stratiform in the SBM compared with only 7% in the bulk simulation. According to observations by Johnson and Hamilton (1988), about 29% of the surface rain came from the stratiform region. Thus, the SBM describes the partitioning of rain into convective and stratiform components better than the one-moment bulk scheme. Another significant difference between the structures of the simulated squall line is the formation of high rainfall rate streaks in the bulk scheme, which extend from the leading convection well into the stratiform region, and the lack of these in the SBM simulations and in the observations. These high surface rainfall streaks are a manifestation of rearward-propagating convective cells. The appearance of a multicell structure of the squall line in the bulkscheme simulations is attributed by Li et al. (2009a,b) to errors in the evaporation rate and fall velocity of graupel, which can be tuned to make the results closer to those of the SBM. Figure 23.17 shows the vertical profiles of terms in the heat budget that are related to different microphysical

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processes. The budgets were calculated for the entire computational area. The bulk scheme leads to higher condensation and stronger evaporation than the SBM, and to similar rates of freezing and melting. In general, despite the very different local precipitation rates and spatial precipitation distributions, the accumulated rain was nearly the same: the area-averaged rain rate was 2.7 mmhr−1 and 2.4 mmhr−1 using the bulk scheme and the SBM respectively.

Fig. 23.17 Components of the microphysical heating profiles simulated by the bulk (dashed lines) and bin (solid lines) schemes for (a) condensation and evaporation, (b) c American Meteorological deposition and sublimation, and (c) melting and freezing.  Society (AMS) 2005, from Fig. 13 of Li et al. (2009a). Reproduced by permission of the AMS.

Note the large spikes in both deposition and sublimation at around 10 km in the bulk simulation, as seen in Fig. 23.17. These spikes are produced artificially by limitations in the saturation-adjustment scheme used by the bulk scheme. Although the spikes largely cancel each other out and have little effect on the total energy budget, the related heating and cooling can substantially affect the structure of the squall line when using the bulk scheme. Morrison et al. (2009b) simulated a squall line similar to that in Li et al. (2009a,b). However, in contrast to the studies of Li et al. (2009a,b), Morrison et al. (2009b) used two-moment bulk schemes, as described by Morrison et al. (2005b) and Morrison and Grabowski (2007), and implemented into the WRF. This scheme includes prognostic variables for the mixing ratio and the number concentration of graupel. The simulations were compared to others using a single-moment scheme. The single-moment scheme was formed from the two-moment one by diagnosing rather than predicting the

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number concentrations of the precipitation species. Thus, intercept parameters were specified for rain, snow, and graupel. A Hovm¨oller plot of the surface rainfall rate for the two-moment and one-moment simulations is presented in Fig. 23.18. The squall lines simulated in this study were much more intense that those in Li et al. (2009a,b). The widths of the convective precipitation zones exceeded 50 km and the fraction of light rain was much less than in the SBM.

Fig. 23.18 Hovm¨ oller plot of the surface rainfall rate for (a) the two-moment and (b) the one-moment simulation. The contour interval is 1 mmhr−1 for rates between 0 and 5 mmhr−1 and is 10 mmhr−1 for rates greater than 10 mmhr−1 . To highlight the stratiform-rain region, moderate precipitation rates between 0.5 and 5 mmhr−1 are c American Meteoroshaded grey. (Figure reproduced from Morrison et al. (2009b).  logical Society. Used with permission.)

The main result of the study was that the two-moment scheme produced a much wider and more prominent region of trailing stratiform precipita-

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tion, relative to the single-moment scheme. This difference was attributed to a decrease in the rain evaporation rate in the stratiform region when using the two-moment scheme. The difference in the rain evaporation rates reflects the difference in the shape of the raindrop distribution expressed in the values of the intercept and slope parameters. In the two-moment scheme where the intercept parameter was calculated, it ranged from 105 to 107 m−4 in the stratiform region, and from 107 to 109 m−4 in the convective region, compared with a constant value of 107 m−4 specified in the single-moment scheme. Larger values of the intercept parameter in the convective region are related to higher collision rates. In contrast, rain in the stratiform region was primarily produced by the melting of snow. The key point is that no single value for the intercept parameter for raindrops in the one-moment scheme was able to reproduce the results of the two-moment scheme. These results show again that a two-moment scheme produced more realistic results for the spatial distribution of precipitation than a single-moment one. They also show that the spatial precipitation distribution dramatically depends on the calculation of parameters determining the shape of PSDs. Khain et al. (2009) simulated the same squall line as Morrison et al. (2009b), but using WRF with two other microphysical schemes: the SBM scheme similar to that used by Khain et al. (2004b) and Li et al. (2009a,b), and the two-moment bulk scheme of Thompson et al. (2004, 2008). This two-moment scheme predicts mixing ratios of cloud water, rain, cloud ice, snow, and graupel. The components of the scheme were designed using a bin-microphysics parcel model. Droplet nucleation is diagnosed from the supersaturation, which is parameterized based on work by Abdul-Razzak et al. (1998). Autoconversion is treated following Berry and Reinhardt (1974b). Simulations were carried out with different CCN (and cloud droplet) concentrations. The dynamical and microphysical structure of the squall line simulated with WRF/SBM is quite similar to that simulated by Li et al. (2009a,b) using GCE/SBM. The bulk scheme produced substantially lower CWC in the case of low CCN concentration and substantially lower RWC in the case of high CCN concentration. Similar differences in the structure of squall lines simulated with SBM and with the bulk scheme were found as in the studies by Li et al. (2009a,b). Accumulated rain predicted by the SBM scheme was close to that predicted by the SBM scheme in the studies by Li et al. (2009a,b), but half that produced by the bulk scheme of Thompson et al. (2008). Both schemes predicted rain rates substantially lower than were obtained by Morrison

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et al. (2009b). These results support the conclusion that two-moment bulk schemes predict the spatial structure of rain in squall lines better than single-moment ones, and that the spatial distribution substantially depends on parameters determining the DSD shape. 6.6

Supercell storms

Khain and Lynn (2009) simulated a supercell storm using WRF with 2 km grid length and using the SBM (Khain et al., 2004b), as well as the Thompson bulk scheme. The computational area was (252 km)2 , and the maximum timestep was 10 s. Simulations were performed under clean, semi-polluted, and polluted conditions. To show that aerosol effects on precipitation depend on environmental conditions and to compare aerosol effects with the impact of other factors, the simulations were carried out with moderate environmental relative humidity (typical of the Great Plains) and high relative humidity (typical of the southern Gulf Coast). The relative humidity (RH) difference was about 10%. Maximum vertical velocities in the SBM simulations ranged from 25 to 40 ms−1 , whilst the Thompson scheme produced maximum vertical velocities ranging from 45 ms−1 to 65 ms−1 . Both schemes produced two branches of precipitation but in the SBM simulations the right-hand branch was dominant as opposed to the left-hand branch in the Thompson simulations (Fig. 23.19). Analysis shows that these differences in the precipitation structures were caused by differences in the vertical velocities, which led to the ascent of hydrometeors to different altitudes with different directions of the background flow. The bulk scheme produced values of accumulated rain twice as high as the SBM. The change in humidity led to a dramatic change in both the precipitation amount and distribution (Fig. 23.20a,b). A similar study dealing with the same supercell storm was carried out by Lebo and Seinfeld (2011). In their study, the results obtained using the WRF/MMM bin-microphysics scheme based on Reisin et al. (1996a) were compared with those obtained using the two-moment bulk scheme developed by Morrison et al. (2005a). Figure 23.20c,d shows the domain-average cumulative surface precipitation for these simulations. The cumulative surface rain was similar to the BM and the MMM at both high and low relative humidities. The accumulated rain amount produced by the Morrison et al. (2005a) bulk scheme was twice as high as that in the MMM. In both cases the responses of the accumulated rain amount to the changes in the CCN concentration were opposite in the bulk schemes and the SBM schemes.

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Fig. 23.19 A top view of accumulated surface rain simulated using the SBM (left) and the Thompson scheme (right) after 240 min under relatively high humidity conditions, for the clean (upper row) and polluted (lower row) cases. Note the different grey scales c American Geophysical Union 2009, from Figs. 6 for SBM and for the bulk scheme.  and 13 of Khain and Lynn (2009).

Similarly to the results of Khain and Lynn (2009), Lebo and Seinfeld (2011) explained the different responses of the SBM and the bulk schemes to changes in the aerosol loading through the different impacts of the microphysical processes on the latent heat release and subsequently on the vertical velocity, which in turn modifies the partitioning between liquid and ice phases.

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Fig. 23.20 Cumulative surface precipitation for (a) the SBM and (b) the two-moment c American Geophysical Union 2009, from Fig. 5 of Khain and Lynn bulk scheme.  (2009). Also shown is the domain-averaged cumulative precipitation in simulations (c) with the lower RH and (d) with higher RH using the bulk (black) and bin (red) microphysics models. The simulations were performed for the “Clean” (solid), “Semi-Polluted” (dashed), and “Polluted” (dotted) cases. From Figs. 3 and 9 of Lebo and Seinfeld (2011), c the authors, 2011, CC Attribution 3.0 License. which is 

Fig. 23.21 Surface accumulated amounts of hail (shaded contours) and rain (blue contours) at the end of the simulation (210 min) for (left) REG2M and (right) 3MHAIL. The contoured values are 1, 5, 10, 15, 20, 25, 30, and 35 kgm−2 . The CCN concentration is 600 cm−3 . (Reprinted from Fig. 25 of Loftus et al. (2014b), with permission from Elsevier.)

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Hailstorms

Hailstorms pose a serious hazard to agriculture and property in many places around the world. The scale of the damage rapidly increases with hail size. Since hail grows largely by accretion of supercooled water, the mass of supercooled water should substantially affect the size of hail particles. Hail suppression hypotheses suggest that in order to prevent the growth of initial hail embryos to hailstone size, the amount of supercooled water should be decreased (e.g., Heymsfield, 1982; Krauss, 1999; Wisner et al., 1972). Since an increase in the CCN concentration typically leads to an increase in supercooled water content, one can expect a substantial effect of the CCN on the mass and size of hail particles. Simulation of large hail requires an accurate reproduction of the tail of a wide hail PSD. Wisner et al. (1972) were the first modellers to develop a hybrid bulk/bin formulation in which hail is described using a significant number of bins. Here, the results of three advanced microphysical models are compared that were used for simulation of hailstorms producing hailstones with diameters of up to 5 cm and radar reflectivities of up to 70 dBz. A hailstorm in Villingen–Schwenningen, southwest Germany, on 28 June 2006 was simulated using two models: the two-dimensional SBM model HUCM (Ilotoviz et al., 2014; Khain et al., 2011; Kumjian et al., 2014), and the three-dimensional weather prediction model COSMO, which used the two-moment bulk microphysical scheme of Seifert and Beheng (2006a,b). To simulate big hail, a special hydrometeor class of big hail was introduced into the bulk scheme (Noppel et al., 2010a). Both models were able to reproduce the hailstorm. At the same time the dependence of hail parameters on aerosols turned out to be quite different. The radar reflectivity field simulated by HUCM with high CCN concentration agrees well with observations: the maximum radar reflectivity reaches 70 dBz, and high values of reflectivity occur at altitudes of up to 10–11 km (not shown). The SBM-simulated maximum reflectivity in the case of high CCN concentration is substantially higher than in the case of low CCN concentration. In contrast, the two-moment bulk scheme in COSMO predicts higher values of reflectivity and larger areas of high reflectivity in the case of low CCN concentration. In these simulations with the bulk schemes, radar reflectivity rapidly decreases with height above 6 km. Actually, the SBM and the two-moment schemes simulate different mechanisms of hail growth. While in the SBM for high CCN concentration hail grows largely by the accretion of high supercooled water content,

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in the two-moment bulk scheme hail forms largely through the freezing of raindrops at comparatively low levels above the freezing level. Both in the study of Khain et al. (2011) and in the simulations of the same storm using a new version of HUCM with explicit treatment of wet growth of hail (Ilotoviz et al., 2014), the diameter of hailstones in the case of high CCN concentration reaches 5 cm in agreement with observations. In the clean case, the diameter of hail particles was typically less than 2 cm. In the SBM simulations the hail shaft at the surface was much higher in the polluted case than in the clean one. The reason for a negligibly small hail shaft in the clean-air case is the melting of comparatively small hail particles. By contrast, in simulations with the two-moment schemes, the hail shaft in the clean-air case turned out to be substantially larger than in the polluted case. The inability of two-moment bulk schemes to describe the formation of big hail was also demonstrated by the study of Loftus et al. (2014a,b,c) who simulated a supercell storm that occurred over northwest Kansas on 29 June 2000 during the Severe Thunderstorm and Electrification and Precipitation Study (STEPS). To simulate big hail, a new 3MHAIL bulk scheme was developed, in which hail evolution is described using three PSD moments, while other hydrometeors are described using a two-moment scheme. The new scheme has been implemented in RAMS. The results of the 3MHAIL scheme were compared to two different two-moment schemes, one of which is the original two-moment bin-emulating scheme (REG2M) used in RAMS (Cotton et al., 2003), while the other is a simplification of the 3MHAIL scheme. All schemes managed to reproduce the formation of strong hailstorms, with maximum velocities up to 40 ms−1 , and in all simulations the integrated mass of hail in clouds is larger for lower CCN concentrations. At the same time, all of the two-moment bulk schemes were incapable of reproducing the observed high radar reflectivity at altitudes of 9–11 km. Large hail in the two-moment bulk schemes did not exceed 2 cm in diameter. Figure 23.21 shows surface accumulated amounts of hail and rain at the end of simulation with the original REG2M scheme and with the new 3MHAIL scheme. This shows that the surface precipitation in the case of the threemoment bulk scheme consists largely of hail, while the two-moment scheme actually does not have hail at the surface. The total precipitation is substantially higher in the 3MHAIL case. Comparison of calculated radar reflectivity, differential reflectivity, and hail size with observations indicates that the SBM and 3MHAIL schemes

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reproduced the microphysics of the hailstorm much better than the twomoment schemes. For both the SBM and the 3MHAIL schemes, hailstones were produced with diameters of several cm in the case of high CCN concentration. Moreover, since small hail particles lose a significant amount of their mass during melting, hail precipitation at the surface level in the polluted case was substantially larger than with the two-moment schemes. Simulations of a hailstorm using the three-moment bulk scheme by Milbrandt and Yau (2006) also supported the ability of three-moment schemes to simulate large hail stones. The results of these studies show that the simulation of large hail requires better description of the PSD tail and makes necessary the use of at least three moments of the hail PSD to simulate the formation of hail of several cm in diameter. At the same time, the location of precipitation, as well as the approximate accumulated rain mass can be described using two-moment bulk schemes. 6.8

Simulation of hurricanes

The intensity of hurricanes depends on the rate of latent heat release, as well as on its spatial distribution. Taking into account the dominant role of latent heat release, tropical cyclones are ideal phenomena to test and to compare different microphysical representations (e.g., Tao et al., 2011). The effects of microphysical representations are illustrated using as an example Hurricane Irene, which moved northwards along the US coast during the second half of August 2011. Figure 23.22 shows the evolution of minimum pressure and maximum wind speeds in WRF simulations using different bulk schemes. The spacings of the fine nested grids were 3 km and 1 km. The high diversity of time dependencies is suggestive of the substantial sensitivity of tropical cyclone (TC) intensity to the treatment to cloud microphysics. Variations produced by different bulk schemes are large: up to 30 mb and 40 kt (20 ms−1 ). It is also possible to see that the reduction in grid spacing from 3 km to 1 km led to an intensification of the model hurricane and actually made the intensity forecast worse. The results shown in Fig. 23.22 were obtained in simulations that did not take account of the decrease in sea-surface temperature caused by the hurricane-ocean interaction. Accounting for the ocean coupling led to better agreement with observations in all simulations with 1 km resolution. A specific feature of Irene was that the wind speed reached its maximum

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Fig. 23.22 Evolution of minimum pressure and maximum wind speeds in WRF simulations using different bulk schemes. The resolutions of the fine nested grids are 3 km (upper row) and 1 km (lower low). Observed values are also presented by the black solid line. “Thompson” denotes the bulk scheme of Thompson et al. (2008); “Milbrandt” denotes the scheme of Milbrandt and Yau (2005a,b); “Morrison” denotes the scheme of Morrison et al. (2009b); “SBU-YLin” denotes the scheme of Lin and Colle (2011); “WDM6” denotes the scheme of Lim and Hong (2010); “WSM6” denotes the scheme of Hong and Lim (2006); and, “Goddard” denotes the scheme of Tao et al. (1989) (from Lynn et al., 2014).

Fig. 23.23 The maximum wind plotted against the minimum pressure in 1 km grid length simulations of Hurricane Irene using the WRF model with TC–ocean coupling taken into account. The dependencies are plotted for different microphysical schemes. The dependence calculated using observed data is presented as well (from Lynn et al., 2014).

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about two days earlier than the surface pressure reached its minimum value. Figure 23.23 shows the relationship between maximum wind and minimum pressure in the 1 km ocean-coupled simulations with different bulk schemes. The observed dependence is also presented; the WRF/SBM reproduces the hysteresis reasonably well. Among the bulk schemes, the two-moment Thompson scheme produces the best agreement with observations. The time shift between the maximum wind and the minimum surface pressure is caused by fine changes in the TC structure. A good reproduction of the maximum wind against minimum pressure by WRF/SBM indicates the importance of high-quality microphysical schemes to predict TC intensity (at least in the case tested). Conclusions following from the simulations of hurricanes are that: the intensity of tropical cyclones in numerical models dramatically depends on the microphysical scheme used; the better the model resolution used, the better the model physics that is required to reproduce the observed TC behaviour; and, the use of SBM allows better reproduction of TC intensity and structure than bulk schemes. Simulations of hurricanes performed using different aerosol concentrations (not shown) indicate that an increase in the CCN concentration at the TC periphery lead to TC weakening (e.g., Carri´ o and Cotton, 2011; Cohen and Khain, 2009; Cotton et al., 2012; Hazra et al., 2013; Khain et al., 2010; Rosenfeld et al., 2012; Tao et al., 2011; Zhang et al., 2007, 2009) while an increase in CCN concentration in the inner TC core leads to TC intensification (e.g., Herbener et al., 2014; Lynn et al., 2014). To take these effects into account, as well effects of aerosols on lightning within TCs (Khain et al., 2008b, 2011), it is necessary to advect aerosols in order to predict the location of the zones of enhanced and decreased CCN concentrations. 6.9

Simulation of microstructure of mixed-phase clouds and cloud systems

The important role of satellites and meteorological radars in monitoring, retrievals, and forecasts is well known (cf., Ch. 16). In order to interpret signals obtained from passive and active remote sensing, numerical microphysical models are needed as a connecting link between satellite (or ground-based radar) measurements and microphysical and thermodynamic fields. Many studies have been dedicated to the development of retrieval algorithms and to the investigation of radiation and microphysical processes using atmospheric models of varying complexity from one-dimensional SBM

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models (e.g., Kumjian et al., 2012) to meso- and large-scale models (e.g., Alexandrov et al., 2012; Iguchi et al., 2008, 2012b,a, 2014; Li et al., 2010; Saito et al., 2007; Sato and Okamoto, 2006; Suzuki et al., 2010a, 2011; Tao et al., 2003). Many of these studies use cloud-resolving atmospheric models in which bin microphysics is employed. In some of these studies the results obtained using the SBM are compared with those obtained using bulk schemes, while in others various bulk schemes are compared against observations. Two example studies from the above list are discussed below. Iguchi et al. (2012b) simulated three precipitation events using the Japan Meteorological Agency Non-Hydrostatic Model (JMA-NHM). These events were also observed by ship-borne and space-borne W-band cloud radars of the CloudSat polar-orbiting satellite. The bulk microphysical scheme originally included in JMA-NHM was compared to the SBM (Khain et al., 2004b) analysis. This bulk scheme is single-moment and accounts for two explicit classes of water and three of ice: specifically, cloud water, rain, cloud ice, snow, and graupel (Eito and Aonashi, 2009; Ikawa and Saito, 1991; Lin et al., 1983). Figure 23.24 shows normalized reflectivity-height histograms obtained during the period 1200 UTC 22 May to 1200 UTC 23 May 2001. It can be seen that the SBM reproduces the observed vertical distribution of radar reflectivity much better than the bulk scheme. Preliminary simulations with the SBM showed that to improve the representation of Ze-height histograms it was necessary to take into account an increase of the fall velocity of snow due to riming (Fig. 23.24c). This example illustrates that model microphysics, in this case the SBM, can be improved with regards to the representation of lesser-known processes such as ice–ice and aggregate–drop collisions using radar data. The use of space-borne radar was further demonstrated by Suzuki et al. (2011) who compared various warm-rain process rates from the singlemoment scheme in the NICAM model, the double-moment scheme in the RAMS model, and observational data from the CloudSat cloud precipitation radar (CPR) and MODIS (Moderate resolution Imaging Spectroradiometer). The NICAM (Non-hydrostatic Icosahedral Atmospheric Model) global CRM here includes an implementation by Suzuki and Stephens (2008) of the aerosol transport module from the Spectral Radiation Transport Model for Aerosol Species (SPRINTARS, Takemura et al., 2002) in order to represent cloud–aerosol interactions. Three precipitation categories of no-precipitation, drizzle, and rain were defined according to nonattenuated near-surface radar reflectivity. The fractional occurrence of these precipitation categories and the probability of precipitation were then

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Fig. 23.24 Normalized Ze-height histograms (with Ze in dB) constructed from: (a) radar measurements; and, simulations using (b) bin microphysics (control); (c) bin microphysics with rimed snow; and, (d) the bulk model, during the period 1200 UTC 22 c American Meteorological Society (AMS) 2012, from May to 1200 UTC 23 May 2001.  Fig. 4 of Iguchi et al. (2012a). Reproduced by permission of the AMS.

examined as a function of various cloud characteristics including the droplet size, the liquid-water path (LWP), and the droplet number concentration for both schemes and the observations. The analysis was facilitated through the use of model radar simulators. An example of this analysis is shown in Fig. 23.25.

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Fig. 23.25 Fractional occurrence of no precipitation, drizzle, and rain categories as a function of cloud LWP obtained from: (a) satellite measurements (A-Train, Stephens et al., 2002); (b) NICAM-SPRINTARS; and, (c) RAMS. Also shown is the sum of the contributions from the drizzle and rain categories. (Figure reproduced from Suzuki et al. c American Meteorological Society. Used with permission.) (2011). 

It is evident that both NICAM and RAMS tend to convert cloud water to larger hydrometeors more rapidly than the observations suggest. The drizzle-to-rain transition in RAMS appears to be more realistic than in NICAM, leading to a better rain fraction in RAMS. However, ultimately,

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while the two-moment RAMS scheme performed better than the singlemoment NICAM scheme, both bulk schemes could do better in reproducing the cloud-to-rainwater conversion processes. The too-rapid conversion of cloud water to rain when compared with observations appears to be a common problem with bulk schemes. 6.10

Simulation of microstructure of Arctic stratiform clouds

The investigation of microphysical properties of Arctic stratiform clouds has been the focus of many studies owing to the important role of these clouds in the radiative budget and climate (e.g., Fridlind et al., 2007; Prenni et al., 2007; Shupe and Intrieri, 2004; Verlinde et al., 2007). The degree of sensitivity of the phase composition of such clouds to the concentration of IN varies between models. Model intercomparison studies (e.g., Klein et al., 2009; Luo et al., 2008; Morrison et al., 2009a, 2011; Muhlbauer et al., 2010) reveal a dramatic dispersion of results from different models in predicting the liquid/ice partitioning under the same environmental conditions. Among results obtained using sophisticated LES models, the best agreement with observations was obtained by Avramov and Harrington (2010) using the LES model DHARMA-bin with the BM scheme by Ackerman et al. (2009). Ovchinnikov et al. (2014) performed an intercomparison of seven LES models with different microphysical formulations. The models were used to simulate Arctic clouds observed during the Indirect and Semi-Direct Aerosol Campaign (ISDAC, McFarquhar et al., 2011). Two of the models used SBM, namely SAM-SBM (Khain et al., 2004b) and DHARMA-bin. The large uncertainty in ice nucleation was excluded (the ice concentration was tuned to the observed value), but significant differences between models remained in depositional growth rates and precipitation fluxes, as well in the values of the liquid-water path and ice-water path (IWP). The differences are related to differences in the model representations of the basic processes in mixed-phase stratiform clouds, namely diffusional growth/deposition and evaporation/sublimation. The dispersion in the results is illustrated by Fig. 23.26 which compares the evolution of LWP and IWP in the set of simulations “ice4” with target ice-crystal concentration of 4 l−1 . A wide spread of values of the LWP can be seen, whilst the IWP reaches a stationary state. The values of IWP in SBM simulations are substantially larger than with bulk formulations.

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Fig. 23.26 Time dependencies of LWP and IWP obtained using different models, and with ice-crystal concentrations of 4 l−1 . The models used are SAM-bin (PNNL, with SBM scheme), DHARMA-bin (DHARMA, with bin scheme), WRF-LES-PSU (Penn State), WRF-LES (NOAA), RAMS (Penn State), UCLA-LES-SB (Stockholm University), UCLA-LES (NASA Langley), COSMO (Karlsruhe University), METO (Met. Office, UK), SAM-2M (PNNL, with bulk scheme), and DHARMA-2M (DHARMA, with bulk scheme). Note that SAM-2M, DHARMA-2M, UCLA-LES, and WRF-LES all emc American Geophysical Union 2014, from ploy the same bulk microphysics scheme.  Fig. 3 of Ovchinnikov et al. (2014).

It is interesting to note that the different SBM schemes produced similar results. Ovchinnikov et al. (2014) found that bulk schemes produce better agreement with bin schemes when ice-size spectra are approximated by gamma distributions with widths comparable to those predicted by the bin schemes. Thus, the results show that to simulate the main microphysical parameters of mixed-phase stratocumulus clouds, the accurate simulation of the PSDs of ice crystals and droplets is required.

7 7.1

Discussions Areas of application and ways to improve microphysical schemes

Two approaches to the description of microphysics have arisen almost simultaneously with two different goals in mind. Bulk formulations have been developed to describe microphysics in cloud-resolving simulations, while SBM has been developed to enable detailed investigations of cloud microphysical processes. Originally, these directions in cloud modelling were well separated. However, further developments have been realized by interactions between the two approaches. Individual elements of all modern bulk

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schemes were developed and calibrated against bin analogues. Thus, it would be appropriate to consider ongoing developments as a mutual interaction between the approaches rather than their competition. The comparison between these approaches, as made in this chapter, provides useful information about the strengths and weaknesses of each. In spite of the fact that the overlap in the areas of application of bulk formulations and SBM continually increases, there are still wide areas where the use of one or another method is clearly preferable. Bulk microphysics schemes have been successfully fulfilling their objective of replacing the traditional schemes of convection parameterization in cloud-resolving models. This goal, however, is very far from being accomplished. Most large-scale models are not cloud resolving, and the natural first step with the improvement of model resolution is to use bulk schemes, as being more computationally efficient compared to SBM. In global circulation models, as well operational weather forecasts, the only reasonable solution at present is to use bulk schemes, or else to improve the traditional schemes using formulations coming from components of bin approaches (e.g., Hoose et al., 2010; Khairoutdinov and Kogan, 2000). This is especially true because some important parameters such as temperature, humidity, and, to some extent, precipitation can be predicted by bulk schemes with a precision that is often satisfactory from the practical point of view. The results of bulk/bin comparisons (Li et al., 2009a,b; Morrison et al., 2009b; Seifert et al., 2006; Wang et al., 2013) have shown that practically important quantities for weather forecasting such as accumulated rain under non-extreme conditions are likely determined largely by concentration and mass contents of hydrometeors. At the same time, precipitation rate, type of precipitation (liquid or ice), and spatial distributions of precipitation have been found to be strongly dependent on parameters of the gamma distributions (i.e., on the PSD shape). Comparison of bulk schemes with their bin analogues, as well as between bulk schemes themselves, indicate several main directions to improve the representation of microphysics in cloud-resolving models as well as forecast skill. Among them may be highlighted: • implementation of the aerosol (CCN) budget (i.e., aerosol transport, nucleation scavenging and (desirably) recycling or regeneration); • use of two-moment (with variable parameters of the PSDs) or threemoment schemes that dramatically improve the representation of particle sedimentation and size sorting as compared to one-moment schemes;

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• improvement of the representation of autoconversion and accretion processes; and, • improvement of the representation of melting of hydrometeors. The next important step towards improved accuracy of bulk schemes would be wide implementation of diffusional growth/evaporation (e.g., as is currently incorporated in the bin-emulating bulk scheme of RAMS) instead of using the saturation-adjustment assumption. Note that results of bulk schemes depend on the choice of the moments of the PSD that are used (Milbrandt and McTaggart-Cowan, 2010). The optimum choice of moments is another route to bulk-formulation improvement. Bulk formulation to SBM comparisons indicate that the saturationadjustment assumption exaggerates the intensity of latent heat release and vertical velocities and may lead to an overestimation of convective relative to stratiform precipitation. The assumption of saturation adjustment substantially simplifies the algorithm of diffusional growth, but makes new nucleated droplets indistinguishable from old cloud droplets and may lead to unrealistic DSDs. Moreover, saturation adjustment prevents in-cloud nucleation, which actually plays a very important role in clouds. Thus, avoiding the supersaturation adjustment by implementation of the diffusional growth equation is a promising option. Nonetheless, the use of higher resolutions and smaller timesteps seems to be the fastest way to improve the results of models that use bulk schemes. For instance, the current skill of TC-intensity forecasts using HWRF was attained partially by an increase in the model grid length from 9 to 3 km and partially by improving the bulk microphysical formulation1 . The development of bin-emulating schemes improves the representation of each microphysical process. In principle, most microphysical processes should lead to a deviation of the PSD from the gamma distribution. Restoring a gamma distribution at each timestep after applying bin-emulating procedures leads to a significant decrease in the computer time during the advection substep, but may reduce the benefits of the bin-emulating algorithms. Another important way to improve the skill of models with bulk formulations is the use of empirical data to tune the bulk schemes to certain meteorological conditions or phenomena (e.g., hurricanes), or to eliminate biases by statistical analysis of forecasts in particular geographic regions. A substantial improvement could potentially be reached through the use 1 See

http://www.emc.ncep.noaa.gov/?branch=HWRF.

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of a different set of governing parameters of bulk formulations in different geographical regions (sea, land, urban, etc.). Such tuning of the parameters can also be performed by comparisons with SBM models. SBM schemes have an obvious advantage over the current bulk schemes in a wide range of research problems where knowledge of the PSD shapes is of importance. Analysis of the results of observations and numerical studies shows that the shape of the local PSD often cannot be described by a gamma distribution. Even in cases when particular size distributions can be approximated by gamma or exponential distributions, the parameters of these size distributions often vary dramatically in space and time during cloud evolution so that their values in the models can be determined only approximately. Fixing some of the distribution parameters does not allow reproduction of realistic distributions. Moreover, many microphysical processes foster deviation of the PSD from the gamma or exponential form. At present, there is no firm physical basis for selecting a specific function (gamma, lognormal etc.) to describe particle size distributions in numerical simulations of clouds, although the maximum-entropy principle may prove helpful for constructing such a theory (cf., Liu et al., 1995). The ability of SBM models to calculate PSDs has made them powerful for detailed investigations of cloud microphysical processes. The potential of the current bin-microphysics schemes is far from being exhausted, with the knowledge being accumulated allowing for further elaborations and improvements of the schemes. Large-eddy simulations combined with bin microphysics have been used for developing the expressions for the rates of autoconversion and accretion in mesoscale and large-scale models, and for the formulations for describing haze, homogeneous freezing, etc. (e.g., Hoose et al., 2010; Khairoutdinov and Kogan, 2000). Application of binmicrophysical models has also allowed effects to be discovered that were only found later in observations. Examples include aerosol-induced convective invigoration, an increase in hail size in polluted clouds, and the formation of the first raindrops in undiluted cloud volumes (Khain et al., 2003, 2004b, 2011, 2013). Only SBM models are able to represent the microphysical structure of mixed-phase stratiform clouds (Ovchinnikov et al., 2014). A great number of studies investigating aerosol effects have been performed using bulk schemes (largely two-moment ones). In some studies a similar response of precipitation amount and location to varying CCN amounts was found (e.g., Noppel et al., 2010b). At the same time, it could be said that SBM models can simulate the processes of cloud–aerosol inter-

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action more accurately. Obtaining a response to variations of aerosol size and concentration that is comparable to the SBM response is considered an improvement of the bulk scheme (e.g., Fan et al., 2012a; Wang et al., 2013). Increases in computing power have allowed for multiple month-long cloud-resolving simulations with spectral-bin cloud microphysics that capture the observed macrophysical and microphysical properties of summer convective clouds and precipitation in both the tropics and mid-latitudes (Fan et al., 2011, 2013). These studies provide a comprehensive view of how aerosols affect cloud cover, cloud-top height, and radiative forcing. The study by Fan et al. (2013) clearly indicates the ability of the SBM models to investigate some of the most intricate climatic problems. At present SBM can be successfully used for simulations of the intensity of dangerous phenomena such as storms, hail, and flooding. The SBM approach is indispensable for the development of retrieval algorithms for remote sensing, relating Doppler and polarimetric radar data as well as satellite observations with cloud microphysical parameters. Importantly, radar parameters are highly sensitive to tails of PSDs, and particle shape and phase composition. Note that climate models require higher accuracy in the description of microphysical processes than weather prediction models. Indeed, in several studies a substantial variation of aerosol concentrations led to changes in spatially and time-averaged precipitation by 7–10%. Such evaluations are attributed to the averaged accumulated precipitation: at small timescales, precipitation fluctuations caused by aerosols are much larger. However, even if a 10% effect is not significant in weather forecasts, 7–10% variations of latent heat release (corresponding to the variation in precipitation amounts) for climate problems is of high importance. Although the use of SBM over the entire globe in GCMs is unlikely to take place in the foreseeable future, SBM could be used for climatic studies (say, seasonal prediction of tropical cyclone activity) within limited areas. As discussed in Sec. 4, further progress of both bulk and SBM schemes depends on laboratory, radar, satellite, and in situ measurements that can provide new knowledge on processes in mixed-phase and ice clouds, ice nucleation, mixing, etc. New and better particle probes are clearly required. Shattering of ice crystals is a notable issue that has been resolved only recently. Ice crystals exhibit an extremely wide range of habits. It is well known that details of the shapes of ice have profound impacts not only on their subsequent growth, but also on their interactions with radiation (by

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scattering of the visible light by crystals). In most microphysical schemes, a relatively simple shape (or set of them) for ice particles is assumed. Ice crystals do not at all follow the pure forms listed in textbooks, but rather they are more than often defected. If the ice-crystal shapes have profound impacts, we need very careful computations of ice-crystal growth in order to treat the growth rate correctly. The implication is potentially significant, because if this argument is correct, we not only need to introduce enough bins, but in particular those with enough ice-shape types. It might be for some specific goals that more types of ice crystal are required. Further progress in SBM strongly depends on obtaining in situ microphysical measurements in deep convective clouds and mixed-phase stratiform clouds. These data should include PSDs of different hydrometeors and aerosols. In addition to in situ measurements, satellite remote sensing and ground-based Doppler polarimetric radar measurements are of crucial importance. Since the major microphysical equations are known and used in the SBM, the SBM models can assimilate new findings comparatively easily. As examples of observed data leading to improvements of SBM, reference is made to studies by Freud and Rosenfeld (2012) and Rosenfeld and Gutman (1994) that triggered investigations of the mechanism of formation of the first raindrops in cumulus clouds; by Li et al. (2010) showing the necessity to take into account rimed mass in aggregates and its effect on fall velocity; and, by Kumjian et al. (2012) indicating the necessity of applying time-dependent freezing. It is quite a typical situation that the necessity to interpret remote-sensing signatures produces further development of SBM schemes. A recent example is the development and implementation of a novel, detailed scheme of time-dependent freezing into the HUCM that allowed explanation and reproduction of detailed laboratory experiments of hail wet growth (Phillips et al., 2013), as well as the formation of differential reflectivity columns (Kumjian et al., 2014). 7.2

Computational aspects

The choice of “bulk vs bin” is closely related to the compromise “computer time vs accuracy”. Prescribing the master functions for PSDs makes bulk schemes computationally efficient. At the same time, this assumption imposes limits on the accuracy of the bulk schemes. The comparisons presented in this chapter show that the use of several tens of bins in the SBM allows the reproduction of PSDs better than onemoment and two-moment schemes. It seems that three-moment schemes

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are substantially closer to the SBM. It is possible that using four or five moments of the PSD will produce accuracy similar to that obtained currently using several tens of bins. In this case, however, the computational costs of the SBM and the bulk approaches would become comparable. The requirements of a longer computational time for SBM schemes are related not only to the calculation of PSDs, but also to the use of timesteps which are typically less than 10–15 s (particular microphysical processes such as diffusional growth/evaporation are typically calculated with timesteps as short as 1 s and in some especially accurate schemes even shorter). Typical timesteps used in bulk schemes are of several tens of seconds, with some exceptions (for instance, the current COSMO model of the German weather service works with a timestep of 6 s and a horizontal grid length of 2.8 km). Small timesteps and high spatial resolution are required not only for the numerical stability of SBM schemes, but also for the accurate description of microphysical processes having small characteristic timescales. For instance, the characteristic phase relaxation time (the characteristic time of supersaturation adaptation to a quasi-stationary value) in liquid clouds typically varies from 1 to 10 s (Korolev and Mazin, 2003). It is widely, although possibly wrongly, assumed that models with bulk schemes can be integrated using timesteps substantially larger than those required by SBM models. Note that characteristic timescales (for instance, the timescale of drop evaporation) may be quite small, so the use of explicit finite-difference schemes with timesteps of several minutes may lead to the appearance of negative values of mass contents and/or concentrations (Sednev and Menon, 2012). The replacement of these negative values by zeros during the model integration would break mass conservation. On the other hand, the use of positive-definite schemes also leads to distorting internal fluxes. Sednev and Menon (2012) calculated maximum values of timesteps that can be used in different bulk schemes for the description of particular microphysical processes and showed that at certain mixing ratios of hydrometeors the maximum possible timesteps decrease to a few seconds. Similar remarks apply for other processes, such as sedimentation and collisions: too-large timesteps can lead to errors in the estimation of particle size. Note that all the proposed methods for improving bulk schemes (applying more bin-emulating procedures, using more PDF moments, implementing aerosol budgets, etc.) will also increase the computational time. The number of microphysical equations used in SBM is proportional to the number of bins. As a result, the number of equations is larger by

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an order of magnitude than in bulk schemes. Moreover, the number of operations needed for the description of collisions is proportional to the square of the number of bins. There are, however, several ways to decrease the computational expense of bin-microphysics models. One of the main ways is the proper design of parallel calculations. 7.3

Design of microphysics in large-scale models

Improvement of the representation of convection and microphysics in largescale and climate models can be accomplished only in association with improvements of cloud representation. In current operational models, the clouds are treated by two separate schemes labelled as “clouds” and “convection”. The former scheme handles those clouds almost resolved by a model, with stratiform-type clouds mostly in mind. A cloud fraction parameter is usually introduced into the scheme so that a cloud does not need to fill a whole grid box. Full microphysics can easily be implemented into such a cloud scheme, if the resolution discrepancy is not an issue. Convective clouds are treated separately by a convection scheme, the main purpose of which is to evaluate the convective transport of total entropy and moisture. The mass-flux formulation, typically adopted for this purpose, can handle the problem very well as long as these two quantities are conserved. The convective fraction is a rather ill-defined quantity within this formulation, and so non-conservative processes, including microphysics, are often treated in a rather ad hoc manner. Furthermore, under a standard steady-plume hypothesis no convective life cycle is considered in these convection schemes (cf., Vol. 1, Ch. 7, Sec. 6.4). Thus, it would be physically inconsistent to implement full microphysics that anticipates time-evolving convective dynamics, although it may be technically possible. The one-dimensional cloud models which are used in traditional schemes for the representation of mass fluxes and the calculation of convective heating, cooling, and precipitation are very simple and often unrealistic. Perhaps the first attempt to implement a reasonable description of microphysical processes into such elementary cloud models was undertaken by Zhang (2014). This description uses elements of two-moment bulk schemes. At the same time, some properties of the elementary cloud models remained as in the old schemes: cloud is assumed to be quasi-stationary and new clouds form at each timestep, so that the model has no memory. Even such a simplified scheme allowed substantial improvements in the representation of precipitation in a GCM. An important way to improve traditional

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convection parameterizations could be the tuning of parameters of the elementary cloud models by comparison with the results of LES cloud simulations with SBM or the best bulk schemes appropriately averaged over space and time. Super-parameterization as well as the non-hydrostatic anelastic model with segmentally constant approximation (NAM-SCA, Yano et al., 2010) approaches can circumvent these difficulties. The computing cost is clearly an issue in practical implementations of super-parameterization. By contrast, the SCA approach is computationally relatively cheap and practical. Aerosol advection and cloud–aerosol interaction accompanied by aerosol scavenging due to drop activation should also be taken into account in largescale models. Despite many efforts, the representation of aerosol effects (as well as cloud microphysics altogether) in large-scale and climate models remains a major source of uncertainty (e.g., Carslaw et al., 2013). The sink of cloud water in warm clouds is determined by processes of autoconversion and accretion. While the rate of autoconversion depends on droplet concentration (i.e., it is affected by aerosol concentration), the rate of accretion is determined largely by the mass contents of liquid cloud water and rainwater. Thus, aerosol effects are dependent on the relation between the rates of autoconversion and accretion. This relation is different for stratocumulus and cumulus clouds. The effects of these processes in a GCM was recently investigated by Gettelman et al. (2013) using a stationary bulk-scheme cloud model (Wood et al., 2009) mimicking the microphysics in the GCM. It is desirable to take into account the effects of turbulence on the rate of autoconversion. A close collaboration of scientists involved in large-scale and microphysical problems is required for further progress in this direction. In this chapter, the problems of representation of microphysics in cloudresolving models have been discussed. Comparisons between different methods have been made for comparatively short time and spatial scales. For climate investigations, averaging of such results is desirable to understand the effects on longer timescales and larger spatial scales. It is necessary to understand whether differences are amplified due to internal instabilities (i.e., self-organized criticality) or rather smoothed out due to a self-regulating nature of the system (i.e., homeostasis: Vol. 1, Ch. 4, Sec. 3.5). This is an important question which is yet to be addressed (cf., Vol. 1, Ch. 4). There are many examples showing that for certain environmental conditions the dissipation of one cloud creates downdraughts leading to the formation of new clouds. Note that maritime convective clouds are typically closely

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related. Squall lines represent classical cases of self-organization of convection under certain wind conditions. There are a number of good reasons to think that self-organization is a widespread phenomenon in the atmosphere (cf., Moncrieff, 2013; Yano, 1998). The development of secondary clouds during dissipation of the primary ones can depend on aerosol loading. For instance, Khain et al. (2005) reported the formation of a squall line after the dissipation of a primary cloud in a case with high CCN concentration and the dissipation of a primary cloud without formation of a squall line in the same case but with low CCN concentration. In cases where aerosols are implicated in the triggering of secondary clouds, their effects on precipitation increase substantially. In this sense, the effects of aerosols on single clouds may substantially differ from the effects of aerosols on cloud ensembles. Simulations of cloud ensembles indicate that aerosols tend to make convection more organized, intensify deep convection, and decrease the number of small clouds (e.g., Lee et al., 2008). This problem requires further investigation. It is worth noting, however, that in large-scale models with traditional parameterization of convection the dynamical interaction between clouds at the cloud scale is not taken into account. Simple cloud models used for the representation of convection typically do not take account of wind shear.

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Chapter 24

Cumulus convection as a turbulent flow

A. Grant Editors’ introduction: An alternative approach for understanding cumulus convection is to treat it as a turbulent flow, within which condensation and other microphysical processes occur. Similarity theories are often used as a valuable guide to the study of turbulent flows, and this chapter will describe a similarity theory for cumulus convection. This leads to the idea that a convection parameterization can be formulated by treating it analogously to the formulation of a boundary-layer parameterization. Mass-flux convective parameterizations focus their attention on the clouds and the associated cumulus updraughts and essentially neglect the blue sky in between, which is considered to be a rather passive background environment. Conventional mass-flux schemes also make the assumption that the fractional area covered by updraughts is small, although its magnitude is undetermined (leading to some issues with microphysics, for example, as already extensively discussed). Some longstanding issues for massflux parameterizations concern the determination of the fractional area, of the in-cloud velocity of cumulus updraughts, and of the cloud-base mass flux. The similarity approach, based on scaling the turbulent kinetic energy budget, offers some intriguing possible solutions to these problems.

1

Introduction

The development of large-eddy models has made it possible to study the flows associated with shallow and deep convection in much greater detail than is possible with observations. There have been many studies that have used large-eddy models to diagnose the properties of cumulus updraughts 359

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and downdraughts that form the basis of current mass-flux schemes, and they show that the mass-flux framework provides a good, diagnostic description of cumulus convection (e.g., Kuang and Bretherton, 2006; Romps, 2010; Siebesma and Cuijpers, 1995). However, despite such studies, the parameterization of cumulus convection using mass flux schemes remains a significant problem. The key uncertainties in the parameterization of moist convection can be summarized by the following questions: (1) Why do the cumulus updraughts occupy only a small fraction of the overall area of convection, and what determines this fraction? (convective fraction problem) (2) What determines the typical vertical velocity of cumulus updraughts, which is related to the question as to what determines the magnitude of the entrainment rate? (convective vertical-velocity problem) (3) What determines the cloud-base mass flux, or the intensity of convection? (closure problem) Although there are many problems beyond these three questions that need to be addressed in the development of parameterizations of convection (e.g., how to represent microphysical processes: see Ch. 18), the problems referred to are among the most fundamental. For example, the small fractional area appears to be a fundamental property of moist convection and underlies the mass-flux approach to convective parameterization. However, beyond the assumption that it is small it does not play any significant role in current parameterizations (see Vol. 1, Ch. 7). The cumulus entrainment rate and the mass-flux closure were discussed in detail in Vol. 1, Chs. 10 and 11 respectively, and it is clear that these are also poorly understood areas in convective parameterization. These uncertainties in our understanding of cumulus convection remain some forty years after the publication of Arakawa and Schubert (1974) because the mass-flux approach to parameterization does not provide the theoretical framework required to understand processes occurring in convective flows. Processes such as entrainment are included in parameterizations because observations show that liquid-water contents in cumulus updraughts, for example, are much less than would be expected from adiabatic ascents, even in shallow cumulus clouds (Neggers et al., 2003; Raga et al., 1990). In this chapter, answers to the three questions posed above will be obtained by treating cumulus convection as a turbulent flow. This means that the cumulus updraughts and their environment will be treated as a

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single turbulent flow. In particular, it will not be assumed that the cumulus updraughts are embedded within a passive environment; rather the environment is seen as an important component of the flow.

2

Shallow, non-precipitating convection

The equations for large-scale flow, in horizontally homogeneous conditions can be written as: ′ ∂w′ θvl ∂θv ∂θv +w =− +R ∂z ∂z ∂z ∂w′ qT′ ∂q ∂q +w =− , ∂z ∂z ∂z

(2.1) (2.2)

where θv is the mean virtual potential temperature, q is the mean humidity, w is the mean vertical velocity, θvl is the liquid-water virtual potential temperature, qT is the total liquid-water content1 , and R is the radiative cooling rate. The reason for this particular choice of the variables, in particular θv , will become apparent as the chapter proceeds. The primes in Eqs. 2.1 and 2.2 denote fluctuations about the mean and the overbars an average over space and time. In addition to the assumption of horizontal homogeneity the area-averaged liquid-water content is also assumed to be small (Betts, 1973), which will be true if the fractional area covered by the saturated cumulus clouds is small. ′ and The parameterization problem is to relate the turbulent fluxes w′ θvl ′ ′ w qT in Eqs. 2.1 and 2.2 to the mean flow. Unfortunately, there are few sufficiently detailed observations to aid the development of such a parameterization and so we must turn to large-eddy models. A large-eddy model is a high-resolution numerical model which resolves the main structures of the flow, such as cumulus clouds (the large eddies) and parameterizes the effects of the smaller scales. Because of the high resolution the effects of the small scale motions are mainly to dissipate turbulent energy, with the energy and transports associated with these scales being small compared to the large eddies which are of interest. With adequate resolution, the properties of the large eddies are not sensitive to the details of the subgrid parameterizations used to represent the effects of the small, unresolved scales (e.g., Siebesma et al., 2003). 1 This notation is a departure from that used in most of this set, in which q is a mixing ratio.

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Given the simulated flows generated from the large-eddy model it is necessary to have a framework in which to analyse their properties. Similarity theories provide one method for this. Similarity theories are used extensively to study boundary-layer turbulence and they form the basis for many boundary-layer parameterizations. In this chapter, a similarity theory for cumulus convection will be developed. 2.1

A similarity theory for cumulus convection

The structure of the cumulus-capped boundary layer is illustrated schematically in Fig. 24.1. The main features are as follows: (1) A layer below the cloud base which has a similar structure to the cloudfree convective boundary layer. The turbulent mixing in this layer is driven by the surface buoyancy flux. The top of the mixed layer is close to the lifting condensation level of the near surface air. (2) In the layer between the cloud base and the base of the inversion the virtual potential temperature gradient is less than the virtual moist adiabatic lapse rate of mixed-layer air. The convective available potential energy (CAPE) for the adiabatic ascent of mixed-layer air through this layer is non-zero. (3) The inversion layer separates the cloud layer from the free troposphere. The gradient of the virtual potential temperature in the inversion is greater than the virtual moist adiabatic lapse rate of the mixed-layer air. (4) Mixed-layer air is injected into the cloud layer with a mass flux of ρmb , where ρ is the air density. In what follows, mb will be referred to as the mass flux although it has dimensions of ms−1 . Since the top of the mixed layer is at the lifting condensation level, the air originating near the surface will be saturated and forms the clouds within the cloud layer. The air in these clouds is unstable and forms the cumulus updraughts. The generation of turbulent kinetic energy (TKE) is a fundamental feature of turbulent flows. For cumulus convection the generation of kinetic energy is due to the effects of latent heat release within the cumulus clouds. For steady, horizontally homogeneous conditions, the TKE budget can be written as (Tennekes and Lumley, 1972):  ∂u ∂v ∂ 1 − v ′ w′ + w ′ b′ − (2.3) w′ E + w′ p′ − ǫ = 0, −u′ w′ ∂z ∂z ∂z ρ

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Fig. 24.1 The structure of the cumulus-capped boundary layer. The dashed line shows the virtual moist adiabat for the mixed-layer air.

where u and v are the horizontal wind components,  w is the vertical wind  component, b is buoyancy, E = 0.5 u′2 + v ′2 + w′2 is the turbulent kinetic energy, and p is the pressure. The first two terms in Eq. 2.3 are the production of turbulence by wind shear. In the simulations used here to test the results of the similarity theory, the production of TKE by shear was generally small and can be ignored (Brown, 1999). The third term in Eq. 2.3 is the generation of turbulence by buoyancy, and this is the main production term for TKE in the cloud layer. The fourth term is the turbulent transport term and represents the transport of the turbulent kinetic energy by the turbulence itself. Since it is a flux divergence, the average over the depth of the cloud layer should be zero in the absence of a significant energy flux at the cloud base or loss of TKE through the generation of gravity waves. The final term in Eq. 2.3 is the dissipation of TKE due to molecular effects. The effects of viscosity are usually neglected when studying large-scale atmospheric flows. However, in turbulent flows the effects of dissipation due to the molecular viscosity cannot be neglected (Tennekes and Lumley, 1972). In common with other high Reynolds number flows the dissipation rate for cumulus convection will depend on the characteristics of the large eddies, i.e., the cumulus clouds. It is this feature of high Reynolds number flows that allows a velocity scale for cumulus convection to be derived using the TKE budget. The dissipation rate is related to the characteristic velocity

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and length scale of the large eddies by: w∗3 , (2.4) λ where w∗ is the velocity scale and λ is the length scale. The structure of the cumulus boundary layer shown in Fig. 24.1 suggests that the depth of the layer h between the cloud base and the base of the inversion is the most relevant length scale for the cumulus clouds. The stratification of this layer means that kinetic energy can be generated by the ascent of saturated air. The transfer of saturated air from the mixed layer into the cloud layer generates potential energy at a rate ∼ mb CAPE/h. If this is assumed to be the rate at which kinetic energy is generated through buoyancy, then the buoyancy flux in the TKE budget can be estimated as: ǫ∼

w ′ b′ ∼

mb CAPE . h

(2.5)

Assuming that w′ b′ ∼ ǫ from Eq. 2.4 then:  m 1/2 b 1/3 CAPE1/2 . (2.6) w∗ = (mb CAPE) = w∗ If the ascent of the cumulus updraughts were adiabatic then their ve1/2 locity could be estimated as (2 CAPE) , which is generally much larger than the typical vertical velocities of cumulus clouds (e.g., Siebesma et al., 2003). The difference between the adiabatic estimate of the updraught velocity and the typical velocity is attributed to the effects of mixing between the updraught and the unsaturated environment which reduces the buoyancy of the cumulus updraughts. The velocity scale w∗ is smaller than 1/2 1/2 by a factor of (mb /w∗ ) and Grant and Brown (1999) showed (2CAPE) that w∗ is a reasonable estimate for the velocity of updraughts in simulations of shallow cumulus convection. Since w∗ is derived by assuming that dissipation of TKE limits the TKE of the cumulus clouds, the relatively low vertical velocity of the cumulus updraughts is ultimately due to the action of molecular viscosity. The ratio mb /w∗ can be interpreted as the fractional area covered by cumulus updraughts and because it is non-dimensional it is an important parameter for the similarity theory. Because of this the fractional area cannot be taken to be zero, or as simply having an indeterminate small value. It is the existence of this fractional area parameter that allows a distinction to be drawn between the cumulus updraughts and what is usually considered to be the environment. The fractional area arises because the cloud-base mass flux and updraught velocity are determined by different

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processes. The cloud-base mass flux is a result of the interaction between the sub-cloud layer and the cloud layer, while the vertical velocities of the cumulus updraughts depend on the release of latent heat by condensation, which is reflected in the CAPE within the cloud layer. There is an apparent problem with the derivation of w∗ in that it is based on an estimate of the buoyancy flux that does not take into account the effects of entrainment. However, given that there is a non-dimensional parameter, the buoyancy flux should be written as: mb CAPE  z mb  , (2.7) F , w ′ b′ = h h w∗ where F (mb /w∗ ) is a function of the fractional area that must be determined. The unknown function F (mb /w∗ ) must apply to all of the terms in the TKE budget since if they are made non-dimensional using Eq. 2.7, the relative magnitudes of the various terms must remain the same. Since the unknown function multiplies both the buoyancy flux and dissipation rate in the derivation of w∗ it will cancel. However, it is now necessary to determine what the function F (mb /w∗ ) is. Observations indicate that the dissipation rate tends to be high within the cumulus updraughts (Siebert et al., 2006; Smith and Jonas, 1995) and small in the surrounding environment. If the dissipation rate in the updraughts ǫup can be estimated then the average dissipation rate can be obtained by multiplying by the fractional area mb /w∗ . The dissipation in the updraughts can be estimated as: w∗3 , (2.8) ǫup ∼ λup where λup is the length scale for the cumulus updraughts. Equation 2.8 assumes that the relationship between the dissipation rate and the characteristic velocity and length scales for the updraughts is the same as for an unconstrained turbulent flow, with the length scale being given by the typical size of an updraught (Smith and Jonas, 1995). Assuming that there is one updraught in an area ∼ h2 , and that the updraught occupies a fraction mb /w∗ of this area, the typical size of an updraught is:  m 1/2 b h. (2.9) λup = w∗ Using Eq. 2.9 in Eq. 2.8 and multiplying by mb /w∗ the area-averaged dissipation rate is:  m 1/2 w∗3 b ǫ ∼ Υ∗ = . (2.10) w∗ h

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Thus, the unknown function in Eq. 2.7 is (mb /w∗ )1/2 and Υ∗ should be the correct scale for the terms in the TKE budget. Figure 24.2a is taken from Grant and Lock (2004) and shows the buoyancy flux profiles obtained from a series of large-eddy simulations which are made non-dimensional by Υ∗ . The buoyancy flux is zero at the cloud base and increases rapidly with height over the lowest 20% of the cloud layer. Above this, variations in the buoyancy flux with height are relatively small up to the base of the inversion. The magnitude of the non-dimensional buoyancy flux in the bulk of the cloud layer is ≈ 1.2, so the scale Υ∗ provides a good estimate of the magnitude of the terms in the TKE budget. The variation in the scaled profiles is shown by the error bars which are comparable to the estimated statistical sampling error indicated by the shaded area. Figure 24.2b shows that the variation in the magnitude of the buoyancy fluxes before scaling is much larger than the variability of the non-dimensional profiles. (In the large-eddy simulations used by Grant and Lock (2004), the surface fluxes and the depths of the sub-cloud and cloud layers were varied to produce variations in the cloud-base mass flux and CAPE.)

Fig. 24.2 Buoyancy flux profiles. (a) Profiles scaled by Υ∗ defined in Eq. 2.10. The dotted lines are profiles from individual simulations and the solid line is the average. The shaded area provides an estimate of the sampling error for the buoyancy flux. (b) The buoyancy flux profiles without scaling. (Adapted from Fig. 4 of Grant and Lock c Royal Meteorological Society, 2004.) In this figure, the cloud base at (2004), which is  z = 0 is the lifting condensation level of the mixed-layer air.

Profiles of all of the terms in the TKE budget scaled by Υ∗ are shown in Fig. 24.3. The dissipation rate varies less with height than the buoyancy flux, particularly around the base of the cloud layer. The dissipation rate is comparable in magnitude to the buoyancy flux through most of the cloud

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367

layer. The transport term is smaller than either the buoyancy production or the dissipation, and acts to remove energy through most of the cloud layer.

Fig. 24.3 Profiles of terms in the TKE budget in the cloud layer scaled by Υ∗ . Solid line, the buoyancy flux; dotted line, the dissipation; dashed line, the transport term; dashed-dot line, the time rate of change of TKE. (Adapted from Fig. 8 of Grant and c Royal Meteorological Society, 2004.) Lock (2004), which is 

Figure 24.4 shows the terms in the TKE budget in the inversion layer. The buoyancy flux is positive through the full depth of the inversion, although in the upper half of the inversion this is due to negatively buoyant downdraughts (Grant and Lock, 2004). The negatively buoyant air arises from mixing between the saturated air in the cumulus clouds with unsaturated air from the inversion. The transport term is a source of kinetic energy and is comparable in magnitude with the other terms. The TKE budget in the inversion is consistent with cumulus updraughts overshooting into the inversion as a result of the kinetic energy generated within the cloud layer. The scaling for the TKE budget of shallow cumulus has been derived using a simple model of the structure of the cumulus-capped boundary layer shown in Fig. 24.1. The key processes in this simple model are the generation of buoyancy in the cloud layer due to the release of latent heat and the interaction between the sub-cloud and cloud layers. The combination of these two processes leads to cumulus convection having a characteristic fractional area. Although the fractional area of the updraughts is small, it is not a constant, varying with CAPE and the cloud-base mass flux. This is in contrast to the dry boundary layer where there is only one process that needs to be considered (the transfer of heat from the surface into

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Fig. 24.4 Profiles of terms in the TKE budget in the inversion layer, scaled by Υ∗ . Solid line, the buoyancy flux; dotted line, the dissipation; dashed line, the transport term; dashed-dot line, the time rate of change of TKE. (Adapted from Fig. 9 of Grant c Royal Meteorological Society, 2004.) and Lock (2004), which is 

the boundary layer) and the fractional area of updraughts is always about 40% (e.g., Young, 1988). 2.2

The cloud-base mass flux

For the results from the similarity theory to be useful the cloud-base mass flux must be known. In the picture of the cumulus-capped boundary layer shown in Fig. 24.1, the cloud-base mass flux represents the interaction between the cloud layer and the sub-cloud layer, which suggests that to understand what controls the cloud-base mass flux the region around the cloud base should be investigated. Figure 24.5 shows the terms in the TKE budget in the transition layer between the sub-cloud layer and cloud layer. Height is measured relative to the level at which the buoyancy flux vanishes and is scaled by the depth of the sub-cloud layer. The profiles are scaled using the scaling appropriate to the sub-cloud layer, i.e., w∗3 /hmix , where 1/3  w∗ is the convective velocity scale, defined as w∗ = g w′ b′ 0 hmix /θv (Deardorff, 1970) and hmix is the depth of the sub-cloud layer. Figure 24.5a shows the buoyancy-flux profiles in the transition layer. The buoyancy flux is a minimum at the base of the layer with a magnitude of ≈ −0.2 w′ b′ 0 . This is the same as the entrainment flux in dry convective boundary layers (Otte and Wyngaard, 2001) and also agrees with obser-

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Fig. 24.5 Terms in the TKE budget in the transition layer at the cloud base. Height is normalized by the mixed-layer depth and the terms in the TKE budget are normalized by w∗3 /hmix . (a) Buoyancy flux; (b) dissipation rate; and, (c) transport. (From Fig. 10 c Royal Meteorological Society, 2004.) of Grant and Lock (2004), which is 

vations in the sub-cloud layer in the presence of non-precipitating cumulus convection (Nicholls and Lemone, 1980). The buoyancy flux increases through the transition layer due to latent heat release. The transport term is shown in Fig. 24.5c and acts as a source of TKE within the transition layer, which suggests that the formation of cumulus clouds is associated with the kinetic energy of thermals in the mixed layer. The dissipation rate in Fig. 24.5b is approximately constant with height within the transition zone. The depth of the transition layer in which mixed-layer scaling works is about 0.2hmix (as indicated by the horizontal lines in the figure). In the dry convective boundary layer, without cumulus clouds, the entrainment flux is due to the transport of TKE into the inversion and Fig. 24.5b shows that this also appears to be the case for the formation of cumulus clouds. This suggests that there is a relationship between the cloud-base mass flux and the entrainment velocity of the dry convective boundary layer.

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Figure 24.6 is taken from Grant (2006b) and shows non-dimensional entrainment velocities we /w∗ for the dry convective boundary layer, obtained from a series of large-eddy simulations described by Sullivan et al. (1998), and the cloud-base mass fluxes mb /w∗ taken from the simulations described by Grant and Lock (2004). They are plotted as a function of the Richardson number, defined as: Ri =

Δbhmix , w∗2

(2.11)

where Δb is the change in mean buoyancy across the capping inversion for the dry convective boundary layer, and across the transition layer for the cumulus-capped boundary layer.

Fig. 24.6 Cloud-base mass flux for shallow convection. (a) mb /w ∗ from Grant (2006b) and we /w∗ from Sullivan et al. (1998) as a function of Richardson number. The crosses are entrainment velocities for the dry convective boundary layer and the diamonds are cloud-base mass fluxes. (b) Cloud-base mass flux as a function of the convective velocity c Royal Meteorological scale w∗ . (Adapted from Fig. 5 of Grant (2006b), which is  Society, 2006.)

The entrainment velocities and the cumulus mass fluxes are both consistent, thus: mb we 0.2 . = = w∗ w∗ Ri

(2.12)

Equation 2.12 is a well-known relationship between the entrainment velocity and inversion strength in a dry convective boundary layer (Sullivan et al., 1998). It implies that the rate at which work is done against buoyancy to entrain air into the mixed layer is a fraction of the rate of production of TKE in the mixed layer. This fraction is just the energy that is lost from the

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mixed layer through the transport of TKE. However, unlike the results for the dry convective boundary layer, which show that the Richardson number of the capping inversion varies significantly, the Richardson number of the transition zone in the cumulus-capped boundary layer is approximately constant, with a value around ∼ 5–10. In the dry convective boundary layer the change in the mean buoyancy across the inversion depends on the temperature of the troposphere immediately above the boundary layer, which can vary relative to the mixed-layer temperature. For the transition layer in the cumulus-capped boundary layer the stratification is determined by the heat released by condensation. The small variation in the Richardson number shows that there is a balance achieved between the transport of TKE from the sub-cloud layer into the transition layer and the stabilization due to the latent heat release in the developing cumulus clouds. To see how this arises, consider what would happen if the cloud-base mass flux increased. This would lead to an increase in the condensation rate which would stabilize the transition layer, increasing Ri. The increase in Ri would lead to a decrease in the cloud-base mass flux according to Eq. 2.12. Similarly, a decrease in the cloud-base mass flux will decrease the condensation rate, leading to a reduction in Ri and an increase in the cloud-base mass flux. The result of this feedback between the cloud-base mass flux and the stratification of the transition layer leads to the Richardson number being approximately constant and the cloud-base mass flux just proportional to w∗ . Grant (2006b) shows that: mb ≈ 0.04w∗ .

(2.13)

The TKE budget in the transition layer scales as w∗3 /hmix but in the upper half of the transition zone, cloud-layer scaling given in Eq. 2.10 should also hold, i.e.:    z z w∗3 mb 1/2 w∗3 ′ ′ F G wb = ≡ . (2.14) hmix hmix w∗ hcld hcld The left- and right-hand sides of Eq. 2.14 cannot simply be equated as the functions F and G depend on different non-dimensional heights, so that the choice of a particular height in the transition layer will correspond to different locations in terms of the non-dimensional heights. However, if the gradient of the buoyancy flux is considered, then w∗3 ∂F (ζ)  mb 1/2 w∗3 ∂G (ζ) ∝ . h2mix ∂ζ w∗ h2cld ∂ζ

(2.15)

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Part IV: Unification and consistency

Since the functions F and G describe the same profiles, the derivatives in Eq. 2.15 must be the same. This means that the gradients will be the same if  m 1/2 w∗3 w∗3 b ∼ . (2.16) h2mix w∗ h2cld 1/3

Combining Eqs. 2.14 and 2.12 and recalling that w∗ = (mb CAPE) implies: 5/8   3/2 hmix CAPE . (2.17) Ri ∝ w∗2 hcld The results in Grant and Lock (2004) suggest that the proportionality constant in Eq. 2.17 is about 0.4 and that the Richardson numbers calculated from Eq. 2.17 are in the range 2.8–5.0, which is in reasonable agreement with the range of Richardson numbers in Fig. 24.6a. Equation 2.17 shows that the feedback between the mass flux and stratification at the cloud base described above leads to a relationship amongst the parameters that describe the boundary layer, at least for the steady-state boundary layer. This relationship emphasizes the close connection between the sub-cloud and cloud layers in the cumulus-capped boundary layer. Mapes (2000) suggested that the cloud-base mass flux should be related to the turbulent kinetic energy in the sub-cloud layer and the convective inhibition. Although the functional dependence of the cloud-base mass flux on convective inhibition differs from that in Eq. 2.12, the closure proposed by Mapes (2000) is based on the same physical idea. Fletcher and Bretherton (2010) tested Eq. 2.13 and the Mapes (2000) parameterization against simulations of continental convection that included precipitating convection, finding that the Mapes parameterization performed better than Eq. 2.13. The parameterization proposed by Mapes (2000) is based on plausibility arguments rather than any specific situation, and requires a prediction of the TKE within the sub-cloud layer. In evaluating this parameterization, Fletcher and Bretherton (2010) determined the TKE in the sub-cloud layer from their simulations, which implicitly included the effects of convection on the sub-cloud layer. Equation 2.13 only applies to the case of nonprecipitating convection and is not expected to apply to precipitating convection. However, it does provide a connection with processes that occur in the dry convective boundary layer, which is not the case with Mapes (2000). The extension of the present approach to precipitating convection will be briefly discussed below (cf., Vol. 1, Ch. 11, Sec. 11.5).

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2.3

373

The entrainment rate

The mass-flux approach to parameterization focuses on the cumulus clouds themselves and so introduces the idea of entrainment. Although the entrainment rate is an important parameter in mass-flux parameterizations (see Vol. 1, Ch. 10), its formulation remains an area of considerable uncertainty. In developing and testing the similarity theory, the flow as a whole has been considered and although the effects of entrainment have entered implicitly in the velocity and the scaling of the TKE budget it has not been considered explicitly. The similarity theory should apply equally to the properties of the cumulus clouds as to the properties of the flow as a whole, and in particular to the process of entrainment. Indeed, an early success for the similarity theory was the proposal and testing of an explicit scaling for the entrainment rate (Grant and Brown, 1999). Grant and Brown (1999) assumed that the air which is entrained into a cumulus ensemble must be supplied with sufficient kinetic energy to make it ascend with the rest of the ensemble. This implies that entrainment of air into the cumulus ensemble is controlled by the TKE budget. If the mass entrained into the cumulus updraughts is ∼ εmb , where ε is the fractional entrainment rate, then the rate at which energy must be supplied is εmb w∗2 . Assuming that this is proportional to the rate at which kinetic energy is produced, w∗3 , (2.18) hcld which gives the following scale for the fractional entrainment rate: w∗ 1 ε = Aǫ . (2.19) mb hcld Profiles of the fractional entrainment rate from Grant and Brown (1999) are shown in Fig. 24.7, where the entrainment rates were diagnosed from the properties of the buoyant cumulus updraughts. The magnitudes of the dimensional entrainment rates in Fig. 24.7a vary by almost a factor of two, although the shapes of the profiles are generally similar. Most of the profiles show a rapid decrease in the entrainment rate over a shallow region close to the cloud base, the upper part of the cloud-base transition layer. Through most of the cloud layer the entrainment rate is approximately constant, decreasing with height just below the inversion. Figure 24.7b shows the entrainment rates scaled by (w∗ /mb ) 1/hcld. The scaling is very successful in reducing the variation in the entrainment rates seen in Fig. 24.7a, with the constant Aε = 0.03. εmb w∗2 ∼

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Fig. 24.7 Cumulus entrainment rates. (a) Entrainment rates for cumulus updraughts diagnosed from simulations of shallow convection. (b) Entrainment rates scaled using c Royal MeteoroEq. 2.19. (Adapted from Fig. 9 of Grant and Brown (1999), which is  logical Society, 1999.)

3

Discussion

This chapter started by posing three questions concerning moist convection (see Sec. 1), with answers to these questions obtained by developing a similarity theory to describe shallow, non-precipitating convection. The similarity theory was based on ideas commonly used to study turbulent flows. The answers obtained in this way are as follows: (1) The cumulus updraughts have a characteristic fractional area because there are two independent processes that are important in the cumuluscapped boundary layer: condensation, and the exchange of air between the sub-cloud layer and cloud layer. (2) The velocity scale for the cumulus updraughts was derived using arguments about the TKE budget that are used in the study of turbulent boundary-layer flows. The magnitude of the vertical velocity is determined by the cloud-base mass flux and CAPE. The velocity scale is less than the the velocity expected for undiluted ascent due to entrainment, which reduces the buoyancy of the updraughts. In turn the entrainment rate is determined by the TKE budget through the need to supply kinetic energy to entrained air. (3) The cloud-base mass flux is determined by the interaction between the sub-cloud and cloud layers, and for shallow cumulus convection the

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cumulus mass flux is determined by the same physical process that determines the entrainment velocity in the dry convective boundary layer. Can the similarity approach be extended to precipitating convection, or is the problem of precipitating convection too complex? Grant (2007) showed that the TKE budget for simulations of precipitating convection scales with Υ∗ as defined in Eq. 2.10. The shape of the buoyancy flux profiles are different from the profiles for non-precipitating convection and show sensitivity to the process of homogeneous freezing of supercooled water. Grant (2007) also showed that the cloud-base mass flux can be parameterized by matching the scalings of the sub-cloud layer and cloud layer. The result is:  3/4 2 w′ θv′ 0 g zcld . (3.1) mb ∝ θv (CAPE)5/6 zlcl

This differs from the relationship for non-precipitating convection because the interaction between the sub-cloud and cloud layers is more complex due to downdraughts generated by the cumulus transporting air into the subcloud layer. Consequently, in the presence of deep precipitating convection, the sub-cloud layer is different from the convective boundary layer that occurs in undisturbed conditions. Can treating cumulus convection as a turbulent flow, as in this chapter, help with the parameterization problem? The idea of treating the cloud layer as a turbulent flow is not new, and a number of models of the cumuluscapped boundary layer have been described. Bougeault (1981) and Mellor and Yamada (1982) described high-order closure models for the cumuluscapped boundary layer. These models solve the budgets for the secondorder moments of the turbulent flow. More recently, Bechtold et al. (1995) and Lock and Mailhot (2006) have described closure models based on the TKE budget. In these turbulence-based approaches the key problem is to represent condensation that occurs over a fraction of the area, in order to calculate the buoyancy flux (e.g., Cuijpers and Bechtold, 1995). Grant (2006a) considered the problem of parameterizing cumulus transports, using the similarity theory presented in this chapter to analyse the budgets of the turbulent fluxes associated with cumulus convection. Based on this analysis it is shown that the transports in the cloud layer can be parameterized with a flux–gradient relationship that is analogous to the non-local schemes used to parameterize turbulent transports in the convective boundary layer (Holtslag and Moeng, 1991; Lock et al., 2000). Grant

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(2006a) discussed the relationship between the flux–gradient approach and the mass-flux approach to cumulus parameterization showing that both can be derived from the turbulent flux budgets using different approximations. This does not make the approaches equivalent, but it shows that the massflux approach is just one possible approximation and that other, potentially better, approximations for the parameterization of cumulus transports exist. 4

Editors’ bibliographical note

For general background reading for this chapter, we recommend Tennekes and Lumley (1972).

Chapter 25

Clouds and convection as subgrid-scale distributions

E. Machulskaya Editors’ introduction: A grid-box variable is given by a single value numerically, whereas the actual physical value varies within the given grid box and over the given timestep. This simple point leads to the idea of reducing the problem of the subgrid-scale physical representation to that of defining the subgrid-scale distribution of physical variables. This idea is particularly popular for representing non-convective clouds, and is the focus of this chapter. The time evolution of the subgrid-scale distribution of variables may be formally described by the Liouville equation. For this reason, the first part of the chapter is devoted to a derivation of this equation in the meteorological modelling context. The remainder of the chapter discusses the existing cloud schemes in respect of this formal description. It is stressed that although existing cloud schemes focus on stratiform cloud, the general formulation presented herein is also applicable to convective clouds. The editors have a strong preference for naming the function describing the subgrid-scale distribution as the distribution density function (DDF). However, the author of this chapter prefers to use the term probability density function (PDF) given that this is the familiar, if somewhat loose, terminology that appears throughout the literature. It seems appropriate to emphasize here that the use and consideration of these subgrid-scale functions does not necessarily imply stochasticity in the description, and that the DDF becomes a PDF only as an extra step in the interpretation.

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Part IV: Unification and consistency

Introduction

Clouds are one of the most important parts of weather and climate systems as their presence and amount play a great role in the formation of various atmospheric regimes: they alter planetary albedo and the amount of solar radiation that reaches the Earth’s surface; they redistribute longwave radiation fluxes, changing the thermal regime of the various layers in the atmosphere; they are the source of precipitation. Last but not least, a prominent impact of clouds is on the mixing in the atmosphere because the formation of cloud condensate is accompanied by latent heat release that drastically enhances turbulent activity. Clouds form where phase changes from water vapour to liquid water or ice occur. The transition from water vapour to liquid water is a very fast process, so that an assumption may be made that if the specific humidity of the air exceeds the saturation specific humidity in respect of water at a given air temperature and pressure, the phase changes immediately remove all the excess water vapour producing cloud water, or, vice versa, the excess of cloud water immediately evaporates as the specific humidity falls below its saturation value. Thus, if the phase changes can be considered as instantaneous, the local values of specific humidity and temperature completely determine the presence of clouds and the amount of cloud water. Unfortunately, this is not the case if deposition (the transition from water vapour to ice) takes place, because water vapour can exist at temperatures below freezing point in the oversaturated state for a long time. In what follows, instantaneous phase changes are assumed. It is quite a serious restriction, but the extention of the framework that is described here to ice clouds is far from trivial and is beyond the scope of this chapter. In an atmospheric model, specific humidity, temperature, and pressure are computed as solutions of the governing model equations either discretized on a model grid by means of a finite-difference method or truncated in the spectral representation. In both cases, the model variables represent the quantities that characterize some piece of space, with the horizontal extension of these pieces (grid boxes) ranging in contemporary numerical weather prediction (NWP) and climate models roughly from 1 to 100 km. Using these grid-box mean model variables, one can diagnose cloud cover in a given grid box in the simplest way, by means of the so-called “all-ornothing”, or “0–1”, parameterization. It detects no clouds in the grid box if the grid-scale specific humidity is less than the saturation specific humidity for this box (being controlled by the grid-scale temperature) and

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is overcast otherwise. Since this diagnostic uses only the mean quantities, it does not deal with any subgrid-scale (SGS) variability of temperature and humidity. That is, the whole grid box is either homogeneously subsaturated and clear, or supersaturated and cloudy. This statement may be a reasonable assumption if the size of the grid box is relatively small (e.g., in large-eddy simulations or cloud-resolving models) but may be erroneous for the resolutions used currently in NWP and climate models. This is because the assumption of complete homogeneity of temperature and total humidity within a grid box is not usually valid at these resolutions and there exists horizontal and/or vertical subgrid-scale variability of these quantities. The local excess of the total humidity over its saturation value results in local phase changes and the occurrence of clouds: from the grid-box point of view it means fractional, or partial, cloud cover. Situations are often encountered where the air is undersaturated on average (i.e., the mean specific humidity is less than the mean saturation humidity) but where clouds cover a substantial part of this area. The opposite situation can also occur, when there are broken clouds despite mean oversaturation. Even if the clouds did not interact with other components of the atmospheric system, the errors in cloud representation might be a deficiency of an atmospheric model, because clouds are one of the weather forecast deliverables. But due to the numerous interactions of clouds with other components of weather and climate systems, mentioned at the beginning of the chapter, neglecting partial cloud cover is also detrimental to the representation of many atmospheric processes. A remarkable example is the impact of clouds on turbulent mixing. In the shallow cumulus regime, the air is usually undersaturated in the mean, which means zero cloud cover in the 0–1 approach. However, cumulus-type clouds that form locally despite mean undersaturation are responsible for a significant part of the intense mixing because of the latent heat release within clouds. There exist many cloud parameterization schemes within the currently used NWP and climate models that are capable of estimating partial cloud cover and other related quantities. Apart from the already-mentioned allor-nothing scheme that may also be considered as the simplest cloud-cover parameterization scheme, there are many others that can be clustered into groups. The so-called “relative-humidity schemes” link cloud fraction and cloud condensate primarily to the grid-scale relative humidity of the air and, possibly, to some other quantities (Slingo, 1980; Sundqvist, 1978; Xu and Randall, 1996). There is also a large group called “statistical cloud schemes” (Bechtold et al., 1995; Bougeault, 1982; Fowler et al., 1996; Lar-

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son et al., 2001; LeTreut and Li, 1991; Smith, 1990; Sommeria and Deardorff, 1977; Tompkins, 2002). These use the fact that only information about the frequency of various combinations of temperature and humidity values within the grid box is needed to diagnose cloud fraction and cloud condensate, and not the information about the exact spatial locations of these values. In other words, only the statistics of the subgrid-scale temperature and humidity fluctuations are required; hence the name “statistical”. These schemes make some a priori assumptions about the probability density function (PDF) of the distribution of subgrid-scale fluctuations. If a PDF is chosen then all that is left to do is to predict its parameters. This method, which is described below in more detail, is called the “assumedPDF” approach and it is successfully used in various applications. Finally, there are also cloud schemes that carry prognostic equations for cloud fraction and cloud condensate (Tiedtke, 1993; Wilson et al., 2008). All these types of cloud scheme are described more thoroughly in Sec. 3. Here, it should only be pointed out that although the existing cloud schemes are at first sight based on different principles, they have much more in common than it might seem. Indeed, they all can be obtained from the same general formulation that is discussed in the next section. To derive this general formulation is one of the main aims of this chapter; the other is to show how various cloud schemes follow from it. Before proceeding to this general formulation, let us introduce the variables which will be used, and their notation. The wind velocity vector is denoted by v with components either {v1 , v2 , v3 } or {u, v, w}. The absolute temperature is denoted by T , the specific water-vapour content1 by qv , the specific liquid-water content by ql , and the air density by ρ. The potential temperature is θ = T /Π, where Π = (p/p0 )Rd /Cp is the Exner factor. Here, p is the atmospheric pressure, p0 is the atmospheric reference pressure, Cp is the air heat capacity, and Rd is the universal gas constant for dry air. The ǫ −1)qv +ˆ ǫql ], where ǫˆ = Rd /Rv , virtual potential temperature is θv = θ[1+(ˆ Rv being the universal gas constant for water vapour. These are the variables that are often used in atmospheric models to describe the state of the atmosphere. In some parts of the model (e.g., that responsible for smallscale mixing) it is convenient to work with so-called “quasi-conservative variables” that are approximately conserved during phase changes. For instance, we might choose the liquid-water potential temperature θl and the 1 This notation is a departure from that used in most of this set, in which q is a mixing ratio.

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total specific water content (humidity) qt . They are defined as, Lv ql , θl = θ − Cp Π and qt = qv + ql , respectively, where Lv is the latent heat of vaporization. The liquid-water temperature will also be used, and is defined as Tl = T − (Lv /Cp )ql = Πθl . One further useful quantity is the saturation deficit s˜ that is defined as the difference between the total humidity qt and the saturation specific humidity with respect to liquid water qs (T ): s˜ = qt − qs (T ). Note that if this difference is positive then s˜ is equal to the liquid-water content ql . Note also that the saturation specific humidity is a function of the absolute temperature T . If no fluctuations of the Exner factor about its grid-box mean are assumed (i.e., if Tl  = Π θl ), the saturation deficit can be expressed through the quasi-conservative variables by means of the linearization of s˜ around the grid-box mean liquid-water temperature Tl . This fluctuation can simply be neglected if a fluctuation is considered over a constant pressure surface. This yields a so-called “linearized saturation deficit”, which we denote by s: " #  ∂qs  1 ′ qt − qs ( Tl ) − Π s= θ , (1.1) Q ∂T T = Tl l where

Q=1+

 Lv ∂qs  Cp ∂T T = Tl

(see, e.g., Mellor, 1977b, for the details of the derivation). Here, a gridbox average is denoted by angled brackets, the fluctuations therefrom by primes, and the exact definition of the averaging will be given in the next section. In what follows, only the linearized saturation deficit s will be used. Other notation that is used throughout the chapter: vectors are denoted with bold letters, the vector x determines the spatial location, t is time, θr is a reference temperature, and gi and Ωi are the components of the acceleration vector due to gravity and of the angular velocity pseudo-vector of the Earth’s rotation, respectively. The conventional summation over repeated indices is assumed.

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2 2.1

Part IV: Unification and consistency

General considerations Fine-grained and coarse-grained PDFs

As mentioned in Sec. 1, knowledge of all possible combinations of temperature and humidity values (more precisely, possible values of the saturation deficit s) within a grid box is sufficient to estimate cloud fraction and amount of cloud condensate. Only the information regarding which values of saturation deficit can be encountered across the grid box is needed, not the information about the exact spatial locations of these values. In other words, only the SGS statistics of the saturation deficit are required. The exact meaning may be assigned to the word “statistics” in terms of a filtered density function, probability density function, and their interrelation, as shown, for example, in Colucci et al. (1998); Larson (2004). One starts with the so-called “fine-grained” probability density function for a spatio-temporal field φ(x, t): Pf (ψ; x, t) = δ (ψ − φ(x, t)) ,

(2.1)

where δ(y) denotes the Dirac delta function and ψ is a particular value of the quantity φ. The field φ may be either scalar or vector. The notation Pf (ψ; x, t) is used instead of the more correct Pf (φ(x, t) = ψ; x, t) for convenient reading. As can be seen from the definition, for a given particular value ψ at a point x and time t this PDF goes to infinity if the quantity φ has the value ψ at this point and time instant, and to zero otherwise. Thus, it is a singular PDF and is introduced simply for further developments; so far, the system remains deterministic and the “fine-grained” PDF contains the whole information about its state. Actually, the fine-grained PDF Pf (ψ; x, t) is equivalent to the field φ(x, t) itself in terms of the information they contain (Fig. 25.1). However, it may be desirable to have a function that is more smooth in space than Pf (ψ; x, t). To this end, one can smooth (average) Pf (ψ; x, t) over some area using a spatial filter G(x) (cf., Vol. 1, Ch. 3, Sec. 3.3), thus obtaining a “coarse-grained” PDF P (ψ; x, t), also called “the filtered density function” (FDF): P (ψ; x, t) := Pf (ψ; x, t) :=

∞

−∞

δ (ψ − φ(x′ , t)) G(x, x′ )dx′ .

(2.2)

Here, the filter G(x, x′ ) is a time-independent function of coordinates which is assumed to be spatially invariant (G(x, x′ ) = G(x−x′ )) and even (G(x) =

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383

8 6 4

ψ

2 0 -2 -4 -6 -8 -8

-6

-4

-2

0

2

4

6

8

x

(b)

(a) Fig. 25.1

A field φ(x) (left panel) and its Pf (right panel).

G(−x)). Observe that if G is not a one-to-one mapping and thus is noninvertible (e.g., a box filter in physical space being equal to 1 in a vicinity of x′ and 0 elsewhere), the procedure of filtering not only smooths the original fine-grained PDF but it also allows one to reduce the amount of information about the system state. In other words, it becomes impossible to recover the original field φ using P (ψ; x, t). The resulting FDF P (ψ; x, t) (Fig. 25.2) is now interpreted from the probabilistic point of view, and this is especially easy to understand if the box filter is taken as an example. In this case, we lose the information about the exact spatial locations of the values ψ within a box and start to think that at a particular point x within the box we can encounter all the values that are present therein. The probability of observing a value ψ at the point x is then equal to the frequency with which this value appears across the box. In this way we turn from an FDF to a PDF saying that the value ψ can appear somewhere in the box with a certain probability and forgetting about where it appears precisely. An FDF obtained in this way  ∞ has all the properties of a PDF provided that the filter G is normalized ( −∞ G(x)dx = 1) and is positive (Vreman et al., 1997). In what follows, let us assume the filter G to be the box filter because of its usefulness and for the sake of convenience: it corresponds to the finite-difference discretization of many host atmospheric models, the expressions look simpler, and the generalization to an arbitrary filter is straightforward. For the case of a box filter,  ∞ 1 δ (ψ − φ(x′ , t)) dx′ , δ (ψ − φ(x′ , t)) G(x − x′ )dx′ = V (x) −∞

V (x)

where V (x) is the volume of the box centred at x. From now on, the notion

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of PDF will always be used instead of FDF.

Fig. 25.2 Filtered density function P for the field φ(x) shown in Fig. 25.1, smoothed by means of the box filter.

Before going further, it should be clearly stated that the notions of probability and contingency appear here due to smoothing or averaging only. The exact system of the governing equations, including the initial and boundary conditions, remains deterministic. We are not working with an ensemble of realizations, but with a single one. Only the loss of some piece of information due to averaging leads to a probabilistic description, which is very similar to the situation in statistical mechanics and thermodynamics.

2.2

Link to the cloud-related quantities

It will now be shown that the PDF introduced in the previous subsection is exactly what is needed to diagnose cloud fraction, cloud condensate, and other cloud-related quantities within a given grid box. Indeed, if a distribution function P (s∗ ; x, t) is known for the values s∗ of saturation deficit fluctuations s′ then cloud fraction is the zeroth-order partial moment of the shifted distribution

C=

∞

− s

P ( s + s∗ ; x, t)ds∗ ,

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385

and the amount of cloud condensate within a given grid box is its first partial moment: ∞ ( s + s∗ )P ( s + s∗ ; x, t)ds∗ ,

ql  = − s

i.e., these quantities can be computed by integrating the PDF P ( s + s∗ ; x, t) over its saturated part (s∗ > 0). In a similar way, other statistical moments can be estimated, but some extensions may be needed. For instance, to diagnose the covariances of SGS scalar fluctuations, we should consider the PDF not in respect of the variable s that combines θl and qt , but in respect of temperature and humidity fluctuations separately so that P (θl′ (x, t) = θ∗ , qt′ (x, t) = q ∗ ) is now a joint bivariate PDF. To estimate various vertical fluxes, the PDF should be further extended to account for SGS vertical velocity fluctuations yielding a joint trivariate PDF P (θl′ (x, t) = θ∗ , qt′ (x, t) = q ∗ , w′ (x, t) = w∗ ). Then, for example, one of the third-order moments (used by some turbulence models) is: ∞ ∞ ∞ w∗ θ∗2 P (θ∗ , q ∗ , w∗ ; x, t)dθ∗ dq ∗ dw∗ .

w′ θl′2  = −∞ −∞ −∞

Other moments incorporating w′ , θl′ , and qt′ are computed similarly. The buoyancy flux, as one of the most important quantities among the moments, should also be mentioned here. As stated in Sec. 1, turbulent mixing is greatly affected by latent heat release within clouds. This means that because of the warming of the air parcel during the condensation process, its buoyancy increases. However, as the condensation is not usually uniform across the grid box, and since vertical velocity also fluctuates within the grid box, it does matter which values of vertical velocity w correspond to those air particles where condensation takes place. In the language of turbulent mixing models it is the correlation between vertical velocity fluctuations and virtual potential-temperature fluctuations, in other words, the vertical buoyancy flux g g ′ ′

w θv  = w′ [θ(1 + (ˆ ǫ − 1)qt − ǫˆql )]′  = θr θr g (a w′ θl′  + b w′ qt′  + A w′ ql′ ) (2.3) θr that acts as an amplifier or damper of the turbulent kinetic energy and so controls the turbulent mixing. Here, ǫ − 1) − ǫˆ ql , a = 1 + qt (ˆ

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Part IV: Unification and consistency

b = θ(ˆ ǫ − 1), and A=

1 Lv [1 + qt (ˆ ǫ − 1) − ǫˆ ql ] − ǫˆ θ Π Cp

are combinations of the thermodynamic parameters and grid-box mean fields (the expressions simply follow from the definitions of θv and θl ; for details of the derivation see, e.g., Mironov, 2009). Thus, to find out to what extent clouds affect the turbulent mixing one must know not only the mutual alignment of the fluctuations of humidity, and temperature (which is necessary for the detection of a cloud itself), but also how the vertical velocity, humidity, and temperature fluctuations are co-located within the grid box. According to Eq. 2.3, the buoyancy flux is a linear combination of the three (two full and one partial) second-order statistical moments: ∞ ∞ ∞ ′ ′ w∗ θ∗ P (θ∗ , q ∗ , w∗ ; x, t)dθ∗ dq ∗ dw∗ ,

w θl  =

w′ qt′  =

w′ ql′  =

−∞ −∞ −∞ ∞ ∞ ∞

w∗ q ∗ P (θ∗ , q ∗ , w∗ ; x, t)dθ∗ dq ∗ dw∗ ,

−∞ −∞ −∞ ∞ ∞ ∞

w∗ q ∗ P (θ∗ , q ∗ , w∗ ; x, t)dθ∗ dq ∗ dw∗ .

−∞ qs (Tl ) −∞

Although the representation of buoyancy flux belongs formally to the interface between cloud-cover schemes and turbulence schemes, it may also be regarded as a part of a cloud-cover parameterization scheme (cf., Ch. 26). 2.3

Derivation of the prognostic equation for the coarse-grained PDF

As demonstrated above, the knowledge of P is sufficient to determine all cloud-related quantities needed for an atmospheric model. The next question is how to find P . The most general approach consists of the derivation and solution of a prognostic transport equation for the PDF itself. Then, using such an equation and appropriate initial distribution and boundary conditions, the PDF and thus all derived quantities mentioned above can be determined at an arbitrary time instant. There is no cloud scheme (so far) that operates with such an equation directly. Nevertheless, all existing cloud schemes

Clouds and convection as subgrid-scale distributions

387

should be, in principle, obtained from this equation by means of various simplifications. In this subsection, the derivation is sketched for a general case and in the next subsection it will be applied to atmospheric modelling. The main idea is borrowed from statistical mechanics where Liouville’s theorem is used to describe the phase-space distribution of the system points (Landau and Lifshitz, 1980). This theorem states that the density of points in phase space (including spatial coordinates) remains constant along the system trajectories. Let us look more closely at what it means. Let our physical system be described by means of a spatio-temporal field φ(x, t). Since at each point x of the domain and at each time t the field φ(x, t) has some value ψ (trivial fact), moving points in the extended phase space (phase space together with the spatial coordinates) {x, φ} cannot appear or disappear, and their number is conserved. Hence, the flow of the points in this extended phase space {x, ψ} is incompressible, and the fine-grained PDF Pf (ψ; x, t) that indicates these points with infinite spikes moving in time, is conserved. It obeys the Liouville equation, that is, the conservation law in the form of a continuity equation (Salmon, 1998): ∂Pf ∂Pf dφ dxi ∂Pf + = 0. + ∂t dt ∂xi ∂ψ dt

(2.4)

This equation is the expression of the mass conservation law for the singular probability density Pf , and as such, is a mere formulation for the trivial fact mentioned above. The quantity dφ/dt in the continuity equation may be replaced at this stage with an expression according to the equation that governs the evolution of the field φ(x, t), but it may also be made later. It may be desirable to smooth (filter) the expression in Eq. 2.4 according to the definition of Eq. 2.2 for P . This leads to the transport equation for P (ψ; x, t): $  % dφ  dxi ∂P ∂ ∂P + =− ψ P . (2.5) ∂t dt ∂xi ∂ψ dt  Here, the definition for the conditional filtered value (conditional expectation) of a variable Q(x, t) has been used:  1 ′ ′ ′ V V Q(x , t)Pf (ψ; x , t)dx . (2.6)

Q(x, t)|φ(x, t) = ψ ≡ Q|ψ = P (ψ; x, t)

Now, the expression for dφ/dt should be inserted into Eq. 2.5 to finalize the derivation. Equation 2.5 is the mass conservation law for the non-singular probability density P .

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Part IV: Unification and consistency

Prognostic equation for the coarse-grained PDF applied to atmospheric modelling

Let us consider the phase space consisting of the components of the wind velocity vector v, and two quasi-conservative variables, liquid-water potential temperature θl and total specific humidity qt . In this case, Eq. 2.4 takes the form: dPf (v(x, t) = v∗ , θl (x, t) = θ∗ , qt (x, t) = q ∗ ; x, t) ≡ dt ∂Pf dθl ∂Pf dqt ∂Pf ∂Pf dvi ∂Pf + vi + + = 0, + ∗ ∗ ∂t ∂xi ∂vi dt ∂θ dt ∂q ∗ dt

(2.7)

where the definition dxi /dt = vi has been used. Now we need the prognostic equations for v, θl , and qt that can be inserted into Eq. 2.7. These equations can be obtained, for instance, from the governing equations of the host atmospheric model according to the definitions of θl and qt . They read:  ∂vi ∂vi ∂τij ∂p + vj ρ =− + ρgi − ρΩj vk ǫijk − ρ , ∂t ∂xj ∂xi ∂xj    1 Lv ∂Pil ∂θl ∂Jie ∂Jil ∂Ri ∂θl ρCp = + , + Qh + ǫD − + + vj − ∂t ∂xj Π ∂xi ∂xi Π ∂xi ∂xi  ∂qt ∂qt ∂J v ∂J l ∂P l ρ + vj =− i − i − i , (2.8) ∂t ∂xj ∂xi ∂xi ∂xi where ρΩj vk ǫijk is the Coriolis force,  ∂vi ∂vj τij = −μ + ∂xj ∂xi are the components of stress tensor due to viscosity, μ the dynamic molecular viscosity, Ri the components of radiative flux vector, Pil the components of precipitation flux vector, Jie = −κ∂θl /∂xi the components of the thermal molecular diffusion flux, κ the molecular thermal conductivity, and Jix = −κx ∂q x /∂xi the components of the molecular diffusion flux of the water constituent q x , κx being the corresponding molecular conductivity. Recall that as the constituents of the moist air, only water vapour x = v and liquid water x = l are considered here. The heat source   ∂T  −Cpl Pil − Cpx Jix Qh = ∂xi describes the impact of the precipitation flux and the molecular diffusion of water vapour and liquid water on temperature T and is neglected in what

Clouds and convection as subgrid-scale distributions

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follows since the water constituents contribute very little to the total mass of an air volume. The heating caused by the kinetic energy dissipation due to viscosity ǫD will also be regarded as negligible, and both Qh and ǫD will be dropped for brevity (their optional inclusion would not affect the main results). The precipitation flux Pil consists of three parts describing initial autoconversion of cloud droplets, accretion of rain water that collects cloud droplets, and evaporation of rain water. Their impact on total water and temperature is, of course, different. However, they will be denoted as one term Pil in what follows, for brevity and because they always enter the analysis in the same way. (Of course, in a thorough quantitative analysis they should be regarded separately.) Similar remarks apply to the radiative flux Ri , which consists of short- and long-wave parts. To describe the state of the atmosphere in fully compressible atmospheric models, an additional prognostic equation either for air density or for pressure is solved, but in any case the density or pressure will not be included in the list of arguments of P ; hence, the equations for them need not be considered. Since it is the fluctuations of meteorological quantities about their gridbox mean values that are needed for estimations such as Eq. 2.3, it is convenient to work with the set of governing equations for the fluctuations. The equations for the fluctuations are obtained by means of filtering Eq. 2.8 with the box filter and subtracting the result from Eq. 2.8, as already outlined in Vol. 1, Ch. 3, Sec. 3.3. Henceforth, the primes are dropped for fluctuations, the full fields themselves being denoted with a tilde, and the fluctuations with ordinary letters. The grid-box means are denoted with angled brack˜ ets, so that the subgrid-scale fluctuations of a field φ˜ are φ = φ˜ − φ. Further, for the subgrid-scale fluctuations the Coriolis force is neglected, the Boussinesq approximation is made, the buoyancy term is expressed through the virtual potential temperature, and the molecular conductivities κv and κl are assumed to be equal. After these transformations, the prognostic equations for the fluctuations read:

∂vi ∂vi dvi ∂vi + ˜ uj  = + vj ≡ ∂t ∂xj ∂xj dt ∂p gi ∂ 2 vi ∂ ˜ ui  ∂ vj vi  − vj + − + θv + ν , ∂xj ∂xj ∂xi θr ∂xj ∂xj

(2.9)

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Part IV: Unification and consistency

∂θl ∂θl ∂θl dθl + ˜ uj  = ≡ + vj ∂t ∂xj ∂xj dt   2 ∂ θ˜l  ∂ vj θl  Lv ∂Pil 1 ∂ 2 θl ∂Ri l ∂ ql + + +κ −κ −vj + , − ∂xj ∂xj Π ∂xi ∂xj ∂xj Π ∂xi ∂xj ∂xj (2.10) Cp



∂qt dqt ∂qt ∂qt + ˜ uj  = + vj ≡ ∂t ∂xj ∂xj dt −vj

∂ ˜ qt  ∂ vj qt  ∂Pil ∂ 2 qt + − + κv . ∂xj ∂xj ∂xi ∂xj ∂xj

(2.11)

Substituting the expressions for the substantial time derivatives dvi /dt, dθl /dt, and dqt /dt from Eqs. 2.9–2.11 into Eq. 2.7 yields: ∂Pf ∂Pf ∂Pf + ˜ ui  + vi = ∂t ∂xi ∂xi  ∂ ˜ ui  ∂ vj vi  ∂Pf ∂p gi ∂ 2 vi + − + θ + ν − −v j v ∂vi∗ ∂xj ∂xj ∂xi θr ∂xj ∂xj   1 ∂ 2 θl ∂ θ˜l  ∂ vj θl  ∂Pf ∂Ri −v + + + κ − − j ∂θ∗ ∂xj ∂xj Π ∂xi ∂xj ∂xj  l 2 Lv ∂Pi ∂ ql + − κl Π ∂xi ∂xj ∂xj  2 ∂ ˜ qt  ∂ vj qt  ∂Pil ∂Pf v ∂ qt − + − + κ −v . j ∂q ∗ ∂xj ∂xj ∂xi ∂xj ∂xj

(2.12)

ui /∂xj ) and similar terms involving the The term −(∂Pf /∂vi∗ )vj (∂ ˜ scalars may be rewritten as:    ∗  ∂ ˜ ∂Pf ∂ ˜ ui  ∂ ˜ ui  ui  ∂ ∂ ≡ = vj = Pf vj δ vj − vj vj ∂vi∗ ∂xj ∂vi∗ ∂xj ∂v ∗ ∂xj   i  ∂ ˜  ui  ∂ ˜ ui  ui  ∂ ∂ ∂Pf ∗ ∂ ˜ ≡ = δ vj∗ − vj vj∗ Pf vj∗ vj , (2.13) ∗ ∗ ∗ ∂vi ∂xj ∂vi ∂xj ∂vi ∂xj

where in the first and last equalities the fact that vj ∂ ˜ ui /∂xj does not depend on vj∗ has been used. In the last step, the incompressibility relation ∂ ˜ ui /∂xi = 0 has also been used which yields (∂vj∗ /∂vi∗ )∂ ˜ ui /∂xj = ui /∂xj = 0 (Pope, 2000). For the terms including mean scalar gradiδij ∂ ˜ ents, a similar transformation is valid simply because vj∗ does not depend on θ∗ and q ∗ .

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Clouds and convection as subgrid-scale distributions

In a similar manner, the term vi ∂Pf /∂xi may be transformed as: vi

∂ ∂Pf = Pf vi ≡ ∂xi ∂xi ∂ ∗ v δ (vi∗ − vi ) ≡ ∂xi i

∂ vi δ (vi∗ − vi ) = ∂xi ∂Pf ∂ Pf vi∗ = vi∗ ∂xi ∂xi

(2.14)

by virtue of the incompressibility relation ∂vi /∂xi = 0. Filtering the expression of Eq. 2.12 using Eqs. 2.2 and 2.6 we obtain the transport equation for P (θ∗ , q ∗ , v∗ ; x, t): ∂P ∂P ∂P ∗ ∂ ˜ ui  ∂P ∂ θ˜l  ∂P ∂ ˜ qt  ∂P + ˜ ui  + vi∗ = vj + ∗ vj∗ + ∗ vj∗ ∗ ∂t ∂xi ∂xi ∂vi ∂xj ∂θ ∂xj ∂q ∂xj ∂P ∂ vj θl  ∂P ∂ vj qt  gi ∂ ∂P ∂ vj vi  − ∗ − ∗ −

θv |v∗ , θ∗ , q ∗  P − ∗ ∂vi ∂xj ∂θ ∂xj ∂q ∂xj θr ∂vi∗   % % $ $ ∂p  ∗ ∗ ∗ ∂ ∂ 1 ∂Ri  ∗ ∗ ∗ + ∗ v ,θ ,q P + ∗ v ,θ ,q P ∂vi ∂xi  ∂θ Π ∂xi   % %  $ l $ 2 ∂ Lv ∂Pi  ∗ ∗ ∗ ∂ vj  ∗ ∗ ∗ ∂ ∂ − , θ , q , θ , q P − ν P − v v ∂θ∗ Π ∂q ∗ ∂xi  ∂vj∗ ∂xi ∂xi    % % $ 2 $ 2 ∂ θl  ∗ ∗ ∗ ∂ ql  ∗ ∗ ∗ ∂ κ ∂ L v κl − ∗ v ,θ ,q P + ∗ v ,θ ,q P ∂θ Π ∂xi ∂xi  ∂θ Π ∂xi ∂xi   % $ 2 ∂ ∂ qt  ∗ ∗ ∗ − κv ∗ (2.15) v , θ , q P. ∂q ∂xi ∂xi 

This is a general form of the Liouville equation (cf., Vol. 1, Ch. 2, Eq. 6.3). 2.5

Discussion of the equation for P

Let us now perform an inspection of Eq. 2.15. First, it is noticeable that there are some terms here that have a closed form (i.e., are known) and some that are unclosed and thus should somehow be parameterized (or neglected). The presence of the latter terms is not surprising since some information contained in the original fields has necessarily been lost when passing from the fine-grained PDF to the coarse-grained PDF. The terms on the left-hand side (i.e., the local tendency and the mean and subgridscale transport of the PDF in physical space) appear in closed form. (To understand it, remember that vi∗ are arguments of P , just like xi and t.) The first six terms on the right-hand side represent the transport of the PDF in phase space, and can also be evaluated exactly. As to the remaining terms, they all are written in a form involving conditional means of the type

Q(x, t)|v = v∗ , θl = θ∗ , qt = q ∗ : i.e., the mean value of a quantity Q given

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that v = v∗ , θl = θ∗ and qt = q ∗ . Clearly, if Q(x, t) is a single-valued function of {v, θl , qt }, then the conditional mean can be computed exactly and it is merely Q(x, t)|v = v∗ , θl = θ∗ , qt = q ∗  = Q(v(x, t), θl (x, t), qt (x, t)). This is the case, for instance, with the buoyancy term θv (x, t)|v∗ , θ∗ , q ∗  since  1 Lv ˜ ˜ ˜ ǫ − 1) q˜t − ˆǫq˜l ) θv = θv − θv  = θl + q˜l (1 + (ˆ Π Cp $ % 1 Lv ˜ (2.16) q˜l (1 + (ˆǫ − 1) q˜t − ǫˆq˜l ) , − θl + Π Cp and

⎧

⎨ 1 ˜ qt  + qt − qs ( T˜l ) − Π q˜l = Q ⎩ 0,



∂qs  ∂T 

T = T˜l



θl , q˜t − qs (T˜l )> 0 q˜t − qs (T˜l )≤ 0

(2.17)

(cf., Eq. 1.1 and note the changes in notation). The relation in Eq. 2.17 expresses the fact mentioned at the beginning of the chapter that due to the assumption of instantaneous phase changes, knowing the local values of temperature and specific humidity is sufficient to determine cloud-water content. Thus, all quantities that enter the expression for the virtual potential temperature fluctuation are known (naturally assuming that the grid-box means are known), and hence, the buoyancy term is also closed. On the other hand, if the function Q(x, t) is not a single-valued function of {v, θl , qt }, but there is a certain spread of the values of Q(x, t) given one and the same {v, θl , qt }, then the conditional mean is not known and additional reasoning should be invoked to close (parameterize) it. This is the case, for example, with the divergence of the radiative flux Ri : given one and the same set {v, θl , qt } at a point x, there can clearly be different values of ∂Ri /∂xi at this point. Therefore, the general reason for the unclosed nature of the terms 8–14 on the right-hand side of Eq. 2.15 is the uncertainty in some quantities given one and the same set of local and instantaneous values of the PDF arguments. One can further refine this formulation and divide possible reasons into two groups: an incomplete set of the PDF arguments and the non-locality of atmospheric processes. Let us look at them more closely. First, the uncertainty in the values of a generic function Q may be related to the lack of variables for which the PDF is considered: Q may be a function not only of {v, θl , qt } but, in addition, of some other variables. An example of this type of uncertainty is the eighth term on the right-hand

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side of Eq. 2.15 that involves the pressure gradient ∂p/∂xi |v∗ , θ∗ , q ∗  . This cannot be evaluated, not least because the set of model variables for which the PDF has been written does not include pressure. To this first group also belong the ninth and the tenth terms on the right-hand side of Eq. 2.15: i.e., those involving the sources/sinks of heat due to radiative heating and the sources/sinks of the total water content due to falling precipitation. It should be noted that the knowledge of radiative heating and microphysical processes must be provided in any case; i.e., not only to Eq. 2.15, but also to the governing equations of Eq. 2.8 as well. Without it, the set of the original, unfiltered equations in Eq. 2.8 is also unclosed. However, these source terms cannot obviously be represented as functions of {v, θl , qt } only, to be closed in Eq. 2.15 for the PDF, even if the algorithms for the calculation of these terms are available and they can be computed in the full original model of Eq. 2.8. For example, to determine the rate of radiative heating one has to know the SGS fluctuations of the concentration of all optically active constituents of the atmosphere that are considered by a radiation routine, such as cloud-water droplets, water vapour, ozone, carbon dioxide, and aerosols. One can anticipate the relatively small importance of the SGS fluctuations of all these constituents except for cloud water and water vapour. If that is so, then the local values of the constituents may be regarded as known, because the local values of cloud water and water vapour can be evaluated exactly (see Eq. 2.17), and for other constituents their grid-box mean provides an estimate for the local values that may be taken. There is, however, another problem with the radiation term (the ninth on the right-hand side) that makes it impossible to be determined exactly (see below). The situation regarding the microphysical term (the tenth on the righthand side) is more intricate. If a bulk diagnostic scheme for precipitation (cf., Ch. 23) is used then the expression for the precipitation flux Pil , although being a function of {v, θl , qt }, cannot at the same time be written locally on the subgrid scale because the assumptions required to formulate such a function make the expression valid for the (large) grid-box means only. The necessary assumptions include that the vertical velocity is zero and that horizontal advection is negligible. If, on the other hand, a bulk or spectral prognostic scheme for precipitation is used in which the precipitation flux Pil obeys a prognostic equation, then the flux cannot be uniquely determined through {v, θl , qt } only. The second kind of uncertainty is associated with the one-point statistics being considered. All the terms that have a local formulation and are

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expressed through the values of the quantities in question point-wise, are closed. In contrast, the terms that involve spatial operators like differentiation or integration over physical space are unclosed because given one and the same set of values of model variables at a point, there may not be a single value of, say, their derivative at this point, but a certain spread of possible values. (If a joint PDF were used that were to describe multi-point statistics this problem would disappear, but it would enormously increase the number of degrees of freedom such that the problem would become unsolvable.) This kind of uncertainty applies to all the unclosed terms 8–14: i.e., the four dissipation terms and the terms involving the pressure gradient and the divergences of the radiative and precipitation fluxes. One may suppose that the uncertainty associated with the radiative and precipitation fluxes is mainly the manifestation of the problem of cloud overlap. Indeed, the divergence of the vertical radiative flux at the height z for given local {v, θl , qt } (and thus known {qv , ql }), depends also on the {qv , ql } above and below z, and hence on vertical spatial correlations like

qv (z)qv (z ′ ) and ql (z)ql (z ′ ). Thus, the problem amounts to the determination of the vertical extension of clouds, where z ′ may be thought of as neighbouring grid points in the vertical. The uncertainties in the dissipation terms are of a slightly different nature, because the scales on which dissipation acts and is representable by means of spatial gradients are microscales (surely sub-grid scales), and the discretization of an atmospheric model is inappropriate to represent the dissipation. The representation of dissipation is, however, an aspect of turbulence modelling, so that it does not pose the main challenge as compared with the other unclosed terms. It can also be noted that the pressure as a variable may be eliminated from the full set of prognostic variables by taking the divergence of the equation of motion. This yields a Poisson equation for pressure from which it can be diagnosed. Since the Poisson equation has its solution in integral form, by doing so one removes the problem of closure for the pressure term from the first group (incomplete set of PDF arguments) and shifts it entirely to the second group (non-locality). In summary, the terms in Eq. 2.15 responsible for grid-scale and subgridscale advection, advection in phase space, and the buoyancy term can be evaluated exactly. However, the remaining terms describing radiative heating, microphysical conversion, acceleration due to pressure gradients, and molecular diffusion of momentum, heat, and moisture are unclosed for the first or the second reason, or both: the uncertainties due to the lack of variables for which the PDF is considered, and those associated with the

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one-point statistics. A general remark can be made concerning Eq. 2.15. Although the focus is on the cloud diagnosis, Eq. 2.15 contains in fact much more information. Actually, it describes all subgrid-scale interactions in an atmospheric model and as such, can be considered as the general form of the representation of all subgrid-scale processes (cf., Vol. 1, Ch. 2). With regards to the transport of the PDF in phase space and the subgrid-scale transport in physical space, it might seem a bit surprising that these terms are known exactly given that the information of the precise spatial locations of various values is missing in the PDF as compared to the original exact equations. This apparent confusion can be overcome if one realizes that the knowledge of how the fluctuations of various amplitudes transport and are transported is nevertheless used to determine the time rate-of-change of the PDF, because the governing equations themselves are used to derive it. In this sense, Eq. 2.15 contains in fact the same full information, at least regarding the advection of the quantities in question, as does the system of governing equations in Eq. 2.8. Thus, to solve it as it stands requires no less effort than fully resolving the advection in the original system of governing equations. The total amount of difficulty remains the same, just the form of these difficulties is different. When dealing with the governing equations the main problem is to resolve all scales involved which span many orders of magnitude in space and time. When solving Eq. 2.15 with the same level of accuracy, one must resolve the multi-dimensional joint PDF of the fluctuations sufficiently carefully. More detailed analysis of the difficulties related to Eq. 2.15 and perspectives on how to use this equation directly are found in Sec. 4.

3

Survey of the various types of cloud schemes as obtained from the Liouville equation

In this section, various simplifications of the PDF modelling approach are considered that can help in obtaining a more tractable formulation of the problem. Different combinations of these simplifications lead to different cloud schemes that are currently used in various NWP, climate and onedimensional research models. The simplification types and the corresponding types of cloud scheme are summarized in Table 25.1. As can be seen from the table, the main characteristics that distinguish the cloud schemes from each other are the type of closure of the truncated Liouville equation

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(assumed-PDF vs equations for separate PDF moments), the specific treatment of the unclosed terms (columns in Table 25.1), and the set of variables for which the PDF is considered (rows in Table 25.1). Table 25.1 Classification of cloud schemes. One of the entries is labelled with * for ease of reference in the main text. Compute moments retaining sources due to turbulence, microphysics, convection etc.

Compute moments accounting for turbulence only

Quasi-fixed moments

Assume a PDF (explicit reconstruction of the PDF) {qt } {s}

Tompkins (2002) Watanabe et (2009)

{w, s} {w, qt , θl }

al.

*

Sommeria and Deardorff (1977); Bougeault (1982); LeTreut and Li (1991); Bechtold et al. (1995); Fowler et al. (1996) Mellor (1977a) Lappen and Randall (2001a); Golaz et al. (2002)

Sundqvist (1978) Smith (1990)

Do not assume any PDF (implicit reconstruction of the PDF) {s}

3.1

Wilson et al. (2008)

Moment equations

Let us start with a transformation of the Liouville equation of Eq. 2.15 supposing that we decide not to solve it directly. This transformation, although not always explicitly, stands behind all existing cloud schemes. It consists of the multiplication of Eq. 2.15 with polynomials of increasing order in respect of the PDF’s variables and the integration of the resulting equations over the entire phase space, or some part. In this way, prognostic equations for the full or partial statistical moments of the PDF are obp tained. The equation for a full generic moment of the form vipi vj j vkpk θlq qtr  including velocity component and scalar fluctuations reads:

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p

d vipi vj j vkpk θlq qtr  ≡ dt p p pi pj pk q r ∂ vi vj vk θl qt  ∂ vipi vj j vkpk θlq qtr  ∂ vm vipi vj j vkpk θlq qtr  + ˜ um  + = ∂t ∂xm ∂xm ∂ ˜ ui  ∂ ˜ uj  p p −1 − pi vm vipi −1 vj j vkpk θlq qtr  − pj vm vipi vj j vkpk θlq qtr  ∂xm ∂xm ∂ ˜ uk  ∂ θ˜l  p p − pk vm vipi vj j vkpk −1 θlq qtr  − q vm vipi vj j vkpk θlq−1 qtr  ∂xm ∂xm ∂ v ∂ ˜ q  p p i vm  t + pi vipi −1 vj j vkpk θlq qtr  − r vm vipi vj j vkpk θlq qtr−1  ∂xm ∂xm pi pj −1 pk q r ∂ vj vm  pi pj pk −1 q r ∂ vk vm  vk θl qt  + pj vi vj + pk vi vj vk θl qt  ∂xm ∂xm ∂ θl vm  ∂ qt vm  p p + q vipi vj j vkpk θlq−1 qtr  + r vipi vj j vkpk θlq qtr−1  ∂xm ∂xm gi & pi −1 pj pk q r ' gj & pi pj −1 pk q r ' v v v vk θl qt θv vj vk θl qt θv + + θr i θr i j $ % gk & pi pj pk −1 q r ' pi −1 pj pk q r ∂p v v v vj vk θl qt θl qt θv − vi + θr i j k ∂xi $ % $ % ∂p pi pj −1 pk q r pi pj pk −1 q r ∂p vk θl qt − vi vj θl qt − vi vj vk ∂xj ∂xk % % $ $ 1 ∂R ∂P l L p p v m − + vipi vj j vkpk θlq−1 qtr vipi vj j vkpk θlq−1 qtr m Π ∂xm Π ∂xm $ % $ % 2 l ∂ vi ∂P p p − vipi vj j vkpk θlq qtr−1 m + ν vipi −1 vj j vkpk θlq qtr ∂xm ∂xm ∂xm $ % $ % 2 ∂ vj ∂ 2 vk p −1 p + ν vipi vj j vkpk θlq qtr + ν vipi vj j vkpk −1 θlq qtr ∂xm ∂xm ∂xm ∂xm % % $ $ l 2 κ Lv κ ∂ θl ∂ 2 ql p p + − vipi vj j vkpk θlq−1 qtr vipi vj j vkpk θlq−1 qtr Π ∂xm ∂xm Π ∂xm ∂xm % $ 2 ∂ qt p + κv vipi vj j vkpk θlq qtr−1 (3.1) ∂xm ∂xm as a generalization of Vol. 1, Ch. 2, Eq. 6.1. Note that no Liouville equation was invoked in the derivation in Vol. 1, Ch. 2. Here, the integration is carried out over the entire phase space, and

f  ≡

∞ ∞ ∞

−∞ −∞ −∞

f P (ˆ v, θˆl , qˆt ; x, t)dθˆl dˆ qt dˆ v,

(3.2)

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where f is a generic variable. Note that the two definitions of angled brackets, Eqs. 2.2 and 3.2, although formally differing from each other, are in fact equivalent (Pope, 2000). It should also be noted that if applying the general equation of Eq. 3.1 to a particular moment, in some terms a negative power can appear (e.g., if pi = 0 then pi − 1 = −1), and those terms, which formally contain a negative power, are in fact absent because they integrate to zero. The general form in Eq. 3.1 looks somewhat cumbersome (not least because there are three components of velocity fluctuations, none of which can be neglected in general) but for each of the low-order moments it reduces to a readable equation. For instance, setting pi = pj = r = 0 and pk = q = 1, one obtains the transport equation for the vertical liquid-water potential temperature flux. Under the usual boundary-layer approximation of horizontal homogeneity of the grid-scale flow, it reads: ∂ wθl  ∂ wθl  ∂ w2 θl  d wθl  ≡ + ˜ ui  = + dt ∂t ∂xi ∂z $ % $ % $ % ˜ ∂p gi 1 ∂Rz Lv ∂Pzl 2 ∂ θl  − w 

θl θv  − θl + − w + w ∂z θr ∂z Π ∂z Π ∂z $ % % % $ $ κ L v κl ∂2w ∂ 2 θl ∂ 2 ql + ν θl + − . (3.3) w w ∂xi ∂xi Π ∂xi ∂xi Π ∂xi ∂xi Here, on the left-hand side are the local time rate-of-change, the advection by grid-box mean velocity, and the subgrid-scale advection (third-order turbulent transport). On the right-hand side one can identify the following terms, from left to right: mean gradient production/destruction, impact of buoyancy, pressure-gradient scalar covariance, correlation between vertical velocity and radiative flux divergence, correlation between vertical velocity and precipitation flux divergence, and molecular terms which describe dissipation and molecular diffusion. One more example is the temperature triple correlation (pi = pj = pk = r = 0, q = 3) that characterizes the asymmetry of the temperature distribution. It evolves as follows: d θl3  ∂ θl3  ∂ θl3  ∂ wθl3  + ≡ + ˜ ui  = dt ∂t ∂xi ∂z $ % $ % 1 ∂Rz ∂P l ∂ wθl  ∂ θ˜l  Lv + 3 θl2  − − 3 wθl2  θl2 + θl2 z ∂z ∂z Π ∂z Π ∂z % % $ $ l 2 2 ∂ θl ∂ ql Lv κ κ + − . θl2 θl2 Π ∂xi ∂xi Π ∂xi ∂xi

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One can also integrate the PDF equation over just a part of the phase space, obtaining the prognostic equations for partial statistical moments. The result will be similar to Eq. 3.1, but instead of full moments the partial moments a2 b2 c2 f P (ˆ v, θˆl , qˆt ; x, t)dθˆl dˆ qt dˆ v

f P ≡ a1 b1 c1

should stand, where at least one of the limits of the integrals is finite. More specifically, for our purposes the following partial moments are relevant: the zeroth-order partial moment ∞ ∞ ∞ P (ˆ v, θˆl , qˆt ; x, t)dˆ qt dθˆl dˆ v C = 1P ≡ −∞ −∞ qs (Πθˆl )

(cloud fraction), the first-order partial moment ∞ ∞ ∞

qt P ≡ qˆt P (ˆ v, θˆl , qˆt ; x, t)dˆ qt dθˆl dˆ v −∞ −∞ qs (Πθˆl )

(liquid-water content), and the second-order partial moment ∞ ∞ ∞

wqt P ≡ w ˆ qˆt P (ˆ v, θˆl , qˆt ; x, t)dˆ qt dθˆl dˆ v −∞ −∞ qs (Πθˆl )

(liquid-water flux). Note that in contrast to a full moment equation such as Eq. 3.1, in the equation for a partial moment, which is not provided here, some additional terms appear when integrating by parts, due to the finite integration limit for qt . Now that the propagation in time of the PDF has been replaced with the propagation in time of the infinite number of its statistical moments, there is no need anymore to diagnose the moments, which are prognosed. The PDF itself and, hence, all redundant moments (e.g., partial, if all full moments are prognosed) can be now diagnosed, at least in principle. (In practice, such a reconstruction of the PDF, for example based on the Fourier transform, may be a time-consuming procedure.) So far, the infinite set of equations in Eq. 3.1 for the moments is equivalent to Eq. 2.15 for the PDF considering the amount of information it contains. Once the transformation of the PDF system to the system of moments is complete, there is no longer a need for directly referring to the PDF, nor to

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the Liouville equation. For this reason, the cloud schemes examined below are constructed without any direct reference to the Liouville equation. The infinite set of moments may, however, be truncated (usually moments of order higher than some number n are neglected). In the truncated system of moment equations more terms are unclosed as compared to Eq. 2.15. Indeed, it is well known and can be readily recognized by inspection of the system of Eq. 3.1 that the moment equations are not independent of each other. Each prognostic equation for a moment contains a number of other moments that may not be among the predicted moments, such as those of higher order beyond the truncation number (the closure problem well known in turbulence modelling). For example, in Eq. 3.3 it is the SGS vertical transport of the temperature flux w2 θl  that is unknown if the system is truncated at the second-order level. It can also be partial moments that are unknown if only a set of the full moments is predicted. Specifically, in the case of atmospheric modelling an important example are the covariances involving virtual potential temperature θv (in the example of Eq. 3.3, θl θv ). We have to distinguish the two sets of unclosed terms, viz., the terms that are unclosed in the original PDF equation Eq. 2.15 (correlations with the pressure gradient, short- and long-wave radiation fluxes, autoconversion, accretion and rain evaporation fluxes, and various dissipations) and the moments (higher-order and maybe partial) that become unclosed due to truncation. In what follows, these sets will be referred to as the “intrinsically unclosed” and the “truncated moments”, respectively. In order to specify correctly the truncated moments, knowledge of the complete PDF is required (indeed, to specify correctly the evolution of, for example, the third-order moment, knowledge of the fourth-order moment is needed, and so forth). Thus, the problem of closure amounts to the problem of the reconstruction of PDF. Since the system of moments is truncated, one cannot speak of the exact reconstruction of PDF, but only of an approximation of the PDF that should be made as accurate as possible. At this stage another problem appears that is closely related to the problem of closure. If the moments are not exactly reproduced it can occur that such a sequence of moments does not correspond to any PDF (the simplest example is that there is no PDF with a negative variance). In mathematics, this problem is referred to as the “moment problem”: given a sequence of numbers, does a non-negative PDF exist with moments equal to these numbers? The solution to this problem is the set of inequalities, or constraints, for moments, which they must obey to yield a non-negative PDF. The first subset of relations in this

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chain of constraints concerns second-order moments only and amounts to the requirement of variances to be non-negative and the absolute values of correlation coefficients not to exceed unity. The second subset poses constraints that combine second-, third- and fourth-order moments, and so on (see Akhiezer, 1965, for details). If a model produces moments that obey such constraints, the model is called “realizable” (du Vachat, 1977; Schumann, 1977). The problem of realizability is a serious problem for most (if not all) truncated systems. Clearly, violation of realizability means an unphysical solution; it may be obvious, as with negative variances, but even if it is less obvious, it is dangerous and the consequences are difficult to foresee. We can say now that apart from the set of variables for which the PDF is considered, it is the treatment of the PDF and the truncated moments, the handling of the realizability problem, and the treatment of the intrinsically unclosed moments that make existing cloud schemes different from each other. 3.2

The treatment of the PDF and truncated moments

How can the PDF be estimated from the truncated set of moments? One possibility would be to set the moments beyond the truncation number equal to zero, stating that they do not play any significant role, and then to diagnose the PDF from the full infinite set of moments. However, as mentioned above, such a diagnosis is computationally expensive in general, and in addition, quite a lot of moments might need to be retained: first, to make the complete system of the moments realizable (which may not be possible at all), and second, to match observations with a resulting PDF. Hence, it is not surprising that there is no cloud scheme based on such a diagnosis. There is another presumably more fruitful way to reconstruct a PDF. To obtain the complete PDF from a finite and, preferably, relatively small number of its moments, one can assume a reasonable shape of the PDF with a relatively small number of unknown parameters that may be expressed through its statistical moments. This is referred to as the “assumed-PDF” approach and it was extensively studied and used in various applications (Baurle and Girimaji, 2003; Demoulin and Borghi, 2000; see also references in Colucci et al., 1998). If a PDF shape is assumed, the solution of just a few moment equations completely specifies the PDF. Generally speaking, the moments retained do not necessarily have to be determined from differential

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prognostic equations such as Eq. 3.1; only a parameterization rule should exist that provides the assumed shape with a (realizable) set of moments. Still, the determination of the moments from Eq. 3.1 is the only approach that is completely consistent with the physics. If the PDF is completely specified by some of its moments, then all other moments, including higher-order or partial moments, can be diagnosed from it. (Note, however, that such diagnosis can be performed only if the joint PDF for all involved variables is considered. For example, to be able to diagnose θl θv  the joint distribution of at least {θl , qt } has to be considered; for wθv , at least {w, θl , qt }.) It should be emphasized that, being diagnosed from one and the same PDF, all such quantities are consistent with each other (Larson, 2004). Moreover, the system of moments diagnosed from one and the same PDF is by definition realizable, provided that the moments that serve as input parameters for the PDF are realizable. If that is the case, then the assumed-PDF approach guarantees, at least in principle, the realizability of the model (in practice this remarkable property is often not used, see below). There are a number of cloud schemes based on the assumed-PDF approach. Within this group, the schemes differ from each other in the number of parameters and other properties of the assumed-PDF, in the treatment of the unclosed terms, and in the set of variables for which the PDF is considered. As to the assumed shape, several PDFs have been tested since the pioneering work of Sommeria and Deardorff (1977), where the Gaussian distribution was used. Other symmetric two-parameter distributions were tried, such as triangular (Smith, 1990), uniform (LeTreut and Li, 1991), and two-delta-function (Fowler et al., 1996) PDFs. Bougeault (1982) pointed out that a symmetric PDF is inappropriate for describing the trade-wind, boundary-layer cloud regime. In such a regime, strong localized updraughts where clouds form are accompanied by slow larger-scale, cloud-free downdraughts, and so the distribution of the fluctuations is asymmetric (skewed). It was also shown that given one and the same first and second moments of the saturation deficit distribution, its skewness can be quite different, and a cloud scheme was developed based on the gamma distribution that, although two-parametric, is able to be positively skewed. The need to account for the skewed nature of convective motions motivated Bechtold et al. (1995) to propose a mixture of Gaussian and exponential distributions, equal to Gaussian for mean oversaturation (corresponding to the stratocumulus regime where, indeed, the distribution is close to Gaussian) but with a thick exponential positive tail for strong mean undersaturation,

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403

which is the case in the shallow cumulus regime. Lappen and Randall (2001a) proposed to use a two-delta-function PDF that has three parameters per variable and can have asymmetric peaks. This PDF corresponds to the top-hat character of meteorological fields in a grid box (only two values of each field are observed in a grid box, which are usually identified as the values in updraughts and downdraughts) and thus, amounts to the so-called mass-flux approach (see, e.g., Tiedtke, 1989, for details, cf., also Vol. 1, Part II) formulated in probabilistic terms. The scheme developed by Tompkins (2002) uses a very flexible, although unimodal, beta distribution that has four parameters. In that scheme, an empirical relation is used to reduce the number of parameters to be prognosed to three. In the scheme of Watanabe et al. (2009), an asymmetric three-parameter triangular PDF is taken as a basis. In Lewellen and Yoh (1993), a double Gaussian distribution is considered which has five parameters per variable. It is flexible enough to describe very different regimes sufficiently well. For instance, it allows bimodality like the two-delta-function PDF, but at the same time possesses sufficient small-scale variability around each peak, the feature that is absent in the two-delta-function PDF. In the two opposite limiting cases it reduces to a two-delta-function PDF and to a single Gaussian PDF, with all the variety of possibilities in between. However, it is obviously more expensive and demanding in terms of input than all the abovementioned schemes. This motivated Larson et al. (2001), Lewellen and Yoh (1993), and Naumann et al. (2013) to suggest a double Gaussian PDF with two additional empirical relations between the parameters so that the number of parameters to be determined via moment equations is reduced to three. As was pointed out by Tompkins (2008), the relative-humidity scheme of Sundqvist (1978) may also be regarded as belonging to the group of assumed-PDF cloud schemes, although it might not seem obvious at first sight. In that scheme, cloud fraction is related to relative humidity through  1 − RH , (3.4) C =1− 1 − RHcr where RHcr is the critical relative humidity at which clouds start to form. The 0-1 approach is in fact the limit of this scheme if RHcr → 1, and if RHcr < 1 then fractional cloud cover at mean undersaturation is possible. It is shown (see Tompkins, 2008) that this scheme can be derived assuming an underlying uniform distribution of total humidity. Indeed, if all values of humidity between some qa and qb are equally possible, then the mean value of qs  + qa )/2 (see Fig. 25.3a), specific humidity in the cloud-free part is qe = ( ˜ where the notation ˜ qs  is used for qs ( T˜l ). The critical value of specific

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humidity above which clouds start forming is qcr = ˜ qs  − (qb − qa )/2, and from the similarity of triangles (Fig. 25.3b) it follows that: qb − q˜s C = . 1−C q˜s − qa

These three equalities can be combined to yield the relation qe = C q˜s + qv / ˜ qs , where (1 − C)qcr . Since the grid-box relative humidity RH = ˜ qv  = C ˜ qs  + (1 − C)qe , then

˜ qv  is the mean water-vapour content, and ˜ inserting the expression for qe one obtains: RH = C + (1 − C)[1 − (1 − C)(1 − RHcr )] = 1 − (1 − RHcr )(1 − C)2 , from which the relation of Eq. 3.4 follows.

(a)

(b)

Fig. 25.3 A uniform distribution of qt (left panel) and a schematic illustration of a relative-humidity scheme (right panel).

As mentioned above, the favourable property of the assumed-PDF approach that closure relations for the truncated moments are readily available is usually not exploited by current schemes. In practice, the realization of this property means the coupling between a cloud scheme (more correctly a subgrid-scale PDF reconstruction scheme) and all other schemes that require correlations which might be diagnosed from the PDF. In the models developed to date, such coupling is mostly missing. For instance, turbulence schemes usually do not use the closure relations for the truncated moments that are available within the assumed-PDF approach. Traditionally, turbulence schemes treat the higher-order moments and the buoyancy terms as unclosed (because turbulence schemes developed for the atmosphere usually do not follow the assumed-PDF approach). It means that such decoupled turbulence schemes are forced to introduce some additional assumptions to close the truncated moments which are not necessarily compatible with the assumed-PDF used in the cloud scheme. Two notable exceptions are the

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schemes of Golaz et al. (2002) and Lappen and Randall (2001a) where a tight coupling of a cloud scheme and a turbulence scheme is realized. Let us now return to the question of how the PDF and the truncated moments can be handled. As an alternative to the reconstruction of the PDF assuming its shape and prognosing its moments, one can predict only some of the moments without (formal) reconstruction of the entire PDF. The existence of a PDF possessing the set of predicted moments is, however, not guaranteed. The situation is similar to that in the assumed-PDF approach if the estimates of its input moments are internally and mutually inconsistent. But if separate moments are predicted without an explicit PDF reconstruction, the risk of obtaining a non-realizable model is probably higher than within the assumed-PDF approach, at least when models of comparable complexity are considered. This is because the PDF itself remains unknown and the truncated moments that enter the equations for the retained moments cannot be closed through the diagnosis of the PDF. To close them, additional arguments have to be invoked that do not necessarily lead to a realizable set of moments, so the problem of realizability is more acute here. Furthermore, since the PDF remains unknown, the SGS correlations required for the other schemes cannot be provided, even in principle. This situation is similar to the case of no coupling between different schemes within the assumed-PDF approach: one has to be careful of a possible contradiction between the resulting properties of the (implicit) PDF in the cloud scheme and the assumptions made in the turbulence scheme (cf., Ch. 26, Sec. 6). The Tiedtke scheme (Tiedtke, 1993) and PC2 scheme (Wilson et al., 2008) can be viewed as based on the approach of the prediction of the separate moments. In these schemes, no general assumptions are made about the underlying PDF, and prognostic equations for C and ˜ ql  are carried. The truncated moments, i.e., higher-order turbulent transport and buoyancy terms, are represented as the terms responsible for mixing and are computed using the output of a turbulence scheme. This output consists of the tendencies of total-water content and of liquid-water temperature, from which the tendency of the mean value of linearized saturation deficit

˜ s is calculated. It is assumed that from the point of view of C and ˜ ql , turbulent mixing only shifts the distribution of s to the left or to the right (homogeneous forcing assumption), without affecting any other moments. But, to define properly the impact of this shift on cloud water and cloud fraction, a certain form of the PDF should be nevertheless prescribed, and in this case the two-delta-function PDF for s is assumed. Their idea is, in

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this respect, akin to SCA introduced in Vol. 1, Ch. 7, Sec. 4, though they formulate the problem in a less systematic manner. Since C and ˜ ql  are statistical moments of some PDF, the fact that the PDF is not explicitly computed does not mean that it does not exist. One can see that in the PC2 scheme, the two-delta-function PDF is in fact assumed to parameterize a number of processes. Still, since there are two distinct equations for C and ˜ ql , each having its own, often independent, closure assumptions, it can occur, as in the schemes of Tompkins (2002) and Watanabe et al. (2009), that no PDF exists for a given C and ˜ ql ; for example, if one of the two quantities is zero and the other not. Additional restrictions are thus imposed to avoid such unwanted situations. 3.3

Intrinsically unclosed terms

Let us now turn to the question of how the intrinsically unclosed terms are treated in different cloud schemes (columns in Table 25.1). Recall that these are the terms that stem from the unclosed terms in Eq. 2.15 for the PDF; i.e., the terms 9–20 on the right-hand side of Eq. 3.1. These are correlations involving the pressure gradient, short- and long-wave radiation fluxes, autoconversion, accretion and rain evaporation fluxes, and various dissipations. In the sense of how those terms are treated, there are not many differences between the schemes currently used. The majority of schemes neglect the terms involving the correlations between the SGS fluctuations of microphysical sources or radiative heating and the SGS fluctuations of momentum, temperature, or humidity (middle column in Table 25.1). It should be noted that in the schemes where these terms are not present, it was not because their relative importance was estimated first and it was concluded that their role is insignificant. It is rather that they have not been included yet. The reasons for that seem to be mostly historical, because traditionally the inputs for a statistical cloud scheme (i.e., all necessary moments for an assumed-PDF) are taken from a turbulence scheme, the only place in an atmospheric model where statistical moments are computed. It is common to use as the starting point for the development of a turbulence scheme, whether simplified or sophisticated, the conservation laws for momentum and scalars, where the conservation law for a scalar usually has the form of a balance between storage (time rate-of-change), advection, and molecular diffusion, without right-hand-side sources due to microphysics and radiation. That is why these terms are also absent in

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the subsequently derived moment equations of the system of Eq. 3.1 (that stands behind each turbulence scheme and can be recognized, even if in a heavily truncated form). For example, the third line in the equation for the vertical heat flux, Eq. 3.3, is usually absent in a turbulence model. As an equation of motion, the Navier–Stokes equation with the buoyancy term is considered; that is, the same as in the host atmospheric model. That is why all the terms in Eq. 3.1 that stem from an equation of motion, i.e., those describing the subgrid-scale effects of the inertial force, the pressure-gradient force, buoyancy and dissipation, are usually present. (Note, however, that the grid-scale advection of the moments, i.e., the second term on the left-hand side in Eq. 3.1, although appearing in closed form, is often neglected by turbulence schemes.) Of course, how exactly the retained unknown terms (pressure-gradient correlations and molecular diffusion of momentum and scalars) are closed depends on the turbulence scheme being used, but the discussion of this issue is beyond the scope of this chapter. There are, however, schemes that retain the correlations involving the subgrid-scale microphysical and radiative-flux fluctuations (first column in Table 25.1), although they parameterize these terms in a rather crude way. In the group of the assumed-PDF schemes, the scheme of Tompkins (2002) has a representation of the microphysics term in the equations for the PDF parameters. Recall that in the scheme of Tompkins (2002) the beta distribution of specific humidity is assumed. This is determined by its four parameters, two of which are the lower and upper bounds of the distribution and the other two define its shape. With one additional empirical relation between the two shape parameters there are three irreducible moments that specify the PDF. Those moments should not necessarily be the mean, the variance, and the skewness, i.e., the first three full moments. Instead of these, in the presence of cloud water, mean total-water specific humid∞ qt ; x, t)dˆ qt ), ity, mean cloud water (first partial moment ˜ ql  = qs ( T˜l ) qˆt P (ˆ and the skewness are chosen. Since the host atmospheric model carries a grid-scale prognostic equation for ˜ ql  that contains, among other terms, microphysical sources/sinks, these processes are accounted for. To put it differently, one can say that the closure relations are hidden in the microphysics scheme. The way a microphysics scheme determines the tendency of ˜ ql  means that some assumptions in a microphysics scheme, may be, and likely are, incompatible with a particular underlying PDF; it may not account for any subgrid-scale heterogeneity at all. In other words, there

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may  ∞ be a l rather crude and inconsistent representation of the term (∂Pi /∂xi )P (ˆ qt ; x, t)dˆ qt (see Eq. 3.1). If cloud water is absent (clear qs ( T˜l ) sky), ˜ ql  cannot serve any more as a parameter of the distribution; then, an equation for the variance of q˜t is carried, but in this case the source term due to microphysics vanishes, so that the task is simplified. The way the corresponding microphysical term in the equation for skewness is parameterized is based on the assumption that if in a given grid box rain (the conversion of cloud water to precipitating water) is detected by a microphysics scheme, the right boundary of the total-humidity distribution is shifted to the left (i.e., rain removes the largest values of specific humidity). This means that the positive skewness of the distribution decreases, and a simple linear dependence of this shift on precipitation flux divergence is proposed. Also, the scheme has a term that describes the impact of convection on the variance and skewness of the distribution. Formally speaking, what is called convection is described by the same inertial, pressure, and buoyancy terms in the Navier–Stokes equation as for what is called “turbulence”, so that formally there is no special term responsible for convection besides those that are treated by a turbulence scheme. However, current closures in turbulence schemes are formulated under assumptions that are valid for small-scale turbulence and are not suitable for the description of quasiorganized large eddies. That is why additional terms need to be inserted to represent their impact on variance and skewness. This impact is also parameterized using physically plausible ad hoc considerations. The scheme of Watanabe et al. (2009) is built essentially in the same way as the scheme of Tompkins (2002), except that the distribution of saturation deficit s instead of qt is considered, and the underlying PDF is the three-parameter triangular. Due to its simple representation, the triangular PDF enables analytical expressions for variance and skewness through cloud fraction (zeroth-order partial moment) and mean cloud water (firstorder partial moment). That is why, since the model carries a prognostic equation for cloud fraction C along with the grid-scale equation for ˜ ql , it seems that there is no need for additional arguments of how to represent the impact of microphysics and convection on variance and skewness, but analytical expressions for these impacts can be derived using the tendencies of C and ˜ ql . However, those additional arguments are hidden in the formulations used by the microphysics scheme, and are needed to formulate the prognostic equation for cloud fraction C. Since the assumptions on the unclosed terms in both equations are not necessarily consistent with each

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other, the resulting C and ˜ ql  may be incompatible. In particular, the consistency condition C = 0 ⇔ ˜ ql  = 0 is not guaranteed automatically, so it is additionally imposed to yield the existence of the PDF. In the PC2 scheme (Wilson et al., 2008), one of the predicted moments is again ˜ ql , and source terms due to precipitation and radiation are automatically present in the equation for ˜ ql , with the closure assumptions hidden in the microphysics, convection, radiation, and turbulence schemes. In the equation for cloud fraction, the closure is based on plausible considerations; it is supposed that the processes of autoconversion and accretion do not affect cloud fraction. For other processes such as short- and long-wave radiative heating/cooling and turbulent mixing, a homogeneous forcing assumption is made, which means that the presence of these terms does not alter any statistical moment except for the mean ˜ s, thus only shifting the distribution to the left or to the right. Again, the two-delta-function PDF for s is assumed to define the impact of this shift on cloud water and cloud fraction. Finally, there are schemes where the parameters of the distribution are related to the large-scale environment in a simplified manner (the third column in Table 25.1). The relative-humidity scheme of Sundqvist (1978) and the scheme of Smith (1990) belong to this group. The former scheme, as we have seen, is equivalent to a statistical cloud scheme with a uniform distribution of total humidity. The variance of the uniform distribution is: (qb − qa )2 , 12 where qa and qb are the lower and upper bounds of the possible values qs  − qcr ), of total-water specific humidity. By construction qb − qa = 2 ( ˜ where qcr = RHcr qs  is the humidity at which clouds start forming. Hence, σq2 =

σq2 =

(1 − RHcr )2 (qb − qa )2 = ˜ qs 2 . 12 3

It is assumed that RHcr = 0.75, so that (1/3)(1 − RHcr )2 ≈ 0.02; this means that if the environment value ˜ qs  changes slowly, the variance of the PDF is also a slow function. Smith (1990) explicitly states that accounting for small-scale turbulent effects only is insufficient for the estimate of the parameters of the saturation deficit distribution, because other effects, such as those of microphysics and convection, should also be taken into account. In the scheme proposed by Smith (1990), the variance, or width, of the triangular distribution is thus not taken from the turbulence scheme, but is crudely related to the large-scale conditions through the critical relative

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humidity in a similar manner as in Sundqvist (1978). For the triangular distribution of saturation deficit s it yields: ⎛ ⎞2 1 − RH ⎠ .  cr qs 2 ⎝ σs2 = ˜ Lv ∂qs  1 + Cp ∂T T = T˜l

The coefficient of proportionality is again of order 10−2 , so that the variance of the saturation deficit distribution is nearly constant if the environment changes slowly. 3.4

The set of PDF variables

One more characteristic of the PDF (or of the system of its moments) is the set of variables for which it is written (rows in Table 25.1). As was pointed out in this and the previous sections, the most informative one is the joint PDF of the fluctuations of three components of wind velocity vi , liquid-water potential temperature θl , and total-water specific humidity qt , since it describes the joint variability of all the usual atmospheric prognostic quantities (except pressure and hydrometeors). Currently there is no scheme employing such a joint distribution. A reasonable simplification is to exclude the horizontal velocity from consideration. This does not mean that the horizontal velocity is assumed to be homogeneous; merely that the joint distribution of the SGS fluctuations of the horizontal components with other variables becomes unknown. This joint distribution might be needed to close higher-order moments that include various horizontal fluxes, but currently, it is not the greatest problem of subgrid-scale modelling, and relatively simple (e.g., down-gradient) formulations are employed to close these terms in turbulence schemes. The correlations of the fluctuations of vertical velocity with temperature and humidity fluctuations are, however, of crucial importance, so that the joint distribution of w, θl , and qt can be regarded nowadays as the most desirable (fourth row in Table 25.1). From such a PDF, not only cloud fraction and cloud condensate, but also various SGS vertical fluxes required by a turbulence scheme can be estimated, such as the buoyancy flux wθv  and the vertical flux of potentialtemperature variance wθl2 . The schemes of Lappen and Randall (2001a) and Golaz et al. (2002) are built on such a joint PDF. Recall that the former scheme assumes a two-delta-function distribution, whereas the latter assumes a reduced three-parameter, double-Gaussian PDF. Corresponding to these PDFs, they have to carry twelve and fifteen prognostic equations

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for moments (including means), respectively. In practice, the number of prognostic equations is reduced to seven and ten, respectively, by virtue of empirical algebraic relations between moments. Nevertheless, the schemes are quite expensive in computational time, and both schemes are currently used as one-dimensional parameterization schemes for research purposes only. This disadvantage of being computationally expensive is the price paid for the advantage that the moments are consistent with each other. There are, however, schemes that do not consider the SGS fluctuations of vertical velocity and scalars together. This means that in the above derivations the vertical velocity should be excluded from the arguments of the PDF. In this case the buoyancy flux, as well as other vertical fluxes, cannot be estimated from the PDF, and additional assumptions are required to express them in terms of available moments. Furthermore, if one is interested in the diagnosis of cloud fraction and cloud condensate only, there is no need for a joint bivariate distribution of temperature and humidity. Indeed, it is the local saturation deficit s (i.e., the difference between humidity and saturation humidity, that is a function of temperature) whose sign controls the presence of clouds and whose magnitude (if positive) determines the amount of cloud condensate. Hence, only the distribution of s is required. The schemes of Bechtold et al. (1995); Bougeault (1982); Fowler et al. (1996); LeTreut and Li (1991); Sommeria and Deardorff (1977); Watanabe et al. (2009) are formulated in this way (second row in Table 25.1). The scheme of Sommeria and Deardorff (1977) deserves special consideration. Originally it was formulated under the assumption of a Gaussian distribution for s only. In this case an additional closure relation is needed for the vertical buoyancy flux. Recall that the buoyancy flux (with primes indicating fluctuations) g g g ′ ′

w θv  = w′ [θ(1+(ˆ ǫ −1)qt −ˆ ǫql )]′  = (a w′ θl′  + b w′ qt′  + A w′ ql′ ) θr θr θr is expressed (using the definitions of θv and θl ) through the vertical fluxes of liquid-water potential temperature, total-water content, and liquid-water content. The first two fluxes are usually available from a turbulence scheme. Of course, their formulation is incomplete in the presence of radiative heating/cooling and precipitation. Furthermore, since no joint assumed-PDF for {w, θl , qt } is considered, the turbulence scheme has to make some assumptions as to how to close higher-order and buoyancy terms. However, there is at least an estimate of w′ θl′  and w′ qt′ . In contrast, turbulence schemes provide no estimate of the liquid-water flux w′ ql′  at all, because

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they are formulated for quasi-conservative variables. As was pointed out by Deardorff (1976), in the case that the entire grid box is in local saturated conditions (overcast regime), an exact expression for w′ ql′  holds (because in this case ql′ = s′ ), which in the present notation is: 1

w′ ql′ wet = ( w′ qt′  − P w′ θl′ ) , Q where

 ∂qs  . P=Π ∂T T = T˜l

For partially saturated conditions, Sommeria and Deardorff (1977) proposed a linear interpolation for w′ ql′  and w′ θv′  between cloud-free (with

w′ ql′  = 0) and overcast regimes, with cloud fraction C being an interpolation weight. This results in

w′ ql′  = C w′ ql′ wet =

C ( w′ qt′  − P w′ θl′ ) , Q

and thus, inserting it into Eq. 2.3, we obtain:   AP A C w′ θl′  + b + C w′ qt′ .

w′ θv′  = a − Q Q

(3.5)

(3.6)

Although such an interpolation does not follow from the equations and is purely empirical, in subsequent papers (Mellor, 1977b,a) it was shown that if a joint Gaussian distribution for {w, s} is assumed, the liquid-water flux can be expressed through the fluxes of the quasi-conservative variables by means of direct integration of the PDF as

w′ ql′ 

≡

∞ ∞

sP (w, s; x, t)dsdw =

−∞ 0





ql  1

s2 1 √ exp − 2 2 ( w′ qt′  − P w′ θl′ ) C − , Q 2σs 2π 2Q σs

(3.7)

which would be the same as Eq. 3.5 in the absense of the second term within the square brackets. This term can indeed be negligible in certain circumstances, although generally not. In any case, the scheme of Sommeria and Deardorff (1977) may be regarded as based on the joint PDF for {w, s} if complemented by the considerations in Mellor (1977b,a) (third row in Table 25.1). It can be noticed that for other distributions similar extentions can also be developed. To the best of the author’s knowledge, no analogous

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analytical relations have been derived for other distributions. A partial exception is the scheme of Bougeault (1982), where the relation

s′ ql′ 

w′ ql′  = ,

w′ s′ 

s′2  valid for the joint Gaussian PDF for {w, s}, is used, but at the same time an analytical expression for the quantity s′ ql′  is derived in the same way as Eq. 3.7 using the assumed gamma distribution. It may not be possible to derive a simple analytical expression like Eq. 3.7 for every joint PDF. Nevertheless, for practical purposes it does not matter much whether the result of integration of Eq. 3.7 is analytical or a numerical approximation is used; both are applicable. The only limitation may be that for an advanced PDF not all required moments are readily available. However, usually no assumptions are made as to the distribution of vertical velocity, and for the formulation of buoyancy flux the empirical relations of Eqs. 3.5 and 3.6 are taken as a basis which is supplemented by various non-Gaussian empirical corrections (Bechtold et al., 1995; Cuijpers and Bechtold, 1995; Larson et al., 2001; Naumann et al., 2013). These corrections, in the case that the buoyancy flux is not diagnosed from the joint PDF are indeed very important. As has been shown in many studies (see e.g., Bechtold and Siebesma, 1998), in reality liquid-water flux is a highly non-linear function of cloud fraction. Since at small values of cloud fraction there is a pronounced maximum in liquid-water flux, the linear formula of Eq. 3.6 is inappropriate to correctly represent the buoyancy flux. Returning to the list of the PDF’s arguments, Table 25.1, we observe that, apart from saturation deficit s, there is one more possibility for reducing the number of variables from two (temperature and humidity) to one. The schemes of Sundqvist (1978) and Tompkins (2002) neglect temperature fluctuations and account for humidity variability only (first row in Table 25.1). Whether such a simplification step is justified or not is questionable. Although Tompkins (2002) stated that the temperature fluctuations are less important as compared to the humidity fluctuations, Price and Wood (2002) showed that the neglect of θl′ can lead to errors in cloud fraction of up to 35%. Tompkins (2003) confirmed that the temperature variability and its impact on clouds are not negligible and that it is desirable to account for θl′ . It may also be speculated that the role of temperature variability may increase if cloud ice is taken into consideration, because temperature affects the intensity of phase transitions to ice.

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Discussion and conclusions

In the two preceding sections, it has been shown how the Liouville equation for the joint probability distribution of the wind velocity, temperature, and specific humidity fluctuations can be derived, and how the existing cloud parameterization schemes can be obtained from it. The Liouville equation of Eq. 2.15 offers an opportunity for estimating the probabilities of the fluctuations of different amplitudes about the grid-box mean in a given grid box, thereby enabling estimates of partial cloud cover, liquid-water content, and many other grid box-related quantities that are not linear functions of the local values of the atmospheric variables and thus cannot be computed using the grid-box mean values only. In general, such a prognostic equation for the PDF possesses the favourable property of having a closed form of the grid-scale and subgridscale advection of the PDF in physical and phase spaces, as well as source terms irrespective of their non-linearity and complexity. This statement regarding source terms is, however, valid only if the corresponding source terms in the governing equations are formulated locally in physical space, i.e., if they use one-point statistics. This is the case, for example, in turbulent combustion modelling where the scalar variables are the concentrations of various reacting species and the source terms (except for dissipation and covariances with the pressure gradient) represent chemical reactions whose intensities are uniquely determined by the local concentrations of species (Pope, 1979b). However, the modelling of atmospheric processes is less easy as there is only one locally formulated term: the term involving virtual potential temperature and representing the impact of buoyancy on the vertical component of wind velocity. Other source terms do not have a local formulation, containing spatial derivatives of short- and long-wave radiative fluxes and fluxes of total water due to different microphysical processes. Therefore, additional assumptions are required in order to close not only the dissipation and pressure correlation terms, which is the common problem of turbulence modelling, but also all aforementioned atmosphere-specific, non-local terms. Thus, it seems that there is no benefit in using the Liouville equation to account for these processes represented by the intrinsically unclosed terms as compared with current schemes based on the moment representation, because these terms need closure anyway. There is another problem of a practical nature with the Liouville equation that prevents its direct use. It was mentioned in Sec. 2 that finding the

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solution of the Liouville equation of Eq. 2.15 as it stands implies a problem with a very large number of degrees of freedom. Indeed, as there is no analytical solution of this equation in the general case, it has to be solved numerically (e.g., bin-wise or using the Monte Carlo method). In the first method, the phase space must be divided into a sufficiently large number of small bins (hyper-boxes) representing small intervals of fluctuation amplitudes, and the system of a large number of coupled algebraic equations for them has to be solved for each grid box. These equations are coupled not only in phase space through the partial derivatives of the PDF in respect of atmospheric variables within each grid box, but also in physical space because of the transport of the PDF in physical space represented by the partial derivatives in respect of spatial coordinates. If the PDF is written for three atmospheric variables, and if we want to divide the range of variability of each quantity into, say, ten intervals, or bins, which would be an utterly crude discretization of the phase space, then it results in 103 coupled algebraic equations per grid box, that in addition are connected to the same number of equations in adjacent grid boxes. The Lagrangian Monte Carlo method has been proven to be suitable for the solution of the prognostic equation for the PDF, and is computationally much cheaper than the direct solution method. In the Monte Carlo method, the trajectories of Lagrangian particles are tracked according to the governing equations. It was successfully applied to the turbulent flames modelling problem (Pope, 1981). However, it may still be computationally demanding. For example, in Gicquel et al. (2002), where the PDF equation was used as a subgrid-scale model for a large-eddy simulation (LES) model, estimates of computational requirements are reported. That model was 30 times more expensive than Smagorinsky closure and only 6 times less expensive than direct numerical simulation. The high computational cost of the PDF method is not a problem per se, since we can expect that more and more powerful high-performance computers will be available in the future. But it may appear that, in the sense of time consumed, this method is actually comparable with an increase of the resolution of the host atmospheric model. An increase of the resolution would be preferred since the closure problem might become less vital: for clouds, the simple 0-1 scheme is usually used at high resolutions. The representation of subgrid-scale variability evolution by means of the truncated moment equations seems to be a viable alternative to PDF modelling. Currently, two types of closure for these equations are implemented within different cloud schemes: (i) closures based on assumed shape; and,

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(ii) closures that leave the PDF shape undefined. Analysis of the advantages and disadvantages of having a prognostic assumed PDF from which cloud fraction and cloud water may be diagnosed, as compared to prognostic cloud fraction and cloud water, is given in Larson (2004) and Wilson et al. (2008). Briefly, one of the problems of the assumedPDF approach is that the assumed shape should be realistic and flexible enough, but at the same time sufficiently simple. The more parameters a distribution has, the more expensive the parameterization scheme becomes. In addition, the prognostic treatment of higher-order moments can lead to noisy low-order moments and cause numerical instabilities. But the advantage of this approach is that once the PDF of qt and θl is known, then this knowledge can be transferred to microphysics and radiation schemes which would improve the consistency of the entire model. If only C and

˜ ql  are predicted, then only they are known. As to the prognosing of C and ˜ ql , it may be more difficult to build closure assumptions for these partial moments. One of the essential problems is how to consider the SGS heterogeneity in a completely clear grid box when no information about it is available. Wilson et al. (2008) argued that one of the disadvantages of the assumed-PDF approach is “that it is often difficult to calculate quantitatively how a physical process will affect the shape of the moisture PDF, for example how the skewness of a PDF varies under precipitation processes”. It is also stated that, in contrast, it is straightforward to compute the corresponding increment of ˜ ql , because it is provided by a microphysics scheme. However, as we have seen, the necessity to specify how a process affects statistical moments (i.e., the necessity to close the intrinsically unclosed terms) cannot be regarded as a disadvantage, because it must be done (and it is done, explicitly or implicitly) in any case. In the above example of precipitation, the closure is hidden in the formulations of the microphysics and convection schemes, producing the increment of ˜ ql . It can be also added that the chances of models built upon the assumed PDF being realizable are potentially higher than for models that do not assume the PDF shape. This is because the former models explicitly reconstruct the PDF from which all unclosed truncated moments may be diagnosed. In the latter models, there is no such opportunity because the PDF is not known explicitly, and to close the truncated moments different reasonings are invoked that may contradict each other. Ideally, turbulence schemes that are used together with the assumed-PDF approach should employ the explicitly reconstructed PDF as a closure relation for truncated

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moments, and both microphysics and radiation routines should also use the information about the distribution. A cloud scheme should actually become a common interface, gathering the information about the required input moments and disseminating the output information to particular routines. However, for the time being, the coupling between cloud schemes and other schemes that might use the information about the explicitly reconstructed PDF, is mostly missing. Notable exceptions are the schemes of Golaz et al. (2002) and Lappen and Randall (2001a) where the coupling between the cloud scheme and the turbulence scheme is realized. So the current states of both of the main approaches are comparable in inconsistency. The fact that the distributions realized in the atmosphere may not be completely arbitrary is another consideration in favour of the assumedPDF approach. If this is the case, it makes sense not to spend resources on modelling the most general form of the PDF, but rather to restrict the PDF to being a member of a certain family, if this family can indeed be often observed in reality and if this restriction is accompanied by a substantial reduction in required resources as compared to the PDF modelling. The prerequisite for this method to be successful is, of course, a sufficient flexibility of the PDF family chosen. As mentioned in Sec. 3, symmetric two-parameter distributions are inadequate for the description of skewed convective motions and cumulus regimes (Bougeault, 1981, 1982). In Larson et al. (2002), different assumed shapes are investigated by means of a priori testing, where the input for a PDF, i.e., the set of its statistical moments, is taken from aircraft observations and LES and thus is free from model errors (“ideal input”). In this way the suitability of the functional form of a PDF for describing various states of the atmosphere is tested. It was concluded that the Gaussian and two-delta-function distributions noticeably underestimate the vertical liquid-water flux (and hence, the vertical buoyancy flux). Furthermore, since the two-delta-function distribution represents the state with only two values allowed across the grid box, which is usually thought of as updraught and downdraught characteristics; it does not possess “tails” and so suffers from insufficient variability. One of the consequences shown in Larson et al. (2002) is that if the observed vertical fluxes are used as PDF input parameters, then in the shallow cumulus regime no clouds at all are detected. If, instead, the true cloud fraction and water content are used as PDF parameters, then the diagnosed fluxes are wrong, i.e., perverted fluxes should be used to produce the right cloud cover. The lack of variability of the two-delta-function PDF is an admitted

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drawback of all mass-flux schemes as they use this type of PDF as an underlying assumption. In particular, this drawback forced Lappen and Randall (2001b) to introduce additional sub-plume variability into their mass-flux scheme that employs a joint trivariate two-delta-function PDF. The same reasons motivated the development of the EDMF extension of the conventional mass-flux scheme (Soares et al., 2004, see also Vol. 1, Ch. 7, Sec. 4.4.1). These additions, however, have a rather ad hoc character and do not follow from first principles. That is why a more plausible shape that allows sufficient variability would be preferable. In Larson et al. (2002) it was the five-parameter double-Gaussian distribution that showed the best results in respect of different characteristics among all other PDFs tested. It was also noted in that study that the resulting merits of a scheme are a combination of its good performance and its computational efficiency. Keeping this in mind, one can suppose that among the PDFs proposed to date, the reduced beta distribution and reduced double-Gaussian distribution deserve most attention. The former has bounded support, whereas the latter allows bimodality. Both PDFs have three parameters per variable, so that besides the means that are predicted by the grid-scale equations, one has to compute the (co-)variances and the third moments, with possible empirical relations between some of the moments. It is not that easy and cheap, but it seems to be the minimal level of complexity that allows for a satisfactory description of different cloud regimes. Until now, the selection of an underlying PDF has been based on observational and LES data and common sense, along with estimates of the computational costs of the scheme. It may appear, however, that the PDF selection can be made on more solid theoretical grounds. In statistical mechanics and information theory, the maximum-entropy principle allows the determination of the most probable functional form of a distribution of some microscopic characteristics of a system given certain information about its macroscopic characteristics (Jaynes, 1957; Pope, 1979a, see also Vol. 1, Ch. 4, Sec. 6). Considering the physical realism of the assumed-PDF approach within the entire atmospheric model, it depends not only on the assumed shape itself, but also on the input for the PDF, i.e., on how accurately the required statistical moments are determined. The same problem occurs also for the schemes which are built upon the prediction of separate moments, in terms of the intrinsically unclosed terms that remain unknown in all approaches including direct PDF modelling. Recall that these terms are correlations involving pressure gradients, short- and long-wave radiation

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fluxes, autoconversion, accretion and rain evaporation fluxes, and various dissipations. Dissipation and pressure-correlation terms, whose representation is a part of turbulence modelling, were always present in turbulence closures, although often in a very simplified form. In those turbulence closures that do not carry any prognostic equation for dissipation rate, which is the case for most turbulence closures used in atmospheric models, the modelling of dissipation is confined to the choice of the formulation for the turbulent length scale and of empirical constants in these formulations. Modelling the pressure-correlation terms, although by no means less important than the representation of dissipation, has received far less attention in atmospheric modelling. Some progress in the modelling of pressure-correlation terms has been made in the last three decades although to a greater extent in engineering applications as compared to atmospheric modelling, which has its own specific aspects, such as the effects of buoyancy. In contrast to dissipation and pressure-correlation terms, the other atmospheric non-local source terms such as those due to microphysics and radiation are usually either completely neglected or their representation is based on rather crude ad hoc assumptions. To the best of the author’s knowledge, there exists no systematic investigation of these terms (although some steps have been undertaken; see the analysis of the correlations containing precipitation flux in Khairoutdinov and Randall, 2002). Returning to Table 25.1, we can say that the capabilities of the current approaches are not yet fully explored and exploited. Inspecting the assumed-PDF part of the table, if all its boxes were filled in, it could be said that the consistency and physical content of cloud schemes increases from the upper-right to the lower-left corner, with the upper-left and lower-right corners representing different aspects of partial consistency. Indeed, in the upper-left corner is the scheme of Tompkins (2002), which considers the variability of specific humidity only, but accounts for all possible sources influencing this variability (although not in a rigorously derived form). As an opposite extreme, one can imagine a scheme that considers the joint velocity–temperature–humidity distribution with oversimplified computation of its statistical moments (although it is doubtful that this would produce good results). The lower-left corner, labelled with an asterisk, is empty so far. Bearing in mind the improvements that generally arise from including more variables into the list of PDF arguments and from including more physical processes into the moment equations, one may expect that a hypothetical scheme from the lower-left corner would be not only the

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most physically consistent among the schemes considered, but also quite attractive in terms of its ability to describe different cloud regimes. One more remark can be made concerning the possible use of the Liouville equation in combination with the assumed-PDF approach (cf., Vol. 1, Ch. 2, Sec. 6.2). As mentioned above, there are some terms in Eq. 2.15, viz., the advection in phase space, the sub-grid scale advection in physical space and the buoyancy term, that have a closed form and as such do not require any closure assumptions. In the truncated moments representation, these terms become unclosed. Recall that within the assumed-PDF framework they are (or can be) closed through their diagnosis from the PDF: i.e., the prescribed shape of the PDF is the very closure assumption that is invoked. It might be meaningful to use the opportunity to keep these terms in closed form without any assumptions, and thus a viable option might be to split the physical processes into two groups and calculate them by means of two different approaches. (The situation is similar to the splitting of an equation into two parts in order to solve the parts by means of different numerical methods: see Vol. 1, Ch. 2, Sec. 6.2.3.) One group would include the processes described by terms that require closure in any case (and, possibly, also the subgrid-scale advection in physical space; see below). Those processes would be accounted for by means of the assumed-PDF approach; i.e., the necessary input statistical moments would be provided by the moment equations including these terms only (all terms except for buoyancy and advection in phase space). In this way, a partial update of the PDF is obtained, where the functional form of the PDF remains the same, and only its parameters are changed. After this, the PDF might be discretized and advanced in time according to the Liouville equation with due regard for the buoyancy and the advection in phase space. After this second semi-step, the PDF does not have the assumed shape any longer, but to further propagate the PDF to the next timestep it should have it again. To this end, the updated PDF should be approximated to fit the assumed shape, then the procedure can be repeated at the new timestep. If it is the buoyancy and the advection in phase space only (but not the SGS advection) that enter the Liouville equation apart from the local time rate-of-change, then the algebraic equations representing the discretized Liouville equation become decoupled at least in physical space, which diminishes computational requirements. Moreover, the buoyancy term has a simple form and does not demand much computational effort. But whether or not such splitting is reasonable is still unclear before the added value is estimated.

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5

421

Editors’ supplementary remarks

This chapter derives a generalized Liouville equation in a rigorous manner based on a filtering approach. Recall that Vol. 1, Ch. 3 discussed three approaches for constructing subgrid-scale parameterizations: a cutoff filter based on the numerical grid itself, a more general filtering approach, and a multiscale asymptotic-expansion approach. Here, we outline how the generalized Liouville equation can also be obtained from the perspective of a multiscale asymptotic expansion. From this perspective, the equation for the subgrid-scale distribution of variables simply reduces to that of defining a distribution at a given macroscopic point. By following Vol. 1, Ch. 2, Sec. 6.2.1, we define a single-point physics by: ∂ ϕ = S, (5.1) ∂t with S the total source term. The corresponding Liouville equation for the distribution P is outlined therein, and is given by: ∂ ∂ P (ϕ) = − [P (ϕ)S]. (5.2) ∂t ∂ϕ In generalizing this equation, first note that if the given point is moving relative to a reference frame, the above time derivative must be replaced by a Lagrangian time derivative, D/Dt: ∂ ∂ D P (ϕ) = − [P (ϕ)S] − · [P (ϕ)F], Dt ∂ϕ ∂v

(5.3a)

where ∂ D = + v · ∇, (5.3b) Dt ∂t and where the velocity v also becomes a new dependent variable of the distribution. Here, we have assumed that the velocity equation is given by: ∂ v = F. (5.4) ∂t The symbol ∂/∂v should be understood as analogous to the nabla operator ∇, but the derivative is taken in terms of velocity components rather than the spatial coordinates. For further development, note that the distribution P depends on the spatial coordinates x, also through the dependent variables ϕ and v. Thus: ∂P ∂P + (v · ∇v) . (5.5) v · ∇P = v · (∇P )ϕ,v + (v · ∇ϕ) ∂ϕ ∂v

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Here, the subscripts in the first term on the right-hand side indicate that the spatial derivative is taken by holding ϕ and v constant. The last term is a short handed expression:  ∂P ∂ ∂P vj . = vi (v · ∇v) ∂v ∂vj ∂xi By substituting Eq. 5.5 into Eq. 5.3b, we obtain: DP ∂P ∂P ∂P = + v · (∇P )ϕ,v + (v · ∇ϕ) + (v · ∇v) . Dt ∂t ∂ϕ ∂v

(5.3c)

Equation 5.3a together with Eq. 5.3c is equivalent to Eq. 2.15, once explicit expressions for the source term S and the forcing F are specified. However, also note that the velocity is divided into background and perturbation parts in Eq. 2.15 by setting v = v ¯ + v′ . This derivation follows more directly the presentation in Vol. 1, Chs. 2 and 3 and clarifies how the filtering and multiscale approaches lead to the same equation for the distribution function. Of course, it does not replace the more rigorous derivation in the main text. Finally, we provide a further perspective on a rather cautious remark on the direct use of the Liouville equation within the main text. Ch. 23 suggests a strong need to introduce bins for the microphysical particle size distribution. However, the decompostion into bins could be equally important for subgrid-scale distributions, as suggested in this chapter. Indeed, it may prove that truly accurate microphysical and turbulent calculations for the moist atmsophere require both aspects simultaneoulsy: direct bin calculations for both the microphysical particle size distribution and the subgrid-scale distributions.

Chapter 26

Towards a unified and self-consistent parameterization framework

J.-I. Yano, L. Bengtsson, J.-F. Geleyn, and R. Brozkova Editors’ introduction: Part IV of this set has considered various issues related to a self-consistent treatment of subgrid-scale physical processes. The purpose of the present chapter is to address how the issues of unified and consistent development of parameterizations emerge and may be considered within an operational context. While the previous four chapters have treated specific issues in detail, this chapter presents a more general discussion from the authors’ own perspectives.

1

Introduction

Parameterization development today is a mixture of detailed studies of the actual physical processes, either through observations or process model studies, in combination with relatively crude technical engineering and fitting of unknown parameters, with the aim of building and refining suitable conceptual models. In the context of this set, the most familiar example of such a conceptual model is the one-dimensional mass-flux plume model for deep convection. The convective-plume hypotheses emerged from simple laboratory experiments and conceptual consideration of the tropical heat budget. Cloud-resolving models (CRM) and large-eddy simulations (LES) together with observations are used as the “truth” in order to test the assumptions made in the parameterizations. Traditionally, physical parameterizations are developed independently based on the physical processes that each scheme is envisaged to account for, and the focus is on providing a required thermodynamic forcing from each parameterization, essentially as separate schemes. For instance, there 423

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can be a cloud scheme generating a large-scale cloud cover, which is different from the cloud cover given by deep convection. The convection and cloud schemes may use different microphysical computations for the generation of precipitation, and finally their outputs must be combined somehow in order to give a total cloud cover as seen by the radiation scheme. Thus, unification and consistency of parameterizations become particularly important in operational contexts, because there are so many physical schemes implemented into a numerical model arising from various different needs. These schemes are often developed without much regard to the other pre-existing schemes, and thus many questions around consistency of the schemes naturally arise. In order to avoid having to face this issue, the unification of the physics becomes a key agenda in model development. Another issue in the operational context is the validity of the parameterizations across a range of atmospheric scales. With increased computer resources, the opportunity to go to higher and higher model resolution arises. However, some parameterization schemes, for example, a convection parameterization based on the mass-flux concept, may need to take a different form depending on the horizontal resolution of the numerical model. Today’s general circulation models (GCMs) have a horizontal resolution of the order of 10–100 km. At these scales, vertical transport of heat and moisture due to cumulus convection is certainly of small scales that needs a parameterization. However, at higher horizontal resolution, with grid spacing below 5 km, deep convection is already partly resolved, and thus the underlying assumptions for convection parameterization must be properly adjusted (cf., Ch. 19). Especially, convection may no longer be under quasi-equilibrium against a given grid-box state when the grid-box size is sufficiently small (cf., Vol. 1, Chs. 4 and 11, and Ch. 20). Around the scales of 1 km grid spacing, deep convection may be resolved by the model. However, there remain vertical mixing processes on the subgrid scales which contribute to the thermodynamic forcing, such as shallow convection and turbulence. Thus, a unification between highly parameterized cumulus convection at coarse resolution and cloud-resolving models is required. Here, we emphasize that we approach unification and consistency of parameterizations primarily as an operational issue. In spite of the philosophical connotation behind these words, there are no fundamental theoretical issues to be resolved in order to maintain unification and consistency of parameterizations. To a large extent, it is rather an issue of the management of the model development and maintenance. Nevertheless, an appropriate

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philosophy towards model development must be adopted for this purpose, and this is the main theme of the present chapter. A set of moral dicta is introduced in the next section, because the problems of unification and consistency of parameterization often have more to do with morals and a disciplined approach than any scientific technicalities. Some wisdoms drawn from these dicta are discussed in Sec. 3. The two key words here, consistency and unification, are conceptually examined in Sec. 4. Section 5 presents specific examples of unification issues arising in an operational context. Sometimes, combinations of existing approaches or methodologies are claimed to be a unification in the literature and Sec. 6 examines some of these approaches critically. 2

Basic principles

In order to consider consistency and unification of parameterizations, we may take the following question as a starting point: how can we formulate the parameterization consistently and in unified and general manner? The atmospheric sciences are considered as applications of the basic laws of physics (and chemistry), and in order to maintain the robustness of our scientific endeavours this principle has remained intact ever since the Scandinavian school established modern meteorology. Of course, not all the laws of physics are precisely known for atmospheric processes. Cloud microphysics is a typical example that must be tackled with numerous unknowns. Nonetheless, a clear basic principle for a well-formulated parameterization suite is to respect all of the relevant physical laws in so far as they are known. Thus, the view of the authors is that the above question may be answered by following three dicta: (1) Start from the basic laws of physics (and chemistry). (2) Perform a systematic and logically consistent deduction from these laws. (3) Sometimes it may be necessary to introduce certain approximations and hypotheses to make progress. However, list these carefully so that you can recall later where and why you introduced them. As far as parameterization development is concerned, we insist on starting from robust physics that we can trust. Another way to restate the first dictum above is: never invent an equation. The development must start from sound physics, and any uncertainties in our physical understanding of a given process must properly be accounted for in the development of

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a parameterization. This is a key point of the process at which we should consider making a probabilistic description of a parameterization from a Bayesian point of view (cf., Ch. 20). A parameterization is, by definition, a parametric representation of the full physics on the subgrid scale (cf., Vol. 1, Ch. 2). Thus, a certain process of deduction from the full physics is required in order to arrive at such a parametric representation. Such a deduction process must be self-consistent and logical: a simple moral dictum. A relatively simple representation is required, and easily recognized. However, a completely self-consistent logical deduction to a simple representation is almost always not possible for many complex problems in parameterization. Certain approximations and hypotheses must inevitably be introduced. At a more practical level, those approximations and hypotheses must carefully be listed in the deduction process, with careful notes about the extent of their validity and limits. By doing so, we would be able to say how much generality and consistency is lost in this deduction process. Here, the main lesson is: be honest. Another difficulty in pursuing such an honest path is that the literature has sometimes become confused such that it may be difficult to identify clearly the basic physical principle for a given parameterization. An important wisdom here is: never use an equation unless you know where it came from (i.e., from which physics).

3

Some general wisdom

Before developing more specific discussions, it would be useful to list some general wisdom required in order to proceed in the manner advocated in the previous section. (1) Never go backwards. It is often tempting to add an extra term that appears to be missing from a given parameterization formulation. The addition of a downdraught to mass-flux convection parameterization has been essentially accomplished in this manner so far, and thus it is inherently inconsistent with the basic underlying formulation (cf., Vol. 1, Ch. 13). Instead, a formulation should always be rederived from the basic principles that were used in the original derivation but incorporating from the outset the new process to be included. (2) If you do not know how something is derived, never use it. If a given

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parameterization does not provide a careful derivation, it should be viewed with a great deal of suspicion. Of course, some physical relationships have no mathematical derivation, but they are only supported under a phenomenological basis. In that case, you should study carefully (by means of a critical literature survey) what the range of validity of this given physical law is. The law may not necessarily be applicable to a situation that you are interested in, or may not necessarily be sufficiently accurate for your purposes. (3) Use paper and pencil (this is the most basic wisdom). With the dramatic increase of work based on numerical computations which rely on numerical logistics, we often sit in front of computers all day. Following a given derivation, and repeating this under a slight generalization are basic processes of formulating a problem properly. These are still best done with paper and pencil. This set is also most fruitfully studied by working carefully through some of the key derivations.

4

Consistency and unification

By extending the general wisdom listed in the last section, let us now examine these two key words. 4.1

Consistency

In common scientific discussions, the consistency of a given theoretical formulation refers to either: (1) self-consistency; or, (2) consistency with physics. The first definition refers to the self-consistency of the logic when a formulation is developed in a deductive, systematic manner. Various examples are found throughout this set. Note that self-consistency is a key for following the second dictum in Sec. 2. It must also be realized that a self-consistent formulational development is also subject to the first and third dictums. A self-consistent development of a logic is meaningful only if it starts from a sound set of axioms or a sound set of laws of physics, as the first dictum dictates. As the third dictum dictates, there must always be clarity with introduced hypotheses, assumptions, and approximations: we cannot be clear with our

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whole logic unless we are also clear with these auxiliary matters. However, we should keep in mind that the validity of the hypotheses, assumptions, and approximations introduced goes beyond the issue of self-consistency. In the second meaning of the term, the question is posed as to whether a given formulation is consistent with the given physics, or known physics. Here, note that a self-consistent model can be developed upon an incorrect physical assumption. However, of course, we want to develop a model based on a correct physical assumption. For this purpose, we must carefully define a range of physical processes to be considered in the numerical models. Clearly, all the physical descriptions in current numerical models of the atmosphere do not take direct account of any quantum effects, although they are definitely present ultimately. So are these physical schemes inconsistent? Quantum effects are clearly negligible in a direct sense for all the atmospheric processes insofar as we are aware. For practical purposes, it is therefore sufficient to treat them completely implicitly: for example, as effects that contribute to basic thermodynamic properties (e.g., the heat capacity) of the gases comprising the atmopshere. Thus, this type of inconsistency is not an issue. However, the role of gravity waves, for instance, is more subtle. It is still likely that in many situations they can be neglected, but we can also imagine situations in the atmosphere for which they are relevant. From this point of view, this meaning of the word consistency is better reinterpreted in terms of the accuracy of an approximation, rather than as a fundamental issue of consistency. To a large extent, consistency demanded under this second meaning reduces to what is demanded by the first and the third dictums of Sec. 2. 4.2

Unification

“Unification of physics” is a phrase that we often hear in modelling contexts. However, we have to realize that if we follow strictly the principles outlined in Sec. 2, there would be no issue of unification of physics. We begin with a given single physics, and thus the need for a unification never arises as long as the subsequent deduction is self-consistent. The issue of unification can arise in the process of model development. In such a process, a set of people are often assigned for the development of different physical schemes: one for clouds, another for convection, a third for boundary-layer processes and so on. It is often the case that these developments are made separately because each task requires some specialist

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knowledge and intensive concentration of work. However, a full division is not a sustainable solution, because stratiform clouds are often associated with convection, for example. The treatment of clouds and convection within the boundary layer faces similar issues: should they be treated as part of a boundary-layer scheme, simply because they reside within the boundary layer? It transpires that the issues of unification of physics are usually only revealed in retrospect, and only as a result of uncoordinated efforts of physical parameterization development. If everything were to be developed under a single formulation, such a need should never arise. In this respect, the main issue is more of a matter of morals and a strong collective self-discipline, rather than a real scientific issue. Carefully discuss and agree the steps in the parameterization deduction, and the associated approximations and assumptions, by working with pencil and paper, and writing down everything together, before beginning to type even a single line of code. Coding may require individual work, but you must write down and agree on everything on paper together before getting to that phase.

5

Consistency and unification in an operational model

In order to make the general discussions so far more specific, this section examines a planning strategy for a unification of the physics within a particular operational model. The example taken is the Aire Limit`ee Adaptation/Application de la Recherche l’Operationnel (ALARO) model. The physical parameterizations within ALARO often take a different route compared with the usual framework of development of complex parameterization schemes. For instance, there is hardly any overlap with any of the approaches presented in Sec. 6. The model nevertheless constitutes an appropriate example for the present purposes. ALARO is characterized by a strong concern for a multi-scale operational behaviour (especially in view of grey-zone issues; cf., Ch. 19). More concretely this means that a high priority is given to the complex geometrical considerations that determine each grid point’s cloud cover and cloud condensate, and how such macrophysical processes influence the distribution of the radiative and precipitation fluxes. Furthermore, the turbulent fluxes computed at the previous timestep are used in order to avoid having too many quantities for consideration for a PDF-type approach (cf., Ch. 25). There is also a strong emphasis on the unification

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of computations which do not need a split from parameterization to parameterization (e.g., basic microphysical processes for various sources of condensation). As a consequence, the basic computations are well separated but there is much communication within and between timesteps. Cloudiness plays various parallel roles that should ideally be unified as much as is feasible. This is all displayed in Figs. 26.1 and 26.2. Figure 26.1 is a schematic of the original organization of model physics in ALARO; this version is called ALARO-0. Documentation for the ALARO model can be found in B´enard et al. (2010); Gerard et al. (2009); V´ an ˇa et al. (2008) and references therein. Figure 26.2 is a schematic of anticipated changes made for the goal of unifying some of the physics, especially for clouds and precipitation; this version is called ALARO-1.

Fig. 26.1 A schematic of the original organization of model physics in the numerical weather model ALARO-0.

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Fig. 26.2 A schematic of the changes made for an improved unification of model physics ALARO-1.

5.1

Thermodynamic adjustment and shallow convection

As seen in Fig. 26.1 the thermodynamic adjustment is done in two parts in ALARO-0. The first part is in order to compute a stratiform cloud cover, a target saturation value and a critical relative humidity. The second part is called after the turbulent diffusion computations. In this step the upgraded input variables (temperature and moisture) are adjusted due to latent heating/cooling and drying/moistening in the condensation computation step. In ALARO-1 (Fig. 26.2), the second call to the thermodynamic adjustment now delivers a combined non-convective cloudiness as input to the microphysics. This is called the “resolved cloud cover”. The idea is that the tubulence and shallow convection (which has been substantially upgraded in ALARO-1) can successfully transport the cloud condensates, and thus that the microphyics ought to take it into account. Furthermore, there is now a shallow convective cloud cover computed within the turbulence scheme which is combined with the deep convective cloud cover, and used as input to the thermodynamic adjustment at the next timestep. This is

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done in order to protect the convectively detrained condensates during past timesteps from too quick a re-evaporation. 5.2

Radiative fluxes

In ALARO-0, there was a diagnostic computation of a resolved cloud condensate, and the radiative characteristics of cloud were computed in a purely diagnostic manner also, because the thermodynamic adjustments of the moisture variables (liquid and ice) were done further down in the physics timestep, after the call to turbulent diffusion. In the new formulation of ALARO-1 it is anticipated that the process computations of deep convection, shallow convection and protected thermodynamic adjustment should be realistic enough to allow the removal of the diagnostic computation of a resolved cloud condensate. The hope is that the clouds computed in the first thermodynamics adjustment call will deliver correct inputs to the radiation scheme. 5.3

Turbulence and shallow convection

As seen in Figs. 26.1 and 26.2, in the new formulation a shallow convective cloud cover and corresponding cloud condensates are computed in the turbulence scheme. In ALARO-0 there is no cloud cover generated by shallow convection and no updated fluxes of cloud condensates corresponding to shallow convection. The prognostic liquid and ice condensates are diffused in the same way as temperature and water vapour. The hope is that this will help in providing (together with the updated computations of stratiform condensation) a more coherent picture of clouds and condensates for the microphysical treatment. This should also be true for the radiative computations at the next timestep. 6

Combination of two approaches as a route to unification

In order to follow the principles outlined so far and to develop subgrid-scale parameterization in a unified, consistent manner, a general approach that enables it is also required. Three general approaches for subgrid-scale parameterization development were presnted in Vol. 1, Ch. 2, Sec. 6: moment-based approaches; approaches based on distributions of subgrid-scale variables (DDF or PDF); and, approaches based on mode decomposition. The mass-flux convection

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parameterization is a special case of the mode-decomposition approaches, and is presented in a stepwise manner, essentially following the general wisdoms outlined in Sec. 3, throughout Vol. 1, Part II of this set. It may be tempting to combine some of the above-listed approaches together. In various parameterization studies, two approaches to describe a physical process can sometimes be combined in an attempt to take advantage of both methods. Especially, at the time of writing, methods combining a PDF (DDF) approach and the mass-flux approach are popular. Such a combined approach is often considered as a unification of model physics. However, such a claim must be considered with caution. First of all, we should carefully distinguish between a true unification and a simple combination of two approaches. When two approaches are combined, we also have to be extremely careful with the self-consistency of the final formulation. This section examines several examples. 6.1

EDMF: Eddy diffusion and mass flux

EDMF (see, e.g., Soares et al., 2004) is discussed in Vol. 1, Ch. 7, Sec. 4.4.1 and combines the mass-flux and the eddy-diffusion formulations for describing vertical transport. As outlined there, under SCA, there are some residual terms (eddy transports) that cannot be described in closed form by SCA variables. However, these terms can be described under the eddydiffusion formulation in a straightforward manner. Taking a mass-flux formulation for convective transport, and an eddy-diffusion formulation for environmental eddy transport is simply a combination of the two complementary formulations. There is nothing profound in this process and no true unification. For this reason, Soares et al. (2004) carefully avoid using the term “unification”. Unfortunately, subsequent work adopting the same approach is not as careful, and often a claim of “unification” is made. 6.2

Mass-flux and moment-based approaches

Lappen and Randall (2001a,b,c) noted that once a top-hat distribution is assumed, the various moments can be evaluated in a straightforward manner in terms of the fractional area σi , and the SCA values ϕi , for various physical variables. Once a set of simple analytical formulae for the various moments (as well as some correlations) are obtained, they can be used backwards in order to estimate the top-hat distribution-related variables.

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Lappen and Randall (2001a,b,c) claim that in this manner, a unification between the mass-flux formulation and the moment-based statistical descriptions is achieved. However, their claim should be viewed with some caution. First of all, keep in mind that an assumption of a top-hat distribution does not in itself contain any geometrical implications as for SCA. Thus, linking between top-hat distribution statistics and moments does not automatically mean the latter must be linked with the mass-flux formulation. Various subtle aspects of the mass-flux formulation drop off by simply reducing it to a top-hat distribution (cf., Vol. 1, Ch. 7). More seriously, a set of formulae obtained under this approach is correct only if a physical field satisfies a top-hat distribution, but otherwise, it has no obvious relevance. The question of approximating a general distribution using a top hat is just another issue.

6.3

Mass distribution and subgrid-scale variable distribution

Neggers (2009); Neggers et al. (2009) attempted to estimate the fractional area for convection (in their case further divided into dry and moist convection, to be estimated separately) by assuming a Gaussian distribution of vertical velocity over a grid-box domain. Although this is an interesting approach, one must be careful because the mass-flux formulation and the subgrid-scale variable distribution are described under different approximations. The mass-flux formulation, on the one hand, is based on a top-hat distribution approximation, whereas a Gaussian distribution is assumed for subgrid-scale variables in order to estimate the factional area. This is not necessarily a contradiction or inconsistency. One is free to introduce different approximations at different stages in the development of a formulation, and one may be able to approximate an actual distribution both with a top-hat and a Gaussian consistently. However, it would clearly be desirable to demonstrate this consistency.

7

General strategy

In order to develop a physical representation of subgrid-scale processes in a unified, self-consistent manner, we must first identify a basic general formulation. As recalled at the beginning of Sec. 6, three such approaches

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are suggested in Vol. 1, Ch. 2, Sec. 6. Once such a general formulation is established, the procedures from there mostly reduce to the disciplined application of moral principles. In introducing such moral principles in the context of both operational and climate research, we should also emphasize that the efforts should not, and cannot, be reduced only to individual responsibilities. Model developments are usually coordinated under a team structure. Thus, the team must be properly managed by an experienced scientist in order to maintain the necessary approach. The person who leads such a team must maintain an overall picture as well as take responsibility for many technical details. When either is missing, the model development can easily go astray. In particular, effective communication is crucial, and that is the most important means for maintaining self-consistency of a model developed by multiple people. Thus, from an operational point of view, consistency and unification of model physics reduces to some extent to an issue of team management. Effective team management is a major issue in its own right, but is beyond the scope of this set. A long reading list can easily be found on the subject. 8

Bibliographical notes

This chapter may be considered an expansion of the philosophy behind Yano et al. (2005). Yano et al. (2014) provides complementary reading for this chapter.

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PART V

Theoretical physics perspectives

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Introduction to Part V

In Part V of this set, we present selective materials that we believe may prove important in considering the future direction of convection parameterization studies. A motivation for this is that we wish to demonstrate how a variety of tools from theoretical physics and applied mathematics may have useful roles to play. Such tools may be under-exploited at present, and we hope to suggest to atmospheric scientists that these are tools worth learning and exploring, and also to theoretical physicists and applied mathematicians that there are important and interesting problems in atmospheric modelling to which they may be well placed to contribute. The self-organizing tendency of atmospheric convection suggests that it could in fact be at a state of criticality. For this reason, applications of ideas from self-organized criticality (SOC) to atmospheric convection are discussed in Ch. 27. This chapter also suggests how parameterization approaches can possibly be modified in the light of an SOC regime of atmospheric convection. Chapter 28 explains how the group analysis of differential equations can be valuable for the construction of parameterizations. By requiring the parameterized system to respect fundamental symmetry properties and conservation laws that are important for the full system, admissible forms for the parameterization can be deduced. The chapter outlines the mathematical basis for determining such constraints on parameterizations and includes various examples of its value for some relatively simple but nontrivial problems.

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Chapter 27

Regimes of self-organized criticality in atmospheric convection

F. Spineanu, M. Vlad, and D. Palade Editors’ introduction: Recent, careful, observational analyses have provided solid evidence that tropical convection shares many of the features expected from a system at self-organized criticality (SOC). This is a concept from statistical physics that can describe systems of many degrees of freedom that are slowly driven by some external forcing, with threshold behaviour of the individual degrees of freedom, and interactions between those degrees of freedom. The system’s internal dynamics cause it to evolve towards a critical state, characterized by long-range correlations and large variability over a wide range of space and timescales. SOC offers new perspectives on convection, not least on its organization, but we lack as yet a generally accepted simple model that reproduces the observed characteristics, is representative of the key aspects of convective dynamics, and is suitable for further analytic and numerical study. The physical mechanisms mediating the interactions are also yet to be established. This chapter discusses the requirements and the main elements of a suitable simple model and proposes two specific examples for further study and development: one an algorithmic model of punctuated equilibrium and the other a continuous reaction-diffusion model.

1

Introduction

Many natural systems consist of large collections of identical sub-systems that do not interact with each other. In many cases the ensemble (i.e., the whole system) is subject to an external drive at a rate which is much slower than the corrective reaction of any sub-system when it becomes unstable. 441

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In some circumstances such a system evolves to a type of behaviour called “self-organized criticality” (SOC). To introduce the basic terminology let us start with a simple example: an inflammable gas emerging at a slow rate at the flat surface of a porous material. The burning occurs when the fractional density of inflammable gas in the air reaches a threshold, at the surface of the porous body. Due to the porosity, the gas will accumulate at random at the surface, eventually creating patches of gas that emerged from neighbouring pores. If above one such pore there is ignition, it propagates very rapidly (like an avalanche) to all the sites within the patch, burning the gas and resetting to a stabilized state. Patches that are spatially separated (i.e., non-connected) are not affected. The dimensions of the patches (equivalently, of the avalanches of burn) are arbitrary and can be as large as the whole surface of the porous material. This state is similar to criticality, but without the system undergoing any phase transition. The state is stationary in the statistical sense. The fluctuations (local burning events) are correlated in space and in time with dependences on space and time intervals that exhibit long-range algebraic decays. As simple as it is, this system introduces key basic elements: slow rate of feeding, threshold, fast reaction in each sub-system, and avalanches. These are elements that describe one of the most fundamental means of organization of systems in nature. Each sub-system has a behaviour that is characterized by a threshold. The threshold refers to a parameter (e.g., density of inflammable gas, height of a column in a sand pile, amount of water vapour in a convective column) and separates a regime of quiet equilibrium from an active regime. The activity consists of a fast instability (burning, toppling of sand grains from the column, precipitation from cumulus convection, etc.) that returns the sub-system to the quiet equilibrium regime, in general by removing a certain amount of substance of the same nature as the drive: gas, sand, water, etc. This substance is transmitted to neighbours (at least in part) and those that are close to threshold can themselves switch to an active state, etc. This chain of influences triggered by the instability of one initial subsystem is fast like an avalanche and involves a set of sub-systems that were all close to threshold. It only stops when the neighbouring sub-systems are far from threshold, so that even if they receive a pulse from the active ones they cannot reach the active regime. The substance is redistributed among the sub-systems. This picture can easily be generalized to situations where there is no substance that is transferred between the sub-systems. For example, a sub-system that switches to an active state can undergo

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an internal reorganization that reinstates the quiet equilibrium and simply emits a signal. Neighbouring sub-systems that receive the signal and are themselves close to threshold will switch to active states and undergo internal reorganization, emitting signals, etc. The drive originates in an external source (e.g., the radiative heating of the land or evaporation from the sea surface) and may affect one or several sub-systems, chosen at random. It is also possible that the source can drive the whole ensemble of sub-systems, uniformly or not, but always at a rate which is slow relative to the fast reaction of any sub-system when the threshold is exceeded. Under the weak drive, the system evolves slowly to a state in which most of the sub-systems are close to threshold. The further slow drive will produce avalanches of various sizes that return the sub-systems to the quiet equilibrium state (i.e., all under the threshold). The system as a whole preserves this state as a statistical stationarity. This state is very similar to the state of a system that is on the verge of making a phase transition. The size of an avalanche is similar to the length of correlation of the fluctuations. The avalanches can be extended over all spatial scales, up to the dimension of the system, and in this analogy a correlation length which involves all the spatial extension of the system is the signature of criticality. In the case of SOC the system does not make a phase transition but remains at the critical state. The SOC state is statistically stationary so it is energetically ideal: the activity consists of random transients (avalanches) and the system explores the space of states which is specific to criticality. The system now acts under a rule: minimum rate of entropy production. It just reacts to external excitation so as to keep this statistical equilibrium.

2

Classical view of SOC

During the slow feeding, many sub-systems can reach marginal stability without becoming active. The existence of a threshold means that the state of activity is separated from the last bound state, the marginally stable one, by an interval. A simple fluctuation, with an amplitude comparable with this gap, will switch the sub-system from marginal stability to an active state. There are many systems that exhibit SOC. The models may be: (a) algorithmic, like the sand pile or Bak–Sneppen model (cf., Sec. 4), described by a set of rules for advancing in time the components (sub-systems); or, (b) analytic, like the Kardar–Parisi–Zhang equation describing gradient drive

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v = −∇h for a scalar field h that may represent the local accumulation of a quantity (e.g., sediment falling randomly on the bottom of a river). Analytic models may be studied based on the dynamic renormalization group. Very good general references exist on the subject of SOC (Bak, 1996; Jensen, 1998; Sornette, 2006).

3

The SOC of atmospheric convection processes

The physical processes that take place inside the grid cell of a finiteresolution numerical model have a fundamental role in the success of the large-scale dynamics simulation. This is the problem of parameterization. Part of the difficulty comes from the fact that there is a wide variety of situations at the small scales which have to be represented. The diversity of physical states cannot be simply reduced to a few global characteristics whose formal description would allow the transfer of reliable quantities to the large-scale dynamics. Different physical situations require different formalisms, with various weights placed on the components of the small-scale description: convective cores (either shallow or deep), cloud distribution, entrainment mechanisms (which are dominated by turbulence with different characteristics), detrainment, downdraughts and effects on low-level convergence, etc. Instead of a single formalism for parameterization, one must consider a variety of formalisms, each adapted to the characteristics that are dominant at a certain state of the atmosphere in the small-scale region. However, this is a heavy theoretical task. It first requires a reduction of the diversity of physical situations to a finite number, enumerated and characterized in a systematic way. Second, one should find a way to identify which particular behaviour is manifest in order to activate the appropriate formalism, prepared for that particular behaviour. This is difficult, but one can recognize that stochastic parameterization seems to partially fulfill this task, since by definition it explores a variety of states which belong to the statistical ensemble of realizations of a field of random physical variables (cf., Ch. 20). For example, an interesting dynamics is revealed by the numerical studies based on cellular automata (Bengtsson et al., 2013). This introduction is intended to suggest that in connection with the search for a reliable parameterization (and implicitly for successful largescale numerical modelling) one needs to explore the diversity of physical situations which occur in limited areas, and draw conclusions on how different formalisms can be developed for the description of each of them.

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It is not certain that this approach can be realized in practice. However, it is likely to produce interesting and possibly useful secondary results. This is because it suggests that we abandon the search for a unique formalism and encourages us instead to investigate the typologies of behaviours that can be identified as sufficiently individualized and distinct. Applying this idea and looking for general characteristics of evolution (connecting small scales of the order of a grid cell, and the large scales) one can identify in some circumstances systematic elements which point to large-scale organization of convective events. Many observational studies support this conclusion, but a unifying concept behind this manifestation has not yet been formulated. Obviously, the large-scale organization of convective events is not the unique possible state of the atmosphere. Strong drive induces a large-scale response consisting of strong horizontal pressure gradients, generation of vortical flows, jets and in general displacement of masses of air over large distances. However, such phenomena do not fit into the characteristic state of SOC (at least in its most common, simple form) and need different approaches. The organization of convection in a way that can be described by SOC appears to be just one of the possible states of the atmosphere and as such SOC cannot claim to be the unique, paradigmatic reference of atmosphere dynamics. But it is relevant in certain situations and this is supported by the fact that correlations of fluctuations of physical quantities show algebraic decay both in space and time. The basic elements mentioned before as specific to SOC are a natural component of large-scale convection systems. The quasi-equilibrium hypothesis recognizes the presence of two largely separated timescales: the slow external forcing and the fast convective response, which is similar to the corresponding property of the sub-systems’ dynamics in SOC (cf., Vol. 1, Ch. 8). The mutual influence between the sites of convection is also similar to the avalanches. The important aspects of large-scale organization of convection and precipitation have been revealed, in a more or less explicit form, in many works. In particular, there is accumulated evidence of mesoscale organization of systems of clouds. Leary and Houze Jr (1979) studied the genesis and evolution of a tropical cloud cluster. In the formation phase, the spatial characteristic consists of a line of isolated cumulonimbus cells whose orientation is transversal to the direction of the low-level wind. Furthermore, rain areas within individual cells merge. In this period, new convective cells are generated between and ahead of the existing cells. This is explained by the downdraughts originating from old convection cells, which enhance the

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convergence at low levels. This provides a moisture flux to new cumulonimbus updraughts, enhancing their buoyancy, and producing new convection cells. While the convection cells will eventually dissipate, new cells are generated. This is a propagation, in which the new cells develop in front of the line of the advancing precipitation system, faster than the dissipation of the older cells at the rear. The effect that localized convection exerts on its neighbourhood is compatible with the idea that there can be propagation of an effect in an ensemble of sites, as in an avalanche specific to SOC. The role of the mutual trigger is played by the downdraught from previous active convection sites. The cloud cluster is characterized at later times by the persistence of a large area of precipitation behind the advancing front. From this complex picture, the focus is on the aspects related to the propagation of influences between sites of convection, which bear some analogy with the avalanche phenomena and seem to support the concept of self-organization at criticality. Note that propagation in systems of convection to nearby sites needs a finite time interval, while in many classical realizations of SOC the avalanche is taken as simultaneous in all sites that are involved. The formation of clumps of clouds has been discussed by Lopez (1977), who pointed out the random growth before merging into large-scale mesoscopic formations. Note then that the SOC concept of avalanche, which is a correlated behaviour of a set of sub-systems that are all at or close to the marginal stability and switch to an active state by the effect of an influence coming from a nearby site, must here be adapted and become the signature of the process of generation of large-scale formation of clouds out of isolated active sites of convection. The active convection is stimulated by mutual influences. The paper of Su et al. (2000) on self-aggregation is a study of the generation of large clusters from isolated convection. Various scenarios have been proposed, such as the “gregarious convection” (Mapes, 1993) or, windinduced surface heat exchange (WISHE: Emanuel, 1987; Neelin et al., 1987; Yano and Emanuel, 1991, see also Vol. 1, Ch. 5). The large-scale organization of convection events has a multiple manifestation but at least the first phase seems to strongly suggest SOC. The initially distinct convection cells interact through the subsiding air between them and within an interval of perhaps ten days they may organize into mesoscale patches of rainy air columns. The mesoscale patches of each type (rainy and dry) may then coalesce, generating a single moist patch surrounded by dry subsiding air. Except for the final spatial distribution the large-scale propagation of

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mutual influence is similar to the generation of an avalanche in an SOC system. A description of this propagation is offered by Cruz (1973): “The radarobserved progression of one hot tower is a sequence of growing deep cumuli one ahead of the other in the direction of the cloud motion.” In more precise terms, the influence consists of the change of the environmental properties in the region of the nearby site. This may occur by means of the downdraught and precipitation. There is a minimum time for the process of interaction to take place: the development of convection at the initial site requires of order half an hour, and the decay of this convection of order another half hour. To characterize the duration of coherency of an individual event of convection we can take an interval δt ∼ 1 h. For comparison, the Plant and Craig (2008) stochastic parameterization adopts a duration of an individual plume δt = 45 min. Also defined is a closure timescale Tc , called the “adjustment time” in response to a forcing. This timescale determines the influence of the conditions for the nearby site through modification of the water-vapour content and the air temperature. Over this interval 90% of the convective available potential energy (CAPE) would be removed if the ensemble of plumes were acting on the environment in the absence of any forcing. Plant and Craig (2008) suggested that it is useful to account for a finite lifetime of an individual plume because this provides a physically justified temporal coherence to the noise that seems to play a useful role in terms of its possible upscale effects. The adjustment time used in the closure, on the other hand, is a property of an ensemble of clouds in modifying the largescale environment. By analogy with turbulence modelling, it is perhaps not a big surprise that an eddy turnover time can produce a reasonable approximation in practice to an adjustment time but in principle they are two different timescales (cf., Vol. 1, Ch. 3, Sec. 6; also Ch. 24). Now to focus on an estimation of the distance between nearby sites where convection may potentially arise if the local conditions are favourable. The smallest distance between clouds in densely packed states can be taken to be of the same order as the average dimension of the cloud, assumed to be δl = 2 km. We can use this input to estimate a speed for the propagation of the non-material influence which consists of the fact that the convection at one site may trigger convection at the nearby site: δl = 2 kmhr−1 . (3.1) vprop ∼ δt This speed is certainly different from the velocity of propagation of inertia-

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gravity waves between clouds. What is important is the fact that the SOC picture defines a new type of propagation, which is different from the usual inertia-gravity waves generated through geostrophic adjustment. However, this propagation does not have a unique direction and does not transport momentum or energy; it is just a trigger for possible convection at nearby sites. A distinction should be made between this propagation of influence through the change of the environment properties in the proximity of sites that are at the limit of convective activity, and the direct interaction mediated by gravity waves. The estimate relies on the density of convection sites, assuming the predominance of SOC-like mutual influence when the density of plumes is sufficiently high for the material aspect of interaction to be effective. It may also be worthwhile to note that inertia-gravity waves appear to be responsible for communication between cells in the regime of scattered, disorganized equilibrium convection (cf., Cohen and Craig, 2004). Under this regime the inter-cloud spacing is set essentially by the strength of external forcing because the individual cell strength does not vary much with the forcing, and so the forcing strength then dictates how many cells have to be scattered within unit area. Yano and Plant (2012) discussed the same issue from the point of view of the energy cycle of convection (cf., Vol. 1, Ch. 11, Sec. 10). Several important works have underlined the relevance of SOC for atmospheric convection and have provided solid arguments. Specifically, they consist of identifying algebraic decay of correlations. Peters et al. (2001) examined the statistics of the size of a precipitation event M , defined as  the released water column (in mm) M = t q (t) Δt, with q (t) the rainfall rate (mmhr−1 ). The number of events of size M is: N (M ) ∼ M −1.36 .

(3.2)

Another observation (Peters et al., 2001) regards the duration between precipitation events (inter-occurrence time), which is given as a function of the duration D in minutes: N (D) ∼ D−1.42 .

(3.3)

There is a close similarity with the SOC model of Bak–Sneppen to be discussed in Sec. 4, where the exponent is 3/2. Peters and Neelin (2006) proposed for intense precipitation events a picture inspired from phase transitions, with the water vapour w as the tuning parameter (like the temperature in the magnetization problem) and the precipitation rate P (w) as the order parameter. The slow drive is the

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surface heating and evaporation, while the fast dissipation of buoyancy and rain water is the precipitating convection. Peters and Neelin (2006) noted that this “fast dissipation by moist convection prevents the troposphere from deviating strongly from marginal stability.” This maintains the quasiequilibrium, which is the basic postulate of the Arakawa and Schubert (1974) parameterization (see also Vol. 1, Ch. 4). Above the critical threshold, the statistical averages have variations with the tuning parameter w as: β

P  (w) = a (w − wc ) ,

(3.4)

where β is an universal exponent. After scaling with factors that are imposed by the different climatic regions considered, the variation of the averaged precipitation P  with the difference (w − wc ) shows the same slope: log P  ∼ β log (w − wc ) ,

(3.5)

with β ∼ 0.215. Peters et al. (2009) analysed the mesoscale organization of convective events from the point of view of similarity with the statistics of percolation. The essential element that underlies the large-scale organization is a local property: the sharp increase in the rate of precipitation beyond a certain value of the water-vapour content of the air column. This sharp threshold and the fast reaction that follows (precipitation) are seen as elements composing the usual scenario of a first-order phase transition, in a continuous version. Since the distinctive component of the large-scale organization in a limiting regime is the generation of a cluster (of convective events) of a size comparable to the system’s spatial extension, it has been assumed that this is analogous to the percolation in a two-dimensional lattice of random bonds. Although this is a very solid argument in favour of a kind of phase transition, there remain reservations as to the proposed classification of this state as a phase transition. There are arguments for placing the largescale organization of convective events in the same universality class as percolation in two dimensions. However, we should remember that the SOC itself, as represented for example by the sand pile, belongs to the universality class of the percolation. This is interesting in itself but does not cover all possible interpretations that can be associated with the powerlaw dependence of the correlations of the fluctuating fields in a cluster of convection. It could be considered more appropriate to associate the generation of a correlated cluster of convection that covers almost all the

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space domain investigated with the divergence of the susceptibility, but recognizing simultaneously that correlations on all spatial scales are also possible. Equivalently, we find an approximative state of criticality for the system. Since, however, a phase transition to a completely new phase is not (and cannot) be seen, we identify this state as the statistical stationarity specific to the self-organization at criticality. Many factors, such as the content of water in a column of air, the type of convection (shallow or deep), and the spatial correlation through the effect of downdraughts on the environment contribute to the regime of precipitation. Besides the complexity of the fluid and thermal processes it is still interesting to look for a low-order dynamics that would be able to capture the essential aspects. It is a common procedure in the study of statistical systems to neglect specific particularities of systems and to focus instead on general properties which reveal connections between scale lengths. This allows classifications that go beyond circumstances and appearances, creating families of systems that are grouped into universality classes. It can then be useful to extend knowledge from one system to another, if they belong to the same universality class. An initial suggestion comes from the fact that the statistics of cloud clusters exhibit multiple scale organization, whose prototype is the continuous time random walk (CTRW). The rate of occurrence of earthquakes has a similar property and the similarity between the statistical properties of cloud sizes and of earthquake intensities has been noted and studied (Corral, 2005, 2006). It is simpler to examine the way this statistics results from general properties of the tectonic breaking, with phases of accumulation followed by a sudden release of energy. The events (earthquakes) are well defined, their parameters being recorded and documented in databases. The known property of earthquakes to exhibit self-organization at criticality provides additional support to the existence of the similar property of SOC for the ensemble of random precipitation events. The parallel between these two systems, with completely different manifestations in real life but with identical statistical properties, further allows us to identify a possible quantitative structural model that can be used to explore the SOC of convection: the standard reference for the dynamical behaviour of earthquakes in the state of SOC is the Bak–Sneppen model. It is well suited to describe statistical properties of convection as well.

Regimes of self-organized criticality in atmospheric convection

4

4.1

451

The SOC formulation of convection within the Bak–Sneppen model Why SOC realized in a punctuated equilibrium system may be relevant to the organization of convective precipitation events

The Bak–Snappen model consists of an ensemble of sites (sub-systems), each characterized by a value of a parameter which reflects the degree of adaptation (fitness) to the environment. There is a threshold level of fitness λ. By definition the sites whose current level of fitness is smaller than λ are not in equilibrium relative to the environment and can possibly switch to an active state, which involves update of their fitness parameter with a new, random value extracted with uniform probability from [0, 1]. It is only certain that the site k with the smallest fitness parameter xk (of course, xk < λ) is subject to update and the other sites (both < λ and > λ) are updated only if they are currently in a relationship of mutual influence with the site k (Fig. 27.1).

Fig. 27.1 Simulated avalanche in the Bak–Sneppen model in one dimension with one random site updated. The simulation uses N = 500 sub-systems and the first true avalanche appears after ∼ 103 updates from the initial homogeneous state. The threshold λ = 0.5 (solid line).

The approach to the state of SOC in the atmosphere may be similar to the evolution of the gap in the Bak–Sneppen model (Paczuski et al., 1996). In the initial state, a large number of sub-systems are far below λ and the updates are actually driven by the algorithm of choosing the lowest

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states (see Fig. 27.1). The gap is the distance between zero level of fitness (i.e., total incompatibility relative to the environment) and the current average level of fitness, which in states that are still far from statistical equilibrium is below the threshold λ. The evolution of the gap (which takes the aspect of a “devil’s staircase” known from dynamical systems) is the increase of the adequacy of the sub-systems to their environment (in other words, the rise of the average level of fitness). For the atmosphere, any convective event leaves a trace by modifying the environment in which the other convection will take place. In the Bak–Sneppen model, the fact that the sub-systems with the lowest fitness (the ones that are least compatible with the environment) are systematically updated has the consequence that the average limit of fitness increases, towards the limit λ, the threshold for self-organization (Fig. 27.2). Therefore, this limit (although λ can be calculated a priori) is actually produced by the system itself, and reached asymptotically (Paczuski et al., 1996). The plateaux in the gap evolution are interrupted by fast events when there are avalanches.

Fig. 27.2 Numerical simulation of the critical state in the Bak–Sneppen model in one dimension with one random updated site. The simulation uses N = 500 sub-systems and in the large time limit the threshold λ = 0.5 is reached (solid line).

The time evolution which precedes the SOC state therefore consists of the increase of the average fitness parameter across the system, with a tendency to reach the λ value. A histogram of minimum values of the fitness parameter, along the evolution, indicates that the number of subsystems with fitness lower than λ decreases as we approach the critical λ (see Fig. 27.3). In the atmosphere, this regime corresponds to the time

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interval where there are long periods of inactivity interrupted by precipitation events. The precipitation events remove part of the water vapour from a number of sites. This sub-ensemble of sites which are active almost simultaneously represents an avalanche. Affecting successively various subensembles of sites, the process removes part of the water vapour from almost all sites and the final result is that the differences in the amount of water vapour decrease progressively. The fact that the sites have similar values of this parameter (amount of water vapour) makes possible large-scale responses of the system to a fluctuation of the drive, or, in other words, the avalanches can reach the spatial dimension of the system. This is the signature of criticality. After reaching this state, the system remains in a statistically stationary state.

Fig. 27.3 Histogram of the minimum value of fitness at every timestep in a numerical simulation of the one-dimensional Bak–Sneppen model with one random neighbour. The Heaviside–Theta-like plot has the critical value at 0.5 as theoretically predicted. Simulation made with N = 50 sites and ∼ 106 updates.

This is a model of punctuated equilibrium. By “punctuated equilibrium” one understands the situation where “any given small segment of the sites will experience long periods of inactivity punctuated by brief periods of violent activity” (de Boer et al., 1995). The model has been shown to describe the statistics of earthquakes (Ito, 1995). 4.2

Correlated convection events

The comparison is based on the following mapping. The sites are local atmospheric processes (attached to a small and fixed area) that are close

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to producing convection. “Fitness” is a characteristic that refers to nonactivity. A site fits to the conditions of non-activity when the conditions supporting convection are not realized. The sites (local atmospheric processes) are affected by the other sites during their update. Let us now derive quantitatively some results regarding Bak–Sneppen statistics that seem to be close to the statistics of atmospheric processes. Assume that the system consists of N sites, each characterized by a real number xi , i = 1, N . The dynamical rule is: at each timestep the xi with minimal value is replaced by a random number extracted from the interval [0, 1] with uniform probability distribution. After updating the site i, K − 1 other sites chosen at random must also be updated. These K − 1 other sites are also replaced by random numbers, in the same way as for xi . The model is called “random neighbours” (de Boer et al., 1994, 1995; Paczuski et al., 1996). 4.3

The K = 2 model of random neighbours: The algorithm

At every timestep (update), the site with the minimum barrier xi is chosen along with K − 1 other sites, and the fitness parameters are updated. In this model only one other site, chosen at random, is updated (de Boer et al., 1994). The definition of an avalanche is made up of the following steps: (1) fix a threshold barrier λ; (2) count the number of active sites, that is the number of sites that have fitness less than λ; (3) if there are active sites for T count the number of consecutive temporal steps T for which there exist active sites. This defines an avalanche of temporal duration T , and the duration has a probability with a scaling at large T of: Paval (T ) ∼ T −3/2 .

(4.1)

S (t) ∼ t−3/2 .

(4.2)

This is illustrated numerically in Fig. 27.4. There is also a scaling law governing the time separation of the moments when the same site will again be the minimum of all the sites. The first return probability S (t) is defined as the probability that, if a given site i is the minimum at time t0 , it will be again minimum (but for the first time) at time t0 + t. This means that between t0 and t0 + t the site i has not been minimum. The probability S (t) for the random-neighbour model, scales as: Important results on the Bak–Sneppen model can be derived analytically. Here and in Appendix A, results are presented that have been ob-

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Fig. 27.4 Log–log plot of the probability of avalanche duration P (T ) against T . The results were obtained numerically for a one-dimensional Bak–Sneppen model with one random updated site. The numerically determined exponent is ≃ 1.32 (fit shown by the solid line), less than the analytical one due to finite N and finite time of simulation.

tained by Boettcher and Paczuski (1996) and de Boer et al. (1994). They permit us to make a comparison with the results of Peters and Neelin (2006) for atmospheric convection. Let us fix a real value for the parameter λ. Consider the number n of sites that have the value xi less than λ. Define Pn (t) as the probability that at time t there are n sites that have value xi lower than λ. This probability satisfies the following master equation: N  Mn,m Pm (t) , (4.3) Pn (t + 1) = m=0

where (de Boer et al., 1995)

n−1 N −1   n−1 = 2λ (1 − λ) + 3λ2 − 2λ N −1   n−1 = (1 − λ)2 + −3λ2 + 4λ − 1 N −1 2 n−1 , = (1 − λ) N −1

Mn+1,n = λ2 − λ2 Mn,n Mn−1,n Mn−2,n and

M0,0 = (1 − λ)

2

M1,0 = 2λ (1 − λ) 2

M2,0 = λ ,

(4.4a) (4.4b) (4.4c) (4.4d) (4.5a) (4.5b) (4.5c)

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as derived in Sec. 7. Consider N sites and fix the parameter λ = 1/2. Consider that at time t there are m sites for which xj < λ. The probability that at time t there are m such sites is Pm (t). In terms of a random walk, the start of an avalanche is the start of a walker at x = 0. Later, the walker stops as it reaches again x = 0. This corresponds to the end of an avalanche. One defines P2n ≡ probability that a walk will return to x = 0 at step 2n . To find it, it is necessary to start with a more general quantity: Q2n ≡ probability that a walk started at x = 0 returns to x = 0 irrespective of the fact that it may have returned to x = 0 at several intermediate times. This quantity is known in the theory of random walks (Feller, 1968, his Vol. 1, p. 75): 1 n Q2n = 2n C2n , (4.6) 2 k where Cm are the binomial coefficients; i.e., the number of combinations of m objects in sets of k objects. The sum over all possible situations where the walk ends up at x = 0 at 2, 4, ..., 2n for all n simply gives the probability of closing the path at x = 0 in the simplest way, by any of the  single excursions of lengths 2n, with exactly one return to x = 0, n P2n .  The complementary state has probability 1 − n P2n , meaning that in these evolutions there is no return to x = 0. The way to clean the returns to x = 0 after 2n steps of those paths that do not close to x = 0 is by calculating −1    P2n . P2n 1 − n

n

This should give us all the returns that do not take into account the repeated visits of the final position x = 0. In other words, we have taken the loops out of the total family of return paths. This is actually the sum ∞  Q2n n=1

over all probabilities of returns to x = 0 that do not count the multiple visits to x = 0 as different. This is better expressed as the following relationship between the generating functions of the two sets of probabilities Q2n and P2n : ∞  P2n z 2n ∞  n=1 2n Q2n z = . (4.7) ∞  n=1 1− P2n z 2n n=1

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The left-hand side is explicit: 1+

∞ 

n=1

and leads to:

1−

−1/2  , Q2n z 2n = 1 − z 2

∞ 

n=1

P2n z 2n =

 1 − z2,

(4.8)

from where, using the Stirling’s formula to approximate the factorials:  (2n − 3)!! 2 1 ≈ P2n = (large n) (2n)!! π (2n)3/2 ∼ τ −3/2 .

(4.9)

This is the scaling of the duration of an avalanche, and is not far from the result of Peters et al. (2001) given in Eq. 3.2 for the statistics of atmospheric precipitation. 5

The Gierer–Meinhardt model of clusters and spikes in clusters (slow-activator, fast-inhibitor): Spotty–spiky solutions

There is a well-known difficulty with the analytical description of SOC systems: there is no unique formalism that can be applied to the wide variety of systems that are known to exhibit SOC behaviour. In other words, while intuitively we have the certitude that a system (sand pile, forest fire, river networks, convective events in the atmosphere, etc.) show specific characteristics of SOC, the formalism must each time be reinvented; a universal receipe does not exist. As a statistical physics problem it may be useful to invoke the dynamic renormalization group (DRNG) procedures. In this way we can obtain exponents of the scaling relationships between correlation lengths and the relative distances, and this should confirm that our system belongs to the same universality class, specific for other systems that show SOC behaviour (the percolation class in two dimensions). However, to apply the DRNG procedure one needs a continuous-field model. This is not the case with the Bak–Sneppen (algorithmic) model. Let us now discuss briefly a continuous model that has some similarity to the Bak–Sneppen model. In the realization of the algorithm of Bak– Sneppen SOC for the convection–precipitation case, the function A (the

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activity at the location (x, y)) and the function H (a measure of the chances for instability (potential instability), which is the water-vapour content) are introduced. At any point, the barrier against an update is determined as the inverse of the water-vapour content 1/H. The two functions A and H have a mutual relationship which is similar to the activator and inhibitor in the Gierer–Meinhardt (GM) model (Meinhardt, 1982). The Bak–Sneppen model is discrete and algorithmic, but the Gierer–Meinhardt model is space and time continuous. There are elements that suggest the adoption of the GM model as a low-order, continuous limit of the ensemble of precipitation events. The GM model has spotty–spiky solutions which can be associated with extreme events. The current approach to local convection dynamics has its origin in the mass-flux parameterization, with different developments, as extensively discussed in Vol. 1, Part II. Assuming that there are centres of convection within a grid box, there is a parameter (called the “barrier”) representing the degree of fitness of the local sub-system to the environment. For every point (sub-system), the value of that parameter is a random number. Let us examine the possible similarity by starting from the Bak–Sneppen dynamics. (1) The sub-system with the lowest value of the parameter (barrier) is found. This may be the point with the highest column water vapour (CWV) content, since this means a high chance of instability. A simplified representation is to consider the inverse of the CWV, so that 1 1 ∼ , (5.1) barrier ∼ CWV H which appears in the Gierer–Meinhardt model. The smallest barrier means the highest chance to become unstable and then to be updated. The non-linearity in the GM model has the form: A2 ∼ (barrier) × A2 . (5.2) H (2) The sub-system is mutated into a different state. A new barrier is attributed to this new sub-system. The new value of the barrier is taken at random from the same set (0, 1) with uniform probability. This is equivalent to assuming that some precipitation has been produced and after that the CWV is different in that specific sub-system. (3) Other sub-systems are affected by this mutation, and are updated at the same time. The update is random, with new barriers from the set (0, 1). This may be interpreted as follows: other centres of convection have been influenced by the downdraught from the main sub-system.

Regimes of self-organized criticality in atmospheric convection

459

(4) The update of the connected sub-system is annealed: the sub-systems that are chosen to be updated are chosen anew every time a centre is changed. It can be imagined that other centres of convection are affected by the update of the centre that is active at the current moment. The GM equations are: A2 ∂A = δ 2 ΔA − A + ∂t H ∂H = DΔH − H + A2 , τ ∂t where A, H > 0, with:

(5.3a) (5.3b)

A ≡ concentration of activator at point (x, y, t)

H ≡ concentration of inhibitor at point (x, y, t) .

It is easy to recognize the variables: δ 2 is the diffusion coefficient of the A (∼ activity) field. If a convection event occurs at a site, the local activity field A will diffuse to neighbouring sites, but with small efficiency, so that δ is a small quantity. A small δ means that the activity is localized, a feature of the GM model. D is the diffusion coefficient of the potential for instability H. Since 1/H is the barrier against updates and local evolution, the diffusion of H means that higher H (low barrier) neighbouring sites diffuse their propension to start convection to sites that are safer (they are not close to convection, having high barriers; i.e., small H). The diffusion D from some high H local value acts to fragilize the neighbouring sites that were actually quite reluctant to start convection. D pushes these sites to start their own convection. The factor τ is a scale for the time variable. In general, t/τ ≫ 1, which means that the H (inverse of the barrier height) evolves slowly compared with A. The solutions show the activator function A concentrated in K points in different location of the domain Ω. There is a phenomenon of solution concentration for δ → 0: i.e., the peaks become more and more narrow and at the limit they are the points themselves. Looking for the comparison with the Bak–Sneppen dynamics, we have:  A ∂A = −A 1 − , ∂t H or 

 A t , (5.4) A ∼ exp − 1 − H

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Part V: Theoretical physics perspectives

and so: A ∼ exp [−t] (5.5)   A . Then, the time constant if there is no other term; i.e., no factor 1 − H is

 t A ∼ exp − with ϑ = 1, (5.6) ϑ However, we have:

or

  A A ∼ exp − 1 − t H



1 t > ϑ, where ϑH = A ∼ exp − ϑH 1 − (A/H)

(5.7)

A/H < 1.

(5.8)

so the time of decay of the activator is longer when the nonlinear term is included. Assume for the moment that

If the H function is very large, which means high CWV, then 1−(A/H) → 1 and ϑH ց ϑ, and the decay of the local value of the activator A is again fast. The large amount of vapour induces instability and favours the local reduction of the activator function. Similarly, we can evaluate qualitatively the role of the current value of the activator. If initially there is a high A (but still A/H < 1), then the time of decay ϑH is longer than ϑ. The continuous decay of A makes 1 − (A/H) approach 1 and the decay becomes faster: ϑH ց ϑ. This highlights the non-linear effect of A: high values of the activator are more persistent. The decay of the activator A begins by having a slow rate but the rate accelerates in time. Within the range where A/H < 1,

(5.9)

the roles of large A and H are opposite: large initial A tends to create persistence of A, but large H induces instability and faster decay of A. Now, let us consider the range A/H > 1.

(5.10)

This range can be reached by decreasing H, which means reducing the CWV. Then the activator A increases exponentially. When A grows and enters the regime A/H > 1, the growth is self-accelerating since the coefficient of the exponent grows as A itself. Alternatively, the regime can

Regimes of self-organized criticality in atmospheric convection

461

be obtained for higher A. The same self-acceleration occurs, and leads to high concentrations of the activator A in a few localized regions, where it happened that A > H and growth with self-amplification has occurred. In the other regions, the activator A is smaller than the vapour content H and there is a quiet state, with no dynamics. H may be seen as a passive inhibitor in the sense that its presence means that there are chances for instability but the instability has not yet started. Some numerical examples illustrating these behaviours of the GM model are given in Figs. 27.5–27.7.

Fig. 27.5 Time variation of the two functions A (activity, solid line) and H (column vapour water content, dashed) at a randomly chosen point x0 . Result from the onedimensional Gierer–Meinhardt model. Oscillatory regime.

Assuming that a mapping to atmospheric convection is plausible, the function A was named as the activity at the location (x, y) and the function H as the potential instability. When the chance for instability is larger than the local activity (H > A), the activity has not yet started. The activity A is low. The so-called “inhibitor” H does not suppress the activity but rather it is just a measure indicating that the activity has not yet started. When the potential instability is lower than the activity H < A, this means that the activity has started at that point and the growth of the activity is very efficient. There is a connection between this system and the Bak–Sneppen model. The Bak–Sneppen dynamics raises continuously the level of fitness (the value of the parameter that shows the degree of adaptation to the environment). This means that the points that have been updated are now more

462

Fig. 27.6

Part V: Theoretical physics perspectives

As in Fig. 27.5, but showing a regime of fast compensation and decay.

Fig. 27.7

As in Fig. 27.5, but showing a regime of saturation.

stable than they were immediately after the random initialization. This, however, does not mean stability. It only means that the degree of stability (or fitness) has become rather similar for all of the sub-systems. There are no more large discrepancies that make the update a localized and isolated dynamic. Now a change can affect a large number of sub-systems and creates an avalanche. The system is in a critical state and it has reached that state by its own dynamics (i.e., it is self-organized at criticality). The degree of fitness is the barrier of the sub-system against mutation.

463

Regimes of self-organized criticality in atmospheric convection

It is then the inverse of the CWV: 1 1 = , (5.11) CW V H which is the inverse of the inhibitor. If there is large CWV (H) at a point then this means that there is low fitness, a low barrier against mutation, and the chances of instability are large. However, when H is measured there is no or low activity A at that point. The operation of an update consists of the conversion of some CWV (H) into activity (A). The CWV H decreases and the activity A increases, and the barrier is increased as required by Bak–Sneppen dynamics. The activity A may increase but not sufficiently such that A > H. This results in just another landscape of A (x, y) and H (x, y), but not a spiky solution. After the update, we have a smaller H (less CWV) in the updated locations, which means higher barriers. The activity has also increased to a certain extent at those points. The standard Bak–Sneppen random dynamics has been realized by a random factor of conversion from the CWV H to the activity A. After a sequence of such updates, the activity A will be higher than initially, and with all points being at comparable degrees of activity. At the same time, the CWV has decreased in almost all points (i.e., the barriers have increased for those points). barrier against mutation = fitness =

5.1

Spikes in the solutions of the Gierer–Meinhardt model

The GM system in two dimensions for t > 0 and boundary conditions ∂n A = 0, ∂n H = 0 for x ∈ ∂Ω

(5.12)

exhibits a spiky solution that will tend to ∞ for δ → 0 (Wei and Winter, 1999, 2001, 2002). We have solved these equations numerically and indeed have found that there is concentration of the activity in localized areas where it can have large amplitudes. This should require a more detailed understanding of the role of the diffusion terms. The issue of random neighbours merits a short discussion. Certainly, neither the Bak–Sneppen model nor atmospheric convection are dependent on the geometrical shortest distance. The presumed sites that are at marginal stability can actually be found at distances larger than the strict shortest distance. The connection is defined by mutual influence that consists of changes in the environment and this can be biased by external deterministic factors. The freedom to choose in the set of sites any of the other sites illustrates this influence.

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Part V: Theoretical physics perspectives

The Gierer–Meinhardt model belongs to the wide class of reactiondiffusion systems. It presents interesting similarities with a simplified picture of atmospheric convection in terms of what is related to a statistical manifestation of the self-organization at criticality. The GM model has intrinsic periodic spatial distributions of spikes, as shown by the mathematical analysis of Wei and Winter (2002), which has been reproduced numerically here. It is revealed that the locations of the spikes are governed by an equation whose solution is an elliptic function; i.e., doubly periodic in the plane. Since the GM model has no intrinsic mechanism that would give a unique location to a spike, in contrast to an extreme precipitation event that occurs in some precise site, it can be concluded that more physics must be introduced in the model. Either this model or an improved form should be used as continuum versions of algorithmic SOC models (like Bak–Sneppen), which would allow analytical work in the dynamic renormalization group approach.

6

Conclusions

Atmospheric convection has many manifestations and a unique description of the specific regimes is difficult. However, there are situations where atmospheric convection can be seen as a part of a complex phenomenon with the essential characteristics of self-organized criticality: the slow drive, the fast reaction when there is departure from quasi-equilibrium, the interaction between neighbouring sites, and the formation of large-scale ensembles with a correlated response. The correlations of fluctuations of SOC systems should exhibit universal scaling in space and time and this is indeed found for the SOC regimes of convection. Observational data are compatible with the results of an analytical derivation based on a paradigmatic example of a punctuated equilibrium system, the Bak–Sneppen model. An analytical derivation of the equations for the probabilities of states of the sequential update of the Bak–Sneppen model with K = 2 random neighbours was provided. A possible extension to continuum dynamics, by means of the reactiondiffusion model of Gierer–Meinhardt, was also explored. A numerical investigation confirms the expectation that the system evolves by a continuous mutual control of inhibitor and activator variables. Small-scale parts of the system (which we can see as sub-systems) evolve to marginal stability against fast reaction events and, under fluctuations, produce a correlated

Regimes of self-organized criticality in atmospheric convection

465

large-scale response of the system, similar to an avalanche. The GM system also presents spiky solutions that may be of interest in the investigation of extreme dynamics. In conclusion, SOC appears to be a possible basis for the construction of a coherent perspective on the large-scale organization of convection.

7

7.1

Appendix A: Calculation of the probabilities of the states for Bak–Sneppen dynamics with K = 2 random neighbours Definition and calculation of the transition probabilities

Exact analytic results exist for the time-dependent statistical characteristics of the Bak–Sneppen model. These have been obtained by de Boer et al. (1994, 1995); Paczuski et al. (1996). In this Appendix we start from these original works and provide details of the calculations leading to the system of equations connecting the probabilities of the states before and after update. The case that will be examined has K = 2. This means that besides the site with the smallest value xi , only one other site xl is changed by replacing xl with a new, random value x′l . The site xl is chosen at random. Defining Pn (t) as the probability that at time t there are n sites that have value xi lower than λ, this probability satisfies the master equation: Pn (t + 1) =

N 

Mn,m Pm (t) ,

(A.1)

m=0

where the non-zero matrix elements are given in Eqs. 4.4 and 4.5. At time t there are m sites for which xj < λ and we now apply the algorithmic change at step t. First, identify the site k with the lowest value xk from the set of m sites. This is made with probability 1/m because any of the m sites can be at this state. Next, replace xk by the result x′k of extracting a random number from [0, 1] with uniform probability: site k : xk → x′k

Now let us examine the two possibilities.

A There is a chance that the new x′k value is larger than λ: probability that the site k with minimum xk is updated to a value x′k which is greater than λ = 1 − λ,

(A.2)

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Part V: Theoretical physics perspectives

since the distribution is uniform on [0, 1] and the probability is given by the length of the interval. In this case the number of sites with value x smaller than λ at the next timestep t + 1 is smaller by one unit: m → m − 1. Simultaneously, we have to identify a site l that is in interaction with the site k. This is done under the assumption that this other site is any of the available N − 1 sites remaining, at random, with uniform probability. Therefore, it would be tempting to say that the probability of choosing the site l from the rest of N − 1 sites is 1/ (N − 1). However, we must distinguish between the possibilities that l has its xl smaller or larger than λ. Thus, we divide this step into two branches: A.1 Assume that the secondary, interacting site xl has at t a barrier which is lower than the limit λ: secondary xl < λ.

(A.3)

Then it is one of the m sites that are all characterized at t as having xi < λ. (To this family belongs also the main site, the lowest, xk .) The probability for this is: m−1 (probability to choose l between the m − 1 sites with xi < l) = , N −1 (A.4) since there are m − 1 such sites excluding the site xk . Then we have two possible evolutions: A.1.1 The update of the site l is such that the new value of xl is larger than λ: xl → x′l > λ

(A.5)

with (probability that the secondary site l belongs to the set m − 1 and m−1 × (1 − λ) . (A.6) after update x′l is greater than λ) = N −1

This update of the other (interacting) site reduces the number of sites in the initial set m by one unit. This reduction comes after the first reduction by one unit, made by the the principal site xk , the lowest at t, which we assumed has been updated to x′k > λ.

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Regimes of self-organized criticality in atmospheric convection

We have m sites that are initially under λ and so due to these two updates, the number m of sites having x < λ changes as: xk → x′k : m → m − 1 (the main site escapes to > λ)

xl → x′l : m − 1 → m − 2 (the secondary site escapes to > λ) .

The transition represented by these updates is: Pm (t) → Pm−2 (t + 1) ,

(A.7)

which means a contribution to: Pm−2 (t + 1) ,

(A.8)

and the contribution is: (1 − λ) ×

m−1 m−1 2 × (1 − λ) = (1 − λ) . N −1 N −1

(A.9)

A.1.2 The update of the site l is such that the new value of xl is smaller than λ: xl → x′l < λ

(A.10)

with (probability that the secondary site l belongs to the set m − 1 and m−1 after update x′l is less than λ) = × λ. (A.11) N −1

Due to these two updates:

xk → x′k : m → m − 1 (the main site escapes to > λ)  the secondary site is from the m family xl → x′l : m − 1 → m − 1 . and remains inside this family The transition represented by these updates is: Pm (t) → Pm−1 (t + 1) ,

(A.12)

which means a contribution to: Pm−1 (t + 1) ,

(A.13)

and the contribution is: (1 − λ) ×

m−1 m−1 ×λ= (1 − λ) λ. N −1 N −1

(A.14)

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Part V: Theoretical physics perspectives

A.2 Assume that the secondary, interacting site xl has at t a barrier which is higher than the limit λ: secondary xl > λ.

(A.15)

Then it is one of the N − m sites that are all characterized at t as having xi > λ. The probability for this is: (probability to choose l between the N − m sites with xi > l at t) N −m . = N −1 (A.16) Then we have two possible evolutions: A.2.1 The update of the site l (which belonged to N − m) is such that the new value of xl is larger than λ: xl → x′l > λ

(A.17)

with (probability that the secondary site l belongs to the set N − m and N −m after update x′l is greater than λ) = × (1 − λ) . (A.18) N −1 Due to these two updates: xk → x′k : m → m − 1 (the main site escapes to > λ)  the secondary site does not belong to xl → x′l : m − 1 → m − 1 . m − 1 and it does not enter under λ The transition represented by these updates is: Pm (t) → Pm−1 (t + 1) ,

(A.19)

which means a contribution to: Pm−1 (t + 1) ,

(A.20)

and the contribution is: (1 − λ) ×

N −m N −m × (1 − λ) = (1 − λ)2 . N −1 N −1

(A.21)

A.2.2 The update of the site l (which belonged to N − m) is such that the new value of xl is smaller than λ: xl → x′l < λ

(A.22)

Regimes of self-organized criticality in atmospheric convection

469

with (probability that the secondary site l belongs to the set N − m and N −m × λ. (A.23) after update x′l is less than λ) = N −1 Due to these two updates: xk → x′k : m → m − 1 (the main site escapes to > λ)  the secondary site did not belong to m ′ xl → xl : m − 1 → m − 1 + 1 . family but now comes under λ The transition represented by these updates is: Pm (t) → Pm (t + 1) ,

(A.24)

which means a contribution to: Pm (t + 1) ,

(A.25)

and the contribution is: (1 − λ) ×

N −m N −m ×λ= (1 − λ) λ. N −1 N −1

(A.26)

B There is a chance that the new x′k value is smaller than λ: probability that the site k with minimum xk is updated to a value x′k which is smaller than λ = λ.

(A.27)

In this case, the number of sites with value x smaller than λ at the next timestep t + 1 does not change: m → m.

(A.28)

Simultaneously, we have to identify a site l that is in interaction with the site k. This is done under the assumption that this other site is any of the available N − 1 sites remaining. Therefore, it would be tempting to say that the probability of choosing the site l from the rest of N − 1 sites is 1/ (N − 1). However, we must distinguish between the possibilities that l has its xl smaller or larger than λ. Thus, we divide this step into two branches: B.1 Assume that the secondary, interacting site xl has at t a barrier which is lower than the limit λ: secondary xl < λ.

(A.29)

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Part V: Theoretical physics perspectives

Then it is one of the m sites that are all characterized at t as having xi < λ. The probability for this is: m−1 , (probability to choose l between the m sites with xi < l) = N −1 (A.30) since there are m − 1 such sites excluding the site xk . Then we have two possible evolutions: B.1.1 The update of the site l is such that the new value of xl is larger than λ: xl → x′l > λ

(A.31)

with (probability that the secondary site l belongs to the set m and m−1 × (1 − λ) . (A.32) after update x′l is greater than λ) = N −1 This update of the other (interacting) site reduces the number of sites in the initial set m by one unit. Due to these two updates: xk → x′k : m → m (the main site remains < λ)  the secondary site was in the m family ′ xl → xl : m → m − 1 . but now escapes to > λ

The transition represented by these updates is:

Pm (t) → Pm−1 (t + 1) ,

(A.33)

which means a contribution to: Pm−1 (t + 1) , and the contribution is: m−1 m−1 λ× × (1 − λ) = (1 − λ) λ. N −1 N −1

(A.34)

(A.35)

B.1.2 The update of the site l is such that the new value of xl is smaller than λ: xl → x′l < λ

(A.36)

with (probability that the secondary site l belongs to the set m and m−1 × λ. (A.37) after update x′l is less than λ) = N −1

471

Regimes of self-organized criticality in atmospheric convection

Due to these two updates: xk → x′k : m → m (the main site remains < λ)

xl → x′l : m → m (the secondary site remains < λ) .

The transition represented by these updates is: Pm (t) → Pm (t + 1) ,

(A.38)

which means a contribution to: Pm (t + 1) ,

(A.39)

and the contribution is: λ×

m−1 2 m−1 ×λ= λ . N −1 N −1

(A.40)

B.2 Assume that the secondary, interacting site xl has at t a barrier which is higher than the limit λ: secondary xl > λ.

(A.41)

Then it is one of the N − m sites that are all characterized at t as having xi > λ. The probability for this is: (probability to choose l between the N − m sites with xi > l at t) N −m . = N −1 (A.42) Then we have two possible evolutions: B.2.1 The update of the site l (which belonged to N − m) is such that the new value of xl is larger than λ: xl → x′l > λ

(A.43)

with (probability that the secondary site l belongs to the set N − m and N −m after update x′l is greater than λ) = × (1 − λ) . (A.44) N −1 Due to these two updates: xk → x′k : m → m (the main site remains < λ)  the secondary site does not belong to xl → x′l : m → m . m and it does not enter under λ

472

Part V: Theoretical physics perspectives

The transition represented by these updates is: Pm (t) → Pm (t + 1) ,

(A.45)

Pm (t + 1) ,

(A.46)

which means a contribution to:

and the contribution is: N −m N −m × (1 − λ) = (1 − λ) λ. λ× N −1 N −1

(A.47)

B.2.2 The update of the site l (which belonged to N − m) is such that the new value of xl is smaller than λ: xl → x′l < λ

with

(A.48)

(probability that the secondary site l belongs to the set N − m and N −m after update x′l is less than λ) = × λ. (A.49) N −1 Due to these two updates: xk → x′k : m → m (the main site remains < λ)  the secondary site did not belong to m xl → x′l : m → m + 1 . family but now comes under λ

The transition represented by these updates is:

Pm (t) → Pm+1 (t + 1) ,

(A.50)

Pm+1 (t + 1) ,

(A.51)

which means a contribution to: and the contribution is: λ× 7.2

N −m 2 N −m ×λ= λ . N −1 N −1

(A.52)

Results

Now let us collect the results (i.e., write the expressions for the elements of the transition matrix Mn,m ). These connect the state m at time t (whose probability is Pm (t)) with the state n at time t + 1 (whose probability is Pn (t + 1)). We have seen that the transitions having n as the final state can only originate from states of the small set n, n ± 1, n − 2. Hence, we can replace the generic notation m with the appropriate value from this set. Table 27.1 summarizes the connections that are possible as transitions.

473

Regimes of self-organized criticality in atmospheric convection Table 27.1

Summary of results for the transition matrix elements.

A1. xl < λ A.

x′k

>λ A2. xl > λ B1. xl < λ

B. x′k < λ

7.3

B2. xl > λ

A1.1 A1.2 A2.1 A2.2

x′l > λ x′l < λ x′l > λ x′l < λ

(n − 1) (1 − λ)2 (N − 1)−1 (n − 1)λ (1 − λ) (N − 1)−1 (N − n) (1 − λ)2 (N − 1)−1 (N − n)λ (1 − λ) (N − 1)−1

n−2 n−1 n−1 n

B1.1 B1.2 B2.1 B2.2

x′l > λ x′l < λ x′l > λ x′l < λ

(n − 1)λ (1 − λ) (N − 1)−1 (n − 1)λ2 (N − 1)−1 (N − n)λ (1 − λ) (N − 1)−1 (N − n)λ2 (N − 1)−1

n−1 n n n+1

Calculation of the elements of the transition matrix

Now we use Table 27.1 to obtain the transition probabilities Mn,m , with m from the small set. Given the destination state n, we add the contributions that originate from one or several states of the small set. 7.3.1

The case Mn−2,n

We take the single case of this type: Pn (t) → Pn−2 (t + 1) ,

(A.53)

which is made by A.1.1 with the probability: Mn−2,n = 7.3.2

n−1 2 (1 − λ) . N −1

(A.54)

The case Mn−1,n

The transition is: Pn (t) → Pn−1 (t + 1) , which is obtained from the sum of: A.1.2

→

A.2.1

→

B.1.1

→

n−1 λ (1 − λ) N −1 N −n 2 (1 − λ) N −1 n−1 λ (1 − λ) . N −1

(A.55)

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Part V: Theoretical physics perspectives

Thus: 

N −1−n+1 n−1 2 λ (1 − λ) (1 − λ) + 2 N −1 N −1  n−1  = (1 − λ)2 + − (1 − λ)2 + 2λ (1 − λ) N −1  n−1  2 −1 + 4λ − 3λ2 . = (1 − λ) + N −1

Mn−1,n =

7.3.3

(A.56)

The case Mn,n

The transition is: Pn (t) → Pn (t + 1) ,

(A.57)

which is obtained from the sum of: A.2.2

→

B.1.2

→

B.2.1

→

N −n λ (1 − λ) N −1 n−1 2 λ N −1 N −n (1 − λ) λ. N −1

Thus: Mn,n

7.3.4



N −1−n+1 n−1 2 λ = 2λ (1 − λ) + N −1 N −1 ! 1 = 2λ (1 − λ) + − (n − 1) 2λ (1 − λ) + (n − 1) λ2 N −1  n−1  2 = 2λ (1 − λ) + 3λ − 2λ . (A.58) N −1

The case Mn+1,n

We take the single case of this type: Pn (t) → Pn+1 (t + 1) , which is made by B.2.2 with the probability:  N −1−n+1 Mn+1,n = λ2 N −1 n−1 . = λ2 − λ2 N −1

(A.59)

(A.60)

Regimes of self-organized criticality in atmospheric convection

7.3.5

475

The particular case M0,0

This corresponds to the following situation: at time t there is no site under λ. At time t + 1 the number of sites under λ is not modified, it is zero. This means that the update of xk , the minimum site, takes it from > λ and keeps it somewhere > λ. For the secondary, interacting, site, it was initially > λ and after update it remains > λ. The probability of this combination is the product of two probabilities: • that xk takes after update a value that is greater than λ: 1 − λ,

(A.61)

• and that xl takes after update a value that is greater than λ: 1 − λ.

(A.62)

Their product is the matrix element: 2

M0,0 = (1 − λ) . 7.3.6

(A.63)

The particular case M1,0

This can be obtained in two ways: • the lowest site xk (which, since we start from m = 0, has at time t the value > λ) is updated to the same region, with probability (1 − λ). The secondary site that initially at time t is > λ (since m = 0) after update at time t + 1 takes a value < λ, with probability λ. Then the contribution of this situation to the matrix element M1,0 is: (1)

M1,0 = λ (1 − λ) ;

(A.64)

• the lowest site xk (which, since we start from m = 0, has at time t the value > λ) is updated to the region that is under λ, with probability λ. The secondary site xl that initially at time t is > λ (since m = 0) after update at time t + 1 takes a value in the region > λ, with probability (1 − λ). Then the contribution of this situation to the matrix element M1,0 is: (2)

M1,0 = λ (1 − λ) .

(A.65)

Finally, we get the total matrix element: M1,0 = 2λ (1 − λ) .

(A.66)

476

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Part V: Theoretical physics perspectives

The particular case M2,0

It is easy to see that M2,0 = λ2 . (A.67) This determines completely the set of transition probabilities (matrix elements Mn,m ). 7.4

Equations connecting the probabilities of the states at the update transition

The knowledge of the elements of the transition matrix allows us to write in detail the equations connecting the probabilities: N  Mn,m Pm (t) . (A.68) Pn (t + 1) = m=0

To use the results obtained above, we start with the lowest-n cases P0 (t + 1) = M0,0 P0 (t) + M0,1 P1 (t) + M0,2 P2 (t) (A.69a) P1 (t + 1) = M1,0 P0 (t) + M1,1 P1 (t) + M1,2 P2 (t)

(A.69b)

P2 (t + 1) = M2,0 P0 (t) + M2,1 P1 (t) + M2,2 P2 (t) + M2,3 P3 (t) . (A.69c) For the n = 0 equation: 2 P0 (t + 1) = (1 − λ) P0 (t)

  n−1  2 −1 + 4λ − 3λ2 + (1 − λ) + P1 (t) N −1 n=1

 n−1 2 + (1 − λ) P2 (t) . (A.70) N −1 n=2 In the limit of N → ∞: P0 (t + 1) = (1 − λ)2 [P0 (t) + P1 (t)] . (A.71) For the n = 1 equation P1 (t + 1) = 2λ (1 − λ) P0 (t)

  n−1  2 3λ − 2λ + 2λ (1 − λ) + P1 (t) N −1 n=1 

 n−1  2 P2 (t) + (1 − λ) + −1 + 4λ − 3λ2 N −1 n=2 = 2λ (1 − λ) [P0 (t) + P1 (t)]

  1  + (1 − λ)2 + −1 + 4λ − 3λ2 P2 (t) . N −1

(A.72)

Regimes of self-organized criticality in atmospheric convection

In the limit of N → ∞:

2

P1 (t + 1) = 2λ (1 − λ) [P0 (t) + P1 (t)] + (1 − λ) P2 (t) .

477

(A.73)

For the n = 2 equation: P2 (t + 1) = λ2 P0 (t)

 2 2 n−1 + λ −λ P1 (t) N − 1 n=1

  n−1  2 P2 (t) 3λ − 2λ + 2λ (1 − λ) + N −1 n=2

  n−1  2 −1 + 4λ − 3λ2 + (1 − λ) + P3 (t) N −1 n=3 = λ2 [P0 (t) + P1 (t)] 

 1  2 + 2λ (1 − λ) + 3λ − 2λ P2 (t) N −1

  2  2 + (1 − λ) + −1 + 4λ − 3λ2 P3 (t) . N −1

In the limit of N → ∞:

(A.74)

2

P2 (t + 1) = λ2 [P0 (t) + P1 (t)] + 2λ (1 − λ) P2 (t) + (1 − λ) P3 (t) . (A.75)

Consider now the equation for n ≥ 3. To reach n at t + 1 the following possibilities exist: start from n − 1 : Mn,n−1 start from n : Mn,n

start from n + 1 : Mn,n+1 start from n + 2 : Mn,n+2 . The last of these is obtained from the formula for Mn−2,n in Eq. A.54 and replacing n by n + 2 to obtain: n+1 2 Mn,n+2 = (1 − λ) . (A.76) N −1 The limit N → ∞ makes this transition element vanish. Then: Pn (t + 1) = Mn,n−1 Pn−1 (t) + Mn,n Pn (t) + Mn,n+1 Pn+1 (t) .

(A.77)

The probability of transition Mn,n−1 is calculated from the formula for Mn+1,n in Eq. A.60 with the replacement n → n − 1 so that: Mn,n−1 = λ2 − λ2

n−2 . N −1

(A.78)

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In the limit N → ∞ we obtain:

Mn,n−1 = λ2 .

(A.79)

The probability of transition Mn,n is given in Eq. A.58 and after taking the limit N → ∞ it becomes: Mn,n = 2λ (1 − λ) .

(A.80)

The probability of transition Mn,n+1 is calculated from the formula for Mn−1,n in Eq. A.56 with the replacement n → n + 1 so that:  n  2 (A.81) −1 + 4λ − 3λ2 . Mn,n+1 = (1 − λ) + N −1 In the limit N → ∞ we obtain: 2

Mn,n+1 = (1 − λ) .

(A.82)

Finally, we introduce these results in the expression of Eq. A.77 for Pn (t + 1), which gives: 2

Pn (t + 1) = λ2 Pn−1 (t) + 2λ (1 − λ) Pn (t) + (1 − λ) Pn+1 (t) . 7.5

(A.83)

Avalanches in the Bak–Sneppen model

The avalanches of the Bak–Sneppen model are defined, for example, in Paczuski et al. (1995). de Boer et al. (1995) defined a λ-avalanche as “an evolution taking place between two successive times where the number n of sites lower than λ vanishes.” To make practical this definition, consider that an avalanche has started t temporal steps ago at a time we denote 0. One defines the probability Qn (t) of having n sites with barriers xi < λ, conditioned by the situation that at time t = 0 there was no site less than λ. The probability Qn (t) verifies an equation like that for Pn (t) but with the constraint M0,n → 0,

(A.84)

which means that the transition (matrix element) from the state with n sites < λ to the state with 0 sites less than λ is zero. This condition eliminates the possibility that from the state with a non-zero number of sites less than λ the system cannot evolve to the state with 0 such sites, since if this were possible the avalanche would be terminated. Let us examine under what conditions the avalanche terminates at time t. Consider the probability Qn (t − 1) of having n sites less than λ at

479

Regimes of self-organized criticality in atmospheric convection

time t − 1. The update from t − 1 to t replaces two sites: the lowest xk and the interacting xl , chosen at random. The condition that the avalanche terminates at t is that there is no longer a site lower than λ, so the transition from xk < λ at t − 1 to x′k should move it to higher than λ values. That probability is (1 − λ). Now we recognize that there may be two initial states. 7.5.1

Case A

The initial state at t − 1 consists of only one site less than λ and this inevitably is the lowest, xk . After update it will move to > λ with probability (1 − λ). However, according to the algorithm, we have to update another site: the one which is in interaction with xk . This site xl by assumption is not in the region < λ. Thus, it is in the region > λ but we require it to make a transition in the same region. This is with probability (1 − λ). The probability of transition in this case A is: (1 − λ)

2

(A.85)

from the initial state: Q1 (t − 1) . 7.5.2

(A.86)

Case B

The initial state at t − 1 consists of two sites less than λ, which necessarily are: • the lowest, xk , which after update goes to > λ with probability (1 − λ); • the secondary or interacting site:

– the probability to chose (at random) the appropriate secondary one is uniform over the N − 1 sites which can interact with k; – after update the secondary site will move to > λ, with probability (1 − λ).

It follows that in the case B we have the probability of transition:

from the initial state

1 (1 − λ)2 N −1

(A.87)

Q2 (t − 1) .

(A.88)

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There is no other case. We canot assume the existence of more than two sites < λ since the algorithm requires that two and only two sites be updated. Summing over the two cases: 2

(1 − λ) Q2 (t − 1) . (A.89) N −1 The numerical simulation of this algorithm has led to the result: 1 q (t) ∼ 3/2 , (A.90) t and this is also derived analytically by de Boer et al. (1994, 1995); Paczuski et al. (1996). 2

q (t) = (1 − λ) Q1 (t − 1) +

7.6

The limit of a large number of sites

Here, we re-examine the equations for Pn (t) in the limit N → ∞. 7.6.1

Calculation of the probabilities Pn (t) for N → ∞ and t → ∞. Connection with random walk with reflection

de Boer et al. (1994) calculated the probabilities Pn for the large time limit. The reference model is a random walk with reflection at n = 0. For the convergence of the geometric sum it is assumed that 1 (A.91) λ< . 2 Then the results are: P0 = 1 − 2λ P1 = (1 − 2λ) Pn = (1 + 2λ)

"

1 2

(1 − λ) λ2n−2

(1 − λ)

2n

+1

#

for n ≥ 2.

(A.92a) (A.92b) (A.92c)

de Boer et al. (1994) note that as

1 . (A.93) 2 then all probabilities vanish and this means that n cannot remain finite. The probability that n is finite and not very close to zero is vanishingly small. In physical terms this means that there will be no sites which have the fitness value under λ = 1/2. This is the expression of the fact that the domain of fitness values under λ = 1/2 is now effectively empty. The λ→

Regimes of self-organized criticality in atmospheric convection

481

same final conclusion is reached after calculating the probabilities Pn (t) for t → ∞ at λ > 1/2. The probability that an avalanche starts at 0 and ends at t is q(t) and this is calculated as discussed above by first obtaining the probabilities Qn (t) that, with an avalanche started at time 0, there are at moment t a number of n sites that are still under λ. We find: Q1 (1) = 2λ (1 − λ)

(A.94a)

2

Q2 (1) = λ

(A.94b)

Qn (1) = 0 for n ≥ 3,

(A.94c)

and the result for general t is: Qn (t) =

2n (2t + 1)! t−n+1 λt+n−1 (1 − λ) . (t + n + 1)! (t − n + 1)!

(A.95)

Now it is possible to calculate the probability that an avalanche has duration t. From Eqs. A.89 and A.95 this is: q (t) =

(2t)! t+1 λt−1 (1 − λ) . (t + 1)!t!

(A.96)

The average duration of an avalanche is:

t =

∞ 

tq (t) =

t=1

1 , 1 − 2λ

(A.97)

and we see that it diverges for λ → 1/2. For large t the probability of an avalanche of duration t, q (t) has the asymptotic form: t

q (t) ∼

(1 − λ) [4λ (1 − λ)] 1 √ λ π t3/2

with the limit at λ → 1/2 given by: q (t) ∼

1 t3/2

,

(A.98)

(A.99)

which is taken as the basis for the comparison with the statistics of the observations.

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Chapter 28

Invariant and conservative parameterization schemes

A. Bihlo, E. Dos Santos Cardoso-Bihlo, and R.O. Popovych Editors’ introduction: This chapter constitutes an introduction to the group analysis of differential equations, with a focus on the construction of parameterization schemes. It describes techniques which can be applied in order to ensure that a parameterization respects invariance properties and conservation laws. Such properties and laws that hold for the full system of differential equations under consideration often have a fundamental status as basic requirements for realistic modelling of the system. When a system is partitioned into parameterized and explicitly simulated terms, it is natural to ask whether the modelled system continues to respect the invariance properties and conservation laws of the original. For many physical systems, we can expect that a modelling system that does not respect, say, energy or momentum conservation, is unlikely to perform well. The key point of the chapter is that such matters need not be treated merely as valuable post hoc checks of a given parameterization, but that they can be used in a constructive sense for parameterization design. By insisting that the parameterized system respect certain invariances or conservation laws, it is possible to constrain the parameterization from the outset to take an acceptable form. This approach is potentially extremely powerful, as is demonstrated by some relatively simple but non-trivial examples. For instance, it is illustrated how an eddy-diffusion term can be added as a valid parameterization to some systems and even suggests acceptable dependencies of the eddy diffusion coefficients. Note that the chapter describes the subgrid-scale parameterization problem from the perspective of filtering (cf., Vol. 1, Ch. 3, Sec. 3.3). However, the analysis holds for any method of defining a split of the original system into parameterized and explicitly simulated parts. 483

484

1

Part V: Theoretical physics perspectives

Introduction

The idea of preserving geometric properties of differential equations such as symmetries and conservation laws was only recently introduced in the parameterization problem. Both symmetries and conservation laws play a key role in modern physics and mathematics. It is immediately obvious that if a parameterization scheme violates basic scaling properties (e.g., if the terms in the parameterization do not have the same dimension as the terms that should be closed), it should be regarded as suspicious. However, the violation of symmetries in a closure model can be more subtle and can go unnoticed for a while. For example, it is now established that the Kuo (1974) convection scheme is not compatible with Galilean invariance because it equates a Galilean-invariant quantity (the rain rate) with a noninvariant quantity (the moisture-flux convergence). Here, systematic methods for the construction of parameterization schemes with prescribed symmetry properties are described. That is, the parameterization is constructed in such a manner that selected symmetries from the original system of differential equations will be preserved in the closed model as well. The first principle of symmetry preservation is thus made a constructive requirement for the design of parameterization schemes. The role of conservation laws for constructing parameterization schemes is also discussed. There are several processes in the atmospheric sciences that comply with basic conservation laws, such as energy, momentum, and mass conservation. It is thus sensible to require preservation of these conservation laws also in the case when considering only the resolved part of the flow. That is, if a process is known to be conservative, then also the parameterization for this process should be conservative. It should also be stressed that methods related to the ones to be introduced in this chapter for the construction of geometry-preserving parameterization schemes are already in use in meteorology, namely in the field of geometric numerical integration. This area is devoted to the design of discretization schemes that preserve fundamental properties of the governing equations of hydro-thermodynamics numerically. The properties are, inter alia, mass conservation, energy conservation, axial angular momentum conservation, the absence of spurious Rossby modes, and stability of geostrophic balance (see Staniforth and Thuburn, 2012, for a more complete list along with further explanations). Guaranteeing these properties on the discrete level requires a careful design of the dynamical core of a

Invariant and conservative parameterization schemes

485

numerical model. The methods used to accomplish this goal are referred to as “mimetic discretization” (see Bochev and Hyman, 2006; Thuburn and Cotter, 2012, for a more thorough exposition of this recent field). Drawing a parallel with the field of mimetic discretization, it is fair to say that the methods to be introduced in this chapter could be dubbed “mimetic parameterization”. If used in a numerical model that features a mimetic dynamical core, one could guarantee the preservation of symmetries and conservation laws of the governing equations on the level of the discretized resolved part (using mimetic discretization) as well as on the parameterization of the unresolved part (using mimetic parameterization). This chapter is organized in the following way. In Sec. 2, the necessary background material on symmetries and conservation laws is discussed. This discussion is by no means complete but should help the reader to navigate through the relevant literature. In Secs. 3 and 4, the material presented in Sec. 2 is used to introduce different methods for the construction of invariant and conservative parameterization schemes. Mostly minimal examples are given in these sections that should serve as an illustration of the theory presented, while more realistic examples are discussed in Sec. 5. The chapter concludes with Sec. 6, which contains a short summary and discussion of further problems in the field of geometry-preserving parameterization schemes. Many of the original results reported in this chapter can be found in more detail in the papers Bihlo and Bluman (2013); Bihlo and Popovych (2014); Bihlo et al. (2014); Popovych and Bihlo (2012).

2

Symmetries and conservation laws

In this section, some of the fundamental concepts of symmetries and conservation laws are presented as formulated in the field of group analysis of differential equations. A more thorough presentation of the material covered here can be found in the textbooks by Bluman and Kumei (1989); Bluman et al. (2010); Hydon (2000); Olver (2000); Ovsiannikov (1982).

2.1

Symmetries, invariants, and group classification

In what follows, a system of differential equations is denoted by L : Δl (x, u(n) ) = 0, l = 1, . . . , L, where each Δl is regarded as a function of the independent variables x = (x1 , . . . , xp ), the dependent variables u = (u1 , . . . , uq ), as well as all derivatives of u in respect of x up to or-

486

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der n. For the sake of brevity, all these derivatives (including u itself as the derivative of order zero) are collected in the tuple u(n) . The space of variables z = (x, u) is denoted by M and the extended space of variables z (n) = (x, u(n) ) is denoted by M (n) . Thus, within the local approach, derivatives of u in respect of x up to order n are just assumed as additional dependent variables in the extended space M (n) . Smooth functions defined on domains in M (n) for some n, like Δl , are called “differential functions”. The order of a differential function F is defined to be equal to the highest order of derivatives involved in F . For specific examples, the simpler notation of independent variables as t, x, y, . . . will be used instead of x1 , x2 , x3 , . . . . Example 28.1. In the case of a single dependent variable u of two independent variables t and x (i.e., p = 2 and q = 1), u(2) is the tuple (u; ut , ux ; utt , utx , uxx ), where here and in the following the shorthand notation ut = ∂u/∂t, ux = ∂u/∂x, utt = ∂ 2 u/∂t2 , etc. is used. Definition 28.1. A point symmetry of the system L is a (non-degenerate) point transformation Γ: x˜ = X(x, u), u˜ = U (x, u) that maps the system L to itself. Equivalently, the transformation Γ maps any solution of L to another solution of L. The set of point symmetries of any system of differential equations admits the structure of a group in respect of the composition of transformations. That is, this set contains the composition of its elements, the identity transformation, and the inverse of each element. General point symmetries of systems of differential equations are usually hard to find, as they are determined by solving systems of non-linear partial differential equations, which is often very difficult to do without additional mathematical machinery. This is why most of the literature contents itself by studying so-called continuous (or Lie) symmetries. These are point transformations that are parameterized by one or more continuous parameters and constitute a connected group G. Example 28.2. The (viscous) Burgers equation: ut + uux + uxx = 0

(2.1)

can be regarded as a major simplification of the Navier–Stokes equations, retaining only the non-linear advection and diffusion terms. It admits, inter alia, the continuous symmetry transformation (t, x, u) → (t, x + εt, u + ε),

Invariant and conservative parameterization schemes

487

where ε ∈ R. It is readily checked that this so-called “Galilean transformation” leaves the Burgers equation invariant. However, the transformation (t, x, u) → (t, −x, −u) is also a symmetry of the Burgers equation, but it is not an element of a one-parameter Lie symmetry group of the Burgers equation. This is an example of a “discrete symmetry”. The main advantage of Lie symmetries over other point symmetries is that they can be found algorithmically using infinitesimal techniques, which always boil down to linear problems. For most purposes it suffices to consider the action of the group linearized around the identity element. Moreover, the techniques for finding the infinitesimal action of a group, encoded in the infinitesimal generators of the group transformation, are already implemented in major computer algebra systems such as Mathematica, Maple, or Reduce. What is more, for various important physical systems of differential equations, the Lie symmetries are already computed and can be found in standard handbooks such as Ames et al. (1994); Anderson et al. (1996); Ibragimov et al. (1995). As stated above, solving the determining equations for Lie symmetries of a system of differential equations (either by hand or using a computer algebra system) yields a set of infinitesimal generators, or vector fields, that jointly span the maximal Lie invariance algebra of the system under consideration. Recovering finite group transformations from these vector fields is accomplished by solving a system of first-order ordinary differential equations. Example 28.3. The maximal Lie invariance algebra of the Burgers equation of Eq. 2.1 is spanned by the vector fields: ∂t ,

∂x ,

2

t∂x + ∂u ,

2t∂t + x∂x − u∂u ,

t ∂t + tx∂x + (x − tu)∂u .

(2.2)

More generally, if τ (t, x, u)∂t + ξ(t, x, u)∂x + φ(t, x, u)∂u is a vector field on the space of variables (t, x, u), one can recover the associated oneparameter Lie symmetry by integrating the system of ordinary differential equations: dt˜ = τ (t˜, x ˜, u ˜), dε

d˜ x = ξ(t˜, x ˜, u ˜), dε

d˜ u = φ(t˜, x ˜, u ˜), dε

with the initial conditions t˜|ε=0 = t, x ˜|ε=0 = x, u ˜|ε=0 = u. The extension of this algorithm to the case of several unknown functions of more than two variables is straightforward.

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Part V: Theoretical physics perspectives

Example 28.4. For the Burgers equation of Eq. 2.1, the one-parameter Lie symmetry groups associated with the basis vector fields of Eq. 2.2 consist of the transformations which map (t, x, u) to (t + ε1 , x, u),

(t, x + ε2 , u), (t, x + ε3 t, u + ε3 ),  t x −ε4 ε4 2ε4 , , u(1 − ε5 t) + ε5 x , (e t, e x, e u), 1 − ε5 t 1 − ε5 t

(2.3)

where εi , i = 1, . . . , 5, are arbitrary constants. The physical significance of these symmetries is thus: (i) time translations; (ii) space translations; (iii) Galilean transformations; (iv) scale transformations; and, (v) inversions in time. An important role of symmetries in the study of partial differential equations is that they allow one to find solution ansatzes reducing the number of independent variables in a given system. This is done by computing the invariants of suitable symmetry subgroups of a given invariance group and considering these invariants as the only new variables in the system. Definition 28.2. An invariant of a transformation group G locally acting on M is a function f (z) that satisfies f (g · z) = f (z) for all z ∈ M and all g ∈ G such that the action g · z of g on z is defined. Thus, invariants are functions that do not change their value if their arguments are transformed. Invariants can be found either using infinitesimal techniques or moving frames (Fels and Olver, 1998, 1999). The infinitesimal criterion of invariance of a function f (z) under a group G is: vf (z) = 0 for each infinitesimal generator v of the group G. Example 28.5. The function f (t, x, u) = x−ut is invariant under Galilean transformations (t, x, u) → (t, x + εt, u + ε). The infinitesimal generator of these transformations is v = t∂x + ∂u and thus the function f satisfies the equation vf := tfx + fu = 0. It is meaningful to extend the definition of invariance of a function to functions that also depend on derivatives of the dependent variables. This leads to the definition of differential invariants. Definition 28.3. A differential invariant of a transformation group G acting on M is a differential function f (x, u(n) ) that satisfies f (g (n) · z (n) ) = f (z (n) ) for all z (n) ∈ M (n) and all g ∈ G such that the transformation

Invariant and conservative parameterization schemes

489

of z (n) using the prolongation of g, denoted by g (n) : z˜(n) = g (n) · z (n) , is defined. In practice, the prolongation of a group action to the derivatives of the dependent variables is implemented by repeatedly using the chain rule. Example 28.6. For the Galilean transformation (t˜, x ˜, u˜) = (t, x + εt, u + ε), the partial derivatives in respect of t and x transform as ∂t = t˜t ∂t˜ + ˜x ∂x˜ = ∂x˜ . Therefore, the transformed x˜t ∂x˜ = ∂t˜ + ǫ∂x˜ and ∂x = t˜x ∂t˜ + x partial derivative operators are ∂t˜ = ∂t − ε∂x and ∂x˜ = ∂x . With these derivative operators, it is now possible to determine the action of Galilean transformations on the various derivatives ut , ux , etc. In particular, we have: u˜t˜ = ut − εux ,

u ˜x˜ = ux .

Example 28.7. The function ut +uux is a differential invariant of Galilean ˜x˜ = ut + uux . transformations. Indeed, u ˜t˜ + u˜u Differential invariants can be found using an infinitesimal invariance criterion as well (see, e.g., Ovsiannikov, 1982). It is more convenient though to determine them using the method of equivariant moving frames in the formulation by Fels and Olver (1998). Because this method plays a superior role in the construction of invariant parameterization schemes, it will be introduced shortly here. More in-depth information on moving frames and their applications can be found in the original papers Fels and Olver (1998, 1999); Olver (2003); Olver and Pohjanpelto (2008). For the sake of simplicity, only the case when the group G is finite dimensional is considered here. Definition 28.4. Let G be a finite-dimensional Lie group acting on M . A (right) moving frame ρ is a smooth map ρ : M → G satisfying the equivariance property ρ(g · z) = ρ(z)g −1 for all z ∈ M and g ∈ G. The motivation behind introducing the moving frame ρ is that it allows one to associate to a given function an invariant function. This is accomplished in a process called “invariantization”. Definition 28.5. The invariantization of a function f : M → R using the (right) moving frame ρ is the invariant function ι(f ), which is defined as ι(f )(z) = f (ρ(z) · z).

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It is readily checked that ι(f ) is indeed an invariant function: ι(f )(g · z) = f (ρ(g · z)g · z) = f (ρ(z)g −1 g · z) = f (ρ(z) · z) = ι(f )(z), which is nothing but the definition of an invariant function: the value of ι(f ) is not changed if its argument is transformed. The invariantization of a non-invariant function is the key to one of the methods for the construction of invariant parameterization. One can start with a given parameterization that fails to be invariant and turn it into an invariant scheme by applying the proper moving frame to it. The theorem on moving frames (Fels and Olver, 1998, 1999) guarantees the existence of a moving frame provided that the action of G on M is free and regular. Without going into more details, both requirements are usually satisfied for the groups of interest in physics, although the freeness property often requires the construction of the moving frame on the space M (n) rather than on M . The following is a recipe of how a moving frame can be found through a simple normalization procedure (Cheh et al., 2008). Once again, the construction is entirely algorithmic and to date already implemented in Maple. For the sake of simplicity, it is assumed that G is an r-dimensional Lie group with r < ∞ acting on M although the assumption r < ∞ is not principal. Algorithm 28.1. The construction of a moving frame via normalization. (1) Explicitly write down the transformation formulae for the action of the group G prolonged to M (n) for n sufficiently large: Z n = (X, U (n) ) = g (n) · (x, u(n) ). (2) Choose r normalization constants ci and equate r of the above transformed variables to these constants, i.e.: Zin = ci ,

i = 1, . . . , r.

(3) Solve the arising system of algebraic equations for the group parameters of G in terms of the coordinates z (n) . Example 28.8. We detail the construction of a moving frame for the subgroup G1 of the maximal Lie invariance group G of the Burgers equation of Eq. 2.1 that consists of translations in time and space, Galilean transformations, and scale transformations. Including the inversion symmetry is possible as well but would complicate the resulting computations and formulae without adding substantial information for this introductory example.

Invariant and conservative parameterization schemes

491

If we combine the first four one-parameter symmetry transformations given in Eq. 2.3, we obtain: (2.4) (t˜, x ˜, u ˜) = (e2ε4 (t + ε1 ), eε4 (x + ε2 + ε3 t), e−ε4 (u + ε3 )), where ε1 , ε2 , ε3 , and ε4 are arbitrary constants. Because there are four group parameters but only three variables in the above transformation formula, it is not possible to produce a sufficient number of equations; in other words, the group G1 is not free on the space M = {(t, x, u)}. This is why it is necessary to prolong Eq. 2.4 to the first derivatives of u. Using the chain rule as shown in Example 28.6 for Galilean transformations, for the subgroup G1 we find: u ˜t˜ = e−3ε4 (ut − ε3 ux ),

u ˜x˜ = e−2ε4 ux .

This is step (1) of Algorithm 28.1. The space M (1) is five-dimensional and thus it is possible to single out a hypersurface of dimension 5 − 4 = 1 that allows us to solve for all four group parameters. This hypersurface is defined through the following four equations: t˜ = 0, x˜ = 0, u ˜ = 0, u ˜x˜ = 1, which accomplishes step (2) of Algorithm 28.1. Note that other normalization conditions could be chosen, which would lead to equivalent moving frames. Solving this system of four algebraic equations for the four group parameters accomplishes step (3), and we obtain the moving frame: 1 ε1 = −t, ε2 = −x + ut, ε3 = −u, ε4 = ln ux . 2 This frame can now be used to invariantize any non-invariant function of the variables (x, u(n) ). To realize the invariantization procedure in practice we first transform the function to be invariantized using G and then substitute the computed moving frame for the group parameters that appear. Example 28.9. It is shown here how the moving frame constructed in the previous example can be used to construct an invariant function starting with a non-invariant expression. Consider the function f = uxx . Under the action of the subgroup G1 , this function is transformed to f˜ = e−3ε4 uxx . Thus, f is not invariant under the action of G1 . Let us invariantize f by setting uxx ι(f ) = u˜x˜x˜ |g=ρ(z) =  . u3x

As was shown above, the resulting function ι(f ) now is G1 invariant.

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A moving frame of the group G can also be used to obtain the representation of a G-invariant system of differential equations in terms of differential invariants of G. This is the content of the so-called “replacement theorem” (Cheh et al., 2008). Example 28.10. To obtain the Burgers equation expressed in terms of differential invariants of the subgroup G1 , we invariantize it: ι(ut + uux + uxx ) = ι(ut ) + ι(u)ι(ux ) + ι(uxx ) =

ut + uux + uxx  = 0. u3x

Of course, this expression is equivalent to the original Burgers equation. It is often the case that differential equations contain certain constants or functions that are to be determined externally. The relevance of these constants and functions can be different but they are usually related to the physical properties of the model that is expressed using differential equations. Example 28.11. The incompressible Euler equation in streamfunction form on the β-plane ζt + ψx ζy − ψy ζx + βψx = 0,

ζ = ψxx + ψyy ,

(2.5)

includes the β parameter as constant. Here, the β parameter arises by expanding the Coriolis parameter f as a Taylor expansion to first order as a function of the latitudinal distance: f = f0 + βy. This parameter is externally determined and different choices for β lead to different dynamical properties of solutions for ψ. Henceforth, such constants or functions will be collectively referred to as “arbitrary elements”. Systems of differential equations that include arbitrary elements are called “classes (of systems) of differential equations”. Studying symmetry properties of classes of differential equations is generally more complicated than determining the symmetries of a system of differential equations that does not include arbitrary elements. The reason for this complication is that for different values of the arbitrary elements, the corresponding equations from the class usually admit different symmetry properties. Exhaustively describing the symmetry properties of such classes is the problem of group classification. Example 28.12. For the vorticity equation of Eq. 2.5 there are two essentially different cases, given by β = 0 and β = 0. The former case leads back

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to the f -plane form of the equation. It can be checked by direct computation that the symmetry group for the vorticity equation on the f -plane is wider than that for the β-plane equation (see, e.g., Bihlo and Popovych, 2009). This is understandable as the presence of the β parameter adds an anisotropy to the original f -plane model. There exist different techniques to solve group classification problems and which technique to use largely depends on the class of differential equations under consideration and, in particular, on the form of the arbitrary elements. If the arbitrary elements are constants or functions of a single variable only, then it is often possible to integrate directly the determining equations for Lie symmetries. For more complicated classes of differential equations (as typically arise in the study of invariant parameterization schemes) such a direct integration of the determining equations is generally hopeless. Other techniques that rely on the classification of Lie algebras have to be used. No attempt is made to give an introduction to the various techniques here as this is too large a subject for the scope of this chapter. Rather, the most relevant techniques will be illustrated directly in the course of the invariant parameterization problem in Sec. 3. More background material on group classification can be found in Bihlo et al. (2012); Ibragimov et al. (1991); Nikitin and Popovych (2001); Popovych and Bihlo (2012); Popovych et al. (2010). 2.2

Conservation laws

Let us now turn our attention to conservation laws of systems of partial differential equations. Here, the formal definition of a local conservation law (Bluman et al., 2010; Olver, 2000) is given. Definition 28.6. A local conservation law of the system L is a divergence expression that vanishes on the solutions of the system L (denoted by |L ): Di Φi |L = (D1 Φ1 + · · · + Dp Φp )|L = 0.

(2.6)

The p-tuple of differential functions Φ = (Φi (x, u(m) ), i = 1, . . . , p) with some m ∈ N0 , is called a “conserved vector” of the conservation law. Here, and in the following, the operator Di is the operator of total differentiation in respect of the variable xi , i = 1, . . . , p. It has the coordi|J| α α α u /∂(x1 )j1 · · · ∂(xp )jp , nate expression Di = ∂xi + uα J,i ∂uJ , where uJ = ∂

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α i uα J,i = ∂uJ /∂x , α = 1, . . . , q, J = (j1 , . . . , jp ) is a multi-index, ji ∈ N0 , and |J| = j1 + · · · + jp . The summation convention is used for repeated indices.

Example 28.13. The Burgers equation ut + uux + uxx = 0 can be brought into the form of a conservation law, Eq. 2.6, namely:  1 2 u + ux = 0, Dt (u) + Dx 2 where the operators of total differentiation in respect of t and x are: Dt = ∂t + ut ∂u + utt ∂ut + utx ∂ux + . . . , Dx = ∂x + ux ∂u + utx ∂ut + uxx ∂ux + . . . . It is clear from Definition 28.6 and illustrated in Example 28.13 that the conserved vector associated with a conservation law can depend not only on the independent variables x and the unknown functions u but also on the derivatives of u in respect of x up to any order m. The missing bound on m is the reason why exhaustively describing the space of conservation laws of a system of differential equations is, in general, a complicated problem. In fact, for most equations of hydrodynamics, the conservation laws known are typically of low order. The following two definitions will prove important in Sec. 4 for the construction of conservative parameterization schemes using the direct classification approach. Definition 28.7. A conserved vector Φ is called “trivial” if it is represented ˆ + Φ, ˇ where the components of the p-tuple of differential as the sum Φ = Φ ˆ ˇ is a null divergence, i.e., functions Φ vanish on the solutions of L, and Φ i ˇ = 0 holds identically. Di Φ Definition 28.8. Two conserved vectors Φ and Φ′ are called “equivalent” if their difference Φ − Φ′ is a trivial conserved vector. Trivial conserved vectors satisfy the divergence condition embodied in Definition 28.6 in a trivial manner and thus do not contain any essential physical information. This is why it is important that the computation of conservation laws is carried out by taking into account the possibility of equivalence among conserved vectors. Example 28.14. Consider again the Burgers equation in the form ut + ˆ = (0, ut + uux + uxx)T is trivuux + uxx = 0. The conserved vector Φ ial as it clearly vanishes on solutions of the Burgers equation. Similarly,

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ˇ = (ux uxx , −uxuxt )T is trivial, as Dt (ux uxx ) + the conserved vector Φ Dx (−ux uxt ) = 0 vanishes identically, independent of solutions of the Burgers equation. There are different methods available for finding conservation laws (see for example Wolf, 2002, for an accessible review). Here, the focus is on the multiplier approach to conservation laws (Bluman et al., 2010). This approach uses a reformulation of the definition of a conservation law, Eq. 2.6, in the form: Di Φi (x, u(m) ) ≡ Λl (x, u(s) )Δl (x, u(n) ),

(2.7)

where the tuple of differential functions Λ = (Λl (x, u(s) ), l = 1, . . . , L) with some s ∈ N0 is called the “characteristic” of the conservation law with the conserved vector Φ, and components of Λ are called “conservation law multipliers”. In the case of solutions of the system of differential equations L, the right-hand side of Eq. 2.7 vanishes and thus reduces to the original definition of a conservation law. Example 28.15. We can bring the Burgers equation into the form of Eq. 2.7 by noting that  1 2 u + ux = 1 · (ut + uux + uxx ), Dt (u) + Dx 2 i.e., the multiplier associated with the conservative form of the Burgers equation is Λ = 1. The use of the characteristic form of Eq. 2.7 can aid in the computation of conservation laws. This is done using the Euler operator or variational derivative. Definition 28.9. The Euler operator in respect of the dependent variable uα is the differential operator given by: Euα = ∂uα − Di1 ∂uαi + Di1 Di2 ∂uαi i − . . . = (−D)J ∂uαJ , 1

1 2

(2.8)

where (−D)J = (−D1 )j1 . . . (−Dp )jp . The Euler operators have the property of annihilating any divergence expression Di Φi . That is, applying them to the characteristic form of the conservation law, Eq. 2.7, yields: Euα (Λl Δl ) ≡ 0,

α = 1, . . . , q,

(2.9)

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which leads to a system of determining equations for the conservation law multipliers Λ = (Λl (x, u(s) ), l = 1, . . . , L). It is generally possible to split the system of Eq. 2.9 in respect of derivatives that are not involved in Λ. The result of this splitting is an overdetermined system of linear partial differential equations for Λ. Solving this system leads to the conservation law multipliers for local conservation laws of system L. From these multipliers one can then reconstruct the conserved vectors Φ using either integration by parts or a homotopy formula (Anco and Bluman, 2002a,b; Bluman et al., 2010). Once again, the construction is entirely algorithmic and implemented in various packages for computer algebra systems, such as GeM for Maple (Cheviakov, 2007). Example 28.16. Instead of the running example, the Burgers equation (possessing only one conservation law, which is the equation itself in the conserved form), let us consider the Korteweg–de Vries (KdV) equation ut + uux + uxxx = 0, which models the propagation of waves on shallow-water surfaces. The KdV equation is well known for its solitary wave solution. The aim is to find conservation laws for this equation for multipliers depending only on t, x, and u, i.e., Λ = Λ(t, x, u) (see Bluman et al., 2010, for more details). Thus, Eq. 2.7 reduces to: Dt ρ + Dx Φ = Λ(t, x, u)(ut + uux + uxxx), where ρ is called the “conserved density” and Φ is the “flux” of the conservation law. Applying the Euler operator Eu = ∂u − Dt ∂ut − Dx ∂ux + · · · to this equation leads to: Eu (Λ(ut + uux + uxxx )) = 0. Expanding this equation, we obtain: (Λt + uΛx + Λxxx) + 3Λxxu ux + 3Λxuu u2x + Λuuu u3x + 3Λxu uxx + 3Λuu ux uxx = 0, and as Λ only depends on t, x, and u, we can split this equation in respect of derivatives of u. This leads to the determining equations for conservation law multipliers, which are: Λt + uΛx + Λxxx = 0,

3Λxxu = 0,

Λuuu = 0,

3Λuu = 0.

Λxu = 0,

3Λxuu = 0,

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Solving this linear system, we obtain the parameterized family of multipliers: Λ = c1 + c2 u + c3 (x − ut), where c1 , c2 , and c3 are arbitrary constants. The three independent conservation laws associated with this family are derived by considering the three possibilities (c1 , c2 , c3 ) = (1, 0, 0), (c1 , c2 , c3 ) = (0, 1, 0), and (c1 , c2 , c3 ) = (0, 0, 1). The multiplier form of conservation laws then reads, respectively: Dt ρ1 + Dx Φ1 = 1 · (ut + uux + uxxx ),

Dt ρ2 + Dx Φ2 = u · (ut + uux + uxxx ),

Dt ρ3 + Dx Φ3 = (x − ut) · (ut + uux + uxxx ). From these equations, it is straightforward to recover the conserved densities and fluxes using integration by parts1 , yielding:  1 Dt ρ1 + Dx Φ1 = Dt u + Dx u + uxx , 2   1 1 3 1 u2 + Dx u + uuxx − u2x , Dt ρ2 + Dx Φ2 = Dt 2 3 2   t t x u2 − xu + Dx ux − u2x − (x − ut)uxx − u2 Dt ρ3 + Dx Φ3 = Dt 2 2 2 as three conservation laws of the KdV equation. Using some more elaborate machinery, it can be shown that the KdV equation has infinitely many conservation laws (see, e.g., Olver, 2000). 3

Invariant parameterization schemes

There are different methods for the construction of parameterization schemes with symmetry properties. All of them are based on an intimate relation between the problem of group classification and invariant parameterization. More specifically, Popovych and Bihlo (2012) demonstrated that any problem of finding invariant parameterization schemes is a group classification problem. This statement provides a concise path to the construction of symmetry-preserving closure models. 1 Similarly to this case, the use of the homotopy formula as presented, for example, in Bluman et al. (2010) and Olver (2000) can often be avoided by using integration by parts to construct the conserved vector.

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Let us start with invariant local parameterization schemes. Finding non-local parameterization schemes that preserve symmetry properties of the initial model is a subject that is not well investigated so far. This will be commented on at the end of the chapter. Given a system of differential equations L : Δl (x, u(n) ) = 0, l = 1, . . . , L, let us start with the splitting of the dependent variables u into an averaged and a deviational part u = u ¯ + u′ . The theory that will be outlined below is independent of the form of averaging or filtering method that is used. Introducing this splitting into the system L and averaging the resulting expression for L leads to a system of the form: ˜ l (x, u¯(n) , w) = 0, l = 1, . . . , L, (3.1) Δ ˜ l are smooth functions of their arguments whose explicit form is where Δ determined by the original system of differential equations L and the averaging rule invoked. In the tuple w = (w1 , . . . , wk ), all terms are collected which cannot be related to the resolved grid-scale part in the course of averaging. Thus, the closure problem consists of finding good expressions2 for w in terms of the resolved grid-scale quantities. The system of Eq. 3.1 is closed by establishing a functional relation between the unknown subgrid-scale terms w and the averaged derivatives of u ¯(n) by setting ws = f s (x, u¯(r) ), s = 1, . . . , k, (3.2) 1 k where f = (f , . . . , f ) are the parameterization functions that need to be determined. Introducing this expression in Eq. 3.1 we arrive at: ′ ˜ l (x, u¯(n) , f (x, u Δfl (x, u¯(n ) ) := Δ ¯(r) )) = 0, l = 1, . . . , L, (3.3) which is now a closed system of differential equations Lf that depends on the as yet unspecified form of f . Here, n′ = max{n, r}. In other words, it is a class of differential equation. As stated above, the problem of finding parameterization functions in Eq. 3.2 that lead to a closed system, Eq. 3.3, preserving prescribed symmetry properties is thus solved as a group classification problem. There follows an examination of two principal ways of solving the group classification problem and hence the invariantization problem, which are inverse and direct group classifications. 2 It is assumed here that the local closure is of first order: i.e., that the unknown subgrid-scale quantities w can be determined by u ¯(n) only. More realistically, one might use higher-order parameterizations to close the system of Eq. 3.1. As the description of such higher-order local closure schemes would clutter the presentation they will not be used and only the theory for first-order closures will be outlined. Further comment on invariant higher-order schemes will follow at the end of this chapter; see also Example 28.23.

Invariant and conservative parameterization schemes

3.1

499

Invariant parameterization using inverse group classification

Inverse group classification is done by first fixing a transformation group and then looking for systems of differential equations, each of which admits the selected group as its symmetry group (Ovsiannikov, 1982). This approach to the classification problem is particularly useful as it offers the opportunity of starting with the maximal Lie invariance group of the original, unaveraged system of differential equations and to impose it on the resulting averaged and closed system. In other words, the parameterized system will admit the same symmetries as the original model. This requirement that the closed model should admit the same symmetries as the original, unaveraged model was first advocated in Oberlack (1997) for LES subgrid-scale closure models for the Navier–Stokes equations. There might be physical problems for which it would be overly restrictive to require a closed system of differential equations to admit exactly the same symmetries as the original, unaveraged model. After all, an averaged model only captures the grid-scale part of the solution of the original model, and hence it might be natural that part of the geometry is lost by the averaged model. Mathematically speaking, the associated problem then is to find a parameterization scheme that leads to a closed system of differential equations admitting a subgroup of the maximal Lie invariance group of the original model. To realize invariant parameterization schemes using inverse group classification, it is sufficient to determine the (differential) invariants of the subgroup that the aim is to preserve in the closed model. The replacement theorem discussed in Sec. 2 implies that if we compose the parameterization scheme out of these invariants, it will lead to a system of differential equations with the invariance requested. A natural criterion for selecting which symmetries should be preserved in a subgrid-scale closure model is given by the initial-boundary value problem at hand. When discussing symmetries of differential equations, no relations to the joint consideration with initial-boundary value problems are implied: i.e., we assume the absence of such restraining conditions. Indeed, the maximal Lie invariance group of a system of differential equations is reduced once boundaries are considered (Bluman and Kumei, 1989). This is quite natural as a symmetry transformation then not only has to leave invariant the given system but also these supplementary conditions. On the other hand, in a particular physical model the boundary conditions usually

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constitute an essential part of the problem to be studied. Hence, when constructing a parameterization scheme for such a model, it is natural to at most preserve those symmetries of the system of differential equations that are also compatible with the initial-boundary value problem to be studied. Example 28.17. In order to illustrate the inverse group classification procedure, let us construct invariant parameterization schemes for the Burgers equation. For the sake of simplicity, a Reynolds time filtering operation is invoked to get: 1 (3.4) ¯u¯x + u ¯xx = − (u′ u′ )x = : w u ¯t + u 2 as the corresponding averaged but unclosed model. The momentum flux term on the right-hand side is the subgrid-scale quantity that must be closed in a symmetry-preserving fashion. The one-parameter transformations from the maximal Lie invariance group of the Burgers equation were given in Eq. 2.3. Let us now discuss which of those transformations should be preserved when closing Eq. 3.4. In doing this, it is first necessary to fix the initial-boundary value problem under consideration. Here, periodic boundary conditions in space are assumed, i.e., u(t, L) = u(t, 0) for a channel of length L, and an initial value problem in time u(0, x) = u0 (x). As it stands, it then seems that this initial-boundary value problem is not invariant under the time translations given in Eq. 2.3, because fixing the initial time obviously no longer allows time shifts. On the other hand, from the physical point of view, shifting time just changes the absolute initial time after which one is interested in the evolution of the system. In other words, shifting time simply maps the original initial value problem to another initial value problem of the same kind. That is, time translations do not alter the principal nature of the class of problems under consideration. Time translations therefore act as equivalence transformations in the class of all initial value problems for the Burgers equation. The above discussion is crucial in that it enables the relaxation of the rather rigid condition of point transformations acting as symmetry transformations that leave invariant one fixed problem to equivalence transformations of a class of similar problems. Therefore, as long as a transformation maps a given problem to another problem from a joint class, it should be preserved by the parameterization scheme. It is straightforward to check that the first four transformations from Eq. 2.3 map the class of initial-boundary value problems for the Burgers

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equation with periodic boundary conditions onto the same class. Only the last transformation does not satisfy this requirement, as it reverses the time direction and hence does not preserve the condition t  t0 , and has to be ruled out on physical grounds. Example 28.8 determined the moving frame for the subgroup G1 given by all transformations from the maximal Lie invariance group except for inversions, which were omitted for the above reason. Therefore, the moving frame associated with the subgroup G1 can be used to find the required differential invariants out of which the invariant parameterizations are constructed in order to close Eq. 3.4. So as to determine the maximum order of differential invariants required, it is necessary to select the general form of the parameterization ansatz in Eq. 3.2 first. To keep things simple, the aim is for parameterizations of the form w = f (t, x, u¯, u ¯x , u ¯xx ) subsequently. That is, only the differential invariants of order not higher than two are needed. One can obtain all required differential invariants by invariantizing the arguments of the above function f . That is, by computing ι(t), ι(x), ι(¯ u), ι(¯ ux ), and ι(¯ uxx ). In fact, the required expressions were computed in Examples 28.8 and 28.9. The invariantization of ι(t), ι(x), ι(¯ u), and ι(¯ ux ) just reproduces the normalization conditions, i.e., ι(t) = 0, ι(x) = 0, ι(¯ u) = 0, and ι(¯ ux ) = 1. This is always the case when invariantizing the normalization conditions used to construct a moving frame, which is why these invariants obtain a special name: “phantom invariants”. The only non-phantom invariant is ι(¯ uxx ) and it was computed in Example 28.9. Expressed in terms of the mean variables, it reads: u ¯xx ι(¯ uxx ) =  . u ¯3x Before making use of this differential invariant, it is important to note that the left-hand side of the averaged Burgers equation of Eq. 3.4 is not yet expressed in invariant form. This invariant form is obtained by also invariantizing the left-hand side using the moving frame associated with G1 , which was done in Example 28.10. Thus, an invariant closure model for Eq. 3.4 is given as:   u ¯t + u¯u ¯x + u ¯xx u ¯xx  . =f  u ¯3x u ¯3x

A very simple example for an invariant parameterization is to choose f (z) = kz, leading to ¯x + u ¯xx = k¯ uxx . u¯t + u¯u

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Physically, this boils down to adding a turbulent diffusion term to the viscous diffusion term already present (with viscosity coefficient being equal to −1) and thus a standard down-gradient parameterization for the momentum flux with constant turbulent viscosity. Of course, there is an infinite number of other possible parameterization schemes that are invariant under the subgroup G1 . In the previous example, the typical steps required for the construction of invariant parameterization schemes were discussed. For the sake of convenience, these steps are summarized here again: Algorithm 28.2. Invariant parameterization via inverse classification. (1) Compute the maximal Lie invariance group of the system of differential equations of interest. (2) Choose an averaging rule and average the initial system of differential equations. (3) Define the functional form (Eq. 3.2) of the parameterization scheme to be invoked. (4) Determine which symmetries of the initial model should be inherited by the averaged closed model. (5) Compute the moving frame associated with the symmetry subgroup selected in the previous step. (6) Compute a suitable set of differential invariants using this moving frame and assemble the required parameterization out of these invariants. Concerning Step (1) of the above algorithm, let us recall once again that for most models of physical interest, the computation of Lie symmetries is already accomplished (Ames et al., 1994; Anderson et al., 1996; Ibragimov et al., 1995). More realistic examples will be considered in Sec. 5. There is another way moving frames can be used to construct invariant parameterization schemes. The original idea was presented in Bihlo et al. (2014) and it consists of invariantizing existing parameterization schemes. That is, rather than starting from scratch with the construction of a symmetry-preserving closure model, one takes an existing parameterization that violates certain symmetries and makes it invariant by applying the appropriate moving frame to it. The construction is illustrated here with an example. Example 28.18. Again, consider the famous KdV equation ut + uux + uxxx = 0. The maximal Lie invariance group G of this equation is four

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dimensional and is generated by the one-parameter Lie symmetry transformations that map (t, x, u) to (t + ε1 , x, u), (t, x + ε2 , u), (t, x + ε3 t, u + ε3 ), (3.5) (e3ε4 t, eε4 x, e−2ε4 u), where ε1 , ε2 , ε3 , and ε4 are arbitrary constants. Let us now see what happens when we average the KdV equation employing the Reynolds rule and close the subgrid-scale term u′ u′ with a simple down-gradient ansatz using a constant diffusion parameter κ. This leads to the closed KdV equation: ¯u¯x + u ¯xxx = κ¯ uxx . (3.6) u¯t + u It is not hard to see that this equation is invariant under the first three-point symmetry transformations of the KdV equation listed in Eq. 3.5 but is no longer scale invariant (the term uxx obviously does not scale like uxxx). We can, however, make the closure of Eq. 3.6 invariant under the same symmetry group G (as admitted by the original KdV equation) by constructing a moving frame for the group G and invariantizing Eq. 3.6 subsequently. Following the recipe for the construction of a moving frame, first determine the most general transformation from the maximal Lie invariance group G by composing the one-parameter transformations of Eq. 3.5: (t˜, x ˜, u ˜) = (e3ε4 (t + ε1 ), eε4 (x + ε2 + ε1 ε3 + ε3 t), e−2ε4 (u + ε3 )). (3.7) Next, set up a system of normalization conditions that subsequently allows us to solve for the group parameters. One possibility for the normalization conditions is: t˜ = 0, x˜ = 0, u ˜ = 0, u ˜x˜ = 1, where using the chain rule one can find from Eq. 3.7 that u ˜x˜ = e−3ε4 ux . Solving these normalization conditions for the group parameters ε1 , ε2 , ε3 , and ε4 , we find 1 ε1 = −t, ε2 = −x, ε3 = −u, ε4 = ln ux 3 to be a moving frame. (See, e.g., Dos Santos Cardoso-Bihlo, 2012, for further details.) Now, applying the associated invariantization map (replacing u with u ¯) to model Eq. 3.6 leads to √ ¯x u ¯xx , ¯u¯x + u ¯xxx = κ 3 u u ¯t + u which now again admits the same maximal Lie invariance group G as the original KdV equation. Stated in another way, in order to preserve scale invariance using a down-gradient parameterization, a variable diffusion parameter has to be used. This is quite typical when invariantizing such parameterization schemes (see also Bihlo et al., 2014).

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Invariant parameterization using direct group classification

Direct group classification is performed by starting with a class of differential equations and subsequently aiming to find those specific equations from this class that admit more symmetries than those admitted by all equations from the class (Ovsiannikov, 1982). In order to enable a systematic approach to this comprehensive task, it is important to point out that the classification is only carried out up to point equivalence. This means that if two systems of differential equations can be related to each other by a point transformation, then it is not necessary to include both systems in the final classification list as they necessarily admit similar symmetry groups. A point transformation that maps any system from a given class to another system of the same class is called an “equivalence transformation” of this class. Finding the equivalence transformations of a given class is therefore an important first step in the direct group classification procedure. Example 28.19. Consider generalized Burgers equations of the form (3.8) ut + uux + f (t, x)uxx = 0. This is a class of differential equations with a single arbitrary element, which is a function of both t and x. Meteorologically, the arbitrary element f can be regarded as a variable diffusion parameter in a simple downgradient parameterization of a Reynolds-averaged advection term. This is why it is important to impose the additional constraint f = 0 for the class of Eq. 3.8 as the diffusion cannot vanish for physical reasons. From the mathematical point of view, it is obvious that the inviscid Burgers equation, for which f = 0, is essentially different in structure from the other equations of the form of Eq. 3.8. In particular, this is the only equation of order one among equations of the form of Eq. 3.8. As we consider the usual group classification problem for the class of Eq. 3.8 without an explicit connection to parameterization, we omit the bar over u. Consider the two equations ut + uux + xuxx = 0, ut + uux + (x + c)uxx = 0, which are both elements of the above class. The point transformation (t, x, u) → (t, x + c, u) maps the first equation to the second equation and hence is an example of an equivalence transformation in the above class of generalized Burgers equations. Note that this transformation is no symmetry transformation of either equation as it maps neither of the two equations back to itself.

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Systematically solving the group classification problem for the class of Eq. 3.8 is not a simple task and involves considerably more machinery than was introduced in this chapter. Therefore, only some of the results of the classification of Eq. 3.8 are given, and the specific features of direct group classification when applied to the invariant parameterization problem are indicated. Example 28.20. Table 28.1 contains some equations from the class of Eq. 3.8 that admit particular symmetry properties and have been obtained within the framework of direct group classification. Table 28.1 Some cases of Lie symmetry extensions for the class of Eq. 3.8. Case

Infinitesimal generators

f (t, x)

ω

(i) (ii) (iii) (iv) (v)

No symmetries ∂t ∂x , t∂x + ∂u t∂t + (t + x)∂x + ∂u see Eq. 2.2

f (t, x) h(ω) h(ω) th(ω) 1

ω=x ω=t ω = x/t − ln t

Let us now interpret the results given in Table 28.1 in light of the parameterization problem. In fact, all of the cases listed in this table are representative of typical results that are obtained when using direct group classification to determine invariant parameterization schemes. Case (i) represents the generic form of equations from the class. In the group classification literature, the symmetry group that is admitted by any equation from the class is referred to as the “kernel” of maximal Lie invariance groups. In the present case, if f is completely arbitrary, then Eq. 3.8 admits no continuous symmetry transformation. In a sense, this case represents the conventional approach to the parameterization problem: no particular care is taken of whether or not the resulting parameterization admits symmetries. In other words, possible invariance characteristics of the parameterization scheme are not determined constructively before the fact. Cases (ii) and (iii) represent physical forms of invariant parameterization schemes. In Case (ii), equations from the class of Eq. 3.8 with f = h(x), i.e., ut + uux + h(x)uxx = 0, are invariant under time translations. Likewise, if f = h(t) as in Case (iii), then the equations of the form ut + uux + h(t)uxx = 0 are invariant under spatial translations and

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Galilean transformations, irrespective of the precise form of h. Both parameterizations make sense physically for certain h. In Case (ii) the diffusion parameter is spatially dependent, whereas in Case (iii) it depends on time; for both cases, associated physical conditions could be formulated. The two cases are also typical in that a resulting closure scheme may still constitute a (narrower) class of differential equations. There is still the requirement to determine the form of h precisely. This could be done by incorporating other desirable properties into the parameterization scheme. Indeed, the situation where preserving symmetries in a parameterization only restricts the initial form of the closure scheme (here, the function f (t, x)) rather than giving one particular closure is very typical (see for example Popovych and Bihlo, 2012). Case (iv) unfortunately represents a typical case as well, namely that of an unphysical parameterization. Although the requirement of preserving symmetries in a parameterization scheme is well grounded physically, of course not all combinations of symmetry transformations lead to a physical model. In Case (iv), the resulting parameterization is invariant under a combination of a scaling and a Galilean transformation but the indicated form of the function f in this case does not give a physical ansatz for the diffusion parameter. As direct group classification always produces a list of equations that admit different symmetry properties, there is a high chance that several equations from this list are not physical. This is one of the disadvantages of the direct classification approach to the invariant parameterization problem. It should not be a surprise that if f = 1 (Case (v)), we are led back to the original Burgers equation. Thus, the resulting equation from the class of Eq. 3.8 has the symmetries given in Example 28.3. It is also worth pointing out that the classification results in Table 28.1 are optimal in the sense that there is no point transformation that maps one particular equation to another equation from the table. That is, the classification is carried out up to point equivalence. From the physical point of view, the equations listed in Table 28.1 should therefore not be regarded as single parameterizations but rather as members of inequivalent classes of parameterizations. To give an example, we have already seen in Example 28.19 that the transformation (t, x, u) → (t, x + c, u) maps one equation from the class of Eq. 3.8 to another equation from the same class. One can then use the equivalence transformations from a class to map a given parameterization scheme to a new one. Although this new parameterization scheme will be mathematically equivalent to the original

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one (as it was obtained from applying a point transformation to the initial scheme), it might still be interesting from the physical point of view. For example, shifting x in Case (ii) leads to the equation ut +uux +h(x+c)uxx = 0. Shifting x can be helpful if the model has to be shifted in respect of the origin. Applying an equivalence transformation to a given parameterization scheme can thus be a powerful way of further customizing the closure model to given physical restrictions. The complete classification of this class of generalized Burgers equations is given in Pocheketa and Popovych (2014). Other examples on group classification of various classes of differential equations can be found in Ames et al. (1994); Dos Santos Cardoso-Bihlo et al. (2011); Ibragimov et al. (1991); Lahno et al. (2006); Meleshko (1994); Popovych and Ivanova (2004); Vaneeva et al. (2012). First examples of the use of group classification in the study of physical parameterization schemes are given in Bihlo et al. (2014); Popovych and Bihlo (2012).

4

Conservative parameterization schemes

In this section, a few methods for the construction of parameterization schemes are introduced that lead to closed equations possessing non-trivial conservation laws. As stated above, conservation laws are important features of physical models and they play a distinctive role in hydrodynamics and geophysical fluid dynamics. However, care must be taken when conservation laws should be preserved in a subgrid-scale model. As with the problem of invariant parameterization, conservative parameterization schemes can be found either using inverse or direct classification techniques. That is, comparable to the group classification problem for classes of differential equations, a classification problem for conservation laws should be solved. Both approaches are summarized below. (For further details, see Bihlo and Bluman, 2013; Bihlo and Popovych, 2014.) Let us begin with a description of what the inverse and the direct approach have in common. In both approaches, it is necessary to fix in the beginning the general functional form for the parameterization of the subgrid-scale terms. This is done in a local fashion, meaning that the unresolved terms at a point are represented by a function of the independent variables, the resolved unknown functions u ¯, and the derivatives of u ¯ up to a certain fixed order r at the same point only. In fact, the procedure is

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the same as outlined above for invariant parameterization schemes, which is repeated here for the sake of convenience. Starting with the averaged unclosed system: ˜ l (x, u¯(n) , w) = 0, Δ

l = 1, . . . , L,

where w as before denotes the k-tuple of unresolved terms, and fixing the parameterization ansatz: ws = f s (x, u¯(r) ),

s = 1, . . . , k,

we arrive at the averaged and closed system: ˜ l (x, u¯(n) , f (x, u Δfl (x, u¯(n ) ) := Δ ¯(r) )) = 0, ′

l = 1, . . . , L,

(4.1)

which is, of course, the same as the system in Eq. 3.3. Here, n′ = max{n, r}. The task is now to specify f in such a manner that the system of Eq. 4.1 admits different non-trivial conservation laws. Both the inverse and direct classification approaches to conservative parameterization schemes can be realized using the characteristic form of conservation laws. That is, if the system of Eq. 4.1 is to possess certain non-trivial conservation laws, then there must exist characteristics Λ and conserved vectors Φ, such that Λl (x, u¯(s) )Δfl (x, u¯(n ) ) = Di Φi (x, u¯(m) ). ′

(4.2)

Applying the Euler operators Euα to this equation leads to the system: Euα (Λl Δfl ) = 0,

i = 1, . . . , q,

(4.3)

which is the starting point for both the inverse and the direct approach. The main difference in the two methods is whether one specifies the multipliers Λ initially (the inverse approach) or not (the direct approach). 4.1

Conservative parameterization via inverse classification of conservation laws

Similarly to solving the invariant parameterization problem using inverse group classification, in this approach one specifies the conservation laws from the initial model that the closed model should admit and constructs the closure scheme accordingly. Specifically, one first determines the conservation laws that are admitted by the original unaveraged system of governing equations for which a parameterization has to be constructed. This is conveniently done using the multiplier approach and boils down to solving a linear system of partial

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differential equations. After this, physical reasoning is used to determine which of the conservation laws of the original model should also be preserved in the parameterized model. The system of Eq. 4.3 is then solved by treating Λ as a tuple of known functions. Indeed, while computation of characteristics for the original, unaveraged system L yielded Λ = Λ(x, u(s) ), replacing u(s) with the mean part ¯(s) ), which may correspond to conservation u¯(s) , we obtain the tuple Λ(x, u laws having the same physical interpretation and which is now expressed in terms of the mean part of u only. As Λ(x, u¯(s) ) is then known in the system of Eq. 4.3, solving this system allows one to find the associated forms of the parameterization functions f . Alternatively, the parameterization functions f could be determined directly from Eq. 4.2 using integration by parts. This is often a convenient alternative if the characteristics of conservation laws include arbitrary functions. An example for this method will be presented in Sec. 5. 4.2

Conservative parameterization via direct classification of conservation laws

A second possibility for constructing conservative parameterization schemes is by treating the system of Eq. 4.3 as a system for both Λ and f . In order to do this efficiently, a classification problem for conservation laws has to be solved. Recall that point transformations mapping one equation from a class of differential equations to another equation from the same class are called “equivalence transformations”. The group formed by these equivalence transformations is denoted with G∼ . The direct classification approach to conservative parameterization essentially uses the following definition: Definition 28.10. Let L|S denote a class of differential equations and Lθ and Lθ′ be two elements (i.e., systems) of this class. Let Lθ and Lθ′ admit conservation laws with conserved vectors Φ and Φ′ , respectively. The pairs (Lθ , Φ) and (Lθ′ , Φ′ ) are called “G∼ -equivalent” if there exists a point transformation Γ ∈ G∼ which transforms the system Lθ to the system Lθ′ ˜ such that and transforms the conserved vector Φ to the conserved vector Φ, ′ ˜ Φ and Φ are equivalent as specified in Definition 28.8. The direct classification approach to conservative parameterization schemes proceeds by first determining those conservation laws that are admitted by all systems from the class of Eq. 4.1. Then, those particular systems

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from the class (corresponding to particular forms of the parameterization functions f ) are found for which more conservation laws are admitted than in the case of general f . In order to make this approach computationally feasible, the classification is carried out only up to G∼ -equivalence. If a point transformation maps a closed system Lf from the class of Eq. 4.1 to another closed system Lf ′ from the same class and if the associated transformed conserved vectors of Lf and the conserved vectors of Lf ′ are equivalent, then in the framework of conservative parameterization Lf can be assumed as essentially the same closed model as Lf ′ . In other words, Lf and Lf ′ represent two different forms of a closed model admitting the same physical conservation laws rather than two different models. Taking into account G∼ -equivalence is therefore a crucial ingredient in optimizing the computations of conservative parameterization schemes. In practice, the conservative parameterization problem in the framework of the direct approach is pursued by solving the system of Eq. 4.3 for both Λ and f upon splitting into various subcases corresponding to different (inequivalent) forms of f leading to systems from the class of Eq. 4.1 that admit non-trivial characteristics of conservation laws Λ. (See Bihlo and Bluman, 2013, for an example of the direct classification procedure.) It is important to stress that it is often the case that the classification problem for Eq. 4.3 cannot be solved completely. The situation is again comparable to the usual group classification problem as arising in invariant parameterization. If the class is chosen to be very wide (i.e., the parameterization functions depend on several arguments), solving the system of Eq. 4.3 exhaustively in order to find all inequivalent, conservatively parameterized models can be computationally impossible. Rather than attempting to find all inequivalent models it is then advisable to concentrate on finding those that appear physically relevant. The result of the direct classification approach to conservative parameterization schemes is then a list of inequivalent, closed models that possess different conservation laws. The constructed conservative closed models can then be tested numerically to assess which of them describes an unresolved process in the most optimal way. 4.3

Conservative and invariant parameterization schemes

We have seen in Sec. 3 that the construction of invariant parameterization schemes leads in general not to a single parameterization but to a class of

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closure models that has to be narrowed down further (see also the examples below in Sec. 5 and the discussions in Sec. 6). Similarly, the closed systems of differential equations found using the methods of conservative parameterization are generally also classes of systems rather than single systems. It is therefore possible to combine the methods for finding invariant parameterization schemes with the techniques for constructing conservative parameterization models. The resulting closed differential equations then admit pre-defined symmetries and conservation laws, which are generally inherited from the original system of governing equations. The construction of such conservative invariant parameterization schemes is desirable for several reasons. First of all, as stated above, it restricts the freedom which is generally typical for both invariant and conservative parameterizations. While it is possible to narrow down a class of either invariant or conservative parameterization schemes using physical reasoning, it is helpful to have this initial class as specific as possible before constructing a particular parameterization scheme to be used operationally. On the other hand, as has been advocated throughout this chapter, both symmetries and conservation laws are linked to the physics of a process that is described using differential equations and hence should be preserved even if it is not possible to explicitly resolve that process. It is therefore quite natural to construct parameterization schemes that share both some symmetries and some conservation laws of the original system of governing equations. A powerful technique for constructing invariant and conservative parameterization schemes rests on the famous Noether theorem. Noether’s theorem states that to each symmetry of a Lagrangian there corresponds a conservation law of the associated Euler–Lagrange equations (see e.g., Bluman et al., 2010; Olver, 2000). That is, if one preserves the Lagrangian structure in a parameterized model and there are symmetries associated with this Lagrangian then the parameterized model will automatically be conservative as well. (See again Bihlo and Bluman, 2013, for a simple example.) The problem with this approach is that most models of fluid mechanics expressed in Eulerian variables are not Lagrangian and hence Noether’s theorem is not applicable. For problems that are not Lagrangian, it is usually best to combine directly the methods for conservative and invariant parameterization (either using direct or inverse classification) in order to obtain invariant conservative schemes.

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Suppose that the system of Eq. 4.1 has been parameterized in an invariant way by expressing the parameterization functions f using differential invariants of the symmetry group G associated with the original system Δl (x, u(n) ) = 0, l = 1, . . . , L. Let these differential invariants be denoted by I 1 , . . . , I N . Thus, f = f (I 1 , . . . , I N ); see Example 28.17. We then require Λ, an L-tuple of differential functions of u¯, to be the characteristic of a local conservation law of the system of Eq. 4.1 for certain values of f . That is, Λl Δfl is a total divergence for appropriately chosen f . Again, using the property of the Euler operators to annihilate any total divergence, we have that: Eua (Λl Δfl ) = 0. Splitting this system in respect of derivatives of u whenever possible, we obtain the determining equations for the parameterization functions f , which should be solved so as to obtain those specific forms for f (as functions of the differential invariants) that admit Λ as a conservation law multiplier. The resulting parameterization scheme is then both invariant and conservative. Example 28.21. In Bihlo et al. (2014), an example was given for an invariant and conservative parameterization scheme for the barotropic vorticity equation on the β-plane. More precisely, a closure for the divergence of the eddy vorticity flux of the form: ∇2 ζ¯7 ¯ 2) ∇ · (v′ ζ ′ ) = ν∇2 ¯ = 7ν∇2 (ζ¯5 ∇2 ζ¯ + 6ζ¯4 (∇ζ) ζ is invariant under the entire maximal Lie invariance group of the vorticity equation on the β-plane, and additionally conserves generalized circulation, momentum in the x-direction, and energy (see Example 28.24 for the mathematical expression of these conservation laws). This example also demonstrates that the requirement of preserving both symmetries and conservation laws can lead to quite specific closure models. If a conservative process is known to be invariant under a specific transformation group then the introduced methods of invariant and conservative parameterization can be an efficient way of constructing a consistent closure for this process.

5

Examples

Three examples follow for the use of the methods introduced above in the study of physical parameterization schemes. The first example is devoted

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513

to the study of a simple boundary-layer parameterization as presented in Stull (1988), for which Lie symmetries are computed. The second example is a higher-order parameterization for geostrophic eddies in the ocean. The third example is the barotropic vorticity equation for which conservative parameterizations are constructed. Example 28.22. In the classical textbook by Stull (1988), simple firstorder closure schemes for the Reynolds-averaged governing equations of a horizontally homogeneous, dry boundary layer with no subsidence were considered. Specifically, the unclosed model reads: v − vg ) − (u′ w′ )z , u ¯t = f (¯

u − ug ) − (v ′ w′ )z , v¯t = −f (¯ θ¯t = −(w′ θ′ )z ,

(5.1a)

where v = (u, v, w) is the wind vector, which is split as v = v ¯ + v′ with ′ w ¯ = 0, θ is the potential temperature split as θ = θ¯ + θ , and f is the Coriolis parameter. The geostrophic wind vector vg = (ug , vg , 0) will be neglected since its components ug and vg can be set to zero by the obvious shift of the averaged horizontal wind components u ¯ − ug → u¯, v¯ − vg → v¯. The dependent variables u¯, v¯, and θ¯ are functions of t and z only. The closure model proposed is a simple down-gradient ansatz of the form: ∂¯ γ , (5.1b) ∂z with Kγ being the respective eddy viscosity parameters. It was argued that in neutrally stable conditions, the various parameters Kγ are proportional, which is the case considered here, i.e., K := Ku = Kv = cKθ , for c = const. As a result, setting ug = vg = 0, the general form of the closure model is: w′ γ ′ = −Kγ

u ¯t = f v¯ − K u ¯zz ,

v¯t = −f u ¯ − K v¯zz ,

θ¯t = −cK θ¯zz ,

(5.2)

¯u where the coefficient K = K(z, θ, ¯z , v¯z , θ¯z ) is still an arbitrary function of its arguments that should be specified in order to complete the parameterization procedure. In Stull (1988, p. 209, Table 6-4), examples for parameterizations of the eddy viscosity parameter were proposed. We now investigate the symmetry properties of the resulting closed models that were derived from the model in Eq. 5.1 upon using different choices for K reported in Stull’s (1988) Table 6-4. K = const. Before computing Lie symmetries of the system of Eq. 5.2 in the case of a constant eddy diffusivity, we can set f = 0 by the use of the

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point transformation3 : u ¯ cos(f t) − v¯ sin(f t) → u¯,

u ¯ sin(f t) + v¯ cos(f t) → v¯,

(5.3)

which transforms the above system to the system of three decoupled linear heat equations: ¯zz , u¯t = −K u

v¯t = −K v¯zz ,

θ¯t = −cK θ¯zz .

(5.4)

The linear heat equation is one of the most studied examples in the group analysis of differential equations. The symmetries of the system of Eq. 5.4 are thus readily inferred. They are generated by the vector fields: ∂t ,

∂z ,

2t∂t + z∂z , u ¯∂u¯ , v¯∂v¯ ,   ¯ ¯ , 2Kt∂z + z u ¯∂u¯ + v¯∂v¯ + c−1 θ∂ θ

u ¯∂v¯ ,

v¯∂u¯ ,

¯ ¯, θ∂ θ

¯ ¯, 4Kt2 ∂t + 4Ktz∂z + (z 2 − 2Kt)(¯ u∂u¯ + v¯∂v¯ ) + (c−1 z 2 − 2Kt)θ∂ θ U (t, z)∂u¯ ,

V (t, z)∂v¯ ,

Θ(t, z)∂θ¯,

where U , V , and Θ run through the solution sets of the first, second, and third equation in the system of Eq. 5.4, respectively. Physically, these vector fields generate the one-parameter transformations of: (i)–(ii) time and space translations; (iii) scalings of the independent variables; (iv)–(vii) general linear transformations in the space of u ¯ and v¯; (vii) scalings in ¯ (ix) Galilean boosts; (x) inversions in time; and, (xi)–(xiii) the linear θ; superposition principle. These computations show that the simple down-gradient ansatz with constant diffusion parameter admits a wide Lie invariance algebra. This is not surprising, as setting K to a constant leads to a linear system of differential equations, which always admits an infinite number of symmetries.  ¯2z + v ¯z2 . Here k is the von K´ arm´an constant. Using the point K = k2 z 2 u transformation of Eq. 5.3, we can again set f = 0. The symmetries admitted by the closed model are then generated by the vector fields: ∂t ,

∂u¯ ,

∂v¯ ,

∂θ¯,

t∂t + z∂z ,

z∂z + u ¯∂u¯ + v¯∂v¯ ,

¯ ¯, θ∂ θ

u ¯∂v¯ − v¯∂u¯ .

The associated one-parameter Lie symmetry transformations are: (i)–(iv) shifts; (v)–(vii) scalings; and, (viii) rotations. 3 A similar point transformation was applied in Chesnokov (2009) to set f = 0 in the shallow-water equations on the f -plane. More generally, such point transformations can also be found using symmetries, which again indicates the important role played by symmetries in the study of differential equations and their applications. See, for example, Bluman et al. (2010) for further details on how to apply symmetries to construct mappings that relate differential equations.

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K = l2 u ¯2z , l = k(z + z0 )/(1 + k(z + z0 )/Λ). In this parameterization, k again denotes the von K´ arm´an constant and Λ is a length scale. The infinitesimal generators of one-parameter Lie symmetry transformations for the closed model employing this parameterization are: ∂t ,

∂u¯ ,

∂v¯ ,

∂θ¯,

2t∂t − u ¯∂u¯ ,

v¯∂v¯ ,

¯ ¯, θ∂ θ

u ¯∂v¯ .

The corresponding finite symmetry transformations are: (i)–(iv) shifts; (v)– (vii) scalings; and, (viii) the modification of v by adding a summand proportional to u. In summing up, it should be stressed that in order to bring this problem into the proper form of a direct group classification problem, it would first be necessary to define a class of differential equations with the arbitrary element being K regarded as a function of a suitable subset of the independent variables, the dependent variables as well as their derivatives. More precisely, to account for all possible forms given in Table 6-4 of Stull (1988), we would have to consider the class of equations with the arbitrary element ¯u K = K(z, θ, ¯z , v¯z , θ¯z ). This would result in a very general class, the complete group classification of which is too cumbersome. There is no attempt to give a partial classification here, as the model is too idealized to be of practical use in the era of supercomputers. Still, this example should serve as an illustration of how the choice of a parameterization scheme critically influences the symmetries admitted by the closed model. Example 28.23. All invariant local parameterization schemes constructed in the literature so far were of first order. That is, for the parameterization of the unclosed terms only the resolved variables and their derivatives have been used. However, the construction of invariant parameterization schemes is not restricted to first-order closure schemes as will be demonstrated in this example. More specifically, we are interested in finding invariant parameterization schemes for geostrophic eddies in the ocean. The initial model consists of the incompressible Euler equations on the β-plane (written in streamfunction form) and the energy equation, i.e.: ηt + ψx ηy − ψy ηx = k · (∇ × F), Et + ∇ · (Bv) = v · F,

(5.5)

where η = ζ + f0 + βy = ψxx + ψyy + f0 + βy is the absolute vorticity, given as the sum of the relative vorticity ζ and the Coriolis parameter f ≈ f0 + βy using the β-plane approximation, k = (0, 0, 1)T is the vertical unit vector,

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F denotes the forcing terms, and B = φ + E is the Bernoulli function, given as the sum of the mass-specific potential energy and the mass-specific kinetic energy E = v2 /2 = (∇ψ)2 /2. This model was recently considered in Marshall and Adcroft (2010). Note that the second equation of the system of Eq. 5.5 is not independent of the first equation as it is a conservation law of this equation. This observation is important as in constructing invariant parameterization schemes for the averaged (time-filtered) system associated with the model of Eq. 5.5 it is enough to consider the symmetries of the first equation in Eq. 5.5. This averaged model is given by: η¯t + ψ¯x η¯y − ψ¯y η¯x = ∇ · (v′ η ′ ), (5.6) ¯ · k × η ′ v′ + ∇ · B ′ v′ = v′ · F′ , kt − v

where k = v′2 /2 = (∇ψ ′ )2 /2 is the turbulent kinetic energy and v′ · F′ is the transient forcing term. Note that the second equation is derived from the transient momentum equation vt′ + k × (ηv′ ) + k × (η ′ v) + k × (η ′ v′ ) + ∇B ′ = F′ ,

upon forming the scalar product with v′ and averaging the resulting equation (see Marshall and Adcroft, 2010, for further details). For the sake of simplicity, and following Marshall and Adcroft (2010), only the case without forcing is considered (i.e., F = F′ = 0). The system of Eq. 5.6 includes three unknown terms that have to be ¯ · k × η ′ v′ , and ∇ · B ′ v′ . Here, paramparameterized, namely ∇ · (v′ η ′ ), v eterization schemes of order one-and-a-half are discussed; that is, possible ¯ η¯, k, and expressions for the unclosed terms will be found as functions of ψ, their derivatives. Note that there is a fundamental difference between the ¯ and k. Whereas ψ¯ has an unaveraged dependent variable ψ¯ (and hence ζ) counterpart ψ, there is no such counterpart for k. It is therefore necessary to find the prolongation of the symmetries of the original vorticity equation on the space spanned by (t, x, y, ψ) to the relevant space for the closed form ¯ k). of the system of Eq. 5.6, which is spanned by (t, x, y, ψ, The symmetries of the barotropic vorticity equation were first computed in Katkov (1965) (see also Bihlo and Popovych, 2009, for a recent discussion). The most general transformation from the maximal Lie invariance group of the vorticity equation on the β-plane is: t˜ = eε3 (t + ε1 ), x˜ = e−ε3 (x + f (t)), y˜ = e−ε3 (y + ε2 ), (5.7) ψ˜ = e−3ε3 (ψ − ft (t)y + g(t)),

Invariant and conservative parameterization schemes

517

where f and g are arbitrary smooth real-valued functions depending on t, while ε1 , ε2 , and ε3 are arbitrary constants. So as to extend this transformation to the turbulent kinetic energy, it is necessary to investigate the transformation properties of ∇ψ ′ . This is readily done by considering the splitting ψx = ψ¯x + ψx′ and by determining the transformation behaviour of the right-hand-side expression. Note that ψx transforms as ψ˜x˜ = e−2ε3 ψx , which is a mere consequence of Eq. 5.7 and the use of the chain rule. Thus, we have: , ,′ ¯ +ψ ψ˜ = e−2ε3 ψ = e−2ε3 (ψ¯ + ψ ′ ) = ψ x ˜

x

x

x

x ˜

x ˜

, ,′ = e−2ε3 ψ ′ . In a similar ¯x˜ = e−2ε3 ψ¯x and ψ from which we find that ψ x x ˜ fashion, we note that: , ,′ , ¯y˜ + ψ ψ˜y˜ = e−2ε3 (ψy − ft ) = e−2ε3 (ψ¯y + ψy′ − ft ) = ψ y˜

, ,′ = e−2ε3 ψ ′ hold. From the ¯y˜ = e−2ε3 (ψ¯y − ft ) and ψ and therefore ψ y y˜ ′ ′ transformation results for ψx and ψy it follows that the turbulent kinetic energy k transforms as: k˜ = e−4ε3 k. We now aim to construct invariant parameterization schemes for the system of Eq. 5.6 using the method of invariantization. For this, we need the moving frame that is associated to the maximal Lie invariance group of the vorticity equation on the β-plane and that is extended to k. Without giving the details of this computation, it should be stressed that due to the presence of the arbitrary functions f and g in the transformation of Eq. 5.7, the symmetry group of the vorticity equation is infinite dimensional. Moving frames can be computed for infinite-dimensional Lie groups as well (see e.g., Olver and Pohjanpelto, 2008). This is done by specifying not only the group parameters εi and the arbitrary functions, but also the derivatives of these arbitrary functions up to any order. For the vorticity equation on the β-plane, the moving frame was constructed in Bihlo et al. (2014). It reads:  ε1 = ln |ψx |, ε2 = −t, ε3 = −y, f = −x, (5.8) dk+1 f dk g = (Dt − ψy Dx )k ψy , = −(Dt − ψy Dx )k ψ, k+1 k dt dt where k = 0, 1, . . . , and Dt , Dx , and Dy denote the total derivative operators in respect of t, x and y. This moving frame can now be used to invariantize any existing parameterization scheme for the averaged unclosed

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model of Eq. 5.6. For this invariantization, the dependent variables as well as their derivatives in Eq. 5.8 should be regarded as mean quantities. As an example, consider the model proposed in Marshall and Adcroft (2010), which is: η ) − A∇4 η¯, η¯t + ψ¯x η¯y − ψ¯y η¯x = ∇ · (κ∇¯ kt + ψ¯x ky − ψ¯y kx = −κ∇ψ¯ · ∇¯ η + ∇ · (λ∇k) − rk,

(5.9)

where A is a constant biharmonic diffusion coefficient and κ, λ, and r are the parameters of the closure model. For κ, the expression κ = 2γTeddy k

(5.10)

was proposed, in which γ denotes a dimensionless constant and Teddy is the eddy turnover timescale. The constant λ is the eddy energy diffusivity and r is an inverse timescale for the eddy energy decay. To simplify this system, we set r = 0, which is relevant for the case of freely decaying turbulence in the ocean. The following consideration could of course be adapted for the case r = 0. It is straightforward to check that, as it stands, the system of Eq. 5.9 preserves all Lie symmetries of the barotropic vorticity equation except for the scale invariance associated with the group parameter ε3 . Specifically, while the terms on the right-hand side scale as e−2ε3 , the term A∇4 η¯ scales as e3ε3 . That is, the constant A cannot be dimensionless.4 This problem was extensively analysed in Bihlo et al. (2014) where it was shown that linear hyperdiffusion cannot preserve the scale invariance of the original vorticity equation. In order to recover this invariance, we can use the invariantization map of Eq. 5.8 and apply it to the first equation in the closed system of Eq. 5.9. This leads to: (5.11) η ) − A˜ |ψ¯x5 |∇4 η¯, η¯t + ψ¯x η¯y − ψ¯y η¯x = ∇ · (κ∇¯ where A˜ is now truly dimensionless. For the case κ = 0, this model was successfully used in Bihlo et al. (2014) to carry out freely decaying turbulence tests that yielded energy and enstrophy spectra in close accordance with the Batchelor–Kraichnan theory of two-dimensional turbulence (see e.g., Vallis, 2006). Turning to the energy equation in the system of Eq. 5.9, note that the terms on the left-hand side scale as e−5ε3 and thus the constants κ and λ have to be chosen in such a manner that the right-hand side also scales 4 The term ∇ · (κ∇¯ η) scales properly as e−2ε3 provided that relation in Eq. 5.10 is used for κ and the eddy turnover time scales in the same way as t, i.e., Teddy ∼ eε3 .

Invariant and conservative parameterization schemes

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as e−5ε3 ; specifically, κ and λ should scale like e−3ε3 . The form of κ was already fixed by using Eq. 5.10 and due to scaling the eddy turnover time was also fixed as Teddy ∼ eε3 , and hence the first term on the right-hand side in the energy equation also scales properly. Then, choosing λ to be of similar form as κ (i.e., λ = 2˜ γ Teddy k), for another dimensionless constant γ˜ , it indeed scales as e−3ε3 as required. It should also be noted that the invariantization of the vorticity equation in Eq. 5.9 leading to Eq. 5.11 is not unique. More specifically, it is always possible to recombine an invariant equation with other differential invariants in order to arrive at a new invariant equation. For example, the equation √ 4 η ) − ∇2 (A˜ k 5 ∇2 η¯). η¯t + ψ¯x η¯y − ψ¯y η¯x = ∇ · (κ∇¯ is also readily checked to be invariant under the maximal Lie invariance √ group of the original vorticity equation. In particular, the term 4 ∇2 (A˜ k 5 ∇2 η¯) also scales like e−2ε3 . From the physical point of view, this parameterization of the eddy-vorticity flux might be desirable as the hyperdiffusion-like term is now in conserved form. In the same way, other invariant equations could be constructed and tested numerically. In particular, with the moving frame of Eq. 5.8 at hand it is straightforward to determine various differential invariants and to recombine them to form subgrid-scale closure models for the three unclosed terms in the system of Eq. 5.6. This is a constructive way for finding all possible invariant parameterization schemes of order one-and-a-half for the model of Eq. 5.5. Example 28.24. Conservative parameterization schemes are constructed for the eddy-vorticity flux in the barotropic vorticity equation on the f plane (see Bihlo and Popovych, 2014, for more details). Extensions to the β-plane equation or the barotropic ocean model discussed in the previous example can be readily realized. The Reynolds-averaged vorticity equation on the f -plane is: (5.12) ζ¯t + ψ¯x ζ¯y − ψ¯y ζ¯x = ∇ · (v′ ζ ′ ), ζ¯ = ψ¯xx + ψ¯yy .

The task is to find a parameterization for the eddy-vorticity flux in such a manner that the closed vorticity equation admits some of the conservation laws of the original vorticity equation. In the following, the focus is on conservation laws of the vorticity equation associated with the characteristics Λ1 = h(t),

Λ2 = f (t)x,

Λ3 = g(t)y,

Λ4 = −ψ.

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Part V: Theoretical physics perspectives

Denoting the left-hand side of the vorticity equation by V , V = ζt + ψx ζy − ψy ζx , the corresponding conservation laws in characteristic form read: hV = (hζ)t + (−hψy ζ − ht ψx )x + (hψx ζ − ht ψy )y ,

f xV = (f xζ)t + (−f xψy ζ + f ψx ψy − ft xψx + ft ψ)x  f 2 2 + f xψx ζ − (ψx − ψy ) − ft xψy , 2 y   g 2 2 gyV = (gyζ)t + −gyψy ζ − (ψx − ψy ) − gt yψx 2 x + (gyψx ζ − gψx ψy − gt yψy + gt ψ)y ,    1 1 1 (∇ψ)2 + −ψψtx − ψ 2 ζy −ψV = + −ψψty + ψ 2 ζx . 2 2 2 x y t Physically, these conservation laws correspond to generalizations of the conservation of: (i) circulation; (ii)–(iii) the momenta in x- and y-directions; and, (iv) the kinetic energy. In Bihlo and Popovych (2014), the statement was proved that if a single differential equation L: Δ(x, u(n) ) = 0 admits characteristics of conservation laws of the form h(x1 ) + f i (x1 )xi , with arbitrary functions h = h(x1 ) and f i = f i (x1 ), i = 2, . . . , p, then the left-hand side Δ of L can be represented as:  Δ= Di1 Di2 F i1 i2 , 2i1 i2