Applicable Atmospheric Dynamics : Techniques for the Exploration of Atmospheric Dynamics 981433569X, 9789814335690

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Applicable

Atmospheric Dynamics Techniques for the Exploration of Atmospheric Dynamics

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Applicable

Atmospheric Dynamics Techniques for the Exploration of Atmospheric Dynamics

Istvan Szunyogh Texas A&M University, USA

World Scientiic NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

•

CHENNAI

Published 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 Szunyogh, Istvan, 1967– Applicable atmospheric dynamics : techniques for the exploration of atmospheric dynamics / Istvan Szunyogh. pages cm Includes bibliographical references and index. ISBN 978-9814335690 (hardcover : alk. paper) -- ISBN 981433569X (hardcover : alk. paper) 1. Atmospheric physics--Mathematical models. I. Title. QC861.3.S88 2014 551.51'5--dc23 2014013309

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2015 by World Scientiic Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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Printed in Singapore

To my wife Gy¨orgyi for her love and support

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Preface

This book is intended for the growing community of researchers who use weather prediction models, or data sets created with the help of weather prediction models, for scientific research. The goal is to provide physical intuition to readers with a background in the mathematical sciences and mathematical tools and references to those with a background in the atmospheric sciences. The specific techniques covered here have either proven or potential value in solving practical problems of atmospheric dynamics. The description of simplified and highly idealized models that have been found useful for the development and the initial testing of such techniques is also included. Scientists interested in adopting techniques from the atmospheric sciences to study other complex systems may also find some of the topics discussed interesting. The first applications of the quantitative laws of mechanics to the atmosphere were included in Isaac Newton’s “Principia”.1 Cleveland Abbe and Wilhelm Bjerknes were the first to point out the possibility of predicting the weather based on a small set of dynamical equations,2 and Lewis Fry Richardson3 made the first attempt to design a numerical scheme for the solution of Bjerknes’ equations. A group of scientists working at the Institute for Advanced Study, Princeton, New Jersey, led by mathematician John von Neumann and meteorologist Jules Charney, made the first 1 Newton

(1687). (1901) and Bjerknes (1904). While Bjerknes’ paper quickly became famous and Abbe’s paper has been almost completely forgotten for a century, the latter provided, in some respects, a more realistic assessment of the related practical challenges. Willis and Hooke (2006) provide an excellent survey of Abbe’s many contribution to the development of modern meteorology. 3 Richardson (1922); for a modern day interpretation of Richardson’s numerical prediction, see Lynch (2006). 2 Abbe

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Applicable Atmospheric Dynamics

experimental computer based numerical weather prediction in 19504 . By the end of the 20th century, numerical models have become the backbone of operational weather forecasting. Thanks to fifty years of continuous model development and the dramatic increase in computing power, today’s models can realistically simulate and predict a wide range of motions within the predictability time limit, which is set by the chaotic dynamics of the atmosphere. The skill of the models is tested every day by using them to produce real-time predictions of the weather for ranges between a few hours and several weeks. The forecasts by the different models are verified and compared to each other on a daily basis, which ensures that numerical weather prediction models provide the consistently most accurate solutions of the atmospheric governing equations for realistic initial and boundary conditions. Currently, due to the availability of cheap computing power and portable state-of-the-art models, experimenting with realistic atmospheric models is an opportunity open to all interested scientists. Even scientists who do not wish to run models can carry out research with high-quality data sets produced by the modeling centers.The most widely used such data sets are those produced by the different reanalysis projects.5 Other data sets have been produced by The Observing System Research and Predictability Experiment (THORPEX) component of the World Weather Research Programme (WWRP) of the World Meteorological Organization (WMO)6 , which is a 10–year international research and development program “to accelerate improvements in the accuracy of one-day to two–week high impact weather forecasts for the benefit of society, the economy and the environment”. In 2005, I was appointed to serve as one of the two co-chairs of the Predictability and Dynamical Processes Working Group (PDP WG) of THORPEX. The main responsibility of the working group has been to bring together academic scientists and operational model developers to carry out basic research that could potentially lead to further improvements of the model based forecast products; including, but not limited to dynamical process studies that can help diagnose problems with the forecast systems, 4 Charney et al. (1950). For a historical account of the developments that led to the first successful numerical weather prediction and the emergence of operational numerical weather prediction see Harper (2008). 5 Kalnay et al. (1996); Uppala et al. (2005); Compo et al. (2006); Schubert et al. (2008); Saha et al. (2010). 6 These data sets include those collected by The THORPEX Interactive Grand Global Ensemble (TIGGE, Bougeault et al., 2010) and the Year of Tropical Convection (YOTC Waliser et al., 2012).

Preface

ix

the development of improved techniques for the generation of model initial conditions based on observations and the prediction of forecast uncertainty due to uncertainties in the initial conditions and the model dynamics. My association with THORPEX was a major of motivation to write this book. The terminology used in the book distinguishes between model and diagnostic calculations. The term ’model calculations’ refers to all calculations carried out by running a numerical model, regardless of its complexity. The term ’diagnostic calculations’ refers to calculations that are based on such model-based products as an analysis or a forecast data set. Diagnostic calculations also often use observations. The main aim of this book is to help the reader to design diagnostic calculations for data sets produced by one of the forecast centers, or locally, with the help of a community, simplified or idealized model. The structure of the book is as follows. Chapter 1 is a summary of the most important model and diagnostic equations. Chapter 2 discusses concepts based on partitioning the fields of the atmospheric state variables into a basic flow component and a perturbation component, including the most often used reduced forms of the atmospheric governing equations, atmospheric wave dynamics, atmospheric instabilities and atmospheric energetics. Chapter 3 describes how the continuous governing equations can be turned into the spatially and temporally discretized equations of the numerical models. The focus of the chapter is more on the general concepts essential to design numerical experiments and interpret their results than on the numerical techniques. Chapter 4 summarizes the state of the art in atmospheric data assimilation, which is the process of the generation of initial conditions for the models based on observations of the atmosphere.7 More generally, data assimilation is the process that provides an estimate of the atmospheric state based on the observations collected up to the time for which the state estimate is prepared, using the dynamics of a numerical model of the atmosphere as a constraint to ensure that the state estimate is consistent with laws of nature. Data assimilation, viewed not long ago, as an engineering aspect of operational numerical weather prediction, is now widely recognized as a challenging, fundamental research issue for all complex dynamical systems: the state of such a system can be observed only partially, and often only indirectly by remote sensing techniques, which observe physical quantities that have a complicated functional dependence on the state variables. 7 E.g.,

Daley (1991); Kalnay (2003); Evensen (2007).

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Applicable Atmospheric Dynamics

While the reader is assumed to have a good command of calculus, vector calculus, linear algebra, probability theory and statistics, the mathematical tools used in the book would not be considered unusual in a peer reviewed journal of the atmospheric sciences. In addition, the most important mathematical concepts are introduced as boxed text when first used in the book. This layout is expected to save significant time for readers who are either not familiar with some of the mathematical tools, or who need to refresh their memory about a particular mathematical concept. The primary purpose of the exercises is to engage the reader rather than to provide home work problems. While the book includes a large number of references, it should not be viewed as a comprehensive review of the literature on the subjects it covers; selecting the references, my main aim was to give due credit to the original sources of the most important ideas and to provide references that I personally found the most accessible and lucid discussion of the particular subject. Most references are provided in footnotes to make the flow of the main line of discussion more seamless. Acknowledgements. Without the unconditional love, support and encouragement provided by my parents I would not have become a professional scientist. I have also been extremely fortunate to have the opportunity to learn from outstanding teachers, professors. mentors and research collaborators. The first important influence on my professional development was Ibolya ´ ad T´oth, my high school mathematics and physics teacher at the Arp´ Gimn´azium, Budapest, Hungary. In addition to being an excellent teacher of the subject matter, she had a special talent to convince a bunch of misbehaving teenagers, one of them myself, that studying mathematics and physics was actually ‘cool’. In my years as a student in the Meteorology program of the E¨otv¨ os Lor´and University, Budapest, Hungary, the most important influence were the classes taught by Tam´ as Pr´ ager. His textbook on the dynamical basis of numerical weather prediction8 was one of the best of its time. The first university course I taught, titled Selected Chapters of Mathematics for Meteorologists, was also developed by Tam´ as. Teaching from his lecture notes, carefully written by hand, turned out to be an important learning experience for me as a teacher. My thesis advisor for both the Diploma and the Doctoral Degree was Dezs˝o D´ev´enyi, who remained a good friend 8 Pr´ ager (1982); unfortunately, this excellent textbook has never been translated to English.

Preface

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and honest critic of my work until his untimely death at the age of 61 on Thanksgiving Day 2009 in Boulder, Colorado. My first mentors in the United States, Eugenia Kalnay and Zoltan Toth, and former colleagues from the Chaos Group of the University of Maryland, Brian Hunt, Edward Ott and Jim Yorke, all had a great influence on my professional development, including the way I approach technical writing. The central role that downstream baroclinic development plays in atmospheric dynamics in the extratropics was first pointed out to me by Anders Persson. Our correspondence contributed to my appreciation of the true genius of the early greats of atmospheric dynamics, most importantly, CarlGustaf Rossby. Discussions with other members of the THORPEX PDP WG have greatly influenced my thinking on some of the topics included in this book. I am particularly grateful to my fellow co-chair, Heini Wernli of ETH Zurich, who provided many helpful comments on the sections on potential vorticity. The other members of the PDP WG over the years were Craig Bishop, Pat Harr, Sarah Jones, Thomas Jung, Shuhei Maeda, John Methven, Mitch Moncrieff, Mark Rodwell and Olivier Talagrand. The activities of our working group were made possible by the leadership and support provided by David Burridge, Jim Caughey, Huw Davies, Tetsuo Nakazawa, Dave Parsons, Mel Shapiro and Alan Thorpe. Shaima Nassiri, a fellow professors at the Department of Atmospheric Sciences at Texas A&M University, provided many useful comments on the sections on atmospheric radiation and satellite-based observation products. Gy¨orgyi Gyarmati provided many useful comments on the manuscript. My former and current graduate students, Elizabeth Satterfield, Christina Holt, Michael Herrera and Michael Battalio carried out some of the computations and prepared some of the figures. They also provided helpful feedback on several parts of the text. At the time of writing, my research projects were funded by the National Science Foundation and the Office of Naval Research. The results of those projects had a major influence on the selection and the discussion of the techniques covered by the book. I thank World Scientific for giving me absolute freedom in deciding on the scope and the structure of the book. I am particularly grateful to Chandra Nugrahu for his hard work and patience as the desk editor of the book. I. Szunyogh, College Station, Texas, 2014

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Contents

Preface 1.

vii

Governing Equations 1.1 1.2

1.3

1.4

1.5

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . Primitive Equations . . . . . . . . . . . . . . . . . . 1.2.1 The Equations . . . . . . . . . . . . . . . . . 1.2.2 Eulerian Form of the Equations . . . . . . . 1.2.3 Scale Analysis of the Momentum Equation . 1.2.4 Diabatic Heating . . . . . . . . . . . . . . . 1.2.5 Atmospheric Constituents . . . . . . . . . . 1.2.6 Boundary and Initial Conditions . . . . . . . Representation of the Location with Coordinates . . 1.3.1 Spherical Coordinates . . . . . . . . . . . . . 1.3.2 Map Projections . . . . . . . . . . . . . . . . 1.3.3 Cartesian Coordinates . . . . . . . . . . . . . Alternate Vertical Coordinates . . . . . . . . . . . . 1.4.1 General Formulation . . . . . . . . . . . . . 1.4.2 Pressure Vertical Coordinate . . . . . . . . . 1.4.3 Sigma Vertical Coordinate . . . . . . . . . . 1.4.4 Isentropic Vertical Coordinate . . . . . . . . 1.4.5 Hybrid Vertical Coordinates . . . . . . . . . 1.4.6 Pseudo-Height and Log-Pressure Vertical Coordinates . . . . . . . . . . . . . . . . . . Vorticity and Divergence Equations . . . . . . . . . . 1.5.1 Vorticity, Absolute Vorticity and Divergence 1.5.2 Vorticity Equations . . . . . . . . . . . . . . xiii

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. 1 . 2 . 3 . 14 . 18 . 28 . 35 . 39 . 46 . 46 . 60 . 70 . 73 . 74 . 80 . 99 . 101 . 110

. . . .

. . . .

. . . .

112 117 118 119

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Applicable Atmospheric Dynamics

1.5.3

1.6

1.7

2.

The Vorticity and the Divergence as Prognostic State Variables . . . . . . . . . . . . . . . . . . . . 1.5.4 The Vorticity and the Divergence Equation in Pressure Coordinate System . . . . . . . . . . . . 1.5.5 Reduced Forms of the Vorticity and the Divergence Equations . . . . . . . . . . . . . . . . . . . . . . Potential Vorticity (PV) . . . . . . . . . . . . . . . . . . . 1.6.1 General Case . . . . . . . . . . . . . . . . . . . . . 1.6.2 Hydrostatic Case . . . . . . . . . . . . . . . . . . 1.6.3 Computation of the Potential Vorticity . . . . . . 1.6.4 Vertical Structure of the Potential Vorticity Field 1.6.5 Potential Vorticity Inversion and “PV-thinking” . Integral Invariants . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Hamiltonian Form of the Governing Equations . . 1.7.2 Energy, Momentum, and Angular Momentum . . 1.7.3 Integrals of the Potential Vorticity . . . . . . . . . 1.7.4 Integral Invariants of the Simplified Equations . .

126 127 135 151 151 155 164 167 174 177 177 183 183 184

Perturbation Dynamics

189

2.1 2.2

189 191 192 193 195 198 199 200 204 206 206 213 214

2.3

2.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Zonal-Mean Structure of the Atmosphere . . . . . . . . . 2.2.1 Zonal-Mean Temperature Field . . . . . . . . . . 2.2.2 Zonal-Mean Potential Temperature Field . . . . . 2.2.3 Zonal-Mean Wind Field . . . . . . . . . . . . . . . 2.2.4 Available Potential Energy . . . . . . . . . . . . . Quasi-Geostrophic Baroclinic Equations . . . . . . . . . . 2.3.1 General Assumptions . . . . . . . . . . . . . . . . 2.3.2 Quasi-Geostrophic Potential Vorticity . . . . . . . 2.3.3 Quasi-Geostrophic ω-Equation . . . . . . . . . . . 2.3.4 Quasi-Geostrophic Baroclinic Model Equations . . Atmospheric Waves . . . . . . . . . . . . . . . . . . . . . . 2.4.1 General Formulation . . . . . . . . . . . . . . . . 2.4.2 Large and Synoptic Scale Waves: Rossby Waves and Unstable Baroclinic Waves . . . . . . . . . . . 2.4.3 Techniques to Detect Synoptic-Scale Wave Packets 2.4.4 Eddy Kinetic Energy Equation . . . . . . . . . . . 2.4.5 Shallow-Water Waves with Constant Amplitude . 2.4.6 Convectively Coupled Equatorial Waves: ShallowWater Waves with Latitude Dependent Amplitude

232 256 263 276 279

Contents

3.

Numerical Models 3.1 3.2 3.3

3.4

3.5

3.6

3.7

3.8

3.9

4.

xv

Introduction . . . . . . . . . . . . . . . . . . . . . . . . Dynamical Cores . . . . . . . . . . . . . . . . . . . . . Spatial Discretization . . . . . . . . . . . . . . . . . . 3.3.1 Nonlinear Interactions in the Horizontally Discretized Equations . . . . . . . . . . . . . . 3.3.2 Three- and Two-Dimensional Turbulence . . . 3.3.3 Spectral Transform Method . . . . . . . . . . 3.3.4 Finite-Difference Methods . . . . . . . . . . . 3.3.5 Finite-Volume and Spectral-Element Schemes Temporal Discretization . . . . . . . . . . . . . . . . . 3.4.1 Explicit and Implicit Schemes . . . . . . . . . 3.4.2 Semi-Implicit Schemes . . . . . . . . . . . . . 3.4.3 Semi-Lagrangian Schemes . . . . . . . . . . . . Parameterization Schemes . . . . . . . . . . . . . . . . 3.5.1 Radiative Processes . . . . . . . . . . . . . . . 3.5.2 Boundary Layer Turbulence and Ocean-Land-Atmosphere Interactions . . . . . 3.5.3 Convective Processes . . . . . . . . . . . . . . 3.5.4 Microphysics . . . . . . . . . . . . . . . . . . . 3.5.5 Orographic Drag . . . . . . . . . . . . . . . . . State-of-the-Art Numerical Models . . . . . . . . . . . 3.6.1 Global Models . . . . . . . . . . . . . . . . . . 3.6.2 Limited Area Models . . . . . . . . . . . . . . Simplified and Idealized Numerical Models . . . . . . . 3.7.1 Simplified Models . . . . . . . . . . . . . . . . 3.7.2 Idealized Models . . . . . . . . . . . . . . . . . Measures of Forecast Error . . . . . . . . . . . . . . . 3.8.1 Root-Mean-Square Error . . . . . . . . . . . . 3.8.2 Anomaly Correlation . . . . . . . . . . . . . . Models as Dynamical Systems . . . . . . . . . . . . . . 3.9.1 Finite-Dimensional State Vector . . . . . . . . 3.9.2 Nonlinear Models . . . . . . . . . . . . . . . . 3.9.3 Linearized Models . . . . . . . . . . . . . . . . 3.9.4 Lyapunov Exponents and Vectors . . . . . . . 3.9.5 Transient Perturbation Growth . . . . . . . . 3.9.6 Forecast Ensembles . . . . . . . . . . . . . . .

Data Assimilation

287 . . 287 . . 289 . . 290 . . . . . . . . . . .

. . . . . . . . . . .

291 301 308 318 322 323 323 326 327 331 332

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

333 333 334 334 335 335 336 338 339 343 348 350 357 362 362 363 368 378 390 394 405

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Applicable Atmospheric Dynamics

4.1 4.2

4.3

4.4

4.5

4.6

4.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . General Formulation for Normally Distributed Observation Errors . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Cost Function . . . . . . . . . . . . . . . . . 4.2.2 Behavior of the Cost Function . . . . . . . . . . . 4.2.3 Sequential Formulation for the Linear Case: Kalman Filter . . . . . . . . . . . . . . . . . . . . 4.2.4 Computation of the Kalman Gain Matrix . . . . . 4.2.5 Sequential Formulation for the Nonlinear Case: Extended Kalman Filter . . . . . . . . . . . . . . 4.2.6 Serial Processing of the Observations . . . . . . . 4.2.7 Sensitivity to Nonlinearities: Simulated Observations Experiments . . . . . . . . . . . . . . . . . . 4.2.8 Robust Statistics . . . . . . . . . . . . . . . . . . 4.2.9 The Sequential Cost Function and Bayes’ Rule . . 3-Dimensional Schemes . . . . . . . . . . . . . . . . . . . 4.3.1 General Formulation . . . . . . . . . . . . . . . . 4.3.2 Optimal Interpolation . . . . . . . . . . . . . . . . 4.3.3 3-Dimensional Variational Schemes . . . . . . . . 4.3.4 Proxies for the Background Error . . . . . . . . . 4.3.5 Balance Constraints . . . . . . . . . . . . . . . . . 4-Dimensional Algorithms . . . . . . . . . . . . . . . . . . 4.4.1 4-Dimensional Variational Schemes . . . . . . . . 4.4.2 Ensemble-based Kalman Filtering (EnKF) . . . . 4.4.3 Hybrid Schemes . . . . . . . . . . . . . . . . . . . Accounting for Model Errors and Observation Bias . . . . 4.5.1 Model Errors . . . . . . . . . . . . . . . . . . . . . 4.5.2 Modifying the Observation Function . . . . . . . . 4.5.3 Modifying the Model Dynamics . . . . . . . . . . 4.5.4 Modifying the Observation Error Statistics . . . . 4.5.5 Sequential Schemes . . . . . . . . . . . . . . . . . 4.5.6 Weak Constraint 4D-Var . . . . . . . . . . . . . . Assimilating Satellite-based Observations . . . . . . . . . 4.6.1 Radiative Transfer in the Infrared and Microwave Ranges . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Assimilating Radiance Observations . . . . . . . . 4.6.3 Assimilating Retrievals . . . . . . . . . . . . . . . Frequently Assimilated Observation Types . . . . . . . . . 4.7.1 In Situ Observations . . . . . . . . . . . . . . . .

405 405 405 409 414 421 424 427 430 440 442 445 445 447 450 458 460 461 462 464 483 484 485 486 488 489 490 501 502 504 507 511 518 518

Contents

4.7.2 4.7.3

4.8

Satellite-based Observations . . . . . . Diagnosing and Predicting the Forecast Observations . . . . . . . . . . . . . . . Reanalysis Data Sets . . . . . . . . . . . . . . . 4.8.1 First Generation Data Sets . . . . . . . 4.8.2 Second Generation Data Sets . . . . . . 4.8.3 Third Generation Data Sets . . . . . .

xvii

. . . . . Effect of . . . . . . . . . . . . . . . . . . . . . . . . .

. 520 . . . . .

527 531 532 534 535

Bibliography

537

Index

557

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Chapter 1

Governing Equations

1.1

Introduction

The atmosphere is a high-Reynolds-number,1 compressible hydrodynamical system. As such, it can be modeled as a compressible fluid with a thin boundary layer at the Earth’s surface: the atmosphere is considered a continuum of fluid (air) parcels whose state is described by the macroscopic laws of mechanics and thermodynamics. The macroscopic state of the air parcels is described by a set of scalar and/or vector state variables. Mathematically, the parcels are considered infinitesimally small. This model allows for two closely related formulations of the atmospheric governing equations: (i) In the Lagrangian form of the equations, the independent variables are the labels a that identify the air parcels. For instance, the parcels can be labeled by their position at initial time t = t0 : a = r(a, t0 ). Here, the scalar t denotes time and r is the vector of position in threedimensional space. The spatiotemporal evolution of the scalar field f (r, t) of the scalar state variable f , or the vector field A(r, t) of the vector state variable A, can be described by combining the information provided by the function f (a), or A(a), which describes the temporal evolution of f , or A, for the particles, and the function r(a, t), which describes the position of the parcels at the different times. 1 The Reynolds number, R , is the ratio of the characteristic time scale of molecular e diffusion and the characteristic time scale of macroscopic advection in the flow. Formally, Re = U L/ν, where U is the characteristic wind speed and L is the characteristic spatial scale in the macroscopic flow, while ν is the kinematic viscosity. In the atmosphere, Re rarely becomes smaller than 100.

1

2

Applicable Atmospheric Dynamics

(ii) In the Eulerian form of the equations, the spatiotemporal evolution of the scalar field f (r, t) of the scalar state variable f , or the vector field A(r, t) of the vector state variable A, is determined without identifying the fluid parcels, which makes the position r and the time t independent variables. In essence, the Eulerian form of the equations serves to solve a more limited problem than their Lagrangian form. Solving the system of partial differential equations that describe the spatiotemporal evolution of the atmosphere in either the Lagrangian or the Eulerian form of the equations requires the knowledge of the initial and boundary conditions. Since the analytic solutions are known only for some special initial and boundary conditions, solving the system of equations for a realistic situation requires a numerical solution strategy. The computer code implementation of a particular numerical solution strategy is called a numerical model. An important component of the solution strategy is the selection of a discretization strategy that transforms the system of partial differential equations into a system of algebraic equations by the discretization of the independent variables. Almost all numerical models are based on a discretization of the Eulerian form of the equations, providing an approximate solution of the equations for a finite number of spatial modes.2 These modes can be a discrete set of grid points, finite volumes, or the basis functions of a truncated function series. Some models, however, use a Lagrangian approach for the computation of the advection terms.3 In addition, some diagnostic techniques to investigate advected properties in a model solution are based on computing the trajectory of a finite number of air parcels. 1.2

Primitive Equations

This section introduces the atmospheric governing equations in a form that follows directly from the application of the laws of hydro- and thermodynamics to the air parcels. This set of equations is called the primitive 2 A notable exception is the geometric model of (e.g., Cullen and Purser, 1984, 1989), which is based on a discretization of a Lagrangian reduced form of the governing equations. In that model, which was even considered for operational frontal analysis at the UK Met Office in the 1990s, the spatial discretization was done by defining a discrete set of finite volume parcels. 3 These schemes are called semi-Lagrangian, because they transport properties between grid points associated with an Eulerian discretization of the state variables. SemiLagrangian schemes will be discussed in detail in Sec. 3.4.3.

Governing Equations

3

equations. The set of state variables that can fully describe the hydroand thermo-dynamical state of the atmosphere is not unique. Our initial choice of the set of state variables consists of the three-dimensional velocity (wind) vector, v, the temperature, T , the density, ρ, and the pressure, p. In addition, we introduce a set of variables to describe changes in the composition of the atmosphere. Such variables have to be introduced only for those atmospheric components that undergo significant spatiotemporal changes over the time scales of interest. These components are called atmospheric constituents and their contribution to the atmospheric composition is described by their density. 1.2.1

The Equations

The general strategy for the derivation of the governing equations from the laws of physics has the following key components (Fig. 1.1): (i) the laws of physics are applied to an air parcel located at location r at time t; (ii) the combined influence of all other parcels on the parcel located at location r at time t is taken into account through the pressure field they create rather than by describing the interactions between the individual parcels.

Pressure Field Built by the Continuum of Parcels ρ(r+dr,t+dt) T(r+dr,t+dt) r+dr(t+dt) vr(r,t) ρ(r,t)

T(r,t)

ρi(r+dr,t+dt) p(r+dr,t+dt)

r(t) p(r,t)

ρi(r,t)

Fig. 1.1 Illustration of the concept of air parcels in two dimensions. Parcels other than the one shown in the figure exert their effect on the parcel through the pressure field.

4

Applicable Atmospheric Dynamics

The notation here is somewhat “sloppy”, because r denotes both an independent variable and its particular value at a given location. This “sloppiness” is the results of a short-cut in notation, because while the laws of physics are first applied to a particular parcel, the resulting equations are then extended to all parcels in the model domain. A reader wondering whether the outlined strategy can be justified based on first principles of physics is in good company. Although thinking of fluids as a continuum of parcels was already considered by Newton,4 more than two and a half centuries later, Einstein still called the justification of the equations of fluid dynamics a “fiction”.5 It should be noted that Einstein did acknowledge the success of the approach in describing many observed phenomena of fluid dynamics. More importantly, the physical model based on macroscopic fluid (air) parcels has stood the test of time ever since. Einstein’s remark illustrates, however, that the justification for the physical model of fluid dynamics is not as obvious as some textbooks make it sound. The modern approach to justify the equations of hydrodynamics is based on a “subdivision of matter down to” molecules, which qualify as “real material points” with the interactions “explicitly given”. In this approach, the equations of hydrodynamics are obtained by averaging the equations that describe the motion of the molecules in a volume (parcel), which includes a sufficiently large number of molecules to provide stable estimates of the averages, but still can be considered infinitesimal compared to the scale of the smallest scale macroscopic motions.6 On a somewhat philosophical note, it is worth to point out that the most influential physicist of our days, Stephen Hawking, argues that even the most fundamental laws of physics should be merely considered as useful mathematical models of the observed world around us.7 4 Proposition

XXIII: Theorem XVIII of Newton (1687). (1936): “A second method of application of mechanics, which avoids the consideration of a subdivision of matter down to “real” material points, is the mechanics of so-called continuous media. This mechanics is characterized by the fiction that the density of matter and speed of matter is dependent in a continuous manner upon coordinates and time, and that the part of the interactions not explicitly given can be considered as surface forces (pressure forces) which again are continuous functions of location. Herein we find the hydrodynamic theory, and the theory of elasticity of solid bodies.” 6 The interested reader is referred to the textbook by Salmon (1998) for a derivation of the equations based on averaging. 7 Hawking and Mlodinow (2010). An earlier great of science who held similar views was John von Neumann. It may not be an accident that he played a central role in the emergence of atmospheric modeling. 5 Einstein

Governing Equations

5

Mathematical Note 1.2.1 (Del Operator). The del operator, ∇, has the properties of both a vector and a differential operator: ∇ can be manipulated a vector, but the differentiation it represents must be applied to all functions on which ∇ operates in the original expression. With the help of the del operator, the gradient of a scalar-valued function f (r) is written as the vector-scalar product grad f (r) = ∇f (r);

(1.1)

the divergence of the vector-valued function f (r) as the dot-product div f (r) = ∇ · f (r);

(1.2)

and the curl of the vector valued function f (r) as the vector product curl f (r) = ∇ × f (r).

(1.3)

We first state the equations, then explain the physical laws they represent. The system of equations that describes the temporal evolution of the state of the air parcel located at r at time t is 1 Fvisc dv = − ∇p − 2Ω × v + g + , (1.4) dt ρ ρ dρ = −ρ∇ · v, (1.5) dt dT RT dp 1 = Qt + , (1.6) dt cp cp p dt p = ρRT, (1.7) dρi = −ρi · ∇ · v + Soi − Sii , i = 1, . . . , n. (1.8) dt The wind vector, v(r, t), is defined by the velocity of the parcel, that is, dr (r, t). (1.9) v(r, t) = dt This definition of the wind vector assumes that the dimension of all coordinates of r measures distance. Example 1.1. When the vector of position is represented by Cartesian coordinates, that is, r = xi + yj + zk,

(1.10)

the coordinates x, y, and z measure the distance in the direction of the unit vectors i, j and k, respectively. The coordinates of the related wind vector, v = ui + vj + wk,

(1.11)

6

Applicable Atmospheric Dynamics

are u=

dx , dt

v=

dy , dt

w=

dz . dt

(1.12)

In general, the vector of position r, does not have to be a geometric vector, as any triplet that can uniquely identify location in three dimensional space can represent r. Example 1.2. A common choice for the representation of the vector of position for a global atmospheric fields is r = (λ, ϕ, r), where λ is the longitude, ϕ is the latitude and r is the distance from the center of the Earth. Under the assumption that the Earth is a perfect sphere, λ, ϕ and r are spherical coordinates.8 Example 1.3. For a given set of unit vectors, the geometric vector r in Eq. (1.10) could also be written as the triplet r = (x, y, z). When some coordinates of r measure physical quantities other than distance, the function F that converts dr/dt into the wind vector v also has to be supplied. Introducing the notation vr (r, t) =

dr (r, t), dt

(1.13)

the function F is defined by v = F(vr ). Example 1.4. For the spherical coordinates, Eq. (1.9) leads to   dλ dϕ dr , , . vr = dt dt dt

(1.14)

(1.15)

The function F for this definition of vr will be discussed in Example 1.6. 1.2.1.1

Momentum equation

Equation (1.4) is Newton’s second law applied to the air parcel. We start the description of this equation with a couple of general comments before turning our attention to the specific terms. First, all coordinate systems we will consider here rotate with Earth. Because a rotating coordinate system is an accelerating one, the coordinate systems we use are not inertia systems. The effects of rotation on the motions observed on Earth are 8 The specific form of the atmospheric governing equations for spherical coordinates will be discussed in Sec. 1.3.1.

Governing Equations

7

Ω

R

|Ω|2R

a=6357 km

g 90o Air Parcel Plane Tangent a=6378 km to Earth’s surface

Fig. 1.2

Illustration of Ω, R, g and a.

taken into account by a couple of inertial forces, called the Coriolis force and the centrifugal force. Second, following the convention of fluid and atmospheric dynamics, the equation is written for the acceleration rather than the momentum of the parcel; that is, the equation is obtained by dividing the momentum equation for the parcel by its mass, ρ dV , where dV is its volume. Assuming that the air parcels have unit volume dV , the momentum of a particular air parcel can be determined by calculating its density ρ in addition to its velocity v. The first term of the right-hand side of the equation is the acceleration due to the pressure gradient force, which represents the total force exerted by all the other air parcels. The second term is the accelerations due to the Coriolis force. The vector Ω in this term is the angular velocity of Earth’s rotation: it points northward in the direction of the Earth’s axis of rotation (Fig. 1.2), and its magnitude, Ω = |Ω| = 7.292 × 10−5 rad · s−1 , is the angular speed of the Earth’s rotation. The Coriolis force affects all parcels with a nonzero velocity. Exercise 1.1. Show that the Coriolis force does not do work on the air parcel. Answer. The work done by the Coriolis force on the parcel over a unit time is proportional to9 v · (−2Ω × v) = 0. (1.16)

9 The work done by the Coriolis force over a unit time could be obtained by multiplying Eq. (1.16) with the mass of the parcel, ρ dV .

8

Applicable Atmospheric Dynamics

Exercise 1.2. Show that the Coriolis force can change only the direction, but not the magnitude, |v|, of the velocity of the parcel. Answer. Since the square of the magnitude of v is |v|2 = v · v,

(1.17)

if the time derivative of v · v is zero, the time derivative of |v| is also zero. The change in |v|2 due to the Coriolis force is dv d|v|2 = 2v · = 2v · (−2Ω × v) = 0. dt dt

(1.18)

The kinetic energy of the parcel at location r is proportional to 1 E = v · v. (1.19) 2 Since the time derivative of E is 1 d|v|2 dE = , (1.20) dt 2 dt Eq. (1.18) also implies that the Coriolis force cannot contribute to the changes in the kinetic energy of the parcel. Mathematical Note 1.2.2 (Vector and Scalar Triple Products). The vector triple product a × (b × c) of the vectors a, b, and c satisfies the identity a × (b × c) = b(a · c) − c(a · b).

(1.21)

For the scalar triple product (a × b) · c, (a × b) · c = (b × c) · a = (c × a) · b

= −(b × a) · c = −(c × b) · a = −(a × c) · b. (1.22)

The third term of the right-hand side of Eq. (1.4) combines the accelerations due to the gravitational and the centrifugal forces. In an inertia system, the latter would be the reaction force of the centripetal force rather than an inertial force. Because the vector g includes the acceleration due to the centrifugal forces, it is affected by the Earth’s rotation. In particular, g = g0 − Ω × (Ω × R),

(1.23)

where g0 is the acceleration due to the gravitational force and R is a vector pointing from the axis of rotation to the air parcel, with |R| being the distance between the axis of rotation and the location of the air parcel (Fig. 1.2).

Governing Equations

9

Exercise 1.3. Show that Eq. (1.23) can be written in the equivalent form g = g0 − Ω2 R,

(1.24)

where Ω = |Ω|. Answer. Equation (1.24) can be obtained by the application of Eq. (1.21) to the second term on the right-hand side of Eq. (1.23). The absolute value of the term Ω2 R at the Earth’s surface is |Ω2 a|, where a is the Earth’s radius. The Earth’s shape is a geoid rather than a perfect sphere: her radius is about a = 6,357 km at the poles and a = 6,378 km at the equator (Fig. 1.2). The numerical value of |Ω2 a| is zero at the poles and increases toward the equator, where it takes its largest value of about 2 3.39 × 10−2 m/s . The magnitude of g0 also changes with the location, because due to the Earth’s equatorial bulge, a body at the Earth’s surface is further away from the center of the Earth at the lower latitudes, leading to a small decrease of |g0 | toward the equator.10 The absolute value of g, g, is sometimes called the effective or apparent gravity, because unlike the g0 absolute value of g0 , it can be observed (mea2 sured). We will refer to g simply as the gravity. Gravity is g = 9.832 m/s 2 at the poles, and g = 9.780 m/s at the equator, while the nominal “average”, called the standard gravity, is 9.80665 m/s2 . Diagnostic calculations 2 usually use g = 9.81 m/s , which is the standard gravity rounded to two decimal digit precision. Equation (1.24) shows that the term Ω2 R has an effect not only on the magnitude, but also on the direction of g. A convenient property of the direction of g is that it is perpendicular to the surface of the geoid, that is, to the Earth’s surface.11 We can take advantage of this property by defining a unit vector, g k=− , (1.25) g which points upward in the direction perpendicular to the Earth’s surface at all locations. The potential energy of an air parcel at a distance Z from the Earth’s surface is proportional to φ(Z) = g(Z)Z, 10 Differences

(1.26)

between topography and geology at the different locations also lead to small differences in |g0 |. 11 The mechanical forces acting on the atmosphere are the same forces that shaped the Earth.

10

Applicable Atmospheric Dynamics

Plane Tangent to Earth’s surface

North Pole

k g

90o

North

Air Parcel West j

East i South

South Pole

Fig. 1.3

The local Cartesian coordinate system.

where the argument Z is added to the notation of g to indicate that gravity decreases with height (over the Earth’s surface). The dependence of g on Z can be eliminated from g by introducing the notion of geopotential height, z, which is defined by z(Z) =

φ(Z) , g

(1.27)

where g is the constant standard gravity. With the change of variables from geometric height, Z, to geopotential height, z, the dependence of g on height is absorbed into the height variable. The function φ(z) = gz

(1.28)

is called the geopotential. The z = 0 surface, which is also the φ = 0 surface, is chosen, by convention, to be at the mean sea level. The difference between the geometric height and the geopotential height in the lower few ten kilometers of the atmosphere is small.12 We can now introduce the usual choice for a coordinate system to represent a vector state variable at location r. This coordinate system is a Cartesian one, in which the three orthogonal unit vectors are defined the following way (Fig. 1.3): (i) i points eastward, (ii) j points northward, (iii) k is defined by Eq. (1.25). 12 The geometric height, Z, is only 19 m higher than the geopotential height, z, at z = 11 km, and by only 162 m at z = 32 km. The significance of the 11 and 32 km levels will be discussed in Sec. 1.2.4.2.

Governing Equations

11

We will refer to this coordinate system as the local Cartesian coordinate system at r. The directions of i, j, and k are called the zonal, the meridional and the vertical direction, respectively. These three directions change with the location r. The vector state variable A can be written as A = Ax i + Ay j + Az k,

(1.29)

where the coordinates Ax , Ay and Az are called, respectively, the zonal, the meridional and the vertical coordinates of A. Example 1.5. The three coordinates of the wind vector v in the local Cartesian coordinate system are defined by Eq. (1.11). The coordinates u, v and w are called the zonal wind, the meridional wind and the vertical wind, respectively. In addition, the coordinate u is sometimes called uwind, and the coordinate v the v-wind. Finally, the wind vector v can be decomposed into a horizontal component, vH = ui + vj,

(1.30)

vH = wk.

(1.31)

and a vertical component,

Example 1.6. When the wind vector is represented by the local Cartesian coordinates and the vector of location is represented by the spherical coordinates, that is, vr is defined by Eq. (1.15), the function F = (Fx , Fy , Fz ) in Eq. (1.14) is defined by u = Fx (vr ) = r cos ϕ

dλ , dt

v = Fy (vr ) = r

dϕ , dt

dr w = Fz (vr ) = . dt

(1.32)

The differentials of the spherical coordinates, dλ, dϕ and dr, and the differentials, dx, dy and dz of the local Cartesian coordinates are related by dx = r cos ϕ dλ,

dy = r dϕ,

dz = dr.

(1.33)

From Eq. (1.33), u=

dλ dx = r cos ϕ , dt dt

v=

which then leads to Eq. (1.32).

dy dϕ =r , dt dt

w=

dz dr = , dt dt

(1.34)

12

Applicable Atmospheric Dynamics

The last term of the right-hand side of Eq. (1.4) describes the deceleration of the parcel due to internal friction, called viscosity, between the air parcels. This macroscopic friction is the result of the diffusion of momentum by the microscopic interactions between the molecules that form the parcels. The viscous force Fvisc is Fvisc = η∇2 v + (η + η ′ )∇(∇ · v),

(1.35)

where ∇2 = ∇ · ∇ is the Laplace operator, also called the Laplacian, and η and η ′ are the coefficients of viscosity. The coefficient η is also called the first coefficient of viscosity or the dynamic viscosity, while η ′ is the second coefficient of viscosity.. The kinematic viscosity, ν, which appears in the definition of the Reynolds number, is defined by ν = η/ρ. All coefficients of viscosity are positive, that is η > 0,

η ′ > 0,

ν > 0.

(1.36)

Substituting Fvisc from Eq. (1.35) into Eq. (1.4) yields the Navier-Stokes equation.13 In a high-Reynolds-number hydrodynamical system such as the atmosphere, the viscous term can be neglected, except for motions at very small scales and motions near to boundaries with solid surfaces. The largest scale where viscosity still has a direct effect on the motions is called the Kolmogorov scale. The Kolmogorov scale for the atmosphere is not larger than a few millimeters. However, because the atmosphere is a highly nonlinear system in which motions of different scales interact, the dissipation of kinetic energy at and below the Kolmogorov scale also has an important influence on the distribution of kinetic energy at the larger scales.14 A fundamental problem of atmospheric modeling is that the effects of viscosity cannot be taken into account directly, because the smallest scale motions resolved by the numerical models are orders of magnitude larger than the Kolmogorov scale; yet, the effects of viscosity on the motions at the resolved scales must be accounted for.15 The only boundary of the atmosphere where the effects of viscosity have to be considered is at the Earth’s surface. There, viscous effects can be introduced by imposing the proper boundary conditions and adding a 13 An applied mathematical analysis of the Navier-Stokes equation is provided, for instance, by Doering and Gibbon (1995). 14 The effect of viscosity on the spectral distribution of kinetic energy will be discussed in detail in Secs. 3.3.2. 15 This problem and the techniques used by the models to address it in practice are discussed in Chapter 3.

Governing Equations

13

boundary layer to the model.16 The atmosphere above the narrow boundary layer is called the free atmosphere. Many important dynamical processes of the free atmosphere can be investigated by dropping the viscous term from the momentum equation. It should always be kept in mind, however, that a realistic model of the atmosphere must be able to account for the effects of viscosity. 1.2.1.2

Continuity equation

Equation (1.5) is the continuity equation, which is a local form of the conservation of mass: the mass of an air parcel can increase only if the convergence of the flow increases the density in the local volume defined by the parcel; while the mass of an air parcel can decrease only if the divergence of the flow reduces the density of air in the local volume. 1.2.1.3

First law of thermodynamics

Equation (1.6) is the particular form of the first law of thermodynamics that is most often used in atmospheric dynamics. In this equation, R = 287 Jkg−1 K−1 is the gas constant, cp = 1004 Jkg−1 K−1 is the specific heat for constant pressure and Qt is the diabatic heating per unit mass and time. The equivalence of Eq. (1.6) and the common differential form of the first law of thermodynamics,17 dI + p dα = Q,

(1.37)

by the following considerations. In Eq. (1.37), dI is the change in the internal energy of the air parcel, p dα is the work done by the parcel against its environment (the pressure field), and Q is the transfer of energy as heat to the parcel from its environment. Considering changes in the state that occur in an infinitesimally short period of time, and taking advantage of dI = cv dT,

(1.38)

Eq. (1.37) can be written as cv 16 This

dT dα +p = Qt , dt dt

(1.39)

topic will be further discussed in Sec. 1.2.6.1. even more general form of he first law of thermodynamics is ∆I + p∆V = Q, where Q is the transfer of heat to the parcel over an arbitrary time period, and ∆I and ∆V are the related changes in the internal energy and the volume of the parcel. 17 An

14

Applicable Atmospheric Dynamics

where cv = 717 Jkg−1 K−1 is the specific heat for constant volume. Rearranging Eq. (1.7) as pα = RT,

(1.40)

where α = 1/ρ is the specific volume (the volume of unit mass), then taking the time derivative of Eq. (1.40) yields dp dT dα +α =R . (1.41) dt dt dt Equation (1.6) can be obtained by substituting p dα/dt from Eq. (1.37) into Eq. (1.41) and taking advantage of the relationship, p

cp = cv + R,

(1.42)

between the physical constants. 1.2.1.4

Equation of state

Equation (1.7) is the ideal gas law, which is the equation of choice for the equation of state in atmospheric dynamics. Including this equation in the system of governing equations is necessary because the governing equations include three thermodynamical variables, p, T and ρ, but the continuity equation and the first law of thermodynamics only provide two equations to determine those variables. With the help of the equation of state, one of the three thermodynamical variables can be eliminated. 1.2.1.5

Constituent equations

Equation (1.8), which is the collection of n independent equations, is the continuity equation for the spatio-temporally varying atmospheric constituents.18 The term Soi (r, t) describes the local sources and the term Sii (r, t) the local sinks for the i-the constituent considered in the model. 1.2.2 1.2.2.1

Eulerian Form of the Equations Lagrangian and Eulerian time derivatives

In Eq. (1.4), dv/dt represents temporal changes in v for the parcel located at r at time t. Likewise, in Eqs. (1.5)–(1.8), dρ/dt, dT /dt, dp/dt and dρi /dt represent temporal changes in the scalar state variables for the same parcel. These time derivatives are often referred to as the total time derivative, but 18 Atmospheric modelers often refer to the constituents as tracers and to Eq. (1.8) as the tracer equation.

Governing Equations

15

here we prefer to use the term Lagrangian time derivative to emphasize that they represent temporal changes in the state of a parcel. At first glance, the Lagrangian time derivative of a state variable is not an Eulerian state variable, because the location depends on time, that is, the location and the time are not independent variables. More precisely, the argument of the functions that describe the Lagrangian evolution of the state variables is [r(t), t] rather than (r, t). As it turns out, however, a Lagrangian time derivative can be expanded in a way that includes only Eulerian variables.19 We first show how to do this expansion for an arbitrary scalar variable f [r(t), t]. Making use of the chain rule, the Lagrangian time derivative of f can be written as20 dr ∂f df = ∇f · + . (1.43) dt dt ∂t If vr = v, that is, the relationship between v and r is described by Eq. (1.9), Eq. (1.43) can be written as ∂f df = + v · ∇f. (1.44) dt ∂t In the more general case, where v 6= vr , Eq. (1.44) has to be replaced by

df ∂f = + F−1 (v) · ∇f, (1.45) dt ∂t where F−1 (v) is the inverse of the function defined by Eq. (1.14). Because all variables on the right-hand side of Eq. (1.44) and Eq. (1.45) are Eulerian variables, df /dt can also be considered an Eulerian variable. Writing the Lagrangian time derivative dA/dt with the help of Eulerian variables for an arbitrary vector variable A [r(t), t] is considerably more complicated than for the scalar variable f [r(t), t]. The calculation requires the selection of a coordinate system for the representation of A. In the local Cartesian coordinate system, the Lagrangian derivative of A can be written as di dj dk dAx dAy dAz dA = i+ j+ k + Ax + Ay + Az . (1.46) dt dt dt dt dt dt dt The last three terms on the right-hand side of Eq. (1.46) can be different from zero only in a coordinate system where the unit vector i, j and k 19 This result is the consequence of our choice to refer to the parcel by its location r instead of a label a in the system of Eqs. (1.4)–(1.8). We can make this choice, because we are interested in the value of the state variables at the different locations and times, but not the movement of the individual parcels. 20 Here, the usual “sloppy” notation is used, that is, the independent variables r and t and their particular value at a given location and time are denoted by the same symbols.

16

Applicable Atmospheric Dynamics

depend on r. Such a situation occurs, for instance, when local Cartesian coordinates are used to represent A and spherical coordinates to represent r.21 When the unit vectors do not depend on r, only the first three terms on the right-hand side of Eq. (1.46) are different from zero. In addition, the Lagrangian derivatives of the coordinates, dAx /dt, dAy /dt and dAz /dt, can be expanded with the help of Eq. (1.44). The computation of the dot product in Eq. (1.44) requires the knowledge of the del operator for the coordinate system used for the representation of v.22 For any Cartesian coordinate system with unit vectors i, j and k, the del operator is ∇=

∂ ∂ ∂ i+ j+ k. ∂x ∂y ∂z

(1.47)

Hence, the Lagrangian derivative of A can be written as       dA ∂Ax ∂Ay ∂Az = + v · ∇Ax i + + v · ∇Ay j + + v · ∇Az k dt ∂t ∂t ∂t   ∂Ax ∂Ax ∂Ax ∂Ax +u +v +w i = ∂t ∂x ∂y ∂z   ∂Ay ∂Ay ∂Ay ∂Ay + +u +v +w j ∂t ∂x ∂y ∂z   ∂Az ∂Az ∂Az ∂Az +u +v +w k. (1.48) + ∂t ∂x ∂y ∂z The short form of Eq. (1.48) is ∂A dA = + (v · ∇) A. dt ∂t

(1.49)

Example 1.7. When the local Cartesian coordinate system is used to represent both the wind vector and the vector of location, the Lagrangian time derivative of the wind vector v is ∂v dv = + (v · ∇) v dt ∂t    ∂u ∂u ∂u ∂u ∂v ∂v ∂v ∂v = +u +v +w +u +v +w i+ j ∂t ∂x ∂y ∂z ∂t ∂x ∂y ∂z   ∂w ∂w ∂w ∂w + +u +v +w k. (1.50) ∂t ∂x ∂y ∂z 21 This

case is discussed in detail in Sec. 1.3.1. the relevant formula for the computation of the Lagrangian time derivative is Eq. (1.45) rather than Eq. (1.44), the del operator has to be determined for the coordinates used for the representation of F−1 . 22 If

Governing Equations

17

The Eulerian form of the atmospheric governing equations can be obtained by expanding the left-hand-side of Eqs. (1.4)–(1.8) using Eqs. (1.44) and (1.46). The Eulerian variables ∂f /∂t and ∂v/∂t are called the local time derivative or Eulerian time derivative of the state variables. In meteorology, the local time derivative of a state variable is called the tendency of that variable.23 In addition, an equation that includes a tendency term for a state variable is called a tendency equation for that variable. Finally, the terms v · ∇f and (v · ∇)A represent the advection of the state variables f and A by the wind field v. Hence, these terms called the advection terms. 1.2.2.2

Prognostic and diagnostic variables

Atmospheric governing equations whose solution requires an integration in time are called prognostic equations, while the rest of the equations are called diagnostic equations. For instance, the system of governing Eqs. (1.4)–(1.8) is composed of four prognostic equations, Eqs. (1.4), (1.5), (1.6) and (1.8), and one diagnostic equation, Eq. (1.7).24 In addition, the state variables whose time derivative appears in the system of governing equations are called prognostic (state) variables, while all other variables are called diagnostic (state) variables. The computation of some of the diagnostic variables may be necessary to obtain a model solution. Such diagnostic variables are called diagnostic model (state) variables. The sets of prognostic and diagnostic model variables, which are both model dependent, are computed by the models. Atmospheric models also compute a large number of diagnostic variables that are not used by the solution algorithms. The sole purpose of the computation of these additional diagnostic variables is to help the interpretation of the model forecasts and the diagnosis of potential problems with the model. Many of these variables are computed in a separate post-processing step after the model solution has been completed. A user interested in a diagnostic state variable that is not part of the model output has to solve the relevant diagnostic equations for the model output. It is usually a good practice, however, to avoid recomputing diagnostic variables that are already available from the model output, not only because it saves time, but also because the post-processing package of a model employs numerical schemes that are fully consistent with the 23 For

example, ∂p/∂t is the pressure tendency. precisely, the number of prognostic equations is n + 3, because Eq. (1.8) consists of n equations. 24 More

18

Applicable Atmospheric Dynamics

numerical solution strategy of the model equations and have access to all model variables at full resolution. Some of this information is often not supplied with the model output. 1.2.3 1.2.3.1

Scale Analysis of the Momentum Equation Basics of scale analysis

Scale analysis is the investigation of the typical scale (order of magnitude) of the terms of the governing equations for a particular form of motion, or for a class of atmospheric motions. It is a powerful tool to introduce phenomenological information about the different forms of motion into the analysis of the equations. The technique takes advantage of the property of the atmospheric motions that a well defined spatial scale, or range of spatial scales, and a typical horizontal wind speed can be identified for many important forms of motion. This property of the atmosphere can be utilized to filter certain forms of motions from the solution of the equations by eliminating or simplifying some of the terms in a systematic way. Jules Charney25 was the first to point out the potentials of filtered equations for numerical weather prediction. Reduced equations also play an important role in dynamical process studies, as the role of the key processes can become more transparent in a filtered model. It should always be kept in mind, however, that the atmosphere is a highly nonlinear system, where motions at different scales interact. Hence, a filtered equation always ignores some interactions that are present in nature. While discarding those interactions eliminates some forms of motion that are irrelevant for the problem at hand, it also limits the potential accuracy of the solutions in describing nature. In other words, while the analysis of the solution of a highly reduced form of the equations can provide invaluable insight into the dynamics of a specific form of motion, a numerical model based on a less severely reduced form of the equations usually provides a more realistic description of the spatiotemporal evolution of the investigated process. The horizontal scale L and the characteristic horizontal wind speed U of some of the most important forms of atmospheric motion are listed in Table 1.1. The estimation of the scale of some terms of the equations also requires the estimation of the scale of additional state variables. The goal is to estimate the scale of those state variables based on the estimates of L and U . 25 Charney

(1948).

Governing Equations

19

The estimates listed in Table 1.1 are for the largest scale L associated with each form of motion. Interactions between motions at the different spatial scales play different roles for the different forms of motion. In this respect, the forms of motion listed in the table fall into two categories. One group includes forms of motion whose dynamics is inherently multidimensional, that is the motion at the largest scale L does not exist without the motions at the smaller scales. Most importantly, convectively coupled equatorial waves and squall-lines belong to the class of motions that are collectively called organized convection, because their smallest building blocks are convective cells. Convection also plays an important role in the genesis of tropical cyclones and in the maintenance of their inner core in the later stages of their life cycle. Table 1.1

The scales associated with some selected forms of atmospheric motion.

Form of motion Extratropical planetary waves

Horizontal

Horizontal

scale (L)

wind speed (U)

(m)

(m/s)

107

1–10

106 –107

10

Synoptic-scale waves

106

1–10

Extratropical cyclones

106

1–10

105

10–102

104 –105

10–102

Convectively Coupled Equatorial

Tropical

cyclones∗

Squall-lines∗

Waves∗

Convective cells

103

10

Tornadoes

102

102

∗ Multi-scale

phenomenon, only the largest scale and the wind speed associated with the motion at that scale are listed.

The other group includes forms of motion, for which the motion at the largest scale L can exist without the support of motions at the smaller scales. For instance, while convective cells can be embedded in the frontal systems associated with an extratropical cyclone and the embedded convection may affect the dynamics of the cyclone, the presence of convection is not a necessary condition for the genesis and the maintenance of the cyclone.26 Filtered equations are much better suited to describing the forms of motions that fall into this second group. 26 The one notable exception is the process in which a tropical cyclone transitions into an extratropical cyclone. This process is called extratropical transition (ET).

20

1.2.3.2

Applicable Atmospheric Dynamics

Estimation of the magnitude of spatial and temporal derivatives

Since the equations include temporal and spatial derivatives of the state variables, a scale analysis of the equations requires a strategy for the estimation of those derivatives. The usual approach is to approximate the derivatives by finite differences. For instance, the magnitude of the divergence ∇f of an arbitrary scalar state variable f is estimated by ∆f /∆r, where r is the distance in the direction of the gradient, ∆r is the distance between two locations in that direction and ∆f is the difference between the values of f at the two locations. An estimate L of the horizontal scale can be used to estimate ∆r, if L describes a distance for which the changes in the state variable f can be assumed to be linear. This condition is often stated by saying that L is the linear scale of the motion. The most important forms of atmospheric motion are either waves or closed vortexes. For such forms of motion, the typical distance between the location of the maximum and the minimum of f provides a good estimate of the linear scale, L. It is important to note that because the goal is to estimate the scale of the derivatives rather than the exact value of the derivatives, the estimate of L does not have to be particularly accurate. Using the estimate F for the scale of the change ∆f in f over a distance L, the estimate of the magnitude of ∇f is ∆f F ∇f ≈ ∼ , (1.51) ∆R L where ≈ indicates approximate equality of the two sides, while ∼ implies equality of the scales (orders of magnitude) of the two sides. Assuming that local processes do not play a more important role in the temporal changes of f than advection, that is, the scale of ∂f /∂t is not larger than the scale of v · ∇f , which is (U F )/L, df UF ∼ vH · ∇f ∼ . (1.52) dt L Introducing the notation T for the time scale of the change F , the scale of the left-hand side of Eq. (1.52) is F df ∼ , (1.53) dt T which leads to the F UF = (1.54) T L equivalent of Eq. (1.52) for the scales. From Eq. (1.54), the estimate of the time scale is L T = . (1.55) U

Governing Equations

21

Ωv

Ω

ΩH ϕ

Ω

ϕ Equator

Fig. 1.4

Decomposition of Ω into a horizontal (meridional) and a vertical component.

Strictly speaking, T is the time scale of advection associated with the processes at the horizontal scale L. But, because of the assumption that the local changes are not more rapid than those due to advection, T also provides an estimate of the Lagrangian time scale. This approximation is expected to break down in situations where local changes dominate the Lagrangian time derivative. 1.2.3.3

Scale analysis of the Coriolis force term

The term that represents the acceleration due to the Coriolis force can be decomposed in the local Cartesian coordinate system as aC = −2Ω×v = −2ΩH ×vH −2ΩH ×vV −2ΩV ×vH −2ΩV ×vV , (1.56) where ΩH and ΩV are the horizontal and the vertical components of Ω, respectively. Because the vector ΩH has only a meridional component (Fig. 1.4), ΩH = |Ω| cos ϕj,

ΩV = |Ω| sin ϕk.

(1.57)

The first term in the last part of Eq. (1.56) is a vertical vector, because it is the cross product of two horizontal vectors; the second and the third terms are horizontal vectors, because they are the cross products of a horizontal

22

Applicable Atmospheric Dynamics

and a vertical vector; while the fourth term is a zero vector, because it is the cross product of two collinear vectors. Thus, aC = −2ΩH × vH − 2ΩH × vV − 2ΩV × vH ,

(1.58)

which has a horizontal component aCH = −2ΩH × vV − 2ΩV × vH ,

(1.59)

and a vertical component aCV = −2ΩH × vH .

(1.60)

Introducing the notations f = 2 | Ω | sin ϕ

and

l = 2 | Ω | cos ϕ,

(1.61)

and computing the cross products, Eqs. (1.59) and (1.60) can be written as aCH = −lj × vV − f k × vH ,

(1.62)

aCV = −lj × vH .

(1.63)

and

The parameter f is called the Coriolis parameter. We introduce the notation W for the typical magnitude of the vertical wind speed, w. Unless convection or mechanical (orographic or boundary layer) forcing is present, w is not larger than ∼ 10−1 m/s and W = 10−2 m/s is a reasonable estimate of its typical magnitude. The values of both f and l change between zero and 1.46 × 10−4 1/s, which motivates the use of the scale estimate f0 ∼ 10−4 1/s for both f and l. It follows immediately that the magnitude of the first term on the right-hand side of Eq. (1.62) is at least two orders of magnitude smaller than the magnitude of the second term, which leads to the usual approximation aCH ≈ −f k × vH .

(1.64)

The order of magnitude of the term −f k × vH in Eq. (1.64) and the term −lj × vH in Eq. (1.63) is f0 U = 10−3 . Exercise 1.4. The solution of Exercise 1.1 shows that the Coriolis force does not do work on the air parcel. Using the approximation given by Eq. (1.64) destroys this property of the Coriolis force. What additional assumptions have to be made when Eq. (1.64) is used to ensure that the approximation to the Coriolis force does not do work?

Governing Equations

23

Answer. Taking the dot product of Eq. (1.58) with v yields −2v · (Ω × v) = −2vV · (ΩH × vH ) − 2vH · (ΩH × vV ) − 2vH · (ΩV × vH )

= −vV · (lj × vH ) − vH · (lj × vV ) − vH · (f k × vH ).

(1.65)

The sum of the three terms in the last part of Eq. (1.65) is zero, because the first two terms cancel out after making use of Eq. (1.22), and the third term is zero, as it is the scalar product of two orthogonal vectors. When Eq. (1.64) is used, the second term, −vH · (lj × vV ), drops out, so it can no longer cancel the first term. Hence, the first term must be eliminated in order to keep the sum zero, which can be achieved by making the assumption that aCV ≈ 0.

(1.66)

Since Eq. (1.66) replaces Eq. (1.63), the additional assumption that has to be made is that the contribution of the Coriolis force to the vertical acceleration is zero. 1.2.3.4

Scale analysis of the horizontal momentum equation

Making use of the approximation given by Eq. (1.64), the horizontal momentum equation can be written as dvH 1 = − ∇H p − f k × v H . dt ρ

(1.67)

The parameter that controls the qualitative dynamics associated with Eq. (1.67) is the Rossby number, which describes the ratio between the time scale of changes in the horizontal wind due to the Coriolis force and the advection. The Tf characteristic time scale of changes in the horizontal wind due to the Coriolis force can be determined from U ∼ f0 U, (1.68) Tf which yields Tf = 1/f0 . Because the typical time scale of advection can be described by Eq. (1.55), the Rossby number ǫRo is ǫRo =

Tf U = . T f0 L

(1.69)

Since U/T is the acceleration associated with advection and U/Tf is the acceleration due to the Coriolis force, a value of ǫRo ≤ 1 shows that the

24

Applicable Atmospheric Dynamics

acceleration associated with the advection is comparable to, or smaller than, the acceleration due to the Coriolis force. Heuristically, ǫRo ≤ 1 indicates that the advection of the parcel takes a long enough time for the Coriolis force to have an effect on the motion of the air parcel. A small Rossby number, ǫRo ≪ 1, indicates that at spatial scale L, the time scale of the total change in the wind speed is at least an order of magnitude smaller than the time scale of the change due to the Coriolis force term. Such a situation can occur only if the scale of the pressure gradient force and the scale of the Coriolis force are equal at spatial scale L. In other words, a small Rossby number indicates an approximate balance between the pressure gradient force and the Coriolis force in Eq. (1.67) at spatial scale L. In addition, the smaller the Rossby number, the stronger the balance must be for Eq. (1.67) to hold. When ǫRo ≫ 1, the total acceleration of the parcel due to the processes at scale L is at least an order of magnitude larger than what could be explained by the Coriolis force. That is, the motion must be dominated by a rapid acceleration due to a strong pressure gradient force. Of the forms of motions listed in Table 1.1, the Rossby number is ǫRo . 10−1 for planetary waves, synoptic scale waves, and extratropical cyclones. The typical time scale of changes for these forms of motion can be estimated by Eq. (1.55), which yields T = 105 –107 s, or T = 30–300 h. The smaller scale forms of motion listed in Table 1.1 are characterized by higher Rossby numbers. As will be argued in the next section, Eq. (1.55) cannot be used for the estimation of the typical time scale of these motions. These forms of motions, however, are relatively rare in the sense that they are not present most of the time at a given time and location. In addition, in the range L = 106 − 104 m, the energy of the atmospheric motions decreases with the spatial scale L as ∼ (1/L)−3 , which implies that U typically decreases as ∼ (1/L)−3/2 .27 Hence, when high-Rossby-number forms of motions are not present, or an attempt is not made to resolve their dynamics, the Rossby number can be reasonably low even at the smaller scales. This argument suggests that it makes more sense to use the time scale U/L rather than the spatial scale L to classify the different forms of atmospheric motions. In particular, using the estimate f0 = 10−4 1/s for the Coriolis parameter, the Rossby number is ǫRo . 10−1 , whenever U/L ≤ 10−5 1/s, that is, T ≥ 105 s ≈ 30 h. This motivates our use of the term large and synoptic scale motions collectively for all motions 27 This

property of atmospheric turbulence will be discussed in Sec. 3.3.2.2.

Governing Equations

25

in the range L = 106 − 104 m provided that they satisfy the condition U/L ≤ 10−5 1/s (T ≥ 105 s ≈ 30 h). It should be noted that for the most intense extratropical cyclones the wind speed can change so rapidly with distance that U/L can reach a value of 10−4 1/s. In other words, the time scale of changes associated with such a cyclone can be as short as 3 h and the Rossby number can approach a value of ǫRo = 1. Thus reduced equations derived under the assumption that ǫRo ≪ 1 have limitations in describing the dynamics of an intense cyclone. It may help, however, that intense cyclones often occur at high latitudes, where the Coriolis parameter f is larger than 10−4 , which leads to an estimate of the Rossby number that is lower than a naive estimate based on the assumption of f0 = 10−4 1/s. Hitherto, it has been assumed that the scale of the Coriolis parameter is f0 = 10−4 1/s. While this value is a reasonable estimate at most latitudes, approaching the equator, the value of f goes to zero, indicating that the Rossby number cannot be small near the equator. This behavior of the Rossby number reflects the fact that the horizontal component of the Coriolis force vanishes at the equator, hence, it cannot balance the horizontal component of the pressure gradient force. 1.2.3.5

Geostrophic wind

The term geostrophic balance refers to the balance of the horizontal components of the pressure gradient force and the Coriolis force. In the state of geostrophic balance, the acceleration of the air parcel is zero. While such a situation rarely occurs in the atmosphere, the concept of geostrophic balance plays a central role in our understanding of synoptic and large scale atmospheric dynamics. The hypothetical wind vector that would maintain geostrophic balance given the pressure and the density fields is called the geostrophic wind and denoted by vg . Exercise 1.5. Show that the geostrophic wind satisfies the equation 1 vg = (k × ∇H p) . (1.70) ρf where ∇H is the horizontal component of the del operator. Answer. Substituting the approximation for the horizontal component of the Coriolis force from Eq. (1.64) into the aCH =

1 ∇H p, ρ

(1.71)

26

Applicable Atmospheric Dynamics

definition of geostrophic balance yields

1 ∇H p. (1.72) ρf Taking the cross product of k and Eq. (1.72)  leads to  1 k × (k × vg ) = −k × ∇H p . (1.73) ρf Since the order of the multiplication by the scalar 1/ρf and the computation of the cross product on the right-hand-side of Eq. (1.73) can be interchanged, and the left-hand side of Eq. (1.73) can be written as k × (k × vg ) = k (k · vg ) − vg (k · k) = −vg , (1.74) Eq. (1.73) is equivalent to Eq. (1.70). k × vg = −

The ageostrophic wind, va , is defined by the difference between the horizontal component of the wind and the geostrophic wind, that is, va = vH − vg . (1.75) Similar to the geostrophic wind, the ageostrophic wind is also a horizontal vector. With the help of the ageostrophic wind, the momentum equation, Eq. (1.67) can be written in the equivalent form dvH = −f k × va . (1.76) dt Since the scale of the contribution of the processes at spatial scale L to the magnitude of dvH /dt can be estimated by U 2 /L, the equivalent form of Eq. (1.76) for the scales is U2 = f0 U a , (1.77) L where Ua is the typical magnitude of va . Thus, U2 = ǫRo U. (1.78) Ua = Lf0 Equation (1.78) shows that the ratio between the ageostrophic wind speed and the wind speed is equal to the Rossby number, ǫRo . This relationship between the wind speeds suggests that for small Rossby numbers (ǫRo ≪ 1), the magnitude of the geostrophic wind must be U . Hence, the ratio between the scales of the ageostrophic and the geostrophic wind speed is equal to the Rossby number. For large Rossby numbers (ǫRo ≫ 1), Eq. (1.78) leads to the paradox that the ageostrophic wind speed is much faster than the total wind speed, which shows that the time scale of advection cannot be used for the estimation of the time scale of dvH /dt for high-Rossby-number motions: when the pressure gradient force is much stronger than the Coriolis force, the time scale of local changes is shorter than the time scale of advection and the local time derivative dominates dvH /dt.

Governing Equations

1.2.3.6

27

Hydrostatic balance

The vertical component of the momentum equation is 1 dvV = − ∇V p − lj × vH + g, dt ρ

(1.79)

where ∇V is the vertical component of the del operator. The hydrostatic balance equation, which is the most important reduced form of Eq. (1.79) can be derived by the following scale analysis. The equivalent of Eq. (1.79) for the scales is W 1 = − ∇V p − f0 U + g, (1.80) TH ρ where TH is the time scale of the motions we are interested in. For large and synoptic scale motions, TH is equal to T = L/U = 1/(f0 ǫRo ), the time scale of advection . Thus Eq. (1.80) becomes 1 W f0 ǫRo = − ∇V p − f0 U + g. ρ

(1.81)

The scale of the gravity term (third term on the right-hand side) is 10, while the scale of the vertical component of the Corioilis force (second term on the right-hand side) is 10−3 . When ǫRo . 1, the term W f0 ǫRo is at least an order of magnitude smaller than the Coriolis term, if W . 10. Since W is at least two orders of magnitude smaller than 10, for motions characterized by ǫRo . 1, the vertical acceleration is at least three orders of magnitude smaller than the Coriolis force term. Thus, the only term that can balance gravity in Eq. (1.79) is the vertical component of the pressure gradient force, whose magnitude, therefore, must be ∇V p ∼ 10. Retaining only the two leading order terms of Eq. (1.79) leads to ∇V p = −ρg.

(1.82)

Equation (1.82) is the hydrostatic balance equation. The scale analysis shows that the leading order error term of the hydrostatic approximation is equal to the magnitude of the Coriolis force term, which introduces a relative error of ∼ 10−3 . Dropping this term from Eq. (1.79) has the additional benefit that, as discussed in Exercise 1.4, it ensures that using Eq. (1.67) for the approximation of the horizontal momentum equation does not introduce an artificial source or sink of the kinetic energy. The hydrostatic balance can break down only for high-Rossby-number processes, for which the time scale of local changes can be much shorter than the time scale of advection. In practice, the vertical acceleration term

28

Applicable Atmospheric Dynamics

is retained in situations, where the goal is to explicitly resolve the dynamics of convective processes. Of the forms of motion listed in Table 1.1, tropical cyclones, squall lines, convective cells and tornadoes owe their existence to convective processes. A realistic simulation of these processes by a model requires a model resolution that is sufficient to resolve the convective cells. Hence, using the full vertical momentum equation rather than the hydrostatic balance equation in a model makes sense, only if the model can resolve motions with a horizontal scale of .1 km. Since boundary layer processes play an important role in the convective forms of motions, a more detailed scale analysis of the vertical momentum equation should also consider the additional terms that represent the boundary layer processes. 1.2.4

Diabatic Heating

At first glance, the diabatic heating term Qt in Eq. (1.6) is a simple forcing term. In fact, the simplest approach to account for diabatic heating (Qt > 0), or diabatic cooling (Qt < 0, is to use a prescribed function Qt (r, t). Numerical weather prediction and climate models, however, reached a level of sophistication a long time ago, where they take most processes that contribute to the spatiotemporal changes in Qt (r, t) into account directly. In particular, (i) the absorption of radiation leads to adiabatic heating, while the emission of radiation leads to radiative cooling; (ii) latent heat released when water vapor condenses into droplets of water, or water vapor or droplets of water freeze into ice crystals in clouds lead to diabatic heating, while evaporation and sublimation lead to diabatic cooling; (iii) the transport of heat from a warmer Earth’s surface by turbulent eddies in the boundary layer leads to diabatic heating, while the transfer heat from a warmer atmosphere to the surface lead to diabatic cooling. The quantitative description of these processes requires the inclusion of additional equations that describe the physics associated with the radiative, the microphysical (cloud) and the boundary layer processes of the atmosphere. Some of these added equations and terms include highly nonlinear expressions of the state variables of Eqs. (1.4)–(1.8). In numerical models, these equations and terms are added in the form parameterization schemes. In addition to the aforementioned processes, parameterization schemes also account for the processes that Eqs. (1.4)–(1.8) could, in principle, describe,

Governing Equations

29

but whose effects are filtered from the solutions by the reductions made to the equations28 and/or the finite spatial and temporal resolution of the numerical solutions. The reader may wonder why heat conduction 29 is not included in the list of processes that can transfer heat between the parcel and its environment. The explanation is that because air is an extremely poor conductor of heat, the effects of heat conduction on the atmospheric processes are negligible. Hence, away from the surface, at times where there are no changes in the phase of water, the assumption Qt (r, t) = 0 can be made for all motions, whose characteristic time scale is shorter than the characteristic time scale of the radiative processes that can affect the parcel. A process for which such an assumption can be made is called adiabatic. An important consequence of the poor heat conductivity of air is that an air parcel should not be expected to adjust its temperature to its environment by heat conduction. Thus changes in the pressure of the environment play a more important role in the changes of the temperature of the parcel than changes in the temperature of the environment. For instance, when the pressure of the environment decreases, the pressure of the parcel also decreases, leading to an expansion of the parcel; because the energy necessary for the work done by the parcel during its expansion is supplied from the internal energy of the parcel, the temperature of the parcel decreases. The next section describes the basics of atmospheric radiation, which has a special place among the atmospheric processes, because it plays an important role in the interpretation of remotely sensed observations. The other processes that are conventionally taken into account by parameterization schemes will be discusses in Sec. 3.5. 1.2.4.1

Basics of atmospheric radiation

The ultimate source of the energy of all forms of atmospheric motions is solar radiation. The average solar radiation flux incident on the Earth is called the solar constant and its value is S = 1360.8 ± 0.5 Wm−2 .30 The actual value of the radiation flux changes with the distance between the Earth and the Sun. Most of these changes are periodic and those with a long period are thought to play an important role in the cyclic changes 28 For

instance, the replacement of the vertical momentum equation by the hydrostatic balance equation. 29 Heat conduction is associated with the microscopic motions of the molecules, which could, in principle, transport heat between the parcel and its environment. 30 Kopp and Lean (2011).

30

Applicable Atmospheric Dynamics

in the Earth’s climate (average temperature). The annual variation is a cyclic change between the minimum, S = 1323 Wm−2 , at aphelion and the maximum, S = 1414 Wm−2 , at perihelion. The solar constant measures the radiation energy per unit surface and time incident on Earth’s surface for a ray that reaches the surface at an angle of 90◦ . Since solar radiation reaches the Earth’s surface at most location at an angle lower than 90◦ , it can be shown that the areal average of the solar radiation flux incident on the Earth is S/4.31 Moreover, clouds and the Earth’s surface reflect a fraction of the incoming solar radiation back into space. The ratio of the portion of the solar radiation directly reflected back to space and the total solar radiation is called the albedo. The albedo varies in both space and time, but its global average, called the planetary albedo, is estimated to be α = 0.29. Thus the average solar flux that can be redistributed by the Earth system is (1 − α)S/4. All bodies with a nonzero absolute temperature emit radiation (electromagnetic waves). The dependence of the intensity spectrum of the radiation on the temperature of the emitting body is described by the Planck function, 2hν 3
, (1.83) Bν (T ) = 2 hν/K T B c e −1 where Bν is the energy emitted per unit area, per unit of solid angle of detection, per unit time and frequency, ν is the frequency of the electromagnetic wave, h = 6.626 × 10−34 J s is Planck’s constant and KB = 1.3806 × 10−23 J/K is Boltzmann’s constant. Since the frequency, ν, and the wavelength, λ, of the waves are related by λ = c/ν, (1.84) where, c = 299,792,458 m/s is the speed of light,32 the Planck function can also be written as C1 λ−5 2hc2
=
, (1.85) Bλ (T ) = 5 hc/KλT λ e −1 π eC2 /λT − 1 where C1 = 2πhc2 = 3.742 × 10−16 W m2 and C2 = hc/KB = 1.439 × 10−2 m K are, respectively, the first and second radiation constants. Exercise 1.6. Show that when hν ≪ KT , the Planck function can be approximated by a linear function of the temperature as 2Kν 2 Bν (T ) ≈ T. (1.86) c2 31 Since

the total surface of the Earth is 4πa2 , where a = 6.371 × 106 is the radius of the Earth, the total energy per unit time that reaches the Earth is Sπa2 = 5.5607 × 1016 W. 32 The speed of light in vacuum provides a good estimate of the speed of light in air.

Governing Equations

31

Equation (1.86) is the Rayleigh-Jeans law. The Planck function takes its maximum at wavelength a (1.87) λmax = , T where a = 2.897 × 10−3 m K is a physical constant. Equation (1.87), called Wien’s displacement law, states that the wavelength where the intensity of the emitted radiance is highest is inversely proportional to the temperature. Since the temperature of the emitting surfaces of the Earth system is much lower than the temperature of the Sun, λmax is much larger for the solar than the terrestrial radiation. In addition, because the Planck function rapidly decreases away from the maximum, solar radiation is composed of much shorter wavelength waves than the terrestrial radiation. This difference in the wavelengths is the basis for the convention to refer to solar radiation as short wave radiation and to terrestrial radiation as long wave radiation. Since the average temperature of the Earth system changes slowly in time, we can assume that the energy of the incoming short wave radiation, Sshort = (1 − α)S/4, and the outgoing long wave radiation, Slong are in balance. For a perfect blackbody, the temperature of the emitted radiation can be computed from the emitting body with the help of the StefanBoltzman law, F = σT 4 .

(1.88)

where F is the flux density of the emitted radiation and σ = 5.67 × 10−8 W m−2 K−4 is the Stefan-Boltzman constant. Substituting Slong = Sshort for F in Eq. (1.88), we obtain the equation  1/4 (1 − α)Sshort Te = . (1.89) 4σ

The temperature Te is called the effective emitting temperature and its value for the atmosphere is Te ≈255 K. The effective emitting temperature can be interpreted as the temperature a hypothetical planet would have if it were the same distance from the Sun as Earth and had the same planetary albedo as Earth, but had a surface that was a perfect blackbody in the long wave and an atmosphere that was transparent across the spectrum. In reality, the temperature at the surface is about 290 K, which is much higher than 255 K (Fig. 1.5). In addition, satellite-based observations of the emission spectrum of the Earth system at the top of the atmosphere show a spectrum that is consistent with the Planck function of a black body

32

Applicable Atmospheric Dynamics

10

20

Pressure [hPa]

30

50 70 100 150 200 250 300 400 500 600 700 850 1000 200

210

220

230

240 250 Temperature [K]

260

270

280

290

Fig. 1.5 Vertical profile of the global mean temperature based on a 30-year mean of the NCEP/NCAR reanalysis data for 1981-2010. The scaling of the y-axis follows the log-pressure vertical coordinate formally introduced in Sec. 1.4.6.3, but the tick labels indicate the actual pressure values.

whose temperature is T ≈ 290 K. At many wave lengths, however, the emitted radiance is much lower than that predicted by the Planck function for T = 290 K. These observed properties of the atmospheric radiation indicate that the atmosphere is not transparent to long wave radiation at some wave lengths: it traps part of the outgoing long wave radiation, leading to an increase of the temperature near to the surface. This process is the so-called greenhouse effect. 1.2.4.2

The vertical thermal structure of the atmosphere

Vertical changes in the temperature are usually described by the lapse rate, ∂T . (1.90) ∂z In atmospheric dynamics, the major vertical layers of the atmosphere are distinguished based on changes in the sign of the lapse rate. The motivation for this practice is that changes in the lapse rate have important effects on the qualitative dynamics of the atmosphere, while changes in the chemical composition of the atmosphere have important effects on the dynamics only if they also have an effect on the lapse rate. The lapse rate γ=−

Governing Equations

33

has an important effect on the qualitative dynamics, because it controls the buoyancy of the air parcels: as will be discussed in Sec. 1.4.4.2, the smaller the lapse rate, the stronger the retaining force that acts on a parcel with an upward momentum. Hence, sustained vertical motions can exist only in an atmospheric layer of sufficiently large positive lapse rate. All classifications based on the lapse rate use the same names for the layers, but the specific definition of the layers they give can be different. In essence, there are two types of classifications. The first group of classifications consists of the so-called standard atmospheres. The most widely used example for such a classification is the International Standard Atmosphere (ISA) of the International Civil Aviation Organization (ICAO). The other type of classification can be found in the different glossaries of meteorology.33 In essence, the goal of the definition of standard atmospheres is to provide a general phenomenological description of the changes in temperature, while the purpose of a meteorological definition is to distinguish between layers of highly different qualitative dynamics. In practical terms, the most important difference between the two types of classifications are in the definition of the interfaces between the deep layers of positive and negative lapse rates. In the standard atmospheres they are considered layers of constant temperature, while in meteorology (and atmospheric dynamics), they are considered sharp boundaries between the deep layers of positive and negative lapse rates, treating the shallow layers of small absolute values of the lapse rate as part of the upper deep layer. While this book, in general, adopts the meteorological definition of the interfaces, this section also provides some information about the ICAO ISA, as it is often used as a reference in the literature. The lowest layer, where the average lapse rate is positive, is called the troposphere. Figure 1.5 shows that for the global average temperature profile, the top of the troposphere is at about 100 hPa (about 16–17 km).34 The figure also shows that up to about 250 hPa, the temperature decreases linearly with height (note that the scale for the y-axis is logarithmic in pressure). The constant lapse rate associated with the linear temperature profile is called the average lapse rate and its value is about 6.5 K/km.The ICAO ISA uses the value of 6.5 K/km for the entire troposphere. The 33 For instance, in the online edition of the Glossary of Meteorology of the American Meteorological Society (AMS). 34 In the ICAO ISA, the top of the troposphere is at 226.32 hPa (11 km geopotential height). The troposphere is higher in Fig. 1.5 than in the ICAO ISA, because the averages in the figure also include data from the Tropics, where the tropopause is higher than in the extratropics.

34

Applicable Atmospheric Dynamics

height of the tropopause changes with both location and time: the height of the tropopause can vary between about 15 and 20 km in the Tropics and between 9 and 12 km in the arctic regions. In addition, negative lapse rates can locally occur in the troposphere due to either a stronger radiative cooling in a lower layer, or the horizontal advection of warmer air in an upper layer. At the top of the troposphere, the lapse rate abruptly drops to a small absolute value, which persists in a narrow layer above the troposphere. This layer appears in Fig. 1.5 as a layer of slight negative lapse rate between 100 and 70 hPa. In meteorology, this layer is considered part of the stratosphere, while in a standard atmosphere, it is called the tropopause and its lapse rate is considered zero.35 In meteorology, the tropopause is simply defined as the boundary between the troposphere and the stratosphere. This definition is more than justified from a dynamical point of view, as the changes in the qualitative dynamics are much more significant when the large positive lapse rate disappears than when the nearly zero lapse rate becomes more negative. Since the investigation of the spatiotemporal evolution of the tropopause plays an important role in modern atmospheric dynamics, we will provide a dynamical definition of the tropopause in Sec. 1.6.4, which will allow for a more quantitative description of the dynamics of the tropopause. Here, we only note that the tropopause is not an impenetrable material surface despite the sudden increase of the strength of the retaining force, which acts on a parcel with an upward momentum, at the tropopause. First, parcels accelerated in a strong convective cell can have sufficient vertical momentum to penetrate the stratosphere.36 Second, tropopause folds can develop due to the folding of the two-dimensional surface between the troposphere and the stratosphere, leading to a stratospheric intrusion into the troposphere, where the tropospheric air and the stratospheric air can mix (Fig. 1.6).37 Figure 1.5 does not show the full depth of the atmosphere, not even the full depth of the stratosphere.38 The phenomena we are primarily concerned in this book take place in the troposphere and at the tropopause and are not influenced directly by the processes that take place above the 35 In the ICAO ISA, the tropopause is defined as the layer between 226.32 and 54.749 hPa (11 and 20 km geopotential height). 36 Such an incident is called a convective overshoot. 37 For instance, Homeyer et al. (2011) showed observational evidence that such mixing can occur when convective clouds penetrate into a stratospheric intrusion. 38 In the ICAO ISA, the top of the stratosphere is at 1.1091 hPa.

Governing Equations

ion

rus

r

Pressure

he

sp ato Str

nt ic I

35

Warmer Troposphere

Tropopause

Colder Troposphere

Horizontal Coordinate

Fig. 1.6 Two-dimensional schematic illustration of a tropopause fold. The troposphere tend to be higher in regions where its vertical mean temperature is higher.

lower stratosphere (above about 10 hP or 32 km).39 For completeness, we note that the deep layer of positive lapse rate above the stratosphere is called the mesosphere and the boundary between the two layers is the stratopause. The top of the mesosphere in the ICAO ISA is at 0.003734 hPa. The next layer is the thermosphere, which is separated by the mesopause from the thermosphere. State-of-the-art global models have their top in the mesosphere and post-processed output from such models is typically provided up to the top of the stratosphere (1 hPa). 1.2.5

Atmospheric Constituents

An atmospheric constituent can be a chemical component of air, or a particular type atmospheric aerosol 40 or cloud particle. The most important atmospheric constituent is water vapor, for which a constituent equation has been included in the operational weather prediction models since the 1960s. Constituents other than those related to the different phases of water play a role in atmospheric dynamics through their effects on atmospheric radiation. The effects of a constituent whose concentration in air is nearly constant can be taken into account without the help of a constituent equation. Thus the addition of a constituent equation is warranted only for a 39 In the ICAO ISA, 32 km is equivalent to 8.6892 hPa and lapse rate between that level and the bottom of the stratosphere is −1 K/km. 40 Atmospheric aerosols are suspensions of small solid and/or liquid particles, excluding cloud particles, in the air.

36

Applicable Atmospheric Dynamics

constituent whose concentration shows significant spatiotemporal changes. The convention has been to use the residence time to decide whether the concentration of a constituent can change or not over the timespan of the model solution. The residence time, τ , also called the average residence time, is the average time spent by the molecules of a given gas in the atmosphere. It is defined by the ratio of the total mass M of the constituent in the atmosphere and the rate F of its removal, that is, τ = M/F . A constituent that spends a sufficiently long time in the atmosphere (about a couple of years) can be efficiently mixed by the atmospheric flow, leading to a nearly constant concentration of the constituent in a deep vertical layer of the entire atmosphere. For instance, even though CO2 is an important greenhouse gas, it has not been included in most weather prediction models, because its residence time is about τ = 3–4 years. As it turns out, however, while the variability of the CO2 concentration is sufficiently small to have negligible effects on the changes in diabatic heating at the time scale of a numerical weather forecast, the same changes are sufficiently large to affect the accuracy of the radiative transfer calculations, which are necessary for the accurate interpretation of satellite based observations of the long wave radiation.41 In most state-of-the-art models, the number of atmospheric constituents is a parameter that can be easily changed. Whenever a new constituent is added to the model, the source and sink terms also have to be provided for the new constituent. These terms represent the rate of change in the density of the tracer due to changes in phase (for cloud particles), chemical and photochemical processes and interactions with other components of the Earth system. The phase changes in the cloud particles are controlled by the cloud physics built into the model. The sources and sinks of the variable chemical components are usually prescribed functions of space and/or time, but in some applications they can be provided by a sophisticated atmospheric chemical model. 1.2.5.1

Water vapor

Water vapor has a larger overall effect on atmospheric dynamics than any other atmospheric constituents because (i) it has important effects on the thermodynamical properties of air; 41 As

will be discussed in Sec. 4.7.2, it has been found recently that including CO2 as a constituent in the model can lead to a more efficient assimilation of satellite-based radiance observations.

Governing Equations

37

(ii) it has a large effect on both the short and the long wave atmospheric radiation processes; (iii) it enables the generation of latent heat, which plays a key role in the transport of heat from Earth’s surface to the atmosphere; (iv) it plays an important direct role in shaping the weather, which would make its prediction important, even if it had no other effects on atmospheric dynamics. The residence time of water vapor is about τ = 10 days. This short residence time and the high spatiotemporal variability in the source and sink terms is consistent with the high spatiotemporal variability of water vapor in the atmosphere. The density ρi for the water vapor is called vapor density and the convention is to denote it by ρv . Exercise 1.7. Show that 1 dq = M, (1.91) dt ρ where q = ρv /ρ and M = Sov − Siv , is equivalent to Eq. (1.8) for the water vapor. Answer. For the water vapor, Eq. (1.8) can be written in the equivalent form ∂ρv /∂t = −∇(ρv v) + M.

(1.92)

Eq. (1.91) can be obtained by multiplying Eq. (1.92) by qi and subtracting the result from Eq. (1.5). [A variable analogue to q can be defined for any atmospheric constituent and Eq. (1.8) can be replaced by an equation analogue to Eq. (1.91).] The ratio q is called the specific humidity. Another frequently used moisture variable is the mixing ratio, r, which is the mass of the water vapor per unit mass of dry air.42 Introducing the notation ρd for the density of dry air, the density of the mixture of air and all variable constituents except for water vapor, the formal definition of the mixing ratio is ρv r= . (1.93) ρd In addition, the formal relationship between specific humidity, vapor density and mixing ratio is ρv r q= = . (1.94) ρd + ρv 1+r 42 A mixing ratio, defined in a similar way, is also often used to describe the concentration of the other constituents.

38

Applicable Atmospheric Dynamics

Water vapor plays a unique role among atmospheric constituents, because unlike other tracers, which have an effect on the hydro- and thermodynamical state variables primarily through the radiative process, it also has a major effect on the thermodynamical properties of air.43 These effects, in principle, could be taken into account by replacing the gas constant R for dry air with the Eulerian variable R′ (r), called the effective gas constant for the mixture of dry air and water vapor. To avoid the complications that arise from replacing a constant with a new Eulerian variable, the conventional approach is to replace the temperature T with the virtual temperature, Tv , which is defined by Rd Tv = R′ T.

(1.95)

The virtual temperature can be computed by the formula Tv = T

1 + r/ǫ ≈ T (1 + 0.608r), 1+r

(1.96)

where ǫ=

R = 0.6220. Rv

(1.97)

Here, Rv = 461.5 Jkg−1 K−1 is the gas constant of water vapor. Equation (1.97) shows that the virtual temperature is always larger than the temperature. Working with data of temperature dimension, special care has to be taken to avoid confusing virtual temperature with temperature. The potential for such confusion is high due to a number of reasons: (i) Since r is usually less than 0.04 in the atmosphere, the typical virtual temperature is only slightly higher than the temperature, making it impossible to judge only from the value whether a data is for temperature or virtual temperature. (ii) While the model variable is usually virtual temperature, the model output is often provided as temperature. (iii) Atmospheric temperature observations are typically transmitted as virtual temperatures, but for a few observing platforms (e.g., commercial airliners) report the temperature. 43 Section 4 of Emanuel (1994) provides an excellent overview of moist thermodynamical processes.

Governing Equations

1.2.5.2

39

Ozone

The second most important chemical constituent is ozone, with a residence time between a few hours and a few days. While tropospheric ozone has an important effect on air quality, it is ozone in the lower stratosphere that has an important effect on atmospheric dynamics: the absorption of solar radiation by stratospheric ozone heats the stratosphere44 and also leads to thermally exited tides in the atmosphere.45 The best known dynamical phenomenon associated with the thermal waves excited by stratospheric ozone is the semi-diurnal tidal wave, which is an about 1 hPa amplitude oscillation in the surface pressure in the Tropics. The semi-diurnal tidal wave is the most regular periodic motion known to exist in the atmosphere. Formally, the absorption of solar radiation affects the dynamical variables through the diabatic heating term Qt . In principle, a model can account for the dynamical effects of ozone by prescribing the periodic temporal changes in Qt due to the absorption of solar radiation by ozone.46 The concentration of ozone, however, is changing, not only with geographical latitude and season, but also with the transient features of the atmospheric flow. The recognition of the importance of flow-dependent changes in the stratospheric ozone concentration and the availability of reasonably good quality satellite-based stratospheric ozone observations, which is necessary for the preparation of the initial condition for the ozone density state variable, have motivated some modeling centers to introduce ozone as a constituent into the models.47 1.2.6

Boundary and Initial Conditions

Solving the equations require the definition of boundary and initial conditions. The boundary conditions define the state of the atmosphere at the boundaries of the model domain, while the initial conditions assign values to the state variables at an initial time. The boundary conditions must be defined for the entire time of the model solution and they also have to be consistent with the initial conditions. While the boundary conditions can be highly complicated, the general 44 In the Martian atmosphere, the same vertical layers exist as in the terrestrial atmosphere, except for the stratosphere, because there is no component of the Martian atmosphere that would play an analogous role to ozone. 45 Chapman and Lindzen (1970) provides a detailed analytical investigation of this process. 46 In fact, this approach still used in some of the most advanced NWP models. 47 Ozone was first introduced as a tracer variable in the mid-1990s.

40

Applicable Atmospheric Dynamics

interpretation of their role in a model is simple: the boundary conditions describe the interactions between the atmosphere in the model domain and its surroundings. All models include the definition of the boundary conditions at the bottom of the atmosphere (at the Earth’s surface) and the “top” of the atmosphere. In addition, a limited area model also includes the definition of the lateral boundary conditions, which describe the conditions at the horizontal boundaries of the limited area domain. 1.2.6.1

Bottom boundary conditions

The bottom boundary conditions prescribe the value of the state variables on the two-dimensional surface S that serves as the boundary between the atmosphere and the Earth’s surface. Formally, the boundary conditions prescribe the evolution of the state variables on S. There are large differences in the level of sophistication of the bottom boundary conditions used by the different models. In an idealized, or reduced, model the boundary conditions can be a simple prescribed function with no space and/or time dependence, while in a state-of-the-art numerical weather prediction model, they can be defined by complex ocean and land models, which can interact with the atmosphere. Regardless of the general level of sophistication of the definition of the boundary conditions, the boundary conditions for the wind vector must be consistent with the assumptions made about the role of viscosity in the momentum equation. The boundary condition consistent with the inviscid atmosphere described by Eq. (1.4) would state that the component of v(rS , t) normal (perpendicular) to S is zero at all times and locations, where rS is the vector of position on S. This condition can be formally written as v(rS , t) · n(rS ) = 0,

(1.98)

where n(rS ) is the vector normal to S at location rS . In the inviscid case, the boundary condition does not impose any constraint on the speed of the wind along S. In other words, the inviscid boundary condition simply states that the flow must be parallel with the surface S. A realistic model of the atmosphere, however, must account for the fact that the atmosphere is a high-Reynolds-number hydrodynamical system rather than an inviscid one. Thus the effect of viscosity can be taken into account by adding a boundary layer to the model. The concept of the boundary layer is based on the observation that the speed of a highReynolds-number flow relative to the speed of the surface of a solid body

Governing Equations

41

immersed in the flow is zero.48 That is, in contrast to the case of an inviscid flow, not only the normal component of the velocity vector, but also the component along the surface must vanish. This boundary condition can be written for the atmosphere, assuming that the Earth is a solid body,49 as v(rS , t) = 0.

(1.99)

Because the Coriolis force is zero for an air parcel whose speed is zero, the boundary condition given by Eq. (1.99) can be satisfied by the horizontal component of the wind, if there exists a force that can exactly balance the pressure gradient force. This force is usually referred to as the surface friction or surface drag force. The stronger the pressure gradient force the stronger must be the surface drag force. The effect of viscosity must be taken into account not only at the surface, but also in the entire boundary layer, where viscosity leads to an irreversible transfer of momentum from the locations of higher wind speeds aloft to the locations of lower wind speed below.50 The boundary layer for the atmosphere is called the planetary boundary layer. The planetary boundary layer is a turbulent boundary layer, where turbulence plays an important role in the transport of heat and atmospheric constituents, most importantly humidity, from the Earth’s surface to the atmosphere. Because many processes in the planetary boundary layer take place at scales that the models cannot explicitly resolve, the effects of those processes on the processes at the resolved scales are taken into account by the implementation of a boundary layer parameterization scheme. In a reduced or idealized model, the boundary layer parameterization terms are usually based on a simple idealized model of turbulence. Independently of the sophistication of the parameterization of the boundary layer processes, two important properties of the planetary boundary layer should always be kept in mind: (i) the boundary layer is not a mathematical fiction invented by scientists to make the inviscid form of the equations work for high-Reynoldsnumber hydrodynamical systems, but an important observable physical phenomenon; 48 Excellent introductions of the concept of the boundary layer for high-Reynolds number fluids can be found in the standard text books by Landau and Lifshitz (1987) and Feynman et al. (2006a). 49 This approach requires some refinement, if the surface of the oceans is not treated as the surface of a solid body. 50 E.g., Landau and Lifshitz (1987).

42

Applicable Atmospheric Dynamics

(ii) the top of the boundary layer is not a sharp surface; hence, it can be assumed that the atmospheric state variables change smoothly between the boundary layer and the free atmosphere.51 1.2.6.2

Top boundary conditions

The atmosphere does not have a well defined top, even though the pressure and the density of air, as well as the wind speed become zero beyond a certain distance from the Earth’s surface. Mathematically, this property of the state variables could be stated by saying that the value of the state variables goes to zero, as the distance from the surface of the Earth goes to infinity. This boundary condition, however, is not very useful when the goal is to design a numerical algorithm, which requires a finite computational domain. There are essentially two options to define the top of the atmosphere in practice. One of them is to define the top of the model atmosphere by a prescribed value of the distance from the Earth’s surface. The other one is to use one of the state variables that become zero with the distance from the Earth’s surface as the vertical component of the vector of position.52 The two approaches can also be combined, using a small, but nonzero constant value of a state variable that goes to zero at the top of the model atmosphere. In either case, the top of the model is a two-dimensional surface T , where location can be described by a two dimensional vector of position rT . Since the sharp surface that defines the top of the model atmosphere is an artifact of modeling, nature does not provide clear guidance on the appropriate boundary condition at T . There is no reason, empirical or theoretical, to believe, however, that there would be a boundary layer at the “top” of the atmosphere. Thus a viscous boundary condition would be hard to justify. In addition, because the total mass of the atmosphere is nearly constant, it is reasonable to assume that mass cannot escape or enter through the “top” of the model atmosphere. Thus a pragmatic choice for the boundary condition is the standard inviscid boundary condition, v(rT , t) · n(rT ) = 0,

(1.100)

where n(rT ) is the vector normal to T at location rT . 51 For an in-depth general discussion of these points, see, e.g., Chapter 4 of Landau and Lifshitz (1987) or Sec. 41.5 of Feynman et al. (2006a). For the specific case of the atmosphere and ocean see Pedlosky (1987). 52 This approach will be discussed in detail in Sec. 1.4.

Governing Equations

1.2.6.3

43

Lateral boundary conditions

The first numerical models of the atmosphere, including the one that produced the first successful experimental numerical weather forecast,53 were all limited area models: in the early days of modeling, the available computing power was not sufficient to solve even a strongly reduced version of the atmospheric governing equations at a resolution that would have been necessary to resolve the main weather systems for the entire globe.54 While the computing power has increased many orders of magnitude since the advent of numerical modeling, both operational centers and academic research groups tend to push the envelope of model resolution on the available computational resources, which explains why limited area models are as popular as ever despite the dramatic increase in computing power. While the early limited area models used time-independent lateral boundary conditions, modern-day limited-area models use time-dependent lateral boundary conditions that are obtained by the interpolation of coarser resolution global model solutions. This approach is called nesting, because the limited area model domain is nested into the global model domain. Most limited area models have capabilities for multiple nesting, a process in which increasingly higher resolution model integrations are carried out for increasingly smaller model domains, using the model solution with the one-step lower resolution to define the time-dependent lateral boundary conditions for each nest. While most limited area models use the coarser resolution model solution of the outer domain only at the boundaries, examples exist for perturbation models, which use the coarser resolution solution of the outer domain to define the solutions at the larger scales in the inner domain as well. These models compute a high resolution perturbation to the lower resolution model fields to obtain a higher resolution solution. There are research models that also allow for two-way nesting, an approach in which the lower resolution solution of the outer nest in the inner domain is obtained by reducing the resolution of the high resolution solution available for the inner domain.55 Such an approach is justified when there is a rea53 Charney

et al. (1950). the first numerical model to study the general circulation of the atmosphere was a limited area model (Phillips, 1956). Lewis (1998) provides an analysis of the significance of Phillips’s 1956 experiment in the history of atmospheric dynamics and modeling. 55 The reduction of the resolution is usually achieved by an interpolation of the high resolution solution of the inner nest onto the lower resolution grid of the outer domain (e.g., Harris and Durran, 2010). 54 Even

44

Applicable Atmospheric Dynamics

son to believe that there is an upscale propagation of information in the inner domain. Finally, there are models that eliminate the lateral boundary conditions by using a variable spatial resolution over the globe, employing higher resolutions in the main regions of interest. At first sight, the differences between the different approaches for limited area modeling may seem to be engineering details. In reality, the underlying scientific issues are among the most fundamental open scientific problems of atmospheric dynamics. When the higher resolution solution cannot affect the lower resolution solution, it is assumed implicitly that the processes at the smaller scales are slaved to the processes at the larger scale; that is, the role of the higher resolution solution is to fill in details at the small scales. The purest form of a practical implementation of this philosophy is a perturbation model. When the coarser resolution solution of the outer domain constrains the higher resolution solution only at the lateral boundaries, there are some scales that are resolved by both the coarse and the high resolution solutions. At those scales, the two solutions can be different. Which one should be trusted more? The approach of two-way nesting puts more faith into the higher resolution solution. It is important to notice, however, that there are larger scale motions that only the coarser resolution model can fully capture, because it provides a solution for a larger domain. If the motions that only the coarser resolution model can fully capture have an important effect on the motions that are resolved by both models, two-way nesting may even have a negative effect. 1.2.6.4

Initial conditions

When a model solution is sought to provide a weather forecast, the initial condition is generated based on observations of the atmosphere. Because an assessment of the atmospheric state that is based on observations has been conventionally called analysis in meteorology, a model initial condition based on observations is usually also called analysis. The modern approach to obtain an analysis is to extract information from the observations with the help of a statistical interpolation and a forecast model. This process is called data assimilation. The data assimilation component of a modern numerical weather prediction system is just as important as the model component. This explains our decision to devote an entire chapter of the present book (Chapter 4) to the topic of data assimilation. Atmospheric simulations, which are model runs to study atmospheric

Governing Equations

45

dynamics without the specific aim of forecasting weather, are also often started from an analysis: although a good model started from a simple initial condition, such as a state of rest with a stable vertical temperature profile, eventually develops realistic atmospheric motions, starting the model from a realistic initial condition significantly shortens the spin-up time, i.e., the time it takes for the model to settle on its attractor.56 Long atmospheric simulations are typically carried out by coupled atmosphere-ocean models, models in which the atmospheric and oceanic dynamics interact. When such a simulation is run for not longer than “only” a few decades, the initial conditions have been observed to have an influence on the simulated atmospheric flow for the entire time of the simulation. This may occur either because the transient time is much longer for a coupled model than for an atmospheric model, or because the model chooses different “attractors” depending on the initial conditions.57 In a limited area model, the initial and the boundary conditions must be consistent. A simple approach to achieve such consistency is to let the global model solution, which provides the lateral boundary conditions, to define the initial state for the limited area model, using the same interpolation scheme at both the boundaries and the interior of the limited area domain. Essentially all limited area models include capabilities to carry out the interpolation of global analyses to obtain the initial conditions. While the conventional approach has been to use interpolated global analyses or short-term global forecasts as the initial conditions of the limited area models, the 21st century brought a proliferation of limited area data assimilation systems. The hope has been that such systems would extract information more efficiently from those observations that are heavily influenced by processes at the scales that only the limited area model can resolve. Limited area models also often use more sophisticated parameterization packages than the global models, which may lead to a more correct interpretation of the effect of the processes at the unresolved scales on the observations. The potentially more accurate interpretation of the contribution of the smaller scale processes to the observed values of the state variables come at the expense of giving up the correct interpretation of the contribution of the global (large) scale motions that the limited area model cannot fully capture. 56 Here, we use the term “attractor” to refer to the collection of states that the model can attain after a long-time integration of the model. 57 A detailed discussion of the dynamics of coupled models is beyond the scope of the present book.

46

1.3

Applicable Atmospheric Dynamics

Representation of the Location with Coordinates

While the laws of physics represented by Eqs. (1.4)–(1.8) are independent of the choice of the three components of the vector of position r, quantitative computations cannot be carried out without the use of a particular coordinate system. Manipulating equations is also often more convenient in a wisely selected coordinate system than in the general vector form of the equations. The selection of the coordinate system determines not only the components of r, but also the components of the vector vr = dr/dt, the Eulerian representation of the total time derivative, d/dt, and the particular form of the del operator, ∇. In what follows, we discuss the coordinate systems and map projections most frequently used in models and diagnostic studies. We pay special attention to the assumptions made to derive the equations in the different coordinate systems. 1.3.1 1.3.1.1

Spherical Coordinates Vector calculus in a spherical coordinate system

Mathematical Note 1.3.1 (Formulas of Vector Calculus). Let A(r) and B(r) two arbitrary vector fields and f (r) an arbitrary scalar field. Then, ∇ × ∇f = 0,

(1.101)

∇ · (f A) = f ∇ · A + A · ∇f,

∇ × (f A) = ∇f × A + f (∇ × A) ,

∇ · (∇ × A) = 0,

∇ (A · B) = (A · ∇) B + (B · ∇) A

+ A × (∇ × B) + B × (∇ × A) ,

∇ · (A × B) = B · (∇ × A) − A · (∇ × B),

∇ × (A × B) = A (∇ · B) − B (∇ · A) − (A · ∇) B + (B · ∇) A,

2

∇ × (∇ × A) = ∇ (∇ · A) − ∇ A.

(1.102) (1.103) (1.104) (1.105) (1.106) (1.107) (1.108)

Exercise 1.8. Show that for any scalar field S, [∇H × (vH × Sk)] · k = −S (∇H · vH ) − (vH · ∇H ) S.

(1.109)

Governing Equations

47

Answer. Making use of Eq. (1.107), ∇H × (vH × Sk) = vH (∇H · Sk) − Sk (∇H · vH ) − (vH · ∇H ) Sk + (Sk · ∇H ) vH

= −Sk (∇H · vH ) − (vH · ∇H ) Sk,

(1.110)

where in the second step, two terms dropped out, because they included the dot product of the orthogonal vectors ∇H and Sk. Taking the dot product of Eq. (1.110) and k leads to Eq. (1.109) Under the assumption that the Earth is a perfect sphere, the spherical coordinates introduced in Example 1.2 provide a natural choice for the representation of global fields of the state variables. The function to compute the wind vector v from the vector vr for the spherical coordinates has already been determined in Example 1.6. Here, we present the specific form of the atmospheric governing equations in the spherical coordinate system. Combining Eqs. (1.47) and (1.33), the del operator for the spherical coordinate system can be written as ∇=

∂ ∂ 1 1 ∂ ∂ ∂ ∂ i+ j+ k= i+ j+ k. ∂x ∂y ∂z r cos ϕ ∂λ r ∂ϕ ∂r

(1.111)

Thus, the gradient of the scalar field f (λ, ϕ, r) is ∇f =

1 ∂f 1 ∂f ∂f i+ j+ k. r cos ϕ ∂λ r ∂ϕ ∂r

(1.112)

The divergence ∇ · A of the vector field58 A(λ, ϕ, r) = Ax (λ, ϕ, r)i(λ, ϕ) + Ay (λ, ϕ, r)j(λ, ϕ) + Az (λ, ϕ, r)k(λ, ϕ).

(1.113)

can be written with the help of Eq. (1.102) as ∇ · A = ∇Ax · i + ∇Ay · j + ∇Az · k + Ax ∇ · i + Ay ∇ · j + Az ∇ · k ∂Ay ∂Az ∂Ax + + + Ax ∇ · i + Ay ∇ · j + Az ∇ · k. (1.114) = ∂x ∂y ∂z Making use of Eq. (1.33), the first three terms of the last part of Eq. (1.114) can be written as 1 ∂Ax ∂Ay 1 ∂Ay ∂Az ∂Az ∂Ax = , = , = . (1.115) ∂x r cos ϕ ∂λ ∂y r ∂ϕ ∂z ∂r The computation of the last three terms on the right-hand side of Eq. (1.114) requires the computation of the divergence of the vector fields 58 Note

that none of the unit vectors depends on the spherical coordinate r.

48

Applicable Atmospheric Dynamics

(a)

(c)

i(x0+δx)

δx δλ r0cosϕ0

δix

δix

i x0

Latitude Circle (ϕ0) δix (b) α

ϕ0 k

y0

k

r0

Longitude Circle (λ0) (d) j

δλ i i(x0+δx)

Fig. 1.7

j

ϕ0 k β y δix 0 β=90o-ϕ 0

Illustration of the derivation of Eq. (1.119).

of the unit vectors, which can be different from zero, because the direction of the unit vectors depends on λ and ϕ. The divergence of the unit vectors at location r0 = (λ0 , ϕ0 , r0 ) can be most conveniently computed in the local Cartesian coordinate system at r0 . To simply notation, in the present discussion, the unit vectors i, j and k denote the unit vectors for the local Cartesian coordinates at r0 . Hence, the vector field i(λ, ϕ) in the neighborhood of r0 can be written as i(λ, ϕ) = ix (λ, ϕ)i + iy (λ, ϕ)j + iz (λ, ϕ)k,

(1.116)

and the divergence of i(λ, ϕ) as ∂ix ∂iy ∂iz + + . (1.117) ∂x ∂y ∂z The change in the direction of the zonal unit vector i over a distance δx along the latitude circle that passes through r0 (Fig. 1.7, panel a) is ∇·i=

δix = i(x0 + δx) − i(x0 ),

(1.118)

where x0 is the location on the latitude circle at r0 . When δx goes to zero, it can be considered an infinitesimal change in the local Cartesian coordinate x at r0 . In addition, the partial derivatives of the components of i satisfy ∂ix ∂iy ∂iz δix = i+ j+ k. (1.119) lim δx→0 δx ∂x ∂x ∂x

Governing Equations

49

The partial derivatives of the components of i can be determined by computing first the magnitude and then the direction of lim δi/δx. δx→0

As δx goes to zero, the sector of the circle associated with arc δx becomes a triangle (Fig. 1.7). By similarity of this triangle and the triangle shown in panel b of Fig. 1.7, δix 1 (1.120) δx = r cos ϕ ,

because |i(x0 )| = |i(x0 + δx)| = 1. Hence, |δix | = (δx/r cos ϕ). In addition, as δx goes to zero, δλ also goes to zero, leading to 180◦ − δλ = 90◦ . δλ→0 2

lim α = lim

δx→0

(1.121)

Equation (1.121) shows that as δx goes to zero, δix turns into the direction perpendicular to i, pointing toward the Earth’s axis of rotation. Thus, δix lies in the plane spanned by j and k (panel c of Fig. 1.7). According to panel d of Fig. 1.7, lim δix =

δx→0

δx δx [cos (π/2 − ϕ)j − cos ϕk] = (tan ϕj − k) . r cos ϕ r

(1.122)

Dividing Eq. (1.122) by δx, 1 1 1 δix = (sin ϕj − cos ϕk) = tan ϕj − k. δx→0 δx r cos ϕ r r lim

(1.123)

Comparing Eqs. (1.123) and (1.119) yields ∂iy 1 = tan ϕ ∂x r

∂ix = 0, ∂x

∂iz 1 =− . ∂x r

(1.124)

Since the direction of the vector field i(λ, ϕ) does not change in the meridional and the vertical directions, lim

δy→0

δi = 0, δy

lim

δz→0

δi = 0, δz

(1.125)

which immediately leads to ∂iy ∂iz ∂ix = = = 0, ∂y ∂y ∂y

∂ix ∂iy ∂iz = = = 0. ∂z ∂z ∂z

(1.126)

Substituting the appropriate partial derivatives from Eqs. (1.124) and (1.126) into Eq. (1.117), the divergence of i is ∇·i=

∂ix ∂iy ∂iz + + = 0, ∂x ∂y ∂z

(1.127)

50

Applicable Atmospheric Dynamics

because according to Eqs. (1.124) and (1.126), all three partial derivatives in Eq. (1.127) are zero. Therefore, the fourth term of the last part of Eq. (1.114), Ax ∇ · i, is also zero. The derivation of the fifth and the sixth terms of the last part of Eq. (1.114), Ay ∇·j and Az ∇·k, requires the computation of the divergence of the vector fields of the unit vectors j and k. Intuition suggests that the divergence of these two vector fields is not zero, as the vector field j diverges moving from the South Pole to the Equator and converges moving from the Equator toward the North Pole, while the vector field k diverges moving away from the Earth’s center. Exercise 1.9. Show that the only nonzero partial derivatives of the components of j and k are ∂jx 1 ∂jz 1 = − tan ϕ, =− (1.128) ∂x r ∂y r and ∂kx 1 ∂ky 1 = , = . (1.129) ∂x r ∂y r Answer. These partial derivatives can be obtained by following a procedure similar to the one that was used for the computation of the partial derivatives of the components of i. From Eq. (1.128), ∇·j=

∂jy ∂jz 1 ∂jx + + = − tan ϕ, ∂x ∂y ∂z r

(1.130)

while from Eq. (1.129), ∂ky ∂kz 2 ∂kx + + = . (1.131) ∂x ∂y ∂z r Substituting ∇·j from Eq. (1.130) and ∇·k from Eq. (1.131) into Eq. (1.114) yields 1 ∂Ay ∂Az 1 2 1 ∂Ax + + − tan ϕAy + Az , (1.132) ∇·A= r cos ϕ ∂λ r ∂ϕ ∂r r r ∇·k=

Exercise 1.10. Show that the curl of the vector field A(λ, ϕ, r) is   ∂Ay 1 1 ∂Az − − Ay i ∇×A = r ∂ϕ ∂r r   1 ∂Az 1 ∂Ax − + Ax j + ∂r r cos ϕ ∂λ r   1 ∂Ax 1 1 ∂Ay − + tan ϕAx k, + (1.133) r cos ϕ ∂λ r ∂ϕ r

Governing Equations

51

Answer. First substituting A from Eq. (1.113) into the left-hand side of Eq. (1.133), then making use of Eq. (1.103) yields ∇ × A = ∇ × Ax i + ∇ × Ay j + ∇ × Az k

= ∇Ax × i + Ax (∇ × i) + ∇Ay × j + Ay (∇ × j)

+ ∇Az × k + Az (∇ × k).

(1.134)

The cross products in the last part of Eq. (1.134) can be expanded as ∂Az ∂Ax 1 ∂Az 1 ∂Ax ∇Ax × i = j− k= j− k, (1.135) ∂z ∂y r ∂r r ∂ϕ ∇Ay × j =

∂Ay ∂Ay 1 ∂Ay ∂Ay k− i= k− i, ∂x ∂z r cos ϕ ∂λ ∂r

(1.136)

∂Az 1 ∂Az 1 ∂Az ∂Az i− j= i− j, (1.137) ∂y ∂x r ∂ϕ r cos ϕ ∂λ   ∂iy ∂iz 1 1 Ax (∇ × i) = Ax k− j = tan ϕAx k + Ax j, (1.138) ∂x ∂x r r ∇Az × k =

Ay (∇ × j) = Ay Az (∇ × k) = 0.

∂jz 1 i = − Ay i, ∂y r

(1.139) (1.140)

Substituting the expansion of the cross products from Eqs. (1.135)–(1.140) into Eq. (1.134) yields Eq. (1.133). Exercise 1.11. Show that the Lagrangian time derivative of the scalar f (λ, ϕ, r) is ∂f 1 ∂f 1 ∂f ∂f df = + u + v +w . dt ∂t r cos ϕ ∂λ r ∂ϕ ∂r Answer. Equation (1.141) can be obtained by substituting the specific form of the del operator from Eq. (1.111) into the right-hand side of Eq. (1.45), then making use of Eq. (1.32). Exercise 1.12. Show that the Lagrangian time derivative of the vector A(λ, ϕ, r) is   dAx 1 1 dA = − tan ϕuAy + uAz i dt dt r r   1 1 dAy + tan ϕuAx + vAz j + dt r r   1 1 dAz − uAx − vAy k. + (1.141) dt r r

52

Applicable Atmospheric Dynamics

Answer. The most general form of the Lagrangian time derivative is given by Eq. (1.46). The computation of the last three terms on the right-hand side of Eq. (1.46) requires the computation of the Lagrangian time derivative of the vector fields of i, j and k. The Lagrangian time derivative of i can be computed by making use of Eqs. (1.123) and (1.125) as di ∂i di di di 1 1 = +u +v +w = u tan ϕj − u k. (1.142) dt ∂t dx dy dz r r Likewise, ∂j dj dj dj 1 1 dj = +u +v +w = −u tan ϕi − v k (1.143) dt ∂t dx dy dz r r and dk ∂k dk dk dk 1 1 = +u +v +w = u i + v j. (1.144) dt ∂t dx dy dz r r Substituting di/dt, dj/dt and dk/dt from Eqs. (1.142)–(1.143) into Eq. (1.46) yields   dA dAx 1 1 = − Ay u tan ϕ + Az u i dt dt r r   dAy 1 1 + + Ax u tan ϕ + Az v j dt r r   1 1 dAz − Ax u − Ay v + k. (1.145) dt r r In Eq. (1.141), the terms dAx /dt, dAy /dt and dAz /dt can be further expanded with the help of Eq. (1.141). Example 1.8. The Lagrangian time derivative of the wind vector can be obtained by substituting v for A in Eq. (1.145), which leads to   du vu wu dv = − tan ϕ + i dt dt r r   wv dv u2 + tan ϕ + j + dt r r   v2 dw u2 + − + k. (1.146) dt r r 1.3.1.2

The governing equations for spherical coordinates

Using the results of Sec. 1.3.1.1, the system of governing equations Eq. (1.4)–(1.8) can be written as du uv uw 1 ∂p − tan ϕ + =− + f v − lw, (1.147) dt r r ρr cos ϕ ∂λ

Governing Equations

vw 1 ∂p dv u2 + tan ϕ + =− − f u, dt r r ρr ∂ϕ dw u2 + v 2 1 ∂p − =− − g + lu, dt r ρ ∂r   dρ v 1 ∂u 1 ∂v 2w ∂w + tan ϕ − = −ρ + + , dt r r r cos ϕ ∂λ r ∂ϕ ∂r

53

(1.148) (1.149) (1.150)

R T dp 1 dT Q, − = dt cp p dt ρcp

(1.151)

1 dq = − M. dt ρ

(1.152)

In these equation, all state variables, parameters, and source and sink terms are considered functions of the spherical coordinates. The terms proportional 1/r on the left-hand side of Eqs. (1.147) and (1.150) are called the curvature terms. Equations (1.147)–(1.152) become singular at the two poles due to the presence of the coefficient tan ϕ in some of the terms.59 This singularity occurs because at the poles the distinction between the zonal and the meridional directions becomes ambiguous.60 The singularity of the spherical coordinates at the poles is due to our choice of the coordinates, because there is no related singularity in the physical model, or for that matter, in nature. The singularity of the horizontal spherical coordinates is part of the so-called pole problem. 1.3.1.3

Scale analysis of the components of the momentum equations in spherical coordinates

The vertical coordinate r of the spherical coordinate system can be written as r = a + z,

(1.153)

where a is the Earth’s radius and z is the geopotential height. Since our focus is on motions for which z . 10 km, the relative error of the approximation

59 At

90◦ .

r≈a

(1.154)

the two poles, |ϕ| = an observer standing at the North Pole, any other location on the globe is to the south, while for an observer standing at the south pole, any other location is to the north. 60 For

54

Applicable Atmospheric Dynamics Table 1.2 Scale estimates of the curvature terms. The estimate tan ϕ / 1 is used for latitudes lower than about 80◦ (|ϕ| / 80◦ ). Term

Scale estimate

Scale estimate for ǫRo ∼ 10−1 s−2

(uv/a) tan ϕ

U 2 /a

∼ 10−5

(uw/a)

U W/a

∼ 10−8

(u2 /a) tan ϕ

U 2 /a

∼ 10−5

(vw/a)

U W/a

∼ 10−7

(v/a) tan ϕ

U/a

∼ 10−8

2w/a

W/a

∼ 10−9

is less than 1%. The vertical derivative of a state variable f can be written, without making any approximation, as ∂f ∂f = . ∂r ∂z

(1.155)

In the modern terminology of atmospheric modeling, the governing equations obtained by (i) replacing r by a in all terms of the atmospheric governing equations where it appears undifferentiated, and (ii) dropping the Coriolis force terms that include the parameter l = 2Ω cos ϕ, are called the shallow atmosphere equations. In addition, the original equations are called the deep atmosphere equations. Making use of the shallow atmosphere approximation61 and Eq. (1.155), the del operator can be written as ∇=

1 1 ∂ ∂ ∂ i+ j+ k. a cos ϕ ∂λ a ∂ϕ ∂z

(1.156)

Introducing this form of the nabla operator simplifies many terms of the governing equations. In addition, some terms of the equations can be eliminated by a scale analysis of the resulting equations. The scale estimates of the curvature terms obtained by replacing r by a are listed in Table 1.2. In the two components of the horizontal momentum equation, the scale of the leading order curvature terms, (1/a) tan ϕuv and (1/a) tan ϕu2 , is two orders of magnitude smaller than the scale of the pressure gradient 61 The shallow atmosphere approximation should not be confused with the shallow water approximation discussed in Sec. 1.5.5.8.

Governing Equations

55

force terms, 1/ (ρa cos ϕ) ∂p/∂λ and 1/(ρa)∂p/∂ϕ, and the Coriolis force terms, f v and f u. Hence, the lowest order approximation to the horizontal momentum equation in spherical coordinates is the geostrophic balance equation, Eq. (1.70), except that now the horizontal component of the del operator is ∇H =

1 1 ∂ ∂ i+ j. r cos ϕ ∂λ r ∂ϕ

(1.157)

For large and synoptic scale motions, the scale of the Lagrangian acceleration terms, du/dt and dv/dt, is also an order of magnitude larger than the scale of (1/a) tan ϕuv and (1/a) tan ϕu2 . Thus, the next low order approximation to the horizontal momentum equation, which retains all terms whose scale is 10−4 or larger, is 1 ∂p du =− + f v, dt ρa cos ϕ ∂λ

(1.158)

1 ∂p dv =− − f u, dt ρa ∂ϕ

(1.159)

where the Lagrangian time derivative of the two coordinates of the horizontal wind vector can be computed by Eq. (1.141). The next approximation, which retains all terms whose scale is 10−5 , also includes the terms (1/a) tan ϕuv and (1/a) tan ϕu2 .62 The resulting equations are du uv 1 ∂p − tan ϕ = − + f v, dt a ρa cos ϕ ∂λ dv u2 1 ∂p + tan ϕ = − − f u, dt a ρa ∂ϕ
dw 1 2 1 ∂p − − g, u + v2 = − dt a ρ ∂z   dρ 1 ∂u 1 ∂v ∂w = −ρ + + . dt a cos ϕ ∂λ a ∂ϕ ∂z

(1.160) (1.161) (1.162) (1.163)

Of course, the Lagrangian time derivatives must be computed by using the del operator defined by Eq. (1.156. This rule applies not only to Eqs. (1.160)–(1.163), but also to the first law of thermodynamics, Eqs. (1.151), and the constituent equations, Eq. (1.152), which are otherwise not affected by the shallow atmosphere approximation. 62 The

next largest term in the governing equations is −lw, with a magnitude of ∼ 10−6 .

56

Applicable Atmospheric Dynamics

Exercise 1.13. Show that the left-hand side of Eqs. (1.160) and (1.161) can be written as du uv 1 dU − tan ϕ = , dt a cos ϕ dt 1 dV u2 + v 2 dv u2 + tan ϕ = + tan ϕ, dt a cos ϕ dt a

(1.164) (1.165)

where U = u cos ϕ and V = v cos ϕ. Answer. Equation (1.164) can be derived as 1 d(u cos ϕ) du dϕ 1 dU = = − u tan ϕ cos ϕ dt cos ϕ dt dt dt du uv − tan ϕ, = dt a

(1.166)

making use of v=a

dϕ ; dt

(1.167)

while Eq. (1.165) can be derived by taking advantage of 1 dV 1 d(v cos ϕ) dv dϕ = = − v tan ϕ cos ϕ dt cos ϕ dt dt dt 2 dv v − tan ϕ. = dt a

(1.168)

The vector VH = (U, V ) = cos ϕvH

(1.169)

is often called the pseudo-wind. We will follow this terminology. For completeness, we note that for the shallow atmosphere approximation, the zonal coordinate of the wind vector is u=

1 dλ . a cos ϕ dt

(1.170)

The transformation that replaces the horizontal wind vector by the pseudowind with the help of Eqs. (1.164) and (1.164) plays an important role in global atmospheric modeling.63 63 This transformation was introduced by Robert (1966). Its role in global atmospheric modeling will be discussed in Sec. 3.3.3.

Governing Equations

1.3.1.4

57

Diagnostic calculations in spherical coordinate system

Gridded atmospheric data sets almost always use spherical coordinates for the representation of the scalar and the vector fields of the state variables in the horizontal direction. These data sets include the reanalysis data sets, all real time numerical weather prediction products, climate simulations and most model based research data sets. For instance, global data sets are typically provided using a grid spacing of 2.5◦ ×2.5◦ , 1.0◦ ×1.0◦ , 0.5◦ ×0.5◦ , or 0.25◦ × 0.25◦ .64 Example 1.9. Let fi,j , i = 1, . . . , I, j = 1, . . . , J, a discretization of the scalar field f (λ, ϕ) on the globe. For a 2.5◦ × 2.5◦ grid, the number of grid points in the zonal direction is 144, while the number of grid points in the meridional direction is 73. In most data sets, the indexes run from west to east, with i = 1 referring to points at the Prime (Greenwich) Meridian, and from south to north, with j = 1 referring to points at the South Pole.65 Note. It is usually more convenient to number the longitudes from 0◦ to 360◦ than to jump from east to west longitudes at the Dateline on a discrete grid. Likewise, latitudes can be numbered from −90◦ at the South Pole to 90◦ at the North Pole. Using this numbering convention, the spherical coordinates λi and ϕj associated with the grid point (λi ,ϕj ) can be computed as λi = (i − 1)∆λ and ϕj = (j − 1)∆ϕ − 90◦ , where ∆λ and ∆ϕ are the grid spacing in the zonal and the meridional directions, respectively. (In Example 1.9, ∆λ = ∆ϕ = 2.5◦ ). It is important to note that the grid used for the distribution of the output from a model is usually not the same as the computational grid of the model: the uniform resolution (in spherical coordinates) data sets are produced by the post-processing step of the model integrations after all numerical calculations have been completed. In general, the resolution of the “post processing” grid is significantly lower than the nominal resolution of the computational grid, which is sometimes referred to as the native grid of the model. The primary motivation to reduce the resolution of the data sets during post-processing is to account for the difference between the formal resolution and the effective resolutions of the models. A detailed discussion of the difference between these two resolutions will be one of the key topics of Ch. 3. Here, we only note that, in practical terms, the effective resolution 64 In

the vertical direction, the discretization of the fields is usually done using the pressure vertical coordinate, which will be introduced in Sec. 1.4.2. 65 Note that at the two poles (j = 1 and j = 73), f i,j is the same for all values of i.

58

Applicable Atmospheric Dynamics

is defined by the smallest scale motions for which the nonlinear interactions can be efficiently represented by the model. The effective resolution tends to be significantly lower than the nominal resolution. Due to the significant difference between the effective and the nominal resolution, transmitting and storing the model fields at their nominal resolution would be a waste of bandwidth and storage space. In addition, the modeling centers are in a much better position than the end-users of the data to make a judgement about the effective resolution of the model, which depends on the numerical techniques and parameterization schemes of the models. Reducing the resolution of the data to the effective resolution prevents unexperienced users from misinterpreting the resolution of the simulations. Unfortunately, discussing model-based diagnostic results in terms of the nominal resolution of the models, without ever mentioning the effective resolution, is a common practice in our days. In reality, the only situation in which computing diagnostic quantities on the native grid of the model is justified, is when the goal is to study the model dynamics rather than the atmospheric dynamics. Since post-processed model fields tend to be spatially smooth, spatial derivatives in the diagnostic calculations can be adequately approximated by first and second order finite difference schemes. Diagnostic calculations also tend to be less sensitive than model calculations to errors introduced by the numerics, because they usually involve much fewer operations, which greatly reduces the risk of the accumulation and/or the amplification of the numerical errors. Example 1.10. The standard approach for the computation of the zonal coordinate ∇λ f of the gradient of the scalar state variable f based on a gridded field fi,j , i = 1, . . . , I, j = 1, . . . , J, is to use the centered difference scheme 1 fi+1,j − fi−1,j (∇λ f )i,j = , i = 1, . . . , I; j = 2, . . . , J − 1; a cos ϕj ∆λ (1.171) where fij is the data for f at zonal grid point i and meridional grid point j and ∆λ is the grid spacing in radii in the zonal direction.66 Because for each value of j, the index i runs along a closed latitude circle, fI+1,j = f1,j and f0,j = fI,j for j = 1, . . . , J. These relationships are necessary for the computation of the right-hand side of Eq. (1.171) for i = I and i = 1. The index j runs from 2 to J − 1 rather than from 1 to J, because at the two 66 For

instance, for a grid spacing of 2.5◦ , ∆λ = 2.5 × (π/180).

Governing Equations

59

poles cos ϕj = 0 and Eq. (1.171) cannot be used for the computation of the zonal component of the gradient. The root of this singularity is that the zonal direction is not defined at the poles.67 The approximation for the meridional gradient ∇ϕ f is (∇ϕ f )i,j =

1 fi,j+1 − fi,j−1 , a ∆ϕ

i = 1, . . . , I;

j = 2, . . . , J − 1, (1.172)

where ∆ϕ is the grid spacing in radii in the meridional direction. In this case, the formal obstacle to computing (∇ϕ f )i,j at the poles is the lack of rules to define the values of fi,j for j = 0 and j = J + 1. Exercise 1.14. Design a centered difference scheme similar to that of Eq. (1.171) for the computation of the divergence ∇ · vH of the horizontal wind vector, vH . Answer. Substituting vH for A in Eq. (1.132), and using the approximation r ≈ a yields ∇ · vH =

1 ∂u 1 ∂v 1 + − tan ϕv. a cos ϕ ∂λ a ∂ϕ a

(1.173)

Replacing the two spatial derivatives with their centered difference approximation leads to 1 ui+1,j − ui−1,j 1 vi,j+1 − vi,j−1 1 (∇ · vH )i,j = + − tan ϕj vi,j . a cos ϕ j ∆λ a ∆ϕ a (1.174) Once again, the formula we obtain cannot be used for the computation of the divergence at the poles, where the first and the third terms on the righthand side of Eq. (1.174) become singular, while the second term cannot be computed, because no rule exists to determine vi,j for j = 0 and j = J + 1. Note. The singularity of the formulas given by Eqs. (1.171), (1.172) and (1.174) at the poles is the result of the choice to describe vectors by a zonal and a meridional component. In a diagnostic calculation based on a model output, such a singularity is not a cause for concern, unless the availability of the diagnostics at the poles is important, which is rarely the case. In addition, because the gradient of the scalar fields and the divergence of the vector fields is well defined at the poles, non-singular formulas for their approximation must exist and can be used when necessary. An example for such a formula is given in Exercise 1.3.2.3. 67 See

earlier discussion on the pole problem in Sec. 1.3.1.2.

60

1.3.2

Applicable Atmospheric Dynamics

Map Projections

While spherical coordinates provide a natural choice for the representation of the atmospheric fields, their use poses a number of challenges for atmospheric modeling. Some of these challenges are related to the pole problem. Although the direct cause of the pole problem is the choice to use the local Cartesian coordinates for the representation of the vector state variables, that choice is motivated by the availability of the simple function F given by Eq. (1.32) to compute the wind vector from the spherical coordinates. Another problem, which can also be considered part of the pole problem, is that the distance associated with a given value of δλ decreases as the latitude, more precisely |ϕ|, increases. Since the stable numerical integration of the governing equations requires using a shorter time step for a shorter grid spacing, the short distance between the neighboring grid points at high latitudes makes it necessary to use a time step that is shorter than what would be required by the distance between the grid points at the lower latitudes.68 Since the length of the time step is inversely proportional to the number of elementary operations69 required to solve the equations, the short grid spacing at high latitudes leads to a significant computational overhead. While there are efficient strategies to cope with the aforementioned challenges, at least in the global setting, as demonstrated by the fact that most global numerical weather prediction and climate models use spherical horizontal coordinates, they have been considered sufficiently serious to motivate a never relenting search for alternative approaches to represent the atmospheric fields. These alternatives are based on different ideas for mapping the surface of the sphere onto the surface of another threedimensional shape, where alternative coordinates can be considered. (A three-dimensional representation of the atmospheric fields can be obtained by leaving the vertical coordinate of the spherical coordinate system unchanged by the mapping.) Map projections map the surface of the sphere on a plane either directly, or by first projecting it on the surface of another 3-dimensional shape, then rolling out the mapped image in the plane. Solving the equations with the help of a map projection have been a popular approach in numerical modeling, because a smart selection of the map projection can considerably 68 This

property of the numerical solution schemes will be discussed in detail in Chapter 3. 69 Usually defined by the number of floating point multiplications.

Governing Equations

61

simplify the representation of the governing equations and often results in a number of important advantageous properties of the spatially discretized equations. Map projections also play an important role in the visualization of the fields of the state variables.70 When the sole purpose of employing a map projection is visualization, an understanding of the particular form of the atmospheric governing equations for that map projection is not necessary. All standard software used for the visualization of atmospheric fields have convenient built-in capabilities to display gridded data sets using a number of different map projections. For such data sets, the diagnostic calculations can be carried out by using spherical horizontal coordinates; then the results can be displayed by using the most appropriate map projection, taking advantage of the built-in capabilities of the visualization software. The selection of the map projection best suited for the visualization of the fields for a given problem, however, requires an understanding of the advantages and the limitations of the map projections offered by the software. 1.3.2.1

Map factor

A map projection is a pair of continuously differentiable functions, xm = xm (λ, ϕ),

ym = ym (λ, ϕ),

(1.175)

which map the horizontal spherical coordinates, λ and ϕ, into a pair of Cartesian coordinates, x and y, in the plane. Map projections use z = r − a as the vertical Cartesian coordinate. A key parameter of a map projection is the map factor, m, which measures the ratio of the distance between a pair of nearby locations on the map and on the sphere. Formally, the map factor is defined by m2 =

d 2 rm , d 2 rH

(1.176)

where drm is the differential of the vector of position on the map and drH is the differential of the vector of position on the sphere. In the general case, m is a function of the location rH and the direction of drH . The vector drm is the image of the vector drH on the map. With the help of the local Cartesian coordinates, the vector drH can be written as drH = dxi + dyj, 70 Despite

(1.177)

the great advances in 3-dimensional visualization technology, two dimensional maps, cross-sections and diagrams have continued to be the main tools of visualization in both operational synoptic meteorology and atmospheric scientific research.

62

Applicable Atmospheric Dynamics

while its image is drm = dxm im + dym jm ,

(1.178)

where im and jm are the unit vectors on the map. Most atmospheric applications use conformal map projections, that is, map projections that preserve angles. For such map projections, the angle between drm and im is equal to the angle between drH and i, while the angle between drm and jm is equal to the angle between drH and j. Hence, the coordinates dxm and dym satisfy the condition dym dxm = = c, dx dy

(1.179)

where c is a scalar constant. With the help of Eq. (1.179), Eq. (1.178) can be written as drm = c dxim + c dyjm ,

(1.180)

The constant c can be determined by substituting Eqs. (1.177) and (1.180) into Eq. (1.176), as the resulting equation leads to m2 =

c2 d 2 x + c2 d 2 y = c2 . d2 x + d2 y

(1.181)

From Eq. (1.181), |c| = |m|, but because the direction of im and jm can always be chosen such that both m and c are positive, the relationship c = m can be used. Then, Eq. (1.179) can be written as dxm dym = =m dx dy

(1.182)

drm = m dxim + m dyjm ,

(1.183)

and Eq. (1.180) becomes

In essence, the conformal map projection of drH onto a plane can be considered a two-step process (Fig. 1.8): (i) a rotation of the horizontal unit vectors of the local Cartesian coordinates into the direction of the unit vectors on the map, which produces the rotated image drr = dxim + dyjm ,

(1.184)

(ii) a stretching of the rotated image drr by a factor of m to obtain drm .

Governing Equations

Step (ii)

Step (i)

drm

drr

drH

dxj

63

j dx m

i dx m

dxi

Fig. 1.8

yi m md

i m= dy m

i dx m

m i m=

dx m

Illustration of a conformal map projection.

It is important to note that, in general, the angle of rotation for a map projection depends on the horizontal location r = (λ, ϕ).71 Exercise 1.15. What are the formulas for the computation of the coordinates dxm and dym for the spherical coordinates? Answer. dxm = m dx = mr cos ϕ dλ,

dym = m dy = m dϕ.

(1.185)

The procedure outlined above can be applied not only to the differential of the vector of position, drH , but also to the horizontal component, AH (λ, ϕ) of any vector state variable A. Thus, denoting the image of AH (λ, ϕ) by A (xm , ym ), after the rotation of AH , Am = mAH .

(1.186)

Equation (1.186) can be written with the coordinates as Axm = mAx ,

Aym = mAy ,

(1.187)

where the coordinates are defined by AH = Ax im + Ay jm ,

(1.188)

Am = Axm im + Aym jm .

(1.189)

and

Exercise 1.16. What is the relationship between the horizontal coordinates of the wind vector on the map and the horizontal coordinates of the wind vector for the local Cartesian coordinate system and the spherical coordinate system? 71 The

vector of position on the map is rm = (xm , ym ).

64

Applicable Atmospheric Dynamics

Answer. Equation (1.185) for the differentials leads to the relationship um = mu = mr cos ϕ

dλ , dt

vm = mv = mr

dϕ , dt

(1.190)

for the horizontal coordinates of the wind vector. The primary appeal of using a conformal map projection for the representation of the fields in the model is that it provides a conceptually simple approach to replace the representation of the vector variables by a representation that uses a global (single) Cartesian coordinate system at all locations. This feature of the projection can be seen by noticing that the rotation step of the projection turns the unit vectors into the same directions at all locations. This observation motivates the general strategy of the remaining part of the present section, which is based on computing the partial derivatives with the help of Eq. (1.184) and representing the horizontal component of the vectors by Eq. (1.188). In particular, the two equations can be written as drH =

1 1 dxm im + dym jm , m m

(1.191)

AH =

1 1 Ax im + Aym jm . m m m

(1.192)

and

From Eq. (1.191), 1 1 , =m dx dxm

1 1 , =m dy dym

(1.193)

and the del operator for the coordinate system defined by the rotated unit vectors, ∇=

∂ ∂ ∂ im + jm + k, ∂x ∂y ∂z

(1.194)

can be written as ∇=m



∂ ∂ im + jm ∂x m ∂y m



+

∂ k. ∂z

(1.195)

The gradient of the scalar field f (λ, ϕ) can be computed from the mapped field f (xm , ym ) as   ∂f ∂f ∂f im + jm + k, (1.196) ∇f = m ∂x m ∂y m ∂z

Governing Equations

65

while the divergence of the vector field AH (λ, ϕ) can be computed from the mapped field A (xm , ym ) as ∂Ay ∂Az ∂Ax + + ∂x ∂y ∂z     ∂ (Aym /m) ∂Az ∂ (Axm /m) +m + . =m ∂xm ∂ym ∂z

∇·A =

(1.197)

In the first step of Eq. (1.197), the divergence is written in the Cartesian coordinate system obtained by the rotation of the unit vectors. Since this rotation removes the dependence of the direction of the unit vectors on the coordinates, those terms of Eqs. (1.114) that include the divergence of the unit vectors, i and j, no longer appear in the equation for the computation of the divergence. The second part of Eq. (1.197) shows, that employing a map projection shifts the dependence on the coordinates from the unit vectors to the scalar map factor, leading to the elimination of the curvature terms. Example 1.11. The divergence of the horizontal component of the wind vector is     ∂ (uxm /m) ∂ (vym /m) ∇ · vH = m +m . (1.198) ∂xm ∂ym Exercise 1.17. Derive the formula for the computation of the curl of the vector field AH (λ, ϕ) from the map projection Am (xm , ym ). Answer. i jm k m ∂ ∇ × A = m ∂x∂m m ∂y∂m ∂z Axm /m Aym /m Az     ∂Aym ∂Az ∂Axm ∂Azm jm − −m im + = m ∂ym ∂z ∂z ∂xm   ∂ (Aym /m) ∂ (Axm /m) k. +m − ∂xm ∂ym

(1.199)

(1.200)

Exercise 1.18. Derive the equation for the computation of the Lagrangian time derivative of the scalar field f (λ, ϕ) from the map projection fm (xm , ym ).

66

Applicable Atmospheric Dynamics

Answer.   df ∂f ∂f df ∂f ∂f 1 i+ j +w = + v · ∇f = + vm · m dt ∂t ∂t m ∂xm ∂ym dz ∂f df ∂f ∂f + vm +w . + um (1.201) = ∂t ∂xm ∂ym dz In the last part of Eq. (1.201), all terms, including the Eulerian time derivative, are computed using xm and ym as the horizontal coordinates of the vector of location. Exercise 1.19. Derive the equation for the computation of the Lagrangian time derivative of the vector field A(λ, ϕ) from the map projection Am (xm , ym ). Answer. First, writing the Lagrangian time derivative with the help of the rotated local Cartesian coordinates, taking advantage of the property of the rotated unit vectors that their Lagrangian time derivative is zero, then introducing the coordinates for the map projection, dAx dAy dAz dA = im + jm + k dt dt dt dt  1 ∂Axm ∂ (Axm /m) Axm ∂ (Axm /m) = + vm +w + um i m ∂t ∂xm ∂ym ∂z   ∂ (Aym /m) Ay ∂ (Aym /m) 1 ∂Aym + vm +w m j + um + m ∂t ∂xm ∂ym ∂z     ∂Azm Az ∂Azm 1 ∂Azm + + w m k. + vm + um (1.202) ∂t m ∂xm ∂ym ∂z Exercise 1.20. Derive the three components of the momentum equation by making use of the specific form of the differential operators given by Eqs. (1.196) and (1.202). Answer. The specific form of Eq. (1.202) for the wind vector is   ∂ (um /m) um 1 ∂um ∂ (um /m) dv + vm +w = + um i dt m ∂t ∂xm ∂ym ∂z   ∂ (vm /m) vm ∂ (vm /m) 1 ∂vm + vm +w + um + j m ∂t ∂xm ∂ym ∂z     w ∂w 1 ∂w ∂w +w + vm + + um k, (1.203) ∂t m ∂xm ∂ym ∂z

Governing Equations

67

while the pressure gradient force term can be written with the help of Eq. (1.196) as   ∂f m ∂f 1 ∂p 1 im + jm − k. (1.204) − ∇p = − ρ ρ ∂xm ∂ym ρ ∂z After the rotation of the horizontal unit vectors, according to Eq. (1.188), the zonal component of the Coriolis force term, aCZ = (f v − lw) i,

(1.205)

aCZ = (f v − lw) im ,

(1.206)

becomes

which can be written with the help of Eq. (1.190) as   1 aCZ = f vm − lw im . m

(1.207)

Likewise, the meridional component of the Coriolis force after the rotation of the unit vectors is 1 (1.208) aCM = f um jm . m While the rotation of the horizontal unit vectors does not affect the vertical component of the Coriolis force, aC V = luk, the coordinate of aC V has to be written with um rather than u. Hence, 1 aC V = lum k. (1.209) m With the help of Eqs. (1.203), (1.204), (1.207), (1.208) and (1.209), the three components of the momentum equation can be written as   1 ∂ (um /m) ∂ (um /m) um ∂um + vm +w + um m2 ∂t ∂xm ∂ym ∂z 1 1 1 ∂p + f vm − lw, (1.210) =− ρ ∂x m2 m   1 ∂ (vm /m) ∂ (vm /m) vm ∂vm + vm +w + um m2 ∂t ∂xm ∂ym ∂z 1 1 ∂p − f um (1.211) =− ρ ∂y m2   w ∂w 1 ∂w ∂w +w + vm + um ∂t m ∂xm ∂ym ∂z 1 1 ∂p − g + lum . (1.212) =− ρ ∂z m

68

Applicable Atmospheric Dynamics

Exercise 1.21. In the model equations, the variables um and vm are often replaced by the variables 1 1 U m = 2 um , Vm = 2 vm . (1.213) m m Show that this change of variables leads to the system of governing equations  
∂m2 1 dUm 2 = − Um + Vm2 dt m 2 ∂xm − 

dVm dt



m

=− −

 dw dt m   dρ dt m   dT dt m   dqi dt m 

where 

d dt



m

1 ∂p ℓ + f Vm − w, ρ ∂xm m

(1.214)


∂m2 1 2 Um + Vm2 2 ∂ym

1 ∂p − f Um , ρ ∂ym

1 ∂p − g + ℓmUm , ρ ∂z   ∂w ∂Vm ∂Um −ρ + , = −ρm2 ∂xm ∂ym ∂z   T R dp 1 Q, =− + pcp dt m ρcp =−

1 = − M, ρ

(1.215) (1.216) (1.217) (1.218) (1.219)

  ∂ ∂ ∂ ∂ 2 +w . + Vm + m Um = ∂t ∂xm ∂ym ∂z

(1.220)

From the solution of Eq. (1.214)–(1.219), the two horizontal components of the wind vector can be obtained by u = mUm , 1.3.2.2

v = mVm .

(1.221)

Mercator projection

The Mercator cylindrical projection (i) maps the sphere on the side of a circular cylinder tangent to the Equator, using the center of the Earth as the projection point; then (ii) stretches the projection in the South-North direction such that the final image preserves the angle between the projected directions.

Governing Equations

69

The projection functions are x = aλ,

y = −a ln

cos ϕ , 1 + sin ϕ

(1.222)

and the map factor is m=

1 . cos ϕ

(1.223)

The Mercator projection preserves distance at the Equator, but approaching the poles, the distortion goes to infinity.72 The Mercator projection is a highly popular map projection for the visualization of global atmospheric fields, because it provides a simple tool to map the entire globe onto the plane (Fig. 1.9). It should always be kept in mind, however, that the images of the fields are strongly distorted near the poles. When a limited area model allows for the definition of the forecast domain using different map projections, the Mercator projection is a particularly good choice for a forecast domain at low latitudes. In particular, using the Mercator projection, the forecast domain can stretch across the Equator.

Fig. 1.9 A snapshot of the global field of the pressure at mean sea level (z = 0) on a Mercator projection.

72 The poles, which are single points in reality, are mapped into a line which is equally long to the image of the Equator on the map.

70

1.3.2.3

Applicable Atmospheric Dynamics

Polar-stereographic projection

The polar-stereographic projection maps a part of the sphere onto a plane tangent to either the North or the South Pole and parallel with the plane of the Equator. When the projection plane is tangent to the North Pole, the projection is called northern polar-stereographic projection and when the plane is tangent to the South Pole, the projection is called southern polar-stereographic projection. The projection functions are x = 2a

cos ϕ cos λ , 1 + sin ϕ

y = 2a

cos ϕ sin λ , 1 + sin ϕ

(1.224)

and the map factor is m=

2 . 1 + sin ϕ

(1.225)

The polar-stereographic projection preserves the distance at the poles and magnifies the distance toward the Equator. The polar-stereographic projection can be used when the region of interest is confined to one of the hemispheres. It is a particularly useful projection to display data at high latitudes near the pole. This property of the projection is illustrated by Figure 1.10. The figure shows that the polar stereographic projection provides a more realistic depiction of the main flow features at high latitudes. For instance, the prominent high pressure feature, indicated by a closed isobar of 1032 hPa around the dateline at high latitudes in the Mercator projection, is an insignificant feature in reality, as correctly shown by the stereographic projection. When a limited area model allows for the definition of the forecast domain using different map projections, the polar stereographic projection is a particularly good choice for a forecast domain that extends into high latitudes. 1.3.3 1.3.3.1

Cartesian Coordinates The system of governing equations

We have seen that a conformal map projection introduces a global Cartesian coordinate system for the representation of the vector fields, which is identical with the coordinate system used for the representation of the location. In the resulting system of equations Eqs. (1.214)–(1.219), the curvature of the Earth is represented by a single location dependent scalar field, the map factor m. An ideal map projection would be one for which

Governing Equations

71

Fig. 1.10 A snapshot of the of the pressure at mean sea level (z = 0) for the Northern Hemisphere on (top) a Northern polar stereographic map projection and (bottom) a Mercator map projection.

m would be one at all locations: m(λ, ϕ) = m(xm , ym ) = 1. While such a map projection does not exist, we can pretend that it does by setting m to its (unattainable) optimal value of one at all locations. The resulting system of equations is du 1 ∂p =− + f v − ℓw, dt ρ ∂x

(1.226)

1 ∂p dv =− − f u, dt ρ ∂y

(1.227)

1 ∂p dw =− − g + ℓu, dt ρ ∂z

(1.228)

72

Applicable Atmospheric Dynamics

dρ = −ρ dt



∂u ∂v ∂w + + ∂x ∂y ∂z



,

(1.229)

T R dp 1 dT Q, − = dt pcp dt ρcp

(1.230)

1 dqi = − M. dt ρ

(1.231)

While this system of equations cannot be used for the quantitative prediction of the atmospheric state, it has been the basis for many reduced models of the atmosphere, some of which have played a crucial role in developing our current understanding of the dynamics of the atmosphere. Another appeal of using the same global Cartesian coordinate system for the representation of the vector state variables and the location is that the equations can be manipulated more conveniently in it than in any other coordinate system. The representation of the results in a new coordinate system can be determined by reverse engineering: first replacing all terms associated with the Cartesian representation of the del operator by the general symbol for the del operator; then substituting the particular form of the del operator for the new coordinate system. This procedure works, because the gradient, the divergence and the curl are independent of the choice of the coordinate system. 1.3.3.2

Idealized studies

The connection between reality and the model atmosphere described by Eqs. (1.226)–(1.231) is usually made by assuming that the forecast domain is a plane tangent to the Earth at the middle of the domain. In the zonal direction, the boundary conditions are typically assumed to be periodic, mimicking a situation in which full latitude circles are projected onto the plane, such that the latitude circles are parallel with the x-axis of the coordinate system. One can envision this projection as one that is similar to the Mercator cylindrical projection, except that in this case (i) the cylinder is tangent to Earth at the latitude in the middle of the model domain rather than at the Equator, and (ii) the distortion associated with Earth’s curvature is not accounted for in the equations. Accordingly, the longitudes are assumed to be parallel with the y-axis of the coordinate system. To keep the error of this approximation at an acceptable

Governing Equations

73

level, the forecast domain in the meridional direction has to be relatively narrow. Due to the much larger length of the resulting model domain in the x than in the y direction, models based on such an approximation are often called channel models. The horizontal model domain of a channel model can be visualized as a latitude band on a map obtained by the Mercator projection. Because the system of Eqs. (1.226)–(1.231), or a reduced version of it, is typically employed to study the qualitative dynamics of the atmosphere rather than quantitatively predict its state, the presence of the southern and northern boundaries poses a more difficult challenge than the limitations of the tangent plane approach. For instance, the assumption of rigid walls can distort the qualitative dynamics of the flow in the entire model domain. One approach to minimize the artificial effects of the northern and southern boundaries is to extend the model domain in the south and the north directions and introduce periodic boundary conditions at the new boundaries. While the extension of the model domain in the meridional direction stretches the tangent plane approach beyond the point where it breaks and the periodic boundary conditions are obviously unrealistic, the approach can lead to realistic results in the middle of the domain for a finite time. The simulated flow further away from the middle of the model domain in the meridional direction can be simply ignored. Some important results later discussed in Chapter 2 were obtained by using such an approach.73

1.4

Alternate Vertical Coordinates

We have hitherto used the geopotential height, z, as the vertical coordinate. The solution of many problems of atmospheric dynamics and numerical modeling can be greatly simplified by choosing a different vertical coordinate. Some important problems that belong to this group are (i) the definition of the bottom and top boundary conditions for the atmosphere; (ii) the simplification of such key terms of the governing equations as the pressure gradient force term; 73 This approach can lead to even more realistic results for simulations of mesoscale atmospheric dynamics and oceanic dynamics, where the scale of the transient motions is smaller compared to the scale of the meridional changes of the map factor for such atmospheric parameters as the Coriolis parameter or the mean temperature.

74

Applicable Atmospheric Dynamics

(iii) the transformation of the governing equations into a form that allows for an analytical investigation of some atmospheric processes; (iv) the formulation of the governing equations that leads to a more efficient representation of the atmospheric processes by the discretized form of the equations in the numerical models. Some of the coordinates discussed here play an important role in both model and diagnostic calculations, while others are primarily used only for either model or diagnostic calculations. 1.4.1

General Formulation

We introduce the coordinate transformation formulas by considering a general new vertical coordinate η.74 The specific form of the equations for the different choices of the vertical coordinate can be obtained by substituting the chosen variable for η in the general formulas. We start the derivation of the governing equations for the different vertical coordinates from the specific form of the equations for the global Cartesian coordinates. Thus the vector of position before the transformation of the equations is r = (x, y, z), and after the transformation of the equations is rη = (x, y, η). Once a physical quantity is established as a proper vertical coordinate, it can be used in place of z to obtain the governing equations for additional vertical coordinates. In the remainder of Sec. 1.4.1, we refer to the two coordinate systems as the z coordinate system and the η coordinate system. To obtain the spherical form of the governing equation with the new vertical coordinate, the horizontal coordinates of the vectors of position r and rη can be replaced by the horizontal spherical coordinates. The same approach can be used to obtain the equations with the new vertical coordinate for a map projection. Not all physical quantities can be used as a vertical coordinate. In particular, the scalar field of η must satisfy the following conditions in order to be proper vertical coordinate: (i) the function η(r, t) is known, (ii) the function η(r, t) is continuously differentiable, and 74 This approach was first suggested by Kasahara (1974). An update to that paper, which also considered the case of non-hydrostatic and deep atmosphere equations, was provided by Staniforth and Wood (2003).

Governing Equations

75

(iii) the function η(z) is one-to-one at all horizontal locations in the model domain at all times. Since the first two conditions are satisfied by most atmospheric state variables, the critical condition is the third one. This condition can be satisfied by choosing η such that η(z) is a monotonic function of z. 1.4.1.1

Transformation formulas

We introduce the notation r4 for the four-dimensional vector composed of the three Cartesian coordinates and time: r4 = (r, t). In addition, we define rη4 for the η coordinates by rη4 = (rη , t). The four components of the function F(r4 ) = rη4 are Fx : xη = x, Fy : yη = y, Fη : η = η(x, y, z, t), Ft : t = t.

(1.232)

The motivation to introduce the subscript η into the notation of the independent variables in the η coordinate system is the following: in the z coordinate system, ∂z ∂z = = 0, ∂x ∂y

(1.233)

because x, y, and z are all considered independent variables, but in the η-coordinate system, ∂z 6= 0, ∂xη

∂z 6= 0, ∂yη

(1.234)

because z is no longer considered independent of x and y. Likewise, ∂η ∂η = = 0, ∂xη ∂yη

(1.235)

but ∂η 6= 0, ∂x

∂η 6= 0. ∂y

(1.236)

Let f (r4 ) be the function that represents the scalar field f in the z coordinate systems and fη (rη4 ) the function that represents the same scalar field in the η coordinate system. The function fη (rη4 ) can be written as fη (rη4 ) = fη [F(r4 )] = fη ◦ F(r4 ),

(1.237)

76

Applicable Atmospheric Dynamics

where fη ◦ F denotes the composite of the functions fη and F. Since f and fη ◦ F are two different forms of the same function, f = fη ◦ F.

(1.238)

Introducing the notations ∂f = ∂r4 and ∂fη = ∂rη4





∂f ∂f ∂f ∂f ∂x ∂y ∂z ∂t



∂fη ∂fη ∂fη ∂fη ∂xη ∂yη ∂η ∂t

(1.239) 

,

(1.240)

the relationship between the partial derivatives of f and fη can be determined by taking the derivative of both sides of Eq. (1.238) with respect to r4 , which yields ∂f ∂fη ∂F = , ∂r4 ∂rη4 ∂r4

(1.241)

where ∂F/∂r4 is the Jacobi matrix of F(r4 ). Taking the derivative of F defined by Eq. (1.232) yields   1 0 0 0 0 1 0 0   ∂F/∂r4 =  ∂η ∂η ∂η ∂η  . (1.242)  ∂x ∂y ∂z ∂t  0 0 0 1 Thus the matrix form of Eq. (1.241) is 

∂f ∂f ∂f ∂f ∂x ∂y ∂z ∂t



=



∂fη ∂fη ∂xη ∂yη

 1 0 0 0   ∂fη ∂fη  0 1 0 0  ∂η ∂η ∂η ∂η  , ∂η ∂t  ∂x ∂y ∂z ∂t  0 0 0 1 

which can also be written as ∂f ∂fη ∂fη ∂η + = , ∂x ∂xη ∂η ∂x

(1.243)

(1.244)

∂fη ∂η ∂fη ∂f + = , ∂y ∂yη ∂η ∂y

(1.245)

∂fη ∂η ∂f = , ∂z ∂η ∂z

(1.246)

∂fη ∂fη ∂η ∂f = + . ∂t ∂t ∂η ∂t

(1.247)

Governing Equations

77

An additional set of equations for the partial derivatives can be obtained by taking advantage of fη = f ◦ F−1 ,

(1.248)

where the components of the F−1 (rη4 ) inverse of F(r4 ) are Fx−1 : x = xη , Fy−1 : y = yη , Fη−1 : z = z(x, y, η, t), Ft−1 : t = t.

(1.249)

Taking the derivative of both sides of Eq. (1.248) with respect to rη4 leads to ∂fη ∂f ∂F−1 = . ∂rη4 ∂r4 ∂rη4

(1.250)

The matrix form of Eq. (1.250) is 

∂fη ∂fη ∂fη ∂fη ∂xη ∂yη ∂η ∂t



=



∂f ∂f ∂x ∂y

∂f ∂z



1  ∂f   0  ∂z ∂t  ∂x η 0

which can also be written as ∂f ∂f ∂z ∂fη = , + ∂xη ∂x ∂z ∂xη

 0 0 0 0  ∂z ∂z ∂z  , (1.251) ∂yη ∂η ∂t  0 0 1 0 1

(1.252)

∂f ∂f ∂z ∂fη = , + ∂yη ∂y ∂z ∂yη

(1.253)

∂f ∂z ∂fη = , ∂z ∂z ∂η

(1.254)

∂f ∂f ∂z ∂fη = + . (1.255) ∂t ∂t ∂z ∂t The transformation formulas most important for the derivation of the governing equations in the η coordinate system are ∂f ∂f ∂z ∂fη − , = ∂x ∂xη ∂z ∂xη

(1.256)

∂f ∂z ∂fη ∂f − , = ∂y ∂yη ∂z ∂yη

(1.257)

which can be obtained by rearranging Eqs. (1.252) and (1.253).

78

Applicable Atmospheric Dynamics

1.4.1.2

Transformation of the wind vector

The relationship between the components of the wind vector in the z and the η coordinate systems can also be determined with the help of the chain rule. For this calculation, we first introduce the function Frη (r4 ) = rη , which obtains the spatial coordinates in the η coordinate system from the spatial coordinates in the z coordinate system and time. The three components of Frη are the same as the first three components of F. In the η-coordinate system, the wind vector, vη , can be written as  ∂Frη dr4 drη d  vη = = . (1.258) Frη (r4 ) = dt dt ∂r4 dt The 3-by-4 Jacobi matrix ∂Frη /∂r4 of the function Frη (r4 ) can be obtained by dropping the last row of ∂F/∂r4 in Eq. (1.242). In addition,
T dr4 = uvw1 . (1.259) dt Computing the matrix-vector product on the right-hand-sides of Eq. (1.258) and equating the components of the resulting vector with the related components of
T v η = uη v η w η (1.260) yields

uη = u,

vη = v,

wη = u

∂η ∂η ∂η ∂η +v +w + , ∂x ∂y ∂z ∂t

(1.261)

where dη . (1.262) dt Equation (1.261) provides the recipe for the computation of the coordinates of the wind vector in the η coordinate system from the components of the wind vector in the z coordinate system. wη =

Exercise 1.22. Intuition suggests that the Lagrangian time derivative of a scalar state variable should be independent of the choice of the spatial coordinates, that is, df dfη = , (1.263) dt dt where, dfη ∂fη ∂fη ∂fη ∂fη = + uη + vη + wη . (1.264) dt ∂t ∂x ∂y ∂η Show that Eq. (1.263) is satisfied for any proper choice of the new vertical coordinate.

Governing Equations

79

Answer. Equation (1.263) can be verified by substituting the appropriate partial derivatives from Eqs. (1.244) into df ∂f ∂f ∂f ∂f = +u +v +w . (1.265) dt ∂t ∂x ∂y ∂z 1.4.1.3

Bottom boundary condition

According to the discussion of Sec. 1.2.6.1, the component of the wind vector perpendicular to the Earth’s surface must vanish at the bottom of the atmosphere. In the η coordinate system, this condition can be formulated with the help of the wη vertical coordinate of the wind vector. Because wη is defined by the Lagrangian time derivative of η, ∂η (ηs ) + vH (ηs ) ∇H η (ηs ) , (1.266) wη (ηs ) = ∂t where ηs (x, y, t) is the two-dimensional surface that defines the bottom of the atmosphere in the η-coordinate system. In a realistic model of the atmosphere, where viscosity is taken into account at the Earth’s surface, vH (ηs ) = 0, thus Eq. (1.266) becomes ∂η wη (ηs ) = (ηs ) . (1.267) ∂t In a model calculation, where ηs (x, y, t) changes in time, imposing the boundary condition given by either Eq. (1.266) or Eq. (1.267) is cumbersome. The only exception is the situation, where ηs is defined by a constant value of the vertical coordinate η. In that case, both Eqs. (1.266) and (1.267) become wη (ηs ) = 0.

(1.268)

A vertical coordinate η that satisfies Eq. (1.268) is called a terrain-following vertical coordinate. Almost all models use a terrain- following vertical coordinate at the bottom of the model atmosphere. 1.4.1.4

Top boundary condition

The top boundary of the model atmosphere is always defined by a constant value, ηt , of η. Hence, by analogy with the situation at the bottom of the atmosphere, the top boundary condition, dη wη (ηt ) = (ηt ) = 0, (1.269) dt is always satisfied. In addition to satisfying Eq. (1.100), the boundary condition given by Eq. (1.269) also ensures that the model does not gain or lose mass at the top of the model domain.

80

1.4.2

Applicable Atmospheric Dynamics

Pressure Vertical Coordinate

In 1949 Arnt Eliassen published a highly influential paper,75 which showed that using pressure as the vertical coordinate, the atmospheric governing equations took a form that had a number of advantageous properties. Ever since the publication of Eliassen’s paper, pressure has been the most widely used vertical coordinate in the atmospheric sciences: (i) pressure has been the favored vertical coordinate to display the atmospheric flow in synoptic meteorology; (ii) pressure and pressure-based coordinates have been the most popular choices for the vertical coordinate of hydrostatic models; (iii) three-dimensional atmospheric data sets have been distributed using pressure as the vertical coordinate; (iv) most diagnostic calculations have been carried out using pressure as the vertical coordinate. 1.4.2.1

The role of hydrostatic balance

Pressure can be used as a vertical coordinate only under the assumption that the atmosphere is in hydrostatic balance, because otherwise p(z) would not be necessarily a monotonic function of z. In particular, using z as the vertical coordinate, the hydrostatic balance equation, Eq. (1.82), can be written as ∂p = −ρg. ∂z

(1.270)

Since both ρ and g are positive, ∂p/∂z is negative, indicating a monotonic decrease of the function of p(z) with z. As discussed in Sec. 1.2.3.6, the assumption of hydrostatic balance is fully justified unless the model can resolve processes at the ∼ 1 km or smaller horizontal scales. From a theoretical point of view, the advantage of a non- hydrostatic model is that it can explicitly resolve convective processes. While experimental non-hydrostatic global models, and research and operational non-hydrostatic limited area models can provide qualitatively better representation of the convective processes than their hydrostatic cousins, it has not been shown that those qualitative improvements would lead to statistically significant quantitative improvements of forecasts. In addition, the hydrostatic approximation has the practical benefit 75 Eliassen (1949). The first paper to use pressure as the vertical coordinate to address a problem of dynamical meteorology was Sutcliffe (1947).

Governing Equations

81

that it automatically filters the sound waves from the solutions.76 While the development of non-hydrostatic models is one of the exciting current areas of research and development in atmospheric modeling, almost all operational global model products and climate simulations that are currently available for research, including the reanalysis data sets, have been produced by hydrostatic models. These data sets are distributed using pressure as the vertical coordinate. Observational data sets also use pressure as the vertical coordinate. 1.4.2.2

The governing equations

Substituting the pressure p for η in Eqs. (1.256) and (1.257) and making use of Eq. (1.235) in the resulting equations leads to ∂p ∂p ∂z , =− ∂x ∂z ∂xp ∂p ∂z ∂p . =− ∂y ∂z ∂yp

(1.271) (1.272)

Substituting ∂p/∂z from Eq. (1.270) into Eqs. (1.271) and (1.272), and making use of Eq. (1.28) yields ∂p ∂φ , =ρ ∂x ∂xp ∂φ ∂p . =ρ ∂y ∂yp

(1.273) (1.274)

The two horizontal components of the momentum equation in pressurecoordinate system, dup ∂φ + f vp , =− dt ∂xp ∂φ dvp − f up , =− dt ∂yp

(1.275) (1.276)

can be obtained by substituting Eq. (1.273) into Eq. (1.226), Eq. (1.274) into Eq. (1.227) and taking advantage of Eq. (1.263). According to Eqs. (1.275) and (1.276), in the pressure coordinate system, the pressure gradient force term is the horizontal gradient of the geopotential. Thus the pressure gradient force term is linear when pressure is used as the vertical coordinate. 76 Sound

waves are irrelevant for a meteorological forecast.

82

Applicable Atmospheric Dynamics

Mathematical Note 1.4.1 (Inverse Function Theorem). If y = f (x) is a continuously differentiable function of x with df /dx (x0 ) 6= 0, then f is invertible in a neighborhood of x0 , the inverse x = f −1 (y) is continuously differentiable and 1 df −1 (y0 ) = (x0 ) , dy df /dx

(1.277)

where y0 = f (x0 ) The hydrostatic balance equation can be obtained by rewriting Eq. (1.270) such that p replaces z as the independent variable. This change of variable can be done by applying the inverse function theorem to the function p(z): since we always assume in atmospheric dynamics that this function is continuously differentiable and according to Eq.(1.270), ∂p/∂z 6= 0 for any value of z, the inverse function theorem implies that 1 ∂p = . ∂z ∂z/∂p

(1.278)

Substituting ∂p/∂z from Eq. (1.278) into Eq. (1.270) leads to ∂φ 1 = − = −α, ∂p ρ

(1.279)

RT ∂φ =− . ∂p p

(1.280)

which can also be written as

Equation (1.279) can be obtained by rearranging Eq. (1.270), while Eq. (1.280) follows from Eq. (1.279) and the equation of state, Eq. (1.7). While the continuity equation takes a simple form in pressure coordinates, the derivation of the equation is somewhat involved. The first step is to rewrite Eq. (1.280) as 1 ∂z = −g , ρ ∂p

(1.281)

and take the time derivative of both sides of Eq. (1.281) to obtain   d ∂z 1 dρ =g . (1.282) ρ2 dt dt ∂p Substituting dρ/dt from Eq. (1.229) into Eq. (1.282) yields     d ∂z 1 ∂u ∂v ∂w + + =g . − ρ ∂x ∂y ∂z dt ∂p

(1.283)

Governing Equations

83

Substituting −1/ρg from Eq. (1.281) into Eq. (1.283) and rearranging the resulting equation gives     d ∂z ∂z ∂u ∂v ∂w + + = . (1.284) ∂p ∂x ∂y ∂z dt ∂p Using the w = dz/dt definition of the vertical coordinate of the wind for the z-coordinate system, then rearranging the resulting equation yields       d ∂z ∂ dz ∂z ∂u ∂v + − = . (1.285) dt ∂p ∂p dt ∂p ∂x ∂y Exercise 1.23. Show that the right-hand-side of Eq. (1.285) can be written in the equivalent form       ∂vp ∂z ∂ω ∂z ∂up ∂z ∂ dz d ∂z , (1.286) + + − =− dt ∂p ∂p dt ∂p ∂xp ∂p ∂yp ∂p ∂xp where the conventional notation, ω = dp/dt,

(1.287)

is used for the vertical coordinate of the wind in the pressure coordinate system rather than wp . Answer. The right-hand-side of Eq. (1.286) can be obtained by expressing the dz/dt total time derivative of the geopotential height in pressure coordinates, then substituting the resulting equation into the left-hand-side of Eq. (1.286). First substituting Eq. (1.286) into Eq. (1.285), then applying Eq (1.252) to du/dx and Eq (1.253) to dv/dy leads to     ∂up ∂z ∂z ∂u ∂v ∂vp ∂z ∂ω ∂z − = + + + (1.288) ∂p ∂xp ∂p ∂yp ∂p ∂xp ∂p ∂x ∂y   ∂up ∂z ∂vp ∂vp ∂z ∂z ∂up − + − = ∂p ∂xp ∂z ∂xp ∂yp ∂z ∂yp     ∂z ∂up ∂up ∂z ∂vp ∂vp ∂z = − . (1.289) + + ∂p ∂xp ∂yp ∂p ∂xp ∂p ∂yp Rearranging Eq. (1.288) yields   ∂vp ∂ω ∂z ∂up + + = 0. ∂p ∂xp ∂yp ∂p

(1.290)

Since ∂z/∂p is non-zero under the assumption of hydrostatic balance, Eq. (1.290) implies that

84

Applicable Atmospheric Dynamics



∂vp ∂ω ∂up + + ∂xp ∂yp ∂p



= 0.

(1.291)

Equation (1.291) is the continuity equation in pressure coordinate system. The first law of thermodynamics can be written as dT 1 TR ω = Q, − dt pcp cp

(1.292)

while the tracer equation takes the form 1 dqi = − M. dt ρ

(1.293)

In summary, the pressure coordinate system has a number of advantageous properties. In particular, (i) the top boundary of the model domain is defined by a finite value, p = pt = 0, of the vertical coordinate; (ii) the pressure gradient force term is linear; (iii) the continuity equation becomes a diagnostic equation, which states that the atmosphere behaves like an incompressible fluid in the pressure coordinate system. For all of its advantageous properties, there is a major problem with the pressure vertical coordinate: the bottom boundary of the atmosphere is defined by the location and time dependent surface pressure rather than a constant surface of pressure. This is usually not a problem in a diagnostic calculation, because the data sets typically include the values of the surface pressure for the horizontal location of all grid points in the data sets, but it makes using pressure as the vertical coordinate near the Earth’s surface in a model virtually impossible. Exercise 1.24. The del operator for pressure coordinates is ∇p = i

∂ ∂ ∂ +j +k , ∂x ∂y ∂p

(1.294)

while the wind vector is vp = ui + vj + ωk.

(1.295)

[In Eqs. (1.294) and (1.295), the subscripts were dropped from the notations of x, y, u and v to simplify notation.] Write the governing equations for the pressure coordinate system with the help of ∇p and vp ?

Governing Equations

85

Answer. The Lagrangian time derivative of a scalar state variable f is df ∂f = + vp · ∇p f, (1.296) dt ∂t while the Lagrangian time derivative of the horizontal wind vector vH is ∂vH dvH = + (vp · ∇p ) vH . (1.297) dt ∂t Thus the horizontal momentum equation, whose two components are Eqs. (1.275) and (1.276), can be written as dvH = −∇H φ − f k × vH , (1.298) dt where the Lagrangian time derivative can be computed by Eq. (1.297). The continuity equation, Eq. (1.291), takes the simple form ∇p vp = 0,

(1.299)

∂ω = −∇H vH . ∂p

(1.300)

which is usually written as

The first law of thermodynamics, Eq (1.292), and the constituent equation, Eq. (1.8) do not change, except that the Lagrangian time derivatives are defined by Eq. (1.296). Exercise 1.25. Write the formula for the computation of the geostrophic wind with the help of ∇p and vp ? Answer. From Eq. (1.298), the geostrophic balance equation for pressure coordinates is f k × vH = −∇H φ.

(1.301)

By analogy with the solution of Exercise 1.5, the geostrophic wind is 1 vg = k × ∇H φ. (1.302) f For a different choice of the horizontal coordinates, the governing equations can be obtained by replacing the horizontal coordinates of ∇p and vp with the appropriate choices of the coordinates for that coordinate system. Example 1.12. The del operator for the coordinate system that uses spherical horizontal coordinates and pressure vertical coordinate is ∇p =

1 1 ∂ ∂ ∂ i+ j+ k, r cos ϕ ∂λ r ∂ϕ ∂p

(1.303)

86

Applicable Atmospheric Dynamics

while the wind vector for the same coordinate system is ∂ϕ ∂ ∂λ j+ k. (1.304) vp = r cos ϕ i + r ∂t ∂t ∂p The computation of r by Eq. (1.153), however, becomes problematic, because z is now a dependent variable that has to be computed from p. Hence, the spherical horizontal coordinates and the pressure vertical coordinate together are used only with the shallow atmosphere approximation, because in that case, Eqs. (1.303) and (1.304) become 1 1 ∂ ∂ ∂ ∇p = i+ j+ k, (1.305) a cos ϕ ∂λ a ∂ϕ ∂p and ∂λ ∂ϕ ∂ vp = a cos ϕ i + a j + k. (1.306) ∂t ∂t ∂p Equations (1.303) and (1.304) are not more complicated than the related formulas for the z coordinate system. Example 1.13. The del operator for the coordinate system that uses a conformal map projection with a map factor m for the horizontal representation of the fields and pressure as the vertical coordinate is   ∂ ∂ ∂ im + jm + k, (1.307) ∇p = m ∂x m ∂y m ∂p while the wind vector for the same coordinate system is vm ∂ um im + r jm + k. (1.308) vp = m m ∂p 1.4.2.3

Extension to the non-hydrostatic case: hydrostatic pressure vertical coordinate

The pressure vertical coordinate has a simple generalization, which can be used even with the non-hydrostatic form of the governing equations. In particular, the hydrostatic pressure, Z ∞ ρ(x, y, z ′ , t)g dz ′ , (1.309) h= z

is always a monotonically decreasing function of height, regardless of whether or not the atmosphere is in hydrostatic balance. Hence, h can be used as a vertical coordinate even if the non-hydrostatic form of the equations is considered.77 Under the assumption that the atmosphere is 77 This property of the hydrostatic pressure was first observed by Laprise (1992), who also gave a detailed analysis of the related system of governing equations.

Governing Equations

87

in hydrostatic balance, the pressure and the hydrostatic pressure are equal (h = p), hence the two coordinates are the same. In the non-hydrostatic case, however, the pressure can be different from the hydrostatic pressure (h 6= p), thus it behaves like any other dependent variable when hydrostatic pressure is used as the vertical coordinate.78 1.4.2.4

Atmospheric data sets

The fields of the state variables are usually included at the standard pressure levels, which are 1000 hPa, 925 hPa, 850 hPa, 700 hPa, 600 hPa, 500 hPa, 400 hPa, 300 hPa, 250 hPa, 200 hPa, 150 hPa, 100 hPa, 70 hPa, 50 hPa, 30 hPa, 20 hPa, 10 hPa. The standard pressure levels provides a good representation of the vertical structure of the atmospheric fields from the surface to the top of the lower stratosphere.79 The newer products include information at additional layers, some of which are added to extend the data coverage to higher altitudes.80 The pressure value pt that defines the top of the model atmosphere is usually much lower than the lowest pressure value included in the data sets. Some data sets also include the fields at the native vertical levels of the model.81 With the exception of the 600 hPa, 30 hPa and 20 hPa levels, the standard pressure levels are also mandatory pressure levels, where all radiosondes are required to report measurements. The availability of observations for each radiosonde at the mandatory levels is an invaluable feature of the radiosonde observations, as the collection of these observations provides a sufficiently large data set to obtain statistically significant verification results for the model forecasts at those levels. There are substantial differences between a model-based data set and an observational data set. In particular, analyses inherit many properties of the model used in the data assimilation process. For instance, analysis fields cannot show spatial variability at scales that the model does not resolve. These effects of the model on the analyses can be considered part of a deliberate filtering of the effects of observational noise from the state estimates. While this filtering process is thought to be beneficial in 78 The system of governing equations, as well as a detailed discussion of the equations for the hydrostatic pressure vertical coordinate h can be found in Laprise (1992). 79 The typical geopotential height at 10 hPa is ≈ 30 km. 80 for instance, the ERA-Interim reanalysis data set. 81 For instance, in the ERA Interim data set the top model level is 0.1 hPa. The data set made available to the research community includes data at 37 pressure levels and 60 model levels. The model levels are given by the hybrid sigma-pressure coordinate described in Sec. 1.4.5. The top of the model atmosphere is at 0.1 hPa.

88

Applicable Atmospheric Dynamics

Geopotential Height at Surface (2.5o Resolution)

Geopotential Height at Surface (1o Resolution)

Surface Pressure (2.5o Resolution)

Surface Pressure (1o Resolution)

Fig. 1.11 (top) The geopotential height field z(λ, ϕ) at the Earth’s surface and (bottom) a snapshot of the surface pressure ps in the Northern Hemisphere. The fields are shown for the same time as in Fig. 1.10 at both (left) 2.5◦ resolution and (right) 1◦ resolution.

general,82 some effects of the model on the analyses are obviously artificial. For instance, the analyses are consistent with the model orography rather than true orography: because the model orography is a reduced resolution representation of the real orography, the valleys are shallower, the ridges are less sharp, and the peaks are lower in the model than in reality (top panels of Fig. 1.11). To help the correct interpretation of the analysis fields, 82 The data assimilation process also fills data voids where direct observations of the state variables of interest are not available.

Governing Equations

89

most analysis and forecast data sets include the field of the geopotential height of the model orography. 1.4.2.5

The computation of ω and the concept of atmospheric balance

Rearranging Eq. (1.291) yields   ∂ω ∂u ∂v =− + , (1.310) ∂p ∂x ∂y which then can be integrated between pressure levels p1 and p2 , p2 > p1 , to obtain  Z p2  ∂u ∂v ω(p2 ) − ω(p1 ) = − + dp. (1.311) ∂x ∂y p1

Because the vertical mean, hf ipp21 , of a scalar-valued function f (p) in the layer between the pressure levels p1 and p2 can be computed as Z p2 1 p2 hf ip1 = f dp′ , (1.312) p2 − p1 p 1

Eq. (1.311) can be written in the equivalent form  p ∂u ∂v 2 ω(p2 ) − ω(p1 ) = −(p2 − p1 ) + ∂x ∂y p1

(1.313)

Knowing either ω(p2 ) or ω(p1 ) and the vertical average of the horizontal divergence between the two levels, the value of ω at the other level can be computed by Eq. (1.313). For instance, the value of ω at the surface is  p ∂u ∂v s ω(ps ) = −(ps − pt ) + , (1.314) ∂x ∂y pt where we made use of the boundary condition ω(pt ) = 0 at the top of the model atmosphere. The value of ω at pressure level p can be computed by replacing ps with p in Eq. (1.314). Imposing the viscous boundary conditions given by Eq. (1.267) at the surface, Eq. 1.314 can be written as  p ∂u ∂v s ∂p ∂ps = (ps ) = ω(ps ) = −(ps − pt ) + . (1.315) ∂t ∂t ∂x ∂y pt When the surface pressure tendency is negative (the surface pressure is dropping), the motion at the surface is upward83 and the vertical mean of 83 Unlike in the z coordinate system, where a positive values of w indicates a rising motion, in the pressure coordinate system a positive value of wp = ω indicates a sinking motion, because the pressure is a monotonically decreasing function of height.

90

Applicable Atmospheric Dynamics

the divergence must be positive according to Eq. (1.315). Hence, a drop of the surface pressure is accompanied by a dominantly divergent horizontal motion aloft. Likewise, increasing pressure at the surface is accompanied by a dominantly convergent horizontal motion at the higher levels. Finally, it is important to note that Eq. (1.315) does not hold in the inviscid case, where Eq. (1.267) must be replaced by Eq. (1.266), which states that the advection of pressure at the surface can also contribute to the changes in the surface pressure. While Eq. (1.311) provides a straightforward approach to compute ω at pressure level p in a model, Eq. (1.315) is famously sensitive to uncertainties in the estimates of the derivatives ∂u/∂x and ∂v/∂y, which makes it a useless formula for the computation of ω or ∂ps /∂t in a diagnostic calculation. Example 1.14. The sensitivity of the calculation of ω can be demonstrated by rewriting Eq. (1.315) as  ps  ps ∂u ∂ps ∂v 1 = + . (1.316) − ps − pt ∂t ∂x pt ∂y pt We assume that at the large and synoptic scales in the extratropics, pt ≈ 200 hPa, where the boundary condition ω(pt ) = 0 is satisfied.84 In addition, we assume that ps = 1000 hPa. Using the usual values of the scale estimates U and L for large and synoptic scale motions in the layer below pt ≈ 200 hPa yields  ps  ps ∂u ∂v 1 10 m/s U = = 10−5 . (1.317) ∼ ∼ ∂x pt ∂y pt L 106 m s Thus the right-hand side of Eq. (1.316) is the sum of two terms with an order of magnitude of 10−5 1/s. For synoptic scale motions, the magnitude of the pressure tendency is |∂p/∂t| . 1 hPa/h ≈ 2.7 × 10−4 . When |∂p/∂t| ≈ 3 hPa/h, which is a high value for a synoptic scale system, the magnitude of the left-hand side of Eq. (1.316) is 10−6 1/s. Thus, the typical scale of the left-hand side of Eq. (1.316) is . 10−6 1/s. Because the order of magnitude of the two sides of Eq. (1.316) must be equal, the scale of the right-hand side is also . 10−6 1/s. That is, the sum of the two terms on the right-hand side of Eq. (1.316) is at least an order of magnitude smaller than p p either h∂u/∂xipst or h∂v/∂yipst , which implies that the two terms must be of opposite signs: the computation of the pressure tendency, or ω in general, 84 Such

a crude approximation could not be used in a realistic model.

Governing Equations

91

involves computing the small difference between two large terms. Hence, a 1% error in the computation of the two horizontal derivatives leads, at minimum, to a 10% error in the calculated pressure tendency or ω. The typical errors are much higher than the estimate of the minimum. Example 1.15. The conclusions of the previous example can be illustrated by computing the two sides of Eq. (1.313) for an operational analysis available in the standard distribution format: the fields are available at the standard pressure levels with a uniform resolution of 2.5◦ × 2.5◦ . We utilize the built-in routines of one of the standard software packages for the visualization of atmospheric fields to carry out the diagnostic calculations necessary for the evaluation of the terms of Eq. (1.313). These routines use centered-differences for the approximation of the derivatives and a numerical integration scheme for the computation of the vertical mean in pressure vertical coordinate. For the calculations, we use the u, v and ω fields of the data set. p p The two lower panels of Fig. 1.12, which show h∂u/∂xipst and h∂v/∂yipst , confirm the results of the scale analysis of the previous example: the order of magnitude of the two terms on the right-hand side of Eqs. (1.313) and (1.316) is 10−5 1/s and the related fields tend to have the same spatial patterns with opposite signs. The top panel of the same figure shows the vertical average of the divergence of the horizontal wind field, that is, the right-hand side of Eqs. (1.313) and (1.316). At first sight, the values shown are in line with those predicted by the scale analysis, as they are at least an order of magnitude smaller than the values in the two lower panels. A more careful examination of the values shows, however, that they are higher than should be. The problem is further illustrated by Figure 1.13, which shows that the computed value of the left-hand-side of Eq. (1.313) tends to be smaller than the computed value of the right-hand-side. In essence, the sensitivity of the computation of the right-hand-side of Eq. (1.313) to numerical errors85 makes the estimate of the left-hand-side of Eq. (1.313) useless. When the goal is the estimation of the pressure tendency rather than ω, additional errors are introduced by the assumptions (of Example 1.14) that ps = 1000 hPa is a reasonable definition of the bottom of the atmosphere and ω(pt ) = ω(200 hPa) = 0. The former assumption introduces significant 85 These errors are due to the interpolation of the u and v fields by the data providing centers from the native grid of the model to the grid used for the distribution of the data, the approximation of ∂u/∂x and ∂v/∂y by finite differences and the numerical integration of the resulting fields over the standard pressure levels.

92

Applicable Atmospheric Dynamics

Divergence [x 106 1/s]

∂u/∂x [x105 1/s]

∂v/∂y [x105 1/s]

Fig. 1.12 A snapshot of (top) h∂u/∂x + ∂v/∂yippst , (middle) h∂u/∂xippst and (bottom) h∂v/∂yippst for ps = 1000 hPa and pt = 200 hPa in the North-Pacific region, based on an operational NCEP analysis.

Governing Equations

93

Left-Hand Side

M

M

T

Right-Hand Side

Fig. 1.13 (Top) The left-hand side and (bottom) the right-hand side of Eq. (1.313) for the same time and region as shown in Fig. 1.12. M indicates the center of regions where the surface pressure is much lower than 1000 hPa, while T marks the center of the zone where the assumption ω(pt ) = ω(200 hP a) = 0 is clearly not satisfied.

errors in regions of high orography, where the surface pressure is much lower than ps = 1000 hPa. It should be noted, that in those regions, the ω(1000 hPa) fields included in the data sets have no plausible physical interpretation either, as they are obtained by an extrapolation of the fields deep below the Earth’s surface. Hence, the prudent approach would be to black-out the field in the upper panel of Fig. 1.13 at all locations where the surface pressure is lower than 1000 hPa by more than a prescribed threshold value (for instance, 20 hPa).86 The region where the assumption ω(pt ) = ω(200 hPa) = 0 is clearly not satisfied is at the northern edge of the tropical circulation, where the 86 The

field in the regions marked by M are shown here only to illustrate the problem.

94

Applicable Atmospheric Dynamics

time mean flow has a downward vertical component in the NH winter.87 The center of this region, which is located between 30◦ N and 40◦ N, is marked by T in Fig. 1.13. The errors introduced by assuming that ω(pt ) = ω(200 hPa) = 0 could be minimized by using the lowest pressure level of the data set to define pt . The sensitivity of the computation of the pressure tendency by the continuity equation was first pointed out by Max Margules, one of the early greats of atmospheric dynamics.88 Since pressure is proportional to the mass of the atmosphere aloft, the sensitivity of the pressure tendency demonstrates that maintaining a proper balance between the mass and the wind fields in an atmospheric calculation is essential. When atmospheric scientists use the term atmospheric balance, or simply balance, they refer to the balance between the mass and the wind fields. Finding mathematical models of atmospheric balance has been one of the most exciting and most challenging theoretical problems of atmospheric dynamics. In addition, the maintenance of atmospheric balance in numerical solutions of the primitive equations has been one of the grand challenges of numerical weather prediction.89 Atmospheric balance and the practical techniques to maintain it in model solutions will be discussed at several places in the remainder of the book. Here, we only note that the most difficult, and also most important, practical challenge is to prepare well balanced analyses. The procedures to improve the balance in the analyses are called initialization. An initialization is carried out either after the data assimilation has been completed, making small adjustments to the analyzed mass and wind fields, or by imposing a balance constraint in the data assimilation algorithm. Operational data assimilation systems have come a long way since the introduction of primitive equation models in producing well balanced analyses. The ω field of a reanalysis or forecast data set is in a good, albeit not perfect, balance with the wind field of the same data set. The reader should not hesitate to use such an ω field in a diagnostic calculation, unless 87 This

is the northern part of the Hadley-cell, which will be described in Sec. 2.2. (1904) carried out a scale analysis similar to the one presented here, except that he used height as the vertical coordinate. The result made Margules rather pessimistic about the possibility of weather forecasting, as he declared that ”any attempt to forecast the weather was immoral and damaging to the character of a meteorologist” (Lynch, 2006). 89 For instance, Lynch (2006) provides a review of the concept of atmospheric balance and the most important initialization techniques. For a review of the state-of-the-art of the theory of balance in geophysical flows, see Vanneste (2013). 88 Margules

Governing Equations

95

the pressure level of the data is below the Earth’s surface in the model. More care should be exercised, however, when ω is computed from interpolated or observed wind data. In that case, the safe approach is to use the ω-equation, which will be discussed in Sec. 2.3.3. 1.4.2.6

The geopotential height field

While the geopotential height field does not provide a full description of the atmospheric state, plotting the geopotential height at different pressure levels is a powerful tool for a qualitative assessment of the atmospheric flow. An example for such a plot is shown in Fig. 1.14, which displays the geopotential height field at four different pressure levels. In this figure, the pressure level closest to the Earth’s surface, characterized by the highest value of the pressure, is the 1000 hPa pressure level. For this level, a map could not be obtained without an extrapolation of the geopotential height values at the many locations where the surface pressure is lower than 1000 hPa.90 The extrapolation uses the hydrostatic balance equation to obtain the hypothetical geopotential height of the 1000 hPa level, using a prescribed vertical temperature gradient 91 for the hypothetical atmospheric layer between the surface and the 1000 hPa level. This approach is applied at all locations and pressure levels, where the pressure is higher than the surface pressure. Some values of the geopotential height at the 850 hPa level have also been obtained by extrapolation below the Earth’s surface. The smooth transition of the geopotential height fields between the regions of actual values and the regions of interpolated values can be viewed as a justification for the practice of producing full geopotential height fields by extrapolation. Notwithstanding the success of the approach in producing smooth fields, the extrapolated values should never be used for purposes other than visualization. To be more specific, the extrapolated values should not be used in diagnostic calculations, because they describe the flow in a layer of the atmosphere that does not exist in the model. The grid points where the geopotential height and the other state variables have been obtained by extrapolation can be identified by comparing either the pressure at the given level to the surface pressure, or the geopotential height to the geopotential height of the surface. For instance, the 90 The

same statement applies to the mean sea level pressure field shown in Fig. 1.10. term “vertical temperature gradient” refers to the vertical derivative of the temfor the pressure vertical perature in meteorology. For instance, it is defined by ∂T ∂p coordinate. Mathematically, it is the vertical coordinate of the temperature gradient. 91 The

96

Applicable Atmospheric Dynamics

1000 hPa

850 hPa

500 hPa

250 hPa

Fig. 1.14 A snapshot of the geopotential height field z(λ, ϕ), in the Northern Hemisphere at four different pressure levels. The fields are shown for the same time as in Fig. 1.10.

negative values in the top-left panel of Fig. 1.14 indicate extrapolated values, because the geopotential height of the surface can never be negative.92 The dominant patterns of the geopotential height field at 1000 hPa are very similar to the dominant patterns of the mean sea level pressure field shown in Fig. 1.10. For instance, the regions where the closed isohypses and isobars indicate local minima of the fields coincide in the two figures.93 92 There

are a few exceptional locations over land, where the surface is below the mean sea level, but these locations do not show up on the maps at the typical horizontal resolution of the atmospheric data sets. 93 An isoline of the geopotential height is called an isohypse, while an isoline of the pressure is called an isobar.

Governing Equations

97

These are the locations of the centers of extratropical cyclones, which are the dominant transient features of the atmospheric flow at the resolution shown here. The related features of the geopotential height field at the higher altitudes (at the levels of lower pressure) are wave like structures, which are usually referred to as troughs in synoptic meteorology, because they are analogous to the troughs on a topographic map of the surface elevation. Accordingly, the patterns of high values of the geopotential height between the troughs are called ridges. Such troughs and ridges can also be present in the upper layers of the troposphere obviously related flow features near the surface. At locations where the geostrophic wind provides a good approximation to the wind vector, a map of the geopotential height also provides useful qualitative information about the wind: since the gradient of a scalar field is a vector that is perpendicular to the direction tangential to the isolines of the scalar, pointing in the direction of the higher values, Eq. (1.302) indicates that (i) the geostrophic wind is parallel to the direction tangent to the isohypses at any location, such that (ii) for an observer looking downstream, the higher values of the geopotential height are on the right-hand side. In practical terms, rule (i) says that the flow is parallel with the isohypses. For instance, isohypses running parallel with the latitude circles indicate a zonal flow. The isohypses also provide qualitative information about the wind speed: since the speed (magnitude) of the geostrophic wind is proportional to the gradient of the geopotential height, z, the closer the isohypses are the stronger the wind. This rule allows for a comparison of the wind speed at the different pressure levels for the same latitude, because the scalar multiplication factor g/f does not depend on altitude. For instance, the comparison of the four panels of Fig. 1.27 suggests that the highest wind speeds occur at the 250 hPa level in the North-Pacific region between latitudes 20◦ N and 40◦ N, and in the North-Atlantic region between 40◦ N and 60◦ N. The quality of the geostrophic inference about the relationship of the wind vector and the isohypses can be assessed by comparing a snapshot of the two fields. Figure 1.15 shows the wind vector fields and isohypses for such a comparison at the 850 hPa and 500 hPa pressure levels. The figure confirms that geostrophic reasoning leads to qualitatively correct results:

98

Applicable Atmospheric Dynamics

500 hPa

m/s

850 hPa

m/s Fig. 1.15 Illustration of the relationship between the geopotential height field and the horizontal wind vector. The geopotential height fields shown here are the same as those in Fig. 1.14, except that the fields south of 20◦ N are not shown. Note the obvious signs of difficulty in restoring the wind field near the pole from the gridded fields of the two horizontal local Cartesian coordinates, u and v, of the wind vector.

Governing Equations

99

the wind vector is dominantly parallel with the isohypses and the wind speed is stronger in the regions where the isohypses are closer to each other. The figure also shows, however, that at some locations the wind vector has a non-negligible ageostrophic (non-geostrophic) component. A similar figure for the 250 Pa pressure level is not shown simply to save place, as it does not show any notable new features compared to the picture shown for the 500 hPa level. In contrast, the fields at the 1000 hPa level are not shown, because the 850 hPa level is about the pressure level closest to the surface, where the geostrophic approximation is still applicable. The bottom layer of the atmosphere where the flow is not in an approximate geostrophic balance is called the Ekman layer. Geostrophic balance breaks down in the Ekman layer due to the viscous effects at the surface. As discussed before, the horizontal wind vector must satisfy the boundary condition vH = 0 at the surface over land, which makes the Coriolis force vanish there. Because the flow at the surface is in a state of rest, the surface drag force must balance the pressure gradient force. In essence, the dynamics of the Ekman layer is characterized by a transition of the flow from a balance between the pressure gradient force and the surface drag force at the surface to an approximate balance between the pressure gradient force and the Corioilis force at the top of the Ekman layer (bottom of the free atmosphere). While analytical models to describe the transition exist,94 the wind profiles predicted by those models match the observations poorly. This mismatch suggests that the turbulent processes of the atmospheric boundary layer are too complex to be captured by a simple analytical model. The failure of the analytical models to explain the wind profile in the Ekman layer does not invalidate the phenomenological result that the Ekman layer is about 1000–1200 m deep in the atmosphere.

1.4.3

Sigma Vertical Coordinate

The sigma vertical coordinate is a pressure-based terrain following coordinate. The sigma coordinate is sometimes called Phillips sigma coordinate, because it was introduced by Norm Phillips.95 A hydrostatic pressure based extensions of the definition of the sigma coordinate for the non94 The first such model, which is also the most famous one, was introduced by Ekman (1905). He investigated the problem for the top layer of the ocean, which plays a role analogous to that of the bottom layer of the atmosphere. 95 Phillips (1957).

100

Applicable Atmospheric Dynamics

hydrostatic form of the equations has already been developed.96 Finally, height-based terrain-following coordinates, which are also applicable to the non-hydrostatic equations, exist as well.97 The definition of sigma removes the time dependence of the pressure coordinate by a normalization of the pressure by the surface pressure. The coordinate was named after the greek letter σ, which has been conventionally used to denote the normalized pressure, that is p (1.318) σ= . ps The bottom of the atmosphere is defined by the vertical level σ = 1, while the top of the atmosphere is defined by the level σ = 0, because σ(rS ) = σ(rH , p = ps ) = 1,

(1.319)

σ(rT ) = σ(rH , p = 0) = 0.

(1.320)

and

Exercise 1.26. Show that for a hydrostatic atmosphere, σ is a proper vertical coordinate. Answer. Because ∂σ 1 = ∂p ps

(1.321)

and ps > 0, σ(p) is a monotonically increasing function of pressure. In addition, in a hydrostatic atmosphere, σ(z) is one-to-one, because σ is a monotonically decreasing function of z. Exercise 1.27. Show that using the sigma vertical coordinate, the atmospheric governing equations take the form ∂π ∂φ duσ − RT + f vσ , =− dt ∂xσ ∂xσ

(1.322)

dvσ ∂π ∂φ − RT − f uσ , =− dt ∂yσ ∂yσ

(1.323)

0=− 96 Laprise

∂φ RT − , ∂σ σ

(1.324)

(1992). This extension of the sigma coordinate was further generalized for the deep atmosphere equations by Wood and Staniforth (2003). 97 Kasahara (1974); Gal-Chen and Somerville (1975); Cullen et al. (1997).

Governing Equations

d H ps = −ps dtσ



∂vσ ∂wσ ∂uσ + + ∂xσ ∂yσ ∂σ

101



,

(1.325)

dT RT dH ps RT 1 = − w σ + Qt dt cp ps dt cp σ cp

(1.326)

dqi 1 = − M, dt ρ

(1.327)

∂ps ∂ps ∂ps d H ps + uσ + vσ . = dt ∂tσ ∂xσ ∂yσ

(1.328)

where π = ln ps and

Answer. The atmospheric governing equations in sigma coordinate system can be derived by the application of the coordinate transformation formulas to the equations in pressure coordinate system. Formally, this can be done by using the pressure coordinate system in place of the z coordinate system, and the sigma coordinate system as the η coordinate system in the transformation formulas. The definition of the sigma coordinate can be extended to the case in which the top of the model atmosphere is defined by a value of the pressure, pt , which is larger than zero. In particular, using the definition p − pt σ= , (1.329) p s − pt of the sigma vertical coordinate, σ = 1 at the bottom of the atmosphere and σ = 0 at the top of the atmosphere, as before, because at the bottom Eq. (1.319) remains valid, while at the top, σ(rT ) = σ(rH , p = pt ) = 0.

(1.330)

The specific choice pt = 0 in Eq. (1.329) leads to the original definition of sigma by Eq. (1.321). 1.4.4

Isentropic Vertical Coordinate

While representing the entire vertical structure of the atmosphere with an isentropic coordinate system is highly problematic, carrying out diagnostic calculations at a few selected isentropic surfaces has been a highly popular and powerful tool of atmospheric dynamics. This popularity of the isentropic diagnostic tools explains why several model-based atmospheric data sets include data at a few preselected isentropic levels, in addition to the full representation of the fields at a larger number of pressure levels.

102

1.4.4.1

Applicable Atmospheric Dynamics

Potential temperature and entropy

According to the equation of state, Eq. (1.7), the knowledge of the pressure without the knowledge of the density is insufficient information for the computation of the temperature. A one-to-one diagnostic relationship between the pressure and the temperature can be obtained, however, by making the assumption that the thermodynamical processes for the air parcels are adiabatic, that is, Q = 0. For an adiabatic process, Eq. (1.39) becomes dI + p dα = 0,

(1.331)

cv dT + p dα = 0

(1.332)

which can be written as

by making use of Eq. (1.40). Eq. (1.332) can be written as cv

For an infinitesimal change of pressure, dα dT +p = 0. dp dp

(1.333)

Taking the derivative of Eq. (1.40) with respect to p yields α+p

dα dT =R . dp dp

(1.334)

Substituting p(dα/dp) from Eq. (1.333) into Eq. (1.334) leads to dT T =κ , dp p

(1.335)

where κ=

R . cp

(1.336)

Since Eq. (1.335) is an ordinary differential equation, the function T (p) is uniquely determined only for a proper initial condition. Such an initial condition is available, if T is known for any value of p. This motivates the assumption that the temperature at pressure level p′ is known and its value is T ′ . The primitive function T (p) that satisfies Eq. (1.335) is T (p) = cpκ ,

(1.337)

where ln c is an integration constant. The constant c can be determined from the initial condition T (p′ ) = T ′ : substituting p′ for p and T ′ for T in Eq. (1.337) yields c=

T′ . p′κ

(1.338)

Governing Equations

103

Substituting c from Eq. (1.338) into Eq. (1.337) leads to  κ p T (p) = T ′ . (1.339) p′ Equation (1.339) is called Poisson’s equation. Alternatively, Poisson’s equation can be written as  ′ κ p . (1.340) T′ = T p A new temperature state variable, θ, called the potential temperature, , can be defined by choosing p′ to be a prescribed reference pressure level pr (usually 1000 hPa); that is,  κ pr θ=T . (1.341) p The potential temperature is the temperature of the air parcel after its hypothetical adiabatic relocation from pressure level p to the reference pressure level pr . For a parcel that is affected only by adiabatic processes, the potential temperature θ is a Lagrangian invariant, that is, dθ = 0. (1.342) dt This property of the potential temperature follows directly from its definition. Exercise 1.28. Show that an alternative form of the thermodynamical equation (first law of thermodynamics) for the pressure coordinate system is dθ = L, (1.343) dt where  κ 1 pr L= Qt (1.344) cp p is the change in the potential temperature due to the adiabatic heating Qt . Answer. In pressure coordinate system, Qt = Qt (rH , p, t). At pressure level p, making use of the definition of specific heat yields dT , (1.345) Qt = cp dt and taking advantage of the definition of potential temperature leads to  κ dT pr dθ . (1.346) = dt dt p Combining Eqs. (1.345) and (1.346) leads to Eq. (1.343).

104

Applicable Atmospheric Dynamics

A state variable closely related to the potential temperature is the entropy, S = cp ln T − Rd ln p.

(1.347)

The equation that relates the potential temperature to the entropy can be derived by taking the logarithm of Eq. (1.341) and rearranging the resulting equation to obtain S = cp ln θ + c,

(1.348)

where the additive constant is c = −Rd ln pr . Since the Lagrangian time derivative of Eq. (1.348) is 1 dθ dS = cp , (1.349) dt θ dt similar to the potential temperature, the entropy is also a Lagrangian invariant for adiabatic processes. 1.4.4.2

Static stability

The potential temperature is the state variable most often used to characterize the vertical temperature stratification of the atmosphere. Assuming that the atmosphere is in hydrostatic balance, we can ask how resilient is that balance to random fluctuations in the atmosphere? One approach to investigate this problem is to consider the vertical acceleration of an air parcel which was previously moved by a hydrodynamical fluctuation from level z1 to to a higher level z2 , where the distance between the two levels, dz = z2 − z1 , is infinitesimal (Fig. 1.16). If the acceleration due to the forces acting on the parcel is positive, the balance is unstable, because a small vertical displacement of the parcel leads to an accelerating upward motion. If the sum of the forces is negative, the balance is stable, because the upward displacement of the parcel induces a restoring force. The key physical assumptions we make to investigate the forces acting on the displaced parcel are that (i) the parcel moves sufficiently fast, so that the changes in the state of the environment are negligible and the thermodynamical processes for the parcel are adiabatic, that is, θpar (z1 ) = θpar (z2 ) = θ(z1 ) during the movement of the parcel from z1 to z2 ; but, (ii) the parcel does not move too fast, so that the pressure of the parcel can fully adjust to the pressure of the environment, that is, ppar (z1 ) = p(z1 ) and ppar (z2 ) = p(z2 ).

Governing Equations

105

θ(z1) p(z2) ρ(z2)

θ(z2)

z2

ρpar(z2) ppar(z2)

dz

θ(z1) p(z1) ρ(z1)

θ(z1)

z1

ρpar(z1) ppar(z1)

Fig. 1.16

Illustration of the physical model of static stability.

Here, the subscript par denotes values of the state variables for the parcel, while the variables without a subscript indicate values of the state variables for the environment. Under our assumptions, the vertical component of Eq. (1.4) for the parcel is d2 z 1 ∂p (z2 ) − g. (z2 ) = − dt2 ρpar (z2 ) ∂z

(1.350)

Making use of the hydrostatic balance equation, Eq. (1.82), Eq. (1.350) can be written as     ρ(z2 ) ρ(z2 ) − ρpar (z2 ) d2 z (z ) = g − 1 = g . (1.351) 2 dt2 ρpar (z2 ) ρpar (z2 ) The right-hand-side of Eq. (1.351) is the buoyancy force for the parcel: when the density of the parcel is lower than the density of the environment, the right hand-side is positive, indicating that the hydrostatic balance is not stable. When the density of the parcel is higher than the density of the environment, the buoyancy force acts to restore the balance. We are not done, yet, as our goal is to obtain a stability criterion that can be expressed by the variables that describe the state of the environment.

106

Applicable Atmospheric Dynamics

We achieve this goal by re-writing Eq. (1.351) so that the thermodynamical state of the parcel and the environment is represented by the potential temperature, θ, rather than the density. The motivation for this strategy is that unlike the ρpar (z2 ) density of the parcel at z2 , the potential temperature of the parcel, θ(p1 ), is given by a state variable of the environment. Introducing the notation p1 = p(z1 ) and p2 = p(z2 ), Eq. (1.341) can be written for the environment as  κ pr p2 , (1.352) ρ(z2 ) = Rθ(z2 ) p2 and for the parcel as p2 ρpar (z2 ) = Rθ(z1 )



pr p2

κ

.

(1.353)

To obtain Eqs. (1.352) and (1.353), we made use of the equation of state, Eq. (1.7). Substituting these two equations into Eq. (1.351) yields   d2 z θ(z1 ) − θ(z2 ) (z2 ) = g . (1.354) dt2 θ(z1 ) With the help of the Taylor’s series expansion of θ(z) about z1 , θ(z2 ) can be approximated as θ(z2 ) = θ(z1 ) +

∂θ (z1 )dz. ∂z

(1.355)

Substituting θ(z2 ) from Eq. (1.355) into Eq. (1.354) leads to d2 z g ∂θ (z2 ) = − (z1 )dz. dt2 θ(z1 ) ∂z

(1.356)

When the hydrostatic balance is stable, Eq. (1.356) is formally identical to the equation of motion for a unit mass attached to a spring (a linear oscillator) with N 2 (z1 ) =

g ∂θ (z1 ) θ(z1 ) ∂z

(1.357)

playing the role of the spring constant. Based on the analogy to the linear oscillator, the motion of the parcel in a stably stratified atmosphere is an oscillating one with frequency N (z1 ). This frequency is called the BruntV¨ ais¨ ala frequency. Since we did not make any special assumption about the level z1 , Eq. (1.358) holds for any vertical level of the atmosphere and we can rewrite Eq. (1.358) as

Governing Equations

107

d2 z (z + dz) = −N 2 (z)dz, (1.358) dt2 where N (z) is the vertical profile of the Brunt-V¨ais¨ ala frequency. While the physical assumptions made here may seem somewhat arbitrary, the Brunt-V¨ais¨ ala frequency will turn out to be one of the key parameters of atmospheric dynamics. Heuristically, it measures the resilience of the atmosphere to forces that can stretch or compress a column of air in the vertical direction. 1.4.4.3

Phenomenology

In an isentropic coordinate system, the vertical coordinate is the potential temperature, θ. A coordinate system using θ as the vertical coordinate is called isentropic, because according to Eq. (1.348), a vertical level defined by a constant value of θ is also a level of constant entropy. The potential temperature is a proper vertical coordinate only under the assumption that the vertical stratification of the atmosphere is stable. In that case θ(z) is a monotonically increasing (one-to-one) function of height. Since in a stably stratified atmosphere the vertical acceleration of a parcel cannot be sustained, making the additional assumption of hydrostatic balance does not add any further restrictions on the type of motions that can be considered using an isentropic vertical coordinate. Exercise 1.29. Show that θ(p) is a monotonically decreasing function of p. Answer. Taking the partial derivative of Eq. (1.341) with respect to p, ∂θ RT 1 1 =− θ=− θ. ∂p cp p cp ρ

(1.359)

Since the right-hand side of Eq. (1.359) is always negative, θ(p) is a monotonically decreasing function of p. Typical examples for the vertical structure of θ in the middle of the cold and warm seasons are shown in Fig. 1.17. The figure should be considered an illustration of the vertical structure of the potential temperature field rather than an evidence that the large scale stratification of the atmosphere is stable, as the model that was used to prepare the analyses shown here was a hydrostatic model, which constrained the analyses to be in hydrostatic balance. As can be expected for an atmosphere in stable hydrostatic balance, the potential temperature increases with height at all latitudes. In addition, the vertical gradient of the potential temperature

108

Applicable Atmospheric Dynamics

0000 UTC 1 January 2010 340 0 310 33 300

34

0

0

33

33

0

600 0 28

800

0 2 9

−80

−60

0

0

290 1000

0

30

0

310 300

0

29

28

400

0

34

330 310 0 30 0 9 2

31

Pressure [hPa]

200

27

280

−40

−20

0 Latitude

20

40

60

80

0000 UTC 1 July 2010

0

0

0

310

0

31

24

−80

300

0

1000

28 0 27 0 25

32

800

310

0

32

0

600

320

30

400

340

340

320

Pressure [hPa]

200

320 340 310 300 290

290

29 −60

−40

−20

0 Latitude

20

40

60

280 80

Fig. 1.17 Snapshots of the vertical-meridional cross-section of the potential temperature at longitude 150◦ E. Shown are the fields at (top) 0000 UTC 1 January 2010 and (bottom) 0000 UTC 1 July 2010 based on the NCEP/NCAR Reanalysis.

is much stronger at the tropopause and in the stratosphere than in the troposphere. This feature of the potential temperature should not come as a surprise, considering that the temperature of an air parcel brought down from the stratosphere to the reference pressure level will be warmer than the temperature of a parcel brought down from a lower layer of the stratosphere or the tropopause due, not only to the more intense adiabatic warming of the collapsing parcel, but also to its higher initial temperature. The potential temperature also has a well organized structure in the meridional direction: it is lowest over the two poles, with somewhat lower values over the winter pole than over the summer pole. Below about 800 hPa, the potential temperature increases monotonically toward the thermal equator, where it reaches its maximum of about 300–310 K. The equatorward potential temperature remains present up to about 400 hPa in the extratropics, but above about 800 hPa in the tropics and about 400 hPa in the extratropics, the meridional temperature gradient is weak.

Governing Equations

1.4.4.4

109

The equations for the isentropic vertical coordinate

Exercise 1.30. Show that using an isentropic vertical coordinate, the atmospheric governing equations take the form duθ ∂γ + f vθ , , (1.360) =− dt ∂xθ dvθ ∂γ − f uθ , , (1.361) =− dt ∂yθ cp T ∂γ + , (1.362) 0=− ∂θ  θ  ∂vθ ∂wθ ∂p ∂uθ d ∂p + + =− , (1.363) dt ∂θ ∂θ ∂xθ ∂yθ ∂θ  R/cp dθ 1 p0 = Qt , (1.364) dt cp T p 1 dq = − M, (1.365) dt ρ where γ(x, y, θ, t) is the Montgomery stream function defined by γ = φ + cp T. (1.366) Answer. The equations for the isentropic vertical coordinate can be most easily derived from the equations for the pressure vertical coordinate, making use of Eq. (1.359). The only equation whose derivation is somewhat involved is the thermodynamical Eq. (1.364). This equation can be obtained by first taking the time derivative of Eq. (1.341), which yields  R/cp   dθ p0 RT dp dT = − , (1.367) dt p dt pcp dt then combining Eqs. (1.367) and (1.6) to obtain Eq. (1.364). A comparison of Eqs. (1.360) and (1.361) to Eqs. (1.275) and (1.276) shows that the Montgomery stream function plays the same role for the isentropic vertical coordinate as the geopotential for the pressure vertical coordinate. In addition, Eq. (1.363), which is the continuity equation for the isentropic coordinate system, is formally identical to the continuity equation for the Cartesian coordinate system, except that ∂p/∂θ replaces the density ρ.98 Under the assumption that the thermodynamical processes in the atmosphere are adiabatic, all motions are two-dimensional in an isentropic coordinate system : since Q = 0 in the adiabatic case, Eq. (1.364) becomes dθ = 0, (1.368) dt 98 We

will take advantage of this analogy between ∂p/∂θ and ρ in Sec. 1.6.2.3.

110

Applicable Atmospheric Dynamics

which implies that wθ = dθ/dt. Hence, using θ as the vertical coordinate reduces the vertical transport through the vertical coordinate surfaces, leading to a more accurate representation of the vertical transport processes in the vertically discretized model equations. In addition, θ provides an adaptive representation of the vertical structure of the atmosphere in the sense that in the regions where high static stability leads to a strong vertical stratification of the thermodynamical state variables and the density of the atmospheric constituents, most importantly the humidity, the constant-θ surfaces move closer to each other, thus providing an increased vertical resolution. Unfortunately, similar to the situation for the pressure coordinate, the bottom of the atmosphere cannot be defined by a constant surface of θ. Making matters even worse, a simple transformation that would result in a potential-temperature-based terrain-following coordinate does not exist. These properties make using θ as a vertical coordinate near the surface highly complicated.

1.4.5

Hybrid Vertical Coordinates

The six decades of experience with atmospheric models show that in the lowest layers of the atmosphere, where orography has a major effect on the flow, a terrain-following coordinate is the best choice for the representation of the fields of the state variables. But, there is no reason to use a terrainfollowing coordinate at higher altitudes, where the constant pressure and isentropic surfaces are nearly flat and largely unaffected by the orography. At those higher altitudes, the sloping surfaces of the terrain-following coordinates increase the truncation errors introduced by the numerical approximations of the horizontal spatial derivatives.99 The recognition of this limitation of the terrain-following coordinates led to the quest for the proper definition of hybrid coordinates that behave like a terrain-following coordinate near the surface and as a pressure, hydrostatic pressure, or potential temperature vertical coordinate higher in the atmosphere.100 Hybrid coordinates are highly popular in in the current practice of atmospheric modeling. For instance, at the time of writing, the global models of ECMWF and NCEP use hybrid sigma-pressure vertical coordinates. 99 In

the context of the sigma vertical coordinate, this effect was demonstrated by Sundqvist (1976). 100 The first paper on the subject was published by Simmons and Burridge (1980).

Governing Equations

1.4.5.1

111

General formulation

We follow the notational convention of the literature and denote the general hybrid coordinate by η. The standard approach to introduce η is to search for a vertical coordinate that satisfies p(x, y, η, t) = A(η) + B(η)ps (x, y, t),

(1.369)

where A(η) and B(η) are functions of η and are to be determined. The general coordinate η becomes σ when the choices A(η) = 0 and B(η) = η are made. For this choices of A(η) and B(η), A(ηs ) = 0,

B(ηs ) = ηs = 1,

(1.370)

where ηs is the value of η at the surface (bottom of the atmosphere).Thus a hybrid coordinate behaves like σ at the surface, if it satisfies the two conditions given by Eq. (1.370). In addition, η becomes the pressure coordinate when A(η) = η,

B(η) = 0.

(1.371)

Because both sigma and pressure are quantities that monotonically decrease with height, under the assumption of hydrostatic balance, the hybrid coordinate η is also expected to decrease monotonically with height. Hence, the condition that η should satisfy Eq. (1.371) at high altitudes can be written as lim A(η) = η,

η→0

lim B(η) = 0.

η→0

(1.372)

This pair of conditions can be satisfied by defining the function A(η) as A(η) = η − B(η),

(1.373)

and choosing Bη such that it goes faster to zero than η in the limit η → 0. The latter condition is satisfied, for instance, by B(η) = η r ,

(1.374)

where r is an integer larger than one.101 For such a choice of B(η), the speed of the transition from sigma to pressure coordinate is controlled by the parameter r. If no value of r can provide the desired speed of the transition, any other function that converges to zero faster than η can be considered for B(η). A hybrid coordinate for the hydrostatic pressure coordinate and its terrain-following version can be designed along the same lines as for the 101 A

similar choice for A(η) and B(η) was first suggested by Laprise and Girard (1990).

112

Applicable Atmospheric Dynamics

pressure and the sigma coordinates.102 Another notable example for a hybrid coordinate, which was used until not long ago in the NOAA Rapid Update Cycle (RUC) operational limited area model, is one that rapidly transitions from sigma to potential temperature in the troposphere.103 1.4.6

Pseudo-Height and Log-Pressure Vertical Coordinates

1.4.6.1

General definition of the pseudo-height

There are different, but closely related, vertical coordinates called pseudoheight in the literature. These coordinates are designed to have heightdimension, while preserving some of the advantageous properties of the pressure vertical coordinate. They have been particularly popular in analytical and idealized numerical investigations of the atmosphere. Pseudoheight and the closely related log-pressure vertical coordinates are also better choices than pressure for the visualization of the fields for deep atmospheric layers, as they can provide a better resolution picture of the fields in the upper part of a deep layer. All pseudo-height coordinates can be written in the general form  Z p0  RTr (p′ ) zˆ(p) = dp′ , (1.375) gp′ p where Tr (p) is a prescribed reference temperature profile and p0 is a constant reference value of the mean sea level pressure.104 Exercise 1.31. What is the functional relationship between the geopotential height, z, and the pseudo-height, zˆ? Answer. The relationship between z and zˆ can be described by the derivative ∂z/∂ zˆ, which can be obtained in two steps. First, it should be noticed that Eq. (1.375) is the solution of the differential equation RTr ∂ zˆ =− ∂p gp 102 Laprise

(1.376)

(1992). be precise, many different hybrid sigma-isentropic coordinates have been considered and tested over the years. A brief review of the different hybrid isentropic-sigma coordinates can be found in Benjamin et al. (2004). 104 The vertical coordinate defined by Eq. (1.375) was introduced by White and Bromley (1995), while its relationship to the more conventional definitions of pseudo-height was pointed out by White and Beare (2005). White and Bromley (1995) called zˆ(p) a “pressure-based but height-like coordinate”. 103 To

Governing Equations

113

for the initial condition zˆ(p0 ) = 0. Then, the desired derivative can be obtained by  −1 ∂z ∂ zˆ ∂p T = , (1.377) = ∂ zˆ ∂p ∂z Tr where in the second step, ∂ zˆ/∂z and ∂p/∂z were substituted from Eq. (1.376) and Eq. (1.270), respectively. Equation (1.377) can also be written in the equivalent forms θ ∂z = , ∂ zˆ θr

and

∂z ρ = , ∂ zˆ ρr

(1.378)

where θr = θr (p) is the potential temperature profile, while ρr = ρr (p) is the density profile associated with the reference temperature profile, Tr (p). Equations (1.377) and (1.378) show that pseudo-height is a proper vertical coordinate, because it is a monotonically increasing function of the geopotential height. Exercise 1.32. Show that the relationship between the vertical coordinate of the wind vector for a pseudo-height coordinate, wzˆ, and the vertical coordinate of the wind vector for the pressure vertical coordinate, ω, is w ˆ=

dˆ z ωRTr =− . dt gp

(1.379)

Answer. The coordinate wzˆ can be computed from the coordinate ω with the help of the last formula in Eq. (1.261). In particular, assuming that the variables with no subscript denote variables in pressure coordinates, and substituting zˆ for η, Eq. (1.261) can be written as w ˆ = up

∂ zˆ ∂ zˆ ∂ zˆ ∂ zˆ ωRTr ∂ zˆ + vp +ω + =ω =− . ∂xp ∂yp ∂p ∂t ∂y gp

(1.380)

In the first step, we made use of the property of the terms which include the partial derivatives of zˆ with respect to xp , yp and t, that they are zero, because zˆ depends on these variables only through p, which is also an independent variable in pressure coordinates. In the second step, we substituted ∂ zˆ/∂p from (1.376). 1.4.6.2

Conventional definitions of pseudo-height

The most common forms of the pseudo-height can be obtained by choosing Tr (p) to be

114

Applicable Atmospheric Dynamics

(i) a constant temperature profile, Tr (p) = T0 ,

(1.381)

or (ii) an isentropic temperature profile, which can be defined by a constant potential temperature, θ0 , that is,  κ p . (1.382) Tr (p) = θ0 p0 We only discuss the pseudo-height coordinate associated with the isothermal temperature profile of Eq. (1.381), which will be used in some of the analytical models discussed in Chapter 2.105 Hereafter in this book, the term pseudo-height refers to the vertical coordinate with the isothermal reference temperature profile. The motivation for this particular definition of the temperature profile can be most easily understood by first considering the hypothetical situation, in which the temperature does not change with height. In that case, the hydrostatic balance equation can be written as dp p , (1.383) =− dz H0 where RT0 (1.384) H0 = g is called the scale height. The solution of the differential equation Eq. (1.383) is p = ce−z/H0 ,

(1.385)

where the constant c can be determined from the initial condition p(z) = p0 at z = 0: substituting 0 for z and p0 for p in Eq. (1.385) yields c = p0 , which leads to p = p0 e−z/H0 .

(1.386)

Dividing Eq. (1.386) by p0 and rearranging the resulting equation after taking its natural logarithm yields p (1.387) z(p) = −H0 ln . p0 105 This

vertical coordinate was introduced by Eliassen (1949) and Phillips (1963), although those papers did not call the resulting coordinate ‘pseudo-height’. The vertical coordinate associated with the isentropic reference temperature profile was introduced by Hoskins and Bretherton (1972) for the mathematical investigation of atmospheric frontogenesis.

Governing Equations

115

It can be easily verified that for Tr (p′ ) = T0 , the right-hand side of Eq. (1.387) is equal to the integral on the right-hand side of Eq. (1.375), that is, p zˆ(p) = −H0 ln . (1.388) p0 Hence, the pseudo-height, zˆ(p), is the hypothetical height of the pressure level p under the assumption that the atmosphere is in hydrostatic balance and the vertical temperature profile is isothermal with temperature T0 . The pseudo-height, zˆ, takes the value 0 for p = p0 , and the value H0 for p = p0 /e. A representative value of the scale height H0 for the atmosphere can be defined by the typical value of the height at which the pressure drops to p0 /e. Then, the value of T0 computed from Eq. (1.384) can be considered a “weighted average” of the temperature of the lowest one-scale-height deep layer of the atmosphere. Example 1.16. Assuming that p0 = 1000 hPa, we obtain p(H0 ) = p0 /e ≈ 370 hPa. That is, the height of the 370 hPa pressure level provides the estimate of the scale height. The typical value of H0 is lower at the higher latitudes, but H0 ≈ 8 km is a good general estimate of its average value. The estimate of the average temperature provided by Eq. (1.384) is T0 ≈ 273 K, which is equal to the global mean temperature at about 685 hPa (Fig. 1.5). Exercise 1.33. Show that for the pseudo-height vertical coordinate defined by an isothermal reference temperature profile, the Brunt-V¨ ais¨ ala frequency takes the form gT0 ∂θ , (1.389) N2 = θT ∂ zˆ where T0 is the temperature that defines the reference profile. Answer. Equation (1.357), which defines the Brunt-V¨ ais¨ ala frequency, can be written in the equivalent form ∂ ln θ. (1.390) N2 = g ∂z Because ∂S ∂S ∂ zˆ ∂S T0 = = (1.391) ∂z ∂ zˆ ∂z ∂ zˆ T for any scalar S, Eq. (1.390) can be further expanded as gT0 ∂ gT0 ∂θ N2 = ln θ = . (1.392) T ∂ zˆ θT ∂ zˆ

116

Applicable Atmospheric Dynamics

Exercise 1.34. Show that the ˆ2 = N



T T0

2

N2

(1.393)

analogue of the Brunt-V¨ ais¨ ala frequency for the pseudo-height vertical coordinate can be written in the equivalent form   ˆ 2 = R ∂T + κT . N (1.394) H0 ∂ zˆ H0 Answer. Comparing Eqs. (1.393) and (1.33) yields ˆ 2 = gT ∂ ln θ; N T0 ∂ zˆ then, writing ln θ as   κ  p0 κˆ z p0 ln θ = ln T , = ln T + = ln T + κ ln p p H0

(1.395)

(1.396)

and substituting the result into Eq. (1.395) leads to       gT ∂ g ∂T R ∂T κT κT κˆ z 2 ˆ N = = = . + + ln T + T0 ∂ zˆ H0 T0 ∂ zˆ H0 H0 ∂ zˆ H0 (1.397) 1.4.6.3

Log-pressure vertical coordinate

The log-pressure vertical coordinate P (p) is defined by106 p P = − ln . p0

(1.398)

From Eq. (1.388), P =

zˆ , H0

(1.399)

which shows that the log-pressure coordinate can be considered a dimensionless form of the pseudo-height, where the unit of height is the scale height. Exercise 1.35. Show that by using log-pressure as the vertical coordinate, the hydrostatic balance equation can be written in the convenient form ∂z RT = . ∂P g 106 The

(1.400)

behavior of P was first investigated by Eliassen (1949). The pseudo-height defined by Eq. (1.384) is also often called pseudo-height (e.g., Holton, 2004).

Governing Equations

117

Answer. ∂z = ∂P



∂z ∂ zˆ ∂ zˆ ∂P



=

RT T H0 = , Tr g

(1.401)

where in the second step, ∂z/∂ zˆ was substituted from Eq. (1.377), while ∂ zˆ/∂P was determined from Eq. (1.399). In the atmospheric sciences, the convention is to use a linear pressure scale in figures where the main features of interest are in the troposphere. Where processes in the higher layers are also considered, the vertical scaling usually follows the log-pressure coordinate, but the ticks are labeled by either the pressure values or by the pseudo-height, zˆ, values, using Eq. (1.398) and a fixed value of the scale height H0 for the computation of zˆ. Exercise 1.36. What is the functional relationship between the wP vertical coordinate of the wind vector for the log-pressure vertical coordinate, P , and the w ˆ vertical coordinate of the wind vector for the pseudo-height vertical coordinate, zˆ, defined by Eq. (1.388)? Answer. wP =

1.5

dP dP dˆ 1 z w. ˆ = = dt dˆ z dt H0

(1.402)

Vorticity and Divergence Equations

The vertical coordinate of the vorticity vector and the divergence of the horizontal component of the wind can replace the horizontal coordinates of the wind vector as the two prognostic variables of the horizontal component of the momentum equation. In the atmospheric sciences, the vertical coordinate of the vorticity vector is called the vorticity, while the divergence of the horizontal component of the wind is called the divergence. The two new equations, which replace the zonal and the meridional component of the horizontal momentum equations, are called the vorticity equation and the divergence equation. The importance of the role the vorticity and the divergence equations have played in atmospheric dynamics would be hard to overstate: (i) They served as the starting point for the derivation of reduced forms of the atmospheric governing equations, which have played a central role in the analytical investigation of atmospheric motions. Systems of

118

Applicable Atmospheric Dynamics

reduced equations have also been used in countless idealized numerical studies of atmospheric motions and were also used in operational numerical weather prediction until the mid-1970s. (ii) Unlike the definition of the zonal and meridional coordinates of the horizontal wind vector, vH , the definition of the vorticity and the divergence, in principle, does not require the use of the local Cartesian coordinate system. Hence, the vorticity and the divergence can be defined at the poles, thus playing an important practical role in coping with the pole problem. This feature of the two variables explains why many of today’s operational and research models, all based on the primitive equations, use vorticity and the divergence as prognostic variables. For all the advantages of using vorticity and divergence as the prognostic variables of the horizontal component of the momentum equation, unlike the two components of vH , the vorticity and divergence are not observable. In this section, we first define the vorticity and the divergence variables; then, provide a detailed analysis of the vorticity equation; finally, introduce the most important sets of reduced equations. 1.5.1

Vorticity, Absolute Vorticity and Divergence

In fluid dynamics, the curl of the wind (velocity) vector, ζ = ∇ × v,

(1.403)

is called the vorticity. We will refer to ζ as the three-dimensional vorticity, because in atmospheric dynamics the term ‘vorticity’ is reserved for the scalar ζ = (∇H × vH ) · k.

(1.404)

The vector ζk is the vertical component of ζ. Likewise, we will call the vector ζ a = 2Ω + ∇ × v

(1.405)

the three-dimensional absolute vorticity, as the term absolute vorticity refers to the scalar ζa = f + ζ,

(1.406)

where f is the Coriolis parameter. The motivation to introduce the absolute vorticity state variable will became clear soon.

Governing Equations

119

Exercise 1.37. What is the mathematical formula for the computation of the vorticity when the global Cartesian coordinate system of Sec. 1.3.3 is used to represent both the wind vector and the horizontal component of the vector of position? Answer. With the help of the Cartesian coordinates, Eq. (1.404) can be written as i j k ∂ ∂ ∂u ∂v − . (1.407) ζ = ∂x ∂y 0 · k = ∂x ∂y u v 0

In the atmospheric sciences, the term divergence refers to the divergence of the horizontal component of the wind. That is, the divergence D is defined by D = ∇H · v H ,

(1.408)

Exercise 1.38. Write the continuity equation in pressure coordinate system with the help of the divergence D? Answer. ∂ω = −D. ∂p

(1.409)

Exercise 1.39. What is the specific form of Eq. 1.409 when spherical coordinates are used for the representation of the horizontal component of the vector of position? Answer. ∂ω =− ∂p 1.5.2 1.5.2.1



1 ∂u 1 ∂v + a ∂λ a cos ϕ ∂ϕ



(1.410)

Vorticity Equations Three-dimensional vorticity equation

We start the discussion of the vorticity equation with the general case by considering the full three-dimensional momentum equation without making the assumption of hydrostatic balance. Exercise 1.40. Show that Eq. (1.4) can be written in the equivalent form   1 1 ∂v 2 = v × ζa − ∇ | v | − ∇p + g. (1.411) ∂t 2 ρ

120

Applicable Atmospheric Dynamics

Answer. Equation (1.411) can be obtained by expanding dv/dt according to Eq. (1.50), then taking advantage of   1 (v · ∇) v = ∇ | v |2 − v × (∇ × v) , (1.412) 2 which follows from Eq. (1.105).

Exercise 1.41. Show that the Lagrangian time derivative of ζ ζ˜ a = a ρ satisfies the equation   1 1 dζ˜ a = ζ˜ a · ∇ v − ∇ × ∇p. dt ρ ρ

(1.413)

(1.414)

Answer. Taking the curl of both sides of Eq. (1.411) yields ∂ζ 1 = ∇ × (v × ζ a ) − ∇ × ∇p. (1.415) ∂t ρ With the help of Eq. (1.107), the first term of the right-hand-side of Eq. (1.415) can be written as ∇ × (v × ζ a ) = v (∇ · ζ a ) − ζ a (∇ · v) − (v · ∇) ζ a + (ζ a · ∇) v. (1.416) The first term of the right-hand-side of Eq. (1.416) is a zero vector, because v (∇ · ζ a ) = v [∇ · (ζ + 2Ω)] = v [∇ · (∇ × v) + 2∇ · Ω] = 0.

(1.417)

In Eq. (1.417), ∇ · (∇ × v) is zero due to Eq. (1.104), while 2 ∇ · Ω is zero, because it is the divergence of a constant vector. The second term of the right-hand-side of Eq. (1.415) can be written, making use of Eq. (1.103), as 1 1 1 1 −∇ × ∇p = −∇ × ∇p − (∇ × ∇p) = −∇ × ∇p. (1.418) ρ ρ ρ ρ In Eq. (1.417), the term ρ1 (∇ × ∇p) is zero due to Eq. (1.101). Substituting Eqs. (1.416), (1.417) and (1.418) into (1.415) leads to ∂ζ 1 = −ζ a (∇ · v) − (v · ∇) ζ a + (ζ a · ∇) v − ∇ × ∇p. (1.419) ∂t ρ Making use of the continuity equation, Eq. (1.5), the first term of the righthand-side of Eq. (1.419) can be written as 1 dρ −ζ a (∇ · v) = . (1.420) ρ dt Substituting Eq. (1.421) into Eq. (1.419), dividing both sides of the resulting equation by ρ, and moving the first term of the right-hand-side yields

Governing Equations

1 dρ 1 dζ a − ζa 2 = ρ dt ρ dt



 ζa 1 · ∇ v − ∇ × ∇p. ρ ρ

121

(1.421)

Equations (1.414) and (1.421) are equivalent. The state variable ζ˜ a is the mass weighted three-dimensional vorticity for a unit volume. The first term on the right-hand-side of Eq. (1.414) is the stretching-twisting term, while the second term is the baroclinic (solenoid ) term. The latter term can be non-zero only if the surfaces of constant density are not surfaces of constant pressure. The importance of the baroclinic term of Eq. (1.414) can be most easily seen by considering the case of a non-rotating coordinate system (Ω = 0), in which Eq. (1.414) becomes  dζ˜  ˜ 1 1 = ζ · ∇ v − ∇ × ∇p. dt ρ ρ

(1.422)

Equation (1.422) shows that the stretching-twisting term cannot be the ultimate source of the vorticity: if the three components of the threedimensional vorticity vector and the baroclinic term are all zero at initial time, the three-dimensional vorticity remains a zero vector at all times. Thus, in a non-rotating coordinate system, vorticity can be generated only if the surfaces of constant pressure are not surfaces of constant density. Since an atmosphere with such a vertical stratification is called a baroclinic atmosphere, the associated processes are called baroclinic processes An atmosphere in which the surfaces of constant pressure are also surfaces of constant density is called a barotropic atmosphere. The processes that can exist in a barotropic atmosphere are called barotropic processes. According to the equation of state, in a barotropic atmosphere, the surfaces of constant pressure and density are also surfaces of constant temperature. Hence, when the temperature is plotted at a pressure level, changes in the temperature indicate a baroclinic atmosphere. In essence, the horizontal temperature gradient for pressure vertical coordinate can be considered a measure of baroclinicity: baroclinic processes can be expected to play a more important role at the locations, where the horizontal temperature gradient is stronger. While the Earth’s atmosphere is baroclinic, barotropic models have been popular tools to investigate large scale atmospheric dynamics. In a barotropic atmosphere, the Earth’s rotation can still generate vorticity and the stretching and twisting of the vorticity field can change the three-dimensional vorticity of the parcel. In addition, local changes in the

122

Applicable Atmospheric Dynamics

vorticity field can also be due to advection, as dζ˜ ∂ ζ˜ ˜ = + (v · ∇) ζ. dt ∂t 1.5.2.2

(1.423)

Incompressible flow

The simplest barotropic models consider an incompressible flow by assuming that the density field is constant in the entire model domain. According to the discussion of the general barotropic case, in an incompressible flow in a non-rotating coordinate system, if the vorticity is zero at initial time, it remains zero at all times. This property can also be verified directly by rewriting Eq. (1.411) as   1 p ∂v 2 =v×ζ−∇ |v| − + g, (1.424) ∂t 2 ρ then taking the curl of both sides of Eq. (1.424) to obtain ∂ζ = ∇ × (v × ζ). (1.425) ∂t According to Eq. (1.425), ζ = 0 leads to ∂ζ/∂t = 0, which implies that if the three-dimensional vorticity is zero at initial time, it will remain zero at all times. Taking the curl of Eq. (1.424) is a standard procedure in fluid dynamics to eliminate the pressure variable from the governing equations. This simple transformation leads to a prognostic equation, Eq. (1.425), that can be used for the prediction of the evolution of the wind field, v(r, t), provided that the wind field, v(r, t0 ), at initial time t0 and the boundary conditions are known. The computational algorithm takes advantage of the fact, that for an incompressible flow, the continuity equation becomes a diagnostic equation. In particular, for a constant density field, Eq. (1.5) becomes ∇ · v = 0.

(1.426)

The outline of the algorithm is the following: (i) the initial three-dimensional vorticity field, ζ(r, t0 ) is computed from the initial wind field by Eq. (1.403); (ii) the three-dimensional vorticity field, ζ(r, t), at times t > t0 , is determined by solving Eq. (1.425); (iii) the wind field, v(r, t), is computed from ζ(r, t) by solving the system of equations composed of Eqs. (1.403) and (1.426). Since Eq. (1.426) is automatically satisfied if v can be written in the general form

Governing Equations

v = ∇ × A,

123

(1.427) 107

this step involves finding A that satisfies Eq. (1.403).

The effects of Earth’s rotation can be introduced into Eq. (1.425) by replacing ζ with ζ a before taking the curl in the derivation of Eq. (1.425), which leads to ∂ζ = ∇ × (v × ζ a ). (1.428) ∂t The main difference between Eqs. (1.425) and (1.428) is that in Eq. (1.428) the Earth’s rotation (Ω 6= 0) serves as a constant source of vorticity: threedimensional relative vorticity develops in the flow even if its three components are zero at initial time at all locations. Equation (1.428) can also be written as dζ a = (ζ a · ∇)v, (1.429) dt which is the analogue of Eq. (1.414) for the incompressible case: the assumption of incompressibility eliminates the solenoidal term and greatly simplifies, but does not eliminate, the stretching-twisting term. Advection of the three-dimensional absolute vorticity can also lead to local changes in the three-dimensional relative vorticity, as ∂ζ dζ a = + (v · ∇) ζ a . (1.430) dt ∂t 1.5.2.3

Two-dimensional vorticity equation

Exercise 1.42. Show that for an incompressible flow at a given vertical level, Eq. (1.425) takes the form dζ = 0. (1.431) dt Answer. Making use of Eqs. (1.404) and (1.109), we obtain ∂ζ = [∇ × (v × ζ)] · k = −ζk (∇H · vH ) − (vH · ∇H ) ζ. (1.432) ∂t Since the parenthetical expression in the first term of the right-hand side of Eq. (1.432) is zero, because 107 A

∇H · vH = 0,

(1.433)

function A that satisfies 1.427 is called a vector potential. The system of Eqs. (1.403) and (1.426) is formally identical to the equations of magnetostatics, with v playing the role of the magnetic field B. A detailed general algorithm for finding the vector potential A that satisfies Eq. (1.403) is provided by Section 14.1 of Feynman et al. (2006a). The same algorithm can be used to compute v.

124

Applicable Atmospheric Dynamics

is the continuity equation for the two-dimensional incompressible flow, Eq. (1.432) becomes ∂ζ = − (vH · ∇H ) ζ. ∂t

(1.434)

This equation is equivalent to Eq. (1.431). Equation (1.431) is called the two-dimensional vorticity equation. The physical interpretation of this equation is simple: each air parcel preserves the vorticity assigned to the parcel at initial time. In other words, the time evolution of the vorticity field is due to the motion of the fluid parcels that transport the vorticity unchanged. While Eq. (1.431) can describe the advection of vorticity, it does not provide any information about the origin of vorticity. The two-dimensional wind field can be predicted by the system of equations that consists of a prognostic equation, Eq. (1.431), and two diagnostic equations, Eqs. (1.404) and (1.433). The general solution strategy is the same as in the three-dimensional case, but the computation of the vector potential is simpler. In particular, we look for a vector potential that can be written in the general form A = −ψk.

(1.435)

Substituting the resulting expression, vH = −∇H × ψk,

(1.436)

for vH into Eq. (1.404) leads to   ζ = − [∇H × (∇H × ψk)] · k = − ∇H (∇H · ψk) − ∇2 (ψk) · k = ∇2 ψ,

(1.437)

where in the first step, we made use of Eq. (1.107). Exercise 1.43. Show that the expression vH = −∇H ψ × k

(1.438)

for the computation of vH is equivalent to Eq. (1.436). Answer. The right-hand-side of Eq. (1.438) can be obtained by expanding the right-hand-side of Eq. (1.436) using Eq. (1.103).

Governing Equations

125

The ∇2 square of the del operator is called the Laplace operator, or the Laplacian, and sometimes denoted by ∆. The equation ζ = ∇2H ψ,

(1.439)

which is a shortened version of (1.437), is mathematically a Poission equation. A formally identical equation describes the relationship between charges and the electrostatic potential: vorticity plays the role of the charge and the stream function plays the role of the electrostatic potential.108 Fully exploiting the analogy with electrostatics is beyond the scope of the present discussion.109 Here, we only note that the vorticity field is the source of the divergence-free part of the horizontal atmospheric flow the same way as electric charges are the source of the electrostatic field: given ζ(x, y) for the entire model domain, we can determine the divergence-free component of the atmospheric flow. Exercise 1.44. Show that Eq. (1.434) can be written in the equivalent form ∂ζ = J (ψ, ζ), (1.440) ∂t where J (ψ, ζ) is the Jacobian of ψ and ζ. Equations (1.440) and (1.439) form a closed system of equations to determine the time evolution of the vorticity and the stream function field for a set of proper boundary and initial conditions. While this system of equations has very limited direct relevance for weather prediction, the importance of the role it has played in the development of our current understanding of atmospheric dynamics would be hard to overstate. 1.5.2.4

The two-dimensional vorticity equation in rotating coordinates

Next, we introduce the effect of Earth’s rotation into the two-dimensional vorticity equation. In a rotating frame, Eq. (1.432) becomes ∂ζ = vH (∇H · ζ a )−ζ a (∇H · vH )−(vH · ∇H ) ζ a +(ζ a · ∇H ) vH . (1.441) ∂t The vertical component of Eq. (1.441) is ∂ζ k = (vH · ∇H ) ζ aV , (1.442) ∂t 108 Considering

the two-dimensional problem instead of the three-dimensional problem leads to equations that are analogue to the equations of electrostatics rather than to the equations of magnetostatics. 109 A prototype for the analysis of the Poission equation can be found in Chapter 6 of Feynman et al. (2006a).

126

Applicable Atmospheric Dynamics

where ζ aV is the vertical component of ζ a . From Eq. (1.405), ζ aV = 2ΩV + ζk = (f + ζ)k = ζa k,

(1.443)

where we made use of 1 f k. (1.444) 2 from Eq. (1.443) into Eq. (1.442) yields

ΩV = sin ϕ | Ω | k =

Substituting the expression for ω V the equation, ∂ζ = (vH · ∇H ) ζa . (1.445) ∂t Since ∂f /∂t = 0, Eq. (1.445) can also be written as dζa = 0. (1.446) dt Equation (1.446) states the conservation of absolute vorticity for each fluid parcel. Since the value of the Coriolis parameter f depends on the latitudes, for a particle that changes its meridional position, the change in f must be compensated by a change in the vorticity, ζ. As discussed later, the dependence of the Coriolis force on the latitude has a profound effect on the dynamics of atmospheric waves that shape the weather in both the extratropics and the Tropics. Since the meridional derivative of f is usually denoted by β, the aforementioned effect is called the beta-effect. 1.5.3

The Vorticity and the Divergence as Prognostic State Variables

The vorticity ζ and the divergence D can replace the two components of the horizontal wind vector, vH , as prognostic variables in the governing equations. This change of variables is made possible by the Helmholtz decomposition theorem, also known as the fundamental theorem of vector calculus, which states that a smooth vector field can be decomposed into a rotational (divergence-free) vector field and a divergent (vorticity-free) vector field. In particular, the Helmholtz decomposition of the horizontal wind vector is vH = vζ + vD ,

(1.447)

where vζ is the rotational (divergence-free) part of the horizontal wind vector and vD is the divergent (vorticity-free) part of the wind vector. The rotational part, vζ , of the horizontal wind vector vH is given by Eq. (1.436), that is, vζ = −∇H × ψk = k × ∇H ψ.

(1.448)

Governing Equations

127

Thus the relationship between vζ and ζ is established by Eqs. (1.448) and (1.439). A similar pair of equations for vD and D can be obtained by first noting that the vorticity-free component of the wind must satisfy ∇H × vD = 0.

(1.449)

vD = ∇H κ.

(1.450)

D = ∇2H κ,

(1.451)

vH = k × ∇H ψ + ∇H κ.

(1.452)

According to Eq. (1.101), a vector field vD automatically satisfies Eq. (1.449), if it can be written as the gradient of a scalar field, that is, The scalar field κ is called the velocity potential. The equation that establishes a relationship between vD and D, can be obtained by combining Eqs. (1.408) and (1.450). With the help of the stream function and the velocity potential, Eq. (1.447) can be written in the equivalent form The variables ζ and D can be easily computed from vH by taking its curl and divergence, respectively. The computation of vH from ζ and D is more involved: first, the stream function and the velocity potential have to be computed by solving Eq. (1.439) for ψ and Eq. (1.439) for κ, then, vζ and vD can be computed with the help of Eqs. (1.448) and (1.450), and finally, vH can be computed by Eq. (1.447). 1.5.4 1.5.4.1

The Vorticity and the Divergence Equation in Pressure Coordinate System The vorticity equation

A source of technical complication in the derivation of the vorticity equation in Sec. 1.5.2.1 was the nonlinearity of the term that represented the pressure gradient force. This complication can be eliminated by deriving the vorticity equation in pressure coordinate system, where the pressure gradient force is represented by a linear term. Because in the pressure coordinate system the vertical component of the momentum equation is replaced by the hydrostatic balance equation, the derivation of the vorticity equation can be started from the horizontal momentum equation, Eq. (1.298). That equation can be written in the equivalent form   ∂vH ∂vH 1 2 = v H × ζa k − ∇ H | vH | +φ − ω (1.453) ∂t 2 ∂p

128

Applicable Atmospheric Dynamics

by making use of the (v · ∇) vH = ∇H



1 | v |2 2



− vH × ζk + ω

∂vH , ∂p

(1.454)

analogue of Eq. (1.412). Exercise 1.45. Show that Eq. (1.453) leads to the vorticity equation   dζa ∂vH = −ζa D − ∇H ω × · k. (1.455) dt ∂p Answer. The left-hand side of Eq. (1.455) can be written as ∂ζa ∂ζ dζa = + (vp · ∇p ) ζa = + (vp · ∇p ) ζa . (1.456) dt ∂t ∂t The term ∂ζ/∂t in Eq. (1.456) can be obtained by taking first the curl of Eq. (1.453), using the horizontal component ∇H of the del operator, then the dot product of the resulting equation with k. The details of this calculation are the following. The curl of the first term on the right-hand side of Eq. (1.453) is ∇H × (vH × ζa k) = vH (∇H · ζa k) − ζa k (∇H · vH )

− (vH · ∇H ) ζa k + (ζa k · ∇H ) vH .

(1.457)

The first and the last terms on the right-hand side of Eq. (1.457) are zero vectors, because the vectors in those terms are multiplied by the dot products of orthogonal vectors. Thus, taking the dot product of Eq. (1.457) and k leads to [∇H × (vH × ζa k)] · k = −ζa D − (vH · ∇H ) ζa .

(1.458)

Because the second term on the right-hand side of Eq. (1.453) is the gradient of a scalar-valued function, the curl of that term is zero, hence, it does not make a contribution to the time-derivative ∂ζ/∂t. The curl of the last term on the right-hand side of Eq. (1.453) is   ∂vH ∂vH ∂vH −∇H × ω = −∇H ω × − ω ∇H × ∂p ∂p ∂p ∂ (∇H × vH ) ∂vH −ω , (1.459) = −∇H ω × ∂p ∂p where in the first step we made use of Eq. (1.103). The order of taking the spatial derivatives could be changed, because the coordinates of vH are assumed to be smooth functions of the coordinates of the vector of position. Taking the dot product of Eq. (1.459) and k yields

Governing Equations

    ∂vH ∂ζa ∂vH , · k = − ∇H ω × ·k−ω − ∇H × ω ∂p ∂p ∂p

129

(1.460)

where we made use of ∂ζa ∂ζ = , ∂p ∂p

(1.461)

which holds because f does not depend on p. The sum of Eqs. (1.458) and (1.460) provides the right-hand side of the equation   ∂ζ ∂ζa ∂vH = − (vH · ∇H ) ζa − ω − ζ a D − ∇H ω × · k, (1.462) ∂t ∂p ∂p which can also be written as   ∂vH ∂ζ = − (vp · ∇H ) ζa − ζa D − ∇H ω × · k. ∂t ∂p

(1.463)

Substituting ∂ζ/∂t from Eq. (1.463) into Eq. (1.456) leads to Eq. (1.455). Equation (1.455) is a restricted form of the three-dimensional vorticity equation, Eq (1.414): it describes the temporal changes in the vertical coordinate of the three-dimensional vorticity rather than the full threedimensional vorticity and also makes the assumption of hydrostatic balance. The latter assumption puts a restriction on the type of processes that can change the vorticity (the vertical coordinate of the three-dimensional vorticity). The first term on the right-hand side of Eq. (1.455) is the the stretchingtwisting term for pressure vertical coordinate. As before, this term cannot be the ultimate source of vorticity, because it cannot generate vorticity in a parcel whose initial vorticity is zero. It can, however, amplify or reduce the vorticity that was created earlier. In particular, when the vorticity of the parcel is positive (cyclonic), convergence increases the vorticity, while divergence decreases the vorticity of the parcel. The second term on the right-hand side of Eq. (1.455) is the baroclinic (solenoid) term for pressure vertical coordinate. As before, this term can generate vorticity, even if the initial vorticity is zero. A detailed analysis of this term is somewhat difficult using only the coordinate-free representation of the horizontal component of the fields. Here, we only note that this term shows that vorticity can be generated when vertical motions and a vertical wind shear are present. Exercise 1.46. As mentioned before an alternative approach for the manipulation of the equations is to carry out the calculations using a convenient coordinate system and then obtain the general form of the equations by

130

Applicable Atmospheric Dynamics

making use of the particular form of the del operator for the selected coordinate. Follow this strategy to derive Eq. (1.455), using the global Cartesian horizontal coordinates and pressure as the vertical coordinate. Answer. Using Cartesian horizontal coordinates, the vorticity tendency can be written as   ∂ ∂v ∂u ∂ ∂u ∂ ∂v ∂ζ = − − , (1.464) = ∂t ∂t ∂x ∂y ∂x ∂t ∂y ∂t where we made use of Eq. (1.407) and the assumed smoothness of the scalar fields of u and v in space and time, which allows for changing the order of the computation of the spatial and temporal derivatives. The equations for ∂u/∂t and ∂v/∂t, ∂u ∂u ∂u ∂u ∂φ = −u −v −ω − + f v, ∂t ∂x ∂y ∂p ∂x

(1.465)

∂v ∂v ∂v ∂v ∂φ = −u −v −ω − − f u, ∂t ∂x ∂y ∂p ∂y

(1.466)

can be obtained by combining Eq. (1.296) with Eqs. (1.275) and (1.276). Taking the partial derivative of Eq. (1.466) with respect to x and the partial derivative of Eq. (1.465) with respect to y, then taking the difference between the resulting two equations leads to ∂ζ ∂ζ ∂ζ ∂f ∂ζ = −u −v −ω − (ζ + f )D − u ∂t ∂x ∂y ∂p ∂x −v

∂u ∂ω ∂v ∂ω ∂f + − . ∂y ∂p ∂y ∂p ∂x

(1.467)

It can be easily verified that the right-hand side of Eq. (1.467) is the specific form of the right-hand side of Eq. (1.463) for Cartesian horizontal coordinates. 1.5.4.2

The divergence equation

Exercise 1.47. Show that using Cartesian horizontal coordinates and pressure as the vertical coordinate, the divergence equation is  2  2 ∂u ∂v ∂u ∂v ∂D ∂D ∂D ∂D −2 = −u −v −ω − − ∂t ∂x ∂y ∂p ∂x ∂y ∂x ∂y −

∂u ∂ω ∂v ∂ω − + f ζ − βu − ∇2H φ ∂p ∂x ∂p ∂y

(1.468)

Governing Equations

131

Answer. Equation (1.468) can be derived by taking the partial derivative of the zonal component of Eq. (1.465) with respect to x and the partial derivative of the meridional component of Eq. (1.466) with respect to y, then adding the resulting two equations. Equation (1.468) can be written in the equivalent form
∂D ∂D ∂D ∂D 1 2 1 2 1 2 = −u −v −ω − D − ζ + λ1 + λ22 ∂t ∂x ∂y ∂p 2 2 2 −

∂u ∂ω ∂v ∂ω − + f ζ − βu − ∇2H φ ∂p ∂x ∂p ∂y

with the help of the Pettersen transformation,  2  2
∂v ∂u ∂v 1 1 1 2 ∂u +2 = D2 + ζ 2 − + λ1 + λ22 , ∂x ∂y ∂x ∂y 2 2 2

(1.469)

(1.470)

where

λ1 =



∂u ∂v + ∂y ∂x



(1.471)



∂v ∂u − ∂y ∂x



(1.472)

is the shearing deformation and λ2 =

is the stretching deformation. The deformation λ is defined by q λ = λ21 + λ22 .

(1.473)

Exercise 1.48. What is the general vector form of the divergence equation, Eq. (1.469)? Answer.
1 ∂D 1 2 1 2 ∂D = −vH · ∇H D − ω − D − λ1 + λ22 + ζ 2 ∂t ∂p 2 2 2 ∂vH · ∇H ω + f ζ − βu − ∇2H φ. − ∂p

(1.474) (1.475)

Exercise 1.49. Show that an alternative vector form of the divergence equation is     ∂D 1 ∂vH 2 2 = k · ∇H × ζ a v − ∇ | vH | +φ − ∇H · ω . (1.476) ∂t 2 ∂p

132

Applicable Atmospheric Dynamics

Answer. The most straightforward approach to derive Eq. (1.476) is to take the dot product of the del operator and Eq. (1.453). This operation leads to the term on the left hand side and the second and third term on the right-hand side of Eq. (1.476). Only the first term on the right-hand side of Eq. (1.476) requires some additional algebra. In particular, we must show that ∇H · (vH × ζa k) = k · ∇H × ζa v.

(1.477)

An application of Eq. (1.106) to the left hand-side of Eq. (1.477) yields ∇H · (vH × ζa k) = ζa k · (∇H × vH ) − vH · (∇H × ζa k) ;

(1.478)

while an application of Eq. (1.103) to the right-hand side of Eq. (1.477) leads to k · ∇H × ζa v = k · [∇H ζa × vH + ζa (∇H × vH )]

= k · ∇H ζa × vH + ζa k · (∇H × vH ) .

(1.479)

Because the first term on the right-hand side of Eq. (1.478) is identical to the second term of the last part of Eq. (1.479), Eq. (1.477) holds if −vH · (∇H × ζa k) = k · ∇H ζa × vH .

(1.480)

An application of Eq. (1.103) to the left hand-side of Eq. (1.480) yields −vH · (∇H × ζa k) = −vH · [∇H ζa × k + ζa (∇H × k)]

= −vH · ∇H ζa × k = k · ∇H ζa × vH , (1.481)

where in the last step, we used Eq. (1.22) to rearrange the scalar triple product. 1.5.4.3

The vorticity and the divergence of the geostrophic wind

In pressure coordinate system, the geostrophic wind is defined by Eq. (1.302). Thus the ageostrophic component of the wind is 1 (1.482) va = vH − vg = vH − k × ∇H φ. f Exercise 1.50. Show that the vorticity associated with the geostrophic wind, ζg = (∇H × vg ) · k

(1.483)

1 1 ∆H φ − (∇H φ · ∇H ) f f

(1.484)

satisfies the equation ζg =

Governing Equations

133

Answer. Substituting vg from Eq. (1.302) into Eq. (1.483),    1 ζ g = ∇H × k × ∇H φ · k. f

(1.485)

The triple vector product within the brackets can be expanded with the help of Eq. (1.21) as ∇H ×



1 k × ∇H φ f



  1 1 k∆H φ − ∇H φ ∇H · k f f   1 1 − k · ∇H ∇H φ + (∇H φ · ∇H ) k. (1.486) f f

=

The second and the third terms on the right-hand side of Eq. (1.486) are zero, because they include the dot products of orthogonal vectors. Taking the dot products of the remaining two terms with k leads to Eq. (1.484). Exercise 1.51. What is the particular form of Eq. (1.484) for Cartesian horizontal coordinates? Answer. ζg = 1.5.4.4

1 f



∂2φ ∂2φ + 2 ∂x2 ∂y



−

β ∂φ . f 2 ∂y

(1.487)

The case of constant f

When the latitude-dependent Coriolis parameter f is approximated by a constant f0 , the last term of the right-hand side of Eq. (1.485) becomes zero and the geostrophic vorticity, ζg , can be written as ζg =

g 1 2 ∇H φ = ∇2H z. f0 f0

(1.488)

In addition, Eq. (1.302) can be written as vg = −∇H ×

1 φk. f0

(1.489)

This equation is formally equivalent to Eq. (1.436), with (1/f0 )φ playing the role of the stream function ψ. Thus, the stream function associated with the geostrophic wind vg is

134

Applicable Atmospheric Dynamics

g 1 φ = z, f0 f0

(1.490)

vg = −∇H × ψg k.

(1.491)

∇H · vg = 0.

(1.492)

D g = ∇H · v H = ∇H · v a ,

(1.493)

ψg = and Eq. (1.489) can be written as

Because vg can be written as the curl of a vector, it is divergence free, that is, Thus the divergence of vH is due to the ageostrophic component of the wind, because and the continuity equation, Eq. (1.291), can be written as ∂ω ∇H · v a = − . ∂p

(1.494)

Exercise 1.52. Show that the divergence of the geostrophic wind is not zero in the general case, where the assumption f ≈ f0 is not made. Answer. In the general case, the geostrophic wind is defined by Eq. (1.302). Hence, the divergence is   1 k × ∇H φ D g = ∇H · v g = ∇H · f   1 1 = ∇H φ · ∇H × k − k (∇H × ∇H φ) , (1.495) f f where we made use of Eq. (1.106). Since the last term of Eq. (1.495) is zero, because it is the curl of a gradient,   1 (1.496) D g = ∇H φ · ∇H × k . f Exercise 1.53. What is the particular form of Eq. (1.496) for Cartesian horizontal coordinates. Answer. Using Cartesian horizontal coordinates, Eq. (1.496) is i j   1 ∂φ ∂ ∂ i· ∇H φ · ∇H × k = f ∂x ∂x ∂y 0 0 =

the right-hand side of k 0 1/f

β ∂φ ∂(1/f ) = − vg , ∂x ∂y f

(1.497)

Governing Equations

135

where vg = −

1 ∂φ f ∂x

(1.498)

is the meridional component of the geostrophic wind. A constant f0 can be an accurate approximation to the latitudedependent f only in a narrow latitude band. Because in such a narrow latitude band a Cartesian coordinate system can provide a sufficiently accurate representation of the governing equations, the model domain is usually assumed to be a plane when f is approximated by a constant f0 . The resulting approximation is called the f-plane approximation. 1.5.5 1.5.5.1

Reduced Forms of the Vorticity and the Divergence Equations The motivation for reduced governing equations

Because reduced forms of the governing equations tend to support fewer types of atmospheric motion than the full system of primitive equations, certain types of motions can be filtered from the solutions by dropping and/or simplifying some terms of the primitive equations. This filtering effect can be highly useful when the goal is to study a particular atmospheric dynamical process in its purest form. Reduced equations once played a central role in the practice of numerical weather prediction. Today’s operational models are all based on the primitive equations and theoretical investigations are also often based on simulations with such models. It would be still hard to overestimate the importance of the role reduced equations have played in modern atmospheric dynamics: (i) analytical investigations are usually based on reduced forms of the equations, because the reduction can greatly simplify the mathematical analysis;110 (ii) numerical investigations of theoretical problems are often based on reduced forms of the equations; (iii) diagnostic calculations often use diagnostics that are based on reduced equations, even if the investigation is based on data that was produced 110 While

this statement is certainly true in general, the numerical solution of some of the diagnostic equations can be more challenging than the solution of the prognostic equations they replace.

136

Applicable Atmospheric Dynamics

by a state-of-the-art primitive equation model and/or its data assimilation system; (iv) some data assimilation schemes employ one of the reduced forms of the divergence equation to ensure a proper balance between the fields of the state variables in the analyses.111 While reduced equations are powerful tools, the results they provide should always be interpreted judiciously: the fact that a system of reduced equations can simulate some aspects of the dynamics of a particular atmospheric process does not guarantee that the filtered processes do not play an important role in that process in nature. For instance, a model based on the barotropic vorticity equation, which will be derived in Sec. 1.5.5.7 can produce a time series of wind fields that look realistic in the extratropics at the 500 hPa pressure level, provided that realistic initial conditions are supplied for the rotational part of the horizontal wind vector. The spatiotemporal evolution of the wind field, however, is not realistic in the sense that extratropical cyclones cannot develop in the model, even though cyclogenesis (the development of cyclones) plays a central role in shaping the wind field at the 500 hPa pressure level in nature. In other words, capturing some of the processes that shape the evolution of an atmospheric field can produce a deceptively realistic looking evolution of the field.112 The potential effects of the assumptions made to carry out the reduction of the equations should always be carefully considered and investigated. One obvious option for such an investigation is to assess the predictive skill of the reduced equations. For instance, the prediction of the wind field at the 500 hPa pressure level by a model based on the barotropic vorticity equation breaks down rapidly with increasing forecast time. The predictive skill of a reduced model should always be assessed by a comparison to the performance of a high quality model. The derivation of reduced equations, and even more so the justification of their use to investigate a specific problem, is a complex process. This section presents some of the fundamental arguments to support the use of reduced models for the investigation of some problems of dynamical meteorology and numerical weather prediction. 111 This

approach is primarily used by the 3-dimensional variational data assimilation schemes, which will be discussed in Sec. 4.3.3. 112 In the case of the barotropic vorticity equation, the realistic looking evolution of the flow is the result of capturing a single process: the horizontal advection of vorticity.

Governing Equations

1.5.5.2

137

General strategy for the reduction

Systems of reduced equations are usually introduced gradually, starting from the full set of primitive equations and eliminating, or simplifying, an increasing number of terms in each step.113 The elimination and the simplification of the terms can be carried out by considerations of the following types: (i) scale analysis: eliminating the terms of the smallest orders of magnitude for the dynamical processes of interest;114 (ii) symmetry arguments: preserving the key symmetries of the equations during the reduction process is essential for the proper conservation of the reduced forms of the invariants of motion;115 (iii) averaging arguments: averaging the fields of the state variables in the vertical direction provides the usual justification for the use of twodimensional barotropic vorticity equations. In the remainder of Sec. (1.5.5), we use considerations of type (i) and (iii). We will complete the derivation of the systems of reduced equations in Sec. 1.7 by employing considerations of type (ii). 1.5.5.3

General strategy for the scale analysis

While scientists normally prefer elegant forms of the equations, which include a minimal number of terms, in a scale analysis, it is often advantageous to break up each term of the original equations into as many new terms as possible, as the larger number of terms allows for a more refined scale analysis. This potential can be realized, only if there is a better separation between the magnitudes of the newly introduced terms than between the magnitudes of the original terms. The key components of a general strategy to increase the number of terms with a good separation of the magnitudes are (i) the replacement of the two horizontal components of the momentum equations by the vorticy and the divergence equations, 113 A

system of governing equations obtained by making only the assumption of hydrostatic balance is still considered a system of primitive equations. 114 A scale analysis of the atmospheric governing equations was first carried out by Jules Charney (Charney, 1948). 115 The importance of the preservation of the energy conserving properties of the equations was first pointed out by Arakawa (1962) and Lorenz (1960a). The relationship between the symmetries of the equations and the conservation laws will be discussed in Sec. 1.7.

138

Applicable Atmospheric Dynamics

(ii) the decomposition of the horizontal wind vector into a rotational part and a divergent part, (iii) decomposition of the thermodynamical variables (density and temperature) into a basic flow component, which depends only on the vertical coordinate of the vector of position, and a perturbation component, which depends on all three spatial coordinates and time. In what follows, we focus on components (i) and (ii), while component (iii) will be the subject of Sec. 2.3. Component (i) of the strategy simply suggests that we use Eqs. (1.455) and (1.469) instead of Eq. (1.298) in the scale analysis. This change of equations is advantageous, because the divergence equation has many more terms than the two components of the horizontal momentum equation. Another motivation to use the vorticity and the divergence equations for the scale analysis is that they allow for the implementation of component (ii), as the rotational and divergent components of the wind vector are fully determined by the vorticity and the divergence. The appeal of component (ii) itself is that replacing the horizontal wind vector by its rotational and divergent parts breaks up each term that includes vH into two terms. In addition, when the Rossby number is ǫRo . 10−1 , the magnitude of the divergent part of the wind, vD , is at least an order of magnitude smaller than U , the typical magnitude of the horizontal wind. To show that this statement is true, we first consider the divergence of the vector va + vgD , where vgD is the divergent part of the geostrophic wind. This sum of the ageostrophic wind and the divergent part of the geostrophic wind is equal to the difference between the horizontal wind and the rotational component of the geostrophic wind. Because the divergence of the rotational component of the geostrophic wind is zero, D = ∇H · vH = ∇H · (va + vgD ) = ∇H · va + ∇H · vgD .

(1.499)

The estimate of the magnitude of D is UD /L, where UD is the magnitude of vD . The magnitude of ∇H · va is (ǫR oU )/L, while the magnitude of ∇H · vgD is (βU )/f . The latter scale estimate can be obtained by making use of Eq. (1.497). Hence, the equivalent of Eq. (1.499) for the scales is UD ǫRo U βU . ∼ + L L f0 Multiplying Eq. (1.500) by L yields   βL U. UD ∼ ǫRo + f0

(1.500)

(1.501)

Governing Equations

139

The second term of Eq. (1.500), βU/f0 , contributes to the divergence only for spatial scales L . 106 , because according to Eq. (1.497), the spatial derivative in ∇H · vgD is in the meridional direction, and the longest linear spatial scale that can occur in the meridional direction is L . 106 . Because β=

2 | Ω | ∂(sin ϕ) 2|Ω| ∂f = = cos ϕ ∼ 10−11 , ∂y a ∂ϕ a

(1.502)

L . 106 implies that βU/f0 . 10−1 . Thus, for ǫRo & 10−1 , Eq. (1.502) can be written as UD ∼ ǫRo U.

−2

(1.503)

For ǫRo . 10 , the contribution of the divergence of the geostrophic wind to the total divergence can be larger than the contribution of the ageostrophic wind. In that case, however, UD will always be at least a magnitude smaller than U , because βU/f0 . 10−1 . Thus, for ǫRo . 10−1 , UD . 10−1 , U

(1.504)

Uζ ∼ U.

(1.505)

which also means that

1.5.5.4

Scale analysis of the vorticity equation

Substituting vH = vζ + vD into the vorticity equation, Eq. (1.462), yields ∂ζ ∂ζ = − [(vζ + vD ) · ∇H ] ζ − β (vζ + vD ) − ω ∂t ∂p   ∂ (vζ + vD ) − ζ a D − ∇H ω × · k. ∂p

(1.506)

The scale estimates of the terms in Eq. (1.506) are listed in Table 1.3. The two terms whose scale estimate may require further explanation are −ω

U UD U 2 UD ∂ζ ∼ = , ∂p L2 L2 U

(1.507)

and     ∂ (vζ + vD ) U 2 UD UD UD ∇H ω × · k ∼ 2 (U + UD ) = 2 1+ . (1.508) ∂p L L U U These scale estimates can be obtained by first rewriting the continuity equation, Eq. (1.291), as ∂ω = −D. ∂p

(1.509)

140

Applicable Atmospheric Dynamics Table 1.3

Scale estimates of the terms of the vorticity equation.

Term

Scale estimate

Scale estimate for ǫRo . 10−1 s−2

∂ζ/∂t

(U/L)2

. 10−10

vζ · ∇H ζ

(U/L)2

. 10−10

(U/L)2 (U

vD · ∇H ζ

D /U )

. 10−11

βvζ

βU

. 10−10

βvD

βU (UD /U )

. 10−11

ω(∂ζ/∂p)

(U/L)2 (UD /U )

. 10−11

(U/L)2 (U

. 10−11

ζD fD

D /U )

f0 UD /L


∇H ω × ∂vζ /∂p · k (∇H ω × ∂vD /∂p) · k

(U/L)2 (U

D /U ) (U/L)2 (UD /U )2

. 10−10 . 10−11 . 10−12

From this equation, UD ∆p, (1.510) L where Ω is the vertical change in ω over a pressure difference ∆p. This pressure difference can be left unspecified, because ∆p cancels out in both Eqs. (1.507) and (1.508), as U ∂ζ ∼ , (1.511) ∂p L∆p and ∂ (vζ + vD ) U + UD ∼ . (1.512) ∂p ∆p Because the goal is to derive reduced equations to describe large and synoptic scale motions, we focus on the case where ǫRo . 10−1 . According to Eq. (1.504), the magnitude of the terms of scale (U/L)2 , (U/L)2 (UD /U ) and (U/L)2 (UD /U )2 decreases by at least an order of magnitude. Equation (1.506) also has terms of order βU , βUD , and (f0 UD )/L. Of these terms, βUD is one order of magnitude smaller than βU , but the difference between the magnitude of these terms and the magnitude of the other terms depends on the magnitude of U and L. For synoptic scale motions, both βU and (f0 UD )/L are . 10−10 , which is the same as the magnitude of U 2 /L2 . Thus the leading order terms are those with magnitude (U/L)2 , βU and (f0 UD )/L. When only these terms of Eq. (1.506) are retained, the vorticity equation becomes ∂ζ = − (vζ · ∇H ) ζa − f D. (1.513) ∂t ω∼

Governing Equations

141

When the terms that are . 10−11 are also retained, the reduced vorticity equation is   ∂ζ ∂ (vζ ) ∂ζ = − (vH · ∇H ) ζa − ω − ζ a D − ∇H ω × · k. (1.514) ∂t ∂p ∂p The only term of Eq. (1.506), which is not included in Eq. (1.514) is   ∂ (vζ ) ∇H ω × · k, (1.515) ∂p

which has a magnitude of . 10−12 . 1.5.5.5

Scale analysis of the divergence equation

Exercise 1.54. Show that the divergence equation in pressure coordinates, Eq. (1.474), can be written in the equivalent form ∂D ∂D = −vH · ∇H D − ω − D2 − J (u, v) ∂t ∂p ∂vH · ∇H ω + f ζ − βu − ∇2H φ. − ∂p

(1.516)

where J (u, v) is the Jacobian of the state variables u and v. Answer. Equations (1.474) and (1.476) are equivalent, if
1 1 1 2 D2 + J (u, v) = − D2 − (1.517) λ1 + λ22 + ζ 2 . 2 2 2 The easiest way to show that Eq. (1.517) holds is to write each term in Cartesian coordinates. Exercise 1.55. Provide an estimate of the magnitude of each term in the equation that results from substituting vH = vζ + vD into Eq. (1.476). Answer. See Table 1.4. Retaining only the two leading order (∼ 10−9 ) terms of Eq. (1.476) yields ζ=

1 2 g ∇ φ = ∇2 z. f f

(1.518)

This equation is an approximate form of Eq. (1.484), which defines the vorticity of the geostrophic wind.116 In addition, Eq. (1.518) is formally identical to Eq. (1.488), the equation for the vorticity of the geostrophic wind for constant Coriolis parameter, except that in Eq (1.518) f depends 116 Equation

(1.518) can be obtained by dropping the second term of Eq. (1.484).

142

Applicable Atmospheric Dynamics Table 1.4

Scale estimates of the terms of the divergence equation.

Term

Scale estimate

Scale estimate for ǫRo . 10−1 s−2

∂D/∂t

(U/L)2 (UD /U )

. 10−11

(U/L)2 (U

. 10−12

∇ω · (∂vD /∂p)

D /U ) 2 D /U ) 2 (U/L) (UD /U )2 (U/L)2 (UD /U )2 (U/L)2 (UD /U ) (U/L)2 (UD /U )2

. 10−11

(U/L)2 (U

fζ

f0 U/L

. 10−9

βu

βU

. 10−10

∇2 φ

f0 U/L

. 10−9

J (uζ , vζ )

(U/L)2

. 10−10

vζ · ∇H D vD · ∇H D ω(∂D/∂p) D2 ∇ω · (∂vζ /∂p)

J (uζ , vD ) J (uD , vζ ) J (uD , vD )

(U/L)2 (U

D /U ) (U/L)2 (UD /U ) (U/L)2 (UD /U )2

. 10−12 . 10−12 . 10−11 . 10−12

. 10−11 . 10−11 . 10−12

on the latitude. In essence, the scale analysis suggests that in the lowest order approximation to the divergence equation, the vorticity can be computed as the flow was in geostrophic balance and the Coriolis parameter did not change with latitude. It should be emphasized, however, that in Eq. (1.518), f is the local value of the Coriolis parameter rather than the constant f0 of Eq. (1.488). Hence, Eq. (1.518) can be used for the computation of the vorticity at any latitude, except for the Equator, where it becomes singular. If the two ∼ 10−10 terms of Eq. (1.476) are also retained, Eq. (1.518) becomes f ζ − βu + 2J (u, v) = ∇2 φ.

(1.519)

Equation (1.519), which is called the nonlinear balance equation, is a diagnostic relationship between the rotational component of the wind vector and the geopotential height. This relationship becomes most transparent, when all wind related variables are expressed by the stream function: making use of Eqs. (1.438) and (1.439), Eq. (1.519) can be written as  2 2 ∂ψ ∂ ψ ∂2ψ ∂2ψ 2 f∇ ψ − β +2 = ∇2 φ. (1.520) −2 2 ∂y ∂x∂ ∂x ∂y 2 In addition, the first two terms on the left-hand side of Eq. (1.520) can be amalgamated into a single term, which leads to

Governing Equations

∇(f ∇ψ) + 2



∂2ψ ∂x∂

2

−2

∂2ψ ∂2ψ = ∇2 φ. ∂x2 ∂y 2

143

(1.521)

With the help of the nonlinear balance equation, at any given time when the geopotential height field is known, the stream function, or more generally, all state variables related to the rotational component of the wind can be computed. Likewise, at any given time when the rotational component of the wind vector is known, the geopotential height field can be computed with the help of the nonlinear balance equation. This equation is the most accurate, and also the most complicated, diagnostic equation between the rotational wind and the geopotential height: once the ∼ 10−11 terms are also added, the tendency of the divergence, ∂D/∂t, appears in the equation, thus destroying the diagnostic nature of the equation. The nonlinear balance equation, however, is only a limited accuracy approximation to the divergence equation. Hence, a time dependent solution of the full equations can be expected to provide a better description of the relationship between the different state variables than the nonlinear balance equation. Because the nonlinear term 2J (u, v) of Eq. (1.519) significantly complicates the solution of the equation, it is often dropped, although the elimination of this term cannot be justified strictly based on a scale analysis. The resulting equation, f ζ − βu = ∇2 φ,

(1.522)

is called the linear balance equation. With the help of the stream function, Eq (1.522) can be written as ∇(f ∇ψ) = ∇2 φ. 1.5.5.6

(1.523)

The quasi-geostrophic approximation

The two components of the horizontal momentum equation can be replaced by a reduced vorticity equation and a reduced divergence equation. The proper pairs of equations are usually determined by considering the scale of the terms in both equations and the symmetry properties of the resulting system of governing equations. The terms with the largest magnitudes in either set of the reduced equations are the two ∼ 10−9 terms of the divergence equation, which suggests that the simplest quantitative model of synoptic scale meteorology is Eq. (1.518). According to this model, the atmosphere can be described by a static geopotential height field generated by a static vorticity field. The relationship between the vorticity and the geopotential height fields

144

Applicable Atmospheric Dynamics

is formally identical to the relationship between the two fields under the assumptions that the atmospheric flow is in geostrophic balance and the Coriolis parameter is constant.117 Thus the wind field associated with Eq. (1.518) is defined by Eq. (1.489), that is, by the geostrophic wind for constant Coriolis parameter. In essence, by leading to Eq. (1.518), the scale analysis suggests that the meridional change in the Coriolis parameter plays a secondary role in the relationship between the geopotential height and the horizontal wind fields.118 According to Eq. (1.489), the wind blows parallel with the isohypses, such that looking downstream, the higher values of z are on the righthand side.119 Figure 1.15 shows that this simple rule also provides a good description of the relationship between the instantaneous wind field and geopotential height field in the free atmosphere. This observation suggests that Eq. (1.489) is a good zeroth order approximation for the description of the relationship between the gepotential height and wind fields, even if the flow changes in time. It also implies that Eq. (1.518) provides a good approximate relationship (diagnostic equation) for the vorticity and the geopotential height field even for a time dependent vorticity field. Using Eqs. (1.489) and (1.518) as diagnostic equations in combination with a prognostic equation for the vorticity is called the quasi-geostrophic approximation. The lowest order reduced form of the vorticity equation is Eq. (1.513). The combination of the quasi-geostrophic approximation and Eq. (1.513) leads to the quasi-geostrophic vorticity equation., ∂ζg = − (vg · ∇H ) (ζg + f ) − fg D. (1.524) ∂t The subscript g indicates state variables and parameters computed according to the quasi-geostrophic approximation. In particular, fg is the Coriolis parameter used in the quasi-geostrophic relationships between ζg , ψg , vg and z. Manipulating quasi-geostrophic equations, fg should always be treated as a constant, but the resulting expressions can be evaluated by either using the actual local value of f or a prescribed constant value f0 , as deemed more appropriate for the particular application. 117 This

property of the model provided by Eq. (1.518) can be seen by comparing Eqs. (1.488) and (1.518). 118 The effect of the meridional change in the Coriolis parameter on the relationship of the two fields is represented by a term of scale ∼ 10−10 . The addition of that term to Eq. (1.518) leads to the linear balance equation, Eq. (1.523). 119 Practitioners of synoptic meteorology have been using this model for decades to read maps of the geopotential height.

Governing Equations

145

The prognostic Eq. (1.524) and the diagnostic Eqs (1.518), Eq. (1.489), (1.490) and (1.491) do not form a closed system of equations, because they do not include a rule for the computation of the divergence D, which is an unknown variable of Eq. (1.524). There are two approaches to eliminate this problem. One of them is to find a set of assumptions that eliminates the term −fg D. This approach is the subject of the next section (Sec. 1.5.5.7). The second approach is based on adding the continuity equation, Eq. (1.409), and a reduced form of the thermodynamical equation to the system of equations: the continuity equation relates the divergence D to the vertical velocity, ω, while the thermodynamical equation relates the stream function, ψg to ω. This approach will be discussed in Sec. 2.3. Finally, it should be noted that while the quasi-geostrophic approximation has proved one of the most fruitful concepts of atmospheric dynamics, it does have some problematic aspects. Most importantly, while Eq. (1.513) consists of terms of order ∼ 10−10 , in the balance equation only terms of order ∼ 10−9 are retained. It would be a more consistent approach to retain all terms of order ∼ 10−10 in the balance equation as well, that is, to use the nonlinear balance equation rather than the geostrophic balance equation. Such change of the balance equation, however, would make the analysis of the resulting system of equations by analytical techniques virtually impossible. Designing numerical schemes and interpreting the results of numerical experiments with the resulting models would also be significantly more difficult than for the quasi-geostrophic equations. Using such a complicated system of reduced equations instead of the more realistic primitive equations would not make much sense. The only reason why such a system of equations had ever been considered is that maintaining the proper atmospheric balance in the primitive equation models was a largely unsolved problem until the 1970’s. 1.5.5.7

The barotropic vorticity equation

Jules Charney120 argued that a vertical averaging of the governing equations for large and synoptic scale motions would lead to the equation ∂ζg = − (vg · ∇H ) (ζg + f ) , ∂t

(1.525)

where ζg and vg are related by Eqs. (1.488)–(1.491). This equation, which is called the quasi-geostrophic barotropic vorticity equation, or simply the 120 Charney

(1949).

146

Applicable Atmospheric Dynamics

barotropic vorticity equation, is identical to (1.524), except that it does not include the term −fg D. The key assumption that Charney made was that the direction of the horizontal wind at a given location did not change with the altitude. In pressure coordinate system, Charney’s assumption can be written as vH (rH , p) = A(p)v(rH ).

(1.526)

Equation (1.526) is not an unreasonable, albeit limited accuracy, approximation for the wind field at the synoptic scales in the free atmosphere.121 A model atmosphere that satisfies Eq. (1.526) is called an equivalent barotropic atmosphere. The atmospheric level where the horizontal wind field is most similar to the vertically averaged wind field of an equivalent barotropic atmosphere is called the equivalent barotropic level.122 The convention of synoptic meteorology is to treat the 500 hPa pressure level as the equivalent barotropic level: when synoptic meteorologists assess the weather situation, they often first look at the geopotential height maps for the 500 hPa pressure level to develop a general sense about the major weather systems and their movements. This practice can be justified based on the concept of the equivalent barotropic atmosphere and the geostrophic approximation. The most common verification scores to measure the accuracy of numerical weather forecasts are also computed at the 500 hPa pressure level.123 For proper initial and boundary conditions, Eq. (1.525) can be solved for z: given the geopotential height field at initial time and a set of proper boundary conditions, the geopotential height field can be computed for all future times. In practice, Eq. (1.525) is usually solved by rewriting the equation as ∂ζg = −J (ζg + f, ψg ) , ∂t ζg = ∇2H ψg ,

(1.527) (1.528)

where the conversion between ψg and z at the beginning and at the end of the model integration is done by Eq. (1.490). A numerical solution strategy 121 Equation

(1.526) is obviously not satisfied in the Ekmam layer. concept of the equivalent barotropic atmosphere and the equivalent barotropic level was introduced by Charney (1947, 1949). Charney’s estimate for the equivalent barotropic level was between 550 and 600 hPa. 123 The standard verification score that has been computed at all numerical weather prediction centers for decades is the anomaly correlation for the geopotential height at the 500 hPa level. This score will be introduced and discussed in Sec. 3.8.2.

122 The

Governing Equations

147

for the barotropic vorticity equation has three major components: (i) a strategy to calculate the Jacobian −J (ζg + f, ψg ) in Eq. (1.527), (ii) an algorithm to solve the elliptic partial differential equation Eq. (1.528) for ψg , and (iii) a time integration algorithm to determine the time evolution of ζg from ∂ζg /∂t. The first successful attempt to design a numerical solution strategy along these lines led to the first successful numerical weather prediction experiment.124 Some authors refer to the two-dimensional vorticity equation, Eq. (1.431), as the barotropic vorticity equation. This use of the terminology is misleading, because the differences between the qualitative dynamics of the two equations are important. The source of this differences is the β-effect, which is included in the barotropic vorticity equation, but not in the two dimensional vorticity equation. For instance, Rossby waves, which play a central role in synoptic scale meteorology, would not exist without the beta-effect.125 1.5.5.8

Shallow-water vorticity and divergence equations

A reduced form of the atmospheric governing equations that has played a central role in the development of the techniques of atmospheric modeling is the system of shallow-water equations. These equations can be obtained by the shallow-water approximation (Fig. 1.18) to the full set of governing equations: (i) the composition and thermodynamical state of the model atmosphere is homogeneous, that is, the composition, the density and the temperature of air are constant in both space and time; (ii) the flow is in hydrostatic balance and purely zonal with no vertical wind shear; (iii) the modeled layer has a free top that can rise or sink in response to the motion of the flow. In essence, the shallow water approximation converts the atmosphere into a classical mechanical system, in which the thermodynamical processes can 124 Charney

et al. (1950). Harper (2008) provides a historical account of the developments that led to the implementation of a baroclinic model as the first operational numerical weather prediction model of the United States. 125 Rossby waves will be discussed in Sec. 2.4.2.

148

Applicable Atmospheric Dynamics

η ρ=const.

h v

Τ=const.

v H

z0

Fig. 1.18 Schematic illustration of a shallow water flow. H is the average height of the free surface, h is the spatiotemporally varying depth of the atmospheric layer and z0 is the elevation of the topography.

be ignored; thus the spatiotemporal evolution of the flow is governed by the two horizontal components of the momentum equation and the continuity equation. Since height is a dependent state variable and the layer is assumed to be in hydrostatic balance, as in the case of the pressure vertical coordinate, by analogy, the horizontal momentum equation can be written as dvH = −f k × v − g∇H η, dt

(1.529)

∇H Φ = g∇H (H + η) = g∇H η.

(1.530)

where we made use of

For the shallow water flow, we can think of the air parcels as atmospheric columns of depth h = H + η − z0 with an infinitesimal horizontal extent. The parcels can travel in the horizontal direction and they can also stretch, or compress. in the vertical direction when the neighboring parcels squeeze them or let them expand in the horizontal direction. That is, the shallow water layer responds to the convergence, or the divergence, of the horizontal flow by changing the elevation of the top surface rather than the density of air.126 These considerations lead to the continuity equation dh = −hD. (1.531) dt An important property of the shallow-water momentum equation is that it is formally similar to the horizontal momentum equation in pressure co126 The

assumption of a free surface is necessary, because otherwise the constant density could not be maintained in a plausible way.

Governing Equations

149

ordinate system. More precisely, the only difference of the horizontal momentum equation in pressure coordinate system and the shallow-water momentum equation is that the latter does not include the vertical advection term. Thus, the vorticity equation, Eq. (1.455), becomes dζa = −ζa D (1.532) dt for a shallow-water flow. That is, changes in the absolute vorticity of the parcels are solely due to stretching and twisting. Making use of the definition of the Lagrangian time derivative, Eq. (1.532) can be written in the equivalent forms ∂ζa = −∇ · ζa vH ∂t

(1.533)

∂ζ = −∇ · ζa vH . ∂t

(1.534)

and

Exercise 1.56. Show that in spherical coordinate system, Eq. (1.534) takes the form ∂ζ 1 1 ∂ (ζa V ) ∂ (ζa U ) =− − . ∂t a cos2 ϕ ∂λ a cos ϕ ∂ϕ

(1.535)

Answer. Making use of the definition of the pseudo-wind and applying Eq. (1.132) to the right-hand side of Eq. (1.534) yields     1 ∂ 1 1 ζ a VH = − ζa U −∇ · ζa vH = −∇ · cos ϕ a cos ϕ ∂λ cos ϕ   1 ∂ tan ϕ 1 − ζa V + ζa V a ∂ϕ cos ϕ a cos ϕ =− − =−

1 1 ∂ (ζa V ) ∂ (ζa U ) − a cos2 ϕ ∂λ a cos ϕ ∂ϕ tan ϕ tan ϕ ζa V + ζa V a cos ϕ a cos ϕ 1 ∂ (ζa V ) ∂ (ζa U ) 1 − . a cos2 ϕ ∂λ a cos ϕ ∂ϕ

(1.536)

Exercise 1.57. Show that replacing ϕ by µ = sin ϕ as the zonal coordinate of the vector of position, Eq. (1.535) becomes ∂ζ 1 ∂ (ζa U ) 1 ∂ (ζa V ) =− − . ∂t a (1 − µ2 ) ∂λ a ∂µ

(1.537)

150

Applicable Atmospheric Dynamics

Answer. The first term of the right-hand side of Eq. (1.535) and the first term of the right-hand side of (1.537) are obviously equal. The equality of the second terms of the right-hand side of the same equations can be shown in the following steps. First, −

1 ∂ (ζa V ) ∂ϕ 1 ∂ (ζa V ) =− a ∂µ a ∂ϕ ∂µ

(1.538)

and taking the derivative of µ = sin ϕ with respect to µ yields 1 = cos ϕ

∂ϕ . ∂µ

(1.539)

Substituting ∂ϕ/∂µ from Eq. (1.539) into the right-hand side of Eq. (1.538) leads to −

1 ∂ (ζa V ) 1 ∂ (ζa V ) =− . a ∂µ a cos ϕ ∂ϕ

(1.540)

The divergence equation for the shallow-water flow can be obtained by dropping the last term on the right-hand side of Eq. (1.476), which yields   1 ∂D 2 2 = k · ∇H × ζ a v − ∇ | vH | +φ . (1.541) ∂t 2 Exercise 1.58. Show that in a spherical coordinate system where µ = sin φ is used as the meridional coordinate, Eq. (1.541) can be written as 1 ∂ (ζa V ) 1 ∂ (ζa U ) ∂D = − ∂t a (1 − µ2 ) ∂λ a ∂µ   U +V 2 + ∇2 +φ . 2 (1 − µ2 )

(1.542)

Answer. Making use of Eq. (1.133), the first term on the right-hand side of Eq. (1.542) can be written as k · ∇H × ζ a v =

1 ∂ (ζa V ) 1 ∂ (ζa U ) 1 − + ζa U tan ϕ. a cos ϕ ∂λ a ∂ϕ a

(1.543)

The first and the second terms on the right-hand side of Eq. (1.542) can be obtained from the right-hand side of Eq. (1.543) by following the steps taken in the solution of Exercises 1.56 and 1.57. The equivalence of the last terms on the right-hand side of Eqs. (1.541) and (1.542) is obvious.

Governing Equations

151

Exercise 1.59. Show that in a spherical coordinate system using µ = sin ϕ as the meridional coordinate, the vorticity and the divergence can be computed from the two coordinates of the pseudo-wind by ζ= D=

1 ∂U ∂V 1 − , 2 a (1 − µ ) ∂λ a ∂µ

1 1 ∂V ∂U + . a (1 − µ2 ) ∂λ a ∂µ

(1.544) (1.545)

Exercise 1.60. Show that the two components of the pseudo-wind satisfy the following equations:
1 − µ2 ∂ψ 1 ∂κ − (1.546) U = a ∂λ a ∂µ
1 − µ2 ∂κ 1 ∂ψ V = + . (1.547) a ∂λ a ∂µ Answer. Equation (1.452) can be written in the equivalent form VH = cos ϕ [k × ∇ψ + ∇κ] . (1.548) Equations (1.546) and (1.547) can be obtained by substituting the proper form of the del operator using λ and µ as the horizontal spherical coordinates. 1.6

Potential Vorticity (PV)

We have seen that for a two-dimensional, purely rotational flow, a single prognostic equation, expressing the Lagrangian conservation of vorticity, can provide a full description of the evolution of the flow. A similar equation and a similar state variable, called the potential vorticity (PV), also exists for a fully three-dimensional atmospheric flow. Potential vorticity is primarily used as a diagnostic variable. 1.6.1 1.6.1.1

General Case Definition

The potential vorticity, q, is defined by ζ · ∇θ . (1.549) q = ζ˜ a · ∇θ = a ρ Assuming that all processes in the atmosphere are adiabatic, Qt = 0, the potential vorticity is a Lagrangian invariant, that is,

152

Applicable Atmospheric Dynamics

dq = 0. dt

(1.550)

Exercise 1.61. Show that Eq. (1.550) follows from Eqs. (1.414) under the assumption that Qt = 0.127 Answer. Substituting the definition of q from Eq. (1.549) into Eq. (1.550) leads to   ˜ · ∇θ d ζ dq dζ˜ d (∇θ) = = ∇θ · a + ζ˜ · . (1.551) dt dt dt dt An equation for the first term on the right-hand side of Eq. (1.551) can be obtained by taking the dot product of ∇θ and Eq. (1.414), which yields      dζ˜ 1 1 ∇ × ∇p . (1.552) ∇θ · a = ∇θ ζ˜ a · ∇ v − ∇θ · dt ρ ρ The second term on the right-hand side of Eq. (1.551) is zero, which can be seen by first expressing θ as pκ 1 θ (1/ρ, p) = r p−κ+1 , (1.553) Rρ by combining Poisson’s equation, Eq. (1.339), and the equation of state, then computing ∇θ from Eq. (1.553) as   pκr −κ+1 1 κ −κ (1.554) ∇ − p ∇p . p ∇θ = R ρ ρ Since ∇θ is a linear combination of ∇(1/ρ) and ∇p, it is orthogonal to the two vectors defined by the vectorial product of the two gradients. Thus, the second term on the right-hand side of Eq. (1.551) is zero, because it is the dot product of two orthogonal vectors. With this result, Eq. (1.552) takes the form    dζ˜ ∇θ · a = ∇θ ζ˜a · ∇ v . (1.555) dt Next, we rewrite the second term on the right-hand side of Eq. (1.551) as     ∂ (∇θ) ∇ (∂θ) d (∇θ) = ζ˜ a · + (v · ∇) ∇θ = ζ˜ a · + (v · ∇) ∇θ ζ˜ a · dt ∂t ∂t       dθ − ∇θ ζ˜ a · ∇ v = −∇θ ζ˜ a · ∇ v , (1.556) = ζ˜a · ∇ dt 127 Because

this form of the potential vorticity was first shown to be a Lagrangian invariant in a series of papers by Hans Ertel (e.g., Ertel, 1942), q is often referred to as Ertel’s potential vorticity. Some contemporary authors (e.g., McIntyre, 2014) prefer using the term Rossby-Ertel potential vorticity, because the material conservation of potential vorticity type invariants was first discussed by Rossby in the 1930s. In addition, Rossby and Ertel obviously interacted at the time when Ertel published his papers on the topic (Samelson, 2003).

Governing Equations

153

where in the last step, we made use of dθ/dt = 0, which follows from Qt = 0. Substituting Eqs. (1.555) and (1.556) into Eq. (1.551) leads to Eq. (1.550). 1.6.1.2

The effects of diabating heating and friction

Exercise 1.62. Show that in the presence of diabatic heating or cooling, the potential vorticity equation takes the form dq dθ = ζ˜ a · ∇ . (1.557) dt dt Answer. In the presence of diabatic heating or cooling, the potential temperature of the air parcel is not constant, thus dθ/dt 6= 0 and the term ζ˜ a · ∇(dθ/dt) does not drop out in the last step of Eq. (1.556). Equation (1.557) shows that diabatic heating or cooling can amplify or weaken the potential vorticity depending on the direction of the gradient of the heating, or cooling, and the direction of the components of the threedimensional absolute vorticity. The equation also shows, however, that if the absolute vorticity ζ˜ a was zero in the atmosphere, diabatic heating or cooling could not produce potential vorticity. Of course, due to Earth’s rotation, the planetary vorticity, 2Ω, is always present and diabatic heating or cooling generates potential vorticity.128 In the boundary layer, the frictional effects become an additional source/sink of potential vorticity. In particular, adding a term Fr to the momentum equation to represent the frictional force and repeating the steps that led to Eq. (1.557), we obtain dθ 1 dq = ζ˜ a · ∇ + ∇ × Fr · ∇θ. (1.558) dt dt ρ Equation (1.558) shows that the frictional force can generate potential vorticity in the boundary layer even in the absence of diabatic heating, provided that the gradient of the potential temperature is not a null vector. Exercise 1.63. Show that the mass-weighted volume integral of the potential vorticity for the entire atmosphere can be written as  Z Z  d dθ + θ∇ × Fr · dS, (1.559) ρq dV = ζa dt V dt S R where the surface integral S is computed at the bottom of the atmosphere for the whole globe, and dS = n dS, where n is the vector normal to the Earth’s surface. 128 In

theory, the case ζ = −Ω would be an exception, but it has no significance in practice.

154

Applicable Atmospheric Dynamics

Answer. Multiplying Eq. (1.557) by ρ yields ρ

dθ dq = ζa · ∇ + ∇ × Fr · ∇θ. dt dt

(1.560)

Next, we show that the right-hand side of Eq. (1.560) can be written as     dθ dθ + ∇ × Fr · ∇θ = ∇ ζ a + (∇ × Fr) θ . (1.561) ζa · ∇ dt dt The right-hand side of Eq. (1.561) can be obtained by the repeated use of Eqs. (1.102) and (1.104), as    dθ dθ dθ ζa · ∇ = ∇ · ζa ∇ · ζa, (1.562) − dt dt dt where the second term of the right-hand side is zero, because ∇ · ζ a is the divergence of a vector that is defined by the sum of the curl of a vector and a constant vector; and ∇ × Fr · ∇θ = ∇ · [(∇ × Fr) θ] − θ [∇ · (∇ × Fr)] ,

(1.563)

where the second term of the right-hand side is zero, because it is the divergence of the curl of a vector. Substituting the definition of q into the left-hand side of Eq. (1.560) leads to d (ζ a · ∇θ) 1 dρ dq = + (ζ a · ∇θ) dt dt ρ dt ∂ (ζ a · ∇θ) = + (v · ∇) (ζ a · ∇θ) + (ζ a · ∇θ) ∇ · v ∂t ∂ (ζ a · ∇θ) + ∇ [v · (ζ a · ∇θ)] = ∂t ∂ (ρq) + ∇ [v · (ζ a · ∇θ)] . (1.564) = ∂t Substituting the expressions we obtained for the left- and right-hand sides of Eq. (1.558) from Eqs. (1.561) and (1.564) yields     ∂ (ρq) dθ + ∇ [v (ζ a · ∇θ)] = ∇ ζ a + (∇ × Fr) θ . (1.565) ∂t dt ρ

Taking the volume integral of both sides of Eq. (1.565) and making use of the Gauss-theorem, we obtain  Z Z Z    d dθ + (∇ × Fr) θ dS. (ρq) dV + [v (ζ a · ∇θ)] dS = ζa dt V dt S S (1.566)

Governing Equations

155

We were allowed to change the order of the computation of the volume integral and the time derivative, because the state variables are assumed to be continuously differentiable functions of space and time. In addition, we replaced the partial time derivative with a full time derivative, because the volume integral depends only on time, thus the two time derivatives are the same. The surface integral on the left-hand side vanishes, because v = 0 at both the bottom and the top of the atmosphere. The surface integral vanishes only at the top of the atmosphere, where the frictional force is zero, there is no source for diabatic heating or cooling and the vorticity is a null vector. Dropping the vanishing term from the left-hand side, Eq. (1.565) becomes identical to Eq. (1.559). Equation (1.559) shows that diabatic heating and frictional forces cannot be the net source of potential vorticity in the free atmosphere. Since frictional forces can be neglected in the free atmosphere,129 Eq. (1.559) indicates that whenever diabatic processes create positive potential vorticity, they must also create the same amount of negative potential vorticity in an adjacent region. Thus one interpretation of the role of diabatic processes in the free atmosphere is that they help redistribute potential vorticity by changing the potential vorticity of the parcels. Equation (1.559) also shows that the surface is the only location where a net positive or negative potential vorticity can be created. In particular, over a flat surface (e.g., over the ocean), n is equal to the unit vector of the vertical direction, k, and the source term due to diabatic heating can be written as  Z Z Z  dθ dθ dθ (1.567) (f + ζ) dS. · dS = ζa k · kdS, = ζa dt dt dt S S S That is, over a flat surface, the generation of the potential vorticity is solely due to the vertical component of the three-dimensional absolute vorticity. 1.6.2

Hydrostatic Case

Since global atmospheric data sets are usually produced by hydrostatic models and many limited area models are also integrated in hydrostatic mode, the question naturally arises whether the potential vorticity defined by Eq. (1.549) remains a Lagrangian invariant in the hydrostatic case. 129 The

situation is somewhat different in a numerical model, where the diffusion terms that are included to control the effects of the truncation errors introduce dissipative effects into the dynamics at scales where such effects do not exist in nature. This issue will be discussed in detail in Sec. 3.3.1.

156

Applicable Atmospheric Dynamics

While the strict answer is ‘no’, there exists an analogous state variable, which is a Lagrangian invariant. For this variable, the absolute vorticity plays a role analogous to that played by the three-dimensional absolute vorticity in the potential vorticity. This state variable is called the isentropic potential vorticity (IPV). The Lagrangian conservation of this quantity was discovered by Rossby130 and our development of the subject, by first discussing the conservation of potential vorticity for a shallow-water flow, loosely follows the logic of Rossby’s original paper. 1.6.2.1

Shallow water potential vorticity

The expression for the potential vorticity can be greatly simplified by using the shallow-water approximation. The shallow water potential vorticity is qh =

ζa f +ζ = . h h

(1.568)

Exercise 1.64. Show that in a shallow water flow the shallow water potential vorticity is conserved by the air parcels. Answer. Because the horizontal momentum equation of the shallow water equations is formally identical to the horizontal momentum equation for the pressure vertical coordinate, except that the vertical coordinate of the velocity is zero, the vorticity equation for the shallow water flow can be obtained by dropping the second term on the right-hand side of Eq. (1.455); that is, d (ζ + f ) = − (ζ + f ) D. dt Then, the Lagrangian time derivative of qh can be written as

(1.569)

dqh 1 d(ζ + f ) ζ + f dh = − dt h dt h2 dt ζ +f ζ +f =− D+ D = 0, (1.570) h h where, in the second step, we substituted the Lagrangian time derivative of h from Eq. (1.531). The conservation of the potential vorticity defined by Eq. (1.568) implies that when the parcel is stretched in the vertical direction (h is increased) the vorticity ζ must also increase to preserve the potential vorticity. In other words, potential vorticity controls the amount of vorticity that can 130 Rossby

(1940).

Governing Equations

157

ω

r

h

k

j i

Fig. 1.19 Schematic illustration of the air parcel and coordinate system considered in the discussion of Sec. 1.6.2.2.

be created by the horizontal convergence by stretching the parcel in the vertical direction. One way to apply the shallow water approximation to the atmosphere is to think of the atmosphere as a collection of narrow incompressible (constant density) layers. The dynamics of such layers is governed by Eqs. (1.529) and (1.531). This line of thinking provided generations of atmospheric scientists much needed intuition to address a number of problems of atmospheric dynamics. In particular, Rossby131 used this approach to guide his intuition when he introduced an early form of the atmospheric variable today known as the isentropic potential vorticity. 1.6.2.2

Physical interpretation

The air parcel defined by a column of air can be considered a body in solid body rotation around a vertical axis at angular velocity (1/2)ζa k. To show that such an interpretation of the role of the absolute vorticity, ζa = ζ + f , is possible, consider an atmospheric column in solid body rotation at an ˆ =ω angular velocity ω ˆ k (Fig. 1.19). Using a Cartesian coordinate system, 131 Rossby

(1940).

158

Applicable Atmospheric Dynamics

whose origin falls on the axis of rotation, the velocity vector, vr , of a material point of the air parcel is ˆ × r = −y ω vr = ω ˆ i + xˆ ω j;

(1.571)

ˆ ∇ × vr = 2ˆ ω k = 2ω,

(1.572)

hence,

which yields 1 k · (∇ × vr ) . (1.573) 2 The coordinate system used in Eqs. (1.571)–(1.573) is not the same as the one used for the representation of the atmospheric governing equations. Most importantly, the motion of the solid body is described in an inertial frame, while the atmospheric motions are described in a rotating frame. The relationship between the wind vectors in the two frames is given by ω ˆ=

vr = v + Ω × R.

(1.574)

Substituting vr from Eq. (1.574) into Eq. (1.573) yields 1 (1.575) ω ˆ = k · [∇ × v + ∇ × (Ω × R)] = (1/2)(ζ + f ). 2 The angular momentum of the air parcel about the vertical axis is 1 L = Iω ˆ = I (ζ + f ), (1.576) 2 where I is the moment of inertia defined by Z r r′2 M (r′ )dr′ (1.577) I= 0

where M (r′ ) describes the distribution of mass as function of the distance, r′ , from the axis. For the atmospheric column considered here, M (r′ ) = ρ2πhr′ dr′ ,

(1.578)

which leads to I = ρ2πh

Z

r

r′3 dr′ = 0

1 ρπhr4 . 2

Because the mass of the parcel, Z r M (r′ )dr′ = ρπhr2 , M=

(1.579)

(1.580)

0

is conserved, the moment of inertia can be written as I=

2c M2 = , 2ρπh h

(1.581)

Governing Equations

159

where c = M 2 /4ρπ is a constant. Equation (1.581) shows that the moment of inertia is inversely proportional to the layer depth, h. Substituting I from Eq. (1.581) into Eq. (1.576) yields ζ +f . (1.582) h The law of the conservation of angular momentum states that the rate of change of the total angular momentum about any axis is equal to the total torque, τ , exerted by the external forces about that axis,132 that is L=c

dL = τ. (1.583) dt Because the motion of the air parcel is now described in an inertial frame, the Earth’s rotation has been taken into account through Eq. (1.574) rather than an inertial (Coriolis) force term. Thus the only force acting on the parcel that has a nonzero horizontal component is the pressure gradient force. That force, however, has no tangential component, which implies that, the angular momentum of the parcel, L, is conserved, because Eq. (1.583) becomes dL = 0. (1.584) dt Combining Eqs. (1.576) and (1.584) leads to the conclusion that the vorticity of the parcel can be increased only by decreasing the moment of inertia of the parcel. In addition, substituting L from Eq. (1.582) into Eq. (1.584) leads to dL dqh = = 0. (1.585) dt dt In summary, the conservation of the shallow water potential vorticity is a consequence of (i) the conservation of angular momentum, (ii) the conservation of mass, and (iii) the property of the pressure gradient force that it does not exert torque on the air parcels. The standard textbook example for the illustration of the conservation of angular momentum is a spinning figure skater, whose goal is to spin as 132 The

total torque acting on the parcel is equal to the torques exerted on the material points (molecules) in the air parcel. The torque on a material point of the parcel is the tangential component of the sum of the horizontal forces acting on the material point times the distance of the material point from the axis of rotation.

160

Applicable Atmospheric Dynamics

fast as she can. She achieves this goal by generating an initial torque with her arms spread wide, then pulling her arms close to her body. By doing so, she takes advantage of the conservation of angular momentum, as pulling in the arms reduces the moment of inertia, which leads to a rapid increase of the angular velocity. The conservation of the mass of her body, however, imposes a strong constraint on the maximum reduction of the moment of inertia she can potentially achieve. The potential for the generation of vorticity in the atmosphere is high in situations where the potential vorticity is high, but the air parcels are still un-stretched. The air parcel at this point is in the same stage of its dynamical evolution, as a figure skater who has already generated a lot of angular momentum by a large initial torque, but who still has her arms spread wide. We do not know, yet, whether she will materialize the potential to spin fast later in her move, but she certainly has the potential to do so. In contrast, a parcel that has high potential vorticity, but is already stretched in the vertical direction is in a situation similar to that of a figure skater who generated a large initial torque, but forgot to spread her arms at the beginning of her spin: she has no potential to spin faster in the same move. One crucial difference between the figure skater and the air parcel is that the figure skater uses her muscles to change the shape of her body (to decrease her moment of inertia), while the air parcel relies on its environment to change its moment of inertia: while the pressure gradient force cannot exert torque on the parcel, it can stretch the parcel in the vertical direction by squeezing it in the horizontal direction. According to Eq. (1.531), this process occurs in regions of horizontal convergence. In contrast, in regions of horizontal divergence, the moment of inertia increases as the parcel expands in the horizontal direction, leading to a contraction in the vertical direction. Because the total mass must be conserved for the entire model domain, convergence at one location must be balanced by divergence at another. The balance between convergence and divergence leads to a balance between the generation and the destruction of the potential vorticity. The quantitative laws that describe the conservation of the volume integral of the functions of potential vorticity will be introduced in Sec. 1.7. The other important difference between the figure skater and the air parcel is that the characteristic time scale of her motion is orders of magnitude smaller. While the motion of both the figure skater and air parcel is affected by the Earth’s rotating motion, the Coriolis force has no sufficient time to have a visible effect on the angular velocity of her spinning motion.

Governing Equations

161

But, it has a major effect on the parcels in the slower rotating large and synoptic scale vorteces. We close our discussion with a brief note on the energy conversion processes that take place during the generation of vorticity. Since the center of gravity of the parcel is increased when the parcel is stretched in the vertical direction, the horizontal convergence increases the potential energy of the parcel; that is, work is done by the environment (the pressure gradient force) on the parcel, resulting in a conversion of the divergent kinetic energy to potential energy. Because the vorticity (magnitude of the angular velocity) is also increased, the kinetic energy of the rotational motion is also increased in the process. 1.6.2.3

Isentropic potential vorticity (IPV)

The shallow water case is a special one, because due to the constant density and the lack of changes in the thermodynamical state of the parcel, changes in the mass distribution are solely due to changes in the shape of the parcel. The general case is not so simple, because the density of the parcel can change in response to convergence, or divergence of the flow, and to changes in the thermodynamical state of the parcel. The most natural choice of the vertical coordinate for the investigation of the potential vorticity is the isentropic coordinate. The isentropic potential vorticity is defined by g (f + ζθ ) qθ = − , (1.586) ∂p/∂θ where the vorticity, ζθ , is calculated on isentropic surfaces, that is ζθ = k · (∇θ × vH ) . (1.587) Exercise 1.65. Show that the isentropic potential vorticity satisfies Eq. (1.550) for the hydrostatic primitive equation, if all processes are assumed to be adiabatic. Answer. We first show that when all processes are adiabatic, the vorticity equation in isentropic coordinate system can be written as d (ζθ + f ) = − (ζθ + f ) Dθ , (1.588) dt where the divergence Dθ is computed on isentropic surfaces. Equation (1.588) can be obtained by comparing the horizontal momentum equations for pressure and isentropic vertical coordinates. The difference between the two equations that the pressure gradient force is represented by the gradient of two different potential functions has no effect on

162

Applicable Atmospheric Dynamics

the vorticity equation, because the curl of the gradient of a scalar field is always zero. The only important difference is that the vertical coordinate of the wind vector in the pressure coordinate system, ω, is nonzero, while the vertical coordinate of the wind vector in the isentropic coordinate system, wθ = dθ/dt, is zero due to the assumption that all processes are adiabatic. Thus the second term on the right-hand side of Eq. (1.455), which is the vorticity equation in the pressure coordinate system, vanishes in the case of the isentropic coordinate system, if all processes are assumed to be adiabatic. The continuity equation, Eq (1.363), can be written as ∂p d ∂p = − Dθ . dt ∂θ ∂θ Then, the Lagrangian time derivative of qθ can be written as g ζθ + f g (ζθ + f ) d ∂p dqθ =− − 2 dt ∂p/∂θ dt (∂p/∂θ) dt ∂θ −g (ζθ + f ) Dθ g (ζθ + f ) Dθ = + = 0, ∂p/∂θ ∂p/∂θ

(1.589)

(1.590)

where in the second step, we substituted the time derivatives from Eqs. (1.588) and (1.589). The isentropic potential vorticity, qθ , is the product of the absolute vorticity calculated on an isentropic surface and −g/(∂p/∂θ). To provide a physical interpretation of the conservation of qθ , we first introduce the finite difference approximation −

g g∆θ ≈− . ∂p/∂θ ∆p

(1.591)

Because θ is the vertical coordinate, ∆θ is the constant difference between the potential temperature of the two isentropic surfaces used for the calculations and ∆p is the spatiotemporally varying difference in the pressure between those two surfaces. Since ∆θ is constant, combining Eqs. (1.586) and (1.591) yields the approximate conservation law d ζθ + f ≈ 0, dt hθ

(1.592)

hθ = −∆p/g

(1.593)

where

is the layer thickness. Considering the definition of pressure, hθ is the ratio of the mass in the atmospheric column between the two isentropic surfaces and the area of the projection of the column on the horizontal plane. The

Governing Equations

163

layer thickness, hθ , plays a role analogous to that of the layer depth, h, in the case of the shallow water flow.133 Equation (1.592) would be an exact equality, if ∂p/∂θ was a linear function of p in the layer defined by the two isentropic surfaces. [This statement can be verified by inverting the function θ(p) defined by Eq. (1.341), then taking the derivative of the resulting function p(θ) with respect to p.] It is often argued, therefore, that Eq. (1.592) is an exact equality for a layer of constant potential temperature, independently of the layer thickness, hθ . This argument, however, is not self consistent, because the isentropic vertical coordinate could not be used for the representation of the governing equations, if layers of constant potential temperature were allowed,134 1.6.2.4

A note on the general case

On the one hand, the conservation of isentropic potential vorticity is a weaker conservation law than the conservation of potential vorticity, as it constrains only one component of the three-dimensional vorticity vector. On the other hand, the conservation of the isentropic potential vorticity is a stronger conservation law, because it imposes a stronger constraint on the vertical component of the vorticity, which is by far the most important component for synoptic and large scale motions. This line of argument leads to the following question: is it possible to say anything about the constraint imposed by the conservation of potential vorticity on the vertical component of the vorticity in the more general case? An answer to the aforementioned question can be given by considering the phenomenological result that for large- and synoptic-scale motions, the inner product that defines the potential vorticity in Eq. (1.549) is dominated by the product of the vertical component of the absolute vorticity and the vertical gradient of the potential temperature. That is, q ≈ qV = 133 The

ζa ∂θ f + ζ ∂θ = . ρ ∂z ρ ∂z

(1.594)

similarity between the role of hθ and h is not an accident: the observation that h was equal to the ratio of the mass of the atmospheric column and its projection on the horizontal plane motivated Rossby (1940) to introduce to the more general hθ . 134 Recall that θ is a proper vertical coordinate only if θ(p) is a monotonic function of p. Rossby (1940) introduced Eq. (1.592) as an exact equality, using the argument based on the assumption of a layer of constant potential temperature. He also noted that “It is possible to derive corresponding results also for an atmosphere in which the potential temperature varies continuously . . . ” but he did not present the equation for the continuous case. We can only guess that he was referring to Eq. (1.586).

164

Applicable Atmospheric Dynamics

The approximation provided by Eq. (1.594) leads to the result, by a comparison to Eq. (1.568), that in the general case, the state variable ρ/(∂θ/∂z) plays the same role as h in the shallow water case and −g/(∂p/∂θ) in the isentropic coordinate system. We repeat the assumptions made during the investigation of the static stability in Sec. 1.4.4.2 and suppose that the stratification of the atmosphere is stable (∂θ/∂z > 0). These assumptions allow us to investigate the short term dynamics of the parcel as the gradient ∂θ/∂z, which describes the thermodynamical state of the environment of the parcel, is constant in time at a fixed location. That is, the changes in ∂θ/∂z for a parcel are solely due to the changes in the position of the parcel. Similar to the shallow water case, convergence leads to an increase of the vorticity, because according to the continuity equation, Eq. (1.5), convergence leads to an increase of the density of the parcel; hence, ζ must increase to compensate the increase of the density, ρ, to preserve the potential vorticity, q. The thermodynamical state (vertical stratification) of the environment of the parcel, by determining ∂θ/∂z, constrains the amount of vorticity that can be created by the convergence of the flow. This role of the environment becomes even more transparent by considering the change, δζ, in the vorticity, ζ, due to a change δρ in the density, ρ. Based on Eq. (1.594), δζ =

q δρ. ∂θ/∂z

(1.595)

The weaker the static stability (the smaller ∂θ/∂z) the larger the change δζ in the vorticity caused by the same amount of change δρ in the density. Since the air parcels conserve their potential vorticity, the same amount of convergence increases the vorticity of a parcel by a larger amount in a region of lower static stability. The vorticity also increases when a parcel is transported by the wind field into a region of low static stability. 1.6.3

Computation of the Potential Vorticity

For most diagnostic applications, the ideal potential vorticity type variable is the isentropic potential vorticity, qθ : under the assumption that the dynamical processes are adiabatic, qθ is an exact Lagrangian invariant, which is advected on two-dimensional isentropic surfaces, making the visualization of the field straightforward. Most data sets produced by the operational centers include the isentropic potential vorticity field at one or more isentropic levels.

Governing Equations

165

PVU

Fig. 1.20 A snapshot of the isentropic potential vorticity at the 320 K isentropic surface in the NH Winter season. The plot is based on a 0.5◦ × 0.5◦ resolution operational ECMWF analysis from the TIGGE data set.

Figure 1.20 is an illustration of the isentropic potential vorticity based on data from the TIGGE data set. This data set provides the isentropic potential vorticity field at a single (320 K) isentropic level for all model products included in the data set. Following the convention, the field is shown using the potential vorticity unit (PVU), whose definition is 1 PVU = 10−6 Km2 /kgs. The field is shown at a relatively high, 0.5◦ resolution to illustrate that the (isentropic) potential vorticity field tends to produce significant variability at the smaller scales at high latitudes. The general structure of the field shown in Fig. 1.20 is rather typical for the atmosphere: high values of the potential vorticity tend to occur at high latitudes, forming a pattern that is reminiscent of an octopus with its arms reaching deep into the midlatitudes. The spatiotemporal evolution of the pattern (not shown) is also not unlike the motion of an octopus, with the arms slowly moving, sometimes retreating, while at other times reaching further into the lower latitudes.

166

Applicable Atmospheric Dynamics

Highly localized patterns of high potential vorticity, usually referred to as positive potential vorticity (PV) anomalies, occur at the lower latitudes. In Fig. 1.20, a couple of such anomalies can be seen: one over the middle of the United States, and another over southeast Europe. Since such anomalies usually play an important role in shaping the local weather, they have been the subject of intense research. Here, we only note that such anomalies can be cut-off structures originating from the main pattern of high potential vorticity, or structures locally generated by diabatic heating in a lower latitude region. The negative potential vorticity (PV) anomalies at the higher latitudes indicate anticyclonic circulation. Not all data sets include information on isentropic surfaces, and even those that include such information may do so only for a single isentropic surface (e.g., the TIGGEE data set). For such data sets, all fields required for the computation of qθ at the desired isentropic level have to be interpolated to that level before qθ can be computed by a finite-difference approximation of Eq. (1.586). Sometimes the goal is to study the spatial structure of the potential vorticity field in height or pressure coordinate system. The usual approach in such situations is to use a finite difference form of Eq. (1.594), or q ≈ qV = −g(f + ζ)

∂θ , ∂p

(1.596)

if the computation is done in pressure coordinate system.135 In principle, Eq. (1.594) can also be used in the non-hydrostatic case; but because the horizontal components of the vorticity and the potential temperature gradient may play a more important role in that case, the prudent approach is to use Eq. (1.549). The changes in the potential vorticity due to diabatic heating or cooling can be estimated by a finite-difference form of dq f + ζ ∂(dθ/dt) ≈ . dt ρ ∂z

(1.597)

∂(dθ/dt) dq ≈ −g(f + ζ) . dt ∂p

(1.598)

or

These two equations are approximations to Eq. (1.557), which usually provide sufficiently accurate diagnostic estimates of dq/dt. 135 Equation

(1.596) can be obtained by applying the transformation rule Eq. (1.246) to the vertical gradient ∂θ/∂z in Eq. (1.594) and making use of the hydrostatic balance equation.

Governing Equations

1.6.4

167

Vertical Structure of the Potential Vorticity Field

1.6.4.1

The Structure in Pressure Coordinate System

We describe the vertical structure of the potential vorticity field in pressure coordinate system by first considering the hypothetical situation, in which the (relative) vorticity is assumed to be zero (ζ = 0) at all locations and the potential temperature field is defined by a seasonal mean of the analyzed potential temperature. The resulting potential vorticity field is shown in Fig. 1.21. Results are shown for the winter and the summer of the Northern Hemisphere separately, because there are important differences in the structure of the potential temperature fields between the two seasons. The structure of the potential vorticity field in the Southern Hemisphere

December−January−February 200

2

1

350

Pressure [hPa]

300

350 330

330

6

4315 300 2

300 2 1

400 500

350 330

6 4315

1

5

31

600 700

0

1

30

800

1

900 0

10

20

30

40 50 Latitude [0N]

60

70

80

90

June−July−August 200

12

350

46 2 1

350

Pressure [hPa]

300 400 500

46 2

315

330

330

350 330

1 300

600

315

315

700

300

800 900 0

10

20

30

40 50 Latitude [0N]

60

70

1

80

90

Fig. 1.21 The vertical-meridional structure of the baseline potential vorticity field (thick contour lines) in the Northern Hemisphere. The vorticity, ζ, is assumed to be zero at all locations. The two-dimensional potential temperature field (thin contour lines) is obtained by computing the temporal mean of the field for (upper panel) December, January and February, and (lower panel) June, July and August for the 30-year period between 1981 and 2010 based on the NCEP/NCAR reanalysis, and also taking the zonal mean (the mean along the latitudes) of the field.

168

Applicable Atmospheric Dynamics

(not shown) is similar, except that the values are of the opposite sign. We will refer to the values of the potential vorticity shown in Fig. 1.21 as the baseline values of the potential vorticity. Because f increases monotonically toward the North Pole, the baseline values of the potential vorticity also tend to increase toward the pole. In addition to this effect of the Coriolis parameter, the vertical structure of the baseline potential vorticity field reflects the vertical structure of the potential temperature field. In the winter, there is a pattern of local maximum, as indicated by the 1 PVU contour line, in the lower troposphere north of 65◦ N. A similar pattern of local maximum also exists in the summer, but it is weaker and located further to the north than the one in the winter. The baseline potential vorticity changes little with height in the troposphere, but it starts to increase rapidly at the tropopause due to the sudden increase of the magnitude of the vertical potential temperature gradient. The latter property of the potential vorticity is often utilized to provide a dynamical definition of the tropopause. Because theory does not provide guidance on the optimal choice of the value of the potential vorticity for the dynamical definition of the tropopause, scientists have considerable freedom in choosing a particular value based on their own phenomenological considerations.136 Different authors use different values of potential vorticity to define the tropopause, but the most common choice is 1.5 or 2.0 PVU in the Northern Hemisphere, and −1.5 or −2.0 PVU in the Southern Hemisphere. 1.6.4.2

Meridional transport on isentropic surfaces

In addition to the base line values of the potential vorticity, Fig. 1.21 also shows the contour lines of he potential temperature field used for the calculation of the potential vorticity. The figure shows that in the winter the isentropic surfaces are nearly parallel to the isobaric surfaces in the tropics and the extratopics, but are highly sloped in the transitional zone between about 20◦ N and 50◦ N . In the summer, the slop of the isentropic surfaces in the transitional zone is less steep and the southern boundary of the zone is shifted to about 35◦ N . This property of the isentropic surfaces is a manifestation of the poleward shift of the tropical circulation in the summer. Where the isentropic and isobaric surfaces are nearly parallel, qV computed at the isobaric surface p = p0 provides an estimate of qθ at the isentropic surface, θ0 = θ(p0 ). This relationship between qV and qθ can 136 The

potential vorticity based definition of the tropopause does not work in the Tropics, where f goes to zero and the ±2 PVU surfaces diverge.

Governing Equations

169

be seen by noting that if the isentropic and isobaric surfaces were exactly parallel, the vorticity computed at θ0 = θ(p0 ) would be equal to the vorticity computed at p0 ; and making use of the inverse function theorem to replace ∂θ/∂p at p0 by 1/ (∂p/∂θ) at θ0 . While the inverse function theorem holds even if the isolines of ∂θ/∂p are sloped, the vorticity computed on an isobaric surface is no longer equal to the vorticity at an isentropic surface. Let assume for a moment that the seasonal mean fields of the potential temperature are identical to the actual potential temperature field for the time period considered. Where the isentropic and isobaric surfaces are nearly parallel, we can assume that qV (θ0 ) ≈ qθ (θ0 ) is advected as a Lagrangian invariant on the isobaric surface p0 . Where the isentropic surfaces are sloped, however, the potential vorticity transport has a vertical component, because the transport is on the isentropic surfaces. Figure 1.21 illustrates that the results of an isentropic potential vorticity based diagnostic study strongly depend on the choice of the isentropic level. In particular, the isentropic levels defined by values of the potential temperature between about 300 K and 330 K are tropospheric levels in the tropics, but they are located in the stratosphere in the extratropics. Air parcels traveling southward on such isentropic surfaces can transport high potential vorticity into an environment where the potential temperature gradient (static stability) is weaker, leading to an increase of the vorticity. For the southward moving parcels, the increase in the vorticity is made even bigger by the decreasing value of f along the path of the parcel. The sloping isentropic surfaces also allow for the transport of potential vorticity from lower layers of the tropics to higher layers of the extratropics. Such a transport can be particularly important for isentropic surfaces that reach the surface (isentropic surfaces of about 300 K), because as mentioned before, diabatic processes at the Earth’s surface can be the net source of the potential vorticity. Since unsaturated water vapor is also transported on isentropic surfaces from the Earth’s surface, the potential vorticity of the upward moving parcel can be changed by latent heat release. The instantaneous potential temperature field shows more spatial variability than the zonally and temporally averaged field that was used to obtain Fig. 1.21. The upper panel of Fig. 1.22 was obtained by replacing the average potential temperature field by the potential temperature field shown in Fig. 1.17 in the computation of the potential vorticity.137 While 137 The

results for the summer (not shown) are very similar.

170

Applicable Atmospheric Dynamics

345

300

6

345 2 330

1

15

330

500

2

315 4 2 300

5

285

1

400

1

3

600

28

0

315

700

6

345 330

45 31 30 0

Pressure [hPa]

0000 UTC 1 January 2010, ζ=0

2

200

30

1

800 900 0

300

10

20

30

40 50 Latitude [0N]

1

60

70

1 80

90

0000 UTC 1 January 2010

345

330

15

6

345 330

1

2 1

0

400

6 315 4 2 300 1

345 330

45 31

500

30

Pressure [hPa]

300

2

200

3

600

1

30

800 900 0

300

285

5

28

0

315

700

1 10

20

30

40 50 Latitude [0N]

60

1 70

80

90

Fig. 1.22 A snapshot of the vertical-meridional structure of the baseline potential vorticity field (thick contour lines) in the Northern Hemisphere. The potential temperature field used for the calculation of the potential vorticity is identical to that shown in the upper panel of Figs. 1.17. Results are shown for both (upper panel) the hypothetical case of ζ = 0 and (lower panel) the actual field of ζ computed by a finite-difference scheme based on the NCEP/NCAR reanalysis for 0000 UTC 1 January 2010.

the general structure of the resulting field in Fig. 1.22 is similar to that in Fig. 1.21, there are important differences in the details. Most importantly, the meridional changes in the potential temperature are no longer monotonic; hence, a short-distance movement of an air parcel on an isentropic surface toward the south is not necessarily downward. The longer distance meridional transport is not affected, because the “bumps” in the potential temperature field are small. The lower panel of Fig. 1.22 was obtained by replacing the vorticity field ζ = 0 by the actual instantaneous vorticity field in the computation of the potential vorticity. A comparison of the two panels of Fig. 1.22 shows that replacing the ζ = 0 vorticity field with the actual vorticity field mainly affects the contour lines of the smaller values of the potential vorticity. In order to make these differences more transparent, Fig. 1.23 shows the same

Governing Equations

0000 UTC 1 January 2010, ζ=0

400

600

315

900 0

0

300

0.6

30 0.2 10

20

30

40 50 Latitude [0N] 0000 UTC 1 January 2010

400

31

5

0.2 20

30

0.6

40 50 Latitude [0N]

60

11.2 70

90

285

1

10

0 300.2

6

1

300

4

0.

900 0

0.

0.28

315

800

1.2

0.8 1 80

0.8

700

1 0.8

85

300

0.6 0.4

Pressure [hPa]

0.4

600

70

0.8

0.4

1

60

0.4

500

330

285

0.8

800

0.6

28

0.6 0.4

0.4

700

0.6 5 0.8

300

1

0.2

0.4

1 0.4 0.8

31 5

330

0.4

Pressure [hPa]

500

171

80

0.81

90

Fig. 1.23 Same as Fig. 1.22 except that the contour interval for the potential vorticity is 0.2 PVU instead of 1 PVU and the fields are shown only below 400 hPa.

fields as Fig. 1.22, but with more contour lines and only below the 400 hPa level. It should be noted that using a higher resolution data set would indicate larger differences between the two panels of Fig. 1.23.138 1.6.4.3

Visualization of the atmospheric fields at the dynamical tropopause

The potential vorticity based dynamical definition of the tropopause is often used for the visualization of the different atmospheric state variables (e.g., potential temperature, horizontal components of the wind vector), as it can provide a transparent picture of several synoptic and large scale features 138 Many

data sets, including the NCEP/NCAR reanalysis, use a 2.5◦ × 2.5◦ horizontal resolution, because its is adequate to display the geopotential height (stream function) field. But, as will be formally explained in Exercise 2.9, the potential vorticity has a much finer spatial structure than the geopotential height. This property of the vorticity field makes the older data sets, which use a 2.5◦ × 2.5◦ horizontal resolution, less amenable to a potential vorticity based analysis, and also motivated us to show the horizontal structure of the isentropic potential vorticity at a 0.5◦ × 0.5◦ resolution in Fig. 1.20.

172

Applicable Atmospheric Dynamics

m/s K Fig. 1.24 A snapshot of the (shaded) potential temperature and (vectors) horizontal wind at the 2 PVU isentropic surface in the NH Winter season. The plot is based on a 0.5◦ × 0.5◦ resolution post-processed operational ECMWF analysis from the TIGGE data set.

present in the atmosphere in a single map.139 Hence, some atmospheric data sets include data at one or two potential vorticity levels. An example of a map based on such data is shown in Fig. 1.24. When the atmospheric fields are displayed at the tropopause defined by a constant value of the isentropic potential vorticity, in essence, the isentropic potential vorticity becomes the vertical coordinate. Using the isentropic potential vorticity as the vertical coordinate, however, is problematic from a theoretical point of view. First, the isentropic potential 139 A

description of the different approaches to construct tropopause maps and a general discussion of how such maps can be used to diagnose midlatitude weather systems can be found in Morgan and Nielsen-Gammon (1998).

Governing Equations

173

vorticity is obviously not a proper vertical coordinate, because it is not a one-to-one (monotonic) function of the height: values of 1.5 or 2 PVU can occur, not only at the tropopause, but also in the lower troposphere. Second, the potential vorticity is not a monotonic function of the height even in the neighborhood of the tropopause at the location of tropopause folds. Finally, even if tropopause folds did not exist, potential vorticity would be a poor choice for the vertical coordinate of a numerical model near the tropopause, because the atmospheric state variables can have sharp gradients on the 1.5 and 2 PVU surfaces. Resolving the steep gradients would require a horizontal resolution much higher than what is used for the common choices of the vertical coordinate. What is a disadvantage for numerical modeling is a great advantage for a visual diagnostic technique: the sharp gradients separate regions with important differences in the qualitative dynamics of the flow. It should always be kept in mind, however, that the definition of the surface is not unique in the region of a tropopause folds: in such a region, a downward search started from the stratosphere (e.g., at the 10 hPa level) for the vertical location of the dynamical tropopause returns a different location than an upward search started from the mid-tropopause (e.g., at the 600 hPa level). As already discussed, under the assumption that the processes are adiabatic, both the potential temperature and the isentropic potential vorticity are Lagrangian invariants. The Lagrangian invariance of the two quantities have the following important consequences: (i) all motions are confined to the two-dimensional surfaces of constant isentropic potential vorticity, because the vertical coordinate of the wind vector is equal to the Lagrangian time derivative of the isentropic potential vorticity; (ii) the spatiotemporal evolution of the potential temperature field is solely due to the transport of the potential temperature by the air parcels.140 The potential temperature at the dynamical tropopause reflects the past spatiotemporal evolution of the atmospheric flow. More precisely, due to the presence of diabatic heating and cooling, and the chaotic nature of the atmospheric flow, the potential temperature field reflects the “near past” evolution of the flow. In particular, the color shades, which show the potential temperature field in Fig. 1.24, provide useful information about 140 Of

course, the transport of the potential temperature cannot be followed on a twodimensional map at the tropopause folds.

174

Applicable Atmospheric Dynamics

the origin of the air parcels: we can assume, to a good approximation, that the parcels that occupy the regions colored purple and blue originate from the Arctics, the parcels in the regions colored green and yellow originate from the midlatitudes, while those in the red regions originate from the Tropics. Some of the most interesting features of the potential temperature field in Fig. 1.24 are the circular pattern of midlatitude air over the Northeast Pacific, which extends into both the arctic and the tropical regions; the fingers of arctic air in the midlatitudes over North-America, the North Atlantic and Europe; and the sharp interface between the arctic and tropical air masses over East Asia and the Northwest Pacific. Comparing these patterns of the potential temperature to the wind field in the same regions, it can be seen that the patch of midlatitude air over he Northeast Pacific is associated with a large scale vortex, the fingers of arctic air are related to wave-like motions on a dominantly zonal westerly flow, while at the sharp interface between the arctic and tropical flow, there is a very strong westerly flow that prevents the meridional mixing of the air masses. The strong westerly flow is called the jet stream, or simply the jet. Where the jet reaches the large-scale vortex over the Northeast Pacific, it splits into a northern and a southern branch and do not reunite until reaching East Asia. The northern branch, which runs along the boundary of the midlatitude and arctic air masses, is called the polar jet, while the southern branch, which runs along the boundary of the midlatitude and tropical air masses, is called the subtropical jet.141 The wave-like patterns along the polar jet142 play a central role in shaping the weather in the midlatitudes. 1.6.5 1.6.5.1

Potential Vorticity Inversion and “PV-thinking” Invertibility principle

The invertibility principle for potential vorticity states that at any given time all prognostic state variables of the primitive equations143 can be determined from the three dimensional potential vorticity field. In other words, the invertibility principle states that there exists a set of diagnostic equations to compute the hydrodynamical and the thermodynamical 141 One

of the advantageous properties of a map showing the wind field at the dynamical tropopause is that it can display the polar and the subtropical jets in a single figure. 142 The dynamics of these waves will be the subject of Secs. 2.4.2. 143 Except for the density of the atmospheric constituents.

Governing Equations

175

state variables from the potential vorticity. Because the formal problem to be solved is an elliptic partial differential equation, the invertibility principle assumes that the proper boundary conditions are available. In the most general case, the availability of the boundary conditions requires the knowledge of the potential temperature at the bottom of the atmosphere. Another way to state the invertibility principle is to say that the system of primitive equations can be replaced by a system of equations that includes a single prognostic equation, the potential vorticity equation. Because such a reduction of the number of prognostic variables is not possible without making some assumptions about the atmospheric motions, the atmospheric fields obtained by the inversion reflect all constraints imposed by those assumptions. The hope is that the constraints can be relaxed to the point where they filter only those solutions of the primitive equations that are irrelevant for weather prediction. The concept of the invertibility principle was developed over a period of more than six decades,144 by designing inversion algorithms for gradually less restrictive sets of assumptions about the atmospheric motions. At the heart of the theory behind the invertibility principle is the concept of atmospheric balance. In practical terms, the key diagnostic equation of an inversion algorithm is the balance equation it uses. For instance, the barotropic vorticity equation, Eq. (1.525), can be considered an extreme simplified form of the potential vorticity equation. In the case of the barotropic vorticity equation, the balance equation is the geostrophic balance equation, Eq. (1.518), which can be used to compute the geopotential, φ, or equivalently, the geopotential height, z from the vorticity, ζ. In addition, the stream function, ψ, and the horizontal wind, vH , can be computed from z by taking advantage of ψ = ψg and vH = vg . The other state variables of the primitive equations do not have to be computed, because they are undetermined for the vertically averaged, two-dimensional, homogeneous flow described by the barotropic vorticity equation. As mentioned in Sec. 1.5.5.6, the barotropic vorticity equation is not the only reduced form of the governing equations that uses the quasigeostrophic approximation. The prognostic variable of a quasi-geostrophic baroclinic model is a form of the potential vorticity that we have not discussed, yet. That form of the potential vorticity, called the quasi144 The

pioneering ideas published in a series of paper by Kleinschmidt in the early 1950s (Kleinschmidt, 1950a,b, 1951, 1955) about the role of potential vorticity in midlatitude atmospheric dynamics as a whole is usually considered an early intuitive form of the invertibility principle.

176

Applicable Atmospheric Dynamics

geostrophic potential vorticity (QGPV), will be introduced in Sec. 2.3.2. Here, we only note that the invertibility principle was first shown formally to hold for the quasi-geostrophic potential vorticity.145 1.6.5.2

“PV-thinking”

An important role of a potential vorticity inversion algorithm is to provide a constructive proof for the invertibility principle, as it justifies the use of a single state variable (the potential vorticity) to describe the spatiotemporal evolution of the atmospheric flow.146 The atmospheric science slang for the approach is “PV-thinking”, although it has been pointed out that “PVreasoning” would probably be a more accurate informal term. 1.6.5.3

PV-modification

While inversion algorithms have been used in countless atmospheric dynamical process studies, they are rarely used in the operational practice. One of the notable exceptions is the procedure called PV-modification, which was implemented at some of the forecast centers of Europe, and is still in use at M´et´eo France. The general idea of PV-modification is that an expert meteorologist can recognize a mismatch between the potential vorticity field and the observations. Because the potential vorticity, to a good approximation, is a Lagrangian invariant, the ideal observations for the comparison are those of the density of atmospheric constituents with weak source and sink terms. In the upper troposphere, satellite images of the density of water vapor and ozone satisfy this criterium. While it would be impossible for the meteorologist to change the wind, temperature and pressure fields independently without disturbing the delicate balance between them in the analysis, the potential vorticity inversion, which makes the appropriate changes to the different variables based on the modified potential vorticity field, results in balanced fields of the state variables. The numerical forecasts are started from the modified analyses. Recent results from M´et´eo France suggest that PV-modification can improve the forecast of some high-impact weather events.147 While future improvements in the data assimilation systems will make PV-modification an obsolete operational technique eventually, the fact that it can still im145 Charney

and Stern (1962). point was first spelled out by Hoskins et al. (1985b), in one of the seminal papers on potential vorticity. 147 Arbogast et al. (2012).

146 This

Governing Equations

177

prove state-of-the-art operational forecasts of our days provides strong support for the continued use of “PV-thinking” as a research and diagnostic tool.

1.7

Integral Invariants

Atmospheric motions preserve several integrals of the state variables under the assumptions that (i) all processes are adiabatic: Q(r, t) = 0; (ii) the composition of the atmosphere does not change: ρi (r, t) = constant, i = 1, . . . , n; (iii) the boundary conditions are conservative, that is, air parcels cannot leave or enter the model domain. The last assumption is automatically satisfied for a global model, but it is usually not satisfied for a limited area model. Theoretical investigations also often use conservative boundary conditions, even if the solution domain is not the entire atmosphere. In the present section, Sec. 1.7, we always assume that the aforementioned three assumptions are satisfied. While the integral invariants can always be derived by direct manipulations of the governing equations, we follow an approach based on finding the Hamiltonian form of the equations. While the popularity of this approach has been rapidly fading since its heyday in the mid-1990s, it remains the only systematic method to find the integral invariants, investigate their origin and relationship, and to examine the effect of the numerical solution strategies on the conservation laws.148 1.7.1

Hamiltonian Form of the Governing Equations

We introduce the notation u(r, t) for the atmospheric state vector, whose components u1 (r, t), . . . , uv (r, t) are the scalar prognostic variables for the particular set of atmospheric governing equations. With this notation, the 148 The

fading popularity of Hamiltonian geophysical fluid dynamics is usually attributed to the fact that it has not led to truly new insights into the dynamics of the atmosphere. We still believe that the Hamiltonian formulation makes some important properties of the atmospheric governing equations particularly transparent. Shepherd (1990) and Salmon (1998) provide excellent summaries of the main concepts and results of Hamiltonian geophysical fluid dynamics.

178

Applicable Atmospheric Dynamics

atmospheric governing equations can be written as ∂u (r, t) = F(u, t), u(r, 0) = u0 (r). (1.599) ∂t For instance, when a model is based on the non-hydrostatic primitive equations Eqs. (1.4)–(1.6), the components of u(r, t) are the three components of the wind vector, the temperature, the density and the pressure. 1.7.1.1

General formulation

Writing the atmospheric governing equations in Hamiltonian form involves finding the functional H(u), called the Hamiltonian, and the transformation J(u), for which Eq. (1.599) takes the form δH ∂u =J , ∂t δu where J(u) is skew-symmetric, that is, hu, Jviu = −hJu, viu ,

(1.600)

(1.601)

and satisfies an additional strong symmetry condition, called the Jacobi identity. The symbol δH/δu in Eq. (1.600) indicates the functional derivative of H(u) with respect to u, while h., .iu is the inner product Z hu, viu = hu, viv dr (1.602) V

R on the function space of the functions u(r, t), where V ·dr is the volume integral for the whole atmosphere. The symbol hu, viv represent the inner product for the v dimensional space of the atmospheric state vector at a given location r and time t, which is typically defined by hu, viv =

v X

= ui v i .

(1.603)

i=1

Formally stating the Jacobi identity in a reasonably simple form would require the introduction of additional mathematical terminology, which otherwise would not be used in the remainder of the book. In addition, verifying the Jacobi identity for the full set of atmospheric governing equations is a highly involved calculation. Here we only note that the Jacobi condition must be satisfied by J(u) to ensure that the Casimir invariants, which will be introduced in the next section, are correctly identified. Fortunately, we can take advantage of the hard work of others, who have already verified the Jacobi identity for the atmospheric governing equations.

Governing Equations

179

Mathematical Note 1.7.1 (Functionals and Their Derivatives). Functionals are functions whose domain is an infinite-dimensional space, e.g., the {u} space of the atmospheric states u(r, t). The functional or variational derivative δF/δu of the functional F is defined by  
δF , δu + O δu2 . (1.604) δF = F(u + δu) = δu The linear part of the increment, hδF/δu, δui, is called the differential or variation of F, while δu is the variation of u(r, t).

1.7.1.2

The integral invariants of Hamiltonian systems

In our applications, H(u) is the scalar-valued function that returns the value of the total energy of the atmosphere given the state u(r, t). The conservation of the total energy is the direct consequence of the skew-symmetry of J(u), because       dH δH ∂u δH δH δH δH = , ,J ,J = =− , (1.605) dt δu ∂t u δu δu u δu δu u which implies that   δH δH ,J =0 (1.606) δu δu u and dH = 0. (1.607) dt In the last step of Eq. (1.605), we took advantage of the property of the inner product h., .iu that it is symmetric, that is, hu, viu = hv, uiu . The conservation of energy belongs to a class of conservation laws that are each related to a symmetry of the dynamical system defined by Eq. (1.600). In particular, the conservation of energy is related to the invariance of H(u) under translation in time (temporal symmetry), that is, H [u(r, t + ε)] = H [u(r, t)] .

(1.608)

In mathematical terms, the relationship between the symmetries of the system and its integral invariants is stated by Noether’s theorem.149 With 149 Amalie

Emmy Noether (1882–1935) was one of the first women to make seminal contributions to mathematics. She proved her theorem on the connection between symmetries and invariants in 1915 and published it in 1918 (Noether, 1918). An English translation of Noether (1918) by Morton A. Tavel was published as Noether (1971).

180

Applicable Atmospheric Dynamics

the help of Noether’s theorem, the conservation of energy can be shown by substituting t for x in the theorem [in Eq. (1.609)]. As will be shown, additional invariants can be identified, for instance, by substituting one of the independent variables used for the representation of the location (one of the components of r) for x in Eq. (1.609). Mathematical Note 1.7.2 (Noether’s Theorem). If H is invariant under translation in a coordinate x and the functional M satisfies J

∂u δM =− , δu ∂x

(1.609)

then M is invariant in time. A simple proof of this form of the theorem can be found in Shepherd (1990). Not all integral invariants of the system defined by Eq. (1.600) can be identified by Noether’s theorem. These invariants, which we denote by C, satisfy the condition J

δC = 0, δu

(1.610)

and are called Casimir invariants. Their invariance in time follows from       δC ∂u δC δH δH δC δC = , ,J ,J = =− = 0. (1.611) dt δu ∂t du δu δu δu The Casimir invariants are hidden from Noether’s theorem. In particular, when Casimir invariants exist for a system, the Hamiltonian is only defined to within a Casimir, as adding a Casimir invariant to the Hamiltonian does not change Eq. (1.600). 1.7.1.3

Hamiltonian form of the governing equations

To find H(u) and J(u) for the atmospheric governing equations rewrite Eqs. (1.4)–(1.6) as   ∂v 1 1 = v×ω−∇ | v |2 − ∇p + g, ∂t 2 ρ ∂ρ = −∇ (ρv) , ∂t ∂S = v · ∇S. ∂t

we first

(1.612) (1.613) (1.614)

Governing Equations

The total energy for the system of Eqs. (1.612)–(1.614) is  Z  1 2 ρv + ρcv T + ρgz dr, H(u) = 2 V where the three components of the energy are the kinetic energy, Z 1 2 K(u) = ρv dr, V 2

181

(1.615)

(1.616)

the internal energy, I(u) =

Z

ρcv T dr,

(1.617)

ρgz dr.

(1.618)

V

and the potential energy, P(u) =

Z

V

The expression Eq. (1.617) can be used for the computation of the internal energy because the atmosphere behaves, to a good approximation, as an ideal gas. In the atmospheric sciences, the sum of the internal and the potential energy, I(u)+P(u), is usually called the total potential energy.150 Exercise 1.66. Show that for an atmosphere in hydrostatic balance, the ratio of the potential and the internal energy in an atmospheric column is constant.151 To be precise, (cp − cV ) P = κ. = I cV

(1.619)

Exercise 1.67. Show that using pressure as the vertical coordinate, the total potential energy of an atmospheric column of unit area can be written as Z cp p0 P +I = T dp. (1.620) g 0 Hint: Equation (1.620) follows directly from Eq. (1.619) and the hydrostatic balance equation. An important consequence of Eq. (1.619) is that the potential and the internal energy cannot be changed independently. That is, the generation of kinetic energy requires the simultaneous conversion of potential and internal energy into kinetic energy. 150 This 151 This

naming convention was introduced by Margules (1903). property was first shown by Haurwitz (1941).

182

Applicable Atmospheric Dynamics

The transformation J(u) that satisfies Eq. (1.600) for the Hamiltonian given by Eq. (1.615) is  − ρ1 ω× −∇· ρ1 ∇S   J =  ∇· 0 0 . 1 0 − ρ ∇S 0 

(1.621)

The transformation defined by Eq. (1.621), which is a matrix of vector, differential and scalar operations, may seem unusual for the reader. In practice, applying the transformation is not particularly difficult: it involves first left-multiplying the vector on which J(u) operates (the functional derivative of H(u) in our case) and carrying out the operations represented by the entries of the matrix. Before we turn our attention to this calculation, we show that the J(u) defined by Eq. (1.621) is skew-symmetric, as required by the definition of Hamiltonian systems.152 We first note that J(u) can be skew-symmetric if and only if the diagonal entries of J(u) are skewsymmetric operations, while the operations by the pairs of off-diagonal entries Ji,j (u) and Ji,j (u), i 6= j lead to the same results with opposite signs. Since multiplication by the scalar 0 is a trivial skew-symmetric operation and the off-diagonal elements satisfy the aforementioned symmetry condition, the only task left is to show that the operation −(1/ρ)ω× is skew symmetric. This property follows from the symmetry properties of the scalar triple product, as       1 1 1 u, − ω × v = u · − ω × v = −v · − ω × u (1.622) ρ ρ ρ v   1 (1.623) = − − ω × u, v . ρ v To show that Eq. (1.621) leads to Eqs. (1.612)–(1.614), we first have to calculate the functional derivative  δH  δv

δH  δH  =  δρ  . δu δH δS

of H(u). 152 The

Jacobi identity for this J(u) was first verified by Morrison (1982).

(1.624)

Governing Equations

Exercise 1.68. Show that δH = ρv, δv δH 1 = v2 + gz + cp T, δρ 2 δH = ρT. δS

183

(1.625) (1.626) (1.627)

Exercise 1.69. Show that for Cartesian coordinates, Eq. (1.621) takes the form   1 1 ∂· 1 ∂S 0 ρ ω3 − ρ ω2 − ∂x ρ ∂x  1 1 ∂· 1 ∂S   − ρ ω3 0  ρ ω1 − ∂y ρ ∂y    1 1 ∂· 1 ∂S  J =  ρ ω2 − ρ ω1 0 − ∂x ρ ∂z  . (1.628)    − ∂· − ∂· − ∂·  0 0   ∂x ∂y ∂z ∂S ∂S ∂S − ∂x − ∂y − ∂z 0 0 1.7.2

Energy, Momentum, and Angular Momentum

The conservation of the energy H(u) for Eqs. (1.612)–(1.614) follows from the skew-symmetry of J(u). Exercise 1.70. Show that in Cartesian coordinates the conservation of the absolute zonal momentum Z M= ρ(u − f y)dx dy dz (1.629) V

follows from the symmetry of the equations under translation in the xdirection, while the conservation of the absolute meridional momentum Z M= ρ(v + f x)dx dy dz (1.630) V

follows from the symmetry under translation in the y-direction. 1.7.3

Integrals of the Potential Vorticity

The Casimir invariants C(u) for Eqs. (1.612)–(1.614) can be written in the general form Z C= ρC(S, q)dr, (1.631) V

where C(S, q) is an arbitrary function of the entropy S and the potential vorticity. The integral defined by Eq. (1.631) is invariant, because the entropy

184

Applicable Atmospheric Dynamics

(potential temperature) and the potential vorticity are both conserved for each fluid parcels: while the flow can rearrange the spatial distribution of the entropy and the potential vorticity, it cannot change their volume integrals. The invariant defined by Eq. (1.631) is hidden from Noerther’s theorem, because the associated symmetry, called the particle relabeling symmetry, is hidden in the Eulerian description of fluid dynamics. To explain the origin of this symmetry and the reason why it becomes hidden, we first recall that in the Lagrangian description of fluid dynamics that fluid parcels are labeled,153 so the motion of each parcel can be followed during the evolution of the fluid dynamical system. The term particle relabeling symmetry refers to the property of the system that switching the location of particles whose potential vorticity is the same, or equivalently, switching the labels of particles whose potential vorticity is the same, does not change the evolution of the system. In the Eulerian description, where the fluid parcels are no longer identified, a property associated with “relabeling” cannot be detected. This property can also be stated by saying that motions along surfaces of constant potential vorticity cannot be detected in the Eulerian description. 1.7.4 1.7.4.1

Integral Invariants of the Simplified Equations General comments

In principle, the Hamiltonian formulation provides a straightforward framework to investigate the invariants of the reduced equations: because the equations in their Hamiltonian form can be reduced by reducing the matrix operator J(u) and/or the functional H(u), the conservation laws for the reduced system can be investigated by examining the symmetry properties of the reduced form of J(u). First, if the new J(u) is skew-symmetric, the energy defined by the reduced version of H(u) is conserved. Next, the Jacobi identity has to be verified for the new J(u); then, the rest of the invariants can be determined. In practice, we usually know in advance how the energy is defined for the the system of reduced equations, so we can easily determine H(u) and its functional derivative, which than can be used to determine the reduced J(u). The most difficult step is the verification of the Jacobi identity, but that work has already been done by others for the reduced equations most 153 E.g.,

by the coordinates of the initial locations of the parcels.

Governing Equations

185

often considered in atmospheric dynamics. As it turns out, while J(u) is skew-symmetric for most reduced equations, it often does not satisfy the Jacobi identity. Consequently, while most reduced equations conserve energy, they often do not have Casimir invariants. Since the Casimir invariants are integrals of the (potential) vorticity, the equations for which J(u) does not satisfy the Jacobi identity, does not conserve the integrals of the (potential) vorticity either. Because the conservation of the integrals of the potential vorticity is due to the Lagrangian conservation of the (potential) vorticity, we can expect the Jacobi identity to be satisfied for those systems of reduced equations that conserve a form of the (potential) vorticity for the air parcels. Most importantly, if the only reduction is the assumption of hydrostatic balance,, the integrals of the potential vorticity are not conserved, because the potential vorticity is not a Lagrangian invariant in that case.154 The Hamiltonian structure can be restored by making additional reductions, which eliminate the contribution of the two horizontal components of the three-dimensional absolute vorticity to the potential vorticity and ensure the Lagrangian conservation of the contribution of the vertical component. Two reduced forms of the governing equations that satisfy this condition are the shallow water equations and the two-dimensional vorticity equation. 1.7.4.2

Shallow water equations

For the system of Eqs. (1.529) and (1.531), u = (vH , h) , H(u) =

Z

V

and


h|vH |2 + gh2 drH ,

 0 q −∂/∂x J(u) =  −q 0 −∂/∂y  . −∂/∂x −∂/∂y 0 

(1.632) (1.633)

(1.634)

In Cartesian coordinates, the invariant associated with the coordinate x is the absolute zonal momentum, Z M= h(u − f y)dx dy, (1.635) V

154 We

recall that the conservation of the isentropic potential vorticity is a Lagrangian conservation law only for the vertical component of the potential vorticity.

186

Applicable Atmospheric Dynamics

the invariant associated with the coordinate y is the absolute meridional momentum, Z M= h(v + f x)dx dy, (1.636) V

The Casimir invariants have the general form Z C= hC(q)drH ,

(1.637)

V

where C(q) is an arbitrary function of the shallow-water potential vorticity defined by Eq. (1.568). Some of the notable Casimir invariants are the total mass, Z C0 = h drH , (1.638) V

the circulation,

C1 =

Z

1 C2 = 2

Z

and the potential enstrophy

ζ drH ,

(1.639)

q2 drH , h

(1.640)

V

V

which arise for C(q) = 1, C(q) = q and C(q) = q 2 , respectively. 1.7.4.3

Two-dimensional vorticity equation

Equation (1.440) can be written in Hamiltonian form by making the choices

H(u) = and

1 2

u = ζ,

(1.641)

Z

(1.642)

V

|∇H ψ|2 drH ,

J(u) = J (ζ, ·).

(1.643)

Notice that here the Jacobian is used as a differential operator. Exercise 1.71. Show that δH = −ψ. δu Hint: Make use of the conservative boundary conditions.

(1.644)

Governing Equations

187

In Cartesian coordinates, the invariant associated with the x coordinate is the zonal component of Kelvin’s impulse, Z Mx = yζ dx dy, (1.645) V

while the invariant associated with the y coordinate is the meridional component of Kelvin’s impulse, Z My = − yζ dx dy. (1.646) V

Exercise 1.72. Show that the conservation of Mx implies the conservation of the zonal momentum, Z yu dx dy, (1.647) V

while the conservation of My leads to the conservation of the meridional momentum Z yv dx dy. (1.648) V

Hint: The conservation of the two components of the momentum can be shown by an integration by parts of the integrals that define the two components of Kelvin’s impulse. It can be shown that the two components of the momentum and the angular momentum are invariants that can be identified by Noether’s theorem. The Casimir invariants can be written in the general form Z C= C(ζ)drH . (1.649) V

Two of the invariants that play a central role in our understanding of atmospheric dynamics are the circulation Z C1 = ζ drH , (1.650) V

and the enstrophy

C2 =

1 2

Z

ζ 2 drH . V

(1.651)

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Chapter 2

Perturbation Dynamics

2.1

Introduction

Many important concepts and diagnostic techniques of atmospheric dynamics are based on decomposing the Eulerian state variables into a basic flow component and a perturbation component. There are a number of potential choices for the basic flow component depending on the type of the problem investigated: (i) Definition of the thermodynamical variables of the reduced systems of governing equations. Designing closed systems of reduced three-dimensional atmospheric governing equations requires the availability of reduced forms of the thermodynamical equation. Such forms of the thermodynamical equation can be derived by decomposing the thermodynamical variables into the sum of a basic flow component, which depends only on the vertical coordinate, and a perturbation component, which depends on all three spatial coordinates and time. Since the information about the different components of the time-independent basic flow can be amalgamated into constant parameters, which fully describe the vertical stratification of the model atmosphere, the perturbation components take the role of the original thermodynamical variables in the reduced systems of equations. (ii) Analytical wave solutions and linear stability analysis. In analytical investigations of atmospheric waves and linear stability, the basic flow is usually defined by an idealized flow. In such an investigation, the magnitude of the perturbation is assumed to be infinitesimally small, which allows for a linearization of the governing equations about the basic flow. The origin of a wave observed in the atmosphere can be explained by finding the idealized basic flow that can support a wave 189

190

(iii)

(iv)

(v)

(vi)

Applicable Atmospheric Dynamics

solution with the observed characteristics of the wave (e.g., wavelength and frequency). The linear stability of a steady basic state can be investigated by examining the time dependence of the amplitude of the wave solutions for that basic state. Nonlinear stability analysis. When the basic flow is a steady state solution of the equations and the perturbations are assumed to have finite magnitude, the spatiotemporal evolution of the perturbations is nonlinear. In some cases, the bounds for the magnitude of the nonlinearly evolving perturbations can be determined.1 Global energetics. Diagnostic calculations often use the zonal-mean flow as the basic flow. For example, the concept of the Lorenz’s energy cycle is based on such a decomposition. Local diagnostics. In local diagnostic calculations, the basic flow is often defined by the time-mean flow for a finite time interval (e.g., for a month or a season). For such a description of the atmospheric flow, the evolution of the finite-magnitude perturbations is nonlinear. An example for a local diagnostic tool is the eddy kinetic energy equation, which allows for the investigation of the spatio-temporal changes in the energy conversion processes. Another common choice for the basic flow in a local diagnostic calculation is the ensemble mean, which can be used when an ensemble of fields (e.g., from a set of analyses and/or forecasts from different forecast centers) is available for the same locations and times. Sensitivity of a state space trajectory to perturbations to the initial condition. A state space trajectory associated with the spatiotemporally evolving atmospheric flow can define a time-dependent basic flow. The investigation of the evolution of perturbations to the initial condition associated with a particular state space trajectory plays an important role in both the qualitative and quantitative description of the evolution of forecast uncertainty. Some techniques assume that the initial magnitude of the perturbations is infinitesimal, while others allow for finite-magnitude initial perturbations.

The derivation of reduced forms of the equations and the investigation of wave dynamics are closely related problems: on the one hand, finding the reduced set of equations with a wave solution that matches the properties of an observed wave can provide important qualitative information about the 1 Nonlinear stability analysis is not discussed in this book. The interested reader is referred to Shepherd (1990, 1992).

Perturbation Dynamics

191

atmospheric flow; on the other hand, reduced equations can be employed to filter waves that are thought to be irrelevant from the model solutions.

2.2

Zonal-Mean Structure of the Atmosphere

Since phenomenological considerations play a central role in the selection of the basic state, it is useful to start the discussion of perturbation dynamics with a brief review of the relevant phenomenological results.2 Cyclic changes in the diabatic heating Qt introduce cyclic components into the temporal changes of both the qualitative and the quantitative dynamics of the atmosphere. For instance, there are obvious seasonal and diurnal components of the variability of the atmospheric state variables. In phenomenological studies of atmospheric dynamics, the cold and the warm seasons of the Northern Hemisphere are usually defined by the threemonth periods of December, January and February (DJF), and June, July and August (JJA), respectively. For the Southern Hemisphere, December, January and February (DJF) defines the warm season, and June, July and August (JJA) the cold season. While the processes that generate the kinetic energy of atmospheric motion from the incoming solar radiation show significant zonal variability, the zonal mean of the different state variables provide important information about the effects of the meridional changes in the incoming solar radiation. In both hemispheres, the time when the incoming solar radiation has its maximum is in the early part of the warm season. Likewise, the time when the incoming solar radiation has its minimum is in the early part of the cold season. This delay in the warming and the cooling of the atmosphere in response to changes in the solar radiation are primarily due to the thermal inertia of the oceans. The fact that a significant part of the heat is transported as latent heat from the Earth’s surface to the atmosphere also contributes to the delay. In the absence of oceans and a significant amount of water vapor in the atmosphere, there would be no seasons, as the annual changes of the temperature would much more closely follow the changes in the incoming solar radiation. The Martian atmosphere provides empirical evidence to support this point: because there are no oceans and significant transport of latent heat on Mars, the changes in temperature closely follow the seasonal and the diurnal changes in the incoming solar radiation.3 2 The reader interested in techniques for the phenomenological investigation of atmospheric dynamics is referred to the textbook on the topic James (1994). 3 E.g., Read and Lewis (2004).

192

Applicable Atmospheric Dynamics

December−January−February 225 225 215 205 215 195 205 215 225 235 225 245 255 235 45 2 265 55 52 275 6 2

245235

Pressure [hPa]

200

225 225

400

235

245

600

25

5

−60

28

5 28

5 27

265

−80

Pressure [hPa]

235 245 255 26 5

600

1000

0 Latitude

20

40

June−July−August 225 215 205 205 215 225 225 235 245 255 265

−80

28

27

5

5

−60

−40

60

225

−20

80

235

235 245

255

275

285 235

800

−20

215

205 215 225

400

295

−40

195

200

215

5

800 1000

5

20

275

265

295 0 Latitude

20

40

60

80

Fig. 2.1 Zonal mean of the 30-year mean of the temperature [K] for the period 19812010. The December, January, February (top); and June, July and August (bottom) mean fields are computed based on the NCEP/NCAR reanalysis.

2.2.1

Zonal-Mean Temperature Field

The state variable most directly affected by the meridional changes in the solar radiation is the temperature (Fig. 2.1). In the troposphere the zonal mean temperature decreases with height. While it also decreases toward the poles, the location of the highest temperature is not exactly at the Equator: the longitude of the highest temperature is slightly shifted in the direction of the hemisphere of the warm season. This shift is more pronounced in the Northern Hemisphere in the JJA season than in the Southern Hemisphere in the DJF season. This asymmetry is due to the difference between the two hemispheres in the ratio of the areas covered by ocean and land: because the ratio is higher in the Southern Hemisphere, the temperature follows the changes in the incoming solar radiation with a larger inertia there. There is a reversal of the meridional temperature gradient near the boundary between the upper troposphere and the lower stratosphere in both seasons of the Northern Hemisphere and in the warm season of the Southern Hemisphere.

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December−January−February

400

340

340 320 310 300

33

2

90

32

280

0 31

600

0

0

0

0 31

28

0 27

30

0

800

300

270 1000

330 320 310 300 290

33

340

0 32

Pressure [hPa]

200

−80

−60

−40

0

29

0

−20

0 Latitude

20

40

60

80

June−July−August

400 600

1000

0 28 270 250

3

320

300 290

0

30

240 −80

320 310

0

32 10

0 29 −60

−40

310

800

340

340

300 290

310

Pressure [hPa]

200 320

340

−20

0 Latitude

20

280

40

60

80

Fig. 2.2 Vertical-meridional cross-section of the 30-year mean of the potential temperature for the period 1981-2010 at longitude 150◦ E. Shown are the mean fields for (top) January, February and March, and (bottom) June, July and August.

2.2.2

Zonal-Mean Potential Temperature Field

According to the discussion of Sec. 1.4.4.1, the temperature variable that provides the most direct information about the stability of the vertical stratification of the atmosphere is the potential temperature. Figure 2.2 shows the vertical-meridional cross section of the potential temperature field for the DJF and JJA seasons. As can be expected for an atmosphere which is in stable hydrostatic balance at the large and synoptic scales, the potential temperature increases with height at all locations. It should be kept in mind, however, that the data used for the preparation of the figure were produced by an analysis system that used a hydrostatic model to constrain the analyzed atmospheric states. As can be expected, the vertical gradient of the potential temperature is much stronger in the stratosphere than in the troposphere. A particularly important feature of the zonal-mean potential temperature field is the poleward decrease of the potential temperature in the lower troposphere of the extratropics, because it supports the development

194

Applicable Atmospheric Dynamics

θ1

θ1

θ>θ1

θ

θ1

grad θ

α

90−α

Fig. 2.4 Illustration of the relationship between the angle of the isolines and the gradient of the potential temperature.

of baroclinic instability: 4 a parcel displaced in the meridional direction by some disturbance will rise, because its potential temperature will be higher than that of its environment. This situation is illustrated by Fig. 2.3: the larger the angle between the surface and the isentropes the stronger the instability, because for a larger angle, the parcel can accelerate for a longer vertical distance before encountering an environment whose potential temperature is higher than its own potential temperature. The angle between the surface and the isentropes is determined by the meridional and the vertical gradient of the potential temperature (Fig. 2.4). Because the vertical gradient of the potential temperature determines the static stability of the vertical stratification of the atmosphere, the Brunt4 The

dynamics of baroclinically unstable perturbations will be the subject of Sec. 2.4.2.

Perturbation Dynamics

ï5 5

0

DecemberïJanuaryïFebruary 5 0 ï5

0

15 25

25

15

15

5 0 0

ï40

ï20

0

20

40

60

80

ï5 5

0

0

25

200

0 Latitude JuneïJulyïAugust ï5 50

15

35 25

35

5

15

ï60

ï5

ï5

ï80

0

5

5

15

0

1000

5

15

600 800

15

ï5

400

5

15 5

600

0

0

0

5

1000

0

5

ï5

0

800

0

Pressure [hPa]

25 35

400 0

Pressure [hPa]

200

195

ï80

ï60

ï40

ï20

0 Latitude

20

40

60

80

Fig. 2.5 Zonal mean of the 30-year mean of the zonal wind [m/s] for the period 19812010. The (top) January, February, and March and (bottom) June, July and August mean fields are computed based on the NCEP/NCAR reanalysis.

V¨ais¨ ala frequency provides important information about the potential for the development of baroclinic instability. For a given value of the meridional potential temperature gradient, the smaller the vertical gradient of the potential temperature (the Brunt-V¨ais¨ ala frequency), the larger the angle between the isentropic surfaces and the horizontal plane. Thus a weak static stability is conducive for the development of baroclinic instability. 2.2.3

Zonal-Mean Wind Field

The zonal-temporal mean of the zonal wind is shown in Fig. 2.5. The most striking feature of this figure is the dominance of westerly winds, 5 as weak easterly zonal mean winds are present only in the Tropics. There are two strong local maxima of the westerly wind, one in each hemisphere, in both seasons at a pressure level of about 200-250 hPa. These are the locations of the cores of the jet streams.. The jet streams, or simply jets are high speed westerly flows in the upper troposphere in the extratropics. The figure 5 Winds

that blow from west to east.

196

Applicable Atmospheric Dynamics

shows that the jet stream is stronger and located closer to the Equator in the cold season. In general, the zonal wind speed increases from the surface to the level of the core of the jets. This vertical profile of the zonal wind field, together with the meridional changes in the temperature, suggests that the zonal mean circulation is in a state of near hydrostatic and qeostrophic balance. Exercise 2.1. Show that for an atmosphere in hydrostatic and geostrophic balance, the vertical shear (derivative) of the zonal wind is proportional to the meridional temperature gradient (the meridional derivative of the temperature). Answer. In the local Cartesian coordinate system at an arbitrary location, the two coordinates of the geostrophic wind defined by Eq. (1.302) are 1 ∂φ , f ∂y 1 ∂φ = . f ∂x

ug = −

(2.1)

vg

(2.2)

Taking the derivative of Eqs. (2.1) and (2.2) with respect to pressure and substituting ∂φ/∂p from Eq. (1.280) into the resulting two equations leads to R ∂T ∂ug = , (2.3) ∂p pf ∂y R ∂T ∂vg =− . (2.4) ∂p pf ∂x Equation (2.3) proves the statement. Equations (2.3) and (2.4) are the coordinates of the thermal wind balance equation, ∂vg R = (∇H T × k) . ∂p pf

(2.5)

The zonal-temporal mean of the meridional and the vertical components of the wind is shown in a single figure (Fig. 2.6). In order to better illustrate the zonal mean circulation in the meridional-vertical plane, the figure shows the two components of the wind field by both vectors and streamlines. The most striking feature of the figure is the presence of closed cells of circulation. In the DJF season, there are two cells in the tropics: the air rises in the region between about 20◦ S and 10◦ N, travels toward the poles in both cells in the layer between 300 and 100 hPa, sinks between about

Perturbation Dynamics

197

December−January−February 200 Pressure [hPa]

300 400 500 600 700 800 900 −80

−60

−40

−20

0 Latitude

20

40

60

80

June−July−August

0

Pressure [hPa]

200 400 600 800 1000 1200 −100

−80

−60

−40

−20

0 Latitude

20

40

60

80

100

Fig. 2.6 Zonal mean of the 30-year mean of the vertical-meridional circulation for the period 1981-2010. Values and dimensions are not provided, because the fields are distorted for the sake of better transparency. The (top) January, February, and March and (bottom) June, July and August (bottom) mean fields are computed based on the NCEP/NCAR reanalysis.

45◦ S and 30◦ S in the southern cell and between about 15◦ N and 40◦ N in the northern cell, and then flows back toward the low latitudes in both cells. Such a circulation is called a thermally direct circulation and the two cells are called Hadley cells. The highest speed of the rising and sinking motion in the Hadley cells is about 0.15–0.25 Pa/s, while the highest speed of the meridional part of the circulation is about 2–3 m/s. The two Hadley cells also exist in the JJA season, but the northern cell is much weaker and the southern cell is centered at the equator, leading to a cross equatorial flow in the Hadley cell (from NH to SH at the surface and from SH to NH in the upper troposphere). In the two midlatitude cells, which are called Ferrel cells, the air rises in a latitude band centered around 60◦ in both hemispheres, travels toward the equator in the upper troposphere, joins the sinking flow of the Hadley cells and flows toward the poles at the surface. Such a circulation is called a thermally indirect circulation, because the air is rising in a region where

198

Applicable Atmospheric Dynamics

the zonal mean temperature is lower and the air is sinking in a region where the zonal mean temperature is higher. This naming convention, however, is somewhat misleading, because the rising motion in the Ferrel cells is due to baroclinic instability; that is, due to the rising motion of air parcels that are warmer than their environment. Likewise, the sinking motion in the Ferrel cells is due to the sinking motion of air parcels that are colder than their environment. The least coherent cells are the two polar cells. These polar cells, except for the NH cell in the JJA season, are characterized by strong sinking motions near the surface (below about 850 hPa).

2.2.4

Available Potential Energy

2.2.4.1

Definition

Lorenz’s energy cycle6 describes the energy conversion processes for a flow that is decomposed into a zonal mean component and a perturbation component. A central concept of Lorenz’s energy cycle is available potential energy, which has its origin in Max Margules’ work on the energy of storms.7 The available potential energy is the part of the total potential energy that is available for conversion into kinetic energy by baroclinic instability. It is defined by the difference between the total potential energy of the actual state and the total potential energy of the hypothetical state obtained by rearranging the air parcels such, that no potential energy is available for baroclinic instability. Substituting T from Eq. (1.341) into Eq. (1.620) and integrating the resulting equation by parts yields Z ∞ cp P + I = (1 + κ)−1 p−κ p1+κ dθ. (2.6) g 0 0 Potential energy is no longer available for baroclinic instability once the air parcels are rearranged such that the surfaces of constant pressure are also surfaces of constant potential temperature. For the related state, the pressure on an isentropic surface is equal to the areal mean of the pressure for that surfaces. This motivates the notation p¯(θ) for the related vertical profile of the pressure in isentropic coordinate system, where the bar denotes areal average. The related reference value of Zthe total potential energy is ∞ cp p¯1+κ dθ. (2.7) (P + I)r = (1 + κ)−1 p−κ g 0 0 6 Lorenz

(1955). (1903). He called the related quantity available kinetic energy.

7 Margules

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199

and the available potential energy is A = P + I − (P + I)r = (1 +

cp κ)−1 p−κ 0 g

Z

∞ 0


p1+κ − p¯1+κ dθ.

(2.8)

As observed by Lorenz in his 1955 paper, while the available potential energy is a very small portion of the total potential energy,8 it is still an order of magnitude larger than the kinetic energy. In other words, atmospheric motions cannot tap into the pool of available potential energy easily. 2.2.4.2

Lorenz’s energy cycle

We do not describe Lorenz’s energy cycle in detail here.9 Instead, we will provide a detailed description of the eddy kinetic energy equation in Sec. 2.4.4 for a basic flow that is defined by a temporal mean instead of the zonal mean of Lorenz’s energy cycle. The advantage of that formulation is that it allows for an investigation of the spatiotemporal evolution of the energy conversion processes. Here, we only note that in Lorenz’s energy cycle, the conversion of available potential energy of the basic flow into eddy kinetic energy is called baroclinic energy conversion, while the conversion of kinetic energy of the zonal flow into eddy kinetic energy is called barotropic energy conversion. In the extratropics, baroclinic energy conversion generates eddy kinetic energy and negative barotropic energy conversion transfers kinetic energy of the eddies to kinetic energy of the zonal mean flow. In the Tropics, potential energy is converted directly into kinetic energy of the zonal mean flow by the direct circulation of the Hadley cells. As the examples of Sec. 2.4.4 will illustrate, the picture becomes much more complicated once we consider the spatiotemporal changes in the energy conversion processes. Most importantly, barotropic energy conversion can have both large positive and negative local values; and geopotential flux convergence, which is neither a net source or a sink of kinetic energy, can be the dominant local source or sink of kinetic energy. 2.3

Quasi-Geostrophic Baroclinic Equations

As discussed in Sec. 1.5.5.6, the combination of the lowest order reduced form of the vorticity and the divergence equations leads to the quasi8 Less

than 1 per cent. concise description of Lorenz’s energy cycle can be found in James (1994), while estimates of the different components of the cycle based on reanalysis data can be found in Kim and Kim (2013). 9A

200

Applicable Atmospheric Dynamics

geostrophic vorticity equation, Eq. (1.524). One approach to build a mathematical model of the atmospheric dynamics based on the quasi-geostrophic vorticity equation is to eliminate the term −fg D by a vertical averaging of the atmospheric state.10 The resulting equation is the barotropic vorticity equation, which is a two-dimensional model of the atmosphere. This model cannot be used for the investigation of such inherently three-dimensional dynamical processes as baroclinic instability. The approach that leads to a closed system of three-dimensional quasigeostrophic equations requires adding the continuity equation and a reduced form of the thermodynamical equation to the system. The resulting equations can be combined into a single prognostic equation, which represents a Lagrangian conservation law for a single scalar variable. This scalar variable is called the quasi-geostrophic potential vorticity. As can be expected from a potential vorticity type state variable, the quasi-geostrophic potential vorticity satisfies the invertibility principle: at any given time, all hydrodynamical and thermodynamical state variables can be determined from the quasi-geostrophic potential vorticity. 2.3.1

General Assumptions

Substituting the divergence, D, from the continuity equation, Eq. (1.409), into the quasi-geostrophic vorticity equation, Eqs. (1.513), yields ∂ω ∂ζ = − (vg · ∇H ) (ζ + f ) − fg , ∂t ∂p

(2.9)

which is the usual form of the vorticity equation used in a quasi-geostrophic model. The standard assumption made to derive the thermodynamical equation that completes the quasi-geostrophic system of equations is that the potential temperature field can be decomposed such that θ(x, y, p, t) = θb (p) + θ′ (x, y, p, t),

(2.10)

where ′ ∂θb ≫ ∂θ . ∂p ∂p

(2.11)

Here, θb (p) is the basic state (flow) component, while θ′ (x, y, p, t) is the perturbation component of the potential temperature field. A comparison of Figs. 1.17 and 2.2 suggests that the condition given by Eq. (2.11) is not unreasonable, if θb (p) is defined by the climatological-zonal mean of 10 This

approach was discussed in Sec. 1.5.5.7.

Perturbation Dynamics

201

the potential temperature. For instance, if the goal is to obtain model equations that represent the atmospheric dynamics for the cold season in the NH extratropics, θb (p) can be defined by the field shown in the upper panel of Fig. 2.2 for a given latitude (e.g., 50◦ N), or by the average of the same field for a latitude band (e.g., 40–50◦ N). We start the derivation of the reduced form of the thermodynamical equation by rewriting Eq. (1.343) as ∂θ ∂θ + vH · ∇H θ = − ω + L. (2.12) ∂t ∂p The quasi-geostrophic approximation can be applied to Eq. (2.12) by replacing the horizontal wind, vH , in the horizontal advection term by the quasi-geostrophic wind, vg , which is defined by Eqs. (1.490) and (1.491). Because the basic flow component of the potential temperature does not depend on the horizontal coordinates, ∇H θ = ∇ H θ ′ ,

(2.13)

v H · ∇ H θ ≈ v g · ∇ H θ = v g · ∇H θ ′ .

(2.14)

∂θ′ ∂θb + v g · ∇H θ ′ = − ω + L. ∂t ∂p

(2.16)

and the horizontal advection term of Eq. (2.12) becomes In addition, making use of Eq. (2.11), the vertical advection term of Eq. (2.12) can be approximated as ∂θb ∂θ ω≈ ω. (2.15) ∂p ∂p Substituting the approximate forms of the horizontal and the vertical advection terms into Eq. (2.12) yields

Exercise 2.2. Show that the Brunt-V¨ ais¨ ala frequency for the basic state, Nb , satisfies the equation ∂θb = −hNb2 , (2.17) ∂p where  κ R p . (2.18) h= p pr Answer. ∂θb ∂θb ∂z 1 ∂θb RTb ∂θb hθb ∂θb = =− =− = = −hNb2 , (2.19) ∂p ∂z ∂p ρb g ∂z pb g ∂z g ∂z where the subscript b indicates basic field components of the state variables.

202

Applicable Atmospheric Dynamics

With the help of Nb , Eq. (2.16) can be written as ∂θ′ + vg · ∇H θ′ = hNb2 ω + L. (2.20) ∂t Introducing the notation dg S ∂S = + (vg · ∇H ) S, (2.21) dt ∂t where S is an arbitrary scalar state variable, Eqs. (1.513) and (2.20) can be written as d g ζa ∂ω = −fg , (2.22) dt ∂p dg θ ′ = hNb2 ω + L. (2.23) dt In this system of equations, the state variables ζb and θ are not independent, as they can be both written with the help of the stream function ψg . In particular, combining Eqs. (1.488) and (1.490) yields ζa = ∇2H ψg + f,

(2.24)

while taking the partial derivative of Eq. (1.490) with respect to p and substituting ∂φ/∂p from Eq. (1.280) into the resulting equation leads to h ∂ψg (2.25) = − θ′ , ∂p fg Thus the two independent unknown state variables of the system of Eqs. (2.22) and (2.23) are ψg and ω. Exercise 2.3. Show that for the pseudo-height vertical coordinate, zˆ, using a constant reference temperature profile with temperature T0 , Eqs. (2.22) and (2.23) can be written as dg ζ = dt   dg ∂ψg = dt ∂ zˆ

ˆ fg ∂ (ρr w) , ρr ∂ zˆ Nˆb2 g w ˆ+ L, fg fg θ0

(2.26) (2.27)

where ρr (ˆ z ) is the vertical density profile associated with the reference temperature T0 , that is, ρr (ˆ z ) = ρr (0)e−ˆz/H0 , ˆ 2 is defined by Eq. (1.394). and the Brunt-V¨ ais¨ ala frequency N b

(2.28)

Answer. First, we show how to derive Eq. (2.26) from Eqs. (2.22). With the help of zˆ and w, ˆ the vertical derivative of ω can be written as

Perturbation Dynamics

203

w ˆ p ∂w ∂w ˆ ∂ zˆ ∂w ˆ ∂ wp w ˆ w ˆ ˆ ∂ω =− − − + =− =− =− , (2.29) ∂p ∂p H0 H0 H0 ∂p H0 ∂ zˆ ∂p H0 ∂ zˆ where we made use of ∂ zˆ H0 =− , ∂p p

(2.30)

which follows from Eq. (1.376). Equation (2.29) can be written in the equivalent form 1 ∂ (cw) ˆ ∂ω = , ∂p c ∂ zˆ

(2.31)

where c is any state variable whose dependence on the pseudo-height, zˆ, can be described by c (ˆ z ) = c(0)e−ˆz/H0 .

(2.32)

Equation (2.26) can be obtained from Eq. (2.31) by noticing that the reference density profile, ρr (ˆ z ), which is defined by Eq. (2.28), has the right functional dependence on zˆ to play the role of c(ˆ z ). We derive Eq. (2.27) directly from Eq. (2.16) rather than Eq. (2.23). Taking the partial derivative of Eq. (1.491) with respect to zˆ yields g ∂z g θ′ ∂ψ , = = ∂ zˆ fg ∂ zˆ f g θ0

(2.33)

where θ0 is the reference vertical potential temperature profile that corresponds to T0 , and in the last step, we made use of Eq. (1.378). The first term on the right-hand side of Eq. (2.16) can be written as −

∂θb ∂θb ∂ zˆ ∂θb H0 ω ∂θb ω=− ω= = w, ˆ ∂p ∂ zˆ ∂p ∂ zˆ p ∂ zˆ

(2.34)

where in the last step, we made use of Eqs. (1.379) and (1.384). Substituting the result into Eq. (2.16) and multiplying the resulting equation by g/ (fg θ0 ) yields   g ∂θb dg ∂ψg w. ˆ (2.35) = dt ∂ zˆ fg θ0 ∂ zˆ Equation (2.27) can be obtained by further expanding the right-hand side of Eq. (2.35) as Nˆ2 1 θb ∂ ln θb g ∂θb ˆ w ˆ= w ˆ = b w. fg θ0 ∂ zˆ fg θ0 ∂ zˆ fg

(2.36)

204

Applicable Atmospheric Dynamics

where we made use of θb Tb = . θ0 T0 2.3.2

(2.37)

Quasi-Geostrophic Potential Vorticity

Rearranging Eq. (2.23) as dg ω= dt



1 θ′ h Nb2



−

1 L , h Nb2

(2.38)

then taking the partial derivative of Eq. (2.38) with respect to p leads to    ′   ∂ω 1 ∂ dg 1 ∂ θ L − . (2.39) = ∂p dt h ∂p Nb2 h ∂p Nb2

While normally the order of computing the Lagrangian time derivative and the partial derivative with respect to one of the spatial coordinates cannot be reversed, in the case of Eq. (2.39) that rule does not apply, because the Eulerian form of dg /dt does not include a partial derivative with respect to p. Substituting ∂ω/∂p from Eq. (2.39) into Eq. (2.22) yields    ′   fg ∂ θ L fg ∂ dg = . (2.40) ζ +f − dt h ∂p Nb2 h ∂p Nb2 Introducing the notations

fg ∂ qg = ζ + f − h ∂p and fg ∂ S= h ∂p





L Nb2

θ′ Nb2



,



,

(2.41)

(2.42)

Eq. (2.40) can be written as d g qg = S. (2.43) dt The state variable qg is called the quasi-geostrophic potential vorticity. When all processes are adiabatic, L = 0 (S = 0), Eq. (2.43) becomes

d g qg = 0. (2.44) dt As all other forms of the potential vorticity, the quasi-geostrophic potential vorticity is a Lagrangian invariant, if all processes are adiabatic.

Exercise 2.4. What is the diagnostic equation for the relationship between the quasi-geostrophic potential vorticity, qg , and the stream function, ψg ?

Perturbation Dynamics

205

Answer. Substituting θ′ from Eq. (2.25) into Eq. (2.41) yields qq = ∇2H ψg + f +

fg2 ∂ h2 ∂p



1 ψg Nb2 ∂p



,

(2.45)

Exercise 2.5. Show that Eq. (2.44) remains valid for the pseudo-height vertical coordinate for constant reference temperature, with the potential vorticity defined by ! fg2 ∂ ρr ∂ψ 2 qq = ∇H ψg + f + . (2.46) ˆ 2 ∂ zˆ ρr ∂ zˆ N b Answer. First, rearranging Eq. (2.27) for L = 0 leads to ! dg fg ∂ψg , w ˆ= dt Nˆ2 ∂ zˆ

(2.47)

b

then, substituting w ˆ from Eq. (2.47) into Eq. (2.38) yields Eq. (2.46). When Nb2 is constant (does not depend on p), Eq. (2.45) can be written as qq = ∇2H ψg + f +

fg2 ∂ 2 ψ . h2 Nb2 ∂p2

(2.48)

Since Eq. (2.44) can be written in the equivalent form ∂ζg = J (qg , ψg ) , ∂t

(2.49)

the system of equations that governs the evolution of a quasi-geostrophic flow of constant static stability Nb2 is formally similar to he barotropic vorticity equation given by Eqs. (1.525) and (1.528). In the general case, however, the model defined by Eqs. (2.45) and (2.49), or Eqs. (2.46) and (2.49), is more complicated than the model based on the barotropic vorticity equation due to the nonlinearity of the second term in the definition of the potential vorticity. The standard approach to obtain a system of equations, in which the diagnostic equation is linear, but can still account for the most important changes in the static stability is the discretization of the equations in he vertical direction by using finite differences for the approximation of the vertical derivatives. Section 2.3.4 will provide an example for the derivation of a set of model equations along these lines.

206

2.3.3

Applicable Atmospheric Dynamics

Quasi-Geostrophic ω-Equation

In Sec. 1.4.2.5, we saw that the continuity equation cannot be used for the calculation of the vertical velocity ω in a practical diagnostic calculation. There, we promised that a safe approach for the computation of ω will be presented later. This safe approach is based on computing ω by the help of the quasi-geostrophic ω-equation. While the derivation of an ωequation does not require the assumption of quasi-geostrophic balance, the more complex forms of the equation can rarely be used for a diagnostic calculation, because they include the time derivative of the vorticity, which is almost never available for an archived data set. Exercise 2.6. The quasi-geostrophic ω-equation is ! fg2 ∂ fg2 ∂ 2 2 [vg · ∇H (ζ + f )] − ∇2H (vg · ∇H θ′ ) . ∇H + 2 2 2 ω = 2 2 Nb h ∂p h Nb ∂p (2.50) Show that Eq. (2.50) follows from Eqs. (2.22) and (2.23). Hint: Equation (2.50) can be obtained by eliminating ∂ζ/∂t from the system of Eqs. (2.22) and (2.23). Equation (2.50) is an elliptical partial differential equation for ω, which can be solved only numerically in the general case. 2.3.4

Quasi-Geostrophic Baroclinic Model Equations

Different vertical discretization strategies lead to formally slightly different quasi-geostrophic model equations.11 The general structure of the equations and the qualitative dynamics of the related models, however, is very similar. In what follows, we obtain a prototype system of equations, by discretizing Eqs. (2.49) and (2.46), which use pseudo-height as the vertical coordinate. 2.3.4.1

Derivation of the vertically discretized equations

We consider a model atmosphere that consists of three internal levels and two levels that represent the top and the bottom of the model atmosphere (Fig. 2.7). The levels are chosen such that the distance between the internal layers is ∆ˆ z , while the distance between the top internal layer and 11 Some of these vertically discretized equations can be found in such standard textbooks as Pedlosky (1987); Holton (2004).

Perturbation Dynamics

ρr1+

θt=0

Top Boundary

ψ1, q1, ρr1

Top Internal Level Full State Pseudo-Height

207

Basic State N12

ρr1-, ρr2+

N22

ρr2-, ρr3+

Δz Lower Layer

ψ3, q3, ρr3 Δz/2

Bottom Boundary Potential Temperature Fig. 2.7 model.

Δz Upper Layer

ψ2, q2, ρr2

Midle Internal Level

Bottom Internal Level

Δz/2

θb=0

Schematic illustration of the vertical structure of the 3-level quasi-geostrophic

the top boundary, as well as between the bottom internal layer and the bottom boundary, is ∆ˆ z /2. A common choice for the vertical spacing of the levels is ∆ˆ z = H0 /2: choosing 800 hPa to be the lowest internal level, the middle internal level is at about 500 hPa, while the top internal level is at 300 hPa.12 The variables at the three internal levels are denoted by the subscripts 1, 2, 3, such that a larger integer denotes a lower level. Variables at the top and the bottom boundaries are denoted by the subscripts t and b, respectively. The three internal levels define two internal layers of depth ∆ˆ z . The vertical structure of the basic state is solely represented by two local values of the ˆ 2 and N ˆ 2 , which are, respectively, the values of Brunt-V¨ais¨ ala frequency: N 1 2 ˆ 2 in the middle of the top and the bottom layers. N b Another constant parameter that appears in the model equations is ρri− (∆ρ)ri = = e−∆ˆz/2H0 , i = 1, 2, 3, (2.51) ρri where ρri− and ρri denote the density in the middle and at the bottom of the i − th layer, respectively, for the reference density profile associated 12 This

property follows from Eq (1.386).

208

Applicable Atmospheric Dynamics

with the pseudo-height vertical coordinate. It should be kept in mind that this ratio is a property of the definition of the vertical coordinate rather than the basic state. The top and the bottom boundary conditions are defined by the assumption that the potential temperature perturbation must vanish at the two boundaries, that is, θt′ = θb′ = 0,

(2.52)

which implies via Eq. (2.25) that     ∂ψ ∂ψ = = 0. ∂ zˆ t ∂ zˆ b

(2.53)

The outer derivative in the second term of the right-hand side of Eq. (2.46) is approximated by the centered difference scheme !# " ! ! # " 1 ρr ∂ψ ρr ∂ψ ρr ∂ψ ∂ = − , (2.54) ˆ 2 ∂ zˆ ˆ 2 ∂ zˆ ˆ 2 ∂ zˆ ∂ zˆ N ∆ˆ z N N b b b i

i+

i−

where i = 1, 2, 3, and the subscripts i+ and i− denote values of the variables at the half levels above and below level i, respectively. The two terms at the half levels in Eq (2.54) are computed by !   ρr ∂ψ ρri+ ∂ψ (2.55) = ˆ 2 ∂ zˆ ˆ2 ∂ zˆ i+ N N i+1 b i+ !   ρr ∂ψ ρri− ∂ψ . (2.56) = ˆ 2 ∂ zˆ ˆ2 ∂ zˆ i− N N i−1 b i−

Making use of Eqs. (2.55) and (2.56), and the boundary conditions in the form of     ∂ψ ∂ψ = = 0, (2.57) ∂ zˆ 1+ ∂ zˆ t leads to the approximation q1 =

∇2H ψ1

fg2 (∆ρ)r1 − ˆ 2 ∆ˆ N z 1



∂ψ ∂ zˆ



.

(2.58)

1−

of Eq. (2.46) for the top internal level. Using the centered difference approximation   ψ1 − ψ2 ∂ψ , (2.59) = ∂ zˆ 1− ∆ˆ z

Perturbation Dynamics

209

Eq. (2.59) can be written as q1 = ∇2H ψ1 −

fg2 (∆ρ)r1 2 (ψ1 − ψ2 ) . ˆ 2 (∆ˆ N z)

(2.60)

1

Amalgamating all constant parameters of Eq. (2.60) into a single parameter, R12 =

2 ˆ 2 (∆ˆ N z) 1 , fg2 (∆ρ)r1

(2.61)

the equation can be written in the elegant form q1 = ∇2H ψ1 − R1−2 (ψ1 − ψ2 ) .

(2.62)

Following similar arguments, the diagnostic equation for the potential vorticity and the stream function at the lowest internal level can be written as q3 = ∇2H ψ3 − R2−2 (ψ2 − ψ3 ) ,

(2.63)

where R22 =

2 ˆ 2 (∆ˆ N z) 2 2 fg (∆ρ)r2

(2.64)

Exercise 2.7. Show that using centered differences for the approximation of the vertical derivatives in Eq. (2.46), the equation for the middle internal level is q2 = ∇2H ψ2 − R1−2 (ψ1 − ψ2 ) + R2−2 (ψ2 − ψ3 ) .

(2.65)

Equations (2.62), (2.63) and (2.65) provide the vertical discretization of the diagnostic Eq. (2.46). The related discretization of the prognostic Eq. (2.49) is ∂ζi = J (qi , ψi ) , i = 1, 2, 3. (2.66) ∂t The three equations of Eq. (2.66) state that the potential vorticity is a Lagrangian invariant at each of the three levels. The horizontal advection of the quasi-geostrophic potential vorticity at the different levels is a nonlinear process, while the interactions between the neighboring levels are linear. The latter interactions are described by the second terms of the right-hand side of Eq. (2.62), (2.63) and (2.65). Forcing and dissipative effects can be introduced by adding the appropriate terms to the right-hand side of the three equations of Eq. (2.66). The strength of the linear interactions is controlled by the parameters R1 and R2 . Some authors refer to them as the Rossby deformation radii

210

Applicable Atmospheric Dynamics

for the two layers of the model. We reserve the term Rossby deformation radius for the ratio ˆ ˆ ) = N H0 . (2.67) R(N fg With the help of R, R1 and R2 can be written as   ∆ˆ z −∆ˆz/2H0 ˆ1 R1 = R N e , H0   ∆ˆ z −∆ˆz/2H0 ˆ2 R2 = R N e , H0

or with the help of the log-pressure P defined by Eq. (1.398) as   ˆ1 ∆P e−∆P , R1 = R N   ˆ2 ∆P e−∆P . R2 = R N

(2.68) (2.69)

(2.70) (2.71)

Since the factor ∆P e−∆P is a parameter of the vertical discretization, the strength of the linear coupling parameters between two levels is determined by the Brunt-V¨ais¨ ala frequency in the layer defined by those two levels: the lower the Brunt-V¨ais¨ ala frequency the stronger the coupling between the two layers. We recall from Sec. 1.4.4.2 that the Brunt-V¨ais¨ ala frequency is a measure of the resilience of the atmosphere to forces that can stretch and compress a column of air in the vertical direction. The inverse relationship between the strength of the coupling and the Brunt-V¨ais¨ ala frequency is the manifestation of this role of the Brunt-V¨ais¨ ala frequency for quasi-geostrophic dynamics. Since the Brunt-V¨ais¨ ala frequency plays such a central role in quasi-geostrophic dynamics, next we review its most important phenomenological properties. 2.3.4.2

The spatial structure of the static stability field

The spatial structure of the Brunt-V¨ais¨ ala frequency field associated with the climatological mean of the potential temperature fields of Fig. 2.2 is shown in Fig. 2.8. The largest changes occur at the bottom of the tropopause. As can be expected based on the vertical stratification of the potential temperature, the static stability is stronger in the tropopause and the stratosphere than in the troposphere. The regions of strongest static stability in the polar region of the winter hemisphere, that is, over the north pole in the January, February, March period and over the south pole in the June, July, August period.

Perturbation Dynamics

211

December−January−February 0.018 0.0 14

Pressure [hPa]

200

0.012

0.018 400

0.01

0.01

4

0.012

0.0

0.012

0.01

2

2

0.01 −80

8

1 0.104 0.0

12

600 800

2

0.01

0.01

−60

−40

−20

20

0.012 40

0.012 1 0.0

8 0.014 0.01

0 Latitude

0.014

60

80

June−July−August

00.01 .0148 0.018 0.014

0.0

600

12

0.0

0.01

2

12

0.0

0.018

0.012 1

0.014

0.012

800

4 0.018 0.01

0.01

12

400

0.0

Pressure [hPa]

200

14

0.0

0.012 −80

−60

−40

−20

0 Latitude

20

40

60

80

Fig. 2.8 The Brunt-V¨ ais¨ ala frequency N [1/s] for the climatological mean potential temperature fields shown in Fig. 2.2. Shown are the fields for (top) January, February, and March and (bottom) June, July and August.

It is important to emphasize that the seemingly modest changes in the static stability in the troposphere have important consequences for the atmospheric motions. Particularly important are the horizontal changes in the static stability of the lower troposphere, which are shown in Fig. 2.9. This figure shows that the static stability of the lower troposphere is closely related to the temperature of the Earth’s surface: (i) the regions of highest static stability are located in the polar region of the winter hemisphere; (ii) in the winter hemisphere, the static stability tends to be lower over the oceans and higher over the continents; while in the summer hemisphere, the static stability tends to be higher over the oceans and lower over the continents; (iii) the static stability tends to be high off the west coast of the continents,

212

Applicable Atmospheric Dynamics

December−January−February

Latitude

50

0.02

0

0.015

−50

0.01

0

50

100

150

200 Longitude

250

300

350

June−July−August 0.025

Latitude

0.02 0.015 0.01 0

50

100

150

200 Longitude

250

300

350

Fig. 2.9 Horizontal distribution of the climatological value of the Brunt-V¨ ais¨ ala frequency, N , at 925 hPa, where. The field is computed based on the 30-year mean of the potential temperature in the NCEP-NCAR Reanalysis for the period 1981-2010. Shown are the mean fields for (top) January, February, and March; and (bottom) June, July and August.

where the cold ocean currents keep the temperature at the surface low, and low off the east coast of the continents; (iv) in the tropics, in particularly over the oceans, the static stability tends to be low. 2.3.4.3

Limitations of the quasi-geostrophic models

The quasi-geostrophic approximation has a number of important limitations in reproducing either qualitatively or quantitatively the atmospheric dynamics. As for the quantitative limitations, we only note that even the most advanced versions of these models were abandoned as operational forecast models 30–40 years ago. Since that time, these models have been used solely for research purposes.

Perturbation Dynamics

213

Quasi-geostrophic models account for a single nonlinear process: the horizontal advection of potential vorticity.13 This property represents a major reduction of the complexity of the interactions compared to the full system of primitive equations. For instance, the assumption made by Eq. (2.11) eliminates the nonlinearity from the vertical advection process: air parcels can move in the vertical direction, but their motion cannot change the static stability (vertical stratification of the atmosphere). In addition, if all terms of scale ∼ 10−10 would be retained in he vorticity and the divergence equations, the resulting model would also include the nonlinear term of the nonlinear balance equation. Finally, if of all terms would be retained in the vorticity and the divergence equations, which would be equivalent to using the full primitive equations, many more nonlinear terms would appear in the system of governing equations. While the effects of this radical reduction of the nonlinear processes is not fully understood, experience suggests that initially small perturbations to a realistic atmospheric state evolve quasi-linearly for a longer time in a quasi-geostrophic than in a primitive equations model.14 Another important limitation of a quasi-geostrophic model is that it cannot account for the horizontal and temporal changes in the coupling parameters,15 although with the help of smart strategies for the definition of the source and dissipative terms, the most important spatial and temporal changes in the climatology can be surprisingly accurately reproduced by a quasi-geostrophic model. 2.4

Atmospheric Waves

Atmospheric waves are periodic spatial and/or temporal changes in the state variables.16 Studying atmospheric waves have played a central role in atmospheric dynamics since the 1930’s. Understanding their dynamics also turned out to be essential for the understanding of the dynamics of numerical models: (i) certain types of waves are prominently present in both the atmosphere and the numerical model solutions; 13 Or the advection of absolute vorticity, if the less general barotropic quasi-geostrophic dynamics is considered. 14 The significance of this property will become clear by the end of Chapter 3. 15 R and R for the three-level model described in Sec. 2.3.4.1. 1 2 16 Waves play an important role in many areas of physics. An excellent general description of waves is provided by Sec. 47–51 of Feynman et al. (2006b).

214

Applicable Atmospheric Dynamics

(ii) some important atmospheric wave equations have analytical solutions; (iii) wave equations are powerful analytical tools to study the propagation of hydrodynamical influences in the atmosphere. Wave equations are usually obtained by the linearization of the governing equations about a steady state solution. The linearization transforms the original nonlinear partial differential equations into linear partial differential equations. Some of these equations have known analytical wave solutions. The governing equations selected for linearization can be as radically reduced as the barotropic vorticity equation, or as complex as the full primitive equations, but usually without the constituent equations. As a general rule, (i) the primitive equations support all forms of wave motions that can exist in the atmosphere; (ii) the reduced equations support only certain types of wave motion. The use of reduced equations is often rationalized by stating that the selected reduced form of the equations filters the types of wave motion that are irrelevant for the investigated problem. Reduced equations are often called filtered equations because of this filtering effect. For instance, Rossby waves, which will be discussed in Sec. 2.4.2, are always thought to have meteorological relevance, while sound waves are always considered irrelevant in meteorological applications. There are other types of wave motion, however, that may or may not be relevant, or cannot be filtered by a reduction of the governing equations without distorting other important forms of motion. For instance, external gravity waves, which will be discussed in Sec. 2.4.5, fall into this category. Modelers have to cope with such waves by designing numerical solution strategies and procedures for the generation of the initial condition that can properly control their effect on the model solutions. 2.4.1 2.4.1.1

General Formulation Linearized equations

The linearized form of the atmospheric governing equations, ∂u = F(u, t) = 0, ∂t

u(r, 0) = u0 (r),

(2.72)

Perturbation Dynamics

215

¯ (r), describes the spatiotemporal evolution about a steady state solution, u of a perturbation of infinitesimal magnitude, u′ (r, t), to the basic state, ¯ (r). Here, the components of u(r, t) are the prognostic state variables u and the steady state solution is defined as a time-independent solution of Eq. (2.72), that is, ∂u = F(¯ u, t) = 0 ∂t

¯ (r). u(r, 0) = u

(2.73)

The linearized form of Eq. (2.72) is ∂u′ = DF(¯ u)u′ , dt

(2.74)

¯ . The entries of DF(¯ where DF(¯ u) is the Jacobi matrix of F at u u), in ¯ and spatial partial differential opergeneral, include the components of u ators.17 For the investigation of wave solutions, it is often convenient to write Eq. (2.74) in the equivalent form   ∂ − DF(¯ u) u′ = 0. (2.75) ∂t Example 2.1. Assume that the nonlinear equation is ∂u ∂u = −u . ∂t ∂x

(2.76)

The steady state solutions of this equation can be written in the general form u ¯ = a, where a is an arbitrary constant. The Jacobi matrix for u ¯ has a single entry, a∂/∂x, and the specific form of Eq. (2.75) is   ∂ ∂ −a u′ = 0. (2.77) ∂t ∂x Once a solution u′ (r, t) of Eq. (2.74) is obtained, ¯ (r) + u′ (r, t) u(r, t) = u

(2.78)

provides an approximate solution to Eq. (2.72). Phenomenology suggests that Eq. (2.74), which strictly speaking describes the evolution of an infinitesimal perturbation, provides a sufficiently accurate description of the dynamics of several observed (finite-amplitude) forms of wave motion in the atmosphere. It should always be kept in mind, however, that Eq. (2.78) provides only an approximation to the solution of the original system of 17 For instance, ∂/∂x, ∂/∂y, ∂/∂z when the vector of position is represented by Cartesian coordinates.

216

Applicable Atmospheric Dynamics

nonlinear equations. For instance, the linear part of the solution, u′ (r, t), is not constrained by the conservation laws described in Sec. 1.7. Thus the magnitude of the perturbation u′ (r, t) can grow unbounded without de¯ (r). Such a process does not exist in reality, pleting the energy stored in u ′ where the energy of u (r, t) can increase only at the expense of the energy ¯ (r) and u′ (r, t) can also lose energy to the basic state through nonlinof u ear interactions. The linear approximation breaks down once the effects of nonlinear interactions can no longer be neglected. 2.4.1.2

Wave solutions for Cartesian coordinates

A general form of the wave solutions in Cartesian coordinate system is u′ (x, y, z, t) = A(x, y, z, t)ei(mx+ny+lz−νt) ,

(2.79)

The function A(x, y, z, t) is the wave amplitude, while the integers m, n, and l are, respectively, the zonal wave number, the meridional wave number and the vertical wave number, while ν is the frequency. Because the exponent is dimensionless, the dimension of the wave numbers is that of inverse of distance (e.g., 1/m), while the dimension of the frequency is that of inverse of time (e.g., 1/s). The triple k = (m, n, l) is the wave number vector, while k = |k| is the total wave number. We will also use the notation k for the wavenumber when the dynamics of a wave is investigated in one spatial dimension. Solutions of Eq. (2.74) are usually found by making additional assumptions about the dependence of the amplitude and the exponential function on the coordinates. For instance, atmospheric waves are often treated as plane waves, assuming that the amplitude does not depend on any of the coordinates, while the exponent depends only on the two horizontal spatial coordinates. In some other cases, A is assumed to depend only on the vertical coordinate.18 2.4.1.3

Boundary conditions

Because finding wave solutions requires solving linear partial differential equations, a solution strategy must include a proper definition of the boundary conditions. The most common horizontal boundary conditions are periodic boundary conditions. A periodic boundary condition is a natural choice 18 In Sec. 2.4.1.7, we will show that many important properties of a wave with a space dependent A can be deduced from the analytical solutions for constant A.

Perturbation Dynamics

217

for the zonal direction x, if the goal is to find a global wave solution for a full latitude circle. The justification for a periodic boundary conditions in the meridional direction is usually less obvious and its use is usually viewed as a necessary compromise. Analytical investigations of wave dynamics often use open boundary conditions. The boundary conditions are called open when they are assumed to be located at an infinite distance from the location of interest. Using such boundary conditions is appropriate when the goal is to study the local (in space) rather than the global dynamics of waves. An important formal difference between the cases of periodic and open boundary conditions is that in the latter case, the wave number can be non-integer. While open boundary conditions cannot be used in a numerical model, their effects can be simulated for a finite time by using a large model domain with periodic boundary conditions. 2.4.1.4

Unstable, stable and traveling waves solutions

To keep the notion simple, we assume that the equation we linearize has a single state variable and the amplitude of the solution does not depend on the spatial coordinates. Under this assumption, A(x, y, z) becomes a scalar constant, A. In addition, we assume that the spatial coordinates and the exponential function in Eq. (2.79) depend only on the zonal space coordinate x and time t. Then, Eq. (2.79) can be written as u′ (x, t) = Aei(kx−νt) .

(2.80)

Because the perturbation u′ (x, t) is real and the right-hand side of Eq. (2.80) is complex,   u′ (x, t) = ℜ Aei(kx−νt) . (2.81)

When ν is imaginary, the perturbation defined by Eq. (2.80) has the spatial structure of a wave with an exponentially growing or decaying amplitude, because
u′ (x, t) = e−iνt ℜ Aeikx . (2.82)

where e−iνt is real. Introducing the notation σ = iν, the amplitude of the wave is exponentially growing when σ > 0, and exponentially decaying when σ < 0. The parameter σ is called the growth rate. The most general representation of a wave by Eq. (2.82) can be obtained by assuming that A is complex: writing A in polar form,

218

Applicable Atmospheric Dynamics

A = B(cos α + i sin α),

(2.83)

and substituting A from Eq. (2.83) into Eq. (2.82) leads to u′ (x, t) = eσt B [cos α cos (kx) − sin α sin (kx)] = eσt B cos (kx + α). (2.84)

In Eq. (2.84), the parameter α is the phase of the wave. When ν is real, Eq. 2.81 leads to u′ (x, t) = B cos (kx − νt + α)

(2.85)

kλ = 2π.

(2.86)

instead of Eq. (2.84). Because the origin t = 0 can be chosen to be at any time and the phase α simply shifts the wave along the x-axis, the spatial structure of the wave can be studied by setting both t and α to zero in Eq. (2.85). In addition, because the amplitude B is a constant, the spatial structure of the wave is described by the function cos (kx). Hence, the image of the spatial structure of the wave is characterized by a pattern of alternating crests and troughs. The wavelength λ of a wave is the distance between the highest points of two consecutive waves crests, or between the deepest points of two consecutive troughs. If x = 0 is chosen such that it is the location of the highest point of a wave crest, or the location of the deepest point of a trough, λ must satisfy the equation Equation (2.86) can be considered the definition of the wavenumber. For periodic boundary conditions, a dimensionless wave number kˆ can also be defined by the equation ˆ = L, kλ (2.87) where L is the length of the periodic domain. The wave number k can be computed from the dimensionless wavenumber kˆ by the formula 2π k = kˆ . (2.88) L The dimensionless wave number is a popular descriptor of the spatial structure of global waves, because it can be easily determined for an observed wave by counting the number of wave crests, or troughs, over the periodic domain. It also allows for an easy schematic illustration of a wave in a back-of-the-envelope calculation. Example 2.2. A latitude circle is a periodic domain whose length is equal to the circumference of the Earth at that latitude. For instance, at latitude ϕ = 45◦ N , L = 2π × (6370 km × cos 45◦ ) ≈ 28,000 km. At that latitude, the dimensionless wave number for a wave of wavelength λ ≈ 4000 km is

Perturbation Dynamics

219

kˆ ≈ 7, while the wave number is k = 14π/L = 1.57 × 10−3 1/km. Such a wave is usually referred to as a zonal wave number seven wave. The period T of a wave plays the same role in the description of the temporal changes at a fixed location as the wavelength λ in the description of the spatial changes at a fixed time. Based on this analogue, the relationship between the period T and the frequency ν is described by ν=

2π . T

(2.89)

Equation (2.85) can be written in the equivalent form u′ (x, t) = B cos [k(x − ct) + α],

(2.90)

where c=

ν k

(2.91)

is the phase speed of the wave. Thus the perturbation described by Eq. (2.85) is a wave traveling at speed c:19 the periodic temporal changes of frequency ν at a fixed location are the result of the propagation of a wave of wave number k at speed c. The dependence of the phase speed c on the wave number k is described by the function ν = f (k),

(2.92)

which is called the dispersion relation. The dispersion relation can be obtained by substituting u′ (x, t) from Eq. (2.80) into Eq. (2.74), which leads to P(ikx, −iνt)u′ (x, t) = 0,

(2.93)

where P(ikx, −iνt) is a polynomial of ikx and iνt. Equation (2.93) holds for any function u′ (x, t) that satisfies Eq. (2.80), if and only if P(ikx, −iνt) = 0.

(2.94)

The polynomial P(ikx, −iνt) is called the characteristic polynomial and Eq. (2.94) the characteristic equation. The dispersion relationship can be obtained by solving the characteristic equation for ν: when the wavenumber k is not specified, the solutions of Eq. (2.94) is a function of k. 19 In a visualization of the propagation of the wave, the wave crests and troughs travel at the phase speed.

220

Applicable Atmospheric Dynamics

Example 2.3. For the equation discussed in Example 2.2, the characteristic polynomial is P(ikx, −iνt) = −iν − aik,

(2.95)

which yields the characteristic equation −iν − aik = 0.

(2.96)

From this equation, the dispersion relation is ν = −ak.

(2.97)

The dispersion relation is the “DNA” of a wave. From the dispersion relationship, we can determine the wave number range where ν is imaginary (the wave solution is unstable or decaying) and the wave number range where ν is real (the solution is a traveling wave). For a traveling wave solution, we can also determine the phase speed. When ν is complex and its real and imaginary parts are both nonzero, the wave is either an unstable traveling wave (ℑν > 0), or a decaying traveling wave (ℑν < 0). For such waves, the growth properties can be investigated by studying the imaginary part, ℑν of ν, while the propagation properties can be investigated by studying the real part, ℜν of ν. Example 2.4. For the flow discussed in Examples 2.2 and 2.3, the wave solution is a traveling wave, because according to Eq (2.97), ν is real. The phase speed is ν/k = −a. 2.4.1.5

The superposition of wave solutions for a periodic domain

A perturbation field along a latitude circle at a fixed time can be written as u′ (x) =

∞ X

u′k eikx .

(2.98)

k=−∞

Equation (2.98) is the Fourier series expansion of u′ (x), where the Fourier coefficients Ak are defined by Z π 1 u′k = u′ (x)e−ikx . (2.99) 2π −π The only condition that a function u′ (x) must satisfy for the Fourier series to converge at all locations x is that the function is a piecewise smooth

Perturbation Dynamics

221

function.20 Equation (2.103) shows that the zero wave number mode is equal to the zonal mean of the function times a constant. Thus, u′0 = 0, because the zonal mean of a function defined by the deviation from the zonal mean is zero. Mathematical Note 2.4.1 (Fourier Series). Assume that the function f (x) is periodic with period 2π and integrable over the interval [−π, π]. The Fourier series expansion of f (x) is ∞ X

f (x) =

fk eikx ,

(2.100)

k=−∞

where k is integer and the Fourier coefficients are Z π 1 fk = f (x)e−ikx dx. 2π −π

(2.101)

An equivalent form of the expansion is ∞

X 1 (ak cos kx + bk sin kx) , f (x) = a0 + 2

(2.102)

k=1

where

1 a0 = 2f0 = π

Z

π

f (x)dx,

(2.103)

−π

and for k = 1, 2, . . . , ak = fk + f−k =

1 π

bk = i(fk − f−k ) =

Z

π

f (x) cos nx dx, −π

1 π

Z

(k ≥ 0);

(2.104)

π

f (x) sin nx dx, −π

(k ≥ 1); (2.105)

The spatiotemporal evolution of the perturbation, u′ (x, t), can be described by the superposition (linear combination) of the wave modes, that is u′ (x, t) =

∞ X

u′k ei(kx−νt) ,

(2.106)

k=−∞ 20 To be precise, at the locations of a discontinuity, f has to be replaced by the average of its left- and right-hand limit at that location for the Fourier series to converge. In addition, if the Fourier coefficients of two piecewise smoot functions f and g are the same, then f = g. An excellent textbook on the theory and applications of Fourier analysis is Folland (1992).

222

Applicable Atmospheric Dynamics

where the coefficients uk , k = −∞, . . . , −2, −1, 1, 2, . . . , ∞ can be determined from the initial condition u′ (x, 0) by Z π 1 u′ (x, 0)e−ikx . (2.107) u′k = 2π −π We will refer to the waves associated with the different wave numbers as wave modes. The initial condition determines which modes play a role in the evolution of a perturbation, while the dispersion relation determines how those modes evolve. Exercise 2.8. Show that the equation u′ (x, t) = u′ (x, 0) +

∞ X

k=−∞


u′k eikx e−iνt − 1

(2.108)

is equivalent to Eq. (2.106). Solution. Equation (2.108) can be obtained by calculating u′ (x, t)−u′ (x, 0) based on Eq. (2.106), then rearranging the resulting equation. The wave solutions provide a natural framework to study the spatiotemporal evolution of small perturbations to the basic flow. This straightforward strategy, however, usually leads to a rather challenging analytical problem, which is one of the motivations to study the dynamics of small perturbations by numerical experiments. Another motivation is that realistic perturbations have finite rather than infinitesimal magnitude: when the magnitude of a perturbation is growing, its evolution can be followed into the regime where nonlinear interactions become important in a numerical experiment. Being aware of the available analytical results, however, can be highly useful for the interpretation of the numerical solutions. In what follows, we review the analytical results that proved the most useful from this point of view. 2.4.1.6

The superposition of wave solutions for an open domain

The main formal difference between the cases of the finite and the open domains is that in the latter case the Fourier series expansion has to be replaced by a Fourier transform. On an open domain, Eq. (2.106) is replaced by Z ∞ 1 u′ (x, t) = u(k)ei(kx−νt) dk. (2.109) 2π −∞

Perturbation Dynamics

223

where u′ (k) =

Z

∞

u′ (x, 0)e−ikx dx.

(2.110)

−∞

Notice that the wave number spectrum is continuous, rather than discrete: the discrete wave numbers of a Fourier series expansion are replaced by real wave numbers that can take any value between minus infinity and infinity. Moreover, the difference δk between two wave numbers can be infinitesimal. Mathematical Note 2.4.2 (Fourier Transform). While the Fourier series expansion can be applied only to functions that are periodic on a finite domain [−π, π], the Fourier transform can be applied to functions f (x) that are defined on an open domain, [−∞, ∞]. The Fourier transform of f (x) is Z ∞ 1 f (x) = f (k)eikx (2.111) 2π −∞ where f (k) =

2.4.1.7

Z

∞

f (x)e−ikx dx.

(2.112)

−∞

Wave packets

We start with the investigation of the case where ν is real for all wave modes. In that case, the spatiotemporal evolution of the perturbation is determined by the superposition of traveling waves. We will first show that when the perturbation is composed of wave modes from a narrow wave number band, the perturbation component at a fixed time can be written as u′ (x) = A(x)eikc x ,

(2.113)

where the wave number kc is called the carrier wave number. The simplest approach to demonstrate that such a perturbation can be obtained by the superposition of wave modes of similar wave numbers is to first consider a perturbation composed of a pair of wave modes in an open domain. Let the pair of wave modes be u′1 (x) = A1 eik1 x , u′2 (x)

= A2 e

ik2 x

,

(2.114)

224

Applicable Atmospheric Dynamics

where A1 and A2 do not depend on x, and k1 = kc + δk,

k2 = kc − δk.

(2.115)

The superposition (sum) of the two wave modes is
(u′1 + u′2 ) (x) = A1 eik1 x + A2 eik2 x = eikc x A1 eiδkx + A2 e−iδkx . (2.116)

Comparing Eqs. (2.113) and (2.116),


A(x) = A1 eiδkx + A2 e−iδkx .

(2.117)

If δk