Atmospheric Radiation: A Primer with Illustrative Solutions [1 ed.] 3527410988, 9783527410989

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James A. Coakley Jr. and Ping Yang Atmospheric Radiation

Wiley Series in Atmospheric Physics and Remote Sensing Series Editor: Alexander Kokhanovsky Wendisch, M. / Brenguier, J.-L. (eds.)

Stamnes, K. / Stamnes, J. J.

Airborne Measurements for Environmental Research

Radiative Transfer in Coupled Environmental Systems

Methods and Instruments

An Introduction to Forward and Inverse Modeling

2013

Coakley Jr., J. A. / Yang, P.

2015

Atmospheric Radiation

Zhang, Z. et al.

A Primer with Illustrative Solutions

Polarimetric Remote Sensing

2014

Aerosols and Clouds

Kokhanovsky, A. / Natraj, V.

2015

Analytical Methods in Atmospheric Radiative Transfer 2014

Huang, X. / Yang, P.

Radiative Transfer Processes in Weather and Climate Models

North, G. R. / Kim, K.-Y.

2016

Energy Balance Climate Models

Tomasi, C. / Fuzzi, S. / Kokhanovsky, A.

2014

Atmospheric Aerosols Life Cycles and Effects on Air Quality and Climate

Davis, A. B. / Marshak, A. 2016

Multi-dimensional Radiative Transfer Theory, Observation, and Computation 2015

Minnis, P. et al.

Satellite Remote Sensing of Clouds 2015

Weng, F.

Satellite Microwave Remote Sensing Fundamentals and Applications 2016

James A. Coakley Jr. and Ping Yang

Atmospheric Radiation A Primer with Illustrative Solutions

The Authors Prof. James A. Coakley Jr.

Oregon State University College of Earth, Oceanic, and Atmospheric Sciences United States Prof. Ping Yang

Texas A&M University Department of Atmospheric Science United States Cover picture

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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at . © 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Paperback Print ISBN: 978-3-527-41098-9 Hardcover Print ISBN: 978-3-527-41137-5 ePDF ISBN: 978-3-527-68144-0 ePub ISBN: 978-3-527-68146-4 Mobi ISBN: 978-3-527-68145-7 CourseSmart ISBN: 978-3-527-68390-1 WTX ISBN: 978-3-527-68391-8 Cover-Design Grafik-Design Schulz,

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V

Contents Preface 1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10

2

2.1 2.2 2.3 2.4 2.5

3

3.1 3.2 3.3 3.4 3.5

IX

The Earth’s Energy Budget and Climate Change 1 Introduction 1 Radiative Heating of the Atmosphere 2 Global Energy Budget 3 The Window-Gray Approximation and the Greenhouse Effect 6 Climate Sensitivity and Climate Feedbacks 8 Radiative Time Constant 12 Composition of the Earth’s Atmosphere 14 Radiation and the Earth’s Mean Temperature Profile 19 The Spatial Distribution of Radiative Heating and Circulation 32 Summary and Outlook 35 References 39 Radiation and Its Sources 41 Light as an Electromagnetic Wave 41 Radiation from an Oscillating Dipole, Radiance, and Radiative Flux 42 Radiometry 47 Blackbody Radiation: Light as a Photon 50 Incident Sunlight 57 References 63

65 Cross Sections 65 Scattering Cross Section and Scattering Phase Function 68 Elementary Principles of Light Scattering 71 Equation of Radiative Transfer 77 Radiative Transfer Equations for Solar and Terrestrial Radiation 80 References 82

Transfer of Radiation in the Earth’s Atmosphere

VI

Contents

4

Solutions to the Equation of Radiative Transfer 85

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Introduction 85 Formal Solution to the Equation of Radiative Transfer 86 Solution for Thermal Emission 88 Infrared Fluxes and Heating Rates 93 Formal Solution for Scattering and Absorption 102 Single Scattering Approximation 103 Fourier Decomposition of the Radiative Transfer Equation 110 The Legendre Series Representation and the Eddington Approximation 112 Adding Layers in the Eddington Approximation 121 Adding a Surface with a Nonzero Albedo in the Eddington Approximation 123 Clouds in the Thermal Infrared 124 Optional Separation of Direct and Diffuse Radiances 126 Optional Separating the Diffusely Scattered Light from the Direct Beam in the Eddington and Two-Stream Approximations 127 Optional The �-Eddington Approximation 130 Optional The Discrete Ordinate Method and DISORT 135 Optional Adding-Doubling Method 138 Optional Monte Carlo Simulations 140 References 146

4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17

5

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12

Treatment of Molecular Absorption in the Atmosphere 149 Spectrally Averaged Transmissions 149 Molecular Absorption Spectra 151 Positions and Strengths of Absorption Lines within Vibration-Rotation Bands 155 Shapes of Absorption Lines 159 Doppler Broadening and the Voigt Line Shape 162 Average Absorptivity for a Single, Weak Absorption Line 163 Average Absorptivity for a Single, Strong, Pressure-Broadened Absorption Line 164 Treatment of Inhomogeneous Atmospheric Paths 166 Average Transmissivities for Bands of Nonoverlapping Absorption Lines 169 Approximate Treatments of Average Transmissivities for Overlapping Lines 171 Exponential Sum-Fit and Correlated k-Distribution Methods 177 Treatment of Overlapping Molecular Absorption Bands 182 References 184

6

Absorption of Solar Radiation by the Earth’s Atmosphere and Surface 185

6.1

Introduction

185

Contents

6.2 6.3

Absorption of UV and Visible Sunlight by Ozone 186 Absorption of Sunlight by Water Vapor 191 References 201

7

Simplified Estimates of Emission 203 Introduction 203 Emission in the 15 μm Band of CO2 203 Change in Emitted Flux due to Doubling of CO2 209 Changes in Stratospheric Emission and Temperature Caused by a Doubling of CO2 213 Afterthoughts 215 References 217

7.1 7.2 7.3 7.4 7.5

Appendix A Useful Physical and Geophysical Constants 219 Appendix B Solving Differential Equations

B.1 B.2 B.3

Simple Integration 221 Integration Factor 221 Second Order Differential Equations

221

223

Appendix C Integrals of the Planck Function

225

Appendix D Random Model Summations of Absorption Line Parameters for the Infrared Bands of Carbon Dioxide 227

Reference

229

Appendix E Ultraviolet and Visible Absorption Cross Sections of Ozone 231

References Index 233

231

VII

IX

Preface This book is an introduction to atmospheric radiation. The focus is on radiative transfer in planetary atmospheres with particular emphasis on the Earth’s atmosphere, the Earth’s energy budget, and the role that radiation plays in climate sensitivity and climate change. The material is presented at the level expected of entering graduate students in the atmospheric sciences and most upper division undergraduates in the physical sciences. Students will need to have studied physics with calculus and methods for solving linear differential equations. The goal of the book is to provide students with relatively simple physically based methods for calculating radiances and radiative fluxes at the Earth’s surface and the top of the atmosphere and radiative heating rates within the atmosphere. It does so by following the approaches of two classical works: Rodgers and Walshaw [1], a treatment of infrared radiative transfer in the atmosphere, and Lacis and Hansen [2], a treatment of the transfer of solar radiation. Although more sophisticated treatments have appeared, these classical treatments embrace the physics of the problem. The difference in the modern and classical approaches is in the details with which scattering, absorption, and emission are treated and the numerical accuracy of the solutions to the radiative transfer equation. The material presented in the book is intended to help students become familiar with relatively simple techniques that they can use to develop their intuition for the effects of scattering, absorption, and emission. A second goal is to alert students to the sensitivity of the Earth’s climate to seemingly minor perturbations of the radiation budget. A third goal is to exercise a student’s analytical skills. For the most part, the book’s treatment is analytical. The use of large computer programs, while briefly described, is avoided. The emphasis is on helping students build an understanding that allows analytical manipulation rather than relying on computer exercises. Problems at the end of each chapter are meant to be both interesting and instructive. They are intended to help students hone their understanding of the material covered in the chapter as well as in previous chapters. Some of these problems are rather simple but nonetheless helpful aids to learning; others will challenge students. The ordering of the problems is from the simple to the challenging. Occasionally, a problem will call for simple, straightforward numerical calculations that require the use of a spreadsheet or one of the widely

X

Preface

used software programs for interactive computer analysis. These numerical exercises help students develop a sense of magnitudes for various processes. This book grew from the “Class Notes” for a course on atmospheric radiation taught for more than 20 years at Oregon State University. Over the years, the notes evolved to better fit the needs of students and the time constraints of the 10-week quarter system. The book is not meant to be a reference book. Many fine references on atmospheric radiation already exist [3–8]. Teaching introductory courses from these references, however, has often proved difficult. Instructors are forced to select topics from what must seem to students as random pages from different sections of the books. Owing to time constraints, many sections of the books go untouched. In addition, some of the reference books are difficult for students to read. Some assume readers have far more advanced physical and mathematical knowledge and ability than have been acquired by many entering graduate students in the atmospheric sciences and upper division undergraduate students in the physical sciences. In fairness to the students, atmospheric radiation poses special challenges to those encountering the subject for the first time. New students often find bewildering the need for zenith angles, relative azimuth angles, solid angles, radiances, and irradiances. This book is meant to help them over the hurdles of what seems at first the horrendously complicated geometry, strange parameters, and esoteric terminology and units associated with radiative transfer. The book might have been augmented with applications of atmospheric radiation to photochemical reactions, atmospheric optics, and a myriad of remote sensing problems. In addition, the treatment in the book is restricted to plane-parallel radiative transfer. Avoided are the consequences for remote sensing arising from the spatial variability of liquid water and ice within clouds. Also avoided are the 3D radiative effects observed when clouds are present or nearby. Even without these topics, the short 10-week sprint of quarter systems will seem insufficient for dealing with the basics, let alone the potential applications of atmospheric radiation and 3-D radiative transfer effects. The book includes some material such as radiometry (Section 2.3), elementary principles of light scattering (Section 3.3), and a description of molecular spectra (Sections 5.2 and 5.3). More than a few treatises have been written on each of these topics. The material in these sections is meant to provide students with some background in subjects to which they have had little, if any, exposure. Such material can be covered quickly in introductory courses, or even left to students to read on their own. Other additional material, marked optional (Sections 4.12–4.17), need not be covered at all. These sections briefly describe additional, more accurate solutions to the radiative transfer equation and numerical algorithms for solving radiative transfer problems. These sections are intended to serve as an introduction for students who pursue further studies in atmospheric radiation. This additional material might also serve for courses taught within the longer duration of semester systems.

Preface

I (JAC) am indebted to my coauthor for suggesting that the class notes be converted into an introductory textbook. I am even more indebted that he agreed to coauthor the book. Ping’s contributions greatly extended and improved the material in the notes. He made the presentation more readable, more accurate, and more complete. Without Ping’s involvement and the assistance and angelic patience of our editors at Wiley, Ulrike Fuchs and Ulrike Werner, the book would not exist. I am also indebted to the students of atmospheric radiation who struggled with the class notes over the years at Oregon State. Their struggles helped me see the need to seek better approaches for presenting the material. Former students will surely recognize many sections of this book from the notes. Some might even recognize how they contributed to the book through the added explanations, extra steps in derivations, improved diagrams, additional examples, and fewer mistakes. I doubt that this book is as helpful as some of my former students might have liked, but I hope that it satisfies the needs of most students new to atmospheric radiation. Furthermore, I am indebted to the many wonderful colleagues that I have encountered during my career in the atmospheric sciences. They have been immensely valuable in my education and development as a researcher and teacher. Some will surely recognize their contributions to my education in various sections of this book. I am also grateful for the many years of funding support from Oregon State University, the National Science Foundation (NSF), National Aeronautic and Space Administration (NASA), National Oceanic and Atmospheric Administration (NOAA), and the Office of Naval Research (ONR). The funding made it possible for me and the students and postdocs who worked with me to probe techniques for extracting cloud and aerosol properties from satellite observations and pursue evidence for aerosol–cloud interactions. I am particularly grateful for my participation in NSF’s Center for Clouds, Chemistry, and Climate (C4 ) at the Scripps Institution of Oceanography and my many years of participation in NASA’s Earth Radiation Budget Experiment and the Clouds and Earth’s Radiant Energy System project. Without these experiences, this book would not have been possible. I thank the College of Oceanic and Atmospheric Sciences, now the College of Earth, Oceanic, and Atmospheric Sciences, for supporting my writing of the book and allowing me to teach and use a draft of the book in the course on Atmospheric Radiation after I had retired. In addition, I thank Texas A&M University and the Department of Atmospheric Sciences for supporting my visits to College Station while writing the book. Ping and the atmospheric sciences faculty made these visits enjoyable and memorable experiences. I thank my wife and family for putting up with my absences during the past 3 1∕2 years that I spent transforming the notes into a book. Corvallis, Oregon August 2013

Jim Coakley

XI

XII

Preface

One of my goals in the original planning for the Wiley series on Atmospheric Physics was to develop an introductory textbook on atmospheric radiation for senior undergraduate students with majors in the broad area of the geosciences. When Dr Alexander Kokhanovsky of the University of Bremen took over as editor of this series, he encouraged me to pursue this goal. I turned to Prof. James (Jim) Coakley for advice on such a textbook. In response, Jim shared with me the lecture notes he had developed throughout his 20 years of teaching atmospheric radiation at Oregon State University. I was impressed by the unique style of his lecture notes, especially his ability to explain physical concepts clearly without adopting the typical mathematically intensive approach found in most radiative transfer texts. I knew that Jim was thinking of retiring in a few years, and I did not want to see his excellent lecture notes go waste. Therefore, I urged Jim that he develop his lecture notes into a textbook. To my delight, he took my recommendation and invited me to assist him as a coauthor. I am grateful to have had this opportunity to work with Jim for two reasons. First, he is a very pleasant person to work with. He has welcomed the incorporation of many of my suggestions and additions into the final version of the book. Second, through working with him, I have learned a lot about how to teach atmospheric radiation effectively without over-reliance on mathematical equations. In my opinion, this textbook is unique in three aspects. First, many important concepts are explained with simple models. For example, the greenhouse effect and the linkage of climate sensitivity to radiative transfer are demonstrated with a simplified window-gray model. Second, simple diagrams are used to define and explain physical quantities. And third, the materials covered in this textbook are designed to be accessible to readers who may not already have extensive training in physics and mathematics. I am grateful to several researchers and graduate students at Texas A&M University, particularly, Lei Bi, Shouguo Ding, Chao Liu, Chenxi Wang, and Bingqi Yi, for their assistance in the preparation of some diagrams, specifically, Figures 1.7, 1.8, 1.9, 1.11, 1.12, 3.4, 3.5, 4.3, 4.9, 5.3, 5.4, 5.5, 5.12, and 7.6 used in this textbook. I thank my mentors at Texas A&M University, Profs. Gerald North, George Kattawar, and Kenneth Bowman, for supporting my professional development. Over the years, I have learned atmospheric radiation, either theory or practical applications, from a number of people including Prof. Kuo-Nan Liou, Dr Warren Wiscombe, Prof. George Kattawar, Dr Michael King, Prof. Thomas Wilheit, Prof. Thomas Vonder Haar, and Prof. William L. Smith. My academic growth has substantially benefited from research with a number of collaborators, including (in alphabetical order) Anthony Baran, Bryan Baum, Andrew Dessler, Oleg Dubovik, Qiang Fu, Andrew Heidinger, Andrew Heymsfield, Yongxiang Hu, N. Christina Hsu, Hironobu Iwabuchi, Ralph Kahn, Xu Liu, Istvan Laszlo, Patrick Minnis, Michael Mishchenko, Shaima Nasiri, R. Lee Panetta, Steven Platnick, Jerome Riedi, Si-Chee Tsay, Manfred Wendisch, and Fuzhong Weng. While writing this book, my research was supported by the National Science Foundation (NSF), National Aeronautics and Space Administration (NASA), National Oceanic and Atmospheric Administration (NOAA), and the Federal

Preface

Aviation Administration (FAA). I would like to take this opportunity to thank Dr Hal Maring (NASA), Dr Lucia Tsaoussi (NASA), Dr Bradley Smull (NSF), Dr Chungu Lu (NSF), Dr A. Gannet Hallar (NSF), Dr Rangasayi Halthore (FAA), and Dr S. Daniel Jacob (FAA) for their support and encouragement. As Jim already said in his preface, we sincerely thank our editors at Wiley, Ms. Ulrike Fuchs and Ms. Ulrike Werner, for their assistance and for being patient with us. Last but not least, I thank my family for having endured my preoccupation with both working on the book and keeping up with my research over the past several years. College Station, Texas August 2013

Ping Yang

References 5. Liou, K.N. (2002) Introduction to Atmospheric Radiation, 2nd edn, Academic The computation of infra-red cooling Press, New York. rate in planetary atmospheres. Q. J. R. Meteorolog. Soc., 92, 67–92. 6. Bohren, C.F. and Clothiaux, E.E. (2006) Fundamentals of Atmospheric Radiation, 2. Lacis, A.A. and Hansen, J.E. (1974) A Wiley-VCH Verlag GmbH, Weinheim. parameterization for the absorption of solar radiation in the Earth’s atmosphere. 7. Wendisch, M. and Yang, P. (2012) Theory of Atmospheric Radiative Transfer—A J. Atmos. Sci., 31, 118–133. 3. Goody, R.M. and Yung, Y.L. (1989) AtmoComprehensive Introduction, Wiley-VCH spheric Radiation Theoretical Basis, Verlag GmbH, Weinheim. 8. Mishchenko, M.I., Travis, L.D. and Lacis, Oxford University Press, New York. A.A. (2006) Multiple Scattering of Light 4. Thomas, G.E. and Stamnes, K. (1999) by Particles–Radiative Transfer and Radiative Transfer in the Atmosphere Coherent Backscattering, Cambridge and Ocean, Cambridge University Press, University Press, New York. New York.

1. Rodgers, C.D. and Walshaw, C.D. (1966)

XIII

1

1 The Earth’s Energy Budget and Climate Change 1.1 Introduction

The scattering and absorption of sunlight and the absorption and emission of infrared radiation by the atmosphere, land, and ocean determine the Earth’s climate. In studies of the Earth’s atmosphere, the term solar radiation is often used to identify the light from the sun that illuminates the Earth. Most of this radiation occupies wavelengths between 0.2 and 5 μm. Throughout this book the terms “sunlight, solar radiation, and shortwave radiation” are used interchangeably. Light perceivable by the human eye, visible light, occupies wavelengths between 0.4 and 0.7 μm, a small portion of the incident sunlight. Similar to the terminology used for solar radiation, throughout this book the terms “emitted radiation, terrestrial radiation, thermal radiation, infrared radiation, and longwave radiation” are used interchangeably for thermal radiation associated with terrestrial emission by the Earth’s surface and atmosphere. Most of the emission occupies wavelengths between 4 and 100 μm. Sunlight heats the Earth. Annually averaged, the rate at which the Earth absorbs sunlight approximately balances the rate at which the Earth emits infrared radiation to space. This balance sets the Earth’s global average temperature. Most of the incident sunlight falling on the Earth is transmitted by the atmosphere to the surface where a major fraction of the transmitted light is absorbed. Annually averaged, the surface maintains its global average temperature by balancing the rates at which it absorbs radiation and loses energy to the atmosphere. Most of the loss is due to the emission of infrared radiation by the surface. Unlike its relative transparency to sunlight, the atmosphere absorbs most of the infrared radiation emitted by the surface. The atmosphere in turn maintains its average vertical temperature profile through the balance of radiation absorbed; the release of latent heat as water vapor condenses, freezes, and falls as precipitation; the turbulent transfer of energy from the surface; and the radiation emitted by clouds and by the greenhouse gases, gases that absorb infrared radiation. This chapter begins with the global annually averaged balance between the sunlight absorbed by the Earth and the infrared radiation emitted to space. A simple radiative equilibrium model for the Earth renders a global average Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

2

1 The Earth’s Energy Budget and Climate Change

temperature that is well below freezing, even if the Earth absorbed substantially more sunlight than it currently absorbs. The Earth’s atmosphere is modeled as one that allows sunlight to pass through but blocks the infrared radiation that is emitted by the surface. This simple model produces the Earth’s greenhouse effect leading to a warm, habitable surface temperature. The model provides an estimate of the radiative response time for the Earth’s atmosphere, about a month. The month-long response time explains the atmosphere’s relative lack of response to the day–night variation in sunlight and also its sizable response to seasonal shifts as the Earth orbits the sun. The model also leads to estimates of changes in the average temperature caused by changes in the incident sunlight and in atmospheric composition, such as the buildup of carbon dioxide in the atmosphere from the burning of fossil fuels. Some of the change in temperature is due to feedbacks in the atmosphere and surface that alter the amounts of sunlight absorbed and radiation emitted as the Earth’s temperature changes. This chapter then describes more realistic models for the Earth’s atmosphere considered as a column of air with a composition close to the global average composition. It describes how forcing the atmosphere to be in radiative equilibrium leads to turbulent heat exchange between the atmosphere and the surface. The combination of radiation and the exchange with the surface largely explains the vertical structure of the Earth’s global average temperature. In addition, the chapter describes realistic estimates for the various sources of heating and cooling for the surface and the atmosphere and the roles played by clouds and the major radiatively active gases. The chapter ends by noting that the distribution of the incident sunlight with latitude leads to the complex circulations of the atmosphere and oceans. The winds associated with the atmospheric circulation and the accompanying pattern of precipitation and evaporation contribute to the circulation of the oceans. The winds and ocean currents carry energy from the tropics to high latitudes. This transfer of energy moderates the temperatures of both regions.

1.2 Radiative Heating of the Atmosphere

The first law of thermodynamics states that energy is conserved. According to this law, an incremental change in the internal energy of a small volume of the atmosphere dU is equal to the heat added dQ minus the work done by the air dW. dU = dQ − dW

(1.1)

Air behaves similarly to an ideal gas. For an ideal gas the internal energy is proportional to the absolute temperature. An incremental change in internal energy is given by dU = mCV dT with m the mass of air undergoing change and CV the heat capacity of air held at a constant volume. Of course, the atmosphere is free to expand and contract. It is not confined to a volume. Held at constant pressure, air will expand if its temperature rises. The work done by the air is P dV with P the pressure of the air and dV the incremental change in its volume. Combining the

1.3 Global Energy Budget

equation of state for an ideal gas PV = mRT with the conservation of energy gives mCp dT = dQ +

mRT dP P

(1.2)

with Cp = 1005 J kg−1 K−1 being the heat capacity of air held at constant pressure, CP = CV + R, and R = 287 J kg−1 K−1 the gas constant for dry air. Throughout this book, the small changes sometimes applied to the temperature in order to account for the effects of water vapor on the heat capacity and gas constant will be ignored. For a small increment in time, dt, dQ mRT dP dT = + (1.3) dt dt P dt with dT∕dt often referred to as the heating rate and often expressed in units of Kelvin per day, dQ∕dt being the rate at which energy is being added or removed from the air (J s−1 or W), and dP∕dt the rate at which the pressure of the air is changing. The pressure changes give rise to motion. Air responds to heating by changing its temperature and moving. Most of the radiative heating in the atmosphere goes to changing the thermodynamic state, such as the temperature changes experienced in high latitudes in response to changes in solar heating as the Earth orbits the sun. There are, however, some instances in which the radiative heating nearly balances the dynamical response. The rate of downward motion referred to as subsidence approximately matches the radiative cooling in the troposphere of subtropical high pressure systems. For annual mean, global average conditions, there is no net movement of air, dP∕dt = 0. Consequently, mCp

mCp

dQ dT = dt dt

(1.4)

1.3 Global Energy Budget

Annually averaged, the Earth approximately maintains a state of radiative equilibrium. Under such conditions, the annually averaged, global mean temperature remains constant with time. For radiative equilibrium, the rate at which the Earth absorbs sunlight equals the rate at which the Earth emits radiation. Let Q0 be the solar constant in units of power per unit area (W m−2 ), the sunlight reaching the “top” of the Earth’s atmosphere at the average distance between the Earth and the sun, one astronomical unit (AU). There is, of course, no “top” of the atmosphere. Instead, the top represents the surface of an imaginary sphere that contains the Earth and most of the atmosphere. Here, the sphere contains the atmosphere that absorbs or reflects a major fraction of the incident sunlight and also absorbs and emits a major fraction of the infrared radiation. A sphere with a radius that extends to 30 km above the Earth’s surface serves as an imaginary “top.” Such a sphere contains 99% of the Earth’s atmosphere. The solar constant has been measured with high accuracy (∼0.1%) from satellites. The measured values range roughly from 1360 to 1370 W m−2 . The

3

4

1 The Earth’s Energy Budget and Climate Change

Fraction of sunlight relected α, albedo θ0

Incident sunlight Q0 = 1360 Wm–2

Radiation 2 4 emitted = 4πRE × σTe

Area blocked = πRE2

Radiative equilibrium Sunlight absorbed = Radiation emitted 2

2

4

Q0(1 – α) × πRE = 4πRE × σTe Q0 4 (1 – α) = σTe 4 Figure 1.1 Radiative equilibrium for the Earth. The rays from the sun are parallel as they strike the Earth. To a distant viewer in space, the Earth casts a shadow as if it were a circle having a radius equal to that of the Earth. The Earth reflects a fraction of the sunlight that is blocked and absorbs the remainder. In radiative equilibrium, the

absorbed sunlight is balanced by the emitted infrared radiation. The Earth rotates sufficiently rapidly that its radiative equilibrium temperature, Te , is assumed to be the same for both day and night sides. The angle between the incident sunlight and the normal of the Earth’s surface where the sunlight strikes is the solar zenith angle �0 .

most recent measurements [1] are thought to provide the most accurate value, 1361 W m−2 . Moreover, the solar constant is not constant, but varies with the 11 year cycle of sunspots. The peak-to-peak amplitude of the solar constant variation is about 0.2% of the average value. For convenience, Q0 = 1360 W m−2 is used unless otherwise noted. In addition, the distance between the Earth and the sun is far greater than both the radius of the Earth and the radius of the sun. As a result, the solar radiation incident on the Earth appears as if it were collimated so that the rays of incident sunlight are all parallel to a line joining the centers of the sun and the Earth, as shown in Figure 1.1. If the Earth were to absorb all sunlight incident on its surface, the rate of absorption would be Q0 �R2E with RE = 6371 km the Earth’s radius. As shown in Figure 1.1, the area of the circle �R2E is the area blocked by the Earth as it passes between the sun and a distant observer in space. The Earth does not absorb all of the incident sunlight. It reflects a fraction. The albedo � is the fraction of sunlight reflected. It too has been measured from satellites and has been found to be close to 0.3 [2]. The fraction of sunlight absorbed is thus 1 − �. Finally, the Earth is approximately a sphere. Only half is being illuminated at any instant. For that half, the average sunlight incident per unit area is given by Q0 ∕2 = 680 W m−2 . The fractional factor

1.3 Global Energy Budget

“1/2” represents the ratio of the area of the circle that blocks the sunlight to the surface area of the hemisphere having the same radius. The fractional factor also represents the global average cosine of the solar zenith angle for the sunlit side of the Earth. Zenith is the upward direction normal to the Earth’s surface. The solar zenith angle at the surface of the Earth is the angle between the upward normal of the Earth’s surface and the direction to the sun as illustrated in Figure 1.1. The cosine of the solar zenith angle accounts for the slant of the Earth’s surface away from the direction of the incident rays of sunlight. Owing to the slant angle, the solar radiation at the top of the atmosphere is distributed over the larger area of the slanted surface. The power per unit area is diminished by the cosine of the solar zenith angle. For the sunlit side of the Earth an “effective” average solar zenith angle is that which produces the average cosine of the solar zenith angle, 60∘ . As discussed later in this chapter, the Earth spins rapidly on its axis so that on average, the absorbed sunlight is distributed over both the dayside and the nightside. The global average incident sunlight becomes Q0 ∕4 = 340 W m−2 . The division by 4 is obtained by invoking the ratio of the area of a circle with a given radius to that of a sphere with the same radius. The same result is obtained by calculating the average cosine of the solar zenith angle for the daylight side of a sphere illuminated by sunlight and dividing the average cosine by two to obtain the “day–night average” incident solar radiation for a rapidly rotating sphere. The rate of emission by the Earth is assumed to be that of a blackbody and is given by the Stefan–Boltzmann law. Assuming that the Earth’s temperature is everywhere the same, the rate of emission is given by �Te4 with � = 5.67 × 10−8 Wm−2 K−4 the Stefan–Boltzmann constant and Te the radiative equilibrium temperature or the effective radiating temperature of the Earth. In Figure 1.1 the emission is symbolically portrayed by the curved dashed lines emanating from the Earth. In radiative equilibrium, the rate at which radiation is absorbed equals the rate at which radiation is emitted. Q0 (1 − �) = �Te4 4

(1.5)

The radiative equilibrium temperature for the Earth is given by [ Te =

Q0 (1 − �) 4�

]1∕4

[ =

1360 Wm−2 × (1 − 0.3) 4 × 5.67 × 10−8 Wm−2 K−4

]1∕4 = 255 K

(1.6)

The equilibrium temperature is equivalent to an atmospheric temperature at an altitude of about 5 km. Suppose there were no atmosphere and the Earth was entirely covered by oceans. The albedo of the earth would be that of the oceans, � = 0.06. Then the radiative equilibrium temperature would be [ Te =

1360 Wm−2 × (1 − 0.06) 4 × 5.67 × 10−8 Wm−2 K−4

The Earth would be close to freezing.

]1∕4 = 274 K

(1.7)

5

6

1 The Earth’s Energy Budget and Climate Change

1.4 The Window-Gray Approximation and the Greenhouse Effect

The window-gray, radiative equilibrium model of the Earth is the simplest model that contains the greenhouse effect. The greenhouse effect of the Earth’s atmosphere represents the difference between the radiative equilibrium temperature and the Earth’s surface temperature. In this book, 288 K is used for the global, annually averaged surface temperature, approximately its current value. The warm surface arises from the Earth’s atmosphere transmitting most of the incident sunlight to the surface. The surface absorbs most of this transmitted sunlight. Only a small fraction of the incident sunlight is absorbed by the atmosphere. Unlike its transparency to sunlight, the Earth’s atmosphere absorbs a large fraction of the infrared radiation emitted by the surface. In the window-gray, radiative equilibrium model, the atmosphere is transparent for incident sunlight, but equally absorbing at all wavelengths for infrared radiation. The term “gray” means no color, or equivalently, no variation of the radiation with wavelength. Clouds reflect light and transmit light equally at all wavelengths in the visible spectrum and thus appear to be of various shades of gray from black or dark gray to white. In the window-gray approximation, sunlight passes through the atmosphere and the fraction that is not reflected is absorbed by the surface. Thus, the surface albedo is assumed to be the same as the Earth’s albedo, � = 0.3. In addition, the atmosphere is taken to be isothermal, meaning that it has the same temperature at all altitudes. Furthermore, the atmosphere is also assumed to be in radiative equilibrium. For an isothermal atmosphere in radiative equilibrium, the fraction of radiation absorbed, which is given by an absorptivity, is equal to the fraction of radiation emitted, which is given by an emissivity. This conclusion can be derived from the second law of thermodynamics and is known as Kirchoff ’s radiation law after Gustav R. Kirchoff who first noted this relationship in the latter half of the nineteenth century [3]. Only three things can happen to light as it passes through an atmosphere. It can be reflected by objects that scatter light. It can be absorbed, and it can be transmitted. Assuming that none of the infrared radiation is scattered so that there is no reflection, in order to conserve energy at infrared wavelengths, radiation not absorbed is transmitted. Since the fraction of radiation absorbed is given by the emissivity �, the fraction of radiation transmitted is given by 1 − �. Figure 1.2 illustrates the energy exchanges for the Earth, the atmosphere, and the surface in the window-gray approximation. The window-gray, radiative equilibrium model gives rise to two equations with two unknowns, the temperature of the surface TS , and the temperature of the isothermal atmosphere TA . At the top of the atmosphere, radiative equilibrium for the Earth leads to Q0 (1 − �) = ��TA4 + (1 − �)�TS4 (1.8) 4 At the surface, radiative equilibrium of the surface leads to Q0 (1 − �) + ��TA4 = �TS4 4

(1.9)

1.4

The Window-Gray Approximation and the Greenhouse Effect

At the top of the atmosphere Absorbed sunlight = Emitted radiation Q0 4

(1–α) = (1–ε) σT 4S + εσT 4A = σT 4e 4

(1–ε) Q0

α Relected = 4 sunlight Incident sunlight Q0 = 4

Relected sunlight =

σT 4S =

εσT A = Radiation emitted by the Radiation atmosphere transmitted by the atmosphere 4

Emitted = εσT A radiation 4

TA = Atmospheric temperature

εσT S = Absorbed radiation Q0 4

Surface albedo = α

α

σT 4S = Radiation emitted by the surface

ε = Atmospheric emissivity

εσT 4A = Radiation emitted by the atmosphere

At the surface Absorbed sunlight + Radiation emitted by the atmosphere and absorbed by the surface = Radiation emitted by the surface Q0 (1–α) + εσT 4A = σT 4S 4 Figure 1.2 Window-gray, radiative equilibrium model.

Notice that in Equation 1.9, the surface absorbs all of the radiation that it does not reflect. In radiative equilibrium the surface emits at the same rate that it absorbs radiation. The fraction absorbed is equal to the fraction emitted, which in Equation 1.9 is assumed to be unity. For most surfaces on Earth, the emissivity is close to unity. For simplicity, unit surface emissivity is assumed throughout this book. Substituting Equation 1.5 into Equation 1.9 yields �TS4 = �Te4 + ��TA4

(1.10)

Consequently, TS > Te . The surface has a higher temperature owing to the presence of an atmosphere that absorbs infrared radiation. Subtracting Equation 1.8 from Equation 1.9 leads to the radiative equilibrium condition for the atmosphere ��TS4 = 2��TA4

(1.11)

Clearly, the atmosphere is at a lower temperature than the surface. Substituting Equation 1.11 into Equation 1.9 yields the surface temperature as given by

7

8

1 The Earth’s Energy Budget and Climate Change

⎤ ⎡ ⎢ Q0 (1 − �) ⎥ TS = ⎢ ( ) � ⎥ ⎥ ⎢ 4� 1 − ⎣ 2 ⎦

1∕4

(1.12)

For the Earth, TS = 288 K. With this surface temperature, Equation 1.11 gives the temperature of the atmosphere as TA = 242 K. Using TS = 288 K, � = 0.3, and Q0 = 1360 W m−2 , and solving Equation 1.12 to obtain a consistent value for the emissivity yields � = 0.78. Like the Earth’s atmosphere, the window-gray atmosphere is heated by the surface, and the primary source of atmospheric heating is the absorption of infrared radiation emitted by the surface. Clearly from Equation 1.12, if the absorption of longwave radiation in the atmosphere increases, � increases and the surface temperature rises. Owing to the condition of radiative equilibrium, which leads to Equation 1.11, as the surface temperature rises, the atmospheric temperature must also rise. With no atmosphere, the Earth’s surface temperature would equal the radiative equilibrium temperature. If the Earth had an albedo of 0.3 without an atmosphere, then the surface temperature would be 255 K. Because the atmosphere absorbs infrared radiation, the Earth’s surface temperature is 288 K. The difference, 288 − 255 K = 33 K, is the greenhouse effect. The surface temperature is 33 K higher than it would be if the atmosphere were transparent at infrared wavelengths and the Earth had an albedo of 0.3. The difference between emission by the Earth’s surface and that at the top of the atmosphere, 390 − 240 W m−2 = 150 W m−2 , is referred to as the greenhouse forcing. The greenhouse forcing may be considered as a climate forcing similar to those due to the buildup of greenhouse gases in the atmosphere, which is discussed in the next section. Since the radiation emitted at the top of the atmosphere and by the surface are both measurable, and since the 33 K response is observed, the ratio of the greenhouse effect to the greenhouse forcing provides an empirical estimate of climate sensitivity. 33 K ∼ 0.2 K (W m−2 )−1 150 W m−2

(1.13)

Climate sensitivity is the equilibrium response of the surface temperature that results from a constant radiative forcing. For a climate forcing of 1 W m−2 and a sensitivity of 0.2 K (W m−2 )−1 , the response would be 0.2 K. This sensitivity is about a factor of three smaller than the sensitivity expected for the Earth’s climate.

1.5 Climate Sensitivity and Climate Feedbacks

For the Earth’s temperature to remain constant, the Earth must maintain a state of radiative equilibrium; the amount of absorbed sunlight must equal the amount

1.5

Climate Sensitivity and Climate Feedbacks

of radiation emitted, indicated here by a variable F. Q0 (1 − �) = F 4

(1.14)

If the composition of the atmosphere is changed, as when a volcano erupts and the stratosphere is filled with tiny droplets of sulfuric acid and tiny particles of sulfate that reflect sunlight, then the current state of radiative equilibrium is upset. The climate will respond to the change so that a new state of radiative equilibrium is established. The rate at which the Earth is heated is given by ] [ Q0 dT mCP =Δ (1 − �) − F = ΔℛNET dt 4

(1.15)

The term in the brackets in Equation 1.15 is the change in the net radiation budget of the Earth, ΔℛNET . It is the change in the rate at which sunlight is absorbed minus the change in the rate at which the Earth emits. The change in the net radiation budget is referred to as the radiative forcing of the climate. The symbols in Equation 1.15 for the mass and heat capacity are the same as those used for the atmosphere in Equations 1.3 and 1.4. Their product gives the “thermal inertia” of the atmosphere. A more appropriate mass and heat capacity for the Earth, however, are those associated with the elements of the system that undergo substantial temperature change over periods of a few years. For periods of a few years, the largest thermal inertia is associated with the uppermost 50–100 m of the ocean known as the ocean mixed layer. This layer of water is stirred by surface winds so that its temperature is nearly uniform throughout. For decadal and longer scales, the temperature of the deep ocean also changes but such changes will be ignored in this analysis. Since the thermal inertia of the ocean mixed layer is much larger than that of the overlying atmosphere, atmospheric surface temperatures, and thus mean atmospheric temperatures, are tied to ocean surface temperatures. For land surfaces, however, soils, asphalt, concrete, and vegetation are poor heat conductors. As a result, relatively little mass is involved in temperature changes of land surfaces. Lack of heat capacity is the reason that land surface temperatures respond so dramatically to the daily cycle of solar heating. Since the thermal inertia of the atmosphere is much greater than that involved in changing land surface temperatures, land surface temperatures averaged over several years tend to follow the average temperature of the overlying atmosphere. The term in brackets in Equation 1.15 gives the net radiative heating. It is zero when the Earth is in equilibrium. For a volcanic eruption, the equilibrium can be broken. The albedo increases, less sunlight is absorbed, and the Earth cools. Alternatively, as indicated for the window-gray, radiative equilibrium model, if the infrared absorption by the Earth’s atmosphere is suddenly increased, the emission at the top of the atmosphere as given by Equations 1.8, 1.11, and 1.14, F = (1 − �∕2)�TS4 , decreases and the Earth warms.

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1 The Earth’s Energy Budget and Climate Change

The equilibrium response to a radiative forcing is approximately given by a Taylor series expansion of the brackets in Equation 1.15: ( ) ΔQ0 Q0 ∂� || ∂F || ∂� || ∂F || ΔU + ΔU = 0 ΔT (1 − �0 ) − − | | ΔTS − S | | | 4 4 ∂U |TS ∂TS |U ∂TS |U ∂U ||TS (1.16) with �0 the albedo for the current equilibrium climate and ΔQ0 a change in the incident solar radiation. As was noted earlier, the incident solar radiation is not constant. It changes slightly over the course of the 11 year sunspot cycle and is thought to have longer term variations of a few tenths of a percent on century to millennial scales. Such changes are relatively small compared with other sources of forcing. For the present, these changes are taken to be negligibly small, ΔQ0 = 0. The change in albedo due to a change in atmospheric composition, such as that brought about by a volcanic eruption, is given by (∂�∕∂U)ΔU. A similar term could be used for a change in the albedo due to human practices, such as clearing of forests to create croplands. The change in the albedo due to surface temperature is given by (d�∕dTS )ΔTS . The term includes climate feedbacks, such as the decrease in area covered by snow and ice as the Earth’s temperature rises, as is observed [4], and changes in cloud properties with the Earth’s temperature. For the emitted infrared radiation, the terms are similar. Changes in the emitted radiation due to changes in atmospheric composition are given by (dF∕dU)ΔU. Changes in atmospheric composition arise from human activity, such as the buildup of carbon dioxide in the atmosphere from the burning of fossil fuels. The change in the emitted radiation with temperature is given by (dF∕dTS )ΔTS . It includes not only the rise in emission with increasing temperature of the surface and atmosphere but also feedbacks such as, for example, the increase in the concentration of atmospheric water vapor as the Earth’s temperature rises and changes in the emitted radiation due to changes in cloud properties as the Earth’s temperature changes. The emitted radiation also varies from what appear to be natural causes. An example of natural variations is the increase and decrease in carbon dioxide and methane concentrations in the atmosphere as the Earth thaws from an ice age and then cools as it enters another ice age. Such changes are thought to arise from the thawing and freezing of permafrost and other biological changes that could accompany a warming and cooling Earth. Whether to call such changes a forcing or a feedback is usually determined by how rapidly the feedback alters the climate. Evidence from the paleoclimate record suggests that since the start of the industrial revolution, changes in atmospheric composition due to human activity far outpace any changes that occurred during the thousands of years over which the Earth recovered from the last ice age [4]. Also, changes in vegetation surely accompany climate change, such as the transition from forests to grasslands and grasslands to shrubs and deserts. Such transitions, however, are expected to be relatively slow, decades to century scales. They are much slower than the faster changes expected in the hydrologic cycle, the buildup of water vapor with increasing temperature, decreases in seasonal snow and ice cover,

1.5

Climate Sensitivity and Climate Feedbacks

and changes in cloud properties. Changes in the hydrologic cycle have short time scales, typically shorter than seasonal scales. In addition, the biological changes seem to respond to multiyear trends in temperatures and the hydrologic cycle. The feedbacks normally included in estimates of climate sensitivity are those associated with the hydrologic cycle [5]. Consider the equilibrium response of the surface temperature to the eruption of Mt. Pinatubo in June 1991. From Equation 1.16 the change in the net radiation budget becomes ) ( Q0 Δ�U Q0 ∂� ∂F − ΔTS (1.17) + − ΔFU = 4 ∂TS 4 ∂TS with Δ�U = (∂�∕∂U)ΔU representing the change in albedo caused by the buildup of the volcanic haze layer in the stratosphere and ΔFU = (∂F∕∂U)ΔU representing the change in emitted radiation caused by the layer. Initially, the haze layer affected the emitted infrared radiation, but with time, the large ash and clay particles that were part of the initial plume fell from the stratosphere, leaving the small droplets of sulfuric acid and particles of sulfate behind. While the remaining volcanic layer also had an effect on the emitted infrared radiation, the effect was relatively small compared with the effect of the particles on the reflected sunlight. For simplicity, the effects of the haze layer on emitted radiation will be ignored, ΔFU = 0. From Equation 1.17 the change in the equilibrium temperature is given by Q Δ� − 0 U 4 (1.18) ΔTS = Q0 ∂� ∂F + 4 ∂TS ∂TS Aside from the effects of clouds, the emitted radiation is expected to increase with increasing temperatures, ∂F∕∂TS > 0. Likewise, aside from the effects of clouds and because there is less ice and snow as the temperature rises, the albedo will decrease. In addition, as discussed in Section 1.7, the atmosphere appears to maintain a state in which the relative humidity remains approximately constant. Consequently, as the temperature rises, the amount of atmospheric water vapor increases, thereby increasing the absorption of sunlight and further decreasing the albedo. So, as the temperature rises, ∂�∕∂TS < 0. Numerical estimates indicate that Q0 ∕4 ∂�∕∂TS + ∂F∕∂TS > 0. With the denominator in Equation 1.18 greater than zero, an increase in albedo causes a decrease in the surface temperature. A decrease was observed following the Mt. Pinatubo eruption [4]. Following the same strategy starting with Equation 1.16, if carbon dioxide in the atmosphere were to suddenly double, the emission at the top of the atmosphere would fall by about 4 W m−2 , ΔFU = −4 W m−2 in Equation 1.17. Since the increase in carbon dioxide has almost no effect on the absorbed sunlight, Δ�U = 0. As a result, the change in the equilibrium temperature for an increase in carbon dioxide would be given by −ΔFU ΔTS = (1.19) Q0 ∂� ∂F + 4 ∂TS ∂TS

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1 The Earth’s Energy Budget and Climate Change

Thus, the temperature will increase for a doubling of CO2 . For the window-gray, radiative equilibrium model, the planetary albedo, which is the same as the surface albedo, remains constant, ∂�∕∂TS = 0. In addition, the emissivity of the atmosphere remains constant so that ( ( ) ) � � ∂ 1− 4 1− �TS4 �TS4 4 × 240 ∂F 2 2 = = = W m−2 K−1 = 3.3 W m−2 K−1 ∂TS ∂TS TS 288 (1.20) As a result, the response of the surface temperature to a change in atmospheric composition, or for that matter, a change in the solar constant, is given by ΔTS = �ΔℛNET

(1.21)

with ΔℛNET the change in the radiation budget at the top of the atmosphere. The change in the top of the atmosphere radiation budget is the radiative forcing (W m−2 ). The climate sensitivity � is the inverse of the denominator in Equation 1.19. For the window-gray, radiative equilibrium model, � = 0.3 K W−1 m−2 . This value is close to the 0.2 K W−1 m−2 estimated from the greenhouse forcing and the greenhouse effect of the Earth’s atmosphere. For a doubling of CO2 , the response would be ΔTS = 1.2 K. This same response was obtained in an early model for the Earth’s atmosphere in which the amount of water vapor was held fixed so that the emissivity of the atmosphere also remained constant [6]. For the Earth’s atmosphere, the relative humidity appears to remain fairly constant. Numerical simulations in which clouds, surface reflection, and relative humidity in the atmosphere are held constant give Q0 ∕4 ∂�∕∂TS = −0.2 W m−2 K−1 . With the relative humidity fixed, the rise in the atmospheric temperature is accompanied by a rise in water vapor. The change in albedo results from the slight increase in the absorption of sunlight as the amount of water vapor increases. For emitted radiation, ∂F∕∂TS = 2 W m−2 K−1 . The sensitivity of the emitted radiation is smaller than that for the window-gray model owing to the changes in water vapor. This change in water vapor with a change in temperature is referred to as the water vapor feedback in the climate system. With fixed relative humidity, � = 0.6 K W−1 m−2 . For a doubling of CO2 , the response is ΔTS = 2.4 K, twice the value obtained for the window-gray model [7] and three times the sensitivity estimated from the greenhouse forcing and the greenhouse effect.

1.6 Radiative Time Constant

The window-gray model lends itself to a reasonable estimate of how rapidly the atmosphere would cool if the sun were suddenly removed. For example, as the Earth rotates from daytime to nighttime, half the atmosphere is in the dark. If the sun were suddenly “turned off” in the window-gray model, the atmospheric

1.6

Radiative Time Constant

temperature would be given by mCP

dTA = ��TS4 − 2��TA4 dt

(1.22)

with m = 1.034 × 104 kg m−2 the mass per unit area for the atmosphere. If the heat capacity of the surface is zero, then without the sun, the surface energy budget is given by �TS4 = ��TA4

(1.23)

Note that without the sun the surface is colder than the atmosphere, as it often is on a cloud-free night when the absolute humidity is relatively low. Such conditions are common to high latitude continental regions during winter. Without the sun, the surface is heated by the atmosphere. Combining Equations 1.22 and 1.23, the temperature of the atmosphere is given by mCP

dTA = −�(2 − �)�TA4 dt

(1.24)

As expected, the atmospheric temperature will decrease once the sun is removed. The initial temperature trend is obtained by expanding the atmospheric temperature into a constant equilibrium temperature TA0 and a perturbation temperature T ′ ≪ TA0 . The expansion gives mCP

′ dT ′ 4 4 T = −�(2 − �)�TA0 − 4�(2 − �)�TA0 dt TA0

(1.25)

Rearranging the terms yields 4

�(2 − �)�TA0 dT ′ T ′ + =− dt � mCP

(1.26)

with the radiative time constant for the atmosphere given by �=

mCP TA0

(1.27)

4 4�(2 − �)�TA0

Using values derived for the window-gray model, � = 39 days, about a month. This value is close to the radiative time constant estimated for the Earth’s atmosphere. Multiplying Equation 1.26 by the integration factor et∕� gives t

e � dT ′ +

4

�(2 − �)�TA0 t T ′ �t e dt = − e � dt � mCP

(1.28)

The integration factor together with the chain rule for differentiation transforms Equation 1.28 to give 4 ) ( �(2 − �)�TA0 t t e � dt d T ′e � = − mCP

(1.29)

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1 The Earth’s Energy Budget and Climate Change

Integrating both sides of Equation 1.29 from time t = 0, when T ′ = 0, to some arbitrary time t yields the perturbation temperature at time t given by T ′ (t) = −

4 �(2 − �)�TA0

mCP

∫0

t

dt ′ e

t ′ −t �

Performing the integration produces 4 ( ) ) �(2 − �)�TA0 t t T ( 1 − e− � = − A0 1 − e− � T ′ (t) = −� mCP 4

(1.30)

(1.31)

Since the atmosphere spends half a day turned away from the sun, and since for half a day, t ≪ �, the nighttime temperature change is approximately given by TA0 t 242 =− K = −0.8 K (1.32) 4 � 4 × 2 × 39 There is indeed little difference, about 1 K between the daytime and nighttime average temperature of the Earth’s atmosphere. Of course, on the ground, under stable weather conditions in a dry climate, one can experience large swings in air temperature. Under such conditions, however, most of the change is confined to the boundary layer, typically the lowest 100 hPa or so. The temperature of the rest of the atmospheric column hardly changes. T ′ (t) = −

1.7 Composition of the Earth’s Atmosphere

The absorption and emission of radiation in the Earth’s atmosphere shapes the vertical structure of the global average atmospheric temperature profile. The absorption and emission depend on the concentrations of the gases and particles that absorb and emit the radiation. Table 1.1 lists the concentrations and approximate atmospheric residence times for the major gases. Gases such as nitrogen and Table 1.1 sphere.

Concentrations and approximate residence times of gases in the Earth’s atmo-

Gas

Nitrogen, N2 Oxygen, O2 Argon, Ar Water vapor, H2 O Carbon dioxide, CO2 Ozone, O3 Methane, CH4 Nitrous oxide, N2 O CFCs

Concentration

Approximate residence time

78% 21% 1% 0–2% 390 ppmv 0–500 ppbv 1750 ppbv 300 ppbv 1 ppbv

N/A N/A N/A 1 wk 100 yr 1–150 d 10 yr 150 yr 100 yr

1.7

Composition of the Earth’s Atmosphere

oxygen make up such large fractions of the atmosphere that changes in their concentrations from, for example, combustion, oxidation, and nitrogen fixation, are negligibly small. Gases with residence times greater than about a decade are considered to be long-lived. Through the mixing and circulation of the atmosphere, these gases have concentrations that are constant. Their concentrations remain unchanged within a small percentage relative to nitrogen and oxygen, throughout the Earth’s atmosphere from the surface to the stratopause (altitude ∼50 km) and beyond. Gases with short residence times, such as water vapor and ozone, have high concentrations near their sources and low concentrations near their sinks. The source for water vapor is evaporation and evapotranspiration at the surface. The global average concentration of water vapor held in a column of air is approximately 2 g cm−2 , or equivalently 2 cm of precipitable water. Precipitable water is the depth to which a layer of water would cover the Earth’s surface if all of the water vapor in the atmosphere were condensed into liquid. Evaporation is high where surface temperatures are high, as in the tropics, and low where temperatures are low, as at high latitudes. Within latitude belts, the zonal average concentration of water vapor in the atmosphere follows that of the average surface temperature and an average surface relative humidity near 80%. Since temperature generally falls with altitude in the troposphere, the condensation of water vapor in the atmosphere increases with altitude and water vapor is depleted through precipitation. Figure 1.3 shows the normalized pressure of the atmosphere as a function of altitude along with the normalized vapor pressure for water. The global average scale height of the atmosphere is about 8 km. The scale height gives the exponential rate of decrease in pressure with altitude. With a scale height of 8 km, atmospheric pressure is halved every 5 km. Water vapor has a scale height of about 2 km. Vapor pressure is halved about every 1.5 km. The decrease in the concentration of water 20 Air pressure Vapor pressure Altitude (km)

15

(P/Ps)4

10

5

0 0.0

0.2

0.4 0.6 Normalized pressure

Figure 1.3 Normalized air pressure (solid line) and water vapor pressure (dashed line). The normalized pressures are the pressure divided by the surface

0.8

1.0

value. The dashed dotted line gives the normalized vapor pressure for water when the mass mixing ratio is given by Equation 1.36b.

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1 The Earth’s Energy Budget and Climate Change

vapor with altitude approximately follows the decrease of temperature with altitude and a relative humidity that decreases linearly with pressure from the surface to the tropopause [7]. The tropopause is the demarcation between the troposphere and stratosphere. The altitude of the tropopause ranges from as high as 18 km near the equator to as low as 10 km in the polar regions. The ratio of the scale heights for atmospheric pressure (8 km) and that for water vapor pressure (2 km) suggests that water vapor pressure at pressure level P, pVAP (P), normalized by its value at the surface pVAP S , can be approximated by ( )4 pVAP (P) P = (1.33) pVAP S PS with PS the surface pressure. As indicated by the dash–dotted curve in Figure 1.3, the simple power law in Equation 1.33 matches closely the climatological profile of the normalized water vapor pressure. The mass mixing ratio of a gas is the ratio of the density of the gas to that of the air that contains the gas. In this book, mass mixing ratio is expressed as the mass of a gas divided by the mass of dry air that contains the gas. The water vapor pressure is given by the ideal gas law pVAP (P) = �VAP (P)RH2 O T(P)

(1.34)

with RH2 O the gas constant for water vapor. It is related to the gas constant for dry air through the universal gas constant R∗ by m R∗ R∗ mAIR = = RAIR AIR (1.35) RH2 O = mH2 O mAIR mH2 O mH2 O with mAIR = 0.029 kg the molar weight of dry air and mH2 O = 0.018 kg the molar weight of water vapor. In terms of the mass mixing ratio, Equation 1.34 is given by m m (1.36a) pVAP (P) = r(P)�AIR (P)RAIR AIR T(P) = r(P) AIR P mH2 O mH2 O so that r(P) = rS

(

P PS

)3 (1.36b)

with rS = r(PS ) the mass mixing ratio at the surface. Turning to ozone, some is created in the lower atmosphere through photochemical reactions with oxides of nitrogen in the presence of hydrocarbons. Ozone, being highly reactive, is removed from the lower atmosphere through the oxidation of trace gases and contact with surfaces. Most of the atmospheric ozone is produced in the tropical stratosphere where ultraviolet radiation (UV) photodissociates oxygen molecules. Odd oxygen atoms from the split molecules combine with other molecules of oxygen to form ozone. At altitudes in the stratosphere where its concentration peaks, ozone is removed through catalytic reactions, mostly those involving oxides of nitrogen. Ozone is also photodissociated by UV radiation. The photodissociation of ozone prevents harmful UV

1.7

Composition of the Earth’s Atmosphere

50 Polar winter Midlatitude winter

Altitude (km)

40

Tropics 30 20 10 0 0.00

0.01

0.02

0.03

0.04

O3 (cm–STP km−1)

Figure 1.4 Concentrations of ozone derived from Equation 1.37. Values of the parameters are given in Table 1.2. Table 1.2 Parameters used in Equation 1.37 to determine the concentration of ozone shown in Figure 1.4. Profile

a (cm-STP)

b (km)

c (km)

0.25 0.4 0.5

25 20 18

4 5 4

Tropics Midlatitude winter Polar winter

Parameters from Lacis and Hansen [9].

from reaching the surface, making the stratospheric ozone layer the protective shield for the Earth’s biosphere. Figure 1.4 shows vertical profiles for the tropics, midlatitude winters, and polar winters. The profiles are based on analytic fits to observations [8]. The amount of ozone above a given altitude z is given by u(z) =

a[1 + exp(−b∕c)] 1 + exp((z − b)∕c)

(1.37)

with a the total amount of ozone in the atmospheric column, b the altitude at which the ozone concentration peaks, and c the scale height for ozone above the altitude at which the concentration peaks. Table 1.2 gives the values of the parameters used to produce the concentrations shown in Figure 1.4. The concentration of ozone derived from Equation 1.37 has units of density, cm-STP/km, with STP (standard temperature and pressure) 273.15 K and 1 atm = 1013.25 hPa. The unit cm-STP used above to specify the column amounts of ozone is the gaseous analog of precipitable cm used to express the column amount of water vapor in terms of an equivalent amount of liquid water. If all the ozone in the atmosphere was reduced to STP, it would cover the surface of the Earth with a layer that is so many cm-STP thick. The column amount a in Equation 1.37 is typically between 0.2 and 0.5 cm-STP. A very thin layer of ozone protects the biosphere

17

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1 The Earth’s Energy Budget and Climate Change

from UV. To translate cm-STP to other units, recall that 6.022 × 1023 molecules (1 mole) occupy a volume of 22.4 × 103 cm3 at STP. Column amounts of 1 cm-STP and 2.69 × 1023 molecules m−2 are equivalent. As an exercise, the reader may show that the atmospheric scale height, 8 km, is the depth of the atmosphere when reduced to STP. While ozone is generated primarily in the tropical stratosphere, its column concentration is the smallest in the tropics. The ozone that is produced in the tropical stratosphere is transported downwards and polewards in both hemispheres by a stratospheric circulation. Ozone piles up at higher latitudes, especially during the polar winters, where the amount of UV radiation needed for the ozone depleting reactions is low compared with its amount in the tropics. Carbon dioxide has a constant concentration throughout the atmosphere. Its mass mixing ratio is 5.92 × 10−4 g CO2 /g AIR. A volume mixing ratio is the number of molecules of a trace gas to the number of air molecules that contains the gas. For carbon dioxide in the Earth’s atmosphere the equivalent volume mixing ratio as of 2010 is 390 ppmv (parts per million by volume). Finally, aerosols are composed of small particles in the atmosphere. The particles scatter sunlight and make skies look “hazy.” Aerosol particles are often referred to as haze particles or simply “haze.” Similarly to water vapor, aerosols have residence times in the atmosphere of a few days to perhaps a week or so depending on the altitudes to which they can be lofted, primarily through convective updrafts. Some aerosol particles are generated near the surface by combustion and by mechanical lofting. They arise from both human and natural sources, from the burning of fossil fuels, forest fires, windblown dust, and sea spray. Aerosol particles can also have precursors, gases injected into the atmosphere, which through photochemical reactions give rise to particles. Aerosols congregate near their sources and their concentrations diminish with distance from their sources. Particles are removed through settling, deposition on surfaces, and washout by precipitation. Particles from intense forest fires and desert wind storms can be lofted to middle and upper tropospheric levels. Layers of particles at those altitudes have been observed to traverse both the Atlantic and Pacific oceans. Aerosols are known to absorb and reflect a significant amount of sunlight, 1–2%. Their effects on sunlight are often visible as the cause of reduced visibility, a whitening of the blue sky, and a reddening of sunsets. Owing to their relatively short lifetimes, however, their properties and concentrations are highly variable in both space and time. This variability makes their effects on the Earth’s energy budget difficult to assess. Gases in the atmosphere contribute to the Rayleigh scattering of incident sunlight. This type of scattering causes cloud-free skies under relatively aerosol-free conditions to be blue. On the other hand, only a few gases absorb significant amounts of light. As discussed in later chapters, strong absorbers such as water vapor, ozone, and nitrous oxide have permanent electric dipole moments. These dipole moments arise from the asymmetric structures of the molecules coupled with the congregation of electrons around certain atoms within the molecule balanced by the loss of electrons around other atoms. The oscillations of the resulting dipole moments through the vibrations of the atomic bonds and the

1.8

Radiation and the Earth’s Mean Temperature Profile

rotation of the molecules result in strong interactions with radiation, particularly at infrared and microwave wavelengths. Diatomic molecules, such as nitrogen and oxygen, have no permanent electric dipole moments and consequently interact only weakly with radiation. Oxygen molecules undergo electronic transitions creating a magnetic dipole moment and weak absorption (∼2%) of sunlight at visible wavelengths. Symmetrical molecules, such as carbon dioxide and methane, lack permanent electric dipole moments but they attain the required oscillating electric dipole moments when the bonds between the atoms vibrate and bend. The vibrations and bending of the bonds and the rotation of these molecules lead to strong interactions with radiation.

1.8 Radiation and the Earth’s Mean Temperature Profile

If the single-layer, window-gray atmosphere is replaced with two isothermal but different layers, radiative equilibrium requires that the temperature of the lower atmospheric layer be higher and that of the higher atmospheric layer be lower than the atmospheric temperature of the single-layer model. Thus, in the twolayer, window-gray, radiative equilibrium model, as in the Earth’s atmosphere, the temperature decreases with altitude. The proof is left to the reader. Of course, the atmosphere is far more complex than a two-layer system. A simple approach for the temperature profile in a window-gray, radiative equilibrium model is feasible using an approximate solution for the equation of radiative transfer [10, 11]. In this simple solution, the temperature of the atmosphere falls with altitude from the surface until becoming isothermal in the upper regions of the atmosphere. The isothermal layer at the top of the atmosphere has a temperature given by TISO = Te ∕21∕4 = 214 K. This solution was first obtained by Karl Schwarzschild in the early 1900s. It followed balloon-borne measurements around the turn of the twentieth century that had reached altitudes near the tropopause, thereby revealing the existence of a stratosphere. The temperature of the isothermal top layer obtained with the window-gray model was close to those observed at the base of the stratosphere. In radiative equilibrium, the radiative energy absorbed within each layer of the atmosphere is equal to the radiation emitted. In the case of the single-layer, window-gray model, this equilibrium is given by [ ] [ ] Q0 Q0 dT mCP A = (1 − �) − (1 − �)�TS4 − ��TA4 − (1 − �) + ��TA4 − �TS4 dt 4 4 (1.38) The first term in brackets on the right-hand side is recognized as the radiation budget at the top of the atmosphere. It is the sunlight absorbed by the Earth’s atmosphere-surface system minus the infrared radiation emitted by the system. Notice that both the solar and emitted infrared radiation can be broken into upward and downward solar and infrared radiation fluxes. Radiative fluxes have

19

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1 The Earth’s Energy Budget and Climate Change

units of power per unit area (W m−2 ) and are defined in terms of the properties of light in Chapter 2. The upward solar radiative flux is the reflected sunlight given by Q+TOP = �Q0 ∕4

(1.39a)

and the downward flux is the incident sunlight given by Q−TOP = Q0 ∕4

(1.39b)

In atmospheric radiation the upward direction is indicated by a “+” sign and the downward direction by a “−” sign. The net solar radiative flux is the upward flux minus the downward flux, QNET TOP = Q+TOP − Q−TOP . Similarly, the upward radiation emitted at the top of the atmosphere is given by + = (1 − �)�TS4 + ��TA4 FTOP

(1.40a)

the sum of the radiation emitted by the Earth’s surface and transmitted by the atmosphere and the radiation emitted by the atmosphere. The downward infrared flux at the top of the atmosphere is zero: − =0 FTOP

(1.40b)

The net infrared radiative flux at the top of the atmosphere, similarly to the net + − solar radiative flux, is given by the difference FNET TOP = FTOP − FTOP . Consequently, at the top of the atmosphere, the Earth’s radiation budget is given in terms of the net radiative flux by + − − FTOP ] ℛ NET TOP = −ℱNET TOP = −[Q+TOP − Q−TOP + FTOP

= −(QNET TOP + FNET TOP )

(1.41)

with ℱNET TOP the total net radiative flux, the sum of the solar net radiative flux and the net emitted radiative flux at the top of the atmosphere. The second term in brackets in Equation 1.38 is the net radiation budget for the Earth’s surface. In the window-gray model, the net solar radiative flux at the surface is the same as that at the top of the atmosphere. The net infrared flux is given by + − FNET SURF = FSURF − FSURF = �TS4 − ��TA4

(1.42)

As with the top of the atmosphere, the radiation budget at the surface is related to the net radiative flux by ℛ NET SURF = −ℱNET SURF . Combining the terms in Equation 1.38, the rate of change of the atmospheric temperature is given by dTA = −(ℱNET TOP − ℱNET SURF ) = ��TS4 − 2��TA4 (1.43) dt As has already been noted, in the window-gray model the atmosphere is heated by the absorption of infrared radiation emitted by the surface and cooled by the emission of infrared radiation by the atmosphere. In radiative equilibrium, the cooling balances the heating and the net is zero. The temperature remains constant. mCP

1.8

Radiation and the Earth’s Mean Temperature Profile

Consider, however, the rapid increase in the Earth’s albedo after the Mt. Pinatubo eruption. The sunlight absorbed by the Earth decreased while the emitted infrared radiation at the top of the atmosphere remained largely unchanged. The Earth began to cool. In the window-gray, radiative equilibrium model, the surface has no heat capacity. The surface temperature decreases as the albedo increases. Owing to its heat capacity, on the other hand, the atmosphere does not immediately change its temperature. The temperature decreases with time until a new equilibrium is reached. With no heat capacity, the surface temperature is forced to maintain radiative equilibrium. As a result, the surface temperature initially drops but then slowly decreases in response to the decreasing atmospheric temperature. Ultimately, a new equilibrium is reached so that emission by the surface and emission by the atmosphere–surface system balance the radiation that each absorbs. If a greenhouse gas in the atmosphere were to suddenly increase, the absorption of infrared radiation by the gas would cause the emission at the top of the atmosphere to suddenly decrease. For a greenhouse gas that does not absorb sunlight, the amount of sunlight absorbed would remain unchanged. The Earth would warm. At the surface, owing to the increase in the greenhouse gas, the emission downward by the atmosphere at the surface would increase, thereby increasing the absorption of emitted radiation by the surface. The surface temperature would rise so that the rate of emission by the surface balanced the rate of absorption by the surface, the sum of the rates for the absorption of sunlight and the infrared radiation emitted downwards by the atmosphere. The temperature of the atmosphere would also rise partly because of the increase in absorption from the increase in the greenhouse gas concentration and partly because of the increase in radiation emitted by the surface that the atmosphere absorbs. The rate of rise in temperature would be moderated by the heat capacity of the atmosphere. Ultimately, the atmospheric temperature would rise until a new state of radiative equilibrium is reached. At its new equilibrium temperature, the rate at which the atmosphere absorbs infrared radiation would match the sum of the rates at which it emits upward to space and downward to the surface. Similarly, at its new equilibrium temperature, the rate at which the surface absorbs sunlight and infrared radiation emitted downwards by the atmosphere would match the rate at which it emits infrared radiation. Within a multilayered atmosphere, the radiative heating of a layer of thickness Δz is, from Equation 1.43, given in terms of the vertical gradient of the total net radiative flux, 1 ΔℱNET dT =− dt �(z)CP Δz

(1.44)

with �(z)Δz the mass of the layer. In Equation 1.44, −ΔℱNET ∕Δz = ΔℛNET ∕Δz is the change in the net radiation budget of the layer. It is the change in the rate at which the layer absorbs radiation minus the change in the rate at which the layer emits radiation. For infinitesimally thin layers, the radiative

21

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heating rate is given by dT 1 dℱNET =− dt CP �(z) dz

(1.45a)

or equivalently after applying hydrostatic balance by g dℱNET dT = dt CP dP

(1.45b)

For a multilayered atmosphere in radiative equilibrium, the rate of absorption by each layer equals the rate of emission. The rate of absorption is the sum of the rates for the sunlight and emitted radiation absorbed by the layer. The emitted radiation incident on the layer is the radiation emitted downward by layers above the layer and transmitted by the intervening atmosphere to the layer. The radiation is combined with radiation emitted upwards by layers below the layer and emitted by the surface that is transmitted to the layer by the intervening atmosphere. If, for example, an aerosol layer forms in the stratosphere after a volcanic eruption, the rate of sunlight absorbed by the layer decreases and the layer would cool until equilibrium is reached between the rate it absorbs radiation and the rate it emits. Conversely, if the concentration of an infrared absorber in a layer increases, the layer would warm until equilibrium is reached between the absorption and emission of infrared radiation. Unfortunately, calculating the effects of heating in a multilayered, nongray, nonisothermal atmosphere is a formidable undertaking. Figure 1.5 shows the results of early attempts to obtain realistic solutions for the vertical temperature profile of the Earth’s atmosphere in radiative equilibrium [6]. The temperatures at the centers of the model layers are indicated by the large dots in the figure. In these early attempts, the composition of the atmosphere was taken to be close to the global average conditions of the 1960s. The concentrations of gases then differed from those presented in Section 1.7 only for the greenhouse gases that have increased as a result of human activity. Initially, the atmosphere in the model was assumed to be cloud-free. In radiative equilibrium, the Earth’s atmospheric temperature under cloud-free conditions decreased rapidly from a surface temperature of over 330 K to a temperature of 190 K at 10 km, a temperature gradient of −14 K km−1 . A vertical temperature gradient of −14 K km−1 is unsustainable in the Earth’s atmosphere. Such conditions are buoyantly unstable. The temperature of a gas cannot decrease faster than the rate given by the vertical temperature gradient associated with adiabatic processes. This temperature gradient is referred to as the adiabatic lapse rate. If the temperature decreases more rapidly with increasing altitude than that given by the adiabatic lapse rate, then a parcel of air experiencing a small vertical displacement would either rocket upwards or plunge downwards, depending on the initial direction of its displacement. Unstable temperature gradients create rapid vertical motions referred to as convection, which, through the mixing of air, reshapes the temperature profile so that it becomes buoyantly stable. As long as the decrease of temperature with altitude is less than that given by the adiabatic lapse rate, small vertical displacements of an air parcel will lead to

1.8

Radiation and the Earth’s Mean Temperature Profile

2.3 40

10

20 100

Altitude (km)

Pressure (hPa)

30

Pure radiative equilibrium Dry adiabatic adj. 6.5 K km–1 adj.

10

0

1000 180

220

260

300

340

Temperature (K) Figure 1.5 Temperature profiles for global average abundances of water vapor, carbon dioxide, and ozone. The profiles were calculated for cloud-free conditions by Manabe and Strickler [6]. The solid curve is for radiative equilibrium. Curves for the cases

in which the critical lapse rate was taken to be equal to the dry adiabatic lapse rate (dashed) and 6.5 K km−1 (dotted) are also shown. (After Manabe and Strickler [6]. Reproduced by permission of the American Meteorological Society.)

restoring forces that return the parcel to its original altitude. Barring the passage of weather fronts, temperature profiles satisfying buoyantly stable conditions can exist indefinitely. The stratosphere, for example, was so named because its temperature gradient ranges from near zero near the tropopause to positives values at higher altitudes. The stratosphere is thus stably stratified. The adiabatic lapse rate is derived from the conservation of energy and the relationship for hydrostatic balance. Assuming negligible vertical motion, the hydrostatic balance is between the pull of gravity on a volume of air and the change in pressure with altitude, which holds the mass in place. For adiabatic processes, dQ = 0. With hydrostatic balance, conservation of energy as given by Equation 1.2 for adiabatic processes leads to Cp

1 dP dT = = −g dz � dz

(1.46)

23

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1 The Earth’s Energy Budget and Climate Change

The adiabatic lapse rate is given by Γ=−

g dT = dz Cp

(1.47)

The temperature lapse rate is the decrease in temperature with altitude. A positive lapse rate indicates decreasing temperature with altitude. For the Earth, g = 9.8 m s−2 giving Γ = 9.8 K km−1 . This lapse rate is referred to as the dry adiabatic lapse rate because it does not account for the heating of air due to the release of latent heat as water vapor condenses. The release of latent heat raises the surrounding air temperature, leading to a smaller lapse rate, less negative vertical temperature gradient. For saturated air, the lapse rate is referred to as the moist adiabatic lapse rate. With the effects of radiation, release of latent heat, and other dynamical processes, the lapse rate for the Earth’s atmosphere ranges from about 4 K km−1 for moist tropical air to 9.8 K km−1 for dry desert air. In radiative equilibrium, the composition of the Earth’s atmosphere produces a buoyantly unstable vertical temperature profile throughout most of the troposphere as shown in Figure 1.5. A temperature profile that is close to the average profile of the northern hemisphere is obtained when the observed lapse rate for the northern hemisphere, 6.5 K km−1 , is taken to be the “critical lapse rate.” The critical lapse rate distinguishes “buoyantly unstable” layers from stable layers. When the radiative equilibrium solution produces a supercritical lapse rate between model layers, the temperatures of the layers are adjusted so that the difference between the layers is given by the critical lapse rate. This procedure is called convective adjustment. The combination of radiative equilibrium and convective adjustment leads to “radiative–convective equilibrium.” Figure 1.6 shows the resulting net radiative heating rates for radiative–convective equilibrium. Forcing the temperature lapse rate to a critical value whenever the radiative equilibrium lapse rate becomes supercritical leads to a troposphere that is radiatively cooling. In radiative–convective equilibrium, the lapse rate is assumed to maintain a critical value through an unspecified combination of radiative, convective, and latent heating. The troposphere is approximately 80% of the atmosphere. Its radiative cooling rate is about 1 K/day. For global average conditions, large-scale dynamical processes produce no net heating of the atmosphere. Most of the radiative cooling by the atmosphere is compensated by heating due to the release of latent heat when water vapor condenses, freezes, and falls as precipitation. The surface also is not in radiative equilibrium, but must lose energy to the atmosphere, which it does, through turbulent exchange (dry sensible heat), evaporation, and evapotranspiration (latent heat) as shown in Figure 1.7. Much of this heat is transferred through convective processes. Figure 1.7 depicts an approximate breakdown of the heating and cooling for the surface, troposphere, and stratosphere. The data are taken from Hartmann [12]. The values in the figure are expressed as percentages of the incident global average sunlight. The figure depicts the Earth and the stratosphere as being in balance, consistent with the radiative heating rates for radiative–convective equilibrium

1.8

Radiation and the Earth’s Mean Temperature Profile

2.3 NET LO3 LH2O

40

LCO2 10

20 SH2O + SCO2 + SO3

100

Altitude (km)

Pressure (hPa)

30

10

0

1000 –4.0

–2.0

0.0

2.0

4.0

6.0

Rate of temperature change (K/day) Figure 1.6 Radiative heating rates due to absorption of sunlight under average cloudy conditions, SH2 O + SCO2 + SO3 , and absorption and emission of infrared radiation, LH2 O, LO3 , and LCO2 . The “S” indicates solar heating rates and the “L” indicates

longwave heating and cooling rates. The solid curve labeled NET is the net radiative heating rate. In the stratosphere, it is zero. (After Manabe and Strickler [6]. Reproduced by permission of the American Meteorological Society.)

shown in Figure 1.6. At the tropopause, the troposphere and surface together are also in radiative equilibrium. The sunlight absorbed by the troposphere and Earth’s surface balances the net infrared radiation emitted upward at the tropopause. The figure identifies the primary contributors of absorption and emission in the atmosphere. The spectra of the incident and transmitted sunlight and the emitted radiation at the top of the atmosphere coupled with the absorption spectra of the molecules that account for the absorption lead to the identification of the major contributors in Figure 1.7. Figure 1.8a shows calculated spectra for the incident sunlight at the top of the atmosphere and at the surface when cloud-free and when overcast by cirrus. The profiles of water vapor, carbon dioxide, and ozone in these calculations are those of the U.S. Standard Atmosphere. These profiles are close to global, annual average profiles described in Section 1.7. The surface albedo for these calculated spectra is 0.1. The visible optical thickness of the cirrus is 2. Chapter 3

25

26

1

1 The Earth’s Energy Budget and Climate Change

100

Clouds*, surface, molecules, haze Transmitted Reflected atmosphere

10

Stratosphere

30

Absorbed 3 O3

10

Absorbed 2 O3

Troposphere

Pressure (hPa)

54

6

Absorbed 6 CO2 60

Absorbed

Absorbed

Emitted

17 H2O*, Clouds

98 H2O*, Clouds*, CO2

149 H2O*, Clouds*, CO2

110

Emitted 11 CO2

50

Solar Figure 1.7 Global average energy budget of the surface, troposphere, and stratosphere. (Data are taken from Hartmann [12].) Asterisks indicate the major radiatively active contributors to the absorption and scattering of sunlight and the

5

Absorbed 5 H2O*, Clouds*, CO2 Latent heat

Dry heat 89

Absorbed 1000 200 250 300 Temperature (K)

Emitted stratosphere

Transmitted 12

100

Transmitted stratosphere

Net

Net

–21

–29

Longwave

24

5

Non radiative

absorption and emission of thermal radiation. The values are given as a percentage of the global average incident sunlight, Q0 ∕4 = 340 W m−2 ≡ 100%. The temperature at left is the 1976 U.S. Standard Atmosphere profile.

describes the interaction of radiation with matter, which leads to the definition of optical thickness. With a visible optical thickness of 2, the cirrus is relatively thick but not uncommonly so. The cirrus is sufficiently thick to whiten the sky but not so thick that it prevents the sun’s disk from being clearly seen through the cloud by a surface observer. The effective size of the ice crystals used in the calculation is 50 μm, a typical value. In fact, the cirrus optical properties were those used to retrieve visible optical depths and particle sizes from multispectral imagery of reflected sunlight and emitted infrared radiation collected by NASA’s satellite-borne Moderate Resolution Imaging Spectroradiometer (MODIS) [13]. The spectral distribution of radiation emitted by a blackbody with a temperature of 5800 K is also shown in Figure 1.8a. The distribution represents fairly well the spectral variation of the incident sunlight. Chapter 2 describes blackbody radiation. Figure 1.8a identifies the major molecular absorption bands for the downward solar radiation at the surface. Chapter 5 describes molecular absorption bands. Chapter 6 applies the principles described in Chapters 2−5 to account for the

1.8

Radiation and the Earth’s Mean Temperature Profile

Downward solar irradiance (Wm−2 μm−1)

2500

TOA Blackbody,T = 5800 K Cloud−free Cirrus

2000 H2O

1500

O3

H2O and CO2

1000 500 0

0

0.5

1

TOA upward solar irradiance (Wm−2 μm−1)

2

2.5

3

450 Cloud−free Cirrus

400 350 300 250 200 150 100 50 0

(b)

1.5 Wavelength λ (μm)

(a)

0

0.5

1

1.5

2

2.5

3

Wavelength λ (μm)

Figure 1.8 Simulated incident solar radiation at the top of the atmosphere (TOA) and at the surface under cloud-free and cirrus conditions (a) and reflected sunlight at the top of the atmosphere (b). The surface albedo in the calculations is 0.1 and the solar zenith angle is 45∘ . The atmospheric

composition and temperature profile are those of the U.S. Standard Atmosphere. The cirrus cloud lies between 9 and 10 km, has a visible optical depth of 2, and an effective particle size of 50 μm. Also shown in (a) is the spectral distribution of radiation emitted by a blackbody at a temperature of 5800 K.

absorption and scattering of sunlight by the Earth’s atmosphere and surface. The chapter describes how ozone absorbs nearly all of the incident radiation at wavelengths less than 0.3 μm, accounting for the absorption of nearly 2% of the incident sunlight. Ozone also absorbs relatively weakly over the entire range of the visible spectrum 0.4–0.7 μm, accounting for the absorption of another 1% of the incident sunlight. Water vapor absorbs a substantial fraction of the sunlight at wavelengths in the near infrared, wavelengths ranging from 0.7 μm to less than about 5 μm. Figure 1.8b shows the reflected solar spectra at the top of the atmosphere under cloud-free and the same cirrus conditions used for Figure 1.8a. Clearly, the cirrus adds considerably to the reflected sunlight even though the cloud is not so thick as to block the solar disk from being viewed from the surface.

27

1 The Earth’s Energy Budget and Climate Change

TOA upward radiance (W/m2/sr/cm−1)

0.15

Cloud−free Cirrus, τ = 2

CO2 0.1

H2 O O3

0.05 H2O 0

500

1000

1500

2000

2500

Wavenumber (cm−1)

(a) Surface downward radiance (W/m2/sr/cm−1)

28

0.15

Cloud−free Cirrus, τ = 2

0.1

0.05

0

500

1000

(b) Figure 1.9 Simulated emitted radiation at the top of the atmosphere (a) and radiation emitted downward by the atmosphere at the surface (b) for

1500

2000

2500

Wavenumber (cm−1) cloud-free and cirrus conditions. The atmospheric composition, temperature profile, and cirrus properties are the same as those used for Figure 1.8.

Figure 1.9a shows the upwelling infrared spectra at the top of the atmosphere under cloud-free and cirrus conditions. The atmospheric profiles and cirrus optical properties are the same as those used for Figure 1.8. For the infrared wavelengths the surface albedo was set to 0.01. The surface emissivity was, therefore, 0.99. The measure of wavelength used for Figure 1.9 is the wavenumber = 1/wavelength. The unit of wavenumber is “inverse centimeters” (cm−1 ). A wavelength of 10 μm is the same as a wavenumber of 1000 cm−1 . A wavelength of 4 μm is the same as 2500 cm−1 . When speaking, one often uses 1000 “inverse centimeters” and 1000 “wavenumbers” interchangeably. Water vapor absorbs at nearly all wavelengths associated with the Earth’s infrared emission. The only exception is in the 8–12 μm infrared window, wavenumbers 850–1250 cm−1 . In the infrared window, water vapor absorbs only weakly. The major molecular absorption bands in the infrared are the rotation band of water vapor, 0–850 cm−1 ; the 6.3 μm vibration–rotation band of water vapor, centered near 1590 cm−1 ; the 15 μm vibration–rotation band of carbon dioxide, centered at 667 cm−1 ; and the 9.6 μm vibration–rotation band of ozone,

1.8

Radiation and the Earth’s Mean Temperature Profile

centered near 1040 cm−1 . As was mentioned earlier, the properties of rotation and vibration–rotation bands are briefly described in Chapter 5. The effect of the cirrus in the infrared is apparent in emission at the top of the atmosphere primarily in the infrared window. In the infrared window the cloud behaves similarly to a greenhouse gas. It absorbs radiation emitted by the surface and lower atmosphere and then emits radiation at a lower temperature. Just as in the window-gray model, because the cirrus is semitransparent at the window wavelengths, some of the radiation emitted by the surface and lower atmosphere is transmitted through the cirrus. Figure 1.9b shows the infrared spectra emitted downward by the atmosphere at the surface. The downward emission is larger with the cirrus present than it is for cloud-free conditions. The increase in downward emission from cloudy skies during a cold, middle to high latitude winter night warms the surface by emitting radiation downward toward the surface. Under similar meteorological and cloud-free conditions, the surface rapidly cools by emitting radiation through the infrared window region. The cooling that results is so rapid that within a matter of minutes to an hour or so a temperature inversion builds up in the lowest portion of the atmosphere adjacent to the surface. The surface air becomes cold while the air above the inversion, sometimes only tens to a hundred meters from the surface, remains relatively warm. Figure 1.10 presents another rendition of the surface and atmosphere energy budgets depicted by Trenberth and his coworkers [14]. These budgets are more accurate than those shown in Figure 1.7. In this depiction, the Earth is not in radiative equilibrium but absorbs sunlight at a rate that is slightly greater than the rate at which it emits infrared radiation. The additional sunlight heats the oceans at a rate of approximately 0.9 W m−2 . The radiative imbalance is due to the buildup of greenhouse gases in the atmosphere. The Earth is responding to this imbalance. Its temperature is rising, and it will continue to rise as long as the rate at which the Earth absorbs sunlight is greater than the rate at which it emits. In addition to showing the various terms of the energy budget, Figure 1.10 also indicates the effect of clouds on the amount of sunlight reflected and radiation emitted. In the case of the radiation emitted, the 30 W m−2 emanating from clouds represents the net effect of the clouds on the Earth’s emitted radiation. For scenes with no clouds, the global average radiation emitted at the top of the atmosphere is near 269 W m−2 , 30 W m−2 larger than the global average emission. This 30 W m−2 difference between the cloud-free and average emission is called the cloud longwave radiative forcing. Table 1.3 provides estimates of the greenhouse forcing due to greenhouse gases, the infrared active gases in the Earth’s atmosphere. The wavelengths at which these gases absorb overlap to varying degrees. The values in the table include the effects of the overlapping absorption by the gases. The estimates were calculated using a column model of the Earth’s global average atmosphere [15] with a composition close to that described in Section 1.7. Surface emission was based on a global average surface temperature 288 K, giving an emission of 390 W m−2 . The top of the atmosphere emission was based on 2 years of observations from

29

30

1 The Earth’s Energy Budget and Climate Change

Global energy flows W m–2 102

Reflected solar radiation 101.9 W m–2 Reflected by clouds and atmosphere 79

239

Incoming solar radiation 341.3 W m–2

341

79

40 Emitted by 169 atmosphere

Atmospheric window

30 Greenhouse gases

Absorbed by atmosphere

78

Latent 80 heat

17 Reflected by surface 23 161 Absorbed by surface

Outgoing longwave radiation 238.5 W m–2

356

17 Thermals

80 Evapotranspiration

Net absorbed 0.9 W m–2

Figure 1.10 Earth’s radiation budget based on 4 years of Earth radiation budget measurements by the NASA CERES project combined with model estimates based on global climate model radiation schemes

396 Surface radiation

40

333 Back radiation

333 Absorbed by surface

and observations of the surface radiation budget, precipitation, runoff, and analyzed meteorological fields. (From Trenberth et al. [14], reproduced by permission of the American Meteorological Society.)

the NASA Earth’s Radiation Budget Experiment (ERBE), 265 W m−2 for cloudfree scenes and 235 W m−2 for average cloudy conditions. The breakdown in the table shows that the greenhouse forcing of the gases is diminished considerably by the presence of clouds. With clouds present, the gases contribute only 86 W m−2 . The greenhouse forcing is 390 − 235 W m−2 = 155 W m−2 . Consequently, clouds contribute 155 − 86 W m−2 = 69 W m−2 to the forcing, making their contribution larger than any of the greenhouse gases. In Figure 1.10, the 40 W m−2 that emanates from the surface represents the radiation emitted by the surface that is transmitted by the atmosphere. In Figure 1.7, this term is given as 10% of the incident sunlight. Most of this radiation passes through the infrared window at wavelengths between 8 and 12 μm. When present, clouds absorb most of this radiation emitted by the surface. They then emit radiation as a gray body with an emissivity near unity, so approximately similarly to a blackbody, at a temperature near that of the cloud tops. Most of the 30 W m−2 of the longwave cloud radiative forcing stems from the absorption and emission of radiation at wavelengths in the window region. In the case of reflected sunlight, the term associated with the atmosphere and clouds includes scattering by molecules, which is Rayleigh scattering, and scattering by haze. Most of the scattering is due to molecules. Rayleigh scattering

1.8

Radiation and the Earth’s Mean Temperature Profile

Table 1.3 Estimates of the contribution by greenhouse gases and clouds to the greenhouse forcing by the Earth’s atmosphere [15]. Gas H2 O CO2 O3 CH4 and N2 O Total

Cloud free (W m−2 )

Cloudy (W m−2 )

75 32 10 8

51 24 7 4

125

86

In these calculations the surface emission is 390 W m−2 , the top of the atmosphere emission is 235 W m−2 , and the greenhouse forcing is 155 W m−2 .

accounts for approximately 7% of the reflected sunlight, or about 24 W m−2 . Haze accounts for about 1–2% of the reflected sunlight, about 3–7 W m−2 . Clouds contribute the rest, or about 50–55 W m−2 . Adding the ∼29 W m−2 of sunlight reflected by molecules and haze to the 23 W m−2 contributed by the Earth’s surface leads to reflected sunlight of 52 W m−2 , which is observed for cloud-free scenes by the NASA Cloud and Earth’s Radiant Energy System (CERES) project [2]. As is the case for the greenhouse forcing, clouds dominate the Earth’s reflected sunlight. On the basis of the CERES observations, the annual average global cloudy sky reflected solar radiative flux is 99.5 W m−2 [2]. The absorbed solar radiative flux for a cloudy Earth, 340.2 − 99.5 W m−2 = 238.7 W m−2 ; and that for a cloud-free Earth is 340.2 − 52 W m−2 = 287.8 W m−2 . The change in the absorbed radiative flux in going from a cloudy Earth to a cloud-free Earth is 238.7 − 287.8 W m−2 = −49.1 W m−2 , the cloud shortwave radiative forcing. The change in the absorbed sunlight due to the presence of clouds more than offsets the change in the emitted radiative flux. The net cloud radiative forcing is −19.1 W m−2 . Clouds cool the Earth. With a climate sensitivity of 0.6 K W−1 m−2 , the Earth’s annual mean global surface temperature would be more than 10 K higher without the clouds. The large values of the shortwave and longwave cloud radiative forcing make cloud feedbacks responsible for most of the current uncertainty in the Earth’s climate sensitivity [4]. While the values in Figure 1.10 were based on “best estimates” of the various terms in the energy budget at the time the data were analyzed, many of the separate terms are rather uncertain. The 0.9 W m−2 imbalance is thought to be within 15–20% of the actual imbalance. Release of latent heat in the atmosphere and evapotranspiration at the surface are based on observations of precipitation and may be uncertain by as much as 10–20%. The heating by dry turbulent exchange is based on analyzed meteorological data and is within 10%. The downward and reflected solar radiative fluxes and emission by the surface are known to be within 5%. The downward longwave emission from the atmosphere has an uncertainty of about 10%.

31

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1 The Earth’s Energy Budget and Climate Change

On the basis of known observational errors and estimates of uncertainties, 5 years of CERES observations were used to obtain optimal estimates of the top of the atmosphere radiative fluxes for cloud-free and average cloudy conditions [2]. The resulting estimates were constrained so that the net radiative imbalance equaled the rate at which the oceans were storing energy. The optimal estimates yielded values that differed to some extent from those shown in Figure 1.10. First, the incident sunlight, based on recent observations and a growing understanding of the errors in previous observations [1], is 340.2 W m−2 . The reflected sunlight is slightly less than portrayed in Figure 1.10, 99.5 W m−2 , and emission by the Earth is correspondingly larger, 239.6 W m−2 . Most of the uncertainties in these quantities stem from the calibration of the CERES instrument, ±2% for the reflected sunlight and ±1% for the emitted longwave radiation.

1.9 The Spatial Distribution of Radiative Heating and Circulation

Figure 1.11a shows the net shortwave radiative flux and Figure 1.11b shows the net longwave radiative flux at the top of the atmosphere. The fluxes have units of W m−2 and are 30 year averages (1971–2000) from a simulation with the National Center for Atmospheric Research (NCAR) Community Climate System Model 3 (CCSM3). The simulation is the “Climate of the 20th Century Experiment” (20C3M). The simulated data were obtained from the World Climate Research Program’s (WCRP) Climate Model Intercomparison Project 3 (CMIP3) multi-model database [16]. The results in Figure 1.11 show considerable spatial variation in the net shortwave and longwave radiative fluxes, but these variations lie on top of much stronger latitudinal trends from equator to pole in both hemispheres. Consider the consequences of the annual average incident solar radiation being a strong function of latitude. On the basis of solutions for radiative–convective equilibrium, regions at high latitudes with relatively large solar zenith angles are expected to be cold. Regions at low latitudes with relatively small solar zenith angles are expected to be warm. Because air expands as it is heated, at a given altitude the pressure for the cold regions at high latitudes will fall below those for the warm regions at low latitudes. A horizontal pressure gradient builds in the upper troposphere and the system becomes dynamically unstable. Circulation begins. The circulation that arises strongly influences both the temperature and the moisture profile of the atmosphere. In addition to radiative and convective energy transfers, the zonal mean profiles now respond to dynamical contributions that arise from the atmospheric wind fields [17]. The changes to the temperature, moisture, and cloud fields that accompany the winds alter the radiative heating of the atmosphere and surface. Often, these changes work to enhance the circulation that first gave rise to the changes. For example, the Hadley circulation gets an extra

1.9

The Spatial Distribution of Radiative Heating and Circulation

(a)

(b)

60

80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

Figure 1.11 Net shortwave (a) and longwave (b) radiative fluxes in W m−2 at the top of the atmosphere. They were obtained from a 30 year average (1971–2000) climate simulated with the NCAR CCSM3

as part of the “Climate of the 20th Century Experiment.” The simulated data was obtained from the WCRP CMIP3 multi-model database [16].

boost from the release of latent heat at high altitudes in the tropics and the radiative cooling accompanying subsiding air in the subtropics. While complex numerical models appear to capture some of the feedbacks between circulation and radiation, such as in the Hadley circulation, they obviously miss subtler feedbacks, as for example, the maintenance of the large marine stratocumulus systems hanging over the eastern boundaries of the subtropical oceans in both northern and southern hemispheres. As shown by the radiative fluxes in Figure 1.12, annually averaged, the solar radiation absorbed in the tropics is larger than the flux of infrared radiation emitted.

33

1 The Earth’s Energy Budget and Climate Change

600 Incident solar radiation

500 Radiative flux (Wm–2)

34

Absorbed solar radiation Emitted longwave radiation

400 300 200 100 0 –90

–60

–30

0

30

60

90

Latitude Figure 1.12 Annually averaged zonal mean incident shortwave, absorbed shortwave, and emitted longwave radiative fluxes from the 30 year averages for the NCAR CCSM3 20C3M experiment. The excess of absorbed sunlight over emitted infrared radiation in the tropics and the excess of emitted infrared radiation over absorbed

sunlight at high latitudes reflect the poleward transport of energy from the tropics to the poles in both hemispheres as is indicated by the arrows. The dip in absorbed sunlight and emitted infrared radiation in the tropics marks the persistent location of the intertropical convergence zone and the associated deep convective clouds.

Similarly, the sunlight absorbed by the polar regions is smaller than the infrared radiation emitted by the regions. The circulation of the atmosphere and oceans transports energy from the tropics to higher latitudes, moderating the high temperatures in the tropics and the low temperatures in the polar regions. Table 1.4 lists zonal mean radiative fluxes, albedos, and radiating temperatures for selected latitudes in the Northern Hemisphere. The observations are based on 11 years of the CERES Energy Balance and Filled (EBAF) data [2]. The “observed radiating temperatures” listed in the table are those of a blackbody that emits radiation at the rate given by the emitted zonally averaged infrared radiative flux. The radiative equilibrium temperature in the table is that given by a blackbody in radiative equilibrium with the zonally averaged absorbed solar radiation. In the tropics, the excess of the radiative equilibrium temperature over the observed radiating temperature indicates that the temperatures in the tropics are smaller than would be expected for radiative equilibrium. The smaller temperatures reflect the energy being lost to higher latitudes. At high latitudes, the excess of the observed radiating temperature over the radiative equilibrium temperature indicates that the temperatures in high latitudes are higher than would be expected for radiative equilibrium. The higher temperatures reflect the gain of energy being transported from lower latitudes.

1.10

Summary and Outlook

Table 1.4 Annually and zonally averaged incident shortwave, absorbed shortwave, and emitted longwave radiative fluxes, radiative equilibrium temperatures, and observed radiating temperatures for selected Northern Hemisphere latitudes. Latitude

10∘ N 20∘ N 30∘ N 40∘ N 50∘ N 60∘ N 70∘ N

Incident sunlight (W m−2 )

Albedo

Absorbed sunlight (W m−2 )

Emitted infrared (W m−2 )

Equilibrium radiating temperature (K)

Observed radiating temperature (K)

410 391 364 327 283 235 195

0.24 0.24 0.28 0.32 0.38 0.41 0.49

311 297 263 222 176 139 100

247 270 257 235 219 210 201

272 269 261 250 236 222 205

257 262 259 253 249 246 244

The values are averages from 11 years of CERES-EBAF data.

1.10 Summary and Outlook

This chapter focused on simple radiative equilibrium and window-gray models that behave similar to the Earth and its atmosphere. The remainder of the book explores more deeply the nature of radiation, the scattering and absorption of sunlight, the absorption and emission of infrared radiation, and how the radiative terms shown in Figures 1.7 and 1.10 are calculated. It also explores the ways in which the reflected and transmitted sunlight and emitted infrared radiation at selected wavelengths can be used to infer the composition of the atmosphere and its thermodynamic structure. The narrative begins with the nature of light (Chapter 2) and how it interacts with matter as it propagates through scattering, absorbing, and emitting media (Chapter 3). The propagation, tied to a frame of reference, such as the Earth’s surface, leads to the equation of radiative transfer. Simple solutions are then developed for the transfer of infrared radiation and sunlight, including the effects of multiple scattering (Chapter 4). The solutions set the stage for making simple estimates of the radiative terms in Figures 1.7 and 1.10. Obtaining accurate estimates of the terms, however, requires the integration of solutions to the radiative transfer equation over all wavelengths from the short UV wavelengths to the long thermal infrared wavelengths. The calculations are cumbersome and require many details involving molecular absorption spectra. Simplifications are made that capture the essence of the absorption by molecules so that calculations using the simplified solutions can be performed over finite bands of wavelengths (Chapter 5). These simplifications are then applied to calculate the contributions to the Earth’s albedo by haze, clouds, ozone, and water vapor (Chapter 6). They are then applied to calculate the change in emission caused by a doubling of carbon dioxide in the Earth’s atmosphere and how this doubling not

35

36

1 The Earth’s Energy Budget and Climate Change

only leads to a warming of the Earth and its atmosphere, but also a cooling of the stratosphere (Chapter 7).

Problems

1. Based on the values given in Figure 1.7, calculate the following: a. Use the apparent transmissivity of the troposphere to calculate an effective emissivity for longwave radiation. b. Use the emissivity calculated in (a) to calculate an apparent radiating temperature for the downward emission at the surface. c. Use the effective radiating temperature obtained in (b) to calculate the apparent altitude of the emitting layer. Use the emission at the surface to obtain the surface temperature and assume a tropospheric lapse rate of Γ = 6.5 K km−1 to obtain the apparent altitude. d. Repeat the calculations in (b) and (c), but for the upward radiation emitted by the troposphere at the tropopause. e. Explain why the emissivity and temperatures obtained in (a–d) differ from those obtained with the window-gray, radiative equilibrium model. f. Repeat (a–d) for the stratosphere. Consider the stratosphere to start at an altitude of 15 km and the temperature of the lower stratosphere to be 205 K. Assume a stratospheric lapse rate of Γ = −2 K km−1 . A negative lapse rate indicates that temperature rises with altitude in the stratosphere. g. Assume a tropopause pressure of 250 hPa and calculate the net radiative cooling (K/day) for the troposphere. h. The net radiative cooling of the troposphere is approximately compensated by the release of latent heat when water condenses and falls as precipitation. Calculate the rate of precipitation (mm/day) required to just balance the net radiative cooling. Use 2.5 × 106 J kg−1 for the heat of vaporization for water. Compare this approximate estimate of the precipitation rate with the rate of surface evaporation. i. Assume that the stratosphere represents all of the atmosphere above 250 hPa and calculate the heating rate of the stratosphere (K/day) caused by the absorption of sunlight by O3 . j. Assume a global average surface albedo of 0.1 and calculate the incident sunlight (W m−2 ) at the surface. What surface albedo would contribute the 23 W m−2 to the Earth’s reflected sunlight shown in Figure 1.10? Estimate the albedo to two significant figures. 2. By integrating the cosine of the solar zenith angle over the surface of the sunlit side of the Earth, show that the global average cosine for the solar zenith angle of the sunlit Earth is 0.5. Consequently, a suitable value of the “average solar zenith angle” for the sunlit side of the Earth is 60∘ . 3. Construct a window-gray, radiative equilibrium model for the Earth’s atmosphere in which 20% of the incident sunlight is absorbed by the atmosphere.

Problems

Assume that the albedo for the Earth is 0.3 and the surface temperature is 288 K. Assume that the atmosphere absorbs the incident sunlight but reflects none of it. a. Calculate the surface albedo that is consistent with an Earth albedo of 0.3 and a nonreflecting atmosphere that absorbs 20% of the incident sunlight. Account for the absorption of sunlight as it passes through the atmosphere to the surface and the absorption of the sunlight that is reflected by the surface that passes through the atmosphere to space. b. Is the emissivity obtained for an atmosphere that absorbs sunlight greater or smaller than the emissivity for an atmosphere that absorbs none of the incident sunlight? c. Is the atmospheric temperature of the absorbing atmosphere greater than the temperature for an atmosphere that absorbs none of the incident sunlight? 4. Estimate the amplitude of the day–night temperature difference for a window-gray, radiative equilibrium model of Mars. The Earth and Mars have nearly identical rotation rates. Use the following steps: a. Calculate the emissivity of the Martian atmosphere on the basis of the appropriately reduced solar constant, the albedo, and the surface temperature. b. Calculate the radiative time constant as the e-folding time for cooling after the sun has been removed. c. Calculate the temperature perturbation for t = 12 h. d. What does your estimate of the day–night difference for Mars suggest about the likely relationship between diurnally forced temperature changes and weather-related temperature changes on Mars?

Relative distance to the sun Albedo Surface temperature (K) Surface pressure (hPa) Heat capacity, Cp (J kg−1 K−1 ) Acceleration due to gravity, g (m s−2 )

Earth

Mars

1.0 0.30 288 1013.25 1005 9.80

1.52 0.15 240 7 830 3.76

5. Develop a time-dependent version of the global average radiative equilibrium model as given by Equation 1.5 to determine the response of the Earth’s radiating temperature following the 1991 Mt. Pinatubo eruption. The eruption caused a perturbation in the albedo that initially amounted to a 4 W m−2 decrease in the sunlight absorbed by the Earth. Assume that the albedo is

37

38

1 The Earth’s Energy Budget and Climate Change

given by

) ( t � = �0 + � ′ exp − �a

with �0 = 0.30 the global average albedo and � ′ the change in the albedo required to produce the 4 W m−2 decrease in absorbed sunlight at t = 0. Assume �a = 2 years for the residence time of the haze layer that formed in the stratosphere after the eruption. Use the following steps to obtain a time-dependent solution for the Earth’s radiating temperature. a. Expand the Stefan–Boltzmann law in the radiative equilibrium model about its initial equilibrium state. As was done for the window-gray model in Section 1.6, use a first-order perturbation method to obtain the change in the Earth’s radiating temperature as a function of time following the Mt. Pinatubo eruption. Assume that the change in the radiating temperature approximates the changes in the lower atmospheric and surface temperatures. Use the heat capacity of air and the mass per unit area of the atmosphere to represent the thermal inertia of the system. Solve for the temperature change as a function of time after the eruption and use it to determine when the minimum occurred and the decrease in the surface temperature at the minimum. b. Repeat these calculations using the thermal inertia of the ocean mixed layer. Use 50 m as the depth of the mixed layer and 4218 J kg−1 K−1 as the heat capacity of water. The temperature change for the ocean mixed layer produces a more realistic response. c. Repeat (a) and (b) but instead of the Earth’s radiating temperature and the Stefan-Boltzmann law, assume that the emitted longwave flux is given by F = A + BTS with A and B constants and TS the annually averaged global mean surface temperature. Assume that B = 2 W m−2 K−1 . As with the radiating temperature, assume that the change in the average surface temperature approximates the change in the temperature of the lower troposphere. How does including a feedback comparable to that of the water vapor feedback alter the minimum surface temperature and the delay between the eruption and the minimum? The Earth’s surface temperature decreased to approximately 0.4 K and this minimum was reached approximately 1.3 years after the Mt. Pinatubo eruption. 6. Derive equations for a two-layered, window-gray, radiative equilibrium model. Give the layers arbitrary emissivities �1 > 0 and �2 > 0 with �1 ≠ �2 . Show algebraically that for radiative equilibrium, the temperature of the upper atmospheric layer must be less than that of the lower layer and the temperature of the surface must be greater than that of the lower atmospheric layer. Consequently, within the simple window-gray radiative equilibrium model, temperature must fall with increasing altitude.

References

References 1. Kopp, G., Lawrence, G., and Rottman,

2.

3.

4.

5.

6.

G. (2005) The total irradiance monitor (TIM): Science results. Solar Phys., 230, 129–140. Loeb, N.G., Wielicki, B.A., Doelling, D.R., Smith, G.L., Keyes, D.F., Kato, S., Manalo-Smith, N., and Wong, T. (2009) Toward optimal closure of the Earth’s top-of-atmosphere radiation budget. J. Clim., 22, 748–766. Pippard, A.B. (1957) Elements of Classical Thermodynamics for Advanced Students of Physics, Cambridge University Press, Cambridge. Solomon, S., Qin, D., Manning, M., Alley, R.B., Bernstein, T., Bindoff, N.L., Chen, Z., Chidthaisong, A., Gregory, J.M., Nicholls, N., Heimann, J.M., Hewitson, B., Hoskins, B.J., Joos, F., Jouzei, J., Kattsov, V., Lohmnn, U., Matsuno, T., Molina, M., Nicholls, N., Overpeck, J., Raga, G., Ramaswamy, V., Ren, J., Rusicucci, M., Sommerville, R., Stocker, T.F., Whetton, P., Wood, R.A., and Wratt, D. (2007) Technical summary, in Climate Change, 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (eds S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt, M. Tignor, and H.L. Miller), Cambridge University Press, Cambridge, New York. Soden, B.J. and Held, I.M. (2006) An assessment of climate feedbacks in coupled ocean-atmosphere models. J. Clim., 19, 3354–3360. Manabe, S. and Strickler, R.F. (1964) Thermal equilibrium of the atmosphere with convective adjustment. J. Atmos. Sci., 21, 361–385.

7. Manabe, S. and Wetherald, R.T. (1967)

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

Thermal equilibrium of the atmosphere with a given distribution of relative humidity. J. Atmos. Sci., 24, 241–259. Green, A.E.S. (1964) Attenuation by ozone and the Earth’s albedo in the middle ultraviolet. Appl. Opt., 3, 203–208. Lacis, A.A. and Hansen, J.E. (1974) A parameterization for the absorption of solar radiation in the earth’s atmosphere. J. Atmos. Sci., 31, 118–133. Goody, R.M. and Yung, Y.L. (1989) Atmospheric Radiation, Oxford University Press, Oxford. Houghton, J.T. (2002) The Physics of Atmospheres, Cambridge University Press, Cambridge. Hartmann, D.L. (1994) Global Physical Climatology, Academic Press, San Diego, CA. Baum, B.A., Yang, P., Heymsfield, A.J., Platnick, S., King, M.D., Hu, Y.X., and Bedka, S.M. (2005) Bulk scattering properties for the remote sensing of ice clouds. II: narrowband models. J. Appl. Meteorol., 44, 1896–1911. Trenberth, K.E., Fasullo, J.T., and Kiehl, J.T. (2009) Earth’s global energy budget. Bull. Am. Meteorol. Soc., 90, 311–323. Kiehl, J.T. and Trenberth, K.E. (1997) Earth’s annual global mean energy budget. Bull. Am. Meteorol. Soc., 78, 197–208. Information concerning the 20C3M experiment can be found at, http://www-pcmdi.llnl.gov/ projects/cmip/ann_20c3m.php. Trenberth, K.E. and Smith, L. (2009) The three dimensional structure of the atmospheric energy budget: methodology and evaluation. Clim. Dyn., 32, 1065–1079.

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41

2 Radiation and Its Sources 2.1 Light as an Electromagnetic Wave

In 1865 James Clerk Maxwell presented his famous set of equations, which he used to show that (i) electromagnetic waves exist and (ii) as in vacuum, the waves would have speeds near that of light, light must be an electromagnetic wave. Maxwell’s equations are statements of empirically deduced laws concerning the properties of electric and magnetic fields. The properties that Maxwell’s equations give to a plane electromagnetic wave in a vacuum are as follows: 1) The wave is composed of time-varying electric E and magnetic H fields. The electric and magnetic fields are represented by vectors. 2) E and H are perpendicular to each other. 3) The cross product E × H gives the direction of propagation. The propagation is perpendicular to both E and H. 4) The wave is polarized. By convention, the state of polarization is given by the orientation of the electric field vector, E. Dielectric media have dipole moments that alter their orientations in response to the electric fields of electromagnetic waves. Consequently, the light that is transmitted and reflected by dielectrics depends on the state of polarization of the incident light. Materials such as those used to create polarized lenses in sunglasses pass light that has its electric field aligned with one direction but absorbs light that has its electric field perpendicular to the preferred direction. In a vacuum, the frequency of the radiation f and the wavelength � are related by f � = c with c = 3 × 108 m s−1 the speed of light in a vacuum. In a dielectric medium, the wavelength of light is shortened and the speed is reduced proportionately so that the frequency remains constant. Approximately 99% of the incident sunlight lies within the wavelengths 0.2 μm < λ < 5 μm. For the Earth’s emitted radiation, the range is 4 μm < λ < 100 μm. As was noted in Section 1.8, wavelengths are often specified in terms of “wavenumber,” v = 1∕�, in the infrared. Wavenumbers became common early in the development of infrared spectroscopy. Microns Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

42

2 Radiation and Its Sources

(μm) and nanometers (nm) are typically used for the wavelengths of solar radiation. The usages are not exclusive. The use of wavenumbers for short wavelengths and microns for long wavelengths are also common.

2.2 Radiation from an Oscillating Dipole, Radiance, and Radiative Flux

The simplest source of electromagnetic radiation is an accelerating charge. A charge at rest gives rise to an electrostatic field – constant in time. A charge in uniform motion gives rise to a current, which, in turn, gives rise to a static magnetic field – constant in time. An accelerating charge gives rise to the required time-dependent electric and magnetic fields, and consequently, to radiation. Fortunately, isolated accelerating charges are relatively rare near the Earth’s surface. Such charges interact strongly with all matter and damage living organisms. The simplest bounded source of electromagnetic radiation is an oscillating dipole shown schematically in Figure 2.1a. Figure 2.1b illustrates the time-averaged angular pattern of the power per unit area radiated by an oscillating dipole. The pattern

H z +

E θ r

d –

(a)

p

(b) Figure 2.1 Oscillating dipole (a) and its radiation pattern (b). The dipole moment vector is � = e� with e the charge of both the positive and negative charges in the dipole and � the displacement vector between the charges. The direction of propagation is indicated by the radial vector �. The time-dependent electric field vector � lies in the plane of the page and the associated magnetic field vector � is perpendicular to the electric field and pointed

outwards from the plane of the page. The angular distribution of the average radiated power per unit area of a sphere centered on the dipole and having its surface far from the dipole is given by Equation 2.1. The angular distribution in (b) is proportional to sin2 �. It is maximum perpendicular to the dipole and zero off the ends of the dipole. The geometry and the radiation pattern are rotationally invariant about an axis aligned with the dipole moment vector �.

2.2

Radiation from an Oscillating Dipole, Radiance, and Radiative Flux

represents the power at sufficiently large distances where the radial distance from the dipole is much larger than the displacement between the charges in the dipole and also much larger than the wavelength of the light. As might be expected, the illustrations in Figure 2.1 are invariant to rotation about an axis aligned with the dipole moment vector p. Solutions to Maxwell’s equations provide the power per unit area radiated by an oscillating dipole. The time-averaged radiative power per unit area of a sphere with radius r centered on the dipole is given by c2 Z0 � 2 |�|2 sin2 � (2.1) 2�4 r 2 with c the speed of light; � = e� the dipole moment with e the magnitude of the positive and negative charges in the dipole and d the vector displacement between the charges; � the angle for the direction of propagation measured from the direction of p; � the wavelength of the radiation; and √ Z0 the impedance of free space. The impedance of free space is given by Z0 = �0 ∕�0 with �0 the permittivity and �0 the permeability of free space [1]. The permittivity and permeability √ of free space are related through the speed of light in a vacuum, c = 1∕ �0 �0 . The average power in Equation 2.1 is in the mks units (meter, kilogram, second). In mks, the unit of charge, the coulomb, is actually defined through Ampère’s law for magnetic forces. The result is the introduction of the permittivity and permeability of free space so that the laws for both the magnetic and electric forces are consistent with mechanical forces. In Equation 2.1 cZ0 = 1∕�0 is the factor that appears in Coulomb’s electrostatic force law. It is a proportionality constant and has units of force × length squared/charge squared. Force has units of mass × length/time squared (m�t −2 ) giving 1∕�0 units of (m� 3 t −2 q−2 ). For the other factors in Equation 2.1, c has units of (�t −1 ), |�|2 units of (q2 � 2 ), and r 2 �4 units of (� 6 ). Together the factors have units of (m� 2 t −2 × t −1 × � −2 ) which is energy per unit time per unit area, or power per unit area. The radiation emitted by an oscillating dipole has the following properties: F� =

1) Far from the dipole, the power falls with the square of the distance from the dipole, 1∕r 2 . 2) The direction of polarization of the radiation lies in the plane containing both the dipole moment vector p and the radial vector for the direction of propagation r and is perpendicular to r. 3) No power is radiated off the ends of the dipole moment, in the directions of the positive and negative branches of the z axis in Figure 2.1. Sources of radiation may be taken to contain electric dipoles distributed in space with certain orientations so as to give rise to the observed radiation field. The simplest sources, such as the sun and blackbodies, radiate unpolarized light isotropically, meaning that they radiate uniformly in all directions and the radiation has all orientations of polarization perpendicular to the directions of propagation. For such sources, the dipole moments are uniformly distributed in space and have uniformly distributed orientations.

43

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2 Radiation and Its Sources

The energy propagated by radiation generally depends on wavelength and on the direction of propagation, which in turn is the direction in which the radiation left the emitting, reflecting, or transmitting medium. The incremental spectral radiant energy per unit time (W) is given by d�v = Iv dvdAdΩ

(2.2)

with Iv the radiance. Radiance is the energy per unit time per unit wavenumber per unit surface area per unit solid angle at wavenumber �. In Equation 2.2 the incremental power d�v is for wavenumbers lying within an infinitesimal element of wavenumber d� centered on wavenumber �. Radiance is defined so that it is a measure of the radiated power that passes through an element of surface area dA that has its surface aligned normal with the direction of propagation. In addition, the radiation is propagating in the direction specified by the angular vector �, and into an infinitesimal cone of solid angle dΩ. As is discussed below and as shown in Figure 2.3b, this cone bounds the volume that contains the solid angle subtended by area dA at the radiating source. The radiance Iv has units of W m−2 sr−1 (cm−1 )−1 , sometimes written as W m−2 sr−1 cm. With the possible exception of the solid angle, the description of radiant energy in terms of radiance seems intuitive. Light is energy flowing per unit time through a given area. Light has color. The energy is distributed over a certain wavelength or wavenumber interval. The solid angle is required because all light sources can be broken down into point sources. Within a vacuum, the power per unit area radiated by a point source falls as the square of the distance from the source. The radiance, on the other hand, remains unchanged with distance. A solid angle is an angular domain. It is given by the area of a spherical section subtended by the angular domain divided by the square of the radius of the sphere that contains the section as part of its surface. To appreciate the role of solid angle in defining the flow of radiant energy, consider two isotropic light sources. Assume that the two sources are different; one emits with power P1 , so many watts, and the second with power P2 . Assume that the two sources are aligned with a detector at a distance r1 from source 1 and r2 from source 2. The area of the detector subtends different solid angles for the two sources, as illustrated in Figure 2.2. The placement and strengths of the sources are such, however, that the radiant power measured with the detector is the same for both sources. In addition, assume that r12 and r22 are both much greater than the detector’s surface area Δs so that the area of the detector approximately equals the area of the spherical surfaces associated with the solid angles for both sources. The power for each source measured by the detector would then be given by P1 4�r12

Δs =

P2 4�r22

Δs

(2.3)

The placement of the sources, of course, would have to satisfy the condition that P1 ∕P2 = r12 ∕r22 . Clearly, if r2 < r1 , then the solid angle subtended by the detector viewed from source 2 is larger than that subtended for source 1, as is illustrated in Figure 2.2. If the detector is now moved to a new location a distance � from its

2.2

P1

Radiation from an Oscillating Dipole, Radiance, and Radiative Flux

P2

r2 r1 Figure 2.2 The role of solid angle in the definition of radiance. Two isotropic light sources 1 and 2 have different powers P1 and P2 but are placed so that the radiative power measured with a detector, indicated by the black ellipse, is the same for both sources when placed distances r1 and r2 from the sources. The solid angle subtended

by the area of the detector as viewed from source 2 a distance r2 from the detector is larger than the solid angle subtended for source 1. With the detector placed an additional distance � from the sources, the power measured by the detector is greater for source 1 than for source 2.

previous location and remains aligned with the sources, then the power measured for the two sources would be given by P2 P1 Δs and Δs 4�(r1 + �)2 4�(r2 + �)2

(2.4)

The radiative power per unit area is reduced by the ratio of the solid angles subtended by the detector at the source for the original distance and the larger distance. At the new location, the ratio for source 1, (1 + �∕r1 )2 , is smaller than that for source 2, (1 + �∕r2 )2 ; consequently, the power measured by the detector for source 1 would be greater than that measured for source 2. The radiative flux or irradiance is used to calculate the radiative power per unit area passing through an arbitrary surface in the direction aligned with the normal to the surface. In Figure 2.3, the spectral radiative flux F� is passing through a surface with area A′ . Imagine radiation being emitted or sunlight being reflected by this surface. The radiance I� (�, �) coming from the surface is propagating in the direction given by the zenith angle � and azimuth angle �. The radiance can have any value for any zenith angle, azimuth angle, and for that matter, any increment of the surface that is doing the emitting or reflecting. The coordinate system shown in the figure is arbitrary. The only requirement is that the radiative flux flows in the direction aligned with the normal of the surface. The contribution of radiance I� (�, �) to the radiative power per unit area passing through the surface is given by dF� = I� (�, �)(dA∕dA′ )dΩ

(2.5)

with dA the element of area with its normal in the direction of propagation and dΩ the increment of solid angle projected by dA as viewed from the source. The incremental area with its normal in the direction of propagation is related to the incremental area at the source as dA = dA′ cos �. Assuming that the incremental area dA is associated with a surface element for a sphere of radius r, the increment

45

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2 Radiation and Its Sources

Fν dA

dA Iν (θ, ϕ)

θ

θ

r

dA′ A′ ϕ

ϕ

r sin θ

(a)

(b)

Figure 2.3 Radiance I� (�, �) passes through an infinitesimal increment of the shaded area A′ propagating in the direction given by zenith angle � and azimuth angle �. The radiance illuminates an increment of solid angle dΩ and contributes to the upward radiative flux passing through the surface, F� (a). The increment of the shaded area

through which the radiation flows is given by dA′ = dA∕ cos �. The increment of solid angle dΩ is given by the incremental area dA which has its normal aligned with the direction of propagation divided by r 2 (b). dA = r sin � d� × r d� so that dΩ = dA∕r 2 = sin � d�d�.

of solid angle is given by dΩ = sin � d�d� as is illustrated in Figure 2.3b. As a result, the radiative power per unit area contributed by I� (�, �) to that passing through surface dA′ is given by dF� = I� (�, �) cos � sin � d�d�

(2.6)

The spectral radiative flux at a particular wavelength or wavenumber passing through the surface is obtained by integrating over all angles and is given by Fv =

∫0

2�

d� ∫0

�∕2

d� sin � cos � Iv (�, �) =

∫0

2�

1

d� d�� Iv (�, �) (2.7) ∫0

It has units of W m−2 (cm−1 )−1 or W m−2 μm−1 . In the event that Iv is isotropic, it is constant independent of � and �, and Fv is given by Fv = �Iv . In this book, radiative flux and irradiance are used interchangeably for the quantity given by Equation 2.7. Sometimes, however, the quantity given by Equation 2.7 is called a flux density [2]. This terminology can be understood by referring back to the power per unit area radiated by an oscillating dipole (Equation 2.1). The total time-averaged radiated power passing through a sphere centered on the dipole and having its surface at a large distance from the dipole is given by P� =

4� 3 c|�|2 3 �0 �4

(2.8)

2.3

Radiometry

The total power is, of course, independent of the radius of the sphere over which it is measured. In physics, this total power is referred to as a flux of energy passing through the spherical surface. The term “flux density” characterizes the flow of radiant energy as the energy per unit time per unit area, or power per unit area.

2.3 Radiometry

Measuring radiances and radiative fluxes is simple in principle. Obtaining accurate measurements of radiances, on the other hand, requires sophisticated, carefully engineered instruments. Figure 2.4 shows the design of the scanning radiometer that measures broadband radiances for the Clouds and Earth’s Radiant Energy System (CERES). Figure 2.5 illustrates the scale of the CERES radiometer with a cutaway showing its detector assembly. The radiances measured with the radiometer are used in a complex inversion process to produce monthly averages of the hourly reflected solar and emitted longwave radiative fluxes for 100 km scale regions covering the entire Earth. Briefly, the processing accounts for the following: (i) the sensitivity of the radiometer as a function of wavelength, (ii) the angular dependence of the measured radiation, and (iii) the temporal sampling dictated by the orbits of the satellites carrying the instruments. Accounting for both the spectral sensitivity of the radiometer and the angular dependence of the measured radiation depends on the scene being observed. The CERES radiometers are designed to have a field of view of about 20 km across at nadir, the direction looking straight down from a satellite. For CERES, the scene determination is obtained from collocated 1 km resolution multispectral imagery obtained from the moderate-resolution imaging spectroradiometer (MODIS). Ultimately, averaging the monthly regional, hourly scale radiative fluxes over the Earth and over time produces the annual average global albedo, absorbed solar radiation, and emitted longwave radiation presented in Chapter 1. At the time of writing, two CERES instruments are measuring broadband radiances on each of two satellites, NASA’s Terra and Aqua satellites, and one is measuring these radiances on the Suomi National Polar-orbiting Partnership satellite (NPP). NPP is the predecessor to the next generation of polar orbiting operational weather satellites. The plan is to continue observations of the Earth’s radiation budget on the Joint Polar Satellite System (JPSS). The CERES instrument is to be flown on the first JPSS satellite, which is projected to be operational in 2016. The CERES follow-on instrument planned for a subsequent JPSS satellite is to be an improved version of the CERES instruments now flying. The CERES radiometer is a ballistic bolometer [3]. It continuously measures broadband radiances as it scans. The signals from the detector are electronically sampled to create distinct field of view samples. Each sample is ∼20 km across when the instrument is pointing in the nadir direction. Each CERES instrument has three radiometers. The optical axes of the three are aligned to ensure that

47

48

2 Radiation and Its Sources

Baffle

Conic sections

Reflector cap Indium wire

Filter Hub

Secondary mirror mount Secondary mirror

Detector housing

Primary mirror insert

Field stop

Flake substrate

Primary mirror

Filter

Active element sub-assembly Compensator element assembly End cap assembly

Figure 2.4 Diagram of the CERES scanning ballistic bolometer radiometer. (Courtesy of NASA Langley Research Center).

2.3

Figure 2.5 Cutaway of the CERES ballistic bolometer radiometer showing part of the baffle, the secondary mirror mount, and detector housing. The length of the channel

Radiometry

from the top of the baffle to the base of the detector housing is approximately 90 mm. (Courtesy of NASA Langley Research Center).

all three are viewing the same scene. The three scan across the Earth below the satellite as the satellite moves along its orbit. One of the radiometers, the shortwave channel, has a spectral filter that limits measurements to reflected sunlight at wavelengths between 0.3 and 5 μm. A second channel, the total channel, has no spectral filter and is sensitive to radiation between 0.3 and 200 μm. The third radiometer, the window channel, has a filter that limits measurements to emitted radiation in the 8–12 μm infrared window of the Earth’s atmosphere. Instead of this window channel, the follow-on CERES radiometer to be flown on the subsequent JPSS satellite will have a filter that limits measurements to the Earth’s emitted longwave radiation between 5 and 50 μm. The heart of the radiometer is a pair of detectors in a balanced detector assembly. The detectors are identical thin-film, 40–50 μm thick thermistors. The detectors are squares with sides of about 2 mm length. The size of the detectors represents a trade-off between competing goals. One goal is to obtain radiances for sufficiently small regions so that the effects of clouds may be avoided, at least for a representative fraction of the Earth (15–20% within any orbit). The second is to have sufficient signal so that the measurement has high accuracy with 95% confidence intervals of 1% for thermal emission and 2% for reflected sunlight. Both detectors in the balanced detector assembly are coated so that they are black, meaning that they absorb at all wavelengths. The detectors are attached to opposite sides of an aluminum block that serves as a heat sink. The temperature of the heat sink is maintained within a small range of temperatures and its temperature is carefully monitored. One of the detectors is placed at the focal

49

50

2 Radiation and Its Sources

point of a Cassegrain-like telescope that focuses the surface of the Earth onto the detector. A field stop limits the portion of the detector that is illuminated, approximately 1.5 mm in diameter. The second detector serves as a reference and faces the base of the radiometer. The base is at the same temperature as the aluminum block that holds the detector assembly. Thermistors have resistances that change with temperature. As the detector viewing the Earth is scanned, its temperature varies in response to the radiation that it absorbs. The resulting difference in temperature between it and the reference detector gives rise to a difference in resistance within the arms of a bridge circuit. This difference creates a voltage that is amplified and converted to power, based on the sensitivity of the instrument. Instrument calibration is checked for each scan by a view of deep space (zero radiance) on one end of the instrument’s scan. At the other end of the scan, the infrared and total channels view blackbodies at known temperatures and the shortwave channel views a tungsten lamp through a diffuser. The diffuser converts the light from the lamp’s filament to isotropic radiation. To obtain the desired broadband radiance, the measured power is divided by the surface area of the detector and the solid angle subtended by the Earth’s surface as viewed by the detector. The broadband radiances have units of W m−2 sr−1 . The wavelength dependence of any filter used in the broadband measurements is taken into account to convert the measured power to that for the particular bandpass – 0.3–5 μm for the reflected shortwave radiance and 5–200 μm for emitted radiation. The CERES radiometer loses sensitivity at wavelengths beyond about 50 μm. Approximately 5% of the Earth’s energy is emitted at wavelengths longer than 50 μm. Extending the measured power to emitted radiances covering 5–200 μm is a correction made for the wavelength sensitivity of the radiometer. The error in accounting for this wavelength dependence is estimated to affect the Earth’s annual global average longwave radiative flux by 0.2–0.5 W m−2 , or about 0.1–0.2% [4]. In the case of narrow band radiometers, such as the CERES 8 –12 μm window channel, the measured power is also divided by the wavelength interval to produce a radiance with units of W m−2 sr−1 μm−1 . 2.4 Blackbody Radiation: Light as a Photon

Light has wave–particle duality. Studies of the radiation emitted by blackbodies toward the end of the nineteenth century gave birth to the concept of light as discrete quanta of energy, now known as photons. Max Planck discovered that each photon had a unique energy that depended solely on the wavelength of the radiation. His discovery gave birth to quantum mechanics. Late in the nineteenth century, headway toward predicting the radiation emitted by blackbodies was being made through the application of classical electromagnetic theory and thermodynamics. Although originally determined empirically, one of the intellectual highlights of classical thermodynamics was that the total radiative flux emitted by a blackbody at temperature T was a simple power law,

2.4

Blackbody Radiation: Light as a Photon

�B = �T 4 , with �B the radiative flux emitted by a blackbody integrated over all wavelengths and �T 4 the Stefan–Boltzmann law. In addition, the distribution of spectral radiative flux per unit wavelength was known to have its maximum at a wavelength that was given by Wien’s displacement law, �MAX T = 2897 μm K. As its temperature rises, the emission by a blackbody peaks at a shorter wavelength. Concepts of statistical mechanics were being applied to obtain the wavelength dependence of the emitted radiation. The initial attempts failed. From the point of view of classical electrodynamics, within any enclosure the number of wavelengths allowed for radiation is given by L (2.9) � with L the length of the enclosure in the direction of propagation and � the wavelength of the light. The number of wavelengths per unit length of enclosure is given by 1∕� = v. Within a blackbody, the emitted radiation is in equilibrium with the temperature of the blackbody. The radiation is also isotropic. As a result, the number of wavelengths per unit volume for the emitted radiation at wavenumber v lying within an increment of wavenumber d� propagating in all directions is given by n� =

dn� = 8�v2 dv

(2.10)

The number per unit volume is given by twice the volume of a spherical shell in wavenumber space. The factor of 2 accounts for the two degrees of polarization that electromagnetic waves can have when propagating in any direction. In the classical view, each of these “nodes” of radiation can have any energy. Furthermore, from statistical mechanics the distribution of energy is given by a Boltzmann distribution. The average energy of all the nodes is given by

E=

∫0

∞

∫0

E

E e− kT dE ∞

= kT

(2.11)

E

e− kT dE

with T the temperature of the enclosure and k = 1.38 × 10−23 J K−1 Boltzmann’s constant. According to this view, the power per unit area flowing through a spherical surface propagating in all directions, 4� steradians, is given by 4�Bv (T)d� = 8�ckTv2 dv

(2.12)

This expression is the Rayleigh–Jeans approximation. It is better known when expressed in terms of wavelength instead of wavenumber. d� (2.13) �4 The approximation gives the correct spectral distribution in the limit of large wavelengths. It leads to the famous “ultraviolet catastrophe” at short wavelengths, infinite energy as the wavelength approaches zero. Integrating Equation 2.13 over all wavelengths leads to an infinite energy loss. The integral, however, should not 4�B� (T)d� = 8�ckT

51

52

2 Radiation and Its Sources

blow up. Instead, it should produce the Stefan–Boltzmann law for blackbody radiation. In 1900, Planck overcame the ultraviolet catastrophe by assuming that instead of the nodes taking any energy, they were limited to discrete amounts of energy E = nhc� with n = 1, 2, 3, and so on and h a constant, now known as Planck’s constant. With this assumption, the average energy of the nodes is given by ∞ ∑

E=

nhcve−

nhcv kT

n=0 ∞ ∑

(2.14) e

− nhcv kT

n=0

Notice that ∞ ∑

e−

nhcv kT

=

n=0

1 hcv

1 − e− kT

and ∞ ∑

( − nhcv kT

nhcve

n=0

∂ =− ( ) 1 ∂ kT

∞ ∑

) − nhcv kT

e

n=0

hcv

= (

hcve− kT

hcv

1 − e− kT

)2

(2.15)

As a result, the power per unit area flowing through a sphere in equilibrium with the temperature of the enclosure is given by 8�hc2 v3 dv 4�Bv (T)d� = ( hcv ) e kT − 1

(2.16)

Note that as v → 0, exp(hc�∕kT) → 1 + hc�∕kT, and Equation 2.16 becomes the Rayleigh–Jeans approximation Equation 2.12. In addition, as v → ∞, B� (T) → 0. There is no ultraviolet catastrophe. Since the radiation emitted by a blackbody is isotropic, the radiance in any particular direction, as opposed to all directions within 4� sr, is given by 2hc2 v3 dv Bv (T)dv = ( hcv ) e kT − 1

(2.17)

B� (T) is the Planck function. Often Equation 2.17 is written in terms of Planck’s radiation constants, C1 = 2hc2 = 1.19 × 10−5 mW m−2 sr−1 (cm−1 )−4 and C2 = hc∕k = 1.44 K (cm−1 )−1 . The units of C1 are given in terms of milliwatts (mW) as opposed to watts because infrared radiances emitted by the Earth’s surface and atmosphere are of the order 101 –102 mW m−2 sr−1 (cm−1 )−1 , a convenient magnitude. In terms of these constants, the Planck function is

2.4

Blackbody Radiation: Light as a Photon

given by C �3 B� (T) = ( C2 �1 ) e T −1

(2.18)

Planck’s constants are sometimes referred to as Planck’s first and second radiation constants. In some references, the first constant is sometimes given as 2�hc2 , which has units of Wm2 , and sometimes as 8�hc, which has units of J m. Here, the value of C1 and its units facilitate the evaluation of radiances emitted by the Earth and its atmosphere. To calculate irradiances, both sides of Equation 2.18 must be multiplied by �. Blackbody radiation is isotropic. The Planck function gives the spectral dependence of emission for blackbodies – bodies having unit absorptivity and emissivity. The emission by the sun for wavelengths greater than about 0.6 μm follows closely the spectral distribution of a blackbody at 5783 K [5]. In fact, the sun’s temperature is determined by fitting the spectral distribution of sunlight to the Planck function. In turn, once the temperature is known, the solar constant and the distance to the sun from the Earth provide a measure of the size of the sun. Blackbodies, such as those used in CERES to calibrate the total and longwave channels, have holes through which the emitted radiation escapes. The emitted radiation is isotropic. The Stefan–Boltzmann law is obtained by integrating the emitted radiances over all angles and all wavenumbers to obtain the outward radiative flux. �T 4 =

∫0

2�

d�

∫0

�∕2

d� sin � cos �

∫0

∞

d�B� (T) = �

∫0

∞

d�B� (T)

(2.19a)

The integral over all wavenumbers is easily performed using the rules for series summation or the calculus of residues. �

∫0

∞

d�B� (T) = 2� ∞

=

2�k 4 T 4 h3 c2 ∫0

dx

x3 = x e − 1 ∫0

dx

∫0

∞

d�

hc2 � 3 hc�

e kT − 1

x3 ex − 1

(2.19b)

and ∫0

∞

∞

∞ ∞ ∞ ∞ ∑ ∑ ∑ 1 1 �4 3 −x = dx x3 e−nx = dx x e = 6 (2.20) 4 4 ∫ n 0 n 15 n=1 n=1 n=1

Thus, the Stefan–Boltzmann constant is related to Planck’s constant by 2� 5 k 4 = 5.66961 × 10−8 Wm−2 K−4 (2.21) 15 h3 c2 When Planck first derived this expression, k was known through the ideal gas law, c had been measured, and the Stefan–Boltzmann constant had also been determined. Planck used Equation 2.21 to obtain the value, h = 6.626 × 10−34 J s. The units of Planck’s constant are those of angular momentum. Later, Albert Einstein independently determined the value of Planck’s constant. He invoked �=

53

2 Radiation and Its Sources

1010 108 Sun 6000 K

10

104 102

3000 K Visible light

6

1000 K Earth 300 K

100 10

Infrared window

πBλ (W m–2 μm–1)

54

–2

10–4 0.1

1.0

200 K

10.0

100.0

Wavelength (μm)

Figure 2.6 Planck radiation for temperatures ranging from that of the Earth’s surface and atmosphere, 200–300 K, to that of the sun, approximately 6000 K. The visible part of the spectrum,

0.4–0.7 μm, lies between the dashed lines. The Earth’s infrared window, 8–12 μm, lies between the dotted lines.

the quantization of photon energy in his description of the photoelectric effect. The photoelectric effect is associated with the ejection of electrons from a metal surface when the surface is illuminated by monochromatic light. Einstein’s estimate of h agreed with Planck’s, thereby supporting the tenets of quantum mechanics. Figure 2.6 shows the radiative flux �B� per unit wavelength emitted by blackbodies at various temperatures. �B� is related to �B� by �B� d� = �B� dv and is given by �B� =

(

�C1 C2

�5 e �T − 1

)

(2.22)

The visible part of the spectrum, 0.4–0.7 μm, is bracketed by dashed lines. The infrared window, 8–12 μm, is bracketed by dotted lines. Wavelengths shorter than 0.4 μm are referred to as ultraviolet. Wavelengths longer than 0.7 μm and shorter than about 5 μm are referred to as the near infrared. At wavelengths longer than about 5 μm, the radiation is referred to as the thermal infrared. The thermal infrared itself is loosely divided into the middle infrared and far infrared. The terminology is by no means fixed. Infrared is often used when referring to all wavelengths greater than 0.7 μm. Figure 2.6 shows that as the temperature of a blackbody rises above 500 K it becomes visible, first glowing a deep red at wavelengths near 0.7 μm, as do coals in a campfire. As the temperature climbs further, the color includes more yellow,

2.4

5783 K

1500 1000 500 0

Visible light

π Bλ (Wm–2 μm–1)

2000

Blackbody Radiation: Light as a Photon

1

2

3

4

5

Wavelength (μm)

(a)

0.08 0.06 0.04 0.02 0.00

(b)

Visible light

π Bν (Wm–2 cm)

0.10

1

2

3

4

5

Wavenumber (104 cm–1)

Figure 2.7 Distribution of blackbody emission representing the sunlight incident on the Earth as a function of wavelength (a) and as a function of wavenumber (b). The temperature of the blackbody is 5783 K and the irradiance has been adjusted to the top of the Earth’s atmosphere at 1 AU so that Q0 = 1360 W m−2 .

Visible light from 0.4 to 0.7 μm is bracketed by the dashed lines. The wavelength range shown in (a) is 0.2–5 μm and the corresponding wavenumber range in (b) is 2000–50 000 cm−1 . This spectral range covers approximately 99% of the incident sunlight.

mixing light with wavelengths shorter than 0.7 μm, and ultimately white, mixing light at all visible wavelengths. Emission by the sun is close to that for a blackbody at 6000 K. The incident sunlight peaks in the middle of the visible part of the spectrum. Moreover, emission by the Earth’s surface peaks in the infrared window of the Earth’s atmosphere. Visible stars are like the sun. They are at sufficiently high temperatures that they radiate substantial amounts of light at visible wavelengths. Their apparent colors indicate their temperatures. Planets, on the other hand, such as the Earth, emit in the infrared. Their apparent colors are due to reflected sunlight which, is independent of the planet’s radiating temperature. One should recognize that �B� is the radiative flux per unit wavelength while �B� is the radiative flux per unit wavenumber. The different distributions peak in different parts of the spectrum while representing the same phenomena. The difference is reflected in the different Wien’s displacement laws for the two distributions. The different distributions are also shown in Figures 2.7 and 2.8. Values of the two distributions for different temperatures, wavelengths, and corresponding wavenumbers are given in Table 2.1. Despite the differences in the distributions,

55

2 Radiation and Its Sources

288 K

25 20 15 10 5 0

Infrared window

π Bλ (Wm–2 μm–1)

30

20

40

60

80

100

Wavelength (μm)

(a) 0.5 0.4 0.3 0.2 0.1 0.0 (b)

Infrared window

π Bν (Wm–2 cm)

56

500

1000

1500

2000

2500

Wavenumber (cm–1)

Figure 2.8 Distribution of blackbody emission by the Earth’s surface at 288 K as a function of wavelength (a) and as a function of wavenumber (b). The 8–12 μm infrared window is bracketed by dashed lines. The panels show the distribution

from 4 to 100 μm in (a) and the corresponding wavenumbers, 100–2500 cm−1 in (b). This spectral range covers approximately 99% of the emission by the Earth’s surface.

the radiative flux obtained by integrating �B� over a particular wavelength interval is, of course, the same as that obtained by integrating �B� over the corresponding wavenumber interval. Values for the integral of the Planck function are given in Appendix C. Integrals of the Planck distribution, whether B� integrated over a particular wavenumber interval or B� integrated over a wavelength interval, are often used in atmospheric radiation when calculating wavenumber or wavelength averaged radiative fluxes and radiances. The integrals can be performed following the method used to integrate the Planck function over all wavelengths in Equation 2.20. When x = C2 �∕T is of reasonable magnitude, the series converges fairly rapidly so that large values of n are not needed for accurate results. In fact, when the spectral interval is such that throughout the interval x ≫ 1, Bx ∼ C1 x3 exp(−x) so that the series can be terminated at n = 1. The integral is then readily performed through successive integrations by parts. Moreover, when the spectral interval is such that throughout the interval x ≪ 1, then Bx ∼ C1 x2 , which is readily integrated.

2.5

Incident Sunlight

Table 2.1 Blackbody spectral irradiances for selected temperatures, wavelengths, and wavenumbers. Wavelength (�m)

0.25 0.50 0.80 1.00 1.25 2.00 2.50 4.00 5.00 4.0 5.0 8.0 10.0 12.5 20.0 25.0 40.0 50.0 4.0 5.0 8.0 10.0 12.5 20.0 25.0 40.0 50.0

�B� (W m−2 �m−1 )

Corresponding wavenumber (cm−1 )

Incident sunlight TSUN = 6000 K 479.90 40 000 1 837.27 20 000 1 106.35 12 500 690.31 10 000 389.50 8 000 93.19 5 000 43.96 4 000 8.22 2 500 3.59 2 000 Emission by the Earth TEARTH = 255 K 0.27 2 500 1.49 2 000 9.82 1 250 13.24 1 000 13.52 800 7.38 500 4.47 400 1.18 250 0.57 200 Emission by the Earth’s surface TSURFACE = 288 K 1.36 2 500 5.43 2 000 22.07 1 250 25.36 1 000 22.86 800 10.45 500 5.99 400 1.47 250 0.70 200

�B� (mW m−2 cm)

3.00 45.93 70.81 69.03 60.86 37.28 27.48 13.15 8.98 0.43 3.72 62.83 132.36 211.22 295.09 279.13 188.24 142.84 2.18 13.58 141.23 253.61 357.12 417.90 374.49 234.56 174.06

The values for the incident sunlight have been normalized to the top of the Earth’s atmosphere with Q0 = 1360 W m−2 .

2.5 Incident Sunlight

Incident sunlight affects all aspects of our environment. The solar radiative flux determines UV exposure, photosynthesis, photochemical reactions in the atmosphere, seasonal and diurnal temperature ranges, and so on. Incident sunlight clearly is the source of energy that largely determines surface and atmospheric temperatures.

57

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2 Radiation and Its Sources

As was noted in Chapter 1 (Figure 1.1), when the sun is not overhead, the radiative flux incident on a surface at an oblique angle with respect to the direct solar beam is given by F = Q cos �0

(2.23)

with Q the solar irradiance incident at the top of the atmosphere and �0 the solar zenith angle. The cosine of the solar zenith angle relates the area having its normal aligned with the direction of propagation, through which the light must pass, to the area of the surface on which the light falls. For simplicity, as discussed below, throughout this book the Earth’s orbit about the sun will be taken to be circular. Consequently, Q in Equation 2.23 will often be taken as the solar constant Q0 = 1360 W m−2 . Of course, for accurate calculations of solar radiances and irradiances, the departures from a circular orbit, while relatively small, have to be accounted for. The cosine of the solar zenith angle is easily computed for any location, time of day, and time of year. Assume that ̂ � is a unit vector drawn from the center of the Earth to the location on the Earth’s surface where the solar zenith angle is desired. Assume that ̂ �S is the unit vector from the center of the Earth to the center of the sun as shown in Figure 2.9. In the Earth’s coordinate system, the unit vector to any location on the Earth’s surface is given in terms of the latitude � and longitude � of the location as given by ⎛cos � cos �⎞ ̂ � = ⎜ cos � sin � ⎟ ⎜ ⎟ ⎝ sin � ⎠

(2.24)

The unit vector ̂ �S is similarly given in terms of latitude �S and longitude �S of the subsolar point, the location on the surface of Earth where the sun is directly overhead. The cosine of the solar zenith angle is given by the dot product of the unit vectors. cos �0 = ̂ � ⋅ ̂ �S = cos � cos �S cos h + sin � sin �S

(2.25)

with h the difference in longitudes � − �S and referred to as the hour angle. The name arises from the one-to-one correspondence between the time of day and longitude. Assuming that the Earth’s orbit around the sun is circular, the location of the spot on Earth directly underneath the sun is given by sin �S = sin � sin �(t − t0 )

(2.26)

with � = 23.45∘ the obliquity of the Earth’s orbit. It is given by the angle between the Earth’s axis of rotation and the normal to the plane of the Earth’s orbit. In Figure 2.9 the gray section represents the orbital plane as it slices through the Earth. The angular frequency and phase in Equation 2.26 are given by �=

360∘ 365 days

and t0 = 81 days

(2.27)

2.5

Normal to orbital plane

Incident Sunlight

North pole xˆ δ

θ0 xˆ S Direction to sun

ne

l pla

ita Orb

θ

θS

ϕ ϕS

90° East

Equator Greenwich meridian 0°

Figure 2.9 Geometry for determining the solar zenith angle at latitude � and longitude �. The obliquity of the Earth is indicated by the angle �. The unit vector ̂ � is for the direction between the center of the Earth and the location at which the solar

zenith angle is desired. The unit vector ̂ �S is in the direction of the vector that joins the center of the Earth to the center of the sun. The shaded disk is a section of the Earth’s orbital plane as it slices through the Earth.

which approximately captures the sun directly over the equator on the March 21 and September 21 equinoxes. The angular position of the sun in the Earth’s coordinate system is given by a three-axis rotation that allows for the motion of the Earth around the sun to be coupled with the obliquity of the Earth’s orbit and the Earth’s rotation about its axis. Since the direction of the Earth’s rotation axis is approximately fixed in space, the same result is obtained by assuming that the sun orbits the Earth while the Earth spins on its axis. In this case, the Earth’s coordinate system becomes the right ascension–declination coordinate system used in astronomy with �S the declination angle of the sun and �S the sun’s right ascension [6]. The right ascension is measured from the position of the sun at the vernal equinox, March 21. The obliquity of the Earth’s orbit varies slowly with time (∼40 000 year period) from 22.1∘ to 24.5∘ and its periodic variation is in part thought to be responsible for the cyclical ice ages. The last glacial maximum, 20 000 years ago, is thought to have arisen from the small obliquity, 22.2∘ , 25 000 years ago. Additional cyclical variability in the ice ages stems from the Earth’s other orbital parameters. The Earth’s orbit is not circular but slightly elliptical. The eccentricity is small, 0.016 375. The largest it has been is ∼0.06. Its amplitude varies with a period of

59

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2 Radiation and Its Sources

approximately 100 000 years. The Earth is closest to the sun at perihelion in January and furthest from the sun in July. The timing of the perihelion with respect to the Northern Hemisphere’s spring equinox is also cyclical and has a period that ranges between 19 000 and 23 000 years. Owing to the eccentricity of its orbit, the Earth requires less time to travel through the half of the orbit associated with the Northern Hemisphere winter, September 21 through March 21 = 181 days, than through the half of the orbit associated with summer, March 21 through September 21 = 184 days. Ever wondered why February has only 28 days? The expressions given above do not allow for this slight difference in time for the summer half and the winter half years of the two hemispheres and are therefore, only approximate. The circular orbit approximation gives rise to errors in the noontime cosine of the solar zenith angle that are less than 0.02 in absolute value, or less than about 5% for latitudes 60∘ S to 60∘ N. Methods for accurately calculating values of the solar zenith angle are readily available [7]. The day–night average of the incident sunlight is determined by the cosine of the solar zenith angle and the number of daylight hours, which in turn, are determined by the time of year and the latitude at which the incident sunlight is to be calculated. The cosine of the solar zenith angle averaged over day and night is obtained by averaging the cosine of the zenith angle over daylight hours and dividing by 2� for the 24 h in a day. The average is given by H

cos �0 =

1 dh cos � cos �S cos h + sin � sin �S 2� ∫−H

1 [cos � cos �S sin H + H sin � sin �S ] (2.28) � with H the hour angle between sunrise or sunset and noon for the location where the solar zenith angle is desired. At sunrise and sunset, the solar zenith angle �0 = 90∘ so that =

cos �0 = cos � cos �S cos(±H) + sin � sin �S = 0

(2.29)

which yields cos(±H) = − tan � tan �S

(2.30)

When |sin � sin �S | ≥ cos � cos �S , the region is either in total darkness, sin � sin �S < 0, or 24 h of sunlight, sin � sin �S > 0. These conditions occur in the polar regions, latitudes for which |�| ≥ 90∘ − |�S | ≥ 90∘ − � = 90∘ − 23.45∘ =66.55∘ . The solar zenith angle is readily obtained from simple geometry under the rather restrictive conditions of noontime conditions for the equinoxes and solstices. On the equinoxes, the sun is over the equator and the duration of daylight equals the duration of darkness at all latitudes. Likewise, simple geometry dictates that the durations of day and night are always equal at the equator. Interestingly, owing to the duration of daylight, the daily average sunlight incident on the summer pole exceeds that incident on the tropics even for the equinoxes when the sun is directly over the equator. The simple procedures adopted here ignore the impositions of geopolitically defined time zones. Changes in the local timing of sunrise and sunset caused by

Problems

Table 2.2 Day–night average incident sunlight (W m−2 ) and daylight hours for summer and winter solstices. Latitude

Hours of sunlight Summer

10∘ 40∘ 70∘

12.6 14.8 24.0

Winter 11.4 9.2 0.0

Incident sunlight (W m−2 ) Summer

Winter

439 499 509

345 151 0

The quantities were obtained using a circular orbit of the Earth around the sun.

the location within a geopolitical time zone can be calculated straightforwardly. As the apparent sunrise and sunset are also affected by refraction, which in turn is mostly determined by the thermal structure of the lower atmosphere, the apparent times may differ by a matter of minutes from those calculated using even more accurate methods. Table 2.2 gives the daylight hours and day–night average incident sunlight for three latitudes for the summer and winter solstices. The eccentricity of the earth’s orbit alters the distance between the Earth and the sun, thereby modulating the incident solar radiative flux. Nonetheless, the shorter time spent nearest the sun cancels the effect of the longer time spent further from the sun so that, annually averaged, each latitude zone in the two hemispheres receives exactly the same amount of sunlight. The annual averages of the incident sunlight for latitudinal zones of the Southern Hemisphere are identical to those for the corresponding latitudinal zones of the Northern Hemisphere in Table 1.4. In addition, the average zenith angle and the number of daylight hours shown in Table 2.2 for the summer and winter solstices are identical at the corresponding latitudes in both hemispheres.

Problems

1. Sunlight at 0.64 μm reflected by the Earth is absorbed by a nadir viewing satellite sensor at an altitude of 800 km. The scene viewed by the sensor is a circle with radius 1 km and the sensor has a bandwidth of 0.1 μm centered at 0.64 μm. The sensor has a circular detector with a radius of 1 mm. If the scene being viewed has an albedo of 0.25, reflects sunlight isotropically, and the solar zenith angle is 60∘ , what is the power (W) of reflected sunlight absorbed by the sensor’s detector? Assume that the sun radiates similarly to a blackbody at 6000 K and that the sunlight incident at the top of the atmosphere is given by the solar constant Q0 = 1360 W m−2 . 2. Radiation emitted by a cloud layer is measured by a satellite radiometer and found to be 82 mW m−2 sr−1 (cm−1 )−1 at 11 μm. If the cloud is assumed to emit

61

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2 Radiation and Its Sources

similarly to a blackbody and the atmosphere above the cloud is assumed to be transparent at 11 μm, what is the temperature of the cloud top? 3. .a. Calculate numerically to three significant figures the wavelength at which the global, annual average, incident solar radiative flux per unit wavelength, matches the flux of radiation emitted by the Earth’s surface. Assume that the sun radiates similarly to a blackbody at 6000 K and the Earth’s surface emits similarly to a blackbody at 288 K. Assume that the global, annual average, incident sunlight is 340 W m−2 and calculate the crossover wavelength. To perform these calculations, use a spreadsheet or interactive computer software to iterate solutions for the crossover wavelength in order to obtain the desired accuracy. b. Use the large wavelength limit of the Planck function to determine the fraction of the incident sunlight that is at wavelengths larger than the wavelength obtained in (a). The large wavelength limit is probably a crude approximation in this case. Compare your result with that given in Appendix C for integrals of the Planck function. Linearly interpolate the values given in the appendix to obtain the desired accuracy. c. Use the short wavelength limit of the Planck function to determine the fraction of the radiation emitted by the Earth’s surface that is at wavelengths shorter than that given in (a). d. Repeat (a) through (c) but for fluxes per unit wavenumber. 4. .a. Use the Planck function to analytically show that for any temperature T �MAX T = CONSTANT with �MAX the wavelength at which the Planck function B� reaches the maximum radiance emitted by a blackbody at that temperature. This result also applies to Wien’s displacement law. Determine the constant to three significant figures for �MAX given in micrometers and the temperature in kelvin. b. Repeat the analysis in (a) to show that B� reaches the maximum radiance when �MAX = CONSTANT T and determine the constant to three significant figures for �MAX given in cm−1 and the temperature in kelvin. c. Find the wavelengths at which the emission per unit wavelength reaches a maximum for the sun (T = 6000 K) and for the Earth’s surface (T = 288 K). d. What are the wavenumbers associated with the wavelengths of the maxima in (c)? e. Repeat (c) and (d) but for the wavenumbers at which the emission per unit wavenumber reaches a maximum for the sun and for the Earth’s surface. 5. Calculate the fraction of radiation emitted by the surface that passes through the 8–12 μm window. Assume that the surface temperature is 288 K and that the atmosphere is completely transparent in the window.

References

a. Use the large wavenumber approximation of the Planck function to integrate the emitted radiation over the wavelengths spanned by the window. b. Compare the value obtained in (a) with that obtained by assuming that the Planck function is constant, equal to its value at 10 μm in the window region. What error is incurred by assuming a constant value for the Planck function? c. Compare the result in (a) with the transmissivity of the atmosphere as inferred from Figure 1.7. On average, including both cloudy and cloudfree skies, about half the radiation passing through the infrared window is absorbed by clouds that emit at a temperature lower than the surface temperature. d. If the loss through the 8–12 μm window under cloud-free conditions were confined to the lowest 100 m thick atmospheric layer, how rapidly (K/day) would the layer cool? 6. .a. Calculate the maximum and minimum noontime and day–night average incident solar radiative fluxes for your location. b. Compare the maximum and minimum with today’s noontime incident solar radiative flux and today’s day–night average flux. c. Compare the fluxes in (a) and (b) with radiative fluxes emitted by the surface using typical values for the maximum surface temperatures associated with the times in (a) and (b). 7. “Flat plate” radiometers were the first instruments used to measure the Earth’s albedo and the flux of infrared radiation emitted by the Earth from satellites. The field of view of these instruments was completely filled by the Earth’s disk. a. Show that the solid angle subtended by the Earth viewed by a flat plate radiometer from a satellite is given by ( )1 ⎡ 2RE h + h2 2 ⎤ ⎥ Ω = 2� ⎢1 − ⎢ ⎥ RE + h ⎣ ⎦ with h the altitude of the satellite above the Earth’s surface and RE the radius of the Earth. b. If all parts of the Earth emit isotropically and the emitted radiance is everywhere I = CONSTANT, derive an expression in terms of RE , h, and I for the radiative flux absorbed by the flat plate detector. Assume that the detector is black and absorbs all radiation incident on it.

References 1. Jackson, J.D. (1999) Classical Electrody-

namics, John Wiley & Sons, Inc., New York. 2. Bohren, C.F. and Clothiaux, E.E. (2006) Fundamentals of Atmospheric

Radiation: An Introduction with 400 Problems, Wiley-VCH Verlag GmbH, Weinheim. 3. Haeffelin, M.P.A., Mahan, J.R., and Priestley, K.J. (1997) Predicted dynamic

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2 Radiation and Its Sources

electrothermal performance of thermistor bolometer radiometers for Earth radiation budget applications. Appl. Opt., 36, 7129–7142. 4. Loeb, N.G., Wielicki, B.A., Doelling, D.R., Smith, G.L., Keyes, D.F., Kato, S., Manalo-Smith, N., and Wong, T. (2009) Toward optimal closure of the Earth’s top-of-atmosphere radiation budget. J. Clim., 22, 748–766.

5. Goody, R.M. and Yung, Y.L. (1989) Atmo-

spheric Radiation, Oxford University Press, New York. 6. Kidder, S.Q. and Vonder Haar, T.H. (1995) Satellite Meteorology—Introduction, Academic Press, San Diego, CA. 7. Hartmann, D.L. (1994) Global Physical Climatology, Academic Press, New York.

65

3 Transfer of Radiation in the Earth’s Atmosphere For the Earth, the incident sunlight dominates the spectrum for wavelengths less than about 4 μm and the emitted terrestrial radiation dominates the spectrum for wavelengths greater than about 4 μm. As a result, one often encounters two distinct forms of the equation for the transfer of radiative energy: one for the solar part of the spectrum, which accounts for absorption and scattering, and one for the terrestrial part of the spectrum, which accounts for absorption and emission. The two forms depict two limits of a single equation for the transfer of radiation in a medium that scatters, absorbs, and emits.

3.1 Cross Sections

The transfer of radiation through a medium is a statement of what happens to the light. Radiation propagating in a particular direction is diminished by the scattering of light out of the direction of propagation and by absorption. Radiation is enhanced by the scattering of light from other directions into the direction of propagation and by emission. The interaction of radiation with matter is quantified through the use of cross sections. A cross section is the area that an object presents to the radiation field. Large cross sections imply strong interactions with the radiation field; small cross sections imply weak interactions. The equation of radiative transfer presented here is strictly applicable to monochromatic radiation. In particular, the processes included in the equation do not affect the frequency of the radiation. Processes such as Raman scattering, in which the scattered or induced radiation is not at the same frequency as the incident radiation, have cross sections that are generally much smaller than those for absorption, emission, and elastic scattering. The term elastic scattering means that there is no difference between the frequencies of the incident and scattered light. Processes such as Raman scattering are observed in the atmosphere only with detectors specifically designed to measure the weak spectral signatures associated with the particular process. Cross sections must be specified for each object in the medium, such as a particle, a molecule, or a cloud droplet. For an infinitesimal cylindrical volume, the Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

66

3 Transfer of Radiation in the Earth’s Atmosphere

d

I

I + dI

σ(D) dA

Figure 3.1 Extinction cross sections and the attenuation of radiation as it passes through an infinitesimal cylindrical medium. Within the cylinder there are dn = ndAd� targets. Each target has a cross section that blocks area �(D) of the incident light with D, a linear dimension such as diameter that characterizes the physical size of the target. The light passing through the medium can be attenuated through absorption and scattering. Scattering alters the direction of the light. The scattering is suggested by the

random rays of light exiting the cylinder. The light passing through the cylinder can be augmented by light entering it from all directions. For example, imagine the arrows of the exiting rays turned around so that they enter the cylinder. If this light is scattered into the direction of propagation, it augments the radiance in the direction of propagation. The light can also be augmented through emission within the cylinder.

energy lost from the radiation field is proportional to the area blocked by the various objects. The cylinder represents a section of the cone encompassed by the angular domain of the solid angle associated with the propagation of the radiance. In Figure 3.1 the cross sections are indicated by filled circles. The cross-sectional area of each target is represented by �(D) with D representing a linear dimension such as diameter, indicating particle size. The cross section for the interaction of light with particles is dependent on the size and shape of the particle. The relationship, however, is often complex. The area blocked is given by the number of radiatively active targets per unit area of the beam times the average crosssectional area of the targets, �. The average cross-sectional area is given by ∑ 1∑ �= nj � j , with n = nj (3.1) n j j nj being the number of targets per unit volume of type j, such as a particular molecule, or type of haze particles, soil, ash, and so on, or cloud droplets and ice crystals, referred to as hydrometeors; and � j the average cross section associated with the targets of type j. Because they block the radiation, the cross sections are referred to interchangeably as either “total cross sections” or “extinction cross sections.” The number of targets per unit area illuminated by the light is given by the number of targets per unit volume times the length of the medium through

3.1 Cross Sections

which the light passes, nd�. The attenuation of the light is thus proportional to the fraction of the area illuminated by the light that is blocked, �nd�. For thin haze layers on cloudless days, the fraction of the solar beam blocked is typically small, less than 0.1. For rain clouds, the fraction can be huge, 20–100, meaning that the beam is blocked numerous times as it wends its way through the cloud. That light makes it through rain clouds at all is the result of scattering. Ice crystals and water droplets change the direction of visible light while absorbing almost none of it. In Figure 3.1 the scattering is represented by the different paths of light leaving the cylindrical volume. The product of the number of targets per unit volume and the cross sections gives the extinction coefficient. The coefficient has units of inverse length. The extinction coefficient is the fraction of the area of the incident beam that is blocked per unit length of the medium. For an infinitesimally thin medium, the attenuation of the radiation field is given by the product of the incident radiance, the extinction coefficient �, and the length of the medium through which the radiation passes. dI = −I�d�

(3.2)

In the case of a particular type of molecule, the extinction coefficient is simply related to the cross section by �j = nj �j

(3.3)

For hydrometeors or types of haze particles, the coefficient is related to the average cross section by Dmax

�j =

∫Dmin

nj (D′ )�j (D′ )dD′

with Dmax

Nj =

∫Dmin

nj (D′ )dD′

so that �j = Nj � j

(3.4)

In Equation 3.4 Nj is the total number of targets per unit volume of particle type j, nj (D) is the number of particles per unit volume per unit size with a physical dimension of D within an infinitesimal size bin dD. The minimum and maximum sizes of particles responsible for the extinction are Dmin and Dmax . In addition, the average cross section for particle type j in Equation 3.4 is given by D

�j =

max 1 n (D′ )�j (D′ )dD′ Nj ∫Dmin j

(3.5)

From the definition of the extinction coefficient, Equation 3.2 can also be written as dI = −In�d�

(3.6)

67

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3 Transfer of Radiation in the Earth’s Atmosphere

with n representing the total number of targets in the medium and � the average cross section for the medium as in Equation 3.1. For a homogeneous medium, n and � are constant throughout. Integrating Equation 3.6 for a homogeneous medium of length L produces the exponential law for the extinction of monochromatic light. I = I0 exp(−n�L)

(3.7)

with I0 being the incident radiance and exp(−n�L) the transmissivity of the medium. The attenuation of monochromatic light as given by Equation 3.6 is often referred to as the differential form of the “Beer-Bouguer-Lambert law.” The integral form is given by Equation 3.7. As is discussed in Section 3.4, n�L is dimensionless and is referred to as the optical thickness. The attenuation of light is quantified by the products of the extinction cross sections and number densities of the scattering and absorbing targets, whether they are molecules, particles, or hydrometeors. When reporting extinction in the literature, neither the number densities nor the extinction cross sections may be known. Nonetheless, measurements of extinction lead to estimates of extinction coefficients. Typically, the depth of a layer, such as a layer of haze, is known. Measurements of extinction would then provide an estimate of the extinction per unit length. If the number density of particles were measured and the cross sections calculated from theory, then Equations 3.1–3.5 would provide an alternate estimate that could be compared with the extinction coefficient inferred from measurements of extinction. Sometimes, the coefficients are specified per unit mass of the attenuating substance. If number concentrations and density of the targets can be determined along with their extinction cross sections, then in terms of a mass extinction coefficient �m , Equation 3.6 becomes dI = −I��m d�

(3.8)

with density � = nm, m the average mass of the attenuating targets, and �m = �∕m so that ��m = nm�∕m = n�. Table 3.1 lists some typical cross sections encountered in the atmosphere. For comparison of the relative sizes, the geometrical cross section of a molecule is ∼ 10−16 cm2 , much larger than the cross section for scattering of light by molecules or the relatively large cross section for the absorption of infrared radiation by carbon dioxide. Even though the cross sections in Table 3.1 appear to be tiny, the number of targets can be sizeable, thereby causing significant attenuation. Examples of attenuation are listed in Table 3.2.

3.2 Scattering Cross Section and Scattering Phase Function

As for extinction, cross sections and coefficients can be defined for scattering and absorption. The extinction cross section, or total cross section, is simply the sum of the scattering and absorption cross sections. In addition, for scattering

3.2

Table 3.1

Scattering Cross Section and Scattering Phase Function

Typical cross sections.

Target

Cross section (cm2 )

Scattering by molecules, Rayleigh scattering, at 0.5 μm for air at STP Aerosol particles (with radii of r ∼ 0.1–1 μm) at visible wavelengths Cloud droplets (with radii of r ∼ 10 μm) at visible and near infrared wavelengths Absorption in the 15 μm band of CO2

6.1 × 10−27

Table 3.2

� ∼ �r2 ∼ 3 × 10−9 � ∼ 2�r2 ∼ 10−5 4 × 10−21

Examples of optical thickness.

Process

Optical thickness (�nL)

Attenuation due to Rayleigh scattering by the atmosphere at 0.5 μm Attenuation at visible wavelengths due to a 2 km thick layer of haze with 100 particles/cm3 Attenuation for 500 m thick marine stratus with 100 droplets/cm3 Attenuation in 15 μm absorption band of CO2 for an atmospheric concentration of 390 ppm

6.1 × 10−27 cm2 × 2.2 × 1025 mol.cm−2 ∼ 0.13 3 × 10−9 cm2 × 2 × 107 particles cm−2 ∼ 0.06

10−5 cm2 × 5 × 106 droplets cm−2 ∼ 50 4 × 10−21 cm2 × 8.4 × 1021 mol.cm−2 ∼ 34

the strength of the interaction depends on the angle through which the radiation is scattered. In this book, for simplicity, the objects that scatter are limited to spheres. For spheres, the angular dependence of the scattering is specified as a function of scattering angle; the angle between the incident light and the scattered light as is shown in Figure 3.2. The directions of propagation for the scattered light and the incident light define the “plane of scattering,” the xy plane in Figure 3.2a. Scattering by spheres is rotationally invariant about the direction of the incident light and is thus independent of the orientation of the plane of scattering. In general, of course, particles in the atmosphere, such as dust and ice crystals, are not spheres. Random collections of such particles, however, also exhibit the rotational invariance of scattering by spheres. In the case of scattering by oriented nonspherical particles, such as a hexagonal ice crystal with incident light normal to one of its sides, the scattering depends on the angle of the scattering plane, � in Figure 3.2b, with respect to the particle’s axis of symmetry, the z axis in the figure.

69

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3 Transfer of Radiation in the Earth’s Atmosphere

y

Scattering plane Θ x

(a)

Scattered light

Incident light z z′ φ

y′ y

Scattering plane Θ

x, x ′

(b) Figure 3.2 Scattering plane and scattering angle Θ for light scattered by spheres. In (a) the scattering plane is the xy plane, the plane of the page. In (b) an oblique view of a laboratory xyz coordinate system is shown in which the x′ y ′ z′ coordinate system of the scattering plane has been rotated through � about the direction of propagation for

the incident radiance. The direction of propagation is along the x axis of the laboratory system which for simplicity is shown to be aligned with the x ′ axis of the scattering plane. Scattering that is spherically symmetric is unaffected by rotations through arbitrary values of �.

The angular distribution of scattering is accounted for by the differential scattering cross section given by dΩ (3.9) 4� with Θ being the scattering angle, �SCAT the scattering cross section, P(Θ) the scattering phase function, and dΩ an infinitesimal increment of solid angle into which the radiation is scattered. In terms of the scattering phase function, the fraction of the incident light that is scattered from the incident direction into a direction confined within an infinitesimal element solid angle dΩ is given by P(Θ)dΩ∕4�. The scattering phase function is recognized as the probability of scattering through scattering angle Θ per unit solid angle within a solid angle domain d�(Θ) = �SCAT P(Θ)

3.3

Elementary Principles of Light Scattering

of 4�. As such, the phase function in Equation 3.9 is normalized so that the integral of the differential scattering cross section over all scattering angles gives the area blocked by the scattering cross section. The normalization condition is given by ∫4�

dΩ 1 P(Θ) = 4� 4� ∫0

2�

�

1

1 d� d�P(�) = 1 dΘ sin ΘP(Θ) = ∫0 2 ∫−1

(3.10)

with � = cos Θ.

3.3 Elementary Principles of Light Scattering

When a beam of radiation propagates from one continuous medium to another, the direction of propagation is altered according to Snell’s law for refraction given by mk sin �k = m� sin ��

(3.11)

with mk being the index of refraction of the medium through which the incident beam passes, �k the angle of incidence as illustrated in Figure 3.3, m� the index of refraction of the medium into which the beam propagates, and �� the angle of refraction. The angle of refraction is usually given as sin �� = sin �k ∕mREL , with mREL the relative index of refraction given by the ratio of the indices of refraction for the two media, mREL = m� ∕mk . Figure 3.3 shows an example of refraction, light passing from air into water. Since the index of refraction for air at visible wavelengths is close to unity, the relative index of refraction is that for water, mREL = m� = 1.33. Since mREL > 1, �� < �k . In general, the refractive index is a complex quantity, m = mr + imi , with mr being the real part and mi the imaginary part of the refractive index. Depending on the choice of the time evolution of the electromagnetic wave used to represent light, either exp(−i�0 t) or exp(i�0 t), the imaginary part of the index of refraction

θk Air mk = 1 Water m = 1.33 θ

Figure 3.3 Refraction of light passing from air into water.

71

72

3 Transfer of Radiation in the Earth’s Atmosphere

mi is either positive or negative. Of course, once a sign has been chosen, it must be used consistent throughout the analysis. As an example, consider a plane electromagnetic wave propagating along the z axis within an unbounded medium that has a complex refractive index. The solution of Maxwell’s equations gives the time and space dependence for the amplitude of the electric field vector as i(mr + imi )2�z −mi 2�z imr 2�z �0 E(z, t) = E e e−i�0 t = E e �0 e �0 e−i�0 t (3.12) 0

0

with E0 being the amplitude of the electric field vector and �0 the wavelength in vacuum. Because the wave is propagating in the z direction, the electric field vector � is perpendicular to the z axis. According to Equation 3.12, the electric field decays with distance along the z axis. The rate of decay is given by the factor e−mi 2�z∕�0 . The spatial oscillation of the wave is given by eimr 2�z∕�o . In the medium, the wavelength is given by � = �0 ∕mr . The wavelength is shortened compared with its value in vacuum. The power associated with the electromagnetic wave (W m−2 ), is given by Bohren and Huffman [1] −mi 4�z −mi 4�z (√ (√ ) ) 1 � � 1 E02 e �o � ⋅ �∗ = Re = F0 e �o (3.13) F = Re 2 � 2 � with the asterisk meaning complex conjugate, � the permittivity of the medium, and � the permeability of the medium. For absorbing media, the permittivity and permeability are complex. They are related to the speed of light and the index of refraction by √ √ �� (3.14) m = c �� = �0 �0 with �0 and �0 being the permittivity and permeability in free space introduced in Equation 2.1. F0 in Equation 3.13 is the power at the origin z = 0. The imaginary part of the refractive index determines the absorption in the medium. The real part of the refractive index determines the phase of the wave. The scattering, absorption, and polarization properties of a dielectric particle depend not only on the refractive index but also on the size of the particle relative to the incident wavelength. In the case of a sphere, the dimensionless quantity is the “size parameter,” given by x = 2�r∕� with r the radius of the sphere. For the plane wave in Equation 3.12 the equivalent parameter is 2�z∕�. Figure 3.4 shows the angular distribution of the radiative energy scattered by spheres with size parameters x = 0.01, 10, 100, and 1000. The pattern becomes more complicated and asymmetric as the size of the particle increases. Furthermore, in the case of large size parameters, x = 100 and 1000, the scattered energy is strongly peaked in the forward direction. In fact, in order to show any pattern at all for the large particles, the scattered radiances shown in Figure 3.4 have been logarithmically scaled. For large particles, nearly half the radiation is scattered in the direction of the forward peak. The remainder is distributed over the remaining 180∘ of the scattering angle.

3.3

x = 0.01

Elementary Principles of Light Scattering

x = 10 P(Θ) Θ

(a)

(b) x = 100

(c)

x = 1000

(d)

Figure 3.4 (a–d) Distribution of radiation scattered by spheres having size parameters x = 0.01, 10, 100, and 1000. The radial distances of the pattern from the center of the particle are used to indicate the logarithm of the scattered radiances as a function of scattering angle Θ. The logarithmic scale is used to show the scattering patterns for the large particle sizes along with the peak of the scattered radiances in the forward

scattering direction, Θ = 0∘ . Without this scaling the scattering pattern for angles outside the domain of the peak would have to be rendered as tiny dots in order to keep the forward scattering peaks on the same page. The refractive index m = 1.33 was used to calculate the scattering patterns for the spheres. This refractive index is close to that of water at visible wavelengths.

When a particle is introduced into the path of an electromagnetic wave, it partially blocks the wave and the wave is diffracted around the particle. For spheres, the diffraction pattern is similar to that associated with an opaque circular disk, which in turn is the same as that associated with an opening in an opaque screen perpendicular to the incident direction of the electromagnetic wave. This property of diffraction is known as Babinet’s principle [2]. The angular pattern of the diffracted power associated with a circular opening is given by Van de Hulst [3] ] [ 2J1 (x sin �) 2 x4 F (�) = F0 2 2 (3.15) 4k R x sin � with k = 2�∕�0 , x being the size parameter given by x = kr with r the radius of the circular opening, F0 the incident radiant power, R the distance between the center of the circular opening and the point of observation, and J1 the first-order Bessel function. The pattern given by Equation 3.15 is for the far field, R ≫ �0 . The angular distribution of the diffraction is determined by the factor [2J1 (x sin �)∕(x sin �)]2 . Figure 3.5 shows this angular factor for three size parameters x = 10, 100, and 1000. The diffracted energy is increasingly concentrated in the forward direction as the particle size increases. For large particles, the width of the forward peak is approximately proportional to the inverse of the size parameter x.

73

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3 Transfer of Radiation in the Earth’s Atmosphere

1

1

(2 J1 (x sin Θ) /(x sin Θ))2

0.9

0.9

x = 100

0.8

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0 (a)

1

0.9

x = 10

0

10 Scattering angle, Θ (°)

20

0 (b)

0

2 Scattering angle, Θ (°)

4

0 (c)

x = 1000

0

0.5

1

Scattering angle, Θ (°)

Figure 3.5 (a–c) Angular diffraction pattern for a circular opening in an opaque screen. Patterns are shown for three different size parameters.

Diffraction explains the large forward peaks in the scattering patterns shown in Figure 3.4. For large spheres, such as the cloud droplets listed in Table 3.1, the total cross section at visible wavelengths is approximately twice the geometrical cross section, 2�r 2 . Diffraction accounts for approximately half of the cross section. Scattering outside the domain of the diffraction peak accounts for the remaining portion of the cross section. The relationship between the geometric and total cross sections holds even for strongly absorbing spheres. For large strongly absorbing spheres, nearly half of the extinction cross section accounts for absorption and most of the remaining half accounts for diffraction. For isotropic scattering the probability of scattering is the same for all directions. The phase function is constant, P(Θ) = 1. Isotropic scattering is an idealization. It does not occur in the atmosphere. The simplest form of scattering in the atmosphere is the scattering of light by molecules. The angular dependence of scattering by molecules is given by the Rayleigh phase function, 3 P(Θ) = (1 + cos2 Θ) (3.16) 4 The Rayleigh phase function is named in honor of Lord Rayleigh (John William Strutt). In 1871, he provided the first correct explanation for the blue sky. The pattern of scattering given by Equation 3.16 is associated with any object that has a radius r that is much smaller than the wavelength of the radiation, 2�r∕� ≪ 1. For example, the phase function also applies to the scattering of microwave radiation, � ∼ 3 cm, by rain droplets, r ∼ 0.2–2 mm. The angular pattern associated with the Rayleigh phase function is shown in Figure 3.4a. The pattern is obviously symmetric in the forward, Θ < 90∘ , and backward, Θ > 90∘ , scattering directions. Figure 3.6a also shows the Rayleigh phase function along with the constant phase function for isotropic scattering.

3.3

Elementary Principles of Light Scattering

2.0 Rayleigh phase function Isotropic phase function

1.5

1.0

(a) 0.5 Phase function P(Θ)

103

(b)

10

Ave. cont. aerosol Henyey–Greenstein g = 0.684

2

10 1 10–1 104

Cloud droplet, r =10 μm Henyey–Greenstein g = 0.862

102

1 10–2 (c)

0

50

100

150

Scattering angle

Figure 3.6 Phase functions for Rayleigh and isotropic scattering (a), scattering of light at 0.65 μm by particles typical of a continental haze in air at a relative humidity of 70% (b), and by a distribution of cloud droplets having a mode radius of 10 μm (c).

The phase functions in (b) and (c) are calculated assuming distributions of spheres and using Mie theory. Henyey–Greenstein phase functions with the same asymmetry parameters as those of the Mie phase functions are also shown.

Solutions of Maxwell’s equations for the scattering of light by dielectric spheres were obtained in the late nineteenth and early twentieth centuries by Ludvig Lorenz, Gustav Mie, and Peter Debeye. The resulting theory is known as the Lorenz–Mie theory, or more commonly, simply as Mie theory. The scattering of a plane electromagnetic wave by a dielectric sphere is determined by the complex index of refraction for the medium and the size parameter 2�r∕�. The solution is an infinite series of associated Legendre polynomials and Bessel functions [3]. It is used to calculate the scattering and absorption cross sections and the scattering

75

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3 Transfer of Radiation in the Earth’s Atmosphere

phase function. All of these quantities depend on the polarization of the incident and scattered light and the theory accounts for these effects. While Mie theory is often used to calculate the scattering properties of particles in the atmosphere, many particles are not spherical. Attempts to approximate the scattering properties of nonspherical particles in terms of those for “equivalent spheres” have been shown to cause significant errors in the scattered radiances. These errors lead to gross misinterpretations of the particle properties inferred from the scattered light. Consequently, during the past several decades substantial efforts have been devoted to methods for calculating the scattering properties of nonspherical particles, including particles composed of heterogeneous mixtures of dielectric materials [4, 5]. Scattering by nonspherical particles is an advanced topic in scattering theory. In this book, scattering will be treated using a simple, but useful, analytic approximation to the scattering patterns. Instead of resorting to complicated calculations such as those in Mie theory, an analytic approximation known as the Henyey–Greenstein phase function is often used. The Henyey–Greenstein function is given by 1 − g2

P(Θ) =

(3.17)

3

(1 + g 2 − 2g cosΘ) 2 with g the asymmetry parameter. This parameter is a measure of the forward to backward asymmetry of the scattering pattern. The parameter is the average of the cosine of the scattering angle as determined by the scattering phase function. It is given by �

g = ⟨cosΘ⟩ =

1

1 1 cosΘP(Θ) sinΘdΘ = � P(�)d� 2 ∫0 2 ∫−1

(3.18)

with � being the cosine of the scattering angle. For scattering concentrated in the forward direction, Θ → 0∘ , g → 1. Forward scattering is common for atmospheric particles at visible wavelengths and for hydrometeors at both visible and infrared wavelengths. For scattering concentrated in the backward direction, Θ → 180∘ , g → −1. For phase functions that have symmetric forward and backward scattering patterns, such as the Rayleigh phase function, the asymmetry parameter g = 0. At visible wavelengths, typical values of g are 0.85 for scattering by droplets in water clouds and 0.6–0.8 for scattering by aerosol particles and also by ice crystals in glaciated clouds. In the case of ice crystals the asymmetry parameter depends on the crystal habit, as indicated by the shape of the crystal and the degree of surface roughness. The phase function for the scattering of sunlight at 0.65 μm by a distribution of particles typical of a continental haze in air with a relative humidity of 70% is shown in Figure 3.6b. The phase function is calculated using Mie theory. It is calculated for an ensemble of particles ranging in radius from 0.01 μm for soot to 20 μm for airborne dust. Soot and dust are among the particles that make up continental hazes. The phase function shown in the figure is an average of the phase functions for all particle sizes and types that contribute to continental hazes [6]. The averaging of the phase function is similar to that given for cross sections in Equation 3.5

3.4

Equation of Radiative Transfer

(see also Chapter 3, Problem 7). Because the sizes of many of the particles are comparable to the wavelength of the radiation, the phase function is strongly peaked in the direction of forward scattering. The Henyey–Greenstein phase function having the same value of the asymmetry parameter as that for the Mie phase function is also shown in Figure 3.6b. Notice that while the Henyey–Greenstein phase function fails to match the large value of the phase function for forward scattering, Θ = 0∘ , it does a fair job of following the phase function over a range that spans several orders of magnitude. The phase function for the scattering of sunlight at 0.65 μm by a distribution of cloud droplets having a mode radius of 10 μm is shown in Figure 3.6c. As with the phase function for the continental haze, that for the cloud droplets represents an average for the range of droplet radii that contributes to the scattering, 0.1 μm < r < 30 μm. The corresponding Henyey–Greenstein phase function is also shown. Typical cloud droplets have radii that are much larger than the 0.1 μm < r < 1 μm radii typical of aerosol particles that contribute significantly to scattering. Consequently, the phase function for cloud droplets is more strongly peaked in the forward direction than that for aerosol particles. 3.4 Equation of Radiative Transfer

There are two mechanisms that attenuate light as it passes through a medium: scattering and absorption. Likewise, there are two mechanisms that augment light: scattering and emission. For radiation passing through a cylindrical volume of infinitesimal length d�, as in Figure 3.1, the change in the radiance is given by dΩ′ p(Ω′ , Ω)I(Ω) ∫4� 4� dΩ′ p(Ω, Ω′ )I(Ω′ ) + n� ABS d�B(T) + n� SCAT d� ∫4� 4�

dI(Ω) = −n� ABS d�I(Ω) − n� SCAT d�

(3.19)

with Ω representing the direction of the incident radiation, which is along the axis of the cylinder, Ω′ representing an arbitrary direction for radiances either entering or exiting the cylinder depending on the term in which it is used, n the number of objects per unit volume in the medium, � ABS the average absorption cross section for the objects, � SCAT the average total scattering cross section for the objects, p(Ω, Ω′ ) the scattering phase function for which the scattering is from direction Ω′ to direction Ω, and B(T) the emission by a blackbody at the temperature of the medium. The thermal radiation field is assumed to be in “local thermodynamic equilibrium” with the medium. The conditions necessary for local thermodynamic equilibrium are discussed in the section on molecular spectra, section 5.2. These conditions are satisfied in the troposphere and most of the stratosphere. As a result of local thermodynamic equilibrium, the cross section for absorption and the cross section for emission are identical. The terms in the equation represent the following:

77

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3 Transfer of Radiation in the Earth’s Atmosphere

dI(Ω)

=

−n� ABS d�I(Ω)

=

−n� SCAT d�

=

dΩ′ p(Ω′ , Ω)I(Ω) ∫4� 4�

n� ABS d�B(T) n� SCAT d�

=

dΩ′ p(Ω, Ω′ )I(Ω′ ) ∫4� 4�

=

Change in the radiance as it passes through the volume Radiance lost due to absorption in the volume Radiance lost due to scattering out of the direction of propagation Radiance gained due to emission within the volume, and Radiance gained due to scattering into the direction of propagation

In terms of absorption and scattering coefficients, the change in the radiance is given by dΩ′ p(Ω′ , Ω)I(Ω) ∫4� 4� dΩ′ p(Ω, Ω′ )I(Ω′ ) + �ABS d�B(T) + �SCAT d� ∫4� 4�

dI(Ω) = −�ABS d�I(Ω) − �SCAT d�

(3.20)

One notes that owing to the normalization of the phase function the terms in Equations 3.19 and 3.20 for radiation lost due to scattering out of the direction of propagation become −n� SCAT d�I(Ω). In Equations 3.19 and 3.20 the term was written without invoking the normalization to illustrate its symmetry with the term for the radiance gained due to scattering into the direction of propagation. To simplify the form of the radiative transfer equation, a nondimensional measure for the optical extent of the medium, the optical depth, is defined. The optical depth is given by the product of the extinction coefficient, �EXT = �ABS + �SCAT , and the length of the medium through which the radiation passes. If the medium is an atmosphere and the atmosphere is assumed to be planar, then the optical depth � is measured along the normal. Planar atmospheres are commonly assumed in the planetary sciences. The approximation of a horizontally infinite, homogeneous planar medium leads to plane-parallel radiative transfer theory. In general, the optical depth � = 0 at the top of the atmosphere, and � = �S at the bottom of the atmosphere, the surface. This convention arose from early astrophysical studies of radiative transfer in stellar atmospheres. The convention was adopted by early researchers in the atmospheric sciences. An infinitesimal change in the optical depth is given by d� = −�EXT (z)dz = −n(z)�EXT (z)dz

(3.21)

The optical depth between the top of the atmosphere z = ∞ and altitude z is given by z

dz′ n(z′ )�EXT (z′ ) �(z) = − ∫∞

(3.22)

In terms of the optical depth, the equation of radiative transfer for a planeparallel scattering, absorbing, and emitting atmosphere in local thermodynamic

3.4

Equation of Radiative Transfer

equilibrium is given by �

2� 1 dI(�, �) ω = I(�, �) − (1 − �)B(T) − d�′ d�′ p(�, �; �′ , �′ )I(� ′ , �′ ) ∫−1 d� 4� ∫0 (3.23)

with cos � d� = ��� = dz, � the zenith angle for the direction of propagation, � the azimuth angle, and � the single scattering albedo. The single scattering albedo is the fraction of radiation that is scattered when radiation interacts with an object. It is given by �=

� SCAT � SCAT + � ABS

=

�SCAT �SCAT + �ABS

(3.24)

The fraction absorbed is thus given by 1 − �. The remaining challenge is to derive the scattering angle from the directions specified by (�, �; � ′ , �′ ). Figure 3.7 shows the geometry relating the scattering angle to the zenith and azimuth angles of the incident and scattered radiances. The figure shows incident radiance I(� ′ , �′ ) and scattered radiance I(�, �). In the case shown, the incident radiance is in the downward direction, z < 0, and consequently the zenith angle � ′ > 90∘ , �′ < 0. The scattered radiance is in the upward direction, z > 0, � > 0, and � < 90∘ . As was done for the determination of the solar zenith angle in Section 2.5, the cosine of the scattering angle Θ is given by the dot product of the unit vectors ̂ is ̂ and � ̂ ′ . On the basis of the geometry shown in Figure 3.7, the unit vector � � given by ⎛sin � cos �⎞ ̂ = ⎜ sin � sin � ⎟ � ⎜ ⎟ ⎝ cos � ⎠ ̂ ′ has a similar form. and the unit vector � √ √ ̂ ⋅ � ̂ ′ = cos Θ = 1 − �2 1 − �′2 cos(� − �′ ) + ��′ �

(3.25)

(3.26)

To evaluate the scattering angle for sunlight being scattered by a hydrometeor or aerosol particle in the Earth’s atmosphere, assume that I(� ′ , �′ ) represents the incident sunlight with zenith angle � ′ = �SUN and azimuth angle �′ = �SUN with the xy plane representing the Earth’s surface. The incident sunlight would have a zenith angle �SUN > 90∘ . For observers on the Earth, zenith angle refers to the upward direction and is given by the angle between the normal to the Earth’s surface and the direction pointing to the object in space as was shown in Figure 2.9. In Figure 3.7 the zenith angle of the sun, or solar zenith angle, is given by �0 = � − �SUN . The difference between the azimuth angles for the scattered and incident sunlight � − �SUN is referred to as the relative azimuth angle. For scattering by spheres, the cosine of the scattering angle depends only on the cosine of the relative azimuth angle. Since the cosine is an even function of its argument, the sign of � − �SUN is immaterial.

79

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3 Transfer of Radiation in the Earth’s Atmosphere

z

I(θ ′, ϕ′)

I(θ, ϕ) x θ′ θ Θ

y

φ φ′

x′ x Figure 3.7 Relationship between the scattering angle Θ, the zenith and azimuth angles of the incident light I(� ′ , �′ ), and scattered light I(�, �). Note that in this case, the incident light is in the downward direction � ′ > 90∘ and the scattered light is in the upward direction � < 90∘ .

3.5 Radiative Transfer Equations for Solar and Terrestrial Radiation

Radiances at wavelengths less than about 4 μm are dominated by the incident sunlight. The radiative transfer equation for radiation dominated by the incident sunlight is well approximated by �

2� 1 dI(�, �) � d�′ d� ′ p(�, �; �′ , �′ )I(� ′ , �′ ) = I(�, �) − ∫−1 d� 4� ∫0

(3.27)

The equation accounts for the scattering and absorption of sunlight. At infrared wavelengths, Rayleigh scattering cross sections become negligible when compared with molecular absorption cross sections. Furthermore, molecular number concentrations run ∼1018 cm−3 while number concentrations of aerosol particles even in polluted environments run ∼103 cm−3 . For most path lengths in the atmosphere molecular number concentrations combined with molecular absorption cross sections lead to extinction that typically overwhelms that due to aerosol particles. Consequently, effects due to scattering and absorption by aerosol particles are often assumed to be negligible. The single scattering albedo for most path lengths approaches zero. The radiative transfer equation for terrestrial radiation often includes only absorption and emission. �

dI =I−B d�

(3.28)

Problems

Clouds are an exception to this no scattering rule. Within clouds, particularly for the parts of the spectrum outside of the molecular absorption bands, the scattering and absorption cross sections of hydrometeors can have comparable magnitudes. The effects of absorption, emission, and scattering must be dealt with together. Fortunately, as will be shown in Section 4.10, at infrared wavelengths the effects due to scattering by clouds are generally small compared with those due to absorption and emission. As a result, clouds are often treated as having emissivities and transmissivities, but their reflectivities are assumed to be zero. For sufficiently thick clouds, the emissivity approaches unity and the transmissivity approaches zero. Such clouds behave like blackbodies.

Problems

1. Assume that the solar zenith angle is 70∘ and that the only cause for extinction is Rayleigh scattering by molecules. The transmission of the solar beam through the atmosphere at 0.64 μm is 0.65. On the basis of this extinction, calculate the molecular cross section for Rayleigh scattering at 0.64 μm. 2. Use the absorption cross section for ozone at a wavelength of 305 nm, �ABS = 1.9 × 10−19 cm2 , and the ozone column amount, 0.3 cm-STP, to determine the fraction of this radiation that reaches the Earth’s surface. In the calculation, assume that the atmosphere is transparent aside from the absorption by ozone. Assume that the solar zenith angle is 60∘ , the value that gives the average of the cosine of the solar zenith angle for the sunlit side of the Earth. How much larger is the fraction of this radiation reaching the surface for an overhead sun? 3. One possible approach to measuring air temperatures is to use emitted radiances near the center of the 15 μm band of CO2 . The emission comes from air that is approximately an optical depth of unity away from the detector. Use the average mass absorption cross section for CO2 at the center of the 15 μm band, �ABS = 2.2 × 103 cm2 g−1 , to estimate the distance of the source of the emitted radiation from an air-borne detector. In this estimate, assume that a balloon carrying the detector is at an altitude of about 5 km, pressure level of 500 hPa, and the CO2 number concentration is 390 ppmv. Assume that the surface temperature is 288 K and that the lapse rate is 6.5 K km−1 to calculate the density of air at 500 hPa. 4. Most of the downward emission at the surface in the tropics is due to water vapor. Furthermore, the emission at the surface appears to come from a level in the atmosphere where the difference between the optical depth at the surface �S and that of the level � is approximately unity. Within the midst of the 15 μm band of CO2 an average absorption cross section for the water vapor rotation band is �ABS = 3.1 cm2 g−1 . Assume that the mass mixing ratio of water vapor within the atmosphere is given by Equation 1.36b with a surface mixing ratio of rS = 0.04 kgH2 O kgAIR −1 .

81

82

3 Transfer of Radiation in the Earth’s Atmosphere

a. Derive an expression for the optical depth of water vapor as a function of atmospheric pressure so that at the top of the atmosphere the optical depth is zero and at the surface the optical depth is given by �(PS ) = �S . What is �S ? b. Determine the pressure level P at which �S − �(P) = 1. c. Determine the approximate altitude of the apparent emission level. d. Assume a typical tropical lapse rate of 5 K km−1 and surface temperature of 300 K to determine the apparent radiating temperature at 15 μm for the downward radiation emitted by water vapor. 5. From the definition of the scattering phase function, the fraction of radiation scattered with scattering angles less than Θ is given by the cumulative distribution of the phase function as Θ

�(Θ) =

1

1 1 dΘ′ sin Θ′ P(Θ′ ) = d� ′ P(� ′ ) 2 ∫0 2 ∫�

with � = cos Θ. Calculate the scattering angle Θ1∕2 for which �(Θ1∕2 ) = 0.5. The angle Θ1∕2 is the scattering angle for which half of the scattered light is scattered between 0∘ and Θ1∕2 . This calculation provides a measure of the strength and width of the forward scattering peak. Do the calculations for (i) the Rayleigh phase function and (ii) Henyey–Greenstein phase functions with (a) g = 0.7, appropriate for haze, and (b) g = 0.85, appropriate for cloud droplets. 6. The solar zenith angle is 30∘ . The zenith angle to a satellite radiometer is 45∘ . The relative azimuth angle between the direction of the incident sunlight and the direction to the satellite is 15∘ . Calculate the scattering angle for reflected sunlight observed by the satellite, assuming that the observed radiation is scattered only once. Perform the same calculation for the case in which the relative azimuth angle between the satellite and the incident sunlight is 165∘ . 7. For a spatially homogeneous atmospheric layer in which there is a combination of absorbing and scattering constituents, each with different optical properties, use the definitions of average cross section (Equation 3.1), optical depth (Equation 3.21), single scattering albedo (Equation 3.24), and scattering phase function (Equation 3.9) to show that these quantities for the layer are given by ∑ 1∑ 1 ∑ �= �i , � = �i �i , and p(Θ) = � � p (Θ) � i �� i i i i i with �i , �i , and pi (Θ) being the optical depth, single scattering albedo, and scattering phase function for constituent i. References 1. Bohren, C.F. and Huffman, D.R. (1983)

Absorption and Scattering of Light by Small Particles, John Wiley & Sons, Inc., New York.

2. Born, M. and Wolf, E. (1980)

Principles of Optics, 6th edn, Cambridge University Press, Cambridge.

References 3. Van de Hulst, H.C. (1957) Light Scattering

5. Kokhanovsky, A. (ed.) (2006) Light Scat-

by Small Particles, John Wiley & Sons, Inc., New York. 4. Mishchenko, M.I., Travis, L.D., and Lacis, A.A. (2002) Scattering, Absorption and Emission of Light by Small Particles, Cambridge University Press, Cambridge.

tering Reviews: Single and Multiple Light Scattering, Springer-Praxis. 6. Hess, M., Koepke, P., and Schult, I. (1998) Optical properties of aerosols and clouds: the software package OPAC. Bull. Am. Meteorol. Soc., 79, 831–844.

83

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4 Solutions to the Equation of Radiative Transfer 4.1 Introduction

A wealth of literature describes methods for solving the radiative transfer equation [1–4]. The simplest solution is known as the formal solution. In the case of thermal emission under conditions of local thermodynamic equilibrium, the formal solution is the solution. Provided the temperature and composition of the atmosphere are known, the formal solution is integrated to obtain the emitted radiances and radiative fluxes. Although easily stated, the integrations are far from simple and invariably require carefully constructed numerical codes. In this book simple but reasonable approximations of the Planck function and molecular transmissivities will be applied to obtain analytic solutions from the formal solution for absorbing and emitting atmospheres. For atmospheres that scatter and absorb, the formal solution leads to an integral equation that requires knowledge of the radiation field that is being sought. The formal solution is not really a solution, but rather a transformation of a differential equation into an integral equation. One approach to solving the resulting integral equation starts with the incident solar beam. As it passes through the atmosphere the direct beam is attenuated by scattering of the beam and by absorption. At any level in the atmosphere the radiance that remains in the direct beam is unscattered light. This radiance is inserted into the formal solution to calculate radiances that arise from only one scattering. These radiances are then reinserted into the formal solution to calculate radiances arising from two scatterings, and so on. The process is iterated to obtain contributions to the radiation field from all orders of scattering. The process is useful for atmospheres with small optical depths. Scattered light that escapes such atmospheres suffers only a few scatterings. The approach converges slowly for the scattering of visible sunlight by thick clouds. Thick clouds have large optical depths and absorb almost no light. To overcome this limitation, the equation of radiative transfer for a scattering and absorbing atmosphere is solved using a Fourier series expansion for the azimuthal dependence of the radiance and an associated Legendre polynomial series expansion for the zenith angle dependence. In this chapter, these expansions are severely truncated to produce the Eddington approximation. With this approximation, the solution is analytic, Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

86

4 Solutions to the Equation of Radiative Transfer

and it provides reasonable results for the radiant energy propagated in scattering and absorbing atmospheres. Although not covered in this book, similar expansions also provide reasonable results for absorbing and emitting atmospheres. Among the many numerical methods for solving the equation of radiative transfer for scattering, absorbing, and emitting atmospheres, the most popular are the discrete ordinate method [1–3] and the adding-doubling method [5–7]. A modified and more accurate version of the Eddington approximation and a related two-stream approximation along with methods for obtaining numerical solutions and performing Monte Carlo simulations are briefly described in the Optional Sections 4.12–4.17 at the end of the chapter.

4.2 Formal Solution to the Equation of Radiative Transfer

The equation of radiative transfer is given by �

dIv (�v , �, �) = Iv (�v , �, �) − Jv (�v , �, �) d�v

(4.1)

with Jv (�v , �, �) a radiance source function. For thermal emission under the condition of local thermodynamic equilibrium Equation 4.1 becomes Equation 3.28. The radiance source function is given by Jv (�v , �, �) = Bv (�v )

(4.2)

with the Planck function Bv (�v ) evaluated at the temperature associated with optical depth �v . For scattering and absorption Equation 4.1 becomes Equation 3.27. The radiance source function is given by Jv (�v , �, �) =

�v (�v ) 2� ′ 1 ′ d� d� p� (�� , �, �; � ′ , �′ )I� (�� , �′ , �′ ) ∫−1 4� ∫0

(4.3)

When solving the radiative transfer equation, two conventions are common. In one, as was done in Chapter 3 and as is done in Equation 4.3, the cosines � and �′ of the zenith angles � and � ′ range over the intervals −1 ≤ � ≤ 1 and −1 ≤ � ′ ≤ 1. In the second, the direction of propagation is divided into two hemispheres, one for the upward radiance I + (�, �, �) and one for the downward radiance I − (�, �, �). Sources J + (�, �, �) and J − (�, �, �) are for the upward and downward propagating radiances. In this convention, the cosines of the zenith angles both range from 0 to 1 for both hemispheres and the radiative transfer equation becomes two equations given by �

dI + = I+ − J+ d�

(4.4a)

and −�

dI − = I− − J− d�

(4.4b)

4.2

Formal Solution to the Equation of Radiative Transfer

The indices and variables used to indicate the angular, optical depth, and wavenumber dependence of the terms have been dropped to simplify the notation. The two hemisphere convention is used below. An integration factor is used to derive the formal solution of the radiative transfer equation. The use of integration factors in solving differential equations is reviewed in Appendix B. For the above equations, Equation 4.4a is multiplied by exp(−�∕�) and Equation 4.4b is multiplied by exp(�∕�). Upon multiplication and rearrangement Equations 4.4a and b become exp(−�∕�)

J+ dI + I+ − exp(−�∕�) = − exp(−�∕�) d� � �

(4.5a)

J− I− dI − − exp(�∕�) = − exp(�∕�) d� � �

(4.5b)

and − exp(�∕�)

Applying the chain rule for derivatives, d(e−�∕� I + )∕d� and d(e�∕� I − )∕d� are recognized as the left hand sides of Equations 4.5a and b. Integrating Equation 4.5a from �S to � and Equation 4.5b from 0 to � and rearranging, the upward and downward radiances yields I + (�, �, �) = IS+ (�S , �, �) exp((� − �S )∕�) �

− ∫�S

d� ′ + ′ J (� , �, �) exp((� − � ′ )∕�) �

(4.6a)

and I − (�, �, �) = I0− (0, �, �) exp(−�∕�) +

∫0

�

d� ′ − ′ J (� , �, �) exp((� ′ − �)∕�) (4.6b) �

with IS+ (�S , �, �) the upwelling radiance at the bottom boundary, � = �S . For solar radiation, this term represents the effects of reflection by the surface. In the infrared, surfaces generally have emissivities that vary with wavelength and some of the radiation emitted downwards by the atmosphere is reflected by the surface. For most surfaces, however, the emissivities deviate from unity by only small amounts. Consequently, emission by the surface is often assumed to be that of a blackbody. I0− (0, �, �) is the radiance entering the atmosphere at the top boundary, � = 0. For solar radiation, it is the incident sunlight in the direction of the solar beam. For thermal radiation, it is the radiation from empty space, and for practical purposes, it is zero. Figure 4.1 illustrates the physical meanings of the terms in the formal solution for the upward radiance in Equation 4.6a. There are two components: 1) The upward radiance incident at the bottom of the atmosphere transmitted to the level of interest, and 2) The sum of all contributions to the radiance emitted by infinitesimal elements d�∕� within the atmosphere that are transmitted from the level of the element to the level of interest.

87

88

4 Solutions to the Equation of Radiative Transfer

I+ (τ, �, ϕ) = Radiance leaving the surface and transmitted through the atmosphere + the sum of the contributions to the radiance from each infinitesimal layer of the atmosphere beneath the optical depth level τ and transmitted between the levels τ ′ and τ. τ=0 τS

I+ (τ, �) = BS exp ((τ – τS) / �) +

∫

τ

dτ ′ � B (τ ′)exp ((τ – τ ′) / �) τ=τ

exp exp

τ–τ′ �

τ – τS � � = cos θ

dτ ′ + J (τ′, �) �

θ

τ = τS IS+

(τS, �, ϕ)

Figure 4.1 Physical interpretation of the terms in the formal solution of the radiative transfer equation for the upward emitted radiance at optical depth level �. The term IS+ (�S , �, �) is the upward radiance incident on the bottom of the atmosphere at optical depth �S .

4.3 Solution for Thermal Emission

For thermal emission under conditions of local thermodynamic equilibrium, the formal solution is obtained with the source function Jv set to the Planck function Bv at the temperature of the optical depth level �v . Radiances are given by �

I + (�, �) = BS exp((� − �S )∕�) −

∫�S

d� ′ B(� ′ ) exp((� − � ′ )∕�) �

(4.7a)

4.3 Solution for Thermal Emission

and I − (�, �) =

�

∫0

d� ′ B(� ′ ) exp((� ′ − �)∕�) �

(4.7b)

with BS the upward emission by the surface given by the Planck function evaluated at the surface temperature and no infrared radiation incident at the top of the atmosphere � = 0. Notice that because emission by a blackbody is isotropic, the radiance is the same for all azimuth angles and the azimuthal dependence is dropped from Equations 4.7a and b. The radiances are typically obtained by numerically integrating Equations 4.7a and b. Owing to the complexities of the physical relationships between temperature, pressure, gas concentrations, absorption, and emission, no analytic solutions produce realistic values of the upward and downward radiances. Nonetheless, illustrative solutions can be obtained by applying reasonable approximations to the Planck function and transmissivities at infrared wavelengths. Before pursuing these approximate solutions it is useful to consider alternate forms for Equations 4.7a and b. The alternate forms facilitate the identification of the formal solution with the expressions for the upward and downward infrared fluxes used for the window-gray model in Section 1.4. Consider the transmissivity for the path from z′ to z in the direction given by the cosine of the zenith angle �. In Equation 4.7a the transmissivity for the upward radiance is given by T(z′ , z, �) = exp((�(z) − �(z′ ))∕�)

(4.8a)

Clearly, the possible range of the transmissivity is from 0 to 1. The exponent in Equation 4.8a is less than or equal to zero, z′ ≤ z and thus �(z′ ) ≥ �(z). In Equation 4.7b the transmissivity for the downward radiance is given by T(z, z′ , �) = exp((�(z′ ) − �(z))∕�)

with z′ ≥ z and �(z′ ) ≤ �(z). From Equations 4.8a and b dT(z′ , z, �) = −

d�(z′ ) exp((�(z) − �(z′ ))∕�) �

(4.8b)

(4.9a)

and dT(z, z′ , �) =

d�(z′ ) exp((�(z′ ) − �(z))∕�) �

(4.9b)

With Equations 4.9a and b, Equations 4.7a and b become 1

I + (z, �) = BS T(0, z, �) +

∫T(0,z,�)

dT(z′ , z, �)B(z′ )

(4.10a)

and 1

I − (z, �) =

∫T(z,∞,�)

dT(z, z′ , �)B(z′ )

(4.10b)

The use of transmissivities simplifies the notation. In the case of the infrared emission by a window-gray radiative equilibrium model, the atmosphere is isothermal

89

90

4 Solutions to the Equation of Radiative Transfer

so that the upward radiance is given by I + (z, �) = BS T(0, z, �) + BA (1 − T(0, z, �))

(4.11a)

and the downward radiance is given by I − (z, �) = BA (1 − T(z, ∞, �))

(4.11b)

with BA the Planck function at the atmospheric temperature TA . Radiative fluxes are obtained from the radiances using Equation 2.7 and integrating over all wavenumbers, assuming that the transmissivities in Equations 4.11a and b are constant for each wavenumber. The radiative fluxes are given by F + (z) =

∫0

2�

1

1

d� d� � I + (z, �) = 2� d� � I + (z, �) ∫0 ∫0

= �BS TF (0, z) + �BA (1 − TF (0, z))

(4.12a)

and F − (z) =

∫0

2�

1

1

d� � I − (z, �) d� � I − (z, �) = 2� d� ∫0 ∫0

= �BA (1 − TF (z, ∞))

(4.12b)

�TS4 , �BA

= �TA4 . For the upward radiative flux, the “flux transmissivwith �BS = ity,” or “transmittance,” for the layer between z′ and z is given by TF (z′ , z) = 2

∫0

1

d� �T(z′ , z, �)

(4.13)

with the emittance given by �F (z′ , z) = 1 − TF (z′ , z). Radiative fluxes are commonly referred to as irradiances. Consequently, in this book a distinction is made between reflectivities, transmissivities, absorptivities, and emissivities, which are associated with radiances, and reflectances, transmittances, absorptances, and emittances, which in turn are associated with radiative fluxes, irradiances. For an atmosphere with a reasonably constant lapse rate, such as in the Earth’s troposphere, the Planck function at any particular wavenumber can be approximated by B� (P) = B0� (P∕P0 )�

(4.14)

where P is the pressure and B0� the value of the Planck function at the pressure level P0 . The exponent � and the reference pressure level P0 are adjusted to give the best approximation for the dependence of the Planck function on altitude. If in addition, the absorbing gas is well-mixed, as is CO2 in the Earth’s atmosphere, and further, if the mass absorption coefficient � is independent of altitude, then with hydrostatic balance the incremental change in optical depth is given by d� (z) = −�r� (z) dz = �rdP (z) ∕g

(4.15a)

with r the mass mixing ratio of the absorbing gas. The optical depth between the top of the atmosphere and the pressure level P (z) is given by z P(z) ( ) dP �r = �rP (z) ∕g dz′ �r� z′ = � (z) = − ∫0 ∫∞ g

(4.15b)

4.3 Solution for Thermal Emission

With the above relations, the radiance becomes a function of pressure. Setting � = 1 in Equation 4.14 is an acceptable approximation for emission at wavelengths in the 15 μm absorption band of CO2 for the Earth’s troposphere. With these approximations Equation 4.7a for the upward radiance at the top of the atmosphere becomes ( ) ( ( ) ) PS �rPS �rP P dP �r exp − I + (0, �) = BS exp − + (4.16) B0 ∫0 g� g � P0 g� Because the limits of integration have been switched, the sign of the integral in Equation 4.16 differs and is the opposite of that in Equation 4.7a. In Equation 4.16, integrating from 0 to P instead of integrating from 0 to PS gives the emission contributed by the atmosphere above the pressure level P. The integral is obtained using integration by parts. The contribution to the emitted radiance at the top of the atmosphere is given by ( )[ ( ) ( )] IP+ (0, �) = B0 P� ∕P0 1 − 1 + P∕P� exp −P∕P� (4.17) with P� = g�∕�r the pressure at which the length of the optical path to space becomes unity for �, the cosine of the zenith angle. With P = P� the argument of the exponential transmissivity in Equation 4.17 becomes unity, P∕P� = 1. Notice that if the absorption coefficient, the concentration of the gas, or the zenith angle of the emitted radiance increases, P� decreases thereby lowering the pressure and raising the atmospheric level at which the optical path to space becomes unity. Before pursuing emission further, it should be noted that depending on the wavelength of the emission and also on the profile of the absorbing and emitting gas, the exponent � used to relate the Planck function to the pressure in Equation 4.14 can have any value, including fractions. In the case of fractional values, the integral in Equation 4.16 becomes an incomplete gamma function. In many cases, the resulting incomplete gamma function can be approximated with a rapidly converging power series [8]. For arbitrary values of the exponent, the resulting math is more complex than is presented here. Figure 4.2 shows the emission contributed by the atmosphere above pressure level P for a well-mixed absorbing gas that is optically thick so that PS ∕P� ≫ 1. Under such conditions the emission by the surface is absorbed in the atmosphere (and none ) of the radiation emitted by the surface escapes into space, BS exp −PS ∕P� = 0 in Equation 4.16. The contribution by the uppermost layer at the top of the atmosphere to the emitted radiance increases as the depth of the layer increases until a limit is reached. ( )Ultimately, for P∕P� ≫ 1, the emitted radiance is given by IP+ (0, �) = B0 P� ∕P0 = B� . The emission appears to come from a blackbody at the pressure level P = P� . With the model being used for the transmissivity, P = P� is the level at which the length of the optical path for the particular zenith angle is unity, T (P, 0, �) = exp (−1). As indicated by the results shown in Figure 4.2, however, the emission reaching the top of the atmosphere comes from a relatively thick layer. The layer that contributes 60% of the emission occupies levels with pressures that range over a factor of 3. The emitting layer is thus thicker than an atmospheric scale height H = RT∕g ∼8 km.

91

4 Solutions to the Equation of Radiative Transfer

0

P/Pk ~ 0.8

P = Pk

2 P/Pk

92

P/Pk ~ 3.0

4 6 8 0.0

0.5

1.0

1.5

+

I P (0, �) Figure 4.2 Contribution to emission at the top of an optically thick atmosphere by the atmospheric layer above the pressure level P. The abscissa is given as fractions of B� , the Planck emission at the level for which the length of the optical path to space is unity. The ordinate gives the length

of the optical path to space as given by �∕� = P∕P� . Approximately 60% of the emitted radiance is contributed by a layer that lies between the dashed lines. The pressure of this emitting layer varies by more than a factor of 3.

) ( )( ( As P∕P ) � → 0, emission by the uppermost layer is given by B0 ∕2 P∕P0 P∕P� , which is the average of the Planck function between the level P and the ( )( ) top of the atmosphere, B (P) = B0 ∕2 P∕P0 , times the emissivity of the layer between the level P and space along the direction given by the cosine of the zenith emitted along the path is given by ) The emissivity of the layer(for radiation ) (angle �. P∕P� . To obtain this result, exp −P∕P� must be expanded in a Taylor series to the second order ( in P∕P� )because the product of the zeroth and first order terms with the factor 1 + P∕P� in Equation 4.17 cancel. This cancellation results in a value of zero for the term in the brackets. If the atmosphere is optically thin so that PS ∕P� ≪ 1, the radiance leaving the top of the atmosphere is given by ( ) ]} ( ( ( ){ ) )[ PS PS 1 PS 2 PS P� + + I (0, �) = BS 1 − 1− 1+ 1− + B0 P� P0 P� P� 2 P� which simplifies to

) ( )( ) ( PS PS PS 1 + B0 I + (0, �) = BS 1 − P� 2 P0 P�

(4.18)

with B0 PS ∕P0 representing the Planck function at the surface air temperature, which may deviate from the surface temperature. As was the case for the optically ( ) ( )( ) thin region at the top of the atmosphere, the quantity B (PS = B)0 ∕2 PS ∕P0 is recognized as the average of the Planck function and � PS , 0, � = PS ∕P� is the emissivity for the optically thin atmosphere for the path along the direction given by the cosine of the zenith angle �. The emission at the top of an optically thin atmosphere with a well-mixed absorber, as given by Equation 4.18, decreases as the opacity of the atmosphere increases. Such behavior is to be expected as long as the temperature of the

4.4

Infrared Fluxes and Heating Rates

atmosphere decreases with altitude in the region where the absorbing constituent absorbs and emits radiation. The same condition holds for an opaque atmosphere. As the opacity increases, the level of emission reaches higher altitudes at which the temperature is lower and the emitted radiance at the top of the atmosphere decreases. As most of the emission occurs in the Earth’s troposphere where temperature falls with altitude and most of the greenhouse gases also reside in the troposphere, increasing the concentration of greenhouse gases reduces the emission at the top of the atmosphere. The effects of the opacity of the atmosphere on radiances emitted at the top of the atmosphere are illustrated in the spectra shown in Figure 4.3. These spectra were calculated using a detailed numerical radiative transfer code and a compilation of molecular absorption cross sections. The results are for cloud-free conditions and climatological profiles of temperatures and gas concentrations for the tropics, the 1976 U.S. Standard Atmosphere, and subarctic winters. The dashed lines in the figure give radiances for blackbodies at temperatures that range from 300 K, near the highest values for the surface temperatures of the equatorial oceans, to 200 K, near the lowest values for the tropical tropopause. On the wings of the 15 μm CO2 band, 500–600 and 700–800 cm−1 , the emission is typical of the temperatures spanning the troposphere. Careful comparison of such calculated spectra with observations indicate that the calculated spectra are typically within measurement capabilities, within 2% [9]. Near the center of the CO2 band, 667 cm−1 , temperatures associated with the emission appear to rise. The atmosphere is so opaque at these wavelengths that the emission is from levels high in the stratosphere. There the temperatures are comparable to those of the middle troposphere. At 900 cm−1 , the atmosphere is relatively transparent and the temperature associated with the emission is near that of the surface. As a consequence of Equation 4.18, because the opacity of the atmosphere is nonzero, and because the temperature of the atmosphere typically falls with altitude, the emission is slightly less than that at the surface. The connection between emission levels and the brightness of the emitted radiation is used to “invert” spectral radiances, like those in the 15 μm band of CO2 , to deduce layer temperatures at various atmospheric altitudes and also to obtain the Earth’s surface temperature. In the late 1950s, Lewis D. Kaplan [10] was the first to propose using space based observations of infrared spectra to obtain atmospheric and surface temperatures. Following his proposal, the development of weather satellites and advanced instrumentation rapidly evolved and is continuing to evolve not only for weather forecasting and monitoring but also for climate monitoring and climate change research [11, 12].

4.4 Infrared Fluxes and Heating Rates

For both infrared and solar radiative fluxes, the relationship between transmissivity and transmittance given by Equation 4.13 is often simplified through the use

93

4 Solutions to the Equation of Radiative Transfer

0.2

0.15

Tropical

300 K 280 K

0.1

260 K 240 K 220 K

0.05 200 K 0 400

600

800

1000

1200

1400

1600

1800

Wavenumber (cm−1)

(a) 0.2 Radiance (Wm−2 sr−1/cm−1)

94

0.15

U.S. Standard 1976

300 K 280 K

0.1

260 K

240K 220 K

0.05 200 K 0 400

600

800

1000

1200

1400

1600

1800

Wavenumber (cm−1)

(b) 0.2

0.15

Subarctic Winter

300 K 280 K

0.1

260 K 240 K 220 K

0.05 200 K 0 400 (c)

600

800

1000

1200

1400

1600

1800

Wavenumber (cm−1)

Figure 4.3 (a–c) Upwelling radiances at the top of the atmosphere under cloud-free conditions simulated for tropical, 1976 U.S. Standard Atmosphere, and subarctic winter climatological profiles

of temperatures and gas concentrations. Dashed lines show emitted radiances for blackbodies at the indicated temperatures.

4.4

Infrared Fluxes and Heating Rates

of a diffusivity factor. A diffusivity factor is the inverse of the “average” cosine of the zenith angles for diffuse radiances. Ideally, diffuse radiances are isotropic in which case the average cosine of the zenith angles for the upward and downward radiances is 1/2. In practice, diffusivity factors are applied whenever the radiance is distributed over a range of zenith angles, even if the radiance is not isotropic. In terms of optical depths, Equation 4.13 becomes ( ) TF � ′ , � = 2

∫0

( ) E3 y =

( ) dx exp −xy ∕x3 =

1

( ) d��T � ′ , �, � = 2

∫0

1

d�� exp

((

) ) � − � ′ ∕�

(4.19)

with � ′ ≥ �. The integral in Equation 4.19 is a third order exponential integral function which is given by ∫1

∞

∫0

1

d�� exp

) ) (( � − � ′ ∕�

(4.20)

with x = 1∕� and y = � ′ − �. Numerical routines are available for evaluating exponential integral functions, but the results of the integration over angles have been shown to be reasonably well approximated through the use of a diffusivity factor. A diffusivity factor M = 1∕� is multiplied by the optical depth so that the transmissivity for a path with the cosine of its zenith angle given by � is approximately equal to the transmittance given by Equation 4.19. With a diffusivity factor Equation 4.19 becomes ( ( ) ) ( ( )) ) ) ( ) (( TF � ′ , � = 2E3 � ′ − � ∼ T � ′ , �, � = exp � − � ′ ∕� = exp M � − � ′ (4.21) For irradiances emitted by atmospheres with large optical depths, �S ≫ 1, M = 1.66 gives results that are within a small percentage of those obtained by numerically integrating the radiances over the cosine of the zenith angle. Consequently, diffusivity factors are often used in evaluating emitted radiative fluxes. Integrations over the cosine of the zenith angle are performed only when numerical accuracies of better than 1% are required [13]. Applying a diffusivity factor to the formal solution for the upward and downward radiances transforms Equations 4.7a and b into solutions for the upward and downward radiative fluxes at optical depth level �. The upward flux is given by )) ( ( F + (�) = �BS exp M � − �S +

∫�

�S

( ( )) ( ) Md� ′ �B � ′ exp M � − � ′ (4.22a)

and the downward flux is given by F − (�) =

∫0

�

( ( ( ) )) Md� ′ �B � ′ exp M � ′ − �

(4.22b)

In the remainder of this section, as in the previous section, the optical depth is assumed to depend linearly on pressure, as if the absorber is a well-mixed gas with a constant absorption coefficient throughout the atmosphere. In the troposphere, the Planck function is also assumed to vary linearly with pressure from the surface pressure to the pressure associated with the base of the stratosphere,

95

96

4 Solutions to the Equation of Radiative Transfer

the tropopause pressure. Instead of allowing the Planck function to continue its linear decrease with pressure to the top of the atmosphere, the stratosphere is assumed to be isothermal. This approximation is used in Chapter 7 to evaluate the effects of a doubling of carbon dioxide on emitted radiances. Here the approximation is used to investigate the effects of the stratospheric temperature profile on the conclusions drawn in the previous section and to explore the dependence of both the upward and downward irradiances on pressure. This dependence on pressure leads to evaluation of the radiative heating rates associated with emitted radiances. Assume that the optical depth scales linearly with pressure and that the Planck function scales linearly with pressure in the troposphere. Assume that the stratosphere is isothermal at the temperature of the tropopause. Applying a diffusivity factor and using the pressure scaling for the optical depth and the Planck function, the upward radiance I + (�, �) given by Equation 4.7a is transformed to the upward irradiance. For pressure levels in the troposphere, P ≥ PT with PT being the pressure of the tropopause, the upward irradiance is given by F + (P) = �BS exp

(( ) ) P − PS ∕P� +

∫P

PS

( ) (( ) ) dP′ �B0 P′ ∕P0 exp P − P′ ∕P� P� (4.23a)

Similarly, the downward radiance I − (�, �) given by Equation 4.7b is transformed to the downward irradiance given by F − (P) =

∫0

PT

P

+ ∫PT

) (( ) dP′ �BT exp P′ − P ∕P� P� ( ) (( ) ) dP′ �B0 P′ ∕P0 exp P′ − P ∕P� P�

(4.23b)

In Equations 4.23a and b the pressure level for which the optical path length for the transmittance to space is unity is given by P� = g∕Mr�. The integrals are readily performed using substitution of variables and integration by parts. Upon integration, the upward and downward fluxes are given by ) ) [ (( F + (P) = �BS exp P − PS ∕P� + �B0 P∕P0 + P� ∕P0 ) (( ) )] ( (4.24a) − PS ∕P0 + P� ∕P0 exp P − PS ∕P� and [ ) ( (( ))] ( ) F − (P) = �B0 P∕P0 − PT ∕P0 exp −P∕P� − P� ∕P0 1 − exp PT − P ∕P� (4.24b) Assuming that the temperature of at the surface is the same as the ( the atmosphere ) surface temperature so that �B0 PS ∕P0 = �BS , the upward flux simplifies to ( (( ) )) (4.25a) F + (P) = �B (P) + �B� 1 − exp P − PS ∕P� and the downward flux simplifies to )) ( ( ) ( F − (P) = �B (P) − �BT exp −P∕P� − �B� 1 − exp PT − P∕P� (4.25b)

4.4

Infrared Fluxes and Heating Rates

( ) ( ) ( ) with �B (P) = �B0 P∕P0 , �BT = �B0 PT ∕P0 , and �B� = �B0 P� ∕P0 . From Equations 4.25a and b the net infrared radiative flux at pressure P within the troposphere is given by FNET (P) = F + (P) − F − (P) ) [ (( ) ( ) = �BT exp −P∕P� + �B� 2 − exp PT − P ∕P� (( ) )] − exp P − PS ∕P�

(4.26)

For pressure levels in the stratosphere, P < PT , the upward flux is given by ) ) (( F + (P) = �BS exp P − PS ∕P� PS

( ) (( ) ) dP′ �B0 P′ ∕P0 exp P − P′ ∕P� P�

PT

(( ) ) dP ′ �BT exp P − P′ ∕P� P�

+ ∫PT + ∫P

(4.27a)

and the downward flux is given by F − (P) =

∫0

P

) (( ) dP ′ �BT exp P′ − P ∕P� . P�

(4.27b)

Upon integration, the upward flux is given by [ (( ) ) (( ) )] F + (P) = �BT + �B� exp P − PT ∕P� − exp P − PS ∕P�

(4.28a)

and the downward flux is given by ( ( )) F − (P) = �BT 1 − exp −P∕P�

(4.28b)

The net emitted radiative flux is given by ) ( FNET (P) = �BT exp −P∕P� [ (( ) ) (( ) )] +�B� exp P − PT ∕P� − exp P − PS ∕P�

(4.29)

The left column of Figure 4.4 shows the upward and downward infrared radiative fluxes within the atmosphere for cases in which the optical path length from the surface to space for the transmittance has values of M�S = 1,5, and 20. These different optical paths could be for different absorber amounts at one wavenumber or the same absorber amount at different wavenumbers for which the absorption coefficients are different. In the figure the ordinate is the optical path length for the transmittance to space and is given by M� = P∕P� . The abscissa is scaled with �B0 = 1.0. It ranges from 0 to the value at the surface �BS ∕�B0 = 1.54, approximately the ratio of the Planck functions at the center of the 15 μm band of carbon dioxide for the Earth’s midtropospheric temperature 255 K and the surface temperature 288 K. For the results shown in the figure, PT ∕PS is 0.3, which is approximately the ratio of the tropopause and surface pressure for the Earth. For optically thick atmospheres with PS ∕P� ≫ 1 and PT ∕P� ≫ 1, F + (0) → �BT as P → 0. The emission at the top of the atmosphere would appear to come

97

4 Solutions to the Equation of Radiative Transfer

0.0

F+ F– FNET

0.2

(a)

(a)

(c)

(d)

0.4 0.6 0.8 0 Optical path length

98

1 2 3 4 0

(e)

(f)

5 10 15 20

0

0.5

1.0

1.5 0

Radiative luxes Figure 4.4 Upward F + (dashed line), downward F− (dash-dotted line), and net FNET (solid line) infrared radiative fluxes (left column) with the pressure gradient of the net flux dFNET ∕dP (right column). The gradient of the net flux is proportional to the infrared heating rate. The results are for

–0.5

0 dFNET/dP

5.0

1.0

atmospheres with optical path lengths for the transmittance from the surface to space of M�S = PS ∕P� = 1 (a) and (b), 5 (c) and (d), and 20 (e) and (f ). The ratio of the tropopause pressure to the surface pressure is PT ∕PS = 0.3. The scaling of the axes is described in the text.

from the stratosphere. Because the stratosphere is isothermal, there is no single pressure level associated with this emission. On the other hand, if the absorbing gas is constrained to the troposphere so that the stratosphere is transparent, the gas is well-mixed between the surface and the tropopause, and the troposphere ) ( is optically thick, PS , −PT ∕P� ≫ 1, then F + (0) → �BT + �B� . The emission appears to come from the level at which the optical path length to space for the transmittance is unity, consistent with the conclusions drawn in the previous section. Similarly, consistent with Equation 4.25b, for an optically thick atmo( ) sphere the downward flux F − (P) → �BS − �B� = �B0 PS , −P� ∕P0 as P → PS . The downward flux at the surface is equivalent to that of a blackbody at a level with unit optical path length to the surface for the transmittance. Finally, in the limit of an optically thin atmosphere, P� ≫ PS > PT , the upward flux at the top of

4.4

Infrared Fluxes and Heating Rates

the atmosphere is given by )( ) ( ) ( ) 1( �BS + �BT PS − PT ∕P� 1 − PT ∕P� F + (0) = �BS 1 − PS ∕P� + 2 + �BT PT ∕P�

(4.30)

The three terms on the right hand side of Equation 4.30 are recognized as emission by the surface transmitted to the top of the atmosphere, emission by the troposphere transmitted to the stratosphere, and emission by the stratosphere. The emission by the troposphere is given by the average value of the Planck function( for the )surface and the tropopause times the emittance of the troposphere, PS )− PT ∕P� . It is multiplied by the transmittance of the stratosphere, ( 1 − PT ∕P� . Assuming that PS PT ∕P�2 and PS PT ∕P�2 are much less than both PS ∕P� and PT ∕P� , the upward flux given by Equation 4.30 simplifies to ( ) 1 PS 1P + �BT T (4.31) F + (0) = �BS 1 − 2 P� 2 P� which is consistent with the optically thin limit of Equation 4.28a. To obtain this result, the exponentials in Equation 4.28a must be expanded to second order as was the case for the optically thin limit for the upward radiance in the previous section. Owing to the tropospheric lapse rate, Equation 4.31 indicates that the addition of a well-mixed infrared absorber reduces the emitted radiative flux at the top of the atmosphere. In the case of spectral intervals that are optically thick, for example, the center of the 15 μm band of carbon dioxide in the Earth’s atmosphere, increasing the concentration of a well-mixed greenhouse gas would not appear to affect the emitted flux at the top of the atmosphere. As discussed in Chapter 7, however, this outcome is due to the use of an isothermal stratosphere in this simple model. As the concentration of carbon dioxide in the Earth’s atmosphere increases, emitted fluxes at the top of the atmosphere in the center of 15 μm band of carbon dioxide are expected to increase, reflecting the increase in temperature with altitude in the stratosphere. As is suggested by Equation )4.26, deep within an optically thick atmosphere ( where PS ∕P� , P∕P� , and P − PS ∕P� are all much larger than unity, the net radiative flux becomes a constant independent of pressure as is shown in Figure 4.4e. The somewhat symmetrical behavior of the upward and downward radiative fluxes in the troposphere exhibited in Figure 4.4e reflects the linear dependence of the Planck function on pressure and ( ) a similar dependence on the optical path length from the boundaries, PS − P ∕P� for the upward flux and P∕P� for the downward flux. The difference in the upward and downward fluxes results from the difference in the sources at the boundaries, 0 for the downward flux and �BS for the upward flux. Unlike the upward radiance shown in Figure 4.3 in which Planck emission vanishes at the top of the atmosphere, the upward flux shown in Figure 4.4 reaches a limit within the range set by blackbodies at the surface, tropospheric, and stratospheric temperatures. As was discussed in Section 1.8 and as given by Equation 1.45b, the pressure gradient of the net infrared flux is related to the radiative heating rate. For a finite

99

100

4 Solutions to the Equation of Radiative Transfer

spectral interval of width Δ�, the contribution to the heating rate by infrared radiation is given by g ∫Δ� dT = dt CP

d� dFNET (P) dP

(4.32a)

In Chapter 5, methods of accounting for molecular absorption are described. Average molecular transmissivities and transmittances are developed for finite spectral intervals. The average transmittances lead to spectrally averaged radiative fluxes. With the average fluxes, integrals like that in Equation 4.32a are converted to summations g ∑ dF NET i (P) dT = Δ�i dt CP i dP

(4.32b)

with F NET i (P) the average net radiative flux at pressure level P for the spectral interval i with width Δ�i . The right column of Figure 4.4 shows results for the gradient of the net infrared radiative flux. The scale for the gradient ranges from −1.0 to 1.0 with �B0 ∕P0 = 1.0. The physical units of the gradient are W m−2 (cm−1 )−1 hPa−1 . From Equation 4.26 the gradient of the net infrared flux for pressure levels in the troposphere is given by ( ) dFNET (P) 1 { �BT exp −P∕P� =− dP P� [ (( ) ) (( )]} ) +�B� exp P − PS ∕P� − exp PT − P ∕P�

(4.33a)

and for pressure levels in the stratosphere by ) ( dFNET (P) 1 { �BT exp −P∕P� =− dP P� [ (( ) ) (( ) )]} +�B� exp P − PS ∕P� − exp P − PT ∕P�

(4.43b)

For a relatively thin atmosphere as is shown in Figure 4.4b, layers in the lower troposphere cool by emitting more infrared radiation than they absorb. The temperatures of these layers are higher than they would be in radiative equilibrium. The relatively high temperature is somehow maintained by processes other than radiative processes, as in the case of radiative-convective equilibrium described in Section 1.8. Layers in the upper troposphere and stratosphere, on the other hand, are warmed by absorbing more infrared radiation than they emit. The temperatures of these layers are less than they would be in radiative equilibrium. This behavior is somewhat similar to the longwave cooling by ozone in the lower troposphere and warming by ozone in the upper troposphere and lower stratosphere shown in Figure 1.6. This warming is also expected for optically thin trace gases, such as chlorofluorocarbons, that absorb in the Earth’s infrared window.

4.4

Infrared Fluxes and Heating Rates

As the atmosphere becomes optically thick, much the same behavior holds, except that ultimately the upper part of the stratosphere begins to cool. Cooling occurs when the opacity of the atmosphere below the cooling layers becomes sufficient to block emitted radiation from the lower troposphere and surface from reaching the layers as is the case for atmospheres with PS ∕P� = 5 and 20 in Figure 4.4. As the opacity of the atmosphere becomes large, as in Figure 4.4e, the net infrared radiative flux becomes constant and its gradient becomes zero, no infrared heating as shown in Figure 4.4f. Such conditions hold within the optically thick spectral intervals of molecular absorption bands, such as those of water vapor and carbon dioxide. They also occur inside optically thick clouds. Finally, near the top of an optically thick atmosphere, infrared cooling becomes large. The cooling in this part of the atmosphere is like that of carbon dioxide in the upper stratosphere as shown in Figure 1.6. At these upper levels, the optical paths to space become small whereas the optical paths to the surface and lower troposphere are large. The layers absorb little, if any, radiation from the surface and troposphere, but they emit radiation upwards and downwards. The downward emission is virtually cancelled by the upward emission by the lower stratospheric layers. Owing to their small opacities, the stratospheric layers above these cooling layers emit little radiation downward. As a result, the layers cool by emitting radiation to space. This emitted radiation escapes with little attenuation. When such conditions arise, the longwave cooling approaches that given by the “cool to space” approximation [13, 14]. As an example, in Figure 1.9 for cloud-free conditions, the upward radiative flux at the top of the atmosphere and the downward radiative flux at the surface are strongly affected by water vapor. Because water vapor has a scale height of 2 km, the absorption and emission by water vapor is confined to the troposphere. There, absorption by water vapor makes the atmosphere opaque over most of the emission spectrum. Above 2 km, the atmosphere is relatively dry, and if cloud-free, relatively transparent for the spectral regions in which the absorption by water vapor is large. Consequently, conditions for cooling to space prevail for the cloudfree troposphere over a substantial fraction of the emission spectrum. The strong infrared heating of the upper troposphere and lower stratosphere shown in Figures 4.4d and f is unrealistic. Figure 1.6 shows only cooling for the water vapor and carbon dioxide infrared absorption bands. As will be discussed in Chapters 5 and 7, the simple exponential model for the transmittance used in this section is unrealistic. Although the model is suitable for approximating upward fluxes at the top of the atmosphere under certain conditions, it is inadequate for evaluating net fluxes and heating rates within the atmosphere. In addition, the heating shown in Figure 4.4 is for monochromatic radiation. With contributions from all wavelengths for which emission is significant this heating is swamped by the cooling due to absorption and emission by water vapor. As a result, the radiative heating near the tropopause approaches zero. Numerical radiative transfer codes must be used to obtain reliable estimates of the infrared fluxes and heating rates.

101

102

4 Solutions to the Equation of Radiative Transfer

4.5 Formal Solution for Scattering and Absorption

Sunlight in the Earth’s atmosphere is scattered and absorbed. The formal solution to the radiative transfer equation is given by ) (( ( ) ) I + (�, �, �) = IS+ �S , �, � exp � − �S ∕� �S ) ) ) (( d� ′ + ( ′ (4.34a) J � , �, � exp � − � ′ ∕� + ∫� � and ( ) ( ) ( ) I − (�, �, �) = Q exp −�∕�0 � � − �0 � � − �0 � ) (( ) ) d� ′ − ( ′ J � , �, � exp � ′ − � ∕� + ∫0 �

(4.34b)

with Q the incident solar spectral(radiative ) flux at ( the top) of the atmosphere and the wavelength of interest and � � − �0 and � � − �0 (, Dirac )delta functions. The ( Dirac)delta functions have the following properties: � � − �0 = 0 for � ≠ �0 , � � − �0 = 0 for � ≠ �0 , and for any function, f (�, �), ∫0

2�

1

d�′

∫−1

) ( ) ( ) ( ) ( d�′ f � ′ , �′ � � ′ − �0 � �′ − �0 = f �0 , �0

(4.35)

) ( ) ( ) ( In Equation 4.34b Q exp −�∕�0 � � − �0 � � − �0 is the radiance associated with a DIRECT BEAM of sunlight that reaches optical depth level �. The delta functions indicate that this radiance has one direction given by the cosine of the solar zenith angle �0 and the azimuth angle for the solar beam �0 . Owing to the integral over all angles in Equation 4.35, the product of the delta function has units of the inverse solid angle. As a result, the direct beam has units of radiance (W m−2 sr−1 μm−1 ). As the solar beam passes from the top of the atmosphere to the level of optical depth �, it is attenuated by absorption and by the scattering of light out of the solar beam. ( ) In the formal solution, the upward radiance IS+ �S , �, � incident at the bottom of the atmosphere is often set to zero. This condition arises when the surface albedo is zero. An approach for adding nonzero surface albedos will be described in Section 4.10. The source in the case of scattering and absorption is given by Equation 4.3. Inserting Equation 4.3 into Equations 4.34a and b yields �S ) (( ( ) ) ) ) (( d� ′ I + (�, �, �) = IS+ �S , �, � exp � − �S ∕� + exp � − � ′ ∕� ∫� � { ( ) 2� 1 [ ( � �′ ) ( ) d�′ d� ′ p � ′ , �, �; �′ , �′ I + � ′ , �′ , �′ × ∫ ∫ 4� 0 0 } ( ′ ) − ( ′ ′ ′ )] ′ ′ + p � , �, �; −� , � I � , � , � (4.36a)

4.6

Single Scattering Approximation

and � ) ) ( ) ( ) ( ) (( d� ′ I − (�, �, �) = Q exp −�∕�0 � � − �0 � � − �0 + exp � ′ − � ∕� ∫0 � { ( ) 2� 1 [ ( � �′ ) ( ) × d�′ d�′ p � ′ , �, �; −� ′ , �′ I + � ′ , � ′ , �′ ∫0 4� ∫0 } ( ′ ) − ( ′ ′ ′ )] ′ ′ + p � , �, �; � , � I � , � , � (4.36b)

The minus signs in front of � ′ in the phase functions of Equations 4.36a and b signify that while � and � ′ both satisfy 0 ≤ � ≤ 1, radiance I + is directed upward while I − is directed downward. Using this convention, the cosine of the scattering angle in Equation 3.26 becomes cos Θ =

√

√ ) ( 1 − �2 1 − � ′2 cos � − �′ + ��′

(4.37a)

for radiances directed in the same direction, both upward or both downward and cos Θ =

√ √ ) ( 1 − �2 1 − � ′2 cos � − �′ − ��′

(4.37b)

for one of the radiances directed in the upward direction and the other directed in the downward direction. For example, in Equation 4.36a the integral term that contains I − accounts for radiation scattered into the upward direction; the cosine of the scattering angle is thus given by Equation 4.37b. Similarly, the integral term in Equation 4.36b that contains I − accounts for radiation scattered into the downward direction; the cosine of the scattering angle is thus given by Equation 4.37a. With the source term inserted, the formal solution becomes a coupled set of integral equations in which the desired radiances appear on both sides of the equation. In order to obtain the radiances, the radiances have to be known! At first glance, the formal solution for scattering and absorption seems useless. Actually, it sets the stage for obtaining the desired radiances.

4.6 Single Scattering Approximation

The single scattering approximation offers the first insight into the radiative properties of scattering and absorbing atmospheres. The approximation is obtained from the formal solution by calculating the contribution to the radiance as a result of the radiation that has been scattered only once. Because only one scattering is allowed, the approximation becomes reasonably accurate only when the optical depth of the atmosphere is small, � → 0. If the optical depth is appreciable, then the radiances that are reflected and transmitted by the atmosphere have substantial contributions from light that has been scattered more than once.

103

104

4 Solutions to the Equation of Radiative Transfer

The solution for single scattering is obtained by setting IS+ (�, �, �) = 0 and ) ( ) ( (�, �, �) = Qe−�∕�0 � � − �0 � � − �0 inside the integrals of the formal solution. With these substitutions and applying the properties of the Dirac delta functions, the integrations over the azimuth angles and the cosines of the zenith angles give I0−

I + (�, �, �) =

∫�

�S

) ( ) (( ) ) d� ′ � ( p �, �; −�0 , �0 Q exp −� ′ ∕�0 exp � − � ′ ∕� � 4� (4.38a)

and I − (�, �, �) = DIRECT BEAM � ) ) ) ( ) (( d� ′ � ( + p �, �; �0 , �0 Q exp −� ′ ∕�0 exp � ′ − � ∕� ∫0 � 4� (4.38b) For a medium in which the optical properties are constant throughout, the integrals over the optical depth are readily performed. The contributions owing to single scattering are given by ) ( ) � 0 Q0 � ( p �, �; −�0 , �0 exp −�∕�0 � + �0 4� [ (( )( ))] × 1 − exp � − �S 1∕� + 1∕�0

I + (�, �, �) =

(4.39a)

and ) ( ) � 0 Q0 � ( p �, �; �0 , �0 exp −�∕�0 I − (�, �, �) = DIRECT BEAM + �0 − � 4� [ ( ( ))] × 1 − exp � 1∕�0 − 1∕� (4.39b) The terms associated with light that has been scattered only once are represented by the right hand side of Equation 4.39a and the second term on the right hand side of Equation 4.39b. Regardless of the number of scattering processes, the scattered radiances are generally called diffuse radiances, as opposed to “direct radiances,” like the direct beam radiance. In the limit � → 0 the upward and downward radiances are given by I + (�, �, �) =

)( ) � 0 Q0 � ( p �, �; −�0 , �0 �S − � ��0 4�

(4.40a)

and I − (�, �, �) = DIRECT BEAM +

) �0 Q0 �� ( p �, �; �0 , �0 ��0 4�

(4.40b)

In this limit, whether the medium is homogeneous, meaning that the optical properties are constant with altitude, or inhomogeneous, meaning that the optical properties vary with altitude, becomes irrelevant. The radiation is scattered only

4.6

Single Scattering Approximation

once. The scattering properties are simply given as appropriately weighted average properties for the layer as discussed in Section 3.3 (see also Chapter 3, Problem 7). The form of the solution for single scattering suggests definitions for atmospheric reflectivities and transmissivities for the diffuse radiances. At the top of the atmosphere, � = 0, the upward radiance is a diffuse radiance and is given ) in ( + terms of a diffuse reflectivity defined by IDIFF (0, �, �) = �0 QR �, �; �0 , �0 . The reflectivity is given by ( ) ) I + (0, �, �) ��S ( p �, �; −�0 , �0 R �S , �, �; �0 , �0 = DIFF = �0 Q 4���0

(4.41a)

At the surface, � = �S , the diffuse component of the downward radiance is given ) ( − (�, �) = �0 QT �, �; �0 , �0 . in terms of the diffuse transmissivity defined by IDIFF The diffuse transmissivity is given by ) ( ) I + (0, �, �) ��S ( = p �, �; �0 , �0 TDIFF �S , �, �; �0 , �0 = DIFF �0 Q 4���0

(4.41b)

The total transmissivity is given by the sum of ) diffuse transmissivity and the trans( missivity for the direct beam, exp −�S ∕�0 . The direct beam appears, however, only when the viewer looks at the sun. In other directions only the diffusely transmitted light is observed. The reflectivity and diffuse transmissivity are functions of the directions of both the incident and exiting radiances. They are called bidirectional reflection and transmission functions, and are commonly used in remote sensing applications. In general, all objects are bidirectional. They have reflectivities and transmissivities that depend on the directions of both the incident and exiting radiances. Single scattering by spheres leads to scattering that depends only on the angle between the incident and exiting light, the scattering angle. For media in which light undergoes many scatterings the angle between the incident and the exiting radiances no longer becomes a simple function of the scattering angle. They depend on the incident zenith angle, the exiting zenith angle, and the relative azimuth angle. For optically thick nonabsorbing media, the reflected and transmitted radiances approach isotropy. They vary slowly with the incident and exiting zenith and azimuth angles. An albedo is the reflected radiative flux divided by the incident radiative flux. At the top of the atmosphere, the reflected flux is given by 2� 1 ( ) ( ) d�′ d�′ � ′ I + 0, � ′ , �′ F + 0, �0 = ∫0 ∫0 2� 1 ) ( = �0 Q d�′ d�′ �′ R �S , �′ , �′ ; �0 , �0 ∫0 ∫0 ) ( = �0 QR �S , �0 (4.42a)

105

106

4 Solutions to the Equation of Radiative Transfer

( ) with R �S , �0 referred to as the reflectance, planar albedo, and planetary albedo. Similarly, the downward flux at the surface is given by 2� 1 ( ) ( ) d�′ d�′ � ′ I − �S , �′ , �′ F − �S , �0 = ∫0 ∫0 2� 1 [ ( ) ( ) ( ) d�′ d�′ � ′ Q exp −�S ∕� ′ � �′ − �0 � �′ − �0 = ∫0 ∫ (0 )] +�0 QT �S , �′ , �′ ; �0 , �0 ( ) (4.42b) = �0 QT �S , �0 ( ) with T �S , �0 the “total transmittance.” In the limit that �S → 0, the reflectance is given by ( ( ) ) � R �S , �0 = S �� �0 (4.43a) �0 with

2� 1 ) ( ( ) 1 (4.43b) d�′ d�′ p �′ , �′ ; −�0 , �0 � �0 = ∫0 4� ∫0 ( ) The factor � �0 is called the upward scattering fraction. It gives the fraction of radiation scattered into the upward hemisphere [15, 16]. Similarly, in the single scattering approximation, the total transmittance is given by ) ( ( )) � � ( (4.43c) T �S , �0 = 1 − S + S � 1 − � �0 �0 �0 ( ( )) with 1 − �S ∕�0 the direct beam transmittance and ��S 1 − � �0 ∕�0 the ) ) ( ( diffuse transmittance. The absorptance is given by A �S , �0 = 1 − R �S , �0 − ) ( T �S , �0 = (1 − �) �S ∕�0 . Depending on the solar zenith angle, forward scattering can contribute a sizable fraction ( ) of the radiation scattered into the upward hemisphere. Figure 4.5 shows � �0 for isotropic and Rayleigh scattering and for the same phase functions shown in Figure 3.6b,c associated with continental haze particles, asymmetry parameter, g = 0.691, and cloud water droplets, g = 0.862. Regardless of the solar zenith angle, the value of the upward scattering fraction for the Rayleigh phase function is 0.5, the same as that for isotropic scattering. The scattering is symmetric in the forward and backward(directions. With the forward scattering by aerosol ) particles and water droplets, � �0 increases as the solar zenith angle increases. The increase arises from the increasing contribution of forward scattering to the upward scattered light with increasing ( ) solar zenith angle. In the case of cloud water droplets, the sharp rise in � � for solar zenith angles greater than 87∘ 0

reflects the forward scattering peak of the phase function. Because the scattering by spheres is rotationally invariant about the axis of the incident beam, the upward scattering fraction asymptotically approaches 0.5 as the sun nears the horizon. In the single scattering approximation, as in accurate solutions to the planeparallel radiative transfer equation, the planetary albedo increases with increasing solar zenith angle. The increase results partly from the dependence of the upward

4.6

Single Scattering Approximation

0.6 Reyleigh/isotropic Aerosol Cloud

β(θ0)

0.4

0.2

0

0

20

40 60 Solar zenith angle

Figure 4.5 Upward scattering fraction and solar zenith angle for Rayleigh and isotropic scattering (dotted line), scattering by an average continental aerosol (dashed line), and by a distribution of

80

water droplets with a mode radius of 10 μm (solid line). The scattering by the continental aerosol and cloud droplets is for light at 0.65 μm.

scattered fraction for objects with large forward scattering peaks. The albedo also increases as a result of the increase in the number of scattering objects along the slant path of the incident sunlight, the �∕�0 factor in Equation 4.43a. Less sunlight reaches high latitudes and the poles not simply because of the geometry of the Earth’s orbit around the sun, but also because of the thicker atmosphere presented by the slant paths and the contribution of forward scattering to the reflected radiances. The single scattering approximation provides useful estimates of radiances scattered by molecules and aerosols in the Earth’s atmosphere. As long as the optical depths remain small, � ∼ 0.1–0.2, and as long as the solar and view zenith angles are not too large, the calculated radiances are close to those obtained using accurate numerical methods for solving Equations 4.36a and b. Figure 4.6 compares reflectivities of various scattering and absorbing media obtained with the single scattering approximation and reflectivities obtained with a numerically accurate radiative transfer code. The reflectivities are shown for the solar plane. Reflectivities for reflected radiances with an azimuth angle of 0∘ relative to the incident light are indicated by positive values of the zenith angle. Reflectivities for reflected radiances with a relative azimuth angle of 180∘ are indicated by negative values of the zenith angle. Results are presented for optically thin atmospheres with optical depths of 0.1, (left hand panels), and with optical depths of 1.0, 10 times larger than those of the optically thin atmospheres (right hand panels). Clearly, the single scattering approximation performs better for the optically thin atmospheres. The results for both the single scattering approximation and the numerically accurate solutions to the radiative transfer equation shown in Figure 4.6 illustrate some fundamental principles of reflection. First, the accurate numerical solutions include radiances that result from all orders of scattering. For nonabsorbing

107

4 Solutions to the Equation of Radiative Transfer

0.03 0.02

0.20

ϕ–ϕ0 = 0°

ϕ–ϕ0 = 180°

Exact Single scattering

0.02

τ = 1.0

0.05

0.15

θ0 = 35° τ = 0.10 ω = 1.00 g = 0.00

0.10

0.06 θ0 = 35° τ = 0.10 ω = 0.90 g = 0.70

0.04

0.002

(e)

(f)

0 0.05 0.04

θ0 = 35° τ = 0.10 ω = 1.00 g = 0.85

0.03

τ = 1.0

0.02

0.001 0 –60 –40 –20

τ = 1.0

0.02

0 0.005

0.003

(d)

0

0.006

0.004

τ = 1.0

0.05 (c)

0

0.002

(b)

0 0.20

0.01

0.004

ϕ–ϕ0 = 0°

0.10

(a)

0 0.04 0.03

ϕ–ϕ0 = 180°

0.15

θ0 = 35° ω = 1.00 τ = 0.10 g = 0.00

0.01

R(θ, θ0, ϕ–ϕ0)

108

0.01 (g)

0

20

40

60

0 –60 –40 –20

(h)

0

20

40

60

Zenith angle Figure 4.6 Reflectivities for isotropic scattering � = 0.1 (a) and � = 1.0 (b), Rayleigh scattering (c) and (d), scattering by a Henyey-Greenstein phase function typical of an absorbing continental haze (e) and (f ), and typical of a water cloud (g) and (h). The associated single scattering albedos � and asymmetry parameters g are listed. The

reflectivities are for scattering in the solar plane with a solar zenith angle �0 = 35∘ . Negative zenith angles are for light scattered in the backward direction, relative azimuth angle � − �0 = 180∘ ; positive zenith angles are for light scattered in the forward direction, relative azimuth angle � − �0 = 0∘ .

or even moderately absorbing media, the reflectivities derived from the accurate solutions are always larger than or equal to those derived using the single scattering approximation. The probability of more than one scattering within a scattering medium is always finite. The probability becomes vanishingly small only when the

4.6

Single Scattering Approximation

optical depth of the medium becomes vanishingly small. Second, the reflectivities for � = 1 are almost, but not quite, an order of magnitude larger than those for � = 0.1. The increases in reflectivity with increasing optical depth obtained with the single scattering approximation are less than those obtained with accurate solutions. As a medium increases in depth, the probability that the reflected light has scattered only once increases, but only slightly. Light that has scattered only once comes predominantly from the upper regions of the medium. The increases in the reflectivity obtained with the accurate calculations are because of higher orders of scattering. Third, as the medium thickens, the dependence of the reflectivity on the zenith angle diminishes. The reflected radiation becomes more isotropic and less dependent on the zenith angle. Fourth, regardless of the scattering phase function, the reflectivity increases with increasing zenith angle, at least for the range of zenith angles shown. This increase results from the increase in the optical path along the direction of large zenith angles. Consider the case of isotropic scattering shown in Figure 4.6a but with an incident zenith angle of 0∘ . For a given area of illuminated scattering objects, the number of objects that contributes to light reflected at large zenith angles is greater than the number that contributes to light reflected at small zenith angles. The increase in the number of scattering objects aligned with the path of the reflected radiances at large zenith angles increases the likelihood of scattering in those directions. In addition, for isotropic scattering, the reflectivities are the same for all azimuth angles as is shown in Figures 4.6a and b. The reflectivities are azimuthally symmetric regardless of the solar zenith angle. Rayleigh scattering, on the other hand, leads to reflectivities that are clearly sensitive to the solar zenith angle as is shown in Figures 4.6c and d. The Rayleigh phase function reaches a minimum for a scattering angle of 90∘ (see Figures 3.4a and 3.6a). As was discussed in Section 3.4, with a solar zenith angle of 35∘ a scattering angle of 90∘ is obtained in the forward direction with a zenith angle for the reflected light of 55∘ . The Rayleigh phase function reaches maxima for scattering angles of 0∘ and 180∘ . For reflected radiances in the direction of the incident sunlight, the Rayleigh phase function increases only slightly from its minimum to a scattering angle of 95∘ at a zenith angle of 60∘ . The maximum value of the phase function for radiances reflected in the direction of the incident sunlight is for a zenith angle of 0∘ . At 0∘ the scattering angle is 145∘ . For radiances reflected in the opposite direction, a relative azimuth angle of 180∘ so that the radiances are propagating toward the sun, the maximum of the Rayleigh phase function occurs for a zenith angle of 35∘ and a scattering angle of 180∘ . The light is backscattered directly at the sun. Despite the maximum in the phase function, the reflectivity continues to rise for larger zenith angles. The rise is a result of the increase in the optical path for the reflected radiances with large zenith angles. The rise is sufficient to more than offset the decrease in the Rayleigh phase function as the scattering angle falls from 180∘ for a zenith angle of 35∘ to 155∘ for a zenith angle of 60∘ . So, even in a crystal clear atmosphere with the sun above the horizon, the sky appears whitish in the vicinity of the sun and bluer in the opposite direction, away from the sun.

109

110

4 Solutions to the Equation of Radiative Transfer

When the phase function is not symmetric in the directions of forward and backward scattered light, as shown in Figures 4.6e–h, the peak of the phase function for forward scattering becomes evident. Even without absorption, as in the case of the absorbing continental aerosol in Figures 4.6e and f, the reflectivity is much reduced compared with those for isotropic and Rayleigh scattering. This reduction reflects the relative magnitudes of the phase functions for large scattering angles. Consequently, reflectivities for nonabsorbing cloud droplets are smaller than those for the absorbing continental aerosol particles, even though the cloud droplets absorb no light. Clearly, the enhanced scattering in the direction of forward scattering for haze particles, cloud droplets, and ice crystals and even the enhanced reflectivities for Rayleigh scattering in the direction of backscattering explains the hazy appearance and whitening of the sky when looking toward the sun and its relative clarity and darker hues when looking away from the sun. The single scattering approximation serves as the starting point for several numerical approaches to solving the radiative transfer equation. Of these, the most obvious is the successive orders of scattering methods. The solution for each order of scattering is obtained by inserting the solution for the next lowest order of scattering as the radiance source into the formal solution and evaluating the resulting integrals. The solution for the radiance is then given by I=

∞ ∑

In

(4.44)

n=0

where In is the solution obtained for the nth order of scattering. Whereas the procedure would be laborious were one to attempt an analytic solution, the coding of the iterations is straightforward. The problem, however, is that for atmospheres that only scatter and do not absorb, as is the case for clouds at visible wavelengths, the series converges slowly. Consequently, approaches other than successive orders of scattering are more effective. The successive orders approximation, however, is applicable for atmospheres in which there is moderate to strong absorption. For such atmospheres, light is absorbed before it undergoes large numbers of scatterings. Light that escapes absorbing atmospheres will have undergone only a few scatterings.

4.7 Fourier Decomposition of the Radiative Transfer Equation

When multiple scattering occurs, efficient methods are needed to solve the equation of radiative transfer. One approach to solving such equations is to separate the variables. The radiative transfer equation has a structure that allows for the separation of solutions into functions that depend only on azimuth angles, functions that depend only on zenith angles, and functions that depend only on optical depths. This separation is called spherical harmonic decomposition.

4.7

Fourier Decomposition of the Radiative Transfer Equation

Spherical harmonic decomposition was extensively developed during the latter half of the nineteenth century for wave equations such as those for sound and electromagnetic waves. In the mid twentieth century, S. Chandrasekhar applied this decomposition to obtain solutions for the radiative transfer equations [1–3]. In spherical harmonic decompositions, the azimuthal dependence of the radiance is expressed as a Fourier series. The Fourier series representation begins with the scattering phase function. Because scattering by spheres is rotationally symmetric about the axis of the incident beam, the azimuthal dependence of the scattering phase function is an even function of the relative azimuth angle given by � − �′ with � the azimuth angle of the scattered light and �′ the azimuth angle of the incident light. Because cosines are also even functions of their argument, the phase function may be expanded in terms of a cosine series, given by ∞ ( ) ( ) ) ∑ ( P (Θ) = p �, �; ±� ′ , �′ = pm �, ±� ′ cos m � − �′

(4.45)

m=0

( ) In Equation 4.45 pm �, ±� ′ is referred to as the mth azimuthal component of the phase function. Identifying the order of the azimuthal component with a superscript is common in spherical harmonic decompositions. The subscript is reserved for the order of the polynomial that is used to represent the zenith angle dependence of the phase function. Clearly, Equation 4.45 is justified as a representation of the phase function for scattering by spheres. Such scattering depends only on the cosine of the scattering angle which is given by √ √ ) ( cos Θ = 1 − �2 1 − � ′2 cos � − �′ ± ��′ (4.46) The cosine of the scattering angle in turn depends only on the cosine of the relative azimuth angle. In Equation 4.46 the plus or minus sign is a reminder that both the incident and scattered radiation may be propagating upwards or downwards, in which case the plus sign is used, or one may be propagating upwards and the other downwards, in which case the minus sign is used. The azimuthal components of the phase function in Equation 4.45 are obtained using the orthonormality of cosines. The orthonormal conditions are given by 2� ( ) ) ( 1 (4.47) d�′ cos m�′ cos n�′ = �mn 1 + �m0 = 2�mn ∕ 2 − �m0 � ∫0 with �mn the Kronecker delta symbol. The symbol is 1 for m = n and 0 for m ≠ n. With these conditions, the azimuthal components of the phase function in Equation 4.45 are given by

) ( ) ( ) 2 − �m0 2� ′ ( d� p �, �; ±� ′ , �′ cos m � − �′ pm �, ±� ′ = (4.48) 2� ∫0 Owing to the structure of the phase function for scattering by spheres, a similar structure is assumed for the radiances. The radiances are represented by a cosine series. ∞ ∑ (4.49) I m (�, �) cos m� I (�, �, �) = m=0

111

112

4 Solutions to the Equation of Radiative Transfer

The azimuthal components of the radiance, I m (�, �), are derived from Equation 4.49 by applying the orthonormality of cosines as given by Equation 4.47. The components are given by ) ( 2� 2 − �m0 ) ( m d�′ I �, �, �′ cos m�′ (4.50) I (�, �) = ∫ 2� 0 The zeroth order component of the radiance I 0 (�, �) is referred to as the azimuthally averaged radiance. It is the only component required for the calculation of the radiative flux. For example, from Equation 4.50 with m = 0, the upward flux is given by F + (�) =

∫0

2�

d�′

∫0

1

( ) d�′ � ′ I + �, � ′ , �′ = 2�

∫0

1

( ) d�′ � ′ I 0+ �, � ′ . (4.51)

As was noted in Section 4.2, the radiative transfer equation for a scattering and absorbing atmosphere is given by the combination of Equations 4.1 and 4.3 to give �

2� 1 ) ( ) ( dI (�, �, �) � = I (�, �, �) − d�′ d�′ p �, �; �′ , �′ I �, � ′ , �′ (4.52) ∫−1 d� 4� ∫0

As a simplification, in Equation 4.52 the atmosphere is assumed to have the same scattering and absorbing properties throughout. As a result, the single scattering albedo and the scattering phase function are held constant. With the azimuthal dependence of the phase function and the radiance represented by a pair of cosine series, the equation of radiative transfer for the mth azimuthal component of the radiance is obtained by multiplying both sides of Equation 4.52 by cos m�, integrating over � from 0 to 2�, and applying the orthogonality of the cosines. The mth azimuthal component of the radiance is given by ) ( 1 � 1 + �m0 ( ) ( ) dI m (�, �) m (4.53) = I (�, �) − d� ′ pm �, �′ I m �, �′ � ∫−1 d� 4 Notice that the equations for the azimuthal components of the radiance have a common form. If a method can be found for solving any one of the equations, then the same method can be used for solving all of them.

4.8 The Legendre Series Representation and the Eddington Approximation

As was noted in the previous section, the 0th term of the Fourier series representation for the radiance is the only term used in calculations of radiative fluxes. In this section, the focus is on radiative fluxes. The radiative transfer equation for the azimuthally averaged radiance is given by Equation 4.53 with m = 0, ( ) ( ) � dI d�′ p �, �′ I �′ =I− d� 2 ∫−1 1

�

As a simplification, the superscript 0 has been dropped in Equation 4.54.

(4.54)

4.8

The Legendre Series Representation and the Eddington Approximation

Legendre polynomials represent the zenith angle dependence of the azimuthally averaged radiances and phase function in Equation 4.54. Like the sines and cosines in the Fourier series, Legendre polynomials form a complete orthonormal set of functions on the interval −1 ≤ � ≤ 1. The azimuthally averaged radiance is given by I (�, �) =

∞ ∑

Il (�) Pl (�)

(4.55)

l=0

with Pl (�) the lth order Legendre polynomial. The first three Legendre polynomials are given by P0 = 1

(4.56a)

P1 (�) = �

(4.56b)

and

) 1( 2 3� − 1 (4.56c) 2 Recursion relations give the higher order polynomials in terms of lower order polynomials. The recursion relation for Legendre polynomials is given by P2 (�) =

lPl (�) = (2l − 1) �Pl−1 (�) − (l − 1) Pl−2 (�)

(4.57)

The orthonormal relationship among the polynomials is given by 1

∫−1

Pm (�) Pn (�)d� =

2 � 2n + 1 nm

(4.58)

The phase function in Equation 4.54 is given by a series of Legendre polynomial functions of the cosines of the zenith angles for both the incident and scattered light. It is given by ∞ ( ( ) ) ∑ p �, � ′ = �l Pl (�) Pl � ′

(4.59)

l=0

To avoid confusion with the symbol � used for the single scattering albedo, the symbol �l in Equation 4.59 is used to represent(the coefficient for the Legen) dre polynomial of order l. The phase function p �, � ′ in Equation 4.54 is the azimuthally averaged phase function. It is given by Equation 4.48 with m = 0, 2� 2� ) ) ( ( 1 1 d�′ p �, �; � ′ , �′ = d�′ P (cos Θ) p �, �′ = 2� ∫0 2� ∫0

(4.60)

with Θ the scattering angle. The cosine of the scattering angle depends on the relative azimuth angle as given by Equation 4.46. The phase function is similarly represented as a series of Legendre polynomials. The series is given by P (cos Θ) =

∞ ∑ l=0

�l Pl (cos Θ)

(4.61)

113

114

4 Solutions to the Equation of Radiative Transfer

Applying the orthonormality of Legendre polynomials as given in Equations 4.58–4.61, the coefficients are given by �l =

2l + 1 4� ∫0

2�

1

d� d�P (�) Pl (�) ∫−1

(4.62)

with � = cos Θ. For example, with Pl (�) = P0 = 1, Equation 4.62 leads to �0 = 1. Phase functions are normalized with Pl (�) = P1 (�) = �, Equation 4.62 leads to �1 =

3 4� ∫0

2�

1

d� d��P (�) = 3g ∫−1

(4.63)

with g the asymmetry parameter defined in Equation 3.18. In the Eddington approximation, the series for the azimuthally averaged radiance is severely truncated. It is assumed to be given by I (�, �) = I0 (�) + I1 (�) �

(4.64)

The phase function is similarly represented by a severely truncated series. ( ) (4.65) p �, � ′ = 1 + 3g�� ′ The Eddington approximation and its variants have proven useful for obtaining estimates of radiative fluxes and developing insight into the transfer of radiation for scattering and absorbing atmospheres. The approximation was applied by Sir Arthur Eddington in a study of stellar atmospheres early in the twentieth century [17]. In its various forms the approximation is still in common use. In the approximation, the radiance is assumed to vary linearly with the cosine of the zenith angle. Such behavior is typical of optically thick media, such as thick clouds and fogs at visible wavelengths. Inside thick clouds and fogs the radiation field appears to be nearly the same in all directions. Although originally applicable only for thick atmospheres, some have contrived methods for extending the approximation from optically thin to optically thick media [15, 18, 19]. The equations for the approximation are obtained in much the same way that the radiative transfer equation for the different azimuthal components of the radiances and phase function were obtained. Substituting Equations 4.64 and 4.65 into the azimuthally averaged radiative transfer equation (Equation 4.54), multiplying both sides of the resultant equation by either P0 = 1 or P1 = �, and integrating the result over −1 ≤ � ≤ 1, leads to a pair of coupled differential equations given by 1 dI1 (�) = (1 − �) I0 (�) 3 d�

(4.66a)

( ) dI0 = I1 (�) 1 − �g d�

(4.66b)

and

This pair of equations represents the Eddington approximation [17]. In Equations 4.66a and b, I0 and I1 are referred to as the zeroth and first moments of the radiance. They have easily recognized physical meanings. The

4.8

The Legendre Series Representation and the Eddington Approximation

zeroth moment of the azimuthally averaged radiance is the average radiance I0 given by 1

I0 (�) =

1 d� 4� ∫−1 ∫0

2�

1

d�I (�, �, �) =

1

0

1

0

1 d�I 0 (�, �) 2 ∫−1

=

1 1 d�I 0 (�, �) + d�I 0 (�, �) 2 ∫0 2 ∫−1

=

1 1 d�I 0 (�, �) − d�I 0 (�, −�) 2 ∫0 2 ∫1

=

1 1 d�I 0+ (�, �) + d�I 0− (�, �) 2 ∫0 2 ∫0

1

1

(4.67a)

with I 0+ the upward azimuthally averaged radiance and I 0− the downward azimuthally averaged radiance. I0 is the radiance averaged over all solid angles, 4� steradians. It is the rate at which energy per unit area enters or leaves the surface of a sphere. The sphere can have any radius, from infinitely large, like the radius of the Earth’s orbit, to infinitesimal, like the radius of a molecule. In the case of a molecule, the average radiance is used in calculations of photochemical reaction rates, reactions that require light for molecules to react. For photochemical reactions the average radiance is often referred to as the actinic flux. Notice that the actinic flux is not the same as the radiative flux or irradiance discussed in this book. The actinic flux is the rate of energy per unit area flowing into or out of a sphere. The irradiance is the rate of energy per unit area passing through a planar surface. The first moment of the radiance I1 is related to the net radiative flux by FNET (�) = F + (�) − F − (�) = = 2� = 2� = 2�

∫0

1

∫0

1

∫0

1

∫0

2�

1

1

d� d��I (�, �, �) = 2� d��I 0 (�, �) ∫−1 ∫−1 0

d��I 0 (�, �) + 2� d��I 0 (�, �) + 2�

∫−1 ∫1

d��I 0+ (�, �) − 2�

d��I 0 (�, �)

0

∫0

d��I 0 (�, −�) 1

d��I 0− (�, �)

(4.67b)

with F + the upward flux and F − the downward flux. The net flux FNET is the difference between the upward and downward fluxes, both taken to be either zero or positive quantities. With the truncation used in the Eddington approximation, the second moment is related to the zeroth moment, 1 I (�) 1 d�� 2 I (�, �) = 0 2 ∫−1 3

(4.68)

Unlike the zeroth moment, the mean radiance, and the first moment, which is proportional to the net flux, the second moment has no physical interpretation.

115

116

4 Solutions to the Equation of Radiative Transfer

In terms of the average radiance and the net flux, the Eddington approximation given by Equations 4.66a and b becomes dF = 4�I0 (1 − �) d�

(4.69a)

( ) 4� dI0 = F 1 − �g 3 d�

(4.69b)

and

with subscript NET dropped to simplify the notation. Before proceeding to a solution of the coupled differential equations (Equations 4.69a and b), the case of a nonabsorbing atmosphere, single scattering albedo � = 1, deserves special attention. Such atmospheres are said to “scatter conservatively,” meaning that no light is lost in the process of scattering. The light is simply redirected. As was discussed in Section 1.8, the gradient of the net flux with altitude gives the radiative heating rate of an atmospheric layer. 1 dℱNET dT =− dt CP � (z) dz

(4.70)

with ℱNET the net radiative flux integrated over all wavelengths. Assuming that � = 1 and the asymmetry parameter is constant with wavelength, dF dT = 0 so that F = CONSTANT and =0 d� dt

(4.71a)

There is no radiative heating. Since F = CONSTANT, the average radiance as given by Equation 4.69b becomes a linear function of optical depth given by, ( ) 4�I0 =F 1−g � +C 3

(4.71b)

with constants F and C. These constants are determined through the application of boundary conditions to be discussed below. For a homogeneous atmosphere in which � and g are constant, the solution of the coupled differential equations (Equations 4.69a and b) is obtained by taking a second derivative. Doing so gives ( ) d2 F = 3 (1 − �) 1 − �g F = � 2 F d� 2 with

√ ( ) � = ± 3 (1 − �) 1 − �g

(4.72a)

(4.72b)

The solution to Equation 4.72a has the form of F = Ae�� + Be−��

(4.73a)

with constants A and B determined through the application of boundary conditions. From Equations 4.69a and 4.73a 4�I0 (1 − �) = �Ae�� − �Be−��

(4.73b)

4.8

The Legendre Series Representation and the Eddington Approximation

The boundary conditions are as follows: at the top of the atmosphere, � = 0, the incident flux is that due to the sun, which is given by F − (0) = 2�

∫0

1

d��I − (0, �) = 2�

∫0

1

d��I (0, −�) = �I0 (0) −

F (0) = �0 Q 2 (4.74a)

with Q the incident solar radiative flux at the top of the atmosphere and �0 the cosine of the solar zenith angle. At the bottom of the atmosphere, � = �S , with the surface albedo set to zero, the upward flux is zero. ( ) 1 1 ( ) ( ( ) F �S ) ( ) + + =0 F �S = 2� d�� I �S , � = 2� d�� I �S , � = �I0 �S + ∫0 ∫0 2 (4.74b) Substituting the expressions for I0 and F from Equations 4.73a and b into the equations for the boundary conditions (Equations 4.74a and b) leads to coupled algebraic equations for the constants A and B. The solutions are given by A=

2�0 Q (1 − U) e−��S D

(4.75a)

2�0 Q (1 + U) e��S D

(4.75b)

and B=− with D = (1 + U)2 e��S − (1 − U)2 e−��S

(4.75c)

and U=

� 2 (1 − �)

(4.75d)

The problem is now solved. Fluxes and the mean radiance can be determined for any level in the atmosphere by substituting the solutions for A and B back into Equations 4.73a and b. Of interest are the fluxes of radiation reflected F + (0) and transmitted ( particular ) − � F S by the atmosphere. At the top of the atmosphere, F (0) A B = (1 + U) + (1 − U) = �0 QR 2 2 2 with R the reflectance, or in this case, the albedo, and is given by ) ( 2 U − 1 (e��S − e−��S ) R= (1 + U)2 e��S − (1 − U)2 e−��S F + (0) = �I0 (0) +

(4.76a)

(4.76b)

At the bottom of the atmosphere, the downward flux is given by ( ) ( ) F �S A B − F = �I0 �S − = − (1 − U) e��S − (1 + U) e−��S = �0 QT (4.77a) 2 2 2

117

118

4 Solutions to the Equation of Radiative Transfer

with T the transmittance given by T=

(1 +

U)2 e��S

4U − (1 − U)2 e−��S

(4.77b)

The absorptance is given by A = 1 − R − T. In the limit of large optical depths, �S → ∞, the albedo (Equation 4.76b) approaches a constant value independent of the optical depth and the transmittance (Equation 4.77b) approaches zero. R→

U −1 U +1

and

T →0

(4.78)

In the limit of small optical depth, the albedo is linear in optical depth, as it is for the single scattering approximation. The reflectances obtained with the two approximations, however, have different magnitudes. In the Eddington approximation the linear dependence of the radiance on the cosine of the zenith angle is typical of optically thick media. It fails to capture the zenith angle dependence of reflected radiances typical of optically thin media. Nonetheless, the absorptance for both the Eddington and the single scattering approximations is given by A = 2 (1 − �) �S . To derive this result for the single scattering approximation, one notes that the reflectance and transmittance for the Eddington approximation are a “spherical albedo” and the corresponding transmittance. A spherical albedo, sometimes referred to as a Bond albedo is the total radiant power (joules per second) reflected by a sphere illuminated by a distant source divided by the total radiant power incident on the sphere [2]. The distance between the sphere and the source must be much larger than the radii of both the sphere and the source as is, for example, the case for the Earth and the sun. In terms of a planetary albedo such as given by Equation 4.41a, the( total radiant power reflected by ) a sphere that has a constant planetary albedo R �0 everywhere on its surface is given by �+ =

∫0

2�

1 1 ( ) ( ) rd� rd�0 �0 QR �0 = 2�r2 Q d�0 �0 R �0 ∫0 ∫0

(4.79a)

with r the radius of the sphere and � the rotation angle about the axis formed by a line connecting the source and the sphere. The total incident radiant power is given by �− =

∫0

2�

1

rd� rd�0 �0 Q = �r2 Q ∫0

(4.79b)

The spherical albedo R is given by Equation 4.79a divided by Equation 4.79b. R=2

∫0

1

( ) d�0 �0 R �0

(4.79c)

The spherical albedo in the single scattering approximation is given by R=2

∫0

1

( ) d�0 �� �0 �S = 2���S

(4.80a)

4.8

The Legendre Series Representation and the Eddington Approximation

and the corresponding transmittance is given by ) ( T = 1 − 2�S − 2��S 1 − �

(4.80b)

with 1

�=

1 d�0 ∫0 4� ∫0

2�

d�′

∫0

1

) ( d�′ p � ′ , �′ ; −�0 , �0 =

∫0

1

( ) d�0 � �0 (4.80c)

For conservative scattering � = 1 and absorption is zero. The reflectances and transmittances can be obtained from Equations 4.76b and 4.77b by taking the limit � → 1. Doing so requires application of L’Hospital’s rule. Perhaps a simpler approach is to start with the original coupled differential equations (Equations 4.69a and b) and set � = 1 yielding Equations 4.71a and b. The reader may follow the steps outlined above for a scattering and absorbing atmosphere to obtain the reflectance and transmittance for a conservatively scattering atmosphere. They are given by ) 3 ( 1 − g �S 4 (4.81a) R= ( ) 1 + 34 1 − g �S and T =1−R=

1 ( ) 1 + 1 − g �S 3 4

(4.81b)

The solutions for the reflected and transmitted fluxes in the Eddington approximation suggest a length scale for the diffusion of light within scattering and absorbing media. In the case of absorbing media, the diffusion length in units √ ( ) of optical depth is given by �DIFF ∼ 1∕� = 1∕ 3 (1 − �) 1 − �g . As a medium becomes nonabsorbing, � → 1, or if the scattering objects strongly scatter in the forward direction, g → 1, the diffusion length becomes large. Light incident on the boundaries can penetrate many optical depths into the medium before being absorbed. A corollary is that light escaping the medium could have originated from deep within the medium, although the probability of light escaping generally peaks for an optical depth of unity within the medium. For nonabsorbing media, another measure of the diffusion length is the number of scatterings required to “turn the photon around.” Light incident on the boundaries of [ a( nonabsorbing )] medium can penetrate in units of optical depth �DIFF ∼ 4∕ 3 1 − g before being redirected back toward the incident boundary. Figure 4.7 compares spherical albedos and corresponding transmittances calculated using the Eddington approximation (lines) and those calculated using Discrete Ordinate Radiative Transfer (DISORT), a numerical radiative transfer code based on the discrete ordinate method (symbols) [20]. The numerical method is briefly described in Section 4.15. Examples are shown for a Rayleigh atmosphere, an absorbing haze, and a cloud. For Rayleigh and cloud scattering, there is no absorption and the Eddington approximation does remarkably well in approximating accurate numerical calculations of the spherical albedo. For the absorbing

119

4 Solutions to the Equation of Radiative Transfer

1.0

1.0

0.8

0.8

0.6

0.6 Rayleigh g = 0.00 Eddington ω = 1.00 Exact

0.4 0.2 0.0

0

10

20

30

0.4 0.2

(a)

0.0

40

0.4 0.3 0.2

Aerosol g = 0.70 Eddington ω = 0.90 Exact

0.1 0.0

0

1.0

10

20

30

(c) 40

Cloud g = 0.85 ω = 1.00 Eddington Exact

0.8

0

10

20

30

40

0.8 0.6 0.4 0.2 0.0

0.4

0.4 (e) 0

10

20

30

0

20

30

0.2 0.0

0 40 Optical depth

Figure 4.7 Reflectances, left column, and transmittances, right column, for a Rayleigh atmosphere (a) and (b), an absorbing haze (c) and (d), and a cloud (e) and (f ) calculated using the Eddington approximation

10

40

0.8 0.6

0.2

(d)

1.0

0.6

0.0

(b)

1.0 Transmittance

Relectance

120

(f) 10

20

30

40

(line) and DISORT (symbols), a numerical radiative transfer code. The single scattering albedos � and asymmetry parameters g are given.

haze, the reflected light undergoes low-order scattering for which the Eddington approximation, with its severe truncation of the scattering phase function, does relatively poorly. Still, the Eddington approximation offers useful estimates for the reflectances, transmittances, and absorptances of cloud and aerosol layers. As the transmittances shown in Figure 4.7 indicate, with no absorption and large optical depths, some light seems to get through, as it does even for relatively thick rain clouds. On the other hand, even with a little absorption, as in the case of the absorbing aerosol, once the optical depth reaches a certain value, all light is either reflected or absorbed, and none is transmitted. Conditions similar to these were proposed for “Nuclear Winter,” the outcome of an all out nuclear exchange between superpowers. The nuclear blasts were expected to ignite all combustibles within urban areas and set forests afire creating a massive pall of smoke in the middle to the upper troposphere [21, 22]. This smoke layer would scatter and absorb incident sunlight, greatly reducing photosynthesis, dropping surface temperatures so that in mid continents the temperature would be below freezing, even during summer seasons. Any survivors of the nuclear blasts would face the prospect of

4.9

Adding Layers in the Eddington Approximation

widespread famine. Projections suggested that only a small fraction of the human population would survive. The effect of an absorbing haze layer was also invoked to explain the widespread extinction of species that occurred 65 million years ago at the Cretaceous-Tertiary boundary [23]. The source of the absorbing haze was the collision of a ∼10 km diameter asteroid that presumably crashed into the ocean. The energy released in the collision thrust sediments from the ocean floor into the upper troposphere and stratosphere where they quickly spread around the Earth. As with the smoke in nuclear winter, the dust scattered and absorbed incident sunlight, greatly reducing the light that reached the surface, cutting off photosynthesis, disrupting the food chain at its base, and leading to massive extinctions of marine life and dinosaurs.

4.9 Adding Layers in the Eddington Approximation

With a radiative flux �0 Q incident at the top of the atmosphere and with no radiation incident at the bottom of the atmosphere, the Eddington approximation produces the solutions F0+ = �0 QR for the flux of radiation reflected by the atmosphere, with R the reflectance, or albedo, and FS− = �0 QT for the flux of radiation transmitted by the atmosphere, with T the transmittance. Consider two homogeneous atmospheric layers with different radiative properties: layer 1 with reflectance R1 and transmittance T1 , and layer 2 with reflectance R2 and transmittance T2 . Assume that layer 1 lies above layer 2. The problem is to determine the reflectance and transmittance for the combined, two-layer atmosphere. Using the Eddington expressions for upward and downward fluxes given by Equations 4.76a and 4.77a, the upward flux at the top of the atmosphere is given by F0+ = �0 QR1 + F1+ T1

(4.82)

with F1+

the radiative flux incident at the bottom of layer 1. Similarly, at the bottom of layer 1, the downward flux is given by F1− = F1+ R1 + �0 QT1

(4.83)

In order to solve for the upward flux at the top of the atmosphere, and thereby obtain the reflectance for the combined, two-layer atmosphere, solutions must be obtained for the upward and downward fluxes at the interface between the layers. These solutions are readily obtained by applying the Eddington expressions for the upward and downward fluxes. The upward flux at the bottom of layer 1 is given by F1+ = F1− R2 + F2+ T2 F2+ ,

(4.84)

the radiative flux incident at the bottom of layer 2, which For the time being, is also the bottom of the two-layer atmosphere, is assumed to be known. For example, if the surface albedo is zero, F2+ = 0. If the surface albedo is nonzero, as

121

122

4 Solutions to the Equation of Radiative Transfer

discussed in the next section, then F2+ is finite. Combining Equations 4.83 and 4.84 yields F1+ = F1+ R1 R2 + �0 QT1 R2 + F2+ T2

(4.85a)

which leads to

( )−1 ( )−1 + F2+ T2 1 − R1 R2 F1+ = �0 QT1 R2 1 − R1 R2

(4.85b)

Substituting Equation 4.85b into Equation 4.82 gives [ ( )−1 ] ( )−1 F0+ = �0 Q R1 + T1 R2 1 − R1 R2 T1 + F2+ T2 1 − R1 R2 T1 = �0 QR12 + F2+ T12 (4.86) ( )−1 with R12 = R1 + T1 R2 1 − R1 R2 T1 the reflectance of the combined two-layer system for radiation incident from above and layer 1 on top of layer 2, and T12 = ( )−1 T2 1 − R1 R2 T1 is the transmittance for radiation incident from below with layer 2 on the bottom. Figure 4.8 illustrates the multiple reflections between layers 1 and 2 indicated by Equation 4.86. The upward flux F0+ exiting layer 1 is the sum of the incident flux reflected by layer 1 plus the contributions from the flux transmitted by layer 1, reflected multiple times between the two layers, and after a final reflection by layer 2 is transmitted by layer 1 to escape its top boundary. Similarly, the downward flux exiting layer 2 is the sum of the flux transmitted by layers 1 and 2 plus the flux transmitted by layer 1, reflected multiple times between layers 1 and 2, and after a final reflection by layer 1 is transmitted by layer 2 to escape its bottom. The figure depicts the contributions to the flux from light being reflected between the two layers from zero up to N times. The reflectances and transmittances for the combined layers given by Equation 4.86 are obtained by extending the number of reflections between the layers to infinity. This identity is readily recognized by recalling that the sum of the infinite series ∞ ∑

xn = (1 − x)−1

(4.87)

n=0

provided |x| < 1. Setting x = R1 R2 completes the identity. Notice that in the Eddington approximation, if the layers have different scattering properties and one of the layers absorbs radiation, then R12 ≠ R21 . If one of the layers absorbs, then the reflectance of the combined system for radiation incident at the top differs from that for radiation incident at the bottom. Looking down at a smoke layer overlying a cloud deck, or even looking at light reflected by clouds passing behind a layer of smoke, one observes a reduction in the reflected light that passes through the smoke layer. The same layer of smoke beneath or behind a similar cloud would have no noticeable effect on the light reflected by the cloud. Interestingly, because clouds do not absorb radiation at visible wavelengths, a nonabsorbing haze layer, such as a layer composed of sulfates or sulfuric acid droplets, has the same effect whether it is placed above or below the cloud. Under conditions of conservative scattering and no absorption, R12 = R21 . Because our

4.10

F −0

Adding a Surface with a Nonzero Albedo in the Eddington Approximation

+T1R2 (R1R2)T1 +

F0 = F0− [R1

… +T1R2 (R1R2)N T1]

+T1R2 (R1R2)2 T1

+T1R2T1

τ=0 R1, T1, and τ1

τ = τ1

R2, T2, and τ2 τ = τ 1 + τ2 –

F2 = F0−T1[T2

+ (R1R2)2 T2

… +(R1R2)N T2]

+ (R1R2)T2 Figure 4.8 Illustration of the multiple reflections between two homogeneous layers having different optical properties. Layer 1 has reflectance R1 , transmittance T1 , and optical depth �1 and sits atop

layer 2 with properties given by R2 , T2 , and �2 . The figure schematically depicts reflections at the layer interface numbering from 0 to N.

eyes have logarithmic sensitivities to light, as opposed to linear sensitivities, seeing the effects of nonabsorbing hazes above or in front of highly reflective clouds is unlikely. The changes are small with the nonabsorbing haze above or below the cloud. Nonetheless, such small changes can be detected in some cases with sensitive radiometers, which measure radiances linearly. The transmittances for radiation incident at the top boundary or on the bottom boundary of combined layers are identical, T12 = T21 with or without absorption. In accurate numerical solutions to the radiative transfer equation, the same properties apply for the planar albedo and the corresponding transmittance [24]. If the layers are nonabsorbing, the planar albedo is unaffected by the vertical ordering of the layers. If absorption occurs in one of the layers, then the vertical ordering affects the planar albedo.

4.10 Adding a Surface with a Nonzero Albedo in the Eddington Approximation

For simplicity, surfaces that underlie the atmosphere will be assumed to reflect isotropically. Isotropic reflectors are also called Lambertian reflectors after Johann

123

124

4 Solutions to the Equation of Radiative Transfer

Heinrich Lambert who first hypothesized the law in the sixteenth century. Radiances reflected by such reflectors are isotropic and constant regardless of zenith and azimuthal directions. The light is also unpolarized, regardless of the polarization of the incident light. A nearly perfect Lambertian reflector is a white projection screen. It appears uniformly white when viewed from any direction. Consider the treatment for combining layers in the previous section. If layer 2 is replaced by a surface with a nonzero albedo so that R2 = �S , then the albedo of a homogeneous single layer atmosphere over a reflecting surface is given by ( )−1 R′ = R1 + T1 �S 1 − �S R1 T1 (4.88) Similarly, the downward flux at the surface is obtained by noting that the upward flux of the surface is given by F1+ = �S F1− and the downward flux is given by F1− = F1+ R1 + �0 QT1 Substituting flux as

F1+

=

�S F1−

(4.89a)

in Equation 4.89a leads to the solution for the downward

)−1 ( F1− = �0 Q 1 − �S R1 T1

(4.89b)

Notice that Equations 4.88 and 4.89b have been derived assuming that the atmosphere above the surface is homogeneous. If the atmosphere is inhomogeneous and absorption occurs within the atmosphere, then care is needed to use the proper value for atmospheric reflectances appearing in Equations 4.88, 4.89a, and 4.89b. The value of R1 within the parentheses of Equation 4.88 and appearing in Equations 4.89a and b is that for an upward flux incident on the bottom of the atmosphere. The value of R1 in the first term on the right hand side of Equation 4.88 is that for a downward flux incident on the top of the atmosphere.

4.11 Clouds in the Thermal Infrared

Now that the effects of scattering and absorption have been described, the effects of clouds on infrared radiances can be analyzed. Cloud droplets and ice crystals absorb radiation at wavelengths greater than about 1 μm. At thermal infrared wavelengths, � > 8 μm, the absorption is so strong that scattering by cloud particles can, for practical purposes, be neglected. Clouds absorb and transmit light but reflect only a small fraction. Often the reflectance is assumed to be zero. This assumption can be appreciated by considering the reflectances obtained with the Eddington approximation. For optically thick absorbing media the reflectance in the Eddington approximation is positive only if the optical properties are constrained as given by Equation 4.78 ( ) U −1 R= > 0 ⇒ 3 1 − �g > 4 (1 − �) (4.90) U +1 At the wavelengths of the thermal infrared, droplets and ice crystals are still large compared with the wavelength of the radiation 2�r∕� ≫ 1. The absorption

4.11

Clouds in the Thermal Infrared

cross section for large, strongly absorbing particles approaches the geometric cross section while the extinction cross section approaches twice the geometric cross section. As discussed in Section 3.3, the additional area for the extinction cross section is due to the diffraction of light around the object and into the forward scattering direction. As a result, the single scattering albedo approaches 0.5 while the asymmetry parameter is still rather large, greater than 0.7. Consequently, the condition for nonnegative reflectance in the Eddington approximation often cannot be met. Under such conditions, the reflectance can be set to zero. The reflectance will be small in any case, and thus the absorptance and emittance are given by one minus the transmittance. When clouds are present, it is common to treat them as “paper cutouts.” Clouds simply mask the portion of the Earth that they cover. Effects due to three-dimensional radiative transfer are taken to be negligible. For spatial scales as small as a few tens of kilometers, typical of climate scales, three-dimensional effects are sufficiently small that they become difficult to identify in observations of the radiance fields [25]. Consequently, emitted radiances at the top of the atmosphere are assumed to be given by ( ) I = 1 − AC IS + AC IC

(4.91)

with AC the fraction of the region covered by clouds, IS the average emitted radiance for the cloud-free portions of the region as given by, for example, Equation 4.10a with z = ∞, and IC the average emitted radiance associated with the overcast portions. The average radiance emitted by the overcast portions is given by [ ] 1 ) ( ) ( ′) ( ′ IC = 1 − �C IS − dT z , ∞, � B z ∫T (zC ,∞,�) ( ) + �C BC T zC , ∞, � +

1

∫T (zC ,∞,�)

) ( ) ( dT z′ , ∞, � B z′

(4.92)

with BC the Planck function at the temperature of the cloud top with altitude zC . The expression for average radiance emitted by the overcast portions resembles that for the cloud-free radiance. The differences are the first term, which accounts for transmission of radiation emitted beneath the cloud. The radiance emitted beneath the cloud is given in the brackets of the first term as the cloud-free radiance minus the contribution emitted by the atmosphere above the cloud layer. The second difference is that the emissivity of the cloud appears in place of the emissivity of the surface, which for simplicity has been set to unity in this book. The third difference is that the elevation of the emitting surface is that of the cloud top at zC . Algebraic manipulation of Equations 4.91 and 4.92 leads to ) ( IC = 1 − �C AC IS + �C AC IC′

(4.93)

with IC′ the radiance emitted by a cloud that completely covers the field of observation and behaves like a blackbody at altitude zC . The overcast radiance is

125

126

4 Solutions to the Equation of Radiative Transfer

given by ( ) IC′ = BC T zC , ∞, � +

1

∫T (zC ,∞,�)

( ) ( ) dT z′ , ∞, � B z′

(4.94)

The quantity �C AC is often referred to as the effective cloud fraction. As written, Equations 4.93 and 4.94 appear to apply only for a single layer cloud system. Depending on the degree of overlap in multilayered systems, relationships can be derived in terms of the emissivities, transmissivities, emission altitudes, and fractional cloud cover of the layers to render similar forms for the emission at the top of the atmosphere, at the surface, or for that matter, at any atmospheric level. The radiances are given in terms of an “effective” single layer cloud system having an “effective cloud cover fraction” and “effective altitude for emission.” Naturally, the complexity of the algebra increases substantially with the number of layers.

4.12 Optional Separation of Direct and Diffuse Radiances

The solutions for multiple scattering discussed in this chapter allow for relatively simple analytic solutions. The extension of these simple solutions to more accurate solutions is straightforward. The first step to improving accuracy is the separation of the diffusely scattered light from the direct beam. Diffuse and direct radiances were introduced with the single scattering approximation in Section 4.6. As it is linear, the radiative transfer equation allows the superposition of solutions thereby facilitating the separation of the diffuse radiances from the direct solar beam. Consider the radiative transfer equation for the total radiance as given, for example, by Equations 4.36a and b. As the solution for the direct radiance is a straightforward application of the Beer–Bouguer–Lambert law, the radiative transfer equation is often presented for the diffuse radiances only [1–3]. The strong forward peak in the scattering phase function poses challenges for methods that seek highly accurate radiances. Separating the radiance into a direct component, which has suffered attenuation because of the absorption and scattering out of the direct beam, and a diffuse component that has been scattered, greatly enhances the accuracy of the solutions. The improved accuracy, however, comes at the expense of much more complex analytic solutions even for the simplest approximations of the radiative transfer equation. With the direct and diffuse radiances combined, as in Equations 4.36a and b, relatively simple analytic solutions can be obtained for approximate forms of the radiative transfer equation, like those for the Eddington approximation in Section 4.8. When dividing the radiance into direct and diffuse components, the total radiance is given by I (�, �, �) = IDIR (�, �, �) + IDIFF (�, �, �). With sunlight as the source of light within the atmosphere, the direct component at a given atmospheric level is given by the direct beam that reaches the level. The direct beam that reaches the level has been attenuated through absorption and scattering out of the beam by the intervening atmosphere. The attenuated direct beam

4.13

Optional Separating the Diffusely Scattered Light

was the source within the atmosphere for the single scattering approximation in Section 4.6. With the attenuated direct beam as the source of light within the atmosphere, the equation of radiative transfer for the upward diffuse radiance is given by �

+ dIDIFF (�, �, �)

d�

+ = IDIFF (�, �, �)

− ∫0 + ∫0

1 ) ) + ( ( � �, � ′ , �′ [ d� ′ p �, �; � ′ , �′ IDIFF ∫ 4� 0 1 ( ) − ( ) d�′ p �, �; −�′ , �′ IDIFF �, �′ , �′ ] 2�

d�′

) � ( (4.95) p �, �; −�0 , �0 4� The minus signs in front of the cosines of the zenith angles in the phase functions of Equation 4.95 are reminders that one of the radiances is headed downwards and the other radiance is headed upwards. The radiative transfer equation for the downward diffuse radiance is similar to Equation 4.95 but with appropriate sign changes. Differences between Equations 4.36a and 4.95 are as follows: (i) The added term in Equation 4.95 is the source of scattered light at the level � due to the direct solar beam that has reached the level and (ii) for Equation 4.36a the (incident ) radiance ( )at the top of the atmosphere is the incident solar beam Q� � − �0 � � − �0 whereas that for the diffuse radiance is zero. Of course, these two renditions of the radiative transfer equation give the same total radiance, the sum of the direct and diffuse radiances. −Qe−�∕�0

4.13 Optional Separating the Diffusely Scattered Light from the Direct Beam in the Eddington and Two-Stream Approximations

Both the Eddington and a related two-stream approximation begin with the azimuthally averaged form of Equation 4.95 given by [ 1 ] 1 ) ) ( ( dI + � = I+ − � d�′ p �, � ′ I + + d�′ p �, −� ′ I − ∫0 d� 2 ∫0 ) ( −�∕�0 � (4.96a) p �, −�0 −Qe 2 and [ 1 ] 1 ) ) ( ( dI − � = I− − −� d�′ p �, � ′ I − + d�′ p �, −� ′ I + ∫0 d� 2 ∫0 ) ( −�∕�0 � −Qe (4.96b) p �, �0 2 with the subscripts and superscripts indicating that the radiances are diffuse, that the radiances and phase functions are azimuthally averaged components, and the functional dependence of the azimuthally averaged radiances on both the

127

128

4 Solutions to the Equation of Radiative Transfer

optical depth and cosine of the zenith angle have been dropped to simplify the notation. One approach for solving these equations was started by Schuster and Schwarzschild in the first decade of the twentieth century. They assumed that both the upward radiances and downward radiances had different magnitudes but were constant, independent of the zenith angles in their respective hemispheres, thus the term “two-stream approximation [1].” Applying these rules to Equations 4.96a and b, and integrating both sides of the equation over the cosine of the zenith angle, 0 ≤ � ≤ 1, yields a set of coupled equations given by [ ( )] ( ) 1 dI + (4.97a) = 1 − � 1 − � I + − ��I − − �0 Q0 e−�∕�0 �� �0 2 d� and )] [ ( ( ( )) 1 dI − − (4.97b) = 1 − � 1 − � I − − ��I + − �0 Q0 e−�∕�0 � 1 − � �0 2 d� ( ) with � �0 given Equation 4.43b and � given by Equation 4.80c. Performing the same procedures but applying the assumed dependence of the radiance on the cosine of the zenith angle in the Eddington approximation instead of that in the two-stream approximation leads to a different set of equations: dF = 4�I0 (1 − �) − �0 Qe−�∕�0 � d�

(4.98a)

and ( ) 4� dI0 (4.98b) = F 1 − �g + Qe−�∕�0 �g 3 d� which aside from the additional terms are identical to Equations 4.69a and b. Owing to their different assumptions for the dependence of the radiance on the zenith angle, the equations for the Eddington and two-stream approximations are different and produce different results. Nonetheless, they and a half-dozen or so variants of the Eddington and two-stream approximations explored during the latter half of the twentieth century have a similar structure [26]. In terms of the upward F + and downward F − irradiances, the equations for both approximations are given by dF + = �1 F + − �2 F − − �Q0 �3 e−�∕�0 d�

(4.99a)

and

( ) dF − = �2 F + − �1 F − + �Q0 1 − �3 e−�∕�0 (4.99b) d� For the two-stream approximation, the upward and downward fluxes are given by �I + and �I − . The Eddington approximation given by Equations 4.98a and b can be transformed into Equations 4.99a and b by noting that the net flux and the average radiance are given by

and

F = F+ − F−

(4.100a)

) ( �I0 = 2 F + + F −

(4.100b)

4.13

Optional Separating the Diffusely Scattered Light

consistent with Equations 4.74a and b. The boundary conditions at the top of the atmosphere and the surface are that no diffuse light is incident at either boundary. ( ) F + �S = 0 and F − (0) = 0 (4.101) The solutions to Equations 4.99a and b with boundary conditions (Equation 4.101) are given by ] ( ) ( ) ( ) ( )[ F + (0) = Z + �0 1 − T �S e−�S ∕�0 − Z− �0 R �S (4.102) and

with

( )] ( ) ( )[ ( ) ( ) F − �S = Z − �0 e−�S ∕�0 − T �S − Z+ �0 R �S e−�S ∕�0 [ ( ) ] ( ) �0 Q� �3 − �3 �1 − �2 �0 + �2 �0 Z �0 = 1 − k 2 �02

(4.104a)

( ) ] [ �0 Q� 1 − �3 − �3 �1 − �2 �0 + �1 �0 ( ) Z �0 = − 1 − k 2 �02

(4.104b)

+

and

−

with

(4.103)

)( ( 2 ) U − 1 ek�S − e−k�S ( ) R �S = D

and

( ) 4U T �S = D

(4.105a)

D = (1 + U)2 ek�S − (1 − U)2 e−k�S

(4.105b)

( )( ) � − �2 k 2 = �1 − �2 �1 + �2 and U = 1 k

(4.105c)

Expressions for �1 , �2 , and �3 for both the two-stream and the Eddington approximations are listed in Table 4.1 In the case of the Eddington approximation, the expressions for k 2 , U, R, and T are identical to their corresponding values derived for the simpler version of the approximation in Section 4.8. For a conservatively scattering atmosphere, � = 1, �1 = �2 = 3∕4(1 − g), and R and T are given by Equations 4.81a and b. The accuracy of the Eddington approximation with the diffuse and direct radiances separated will be illustrated in the next section. The section introduces another strategy for improving the accuracy of radiative transfer calculations, the inclusion of a portion of the light scattered in the forward direction with the unscattered incident radiance.

129

130

4 Solutions to the Equation of Radiative Transfer

Table 4.1 Parameters in Eddington and two-stream approximations for diffusely reflected and transmitted fluxes [26]. Eddington Parameter �1 �2 �3

1∕4[7 − �(4 + 3g)] −1∕4[1 − �(4 − 3g)] 1∕4(2 − 3g�0 ) Two-stream

Parameter �1 �2 �3

2[1 − �(1 − �)] 2�� �(�0 )

4.14 Optional The �-Eddington Approximation

As described in Section 4.12, the radiance can be divided into two parts: a direct beam radiance and a diffuse radiance. Similarly, in the �-Eddington approximation, pronounced “delta-Eddington approximation,” the unscattered incident radiance is augmented by including a fraction of the radiation scattered into the forward direction. In the delta approximation, the phase function is approximated by P′ (cos Θ) = 2f �(cos Θ − 1) + (1 − f )P(cos Θ)

(4.106)

with both P′ (cos Θ) and P(cos Θ) normalized the standard way as given by Equation 3.10 [27]. The delta function in Equation 4.106 is the Dirac �-function. It is zero for all scattering angles except for scattering in the forward direction, scattering angle Θ = 0∘ . The radiant energy scattered in the forward direction is added to the unscattered portion of the incident radiance. Combining the forward scattering peak of the phase function with the unscattered portion of the incident radiance permits the representation of P(cos Θ) in Equation 4.106 with a relatively low order series of Legendre polynomials, as is illustrated in Figure 4.9. The figure shows values of a scattering phase function typical of ice clouds. An accurate representation of the scattering phase function requires ∼104 polynomials to capture the scattering in the forward scattering peak (Figure 4.9b). Using Equation 4.106 as the representation of the phase function, a reasonably accurate reproduction is obtained with only 64 terms used for P(cos Θ) (Figure 4.9a). Owing to this low order representation, P(cos Θ) is often referred to as a truncated phase function. At small scattering angles, some of the scattering by the forward peak is missed (Figure 4.9a). Also, the values of P(cos Θ) away from the domain of the forward peak oscillate about the more accurate representation. These oscillations can have amplitudes ∼10% of the

4.14

104

102

106

106

Original Reconstructed

104 102 100

0

1 2 3 4 Scattering angle (°)

5

104

102

131

Original, N = 10,000

106 104 102 100

0

1 2 3 4 Scattering angle (°)

5

100

100

10−2

108

Truncated, N = 64 Phase function

Phase function

106 Phase function

108

108

Phase function

108

Optional The �-Eddington Approximation

0

30

60

90

120

150

Scattering angle (°)

(a)

Figure 4.9 The original phase function calculated from scattering theory for an ice cloud (solid line) and the reconstructed phase function, given by P′ (cos Θ) in Equation 4.106 using a delta function for forward scattering (a) and using a Legendre

10−2

180 (b)

0

30

60

90

120

Scattering angle (°)

polynomial series with 10 000 terms (b). The portion of scattering that is missed at small scattering angles by P′ (cos Θ) is represented by the difference between the solid and dashed curves shown in the inset of panel (a).

actual values depending on the number of terms used to represent P(cos Θ). Owing to the logarithmic scaling in Figure 4.9, the oscillations are not evident. These oscillations as well as the missed scattering in the forward direction can be greatly reduced by combining analytic solutions for first and second order scattering with calculations of radiances obtained using Equation 4.106 [2, 28]. In any case, the numerical burden of calculating accurate radiances and irradiances is greatly reduced by reducing the number of terms required to represent the forward peak of the scattering phase function. With Equation 4.106 representing the phase function and including the delta function representation of forward scattering as part of the unscattered incident light, the resulting radiative transfer equation for the azimuthally averaged radiance remains identical in form to Equations 4.96a and b but with p′ (�, �′ ) replacing the phase function p(�, � ′ ), � ′ = (1 − f �)�

(4.107)

replacing the optical depth �, and �′ =

(1 − f )� (1 − f �)

(4.108)

replacing the single scattering albedo �. When the phase function is approximated as a polynomial in the cosine of the scattering angle, the first order term

150

180

132

4 Solutions to the Equation of Radiative Transfer

of the polynomial gives the asymmetry parameter g. van de Hulst [7] noted that for optically thick atmospheres in which the scattering phase function had a large asymmetry parameter, the reflected and transmitted radiances were well approximated by those of an atmosphere composed of isotropic scatterers with the optical depth replaced by � ′ given by Equation 4.107, the single scattering albedo replaced by �′ given by Equation 4.108, and f = g [2–4, 7]. As a result, the optical depths used in the calculations are smaller, and the single scattering albedos are larger, thereby reducing significantly the computational burdens of obtaining radiances and irradiances. The relationships given by Equations 4.107 and 4.108 with f = g are known as van de Hulst’s similarity principle. The next largest contribution to forward scattering in a Legendre polynomial representation of the phase function comes from the second order term. For the Henyey–Greenstein phase function, the expansion coefficient for the second order term is g 2 . Setting f equal to g 2 gives the �-Eddington approximation. With the transformations given by Equations 4.106–4.108 applied, the equations for the �-Eddington approximation become identical to Equations 4.98a and b but with the primed quantities for the optical depth and single scattering albedo replacing the actual values and g′ =

g−f g = 1−f 1+g

(4.109)

replacing the asymmetry parameter g. In the �-Eddington approximation the total downward radiative flux is the sum of the downward diffuse and direct beam fluxes given by �′

− �S

− − (�S , �0 ) = FDIFF (�S′ , �0 ) + �0 Qe FTOT

0

(4.110)

Since the forward peak of the phase function in the �-Eddington approximation has been added to the direct flux, the approximation is unable to properly predict the downward diffuse flux for optically thin media. An improved estimate of the diffuse flux is obtained by subtracting the actual direct flux �0 Qe−�S ∕�0 from the total flux given by Equation 4.110. The contribution of the direct beam to the total flux becomes vanishingly small as the optical depth becomes large. For large opti− cal depths FDIFF (�S′ , �0 ) becomes an accurate estimate of both the downward total and diffuse radiative fluxes. Figure 4.10 shows reflectances and total transmittances for homogeneous layers having various optical properties. The reflectances and total transmittances are derived using the Eddington, �-Eddington, and “exact” radiative transfer calculations. The exact radiative transfer calculations were obtained using the discrete ordinate method coded as an algorithm known as DISORT. The discrete ordinate method and DISORT are briefly described in the next section. The accuracy of the numerical results is better than 0.1%. With the exception of the exact numerical results for the Rayleigh approximation, in which the Rayleigh phase function was used, all of the results were obtained using the Henyey–Greenstein phase functions.

4.14

1.0

Optional The �-Eddington Approximation

0.8 τ = 10

0.8 0.6 0.4

0.4

(a) Aerosol ω = 0.90 g = 0.70 Eddington δ-eddington Exact

0.4

τ=1

0.0 1.0

(c)

Cloud ω = 1.00 g = 0.85

0.8

0.8 0.6

0.2

0.8

0.4

0.6

0.2

0.4

τ=1

(e)

0

20

40

60

τ = 10

0.0 1.2

(d)

1.0

τ = 10

0.0

Aerosol ω = 0.90 g = 0.70

0.4

0.6

–0.2

(b)

τ=1

τ = 10

0.2

τ = 10

0.0 1.0 Transmittance

0.6

Rayleigh ω = 1.00 g = 0.00

0.2

τ=1

0.2 0.8 Relectance

τ=1

0.6

Rayleigh ω = 1.00 g = 0.00 Eddington Exact

0.2

τ=1

τ = 10 Cloud ω = 1.00 g = 0.85 Eddington δ-eddington Exact

0.0 80 0 Solar zenith angle

Figure 4.10 (a–f ) Reflectances and total transmittances as functions of the solar zenith angle. The results are for cases similar to those shown in Figure 4.7 but with the direct and diffuse radiation separated following Equations 4.99a and b. The results

20

40

(f) 60

80

are for the Eddington and �-Eddington approximations. They illustrate the accuracy of the Eddington and �-Eddington approximations through comparison with “exact” results calculated for the reflectances and corresponding total transmittances.

In Figures 4.10a and b the results are for a Rayleigh atmosphere. For Rayleigh scattering the asymmetry parameter g = 0, identical to that for isotropic scattering. Since in the Eddington approximation both the Rayleigh and isotropic scattering phase functions are identical, the Rayleigh and isotropic results are also identical. Furthermore, since the Eddington approximation uses a Henyey–Greenstein phase function, the Eddington and �-Eddington results are also identical. Even with the Rayleigh and isotropic phase functions used in the exact numerical calculations, the differences in the results are finite but small, and vanishingly small for large optical depths and the long optical paths of the large solar zenith angles. The tendency for conservatively scattering atmospheres to become more isotropic as the optical depth increases is clearly evident when comparing the results for � = 1 and � = 10.

133

134

4 Solutions to the Equation of Radiative Transfer

The effects of the asymmetry in the scattering phase function are illustrated with the aerosol g = 0.70 and cloud g = 0.85 phase functions. Clearly, the results for the �-Eddington approximation are superior to those for the Eddington approximation. In the case of the absorbing aerosol � = 0.9, the results of the �-Eddington approximation follow closely those of the exact numerical calculations for small solar zenith angles, at which the effects of the forward scattering become more prominent. The results for the Eddington approximation fall away from those of the exact numerical calculations at large solar zenith angles. At large solar zenith angles, the optical paths within the layer become sufficient to cause appreciable absorption. Owing to absorption, escaping light has been scattered only a few times. As a result, contributions to the transmittances from higher moments of the phase function become significant. Both the Eddington and �-Eddington approximations perform less well for these large optical path lengths. As they were designed for low-order scattering atmospheres, the two-stream approximation described in the previous section and the �-two stream approximation, perform better than the Eddington and �-Eddington approximations when the optical path lengths are small and when absorption is appreciable. The two-stream approximations, however, perform less well for optically thick and weakly absorbing atmospheres. For the cloud, there is no absorption and the phase function is highly asymmetric. As with the absorbing aerosol, the effect of the forward peak in the phase function becomes noticeable for small optical depths and small solar zenith angles. In fact, the phase function used in the source term for the Eddington approximation, Equations 4.65, 4.96a, and 4.96b, and Table 4.1, becomes negative for small solar zenith angles. As a result, when the optical depths are sufficiently small, the source radiances scattered in the upward direction becomes negative, causing the reflectances to become negative, as shown in Figure 4.10e. Similarly, the transmittances become greater than unity. These results are clearly unphysical. The �-Eddington and two-stream approximations avoid such problems. As illustrated in Figures 4.10e and f, for a nonabsorbing or weakly absorbing medium having large optical depth or for long optical paths, the source term contributes little to the reflectances and transmittances and the results for the Eddington and �-Eddington approximation merge. For both the Eddington and �-Eddington approximations the spherical albedos and corresponding transmittances are R and T given by Equations 4.76b and 4.77b for atmospheres that scatter and absorb and given by Equations 4.81a and b for conservatively scattering atmospheres. The accuracy of the spherical albedo for the case of an absorbing atmosphere can be improved but at the expense of numerically integrating the planar albedos and transmittances over the cosine of the solar zenith angles as in Equation 4.79c. Tables 4.2–4.4 lists numerical results for the reflectances and total transmittances for the two optical depths and two of the solar zenith angles shown in Figure 4.10. Results are listed for the Eddington and �-Eddington approximations and for the exact numerical radiative transfer calculations.

4.15

Optional The Discrete Ordinate Method and DISORT

Table 4.2 Single scattering albedo, asymmetry parameter, optical depths, solar zenith angles, reflectances, and transmittances for a Rayleigh and isotropically scattering layers. Rayleigh/isotropic � 1.0

g 0.0 � = 1.0

Spherical albedo Transmittance Solar zenith 0∘ 60∘ 0∘ 60∘

Eddington 0.4286 0.5714 Reflectance 0.3383 0.4903 Transmittance 0.5829 0.5242

Exact 0.4469 0.5532 0.3405 0.4988 0.6596 0.4999 � = 10.0

Spherical albedo Transmittance Solar zenith 0∘ 60∘ 0∘ 60∘

0.8823 0.1176 Reflectance 0.8529 0.8971 Transmittance 0.1470 0.1029

0.8833 0.1167 0.8524 0.8986 0.1476 0.1013

Results are given for the Eddington approximation and “exact” numerical radiative transfer calculations. The accuracy of the numerical results are ∼0.1%. The Rayleigh and isotropic phase functions are used in the exact radiative transfer calculations. The exact results for the Rayleigh phase function are shown in the table. They differ little from the exact results obtained using the isotropic phase function.

4.15 Optional The Discrete Ordinate Method and DISORT

In his classic book, Chandrasekhar [1] presented the solution to the equation of radiative transfer using a spherical harmonic decomposition of the radiance. In this decomposition the azimuthal dependence of the radiance is given by a cosine series. The zenith angle dependence is given by a series of associated Legendre polynomials in the cosine of the zenith angle. The scattering phase function is similarly rendered. For each azimuthal component of the radiance and scattering phase function, Chandrasekhar used Gauss quadrature points and weights to perform the integrals associated with the cosines of the zenith angles. The integrals become sums of the products of the quadrature weights, and the azimuthal components of the radiance multiplied by the scattering phase function evaluated at the quadrature values for the cosines of the zenith angles. The radiance

135

136

4 Solutions to the Equation of Radiative Transfer

Table 4.3

Same as Table 4.2 but for aerosol layers. Aerosol Eddington

�-Eddington

� 0.9000

g 0.7000 � = 1.0

�′ 0.8211 � ′ = 0.5590

g′ 0.4118

Spherical albedo Transmittance Solar zenith 0∘ 60∘

0.1255 0.6942

0.1255 0.6942

Exact 0.1586 0.6682

0.0767 0.1923

0.0702 0.1951

0.7747 0.6394 � ′ = 5.590 0.2496 0.0335

0.8101 0.5967 0.2764 0.0386

0.2018 0.3218

0.1915 0.3141

0.0572 0.0281

0.0577 0.0295

Reflectance 0.0300 0.2126 Transmittance

0∘ 60∘ Spherical albedo Transmittance Solar zenith 0∘ 60∘

0.8236 0.6171 � = 10.0 0.2495 0.0335 Reflectance 0.1555 0.3299

Transmittance 0∘ 60∘

0.0477 0.0288

Results are listed for both the Eddington and �-Eddington approximations. Henyey–Greenstein phase functions are used in the Eddington and exact numerical radiative transfer calculations.

at the quadrature values of the cosine become a vector and the phase function becomes a matrix, thereby transforming equations like Equation 4.95 for each azimuthal component of the radiance into coupled differential matrix equations. With expressions for the homogeneous and particular solutions substituted into the resulting differential equations, the system becomes a set of coupled algebraic equations typical of eigenvalue problems. Chandrasekhar followed the mathematics of the coupled differential equations to develop analytical representations of the solutions for several special cases. The approach pioneered by Chandrasekhar became known as the discrete ordinate method. Kuo-Nan Liou [29] adapted Chandrasekhar’s approach to obtain numerical solutions for reflected and transmitted radiative fluxes. Numerical solutions are obtained by truncating the Legendre polynomial series representations of the radiance and scattering phase after some suitable number of terms given by 2N. For example, in the Eddington approximation N = 1. The phase function in the Eddington approximation is given by Equation 4.65 and

4.15

Table 4.4

Optional The Discrete Ordinate Method and DISORT

Same as Table 4.3 but for cloud layers. Cloud Eddington

�-Eddington

� 1.0000

g 0.8500 � = 1.0

�′ 1.0000 � ′ = 0.2775

g′ 0.4595

Spherical albedo Transmittance Solar zenith 0 60

0.1011 0.8989

0.1101 0.8989

Exact 0.1344 0.8656

0.0467 0.1490

0.0423 0.1637

0.9150 0.8655 �′ 2.7750 0.5294 0.4706

0.9577 0.8239

Reflectance −0.0409 0.1983 Transmittance

0 60

Spherical albedo Transmittance Solar zenith

1.0223 0.8051 � 10.0000 0.5294 0.4706 Reflectance

60∘

� = 10.0 0.4118 0.5882

0∘ 60∘

0.5882 0.4118

0∘

0.5446 0.4554

� ′ = 2.775 0.4191 0.5880 Transmittance 0.5644 0.4125

0.4233 0.6037 0.5778 0.3951

includes P0 (cos Θ) = 1 and P1 (cos Θ) = cos Θ. The application of discrete ordinates for a homogeneous atmosphere yields a system of 2N × 2N coupled differential equations. The numerical procedures used to solve the resulting system were honed to achieve high numerical accuracy and efficiency by Knut Stamnes and his coworkers [20]. They extended the method to obtain accurate solutions of not only irradiances but also radiances. The resulting computer algorithm is known as Discrete Ordinate Radiative Transfer (DISORT). It is readily available on the Internet. The routine solves the radiative transfer equation for large numbers of quadrature points and large values of N. It is also capable of calculating results for planeparallel atmospheres divided into many homogeneous layers, each layer having different optical properties. As such, DISORT provides radiative transfer solutions for multilayered, inhomogeneous atmospheres. It includes the �-M extension in which the representation of the phase function is carried not to a second order, as in the �-Eddington approximation, but to order M − 1 = 2N − 1 [30]. Thomas and Stamnes [2] provide a thorough and readable description of the numerical approach. DISORT is the routine used to calculate the multiple scattering results labeled as “exact” in this book.

137

138

4 Solutions to the Equation of Radiative Transfer

4.16 Optional Adding-Doubling Method

Grant and Hunt [5] noted that starting in the 1940s the problem of plane-parallel radiative transfer was recognized as being treatable as a vector-matrix equation. The radiance within an atmosphere is represented as a vector and is linearly related to the radiances incident on the atmospheric boundaries. For example, the solution for the upward radiance leaving the top of the atmosphere is given by �+ (0) = ��− (0) + ��+ (�S )

(4.111)

with vector �− (0) the downward radiance at the top of the atmosphere and �+ (�S ) the upward radiance at the bottom of the atmosphere. These vectors represent an azimuthal component of the radiance at a set of quadrature points for the cosines of the zenith angles. All azimuthal components satisfy relationships of the form given by Equation 4.111. � and � are matrices giving the azimuthal components of the bidirectional reflectivity and transmissivity of the atmosphere for the cosines of the incident and exiting quadrature zenith angles. In this form, radiative transfer mimics the propagation of electromagnetic waves through transmission lines [31]. Numerical calculations are performed starting with an optically thin but finite layer for which single scattering is appropriate. Based on the solution for single scattering, the matrices for an optically thin layer are given by � � Rij (�) = p(�i , −�j ) aj 2 �i and

( ) � � � + p(�i , �j ) aj Tij (�) = �ij 1 − �i 2 �i

(4.112)

with �i and �j the quadrature values of the cosines for the zenith angles of the incident and exiting radiances and aj the quadrature weight for the jth quadrature point. With the quadrature representation, the phase function becomes a matrix. The first term of the transmissivity is the direct transmissivity with �ij the Kronecker delta symbol. To obtain the reflectivities and transmittivities for homogeneous layers of arbitrarily large optical depths, the method of adding layers, as was done for the Eddington approximation, is used. The radiance leaving the top of a two-layered atmosphere is given by the radiances incident on the top and bottom boundaries of the uppermost layer, layer 1, and is given by �+ (0) = �1 �− (0) + �1 �+ (�1 ).

(4.113)

Similarly, the upward radiance incident on the bottom boundary of layer 1 is given by �+ (�1 ) = �2 �− (�1 ) + �2 �+ (�2 )

(4.114)

The downward radiance leaving the bottom of layer 1 is given by �− (�1 ) = �1 �+ (�1 ) + �1 �− (0)

(4.115)

4.16

Optional Adding-Doubling Method

Substituting Equation 4.115 into Equation 4.114 and rearranging terms gives (� − �2 �1 )�+ (�1 ) = �2 �1 �− (0) + �2 �+ (�2 )

(4.116)

with the symbol � representing the identity matrix, with ones on the diagonal and zeros off the diagonal. The upward radiance at the bottom of layer 1 is thus given by �+ (�1 ) = (� − �2 �1 )−1 �2 �1 �− (0) + (� − �2 �1 )−1 �2 �+ (�2 )

(4.117)

with the products representing matrix products and (� − �2 �1 )−1 representing an inverse matrix. The reflectivity of the combined layers with a downward radiance incident on the top boundary is given by � = �12 = �1 + �1 (� − �2 �1 )−1 �2 �1

(4.118)

The transmissivity for the combined layers with a radiance incident on the top boundary is given by � = �12 = �1 (� − �2 �1 )−1 �2

(4.119)

In the above expressions the convention is to place the layer number associated with the layer for which downward light is incident on its top boundary, layer 1, as the first index of the matrix and the number of the layer for which the transmitted light exits its bottom boundary, layer 2, as the second index. So �12 is the reflectivity for radiation incident at the top of layer 1 with the transmitted light exiting the bottom boundary of layer 2. Unlike the algebraic method for adding layers in the Eddington approximation, the order of vector and matrix manipulations requires attention. In creating a homogeneous layer of arbitrary optical depth, identical layers with �1 = �2 and �1 = �2 are combined, a procedure called doubling. Starting with a thin layer, for example, � = 2−10 ∼ 10−3 , 10 doublings produces a layer with an optical depth of � = 1. With a few more doublings, the optical depth becomes large. Inversion algorithms for matrices are fast and accurate, making doublings to generate homogeneous layers of substantial optical depth relatively rapid. For inhomogeneous atmospheres, the adding rules for two homogeneous layers are also given by Equations 4.118 and 4.119. When adding layers with different properties, not only must attention be paid to the sequence of the matrix and vector multiplications, but also to the direction of the incident light, whether upwards or downwards. The reflectivity and transmissivity matrices for a combination of inhomogeneous layers with downward radiances incident on the top boundary differ from those with upward radiances incident on the bottom boundary. When the direct beam and diffusely scattered components of the radiances are separated and the � − M adjustment is applied to treat the forward peak of the scattering phase function, the accuracy of adding and doubling radiative transfer routines are comparable to that of DISORT. The matrix multiplications and inversions tend to be faster when the atmosphere is homogeneous or when the number of atmospheric layers is relatively small.

139

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4 Solutions to the Equation of Radiative Transfer

4.17 Optional Monte Carlo Simulations

The most intuitive solution to the equation of radiative transfer is to mimic the statistical nature of a photon’s path as it wends its way through an absorbing and scattering medium. These simulations, known as Monte Carlo simulations, are often used for 3-D radiative transfer calculations in which the scattering and absorbing properties vary substantially in vertical and horizontal directions. Clouds, vegetation canopies, and nuclear reactors are typical of the complex spatial structures to which 3-D transfer calculations are applied. Monte Carlo methods, of course, are not the only approaches to account for the transfer of radiation in complex media. Methods, like the discrete ordinate method of the previous section, can be extended to three dimensions. Marshak and Davis [25] provide an excellent introduction to various approaches to 3-D radiative transfer. Consider photons flowing through a scattering and absorbing medium. As a photon traverses the medium, it travels in its direction of propagation until it interacts with an object, for example, a particle or hydrometeor that scatters and absorbs portions of the light. Let P(�)d� be the probability that a photon travels a distance � in the medium and then undergoes interaction in an infinitesimal distance d�. For simplicity, assume that the medium is spatially uniform with a constant number density n of radiatively active objects having extinction cross section �EXT throughout. The change in the probability of a photon interacting between � and � + d� is given by dP(�) = −n�EXT P(�)d�

(4.120)

so that P(L) = Ce−n�EXT L

(4.121)

with L some finite length within the medium and C = n�EXT = �EXT the extinction coefficient and also the normalization constant. The probability of interaction occurring is unity in an infinite medium. The inverse of the extinction coefficient is also referred to as the mean free path for interaction which is given by �MFP = ⟨�⟩ =

∫0

∞

d� �P(�) = 1∕n�EXT = 1∕�EXT

(4.122)

The mean free path is the average distance light travels before undergoing interaction. In Monte Carlo simulations, the paths traveled by photons are selected randomly so that the distribution of lengths with no interaction per unit length of the selected paths is constrained to be given by P(L). This condition is met by taking the cumulative distribution of the probability for paths up to length L. The cumulative distribution is given by �(L) =

∫0

L

d�P(�) = 1 − e−n�EXT L

(4.123)

4.17

Optional Monte Carlo Simulations

Recognizing that 0 ≤ �(L) ≤ 1, the value of L can be chosen randomly by setting �(L) = �� with �� a random variable uniformly distributed on the interval 0 ≤ �� ≤ 1. For random number �� , L is given by L=−

ln(1 − �� ) n�EXT

(4.124)

and is distributed according to P(L) given by Equation 4.121. Upon interaction, the photon will either be absorbed or scattered. The probability of scattering or absorption is again based on a random number uniformly distributed between 0 and 1, �� . The random numbers �� and �� are different and are chosen to be independent of each other. If �� < �, the single scattering albedo, then the photon is scattered; otherwise it is absorbed. For scattering, the scattering angle must be randomly chosen so that the direction of propagation within an increment of solid angle is changed to another direction and increment of solid angle. The probability that the scattering angle is less than Θ is obtained from the scattering phase function. For scattering that is rotationally invariant about the direction of the incident light, the probability is given by Θ

�(Θ) =

1 P(Θ′ ) sin Θ′ dΘ′ 2 ∫0

(4.125)

with P(Θ) the scattering phase function. As with the distance to the interaction with a particle, the value of the scattering angle is chosen by assuming that �(Θ) is a random number �Θ distributed uniformly between 0 and 1. The angle of rotation � associated with the orientation of the scattering plane shown in Figure 3.2 is also chosen randomly. Since the scattering is assumed to be rotationally invariant about the direction of the incident light, the rotational angle is given by � = 2���

(4.126)

with �� uniformly distributed between 0 and 1. Both the scattering angle Θ the rotational angle � are defined in conjunction with the direction of the incident photon. Consequently, rotations are required to specify the directions of the incident and scattered photons in a common coordinate system, the “laboratory system,” the xyz coordinate system shown in Figure 3.2. Given the path of the incident photon, the laboratory system is rotated so that the direction of the incident photon becomes the axes of rotational symmetry in the scattering plane. Once the scattering angle and rotational angle of the scattered photon are randomly chosen, a second rotation is performed to rotate the direction of the scattered photon back into the laboratory system. After the scattering, the scattered photon continues along its direction of propagation until it either interacts, or escapes the medium. If the source of light in the atmosphere is a downward radiance incident on the top of the atmosphere and if the photon escapes the medium in the upward direction, it is counted as reflected. If the photon escapes in the downward direction, it is counted as transmitted. Photons that fail to escape are counted as absorbed.

141

142

4 Solutions to the Equation of Radiative Transfer

The above description of a Monte Carlo scheme is deceptively simple. As they are statistical approaches to radiative transfer, Monte Carlo schemes provide only statistical estimates of the reflected, transmitted, and absorbed light. Estimates of the reflected and transmitted fluxes require integrals of the radiances over the upward and downward directions. Accurate estimates of these fluxes within a percent or so often require millions and even hundreds of millions of photons in a simulation, depending on the complexity of the medium. More efficient approaches are required for applications in remote sensing in which radiances in a particular direction within a narrow increment of solid angle are needed. Methods such as forward and backward Monte Carlo techniques, among others, as well as some approximate methods can be more efficient and provide reasonable accuracy. Again, Marshak and Davis [25] provide an excellent introduction to such techniques.

Problems

1. In this book relatively simple analytic solutions to the radiative transfer equation for emitted radiances have been obtained by assuming that the Planck function is approximately given by ( )� P B� (T(P)) = B0 P0 with B0 the value of the Planck function for the temperature at pressure level P0 . Obtain suitable values for � by taking P0 = 1000 hPa and P = 280 hPa. Use the temperatures given below to obtain �. Test the validity of the approximation by comparing the estimate of the Planck function with its actual value at 500 hPa using the temperature given for the level.

4.3 m Pressure (hPa) 1000 280

Temperature (K) 288 212

Wavelength 11 m

15 m

Radiances (mW m−2 sr−1cm)



Pressure (hPa) 500

Temperature (K) 252

Radiances (mW m−2 sr−1cm) Approximate Exact

500

252

2. At a particular wavenumber, the Planck function within the atmosphere can be approximated by ( ) P B� (P) = B0 P0

Problems

with P0 some arbitrary reference pressure. The radiance observed at the surface by a radiometer looking up is given by Equation 4.7b. Show that for a well-mixed gas the downward radiance with the cosine of the zenith angle � = 1 is given by I�− (�S ) =

B0 g [(� − 1) + e−�S ] �P0 S

Use this result to show that in the limit of an opaque atmosphere, �s ≫ 1, the observed emission is equivalent to the Planck emission at optical depth � = �s − 1. Looking up at the atmosphere, the downward emission appears to come from a level that is at an optical depth unity from the surface. 3. Some of the well-mixed trace gases contributing to global warming absorb in the 8–12 μm infrared window. Use the expression for the upward emitted radiative flux at the top of the atmosphere given by Equation 4.28a to do the following: a. Show that if the absorption by a trace gas is weak, then the emitted flux in the spectral interval containing the absorption by the gas is approximately given by Equation 4.31 [ ( ) ] 1 PS 1 PT F + (0) Δ� = �BS 1 − + �BT Δ� 2 P� 2 P� with Δ� the width of the spectral interval. b. Assuming that this portion of the spectra is otherwise transparent and assuming that the trace gas does not absorb sunlight, obtain an expression for the radiative forcing due to the presence of the gas. 4. Aerosol burdens are derived from a satellite by differencing the observed reflectivity of a cloud-free ocean scene and that expected for cloud-free and aerosol-free conditions. In the single scattering approximation the reflectivity is given by R(�, �0 , �) = �S + RR (�, �0 , �) + RA (�, �0 , �) with the cosine of the satellite zenith angle �, cosine of the solar zenith angle �0 , relative azimuth �, reflectivity for molecular (Rayleigh) scattering given by RR , and that for scattering by the aerosol RA . The reflectivity of the ocean in the direction of backscattered light under cloud-free and aerosol-free conditions is �S = 0.005. Assume that the optical depth for molecular scattering is �R = 0.053. Assume a solar zenith angle �0 = 49∘ , satellite zenith angle � = 16∘ , and relative azimuth angle � = 173∘ . a. What is the scattering angle for the sun-ocean-satellite geometry? b. What is the value of the Rayleigh phase function for this viewing geometry? c. Use the single scattering approximation to calculate the contribution to the reflectivity due to Rayleigh scattering.

143

144

4 Solutions to the Equation of Radiative Transfer

d. Approximate the phase function of the aerosol by a Henyey–Greenstein phase function and let the reflectivity observed by the satellite be 0.02. Use the single scattering approximation to determine the optical depth of a typical maritime aerosol, single scattering albedo � = 1.0 and asymmetry parameter g = 0.77. e. Repeat the determination for a typical continental aerosol, single scattering albedo � = 0.90 and asymmetry parameter g = 0.68. 5. Follow the steps used to solve the radiative transfer equation in the Eddington approximation to show that the reflectance for a nonabsorbing, scattering atmosphere, single scattering albedo � = 1, is given by Equation 4.81a and the transmittance is given by Equation 4.81b. 6. Show that for two conservatively scattering layers with optical depths �1 and �2 both nonzero and �1 ≠ �2 and asymmetry parameters g1 and g2 both nonzero and g1 ≠ g2 , the rules for combining layers in the Eddington approximation are consistent with the solutions of the radiative transfer equation in the approximation as given by Equations 4.81a and b. For the combined layers the optical depth is given by � = �1 + �2 and the asymmetry parameter is given by g = (g1 �1 + g2 �2 )∕�. a. Based on this result, for a nonabsorbing, multilayered atmosphere, what effect does the vertical ordering of the different layers have on the planetary albedo? b. If within a multilayered atmosphere one of the layers absorbed a significant fraction of radiation, what effect would its placement within the ordering of the layers have on the planetary albedo? Consider placing the absorbing layer (i) near the top of the atmosphere and (ii) near the bottom of the atmosphere. The consistency of adding rules with the solution of the radiative transfer equation in the Eddington approximation also applies for atmospheres that absorb and scatter light. With absorption added, however, the complexity of the algebra becomes substantial. The simplest case is adding two identical homogeneous layers to obtain a layer with twice the optical depth. 7. Why don’t precipitating clouds disappear? During winters, clouds in the Pacific Northwest seem to rain without stopping. A typical value for the transmittance of a rain cloud is 0.2. Water is nonabsorbing at the wavelengths of visible light. The radius r of a cloud droplet is typically 10 μm. Assume that the scattering cross section is given by � = 2�r2 and that the concentration of droplets is 300 cm−3 . a. Based on the solution for the transmittance in the Eddington approximation, calculate the amount of liquid water (g cm−2 ) in clouds with a transmittance of 0.2. b. If a cloud is saturated and has a temperature of 273 K, calculate the amount of water vapor in the cloud and compare the amount of vapor to the amount of liquid water.

Problems

c. During winters a typical precipitation rate for the Pacific Northwest is 1 cm day−1 . At this rate, how long would it take to deplete the liquid water in a rain cloud? d. With the value of the precipitation rate given in (c), is it likely that the cloud will disappear? If the cloud should disappear, why do rain clouds persist? 8. Water droplets absorb strongly in the infrared. Within a few optical depths of cloud top, the net flux of thermal radiation approaches zero. The upward and downward emitted fluxes cancel each other. For low-level marine stratus, however, the emission from cloud top passes to space with almost no absorption in the 8–12 μm infrared window. Considering only absorption and emission by the cloud droplets and assuming that the temperature at the top of the cloud is 280 K, calculate the cooling rate (K h−1 ) due to the cloud top emission for the layer of the cloud above which the transmittance to space is greater than ∼0.1. Assume that the droplets have a radius of 10 μm and a concentration of 100 cm−3 . At a wavelength of 10 μm the single scattering albedo of the droplets is � = 0.46 and the asymmetry parameter is g = 0.92. An atmospheric scale height of 8 km may be used to convert the thickness of the emitting layer to the pressure difference across the layer. The Planck function for the window region may be assumed a constant equal to its value at 10 μm when calculating the emitted flux. 9. In the single scattering approximation, the radiance reflected by an atmosphere in the limit � → 0 is approximately given by I (�, �) = �0 Q

�P(cos Θ)� 4���0

with the cosine of the scattering angle given by √ √ cos Θ = 1 − �2 1 − �02 cos(� − �0 ) − ��0 with �0 the cosine of the solar zenith angle, �0 the solar azimuth angle, � and � the cosine of the zenith angle and azimuth angle for the reflected radiance, � the optical depth, � the single scattering albedo, �0 Q the solar flux incident at the top of the atmosphere, and P(cos Θ) the scattering phase function. The single scattering approximation is reasonably accurate for Rayleigh scattering in the Earth’s atmosphere at visible wavelengths. For Rayleigh scattering � = 1 and the phase function is given by 3 (1 + cos2 Θ) 4 Show that the flux of sunlight reflected by molecules in the Earth’s atmosphere is given by P(Θ) =

F + (0) =

3�0 Q� 1 � Q� d�(3 + 3� 2 �02 − �2 − �02 ) = 0 16�0 ∫0 2�0

As a result, the planar albedo due to molecular scattering is given by �(�0 ) = �∕2�0 . The albedo of the Earth increases with latitude in part because scattering by the atmosphere increases with the increasing solar zenith angle.

145

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4 Solutions to the Equation of Radiative Transfer

References 1. Chandrasekhar, S. (1960) Radiative 2.

3.

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Transfer, Dover Publishing, New York. Thomas, G.E. and Stamnes, K. (1999) Radiative Transfer in the Atmosphere and Ocean, Cambridge University Press, New York. Liou, K.-N. (2002) An Introduction to Atmospheric Radiation, 2nd edn, Academic Press, New York. Wendisch, M. and Yang, P. (2012) Theory of Atmospheric Radiative Transfer—A Comprehensive Introduction, Wiley-VCH Verlag GmbH, Weinheim. Grant, I.P. and Hunt, G.E. (1969) Discrete space theory of radiative transfer. I. Fundamentals. Proc. R. Soc. London, Ser. A, 313, 189. Hansen, J. and Travis, L. (1974) Light scattering in planetary atmospheres. Space Sci. Rev., 16, 527–610. van de Hulst, H. (1980) Radiation and Cloud Processes in the Atmosphere. Theory, Observation and Modeling, Multiple Light Scattering, Vol. 2, Academic Press, New York, London, Sydney, Toronto, San Francisco, CA. Soden, B.J. and Bretherton, F.P. (1993) Upper tropospheric relative humidity from GOES 6.7 μm channel: method and climatology for July 1987. J. Geophys. Res., 98, 16,669–16,688. Aumann, H.H., Broberg, S., Elliot, D., Gaiser, S., and Gregorich, D. (2006) Three years of atmospheric infrared sounder radiometric calibration validation using sea surface temperatures. J. Geophys. Res., 111, D16S90. doi: 1029/2005JD006822 Kaplan, L.D. (1959) Inference of atmospheric structure from remote radiation measurement. J. Opt. Soc. Am., 49, 1004–1007. Chahine, M.T., Pagano, T.S. et al. (2006) AIRS improving weather forecasting and providing new data on greenhouse gases. Bull. Am. Meteorol. Soc., 87, 912–926. Wielicki, B.A., Young, D.F., Mlynczak, M.G., Thome, K.J., Leroy, S., Corliss, J. et al. (2013) Climate Absolute Radiance and Refractivity Observatory (CLARREO): achieving climate change

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absolute accuracy in orbit. Bull. Am. Meteorol. Soc., 94, 1519–1539. Rodgers, C.D. and Walshaw, C.D. (1966) The computation of infra-red cooling rate in planetary atmospheres. Q. J. R. Meteorology. Soc., 92, 67–92. Fels, S.B. and Schwarzkopf, M.D. (1975) The simplified exchange approximation: a new method for radiative transfer calculations. J. Atmos. Sci., 32, 1475–1488. Coakley, J.A. Jr., and Ch´ylek, P. (1975) The two-stream approximation in radiative transfer: including the angle of the incident radiation. J. Atmos. Sci., 32, 409–418. Wiscombe, W.J. and Grams, G.W. (1976) The backscattered fraction in twostream approximations. J. Atmos. Sci., 33, 2440–2451. Eddington, A. (1916) On the radiative equilibrium of the stars. Mon. Not. R. Astron. Soc., 77, 16–35. Stephens, G.L. (1978) Radiative properties in extended water clouds. Part II: parameterization schemes. J. Atmos. Sci., 35, 2123–2132. Russell, P.B. et al. (1999) Aerosolinduced radiative flux changes off the United States mid-Atlantic coast: comparison of values calculated from sunphotometer and in situ data with those measured by airborne pyranometer. J. Geophys. Res., 104, 2289–2307. Stamnes, K., Tsay, S.-C., Wiscombe, W., and Jayaweera, K. (1988) A numerically stable algorithm for discrete-ordinatemethod radiative transfer in multiple scattering and emitting layered media. Appl. Opt., 27, 2502–2509. Crutzen, P.J. and Birks, J.W. (1982) The atmosphere after a nuclear war: twilight at noon. Ambio, 11, 115–125. Turco, R.P., Toon, O.B., Ackerman, T.P., Pollack, J.B., and Sagan, C. (1990) Climate and smoke: an appraisal of nuclear winter. Science, 247, 166–176. Alvarez, L.W., Alvarez, W., Asaro, F., and Michel, H.V. (1980) Extraterrestrial cause for the Cretaceous-Tertiary extinction. Science, 208, 1095–1108.

References 28. Nakajima, T. and Tanaka, M. (1988) Algorithms for radiative intensity calcuYurevich, F.B. (1983) The effect of tropolations in moderately thick atmospheres spheric aerosols on the earth’s radiation using a truncation approximation. J. budget: a parameterization for climate Quant. Spectrosc. Radiat. Transfer, 40, models. J. Atmos. Sci., 40, 116–138. 25. Marshak, A. and Davis, A.B. (eds) 51–69. 29. Liou, K.-N. (1973) A numerical exper(2005) 3D Radiative Transfer in Cloudy iment on Chandrasekhar’s discreteAtmospheres, Springer-Verlag, Berlin, ordinate method for radiative transfer: Heidelberg, and New York. applications to cloudy and hazy atmo26. Meador, W. and Weaver, W. (1980) spheres. J. Atmos. Sci., 30, 1303–1326. Two-stream approximations to radiative 30. Wiscombe, W. (1977) The delta-M transfer in planetary atmospheres: a method: rapid yet accurate radiative unified description of existing methods flux calculations. J. Atmos. Sci., 34, and a new improvements. J. Atmos. Sci., 1408–1422. 37, 630–643. 31. Redheffer, R.M. (1962) On the relation 27. Joseph, J.H., Wiscombe, W.J., and of transmission-line theory to scattering Weinman, J.A. (1976) The deltaand transfer. J. Math. Phys., 41, 1–41. Eddington approximation for radiative flux transfer. J. Atmos. Sci., 33, 2452–2459. 24. Coakley, J.A. Jr.,, Cess, R.D., and

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5 Treatment of Molecular Absorption in the Atmosphere 5.1 Spectrally Averaged Transmissions

Calculations of radiative heating require radiative fluxes at all of the wavelengths that contribute to the heating. The radiative transfer equation and its solutions are strictly valid for monochromatic radiation–radiation at a single wavelength. For monochromatic radiation, if an optical path is divided into two segments, the transmissivity for the path is the product of the transmissivities for the individual path segments as given by T(U1 + U2 ) = T(U1 )T(U2 )

(5.1)

with U1 and U2 representing the lengths of the two optical paths. Although lineby-line radiative transfer models solve the transfer equation monochromatically, the calculations are tedious and computationally expensive. Such treatments are not suitable for most modeling studies in the atmospheric sciences. Instead, the radiative transfer equation is typically solved for broad spectral intervals. For instance, the upward flux of emitted radiation at a particular level and for a particular wavenumber is given by 1

F�+ (z) = �BS� TF� (0, z) +

∫TF� (0,z)

dTF� (z′ , z)�B� (z′ )

(5.2)

with TF� (z′ , z) the transmittance given by Equation 4.13. The total upward flux is obtained by integrating the monochromatic flux over all wavenumbers. The integration may be approximated by a sum over a set of spectral intervals in which the average of the upward irradiance for spectral interval i with width Δ�i is given by F +i (z) =

1 d�F�+ (z) Δ�i ∫Δ�i 1

=

1 1 d� �BS� TF� (0, z) + d� dT (z′ , z)�B� (z′ ) Δ�i ∫Δ�i Δ�i ∫Δ�i ∫TF� (0,z) F� 1

= �BSi T Fi (0, z) +

∫T Fi (0,z)

dT Fi (z′ , z)�Bi (z′ ).

(5.3)

In this averaging, the variation with wavelength of the Planck function or for that matter, the incident sunlight, is typically small for the widths of the spectral Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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5 Treatment of Molecular Absorption in the Atmosphere

intervals being used. Their values may be assumed constant equal to their average values for the spectral interval. As constants the Planck function and incident sunlight can be taken outside of the integral in Equation 5.3. The average value for the Planck function is given by �Bi (z) =

1 d��B� (z) Δ�i ∫Δ�i

(5.4)

so that �Bi (z)Δ�i gives the integral of the Planck function for the interval. The integral that remains gives the spectrally averaged transmittance as T Fi (z′ , z) =

1 d�TFv (z′ , z) Δ�i ∫Δ�i

(5.5)

Expressions similar to Equations 5.3–5.5 hold for the spectrally averaged radiances with mean transmissivities replacing the mean transmittances and average radiances emitted by a blackbody replacing the average emitted irradiance. The total upward irradiance at level z is given by the sum of the average irradiances for the spectral intervals times the widths of the intervals, ∑ F +i (z)Δ�i (5.6) F + (z) = i

The irradiance averaged over a particular spectral interval as given by Equation 5.3 satisfies the same radiative transfer equation as that for monochromatic irradiances. As long as the mean transmittance is properly evaluated and the integral of the emission over the optical path is properly performed, obtaining the radiative flux integrated over a particular spectral interval, or for that matter, over all wavelengths would appear to be straightforward. Unfortunately, while the Planck function and the incident sunlight vary slowly with wavelength so that their mean values can be used for finite spectral intervals, absorption by molecules typically varies by orders of magnitude within fractions of a wavenumber. Owing to this variability, the rules that apply for determining the monochromatic transmissivities between two levels of the atmosphere do not apply for the average transmissivities. In fact, for the average transmissivities, T(U1 + U2 ) ≥ T(U1 )T(U2 )

(5.7)

The following example illustrates this behavior. Imagine a spectral interval within which an absorber has two different absorption cross sections and thus two different absorption coefficients �1 and �2 as is illustrated in Figure 5.1. Suppose that these two different absorption coefficients each occupy exactly half the width of the spectral interval. Figure 5.2 shows the average transmissivity for the interval. In the event that the absorber amount is small, so that the absorption is weak for the entire interval, the mean absorptivity increases and the mean transmissivity decreases linearly with the absorber amount. For sufficiently small absorber amounts the average of the two absorption coefficients �AVE approximates the correct average transmissivity (dashed line). With �AVE as a single absorption coefficient for the interval, the mean transmissivity for the optical

5.2

Molecular Absorption Spectra

κ1

κ2

Δν

Transmissivity

Figure 5.1 Hypothetical spectral interval in which a gas has two distinct absorption coefficients �1 and �2 . Each absorption coefficient occupies exactly half of the spectral interval. The entire interval is indicated on the abscissa by Δ�.

1.0

TAVE (κ1, κ2, U)

0.8

κ1 = 5 κ2 = 1

T (κAVE U) = exp(–κAVE U) κAVE = (κ1 + κ2) / 2 = 3

0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Absorber amount Figure 5.2 Average transmissivity and absorber amount for the hypothetical spectral interval illustrated in Figure 5.1. The values of �1 and �2 are given in units of inverse absorber amount. The average transmissivity for the interval (solid line) is generally greater than the transmissivity

obtained using the average of the two absorption coefficients (dashed line). The two are approximately equal only for small absorber amounts. In this optically thin limit the average transmissivity and the average absorptivity become linearly dependent on absorber amount.

path U1 + U2 would be given by the product of the mean transmissivities for the individual optical paths. The correct mean transmissivity, however, is given by the average of the transmissivities for the two half spectral intervals (solid line). As Equation 5.7 suggests, the mean transmissivity is generally larger and thus the mean absorptivity generally smaller than their counterparts obtained using the average of the absorption coefficients for the two half intervals. 5.2 Molecular Absorption Spectra

The rapid variation of molecular absorption coefficients with wavelength greatly complicates accurate calculations of the radiative fluxes and heating rates. This

151

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5 Treatment of Molecular Absorption in the Atmosphere

section and Sections 5.3–5.5 briefly describe the structure of molecular absorption spectra. The remainder of the chapter describes the effects of the absorption spectra on spectrally averaged transmissivities and transmittances as well as the approximations developed to reduce the effort required to obtain accurate radiative fluxes and heating rates. In addition to its kinetic energy, a molecule possesses internal energy. This internal energy is quantized. The discrete energy levels of a molecule have three components: electronic, vibrational, and rotational. The electronic state describes the energy of the electrons in the molecule. The vibrational state describes the energy associated with the spring-like oscillation of the bonds joining the atoms and the bending of the bonds. The rotational state describes the energy associated with angular motions of the molecule about its center of mass. As a molecule absorbs or emits photons, it changes energy levels in accordance with the energy of the photon absorbed or emitted. High energy photons at visible and ultraviolet wavelengths are associated with the electronic transitions. Low energy photons at infrared wavelengths are associated with vibrational and rotational transitions. Because the energy associated with an electronic transition is so much larger than those associated with vibrational and rotational transitions, the electronic transitions are invariably accompanied by a host of vibrational and rotational transitions. The excitation of vibrational and rotational energy states in an electronic transition is nearly unavoidable. When a molecule absorbs a high energy photon, it has to do something with the energy in excess of or less than that required to jump between lower level and upper level states of electronic energy. The associated changes in vibrational and rotational states of energy account for the energy not used in changing the electronic states. Similarly, since the vibration of chemical bonds requires much more energy than is required to spin a molecule, changes in rotational states of energy usually accompany changes in the vibrational states. The accompanying transitions create bands of absorption lines centered around a particular wavelength, as is the case, for example, in the absorption of sunlight by water vapor in the near infrared and the absorption of infrared radiation by the 15 μm band of CO2 . Consider the simplest molecule that has a permanent dipole moment, two different atoms bonded together. In its simplest form, the bond joining the atoms is rigid, thereby forming a structure like a dumbbell. Such molecules are referred to as rigid rotor molecules. No bond, however, is strictly rigid. The bond vibrates, constantly stretching and shrinking. In the simplest approximation molecular bonds vibrate in the same way that ideal springs vibrate, as harmonic oscillators. A molecule with a vibrating bond loses its vibrational energy by emitting a photon so that the molecule undergoes a vibrational transition, say from vibrational energy level n = 1 to a ground state, vibrational level n = 0, with n the vibrational quantum number, n = 0, 1, 2, 3, . . . . The vibrational quantum number gives the energy associated with the vibration of the bond EVn = (n + 1∕2)hcvV , with h the Planck’s constant, c the speed of light, �V the wavenumber associated with the vibrational transition, and c�V the bond’s vibration frequency.

5.2

Molecular Absorption Spectra

Accompanying the vibrational transition are a host of rotational transitions, say from rotational energy level ERJ with J the total angular momentum quantum number associated with the rotational energy level to rotational energy level ERJ−1 . The total angular momentum of the rotation is quantized. The total angular momentum quantum number is given by J = 0, 1, 2, 3, . . . . The energy of the photon associated with this transition is given by hc� = EV1 + ERJ − (EV0 + ERJ−1 ) = hc(�V + �R ) with EV1 − EV0 = hc�V the change in energy associated with the vibrational transition and ERJ − ERJ−1 = hc�R the change in energy associated with the rotational transition. Typically, �V ≫ �R , so that many rotational transitions accompany a vibrational transition. The rotational transitions create absorption lines at various wavenumber distances �R from the band center at �V . The 8.6 μm band of N2 O in Figure 5.3 shows the classic structure of a relatively simple vibration-rotation band for a rigid rotor molecule. As was noted in Section 2.1, time varying electric dipole moments represent the most prominent form of interaction between light and matter. Molecules composed of different atoms, such as CO, H2 O, and N2 O, have permanent electric dipole moments. Owing to its geometry, O3 also has a permanent dipole moment. The three oxygen atoms in ozone occupy the vertices of an isosceles triangle with an angle of 116.8∘ between the two oxygen–oxygen bonds having equal lengths [3]. Molecules with permanent electric dipole moments create oscillating dipoles when undergoing rotation. Transitions arising from changes in 1 0.95 0.9

Transmissivity

0.85 0.8 0.75 0.7 0.65 0.6 P branch

0.55 0.5 1120

1130

1140

1150

R branch

1160

1170

1180

1190

1200

1210

1220

Wavenumber (cm−1) Figure 5.3 Transmissivities calculated for the 8.6 μm vibration–rotation band of N2 O. The calculations were performed using a line-by-line radiative transfer model [1]. They were performed for the 1976 U.S. standard atmospheric profile of temperature and an

N2 O mixing ratio close to the 2010 value, 300 ppbv. The view zenith angle for the calculations is 0∘ . The spectral line parameters are from the HITRAN 2008 molecular database [2].

153

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5 Treatment of Molecular Absorption in the Atmosphere

the rotational energy levels give rise to what is known as a pure rotation spectrum. The pure rotation spectrum of water vapor at wavelengths greater than 12 μm has a major effect on the Earth’s radiation budget. Molecules with permanent dipole moments also create oscillating dipoles when their molecular bonds vibrate and bend. Molecules with symmetrical geometries, like CO2 , a linear molecule with the carbon atom sandwiched between the oxygen atoms, and CH4 , a tetrahedron with the hydrogen atoms surrounding the carbon atom, have no permanent electric dipole moments. They have no pure rotation spectra. Such molecules, however, acquire oscillating dipoles when the bonds between their atoms vibrate and bend. The 15 μm vibration–rotation band of CO2 is a prominent example. Vibrational and rotational states are primarily populated through collisions in the atmosphere at altitudes below 50 km. The population of the energy levels is given by a Boltzmann distribution, which gives the probability per unit energy of a molecule being in a state with energy E falling within an infinitesimal increment of energy, dE. The distribution is given by E

dP(E) ∝ e− kT dE

(5.8)

with k the Boltzmann constant and T the temperature of the atmosphere at the level of absorption and emission. Because the population of states follows the Boltzmann distribution, the emission source is given by the Planck distribution at the same temperature. This condition is called local thermodynamic equilibrium. The need for thermodynamic equilibrium between radiation and the energy levels of the molecules led Einstein to propose the existence of stimulated emission. When light at a wavelength associated with a transition interacts with a molecule in an excited state for the transition, it causes the molecule to emit light at the same wavelength. The emitted light propagates in the same direction as the incident light and has its electric field in phase with the light that caused the emission. This process is called stimulated emission. Einstein recognized the need for stimulated emission in order for the radiation source to be given by the Planck function when the energy levels of the emitting molecules are populated by collisions. Stimulated emission is used to generate coherent laser beams, polarized light at a single wavelength with one phase propagating in a narrow beam. At temperatures typical of the atmosphere, stimulated emission accounts for less than 1% of the emission. At altitudes in the Earth’s atmosphere >50 km, the molecular collision frequency is sufficiently low that the period between collisions becomes comparable and even smaller than the natural lifetimes of the excited vibrational and rotational states of molecules. As a result, the source of emission is no longer given by the Planck function. At high altitudes, the population of molecular levels must be calculated based on the incident radiances, which populate the energy levels through absorption. The decay of the long-lived molecular energy states spawns emission that departs from the Planck distribution for the temperatures associated with these levels. This behavior is referred to as nonlocal thermodynamic equilibrium. The focus of this book is on radiative transfer in the troposphere and stratosphere where the density of air is sufficiently large that local thermodynamic equilibrium prevails.

5.3 Positions and Strengths of Absorption Lines within Vibration-Rotation Bands

5.3 Positions and Strengths of Absorption Lines within Vibration-Rotation Bands

Molecular absorption spectra provide information on the geometry of the molecules. Based on their spectra, CO, CO2 , and N2 O are linear molecules. The atoms lie in a straight line. Such molecules have two, identical moments of inertia which are perpendicular to each other and to the line joining the atoms. The moment of inertia for such molecules is given by ∑ I= mi ri2 (5.9) i

with mi the mass of atom i and ri the distance of the atom from the molecule’s center of mass. The angular motion of a linear molecule about its center of mass is characterized by the molecule’s angular momentum. The angular momentum is given by L = I� with � the angular frequency of rotation. In terms of the angular momentum, the rotational energy is given by L2 (5.10) 2I Angular momentum is quantized so that L2 = ℏ2 J(J + 1) with ℏ, the Planck’s constant divided by 2� and J, the total angular momentum quantum number. As was noted in Section 2.4, Planck’s constant has units of angular momentum. The rotational energy of a molecule is usually expressed in terms of a molecular “rotation constant” B so that ER = hcBJ (J + 1) with the rotation constant having units of cm−1 . For a rigid rotor molecule, quantum mechanics restricts electric dipole transitions between rotational energy levels to changes between levels J and J ± 1. For the transition J + 1 to J, the rotational wavenumber is given by ER =

ERJ+1 − ERJ = hc�R so that �R = B [(J + 1) (J + 2) − J(J + 1)] = 2B(J + 1)

(5.11)

Thus, separation between adjacent lines in the vibration-rotation band of a rigid rotor molecule is given by the rotation constant, Δ�R = 2B. The lines have equal spacing, as illustrated in Figure 5.3. The rotation constant is readily determined by the location of the spectral lines. In turn, the bond lengths for rigid rotor molecules are determined through the relationship between the rotation constant and the moment of inertia. Transitions with ΔJ = −1 are labeled as the P branch whereas transitions with ΔJ = +1 are labeled as the R branch in Figure 5.3. Rotational transitions with ΔJ = 0 would lead to a Q branch, but these branches are absent for “rigid rotors,” as in the example shown in Figure 5.3. A Q branch would have appeared if the moment of inertia aligned with the linear axis of the nitrous oxide molecule had been nonzero. Then rotation about this axis would have given rise to an oscillating dipole. Such a Q branch is prominent in the absorption spectrum of carbon dioxide near 15 μm, as shown in Figure 5.4. Carbon dioxide, like nitrous oxide, is a linear molecule. Unlike nitrous oxide, however,

155

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5 Treatment of Molecular Absorption in the Atmosphere

Transmissivity

1.0 0.8 0.6 Pressure: 150 mb

0.4

R branch

P branch

Temperature: 215 K

0.2

Q branch

Path length: 1 m

0.0 630

640

650

660

670

680

690

670

Wavenumber (cm–1)

(a)

Transmissivity

1.0 0.8 0.6 0.4

Q branch

0.2 0.0 666

(b)

667

669

668

670

Wavenumber (cm–1) Figure 5.4 Transmissivities near 15 μm calculated for a 1 m path through air at a pressure of 150 hPa and a temperature of 215 K, conditions similar to those of the lower stratosphere (a), and an

expanded view of the Q branch (b). The P, Q, and R branches are for the 15 μm vibration–rotation band of CO2 . The concentration of CO2 in these simulations is 355 ppmv.

the lone carbon atom is sandwiched between two oxygen atoms. In nitrous oxide, the lone oxygen atom anchors one end of the molecule. The Q branch in the 15 μm band of carbon dioxide arises because the carbon atom moves up and down while the oxygen atoms on both sides move in directions opposite to that of the carbon atom. The carbon atom appears to be flapping the attached oxygen atoms like birds flap their wings. With this flapping, the molecule develops a nonzero moment of inertia aligned with line joining the atoms. The difference in the charge densities around the oxygen and carbon atoms creates an oscillating dipole when the molecule spins about its linear axis. Figure 5.4a shows the Q branch along with the P and R branches of the CO2 15 μm vibration-rotation band. Figure 5.4b shows an expanded view of the Q branch that lies between the dashed lines in Figure 5.4a. With ΔJ = 0 for the Q branch transitions, the absorption lines arise from changes in the rotational energy about the line joining the atoms. Once the geometry of a molecule is determined from the wavelength distribution of the absorption line positions and strengths, the bond lengths and the angles between the bonds can be deduced. Molecules which are more complex than rigid rotors, such as NH3 , H2 O, and O3 , are known as either symmetric or

5.3 Positions and Strengths of Absorption Lines within Vibration-Rotation Bands

asymmetric “top” molecules. Symmetric top molecules like NH3 have two distinct moments of inertia and therefore resemble toy tops. With two distinct moments of inertia, symmetric top molecules have two rotation constants. Asymmetric top molecules like H2 O and O3 have three distinct moments of inertia and three rotation constants. Whereas the rotational energies for symmetric top molecules have relatively simple algebraic expressions, and consequently, so do the positions and strengths of their absorption lines, those for asymmetric top molecules are not analytic. Classically, the rotation of an asymmetric top is chaotic. Books, like this book, generally have three distinct moments of inertia and exhibit this chaotic rotation when tossed in the air. Quantum mechanically, the rotational energies of asymmetric top molecules appear to take random values and the positions of the absorption lines appear to occur randomly with wavelength. Figure 5.5 shows a classic example of the randomness in line positions. It shows a portion of the pure rotation spectrum of water vapor. The lengths of the hydrogen bonds and the angle between the hydrogen atoms in the water molecule were deduced from the positions and strengths of absorption lines like those shown in the figure. Absorption by a single spectral line is proportional to the absorption cross section times the number of molecules per unit volume in the ground state associated with the particular transition. The “line strength” is the product of these two quantities. In the case of rigid rotor and symmetric top molecules the line strengths vary systematically with wavenumber. They depend on the rotational energy levels associated with the transition. For example, in the case of a rigid rotor molecule and a rotational transition from a ground state involving 1.0

Transmission

0.8

0.6

0.4

0.2

0.0 800

820

840

860

880

900

Wavenumber (cm–1)

Figure 5.5 Transmissivity calculated for a portion of the rotation band of water vapor between 800 and 900 cm−1 . The results are for the entire atmospheric column were obtained using a line-by-line radiative transfer model [1]. The temperature, pressure, and water vapor profiles are the same as those of the 1976 U.S. Standard

Atmosphere. Other gases that absorb in the spectral interval were not included in the calculations. The viewing zenith angle is 0∘ . The water vapor continuum, which is a major contributor to absorption in this spectral region, was not included in order to better show the line structure.

157

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5 Treatment of Molecular Absorption in the Atmosphere

the angular momentum quantum number J, the number of molecules per unit volume in the ground state is given by ERJ

nJ = N

gJ e− kT

(5.12)

QR (T)

In Equation 5.12 the rotational partition function is given by ∑ ∑ − ERJ nJ = N gJ e kT and is defined so that QR (T) = J

(5.13)

J

with N the total number of molecules per unit volume in the ground state associated with the transition and gJ the “degeneracy” of the rotational energy level. The degeneracy is the number of rotational states that have the same rotational energy ERJ = hcBJ(J + 1). These states arise because in addition to the total angular momentum, the component of angular momentum about a fixed axis in space is also quantized. The quantization is indicated by a second angular momentum quantum number m, which takes 2J + 1 values, −J ≤ m ≤ J with m = 0 as one possible value. In Equation 5.12 the degeneracy is given by gJ = 2J + 1 for a rigid rotor molecule. As a result, the strengths of the absorption lines in the N2 O vibrationrotation band diminish as J → 0 and thus m → 0 near the band center. Even when J is small and the Boltzmann factor exp(−ERJ ∕kT) is near unity, there are only a few values that m can take and thus only a few states that the molecule can occupy. At high rotational energies, J is large so that the Boltzmann factor becomes small and the strengths of the absorption lines are also small. Few molecules acquire sufficient energy from collisions to populate the ground states associated with the large J transitions. The peak in the line strengths occurs near the band center where the rotational quantum number J is sufficiently large so that the number of possible states is large but not so large that the associated rotational energy of the ground state is much larger than kT, the translational energy of the molecules. Finally, the much weaker absorption lines that appear to shadow the strong absorption lines shown in Figure 5.3 are rotational transitions associated with excited vibrational energy levels. At the higher vibrational energy levels fewer molecules populate the lower level ground vibrational states associated with these rotational transitions. Consequently, the strengths of the absorption lines for these upper level vibrational transitions are reduced in accordance with a Boltzmann distribution as in Equation 5.8 for the vibrational energies. In addition, if the bonds between the atoms vibrated as a simple harmonic oscillator, there would be no difference in the wavenumbers associated with the band centers. All the absorption lines in the vibration–rotation bands would fall on top of each other. The vibration of the bonds is not simple harmonic oscillation; it is anharmonic. As a result, transitions between elevated vibrational energy levels occur at wavelengths for which the band centers are slightly shifted from the wavelength associated with the transition between the ground vibrational energy level and that of the next lowest vibrational energy level. This shift in the positions of the band centers is noticeable in Figure 5.3. The transitions between upper level vibrational levels are called hot bands [4].

5.4

Shapes of Absorption Lines

In summary, molecular absorption gives rise to bands of absorption lines. The typical spacing between lines is 0–10 cm−1 . The lines are narrow. At standard temperature and pressure (STP), they occupy ∼0.1 cm−1 . At atmospheric temperatures, lines ranging ∼100 cm−1 from the band center have rotational energy levels that are sufficiently populated to give rise to significant absorption. Thus, absorption bands cover a couple of hundred wavenumbers and may be composed of hundreds to thousands of absorption lines, depending on the complexity of the molecule and its abundance in the atmosphere. 5.4 Shapes of Absorption Lines

Section 5.1 described the breakdown of the exponential law for average transmissivities and transmittances. This breakdown became evident in the early days of infrared spectroscopy. As long as the absorber amount was small and the absorption was weak, absorption by gases was observed to grow linearly with increasing absorber amount. Early investigators used this linear dependence to deduce absorption coefficients, equivalent to the average absorption coefficients introduced in the first section. When these coefficients were used in the Beer–Bouguer–Lambert exponential law, the absorption expected for large absorber amounts grossly overestimated the observed absorption. Furthermore, as the amount of the absorbing gas was increased, the absorption began to depend on the square root of the absorber amount. This apparent breakdown in exponential extinction indicated that substantial variations in absorption cross sections were occurring within the spectral resolution of the infrared spectrometers used in the observations. Further complicating matters, for mixtures of an absorbing gas, like CO2 , and a nonabsorbing gas, like Ar or N2 , the absorption depended not only on the partial pressure of the absorbing gas but also on the partial pressure of the nonabsorbing gas. The absorption depended on the square root of the total pressure of the gas, even though the total pressure could be varied by changing the amount of the nonabsorbing gas. To explain these early observations, H.A. Lorentz, the same Lorentz noted for work in special relativity and electrodynamics, suggested that the absorption cross sections varied enormously within tiny spectral intervals. Absorption occurred at discrete wavenumbers, as if oscillation at a single frequency was being excited. He modeled the absorption in terms of a spectrum in which there was a single absorption line with infinitesimal width. The spectra of the single absorption line was assumed to be represented by a Dirac delta function, given by δ(� − �0 ) =

∞ 1 sin(� − �0 )t 1 dt ei(�−�o )t = lim t→∞ � 2� ∫−∞ (� − �0 )

(5.14)

with � = 2�f the angular frequency, f the frequency, and �0 the central angular frequency associated with the oscillation. The representation of the Dirac delta function in Equation 5.14 is recognized as the Fourier transform of an oscillation having a single angular frequency �0 . If the oscillation is allowed to continue

159

160

5 Treatment of Molecular Absorption in the Atmosphere

indefinitely, then the width of the spectrum is infinitesimally narrow, as given by the delta function. An ideal oscillator has only one frequency and maintains oscillation at this frequency indefinitely. If on the other hand, the oscillation is interrupted after a finite time, the width of the spectrum broadens. Lorentz suggested that collisions between molecules interrupt absorption and emission. The width of the absorption features is thus governed by the period between collisions. To determine the effect of a finite period for the duration of an oscillation on the width of the spectral feature, let P(t) be the probability per unit time when a molecule has NOT undergone a collision for a period of time t. Assume that at the end of this period, the molecule undergoes a collision in an infinitesimal interval of time dt. Since collisions are assumed to occur uniformly with time, the likelihood that a collision will occur in time interval dt is given by dt∕�, with � the average period between collisions. The average number of collisions suffered by a molecule in a finite interval of time is given by nCOL = t∕�. The change in the probability that a molecule has NOT suffered a collision after time t and then suffers a collision within an infinitesimal interval dt is given by the product of P(t) and dt∕�, dP(t) = −P(t)

dt �

(5.15)

The resulting probability is given by t

P(t) = Ce− �

(5.16)

with C chosen so that P(t) is properly normalized, C = 1∕�. ∫0

∞

dtP(t) =

∫0

∞

dt − �t e =1 �

(5.17)

This normalization implies that for short periods of time, t∕� ≪ 1, the probability that a molecule has NOT undergone a collision remains relatively large and falls linearly with time P(t) ∼ 1 − t∕�. For long periods of time with t∕� ≫ 1 the probability that a molecule has NOT undergone a collision becomes vanishingly small P(t) ∝ exp(−t∕�). Notice that the distribution P(t) gives � as the average period between collisions. ⟨t⟩ =

∫0

∞

dt tP(t) =

∫0

∞

dt − �t te = � �

(5.18)

The distribution of angular frequencies for oscillations subject to interruption by collisions is given by f� = lim

t→∞

∫0

t→∞ ∫0

= lim

t

t

dt ′ P(t ′ )

′ 1 sin(� − �0 )t � (� − �0 )

1 ′ 1 dt ′ 1 sin(� − �0 )t − t�′ � e = [( ] )2 1 � � (� − �0 ) � � − �0 + 2 �

(5.19)

5.4

Shapes of Absorption Lines

Owing to the collisions, the oscillation is distributed over a range of angular frequencies. The distribution is normalized so that ∫0

∞

d� f� = 1

(5.20)

The probability distribution f� is known as the Lorentz line shape factor and has units of inverse angular frequency (rad s−1 )−1 . As is described later in this section, the broadening of the spectral lines through collisions is directly related to the pressure of the absorbing gas. Hence, the Lorentz line shape factor is also referred to as the shape factor for pressure-broadened lines. The normalization of the shape factor in Equation 5.20 can be readily shown using standard trigonometric substitutions or the calculus or residues. In terms of wavenumbers, angular frequencies are given by c (5.21) � = 2�f = 2� = 2�c� λ and the shape factor is given by f� d� = f� d�, so that fv = 2�cf� . The shape factor f� is thus given by �L 1 f� = (5.22) � [(� − �0 )2 + �L2 ] with 1 (5.23) 2�c� The shape factor f� has units of inverse wavenumbers (cm−1 )−1 . A measure of the width of a pressure-broadened absorption line is given by �L , which is known as the Lorentz half-width. When |� − �0 | = �L , the shape factor is half its maximum value f� = f� max ∕2. The maximum value occurs at the center of the absorption line, � = �0 . The Lorentz half-width is estimated by noting that the average period between collisions is given by the distance traveled by molecules between collisions divided by the average molecular speed V . The distance traveled between collisions is the molecular mean free path �mfp = 1∕(n�COL ) with n the number of molecules per unit volume and �COL the cross sectional area associated with molecular collisions. The average period between collisions is given by �L =

�=

�mfp V

=

1 n�COL V

(5.24)

The average molecular speed is derived from the average kinetic energy of the molecules. Since molecules travel randomly in all directions, the average speed is given by the root mean square of the velocity √ 3kT V = (5.25) m with k the Boltzmann’s constant, m the mass of the molecule, and T the temperature. The pressure of an ideal gas is given by P = nkT

(5.26)

161

162

5 Treatment of Molecular Absorption in the Atmosphere

Manipulation of Equations 5.23–5.26 leads to the Lorentz half-width at STP given by √ P� 3 �L0 = 0 COL (5.27) 2� kT0 mc2 with P0 = 1 atm and T0 = 273.15 K. For absorption in the 15 μm band of CO2 at STP, �L0 ∼ 0.05 cm−1 , which gives a collision cross section �COL ∼ (10−7 cm)2 . For any other pressure and temperature, the half-width is given by √ T0 P (5.28) �L = �L0 P0 T The Lorentz half-width is proportional to the pressure of the gas doing the absorption. As atmospheric pressure runs over 3 orders of magnitude from the surface to the stratopause, the dependence of line width on pressure must be accounted for in evaluating radiances and irradiances. In comparison, the dependence on temperature is relatively small. For the same range in altitude, atmospheric temperatures range between 200 and 300 K. Reasonably accurate (5–10%) estimates of radiances and radiative fluxes can be obtained using a mean temperature for the optical path to evaluate the associated line half-width. 5.5 Doppler Broadening and the Voigt Line Shape

At low pressures, such as those in the middle and upper stratosphere, molecular speeds have a noticeable effect on line shapes. The wavenumbers of the absorbed or emitted light are Doppler shifted in accordance with the molecular velocity. The shift is given by ( ) V � = �0 1 ± (5.29) c with V the speed of the molecule, toward (+) or away (−) from the direction of light propagation and c the speed of light. The distribution of molecular speed is given by the Maxwell distribution. As was done in the case of a pressurebroadened line, the width of the unbroadened Doppler shifted line is taken to be infinitesimal, as given by the Dirac delta function. Consequently, the Doppler broadening shape factor is obtained by weighing the molecular velocities according to Maxwell’s distribution, ( )2 2 ∞ )) ( ( �−� 2 − 21 � 0 mc V − 21 mV kT kT 0 = �e fv = � � � − �0 1 ± (5.30) dV e ∫0 c with � a normalization constant. The normalization is given by ∫0

∞

d�f� = 1

so that fv = √

1 ��D

e

( )2 �−� − � 0 D

(5.31)

5.6

Average Absorptivity for a Single, Weak Absorption Line

The Doppler line width is given by √ 2kT �D = � 0 mc2

(5.32)

√ The half-width of a Doppler-broadened line is given by |� − �0 | = �D ln 2. It is the wavenumber distance from the line center at which the shape factor f� falls to half its maximum value. The maximum value occurs at the line center � = �0 . For CO2 at 15 μm and 220 ∘ K, which is typical of the lower stratosphere, the width of a Doppler-broadened line is 6 × 10−4 cm−1 . Assuming a typical value for the Lorentz half-width of ∼0.05 cm−1 at STP, the Doppler and Lorentz half-widths in the 15 μm band of CO2 have equivalent magnitudes at a pressure of approximately 10 hPa, or an altitude of approximately 30 km. Doppler effects become prominent in the middle to upper stratosphere. The form of the Doppler line shape factor suggests that Doppler and Lorentz line shape factors might be incorporated into a single shape factor. The combination is known as the Voigt line profile and is given by [ ] (v′ −v )2 ∞ − 20 �L � ′ D f� = C (5.33) dv e ∫0 (v − v′ )2 + � 2 L

with C a normalization constant so that ∫0

∞

dvf� = 1

(5.34)

The form of the Voigt profile indicates that the Lorentz line shape will dominate in the wings of an absorption line |� − �0 | ≳ �L . The Doppler line shape will dominate in the center of the line but only when |� − �0 | < �L < �D . There is no analytic form for the Voigt profile.

5.6 Average Absorptivity for a Single, Weak Absorption Line

To appreciate the impact of the rapid variation in molecular absorption cross sections with wavenumber, consider the mean transmissivity of a single absorption line. For a homogeneous path, the mean transmissivity is given by T � (U) =

1 d� exp(−k� U) Δ� ∫Δ�

(5.35)

with the mass absorption coefficient k� given by the product of the line strength SJ and the line shape factor f�J , k� = f�J SJ . The strength of the absorption line gives the product of the absorption cross section and the concentration of molecules in the ground state of the transition. The transition in Equation 5.35 is indicated by the index J. The line strength has units of wavenumbers times cross sectional area per unit absorber amount. With a mass absorption coefficient, the absorber amount is mass per unit area and the line strength has units of cm−1 cm2 g−1 .

163

164

5 Treatment of Molecular Absorption in the Atmosphere

Owing to its dependence on the concentration of molecules in the ground state, the line strength depends on the temperature of the gas as given, for example, by a Boltzmann distribution as in Equation 5.12. The line shape factor accounts for the spreading of the absorption cross section over wavenumbers. Because line shape factors are normalized, when the absorption is weak the mean transmissivity is given by T �J (U) =

SJ U 1 d�(1 − k� U) = 1 − Δ� ∫Δ� Δ�

(5.36)

with the width of the spectral interval Δ� assumed to be much larger than the width of the absorption line. A related quantity is the equivalent width of the absorption line. It is the product of average absorptivity, 1 − T � , and the width of the spectral interval and is given by WJ (U) = Δ�(1 − T �J (U)) = SJ U

(5.37)

The equivalent width gives the width in wavenumbers of a spectral interval over which the absorption would be complete. Alternatively, in the limit of weak absorption, the line strength SJ is recognized as the equivalent width of a spectral line per unit absorber amount. Note that the spectrally averaged radiance emitted by a single absorption line in an isothermal atmosphere, as for example, would be given by the spectrally averaged version of the downward radiance with � = 1 in Equation 4.11b, is the product of the Planck function evaluated at the temperature of the atmosphere and at the wavenumber of the line center times the equivalent width of the absorption line, I − (z, 1) = Bv WJ . From Equation 5.36, the line strength divided by the width of the spectral interval is the average absorption coefficient, SJ (5.38) Δ� Owing to the normalization of line shape factors, the mean transmissivity and the equivalent width of a weak absorbing line are the same regardless of the process causing line broadening, Doppler, Lorentz, or Voigt. k� =

5.7 Average Absorptivity for a Single, Strong, Pressure-Broadened Absorption Line

Another limit that provides a simple, analytic expression for the mean transmissivity is the limit of a strong, pressure-broadened absorption line. For a pressurebroadened line, the absorption coefficient is given by k� = f�J SJ =

SJ �LJ 2 �[(� − �J )2 + �LJ ]

(5.39)

For a strong line, the absorption is large at the line center. SJ U ��LJ

≫1

(5.40)

5.7

Average Absorptivity for a Single, Strong, Pressure-Broadened Absorption Line

with U the absorber amount and �LJ the half-width of the line. The equivalent width for such a line is given by

WJ (U) =

∫0

∞

⎛ ⎛ ⎞⎞ SJ �LJ U ⎜ ⎜ ⎟⎟ d� ⎜1 − exp ⎜− [( ] ⎟⎟ )2 2 ⎟⎟ ⎜ ⎜ � � − �J + �LJ ⎝ ⎝ ⎠⎠

(5.41)

As the absorption at the center of the line is so large, the integral may be approximated by setting �LJ = 0 in the denominator of the exponential argument. As � → �J , the actual value of the absorption coefficient becomes irrelevant. The absorption coefficient is so large that the absorption is complete, regardless of the coefficient’s actual value. The absorption is said to be saturated at the line center. With �LJ set to zero in the denominator, the integral becomes )) ( ( ∞ ∞ √ −2y �� = � 2y (1 − e−� ) (5.42) dx 1 − exp W� (U) = 2�LJ LJ ∫0 ∫0 �3∕2 x2 with x = (� − �J )∕�LJ , y = SJ U∕(2��LJ ), and � = 2y∕x2 . The integral is performed through repeated integration by parts and substitutions to produce √ √ WJ (U) = 2�LJ 2�y = 2 SJ �LJ U (5.43) The equivalent width for a pressure-broadened line can be obtained analytically with no approximations like those used above. The result is expressed in terms of Bessel functions, and in fact, is a tabulated function known as the LadenburgReiche Function [4]. The exact result, however, is of little use in atmospheric radiation. The most commonly used result is the equivalent width in the strong, pressure-broadened line limit, for which Equation 5.43 is sufficiently accurate. As was mentioned in Section 5.4, the mean transmissivity in the strong, pressure-broadened line limit is definitely not exponential in absorber amount. Suppose an absorption chamber is filled with a large amount of a nonabsorbing gas and a small trace of an absorbing gas. Also imagine that the partial pressure of the absorbing gas is small compared with the total pressure of the gas mixture within the chamber. If the amount of the absorbing gas is sufficient, Equation 5.43 suggests that the equivalent width would vary as the square-root of the absorbing gas amount U. Also, because the half-width �LJ is proportional to the total pressure as seen from Equation 5.28, the equivalent width would be proportional to the square root of the pressure of the gas. This behavior is precisely what Lorentz set out to explain by suggesting that spectrally unresolved absorption features had cross sections that varied by orders of magnitude within tiny spectral intervals in the infrared. The absorption coefficient, spectral absorptivity, and average absorptivity of a single, pressure-broadened line are shown in Figure 5.6. The absorptivity obtained using an average absorption coefficient derived from the absorptivity in the weak line limit matches the exact absorptivity in that limit. As the absorber amount increases, however, extrapolation of the weak line absorptivity grossly overestimates the actual absorptivity. Similarly, the absorptivity obtained for the strong,

165

5 Treatment of Molecular Absorption in the Atmosphere

Absorption coeff.

2.0

Absorptivity

(a)

1.5 1.0 0.5 0.0 1.0 0.8 0.6 0.4 0.2 0.0 980

(b)

1000 Wavenumber (cm–1)

1020

1 Ave. absorptivity

166

10–1

10–3 10–1 (c)

Exact 1 – exp(–SU/Δν) 2 Sα U/Δν

10–2

L

1

10

102

Absorber amount

Figure 5.6 Absorption coefficient for a single, pressure-broadened line (a), its spectral absorptivity for four absorber amounts (b) and its average absorptivity (solid line) (c). Also shown is the average absorptivity in the strong line limit (dotted line) and obtained by assuming an average

absorption coefficient � � = S∕Δ� (dashed line). The absorptivity obtained with the average absorption coefficient equals the average absorptivity in the weak line limit. In the example shown, the line strength S = 1 cm−1 per unit absorber amount. The pressure-broadened half-width �L = 0.2 cm−1 .

pressure-broadened line limit overestimates the absorptivity when the absorber amounts are small, but it equals the absorptivity once the absorber amount is sufficient to achieve the strong line limit.

5.8 Treatment of Inhomogeneous Atmospheric Paths

For atmospheric applications, calculations of transmissivities must account for the variations of pressure and temperature over the path traversed by radiation. Owing to the pressure dependence of the line width and the temperature dependence of the line width and line strength, radiation that is emitted low in the

5.8

Treatment of Inhomogeneous Atmospheric Paths

atmosphere encounters a range of cross sections for interaction as it traverses the atmosphere on its way to space or to high altitudes. Changes in temperature and pressure along an atmospheric path are accounted for through the optical depth. The optical depth associated with a particular absorbing gas i is given by d�� (z) = −k�i (z)ρi (z)dz

(5.44)

with k�i the mass absorption coefficient, ρi the density of the absorbing gas, and z the altitude. The mass absorption coefficient in turn is given by k�i (z) = Si (z)f�i (z)

(5.45)

with Si the line strength and fvi the line shape factor evaluated at wavenumber v. The optical depth for the path from pressure level P1 to pressure level P2 is given by P2

�� =

∫P1

dP r (P)Si (T)f�i (T, P) g i

(5.46)

with ri (P) the mass mixing ratio of the absorbing gas. Since the optical depth is obtained from the integral in Equation 5.46, the resulting wavenumber variation of the extinction is no longer given by f�i . It is far more complex. In detailed numerical schemes for the transfer of radiation, the temperature and pressure dependencies of the line strength and line shape factor are accounted for by dividing the atmosphere into a large number of layers so that the integral for the optical depth can be performed for each layer using a simple integration rule, such as the trapezoid rule. In the limits of weak lines and strong, pressure-broadened absorption lines, on the other hand, the effects of varying pressure and, to some extent, temperature over the optical path can be approximately accounted for analytically. For a weak absorbing line, the mean transmissivity is given by ) ( P2 dP 1 r (P)��i (P) T(P1 , P2 , �) = d� exp − ∫P1 �𝑔 i Δ� ∫Δ� ) ( P2 dP 1 r (P)Si (T)f�i (T, P) = d� exp − ∫P1 �𝑔 i Δ� ∫Δ� ) ( P2 dP 1 ri (P) Si (T)f�i (T, P) = d� 1 − ∫P1 �𝑔 Δ� ∫Δ� =1−

Si Ui ∕� Δ�

(5.47)

with P2

Si Ui =

∫P1

P

2 dP dP r (P)Si (T) = Si r (P) ∫P1 g i g i

(5.48)

For approximate estimates some temperature along the path is chosen for the evaluation of the line strength. The line strength is then treated as a constant Si (T) = Si and moved outside of the integral over the path. Since line shape factors

167

168

5 Treatment of Molecular Absorption in the Atmosphere

are normalized, as long as the absorption line is weak, integration over the spectral interval removes the line shape factor from the optical path integration. With a fixed value for the line strength at some suitable temperature, the integral over the optical path simply gives the absorber amount as given by the last integral on the right hand side of Equation 5.48. The path may be treated as a homogeneous path, albeit with some suitable effective temperature. In the case of a strong, pressure-broadened absorption line, the mean transmissivity is given by ( ) P2 Si (T)�Li (P, T) dP 1 r (P) T(P1 , P2 , �) = d� exp − 2 ∫P1 �𝑔 i Δ� ∫Δ� �[(� − �0 )2 + �Li (P, T)] ) ( P2 S (T)�Li (P, T) dP 1 r (P) i d� exp − ≈ ∫P1 �𝑔 i Δ� ∫Δ� �(� − �0 )2 √ P2 dP r (P)Si (T)�Li (P, T) 2 ∫P1 �𝑔 i =1− (5.49) Δ� As was the case for homogeneous paths, the spectral interval is assumed to be large compared with the width of the line. For a strong, pressure-broadened line, P2

Si (T) dP r (P) ≫1 ∫P1 �𝑔 i ��Li (P, T)

(5.50)

Consequently, the line half-width �Li may be set to zero in the denominator of the shape factor when performing the integral over wavenumber. For an inhomogeneous path, √ P2 dP 2 r (P)Si (T)�Li (P, T) ∫P1 �𝑔 i T(P1 , P2 , �) = 1 − Δ� √ P2 dP 2 r (P)Si �L0i (P∕P0 ) ∫P1 �𝑔 i =1− Δ� √ ′ 2 Si �Li0 Ui ∕� (5.51) = 1− Δ� As with the weak line limit, the temperature dependence of the line strength and half-width are approximated by choosing a suitable temperature along the path. The pressure dependence of the half-width is given by Equation 5.28. The absorber amount in Equation 5.51 is given by Ui′ =

P2

∫P1

P dP r (P) g i P0

(5.52)

5.9

Average Transmissivities for Bands of Nonoverlapping Absorption Lines

with P0 = 1 atm and Ui′ the pressure weighted absorber amount. Equivalently, an effective pressure is defined for an equivalent homogeneous path, U′ Pi EFF = i P0 Ui

(5.53)

For a strong, pressure-broadened line the inhomogeneous path can be treated in two ways. One approach is to treat the path as a homogeneous path at some temperature suitable for the path with the absorber amount given by the pressureweighted amount as given by Equation 5.52. The second approach is to treat the path as a homogeneous path with the product of the line strength and absorber amount given by Equation 5.48, but with an effective pressure for the line halfwidth given by Equation 5.53. For well-mixed gases, like CO2 , the effective pressure is the average pressure between the two pressure levels at the ends of the path. For water vapor the climatological mixing ratio in the troposphere is approximately given by ( r(P) = rS

P PS

)3 (5.54)

with rS the mixing ratio at the surface and PS the surface pressure. From Equations 5.48 and 5.52–5.54, the effective pressure for the absorption by water vapor in the strong, pressure-broadened line limit along the path from pressure level P to space is given by PEFF =

4 P 5

(5.55)

The idea of replacing the inhomogeneous path with that of a homogeneous path, in which the absorber amount is obtained in the weak line limit and the effective pressure is obtained in the strong pressure-broadened line limit, was first proposed by H.C. van de Hulst in the 1940s. Later it was independently proposed by A.R. Curtis and W.L. Godson [4]. The approximation is most commonly referred to as the Curtis–Godson approximation. While the approximation is valid only in the weak and strong, pressure-broadened line limits, its behavior for conditions between these limits is satisfactory for the simple calculations considered in this book.

5.9 Average Transmissivities for Bands of Nonoverlapping Absorption Lines

For a homogeneous optical path, the average transmissivity for a band of lines is given by ( ) ∑ 1 d� exp − k�J U (5.56) T � (U) = Δ� ∫Δ� J

169

170

5 Treatment of Molecular Absorption in the Atmosphere

with k�J the mass absorption coefficient for the absorption line specified by index J. In this section, only homogeneous paths are considered. As was noted in the previous section, if the absorption lines are either in the weak line or strong, pressurebroadened line limits, the arguments used in this section for homogeneous paths are readily extended to inhomogeneous paths. If the absorption lines do not overlap so that the equivalent width is the sum of the equivalent widths for the individual lines, then simple expressions can be obtained for the equivalent width of entire bands of lines. Two examples are common, bands of absorption lines in the weak line limit and bands of pressure-broadened absorption lines in the strong, nonoverlapping limit. The absorption coefficient at a particular wavenumber within an absorption band is given by the sum of the coefficients evaluated for the separate lines at the particular wavenumber, ∑ ∑ kv = k�J = SJ fvJ (5.57) J

J

For a homogeneous medium, the optical depth is given by �v = kv U

(5.58)

If the optical depth is small at all wavenumbers, then the equivalent width for the band is equal to the sum of the equivalent widths for the individual lines, ∑ W= SJ U = �U (5.59) J

∑

with � = S referred to as the band strength. Band strengths have been tabulated for many of the prominent absorption bands in the Earth’s atmosphere [4]. The approximation in Equation 5.59 is valid whenever the absorption is weak, or equivalently, the mean transmissivity is near unity. As an example, V. Ramanathan [5] noted that the bands of the chlorofluorocarbons (CFCs), CFCl3 and CF2 Cl2 , were in the weak line limit and used the limit to deduce the radiative forcing due to these gases. He suggested that if the manufacturing of CFCs had continued, the radiative forcing associated with these gases would have become comparable to that of carbon dioxide early in this century. This finding led to the recognition that growing concentrations of other trace gases that have absorption bands in the Earth’s infrared window, such as methane and nitrous oxide, together with the growth of CFCs would lead to a radiative forcing that could surpass that of carbon dioxide. Reasoning similar to that for the weak line limit leads to an expression for bands of strong, nonoverlapping, pressure-broadened absorption lines. If the lines do not overlap, then the equivalent width is again the sum of the equivalent widths for the individual lines. For strong, pressure-broadened lines, absorption is complete at the line center, SJ U ��LJ

≫1

(5.60)

5.10

Approximate Treatments of Average Transmissivities for Overlapping Lines

for all of the lines within the band. For nonoverlapping lines, absorption between the lines is small, which leads to 2�LJ d

≪1

(5.61)

with �LJ the Lorentz half-width for line J and d the average spacing between lines. This limit is often fulfilled. A stronger condition is that the equivalent width of the individual lines must be smaller than the line spacing, so that WJ d

≪1

(5.62)

For a band of nonoverlapping, pressure-broadened lines in the strong line limit the equivalent width is given by ∑ √ 2 SJ �LJ U (5.63) W= J

In the upper troposphere and stratosphere, many of the absorption lines of H2 O and CO2 are in the strong, nonoverlapping, pressure-broadened line limit. At high altitudes, the low pressures give rise to small line widths, but the absorber amounts for the paths to space coupled with the large line strengths lead to saturated absorption at the centers of the lines.

5.10 Approximate Treatments of Average Transmissivities for Overlapping Lines

When absorption becomes large, absorption lines overlap. Figure 5.7 illustrates the problem of overlapping lines. Consistent with the findings in the first section, the absorption in the overlapping limit is less than would be anticipated through extrapolation of the absorption obtained for the weak and the nonoverlapping, strong, pressure-broadened line limits. There are no accurate, analytical models for the equivalent width of a band of overlapping absorption lines. All models for the effects of overlapping lines are approximate and idealistic. Two such treatments are presented in this book. In both treatments, the average transmissivity is assumed to be independent of the positions of the absorption lines within the spectral interval. In one treatment, the absorption lines are assumed to fall randomly within a spectral interval so that the absorption due to any particular line is uncorrelated with any other line within the interval. This assumption leads to the “random model.” It yields a relatively simple form for the mean transmissivity of a spectral interval and has been used widely in calculations of radiances, radiative fluxes, and heating rates in planetary atmospheres. It is described in this section. In the second treatment a series of exponential transmissivities is summed to model the average transmissivity for a spectral interval. Each transmissivity within

171

5 Treatment of Molecular Absorption in the Atmosphere

Absorption coeff.

2.0

Absorptivity

(a)

1.5 1.0 0.5 0.0 1.0 0.8 0.6 0.4 0.2 0.0 980

(b)

1000 Wavenumber (cm–1)

1020

1 Ave. absorptivity

172

10–1

∑ 2 SJαLJU/Δν

10–3

10–1

(c)

Exact 1–exp(–∑SJU/Δν)

10–2

1

10

102

103

104

Absorber amount

Figure 5.7 Absorption coefficients for idealized pressure-broadened lines (a), absorptivity for three absorber amounts (b), and average absorptivity (solid curve) (c). Also shown are average absorptivities obtained by assuming∑ an average absorption coefS∕Δ� (dashed line) and ficient � � = obtained in the strong, nonoverlapping line limit (dotted line). The absorptivity obtained

assuming an average absorption coefficient matches the actual absorptivity in the weak line limit. For these results the line strengths are a progression starting with S = 1 cm−1 per unit absorber amount and each successive strength half of the preceding strength. The pressure-broadened half-width is �L = 0.2 cm−1 .

the series has an absorption coefficient and an associated weight. These absorption coefficients and associated weights are adjusted using fitting procedures so that the resulting average transmissivities yield accurate estimates of the interval’s contribution to average radiances, radiative fluxes, and atmospheric heating rates. The adjustments are based on results of line-by-line radiative transfer codes. The computational burden for a spectral interval is made to be small by limiting the series of exponential transmissivities to a relatively small number of terms. This approach is referred to as the exponential sum-fit and correlated-k methods. They are widely used for accurate narrow band and broadband radiative transfer calculations. These approaches will be described in the next section.

5.10

Approximate Treatments of Average Transmissivities for Overlapping Lines

In the 1950s, noting the randomness of absorption line positions in the pure rotation band of water vapor, Richard Goody proposed a random model for average transmission [4]. He assumed that within a spectral interval having a width Δv that is much larger than the average line spacing within the interval the absorption lines fall randomly with wavenumber. With this assumption, the mean transmissivity for a spectral interval is given by the product of the mean transmissivities of the separate lines. For example, consider a spectral interval with just two absorption lines. Assume that the transmissivity for the two absorption lines within the interval can be expressed as a mean value T and a departure from the mean that depends on the wavenumber within the interval T�′ . The transmissivity of each line within the interval is given by Ti� = T i + Ti�′ with T i =

1 d�Ti� and d�Ti�′ = 0 ∫Δ� Δ� ∫Δ�

(5.64)

For any two lines T=

1 1 ′ ′ )(T 2 + T2� ) = T 1T 2 d� T1� T2� = d�(T 1 + T1� Δ� ∫Δ� Δ� ∫Δ�

provided that 1 ′ ′ d�T1� T2� =0 Δ� ∫Δ�

(5.65)

For a random transmission model, the variations of the absorption with wavenumber within the spectral interval for the two lines must be uncorrelated as given by the last integral relation, the covariance of the transmissivities for the two lines in Equation 5.65. For N lines, the mean transmissivity is given by T=

N ∏ i=1

Ti =

N ∏ i=1

( 1−

Wi Δ�

) (5.66)

with Wi the equivalent width of the ith line. Because the source of radiation, either Planck emission or the incident sunlight, varies slowly with wavenumber, the placement of the absorption lines within the interval is immaterial. As a result, the equivalent widths for the individual lines Wi can be replaced by a mean value W so that the mean transmissivity can be expressed as T=

)N ( )N ( W W 1− = 1− Δ� N�

(5.67)

with � the mean spacing between the lines and N� = Δ�. In the limit as the number of lines becomes very large, the mean transmissivity is given by Goody and Yung [4]. ) ( W (5.68) T = exp − �

173

174

5 Treatment of Molecular Absorption in the Atmosphere

The average equivalent width is obtained by assuming a distribution for the strengths of the lines within the band, W=

∫0

∞

dS P(S) ∫0

∞

dv[1 − exp(−Sfv U)]

(5.69)

with P(S)dS the probability of finding lines with strength S falling within an infinitesimal interval of line strengths dS, f� the line shape factor, and U the absorber amount. Goody proposed that the probability distribution be given by a function with large numbers of lines having small strengths and few with large strengths. He chose P(S) to be given by S

e− � P(S) = �

(5.70)

This distribution is properly normalized. The average line strength is given by ⟨S⟩ =

∫0

∞

S

e− � =� dS S �

(5.71)

For the Goody random model, the average equivalent width is given by ∞

S

∞

e− � W= dS d�[1 − exp(−Sf� U)] ∫ � ∫ 0

(5.72)

0

The integral over line strength is readily performed giving W=

∫0

∞

d�

�f� U 1 + �f� U

(5.73)

The average equivalent width in Equation 5.73 can be calculated for any shape factor, although for all but the pressure-broadened line shape, the integral must be performed numerically. Also, Equation 5.73 allows for the treatment of inhomogeneous paths, although again, the calculations must be performed numerically. For the pressure-broadened line shape the integral over the wavenumber is easily performed and the mean equivalent width for the Goody random model is given by �U W= √ �U 1+ ��L

(5.74)

with �L the effective Lorentz half-width of the lines. The Goody random model returns the proper asymptotic expressions when the absorption is in the weak line and strong, nonoverlapping line limits. In the weak line limit, the absorption at the center of the line is weak, �U∕��L ≪ 1, so that the equivalent width of an average line is much smaller than the half-width of the line. The average equivalent width is W → �U. In addition, the average equivalent width must be smaller than the average line spacing, W ∕� > 1. Since the lines are nonoverlapping, the average equivalent width must be much smaller than the mean line spacing. Consequently, √ W → ���L U and T = exp(−W ∕�) → 1 − W ∕�. The weak and nonoverlapping, strong, pressure-broadened line limits provide a simple means for fitting the random transmission model to tabulations of line strengths and half-widths. In the weak line limit Equations 5.59, 5.68, and 5.74 lead to ∑ SJ J � = (5.75) � Δ� with Δ�, the width of the wavenumber interval over which the line strengths are summed. In the nonoverlapping, strong, pressure-broadened line limit Equations 5.63, 5.68, and 5.74 lead to ∑√ 2 SJ �LJ √ ���L J = (5.76) � Δ� ∑ ∑√ Houghton [6] provides values of S and S�L for the major infrared absorption bands of water vapor, carbon dioxide, and ozone at three different temperatures. The table in Appendix D lists values for the infrared absorption bands of carbon dioxide from 500 to 6000 cm−1 . Combining the expressions for the weak (Equation 5.75) and nonoverlapping, strong, pressure-broadened line (Equation 5.76) limits provides all the parameters required for the random model mean transmission, 2 ∑ ⎡ ⎤ SJ ⎢ ⎥ J � ⎥ = ⎢ ∑√ (5.77) ⎥ ��L ⎢ 2 S � J LJ ⎥ ⎢ ⎣ J ⎦ The random model accounts, at least approximately, for the effects of overlapping lines through the mean transmissivity given by Equation 5.68. The effects of overlap arise as the mean equivalent width for the lines becomes comparable to the average spacing between the lines. Figure 5.8 shows the average absorptivity, 1 − T, as given by a Goody random model for the five pressure-broadened absorption lines shown in Figure 5.7. Even though the spectral interval contains few lines and their positions are clearly not random, the Goody model captures fairly well the mean transmissivity from the limit of weak lines through the limit of nonoverlapping, strong lines, and finally to overlapping lines as the absorber amount increases. The distribution function proposed by Goody was thought to provide too little weight to weak lines. To enhance the weighting of weak lines, Malkmus [7] proposed a distribution function given by S

e− � P(S) = S

(5.78)

175

5 Treatment of Molecular Absorption in the Atmosphere

1 Ave. absorptivity

176

10–1 Exact Goody model 1 – exp(–∑SJU/Δν) ∑ 2 SJαLJU/Δν

10–2 10–3 10–1

1

102

10

103

104

Absorber amount Figure 5.8 Same as Figure 5.7c but including the average absorptivity obtained using the Goody random model (dashed-dotted line) with parameters given by Equations 5.75 and 5.77.

The distribution function is improper in that it cannot be normalized. Nonetheless, it yields � as the mean line strength. The average equivalent width for the Malkmus model is given by W=

∫0

∞

S

e− � dS S ∫0

∞

d�[1 − exp(−Sf� U)]

(5.79)

Rodgers [8] pointed out that the integral over the line strength is readily performed by setting the lower limit of integration to � and taking the limit � → 0, ∫0

S

∞

dS

e− � [1 − exp(−Sf� U)] = lim �→0 ∫� S

∞

− ∫�

∞

S

dS

e− � S

( )) S( exp − 1 + �f� U � dS S

(5.80)

Then with substitution of variables, the integrals over the line strength lead to W=

∫0

∞

d� ln(1 + �f� U)

(5.81)

As with Equation 5.73 for the Goody model, Equation 5.81 is applicable for all line shape factors and inhomogeneous paths. Like the Goody model, however, only the shape factor for pressure-broadened lines leads to analytic results. For pressurebroadened lines, the integral is readily performed. The mean equivalent width is given by [( ] )1 ��L 4�U 2 1+ W= −1 (5.82) 2 ��L with �L the effective half-width of the pressure-broadened lines. The Malkmus model provides the same weak line and nonoverlapping, strong, pressurebroadened line limits as given by the Goody model. As a result, Equations 5.75 and 5.77 can also be used to determine the parameters in the Malkmus model. Owing to the difference in their probability distributions for the line

5.11

Exponential Sum-Fit and Correlated k-Distribution Methods

Ave. absorptivity

1 10–1 Exact Malkmus model 1 – exp(–∑SJU/Δν) ∑ 2 SJαLJU/Δν

10–2 10–3 10–1

1

10

102

103

104

Absorber amount Figure 5.9 Same as Figure 5.8 but for the Malkmus random model (dash-dotted line) with parameters given by Equations 5.75 and 5.77.

strengths, the two models produce somewhat different mean transmissivities. Because it more heavily weighs weak lines, the Malkmus model allows more radiation to pass through a given amount of atmosphere than does the Goody model. Figure 5.9 shows the absorption obtained with the Malkmus model for the five pressure-broadened absorption lines shown in Figure 5.7. Comparison of the results in Figures 5.8 and 5.9 suggests that the Goody and Malkmus models provide similar accuracies, at least for these five absorption lines. The Goody model appears to do a better job at capturing the mean absorptivity in the nonoverlapping, strong, pressure-broadened line limit, but clearly both the Goody and the Malkmus models capture the overlapping line limit reasonably well. The chief distinguishing feature of the Malkmus model is that it provides an analytic expression for the probability distribution of the absorption coefficient within a spectral interval. This distribution function is a key element in the exponential sum-fit and correlated k-distribution models for mean transmissivities to be discussed next.

5.11 Exponential Sum-Fit and Correlated k-Distribution Methods

For a given spectral interval, the average transmissivity for a homogeneous path is independent of how the absorption lines are distributed with wavelength within the interval. This observation inspired Ambartzumian [9] and Lebedinsky [10] to develop the “k-distribution method,” which here is referred to as the exponential sum-fit method [11]. For a homogeneous path the average transmissivity of a spectral interval Δ� is given by Equation 5.56. Disregarding the position of the absorption lines within the interval leads to kmax

T(U) =

∫kmin

dk P(k)e−kU

(5.83)

with P(k)dk the probability of finding within the spectral interval an absorption coefficient k within an infinitesimal increment of values dk. Taking the limits

177

178

5 Treatment of Molecular Absorption in the Atmosphere

kmin → 0 and kmax → ∞ in Equation 5.83, the mean transmissivity is recognized as the Laplace transform of the probability distribution P(k). In the case of the Malkmus model for mean transmissivities, Domoto [12] obtained an analytic expression for the inverse Laplace transform. The inverse transform is given by √ ( )] [ ��L − 3 ��L k� � P(k) = 2− (5.84) k 2 exp − 2� 4� � k� For practical applications, however, finite values are used for kmin and kmax . Furthermore, Equation 5.83 gains practical value when the integral can be accurately approximated by a summation over a few discrete values of the absorption coefficient so that the mean transmissivity is given by ∑ T(U) = Pi e−ki U (5.85) i

with Pi the fractional weight used for absorption coefficient ki . Of course, in treating realistic atmospheric paths, one must account for the effects of temperature and pressure variations along the path. Typically, the assumption is that the ordering of the spectral lines in the wavelength domain is the same for different atmospheric layers. This assumption leads to the correlated k-distribution method [13, 14]. Fu and Liou [14] demonstrated that for (i) weak lines, (ii) single absorption lines with any line shape, (iii) random band models based on Equation 5.73 for the Goody model and Equation 5.81 for the Malkmus model, and (iv) bands of lines in which a single pressure-broadened line is repeated with constant spacing in wavelength satisfy the conditions required for the correlated k-distribution method to produce accurate mean transmissivities for inhomogeneous paths. For actual absorption spectra, however, line strengths and line half-widths vary from line to line and so does the temperature dependence of the line strengths. Consequently, numerical recipes have been developed to adjust the values of the absorption coefficients and their weights so that the mean transmissivities given by Equation 5.85 lead to the same radiative fluxes and heating rates in the atmosphere as those calculated with line-by-line radiative transfer models. The resulting absorption coefficients ki are often made to be relatively simple functions of temperature and pressure within the different atmospheric layers. Their corresponding weights Pi serve as the probability distribution function. In the case of the Malkmus model, decisions must be made concerning the number of absorption coefficients and their values. Once these values are determined then Equation 5.84 gives the corresponding P(k). Figure 5.10 illustrates the determination of the Pi and ki for the five pressurebroadened absorption lines shown in Figure 5.7. Figure 5.10a shows log10 k as a function of wavenumber and Figure 5.10b shows the probability distribution for log10 k. Logarithms of the absorption coefficient are used to highlight the wide range of values that the absorption coefficient can have within very narrow portions of the spectral interval. In Figure 5.10b the peaks and valleys of probability distribution for log10 k are reconciled with the wavenumber distribution of log10 k

5.11

Exponential Sum-Fit and Correlated k-Distribution Methods

Log10 (kν)

1 –1 –3 –5 980

Rel. freq. log10 (kν)

1000

1020

Wavenumber (cm–1)

(a) 10–2 10–3 10–4

Exact Malkmus model

10–5

Cum. freq. log10 (kν)

(b)

(c)

1.2 0.8 Exact Malkmus model Piecewise fit Average ki

0.4 0.0 –4

–3

–2 –1 Log10 (kν)

0

1

Figure 5.10 Logarithm of the absorption coefficients shown in Figure 5.7 (a), relative frequency distribution of log10 k� (b), and cumulative frequency of log10 k� (c). Also shown in (b) and (c) are the relative frequency and cumulative frequency distributions of log10 k� as given by the Malkums model (Equation 5.84) (dashed lines). In (c) the exact cumulative distribution is fit by

a piecewise model that is linear in log10 k� (dotted lines) within the bracketed intervals (closed circles). Also shown is the average value for the absorption coefficient (+) for each piecewise interval. These values and the weights derived from the cumulative distribution are used to calculate the average absorptivity given by the exponential sum-fit model in Figure 5.11.

by noting the number of times that a line at a constant value of the ordinate is crossed by log10 k in going from the smallest wavenumber to the largest wavenumber in the spectral interval. Figure 5.10c shows the cumulative distribution of log10 k. The distribution and cumulative distribution of log10 k derived from the Malkmus model as given by Equation 5.84 are also shown in Figures 5.10c and d (dashed lines). Clearly, the Malkmus model is an approximation. It lacks the structure evident in the exact distribution. Nonetheless, the Malkmus model matches reasonably well the cumulative distribution of log10 k.

179

5 Treatment of Molecular Absorption in the Atmosphere

1 Ave. absorptivity

180

10–1 Exact Exponential fit 1 – exp(–∑SJU/Δν)

10–2 10–3 10–1

∑ 2 SJαLJU/Δν

1

10

102

103

104

Absorber amount Figure 5.11 Same as Figure 5.9 but for the exponential sum-fit model (dash-dotted line). The average values of the absorption coefficients and their weights are shown in Figure 5.10. The absorptivity obtained with

the exponential sum-fit falls on top of the actual absorptivity. To make the curve visible, the exponential sum-fit absorptivity has been reduced by 10%.

Figure 5.11 shows the average absorptivity derived from an approximation to the cumulative distribution for P(log10 k). The approximation uses the seven discrete absorption coefficients shown by the plus symbols in Figure 5.10. The absorption coefficients were chosen by assuming that for finite intervals of log10 k, the cumulative distribution is approximately linear in log10 k as given by the piecewise fits associated with the dotted line segments. Using the resulting absorption coefficients for each of the seven intervals of the cumulative distribution, the average absorptivity of the exponential sum-fit model agrees nearly exactly with the actual absorptivity. The values shown in Figure 5.11 for the exponential sum-fit (dash-dotted line) are reduced by 10% to make the curve visible. Without the 10% reduction the curve would be hidden by the curve showing the exact absorptivity. Of course, the high accuracy comes from the use of 14 parameters, the 7 absorption coefficients and their weights, rather than the 2, the strong and weak line parameters, used to adjust the Goody and Malkmus models. Once the weights and absorption coefficients in the exponential sum-fit have been adjusted, each term in the series can be treated as representing the transmissivity for monochromatic radiation. As a result, the standard methods for solving the equation of radiative transfer can be applied for each of the effective absorption coefficients. The spectrally integrated radiance or irradiance is then obtained by taking the weighted sums of the corresponding values calculated for each of the absorption coefficients. The exponential sum-fit model and the correlated k-distribution models are particularly useful for problems in which both scattering and molecular absorption occurs, as for example, in clouds [15]. In media where both scattering and molecular absorption become significant, broadband transmission models, like the random models, become unsuitable because their mean transmissivities violate the exponential extinction associated with monochromatic radiation.

Absorption coefficient (cm2 g–1)

5.11

Exponential Sum-Fit and Correlated k-Distribution Methods

100

10−2

10−4

10−6 4300

4350

4400

4450

4500

4550

4600

Wavenumber (cm−1)

(a)

Probability density function

106 104 102 100 10−2 10−4 10−6

10−5

Absorption coefficient (cm2 g–1)

10−3

10−2

10−1 2

(b)

(c)

10−4

100

101

–1

Absorption coefficients (cm g ) 10

2

100 10−2 10−4 10−6

0

0.2

0.4

0.6

0.8

1

Cumulative probability

Figure 5.12 Example showing the application of the exponential sum-fit distribution to absorption by water vapor at T = 260 K and P = 400 hPa. Absorption coefficient as a function of wavenumber (a), probability distribution function of the absorption coefficients (b), and the cumulative probability distribution (c).

181

182

5 Treatment of Molecular Absorption in the Atmosphere

As a practical example of the exponential-sum fit, or k-distribution method, Figure 5.12a shows the absorption coefficient as a function of the wavenumber for absorption by water vapor in the near infrared. The spectral interval lies between the 1.9 and 2.7 μm absorption bands of water vapor. Figure 5.12b shows the probability distribution function for the absorption coefficient and Figure 5.12c shows the absorption coefficient as a function of the cumulative probability. Obviously, the absorption coefficient varies far more rapidly with wavenumber than it does with the cumulative probability. Consequently, numerical calculations of the average transmissivity become much easier and more economical using the dependence of the absorption coefficient on its cumulative probability distribution rather than on its wavenumber distribution.

5.12 Treatment of Overlapping Molecular Absorption Bands

Clearly, in many parts of the spectrum, absorption lines from different molecules overlap. One way to account for this overlap is to apply the assumption that the lines of one molecule are randomly distributed with respect to the lines of other molecules. The assumption leads to the same result obtained for random band models, the mean transmissivity for the overlapping bands is given by the product of the mean transmissivities for the individual molecules. This approximation is typically used for the random band models, and it is also commonly used for exponential sum-fit and correlated k-distribution models. In the case of the exponential sum-fit and correlated k-distribution models, the random overlap assumption leads to each absorption coefficient of one molecule being coupled with each absorption coefficient of the second molecule. The weights for the combinations of absorption coefficients are given by the products of the weights associated with the individual absorption coefficients.

Problems

1. Use the Curtis–Godson approximation to show that for a well-mixed gas in the weak line and strong, pressure-broadened line limits, the mean transmissivity obtained with a Goody random model for the path between the pressure level P and a satellite is given by ) ( P T = exp − Pm For the weak line limit Pm = �g�∕�r with �∕� given by Equation 5.75, � is the cosine of the zenith angle to the satellite, and g the acceleration due to √ gravity. For the strong, pressure-broadened line limit Pm = 2� P0 g�∕���L r √ with P0 = 1 atm and ���L ∕� given by Equation 5.76. Note that this expression for the transmissivity was used in Chapter 4 to investigate the relationship

Problems

between emission at the top of the atmosphere and the atmospheric layer that emitted the radiation. 2. Derive diffusivity factors associated with the weak and strong, pressurebroadened absorption line limits. The diffusivity factor was described in Section 4.3. a. Show that in the limit of weak absorbing lines, the correct diffusivity factor is 2.0. b. Show that in the limit of a strong, pressure-broadened line, the diffusivity factor is 16/9 = 1.78. 3. The water vapor rotation band overlaps the 15 μm band of CO2 . In the wings of the 15 μm band, low-level water vapor in the troposphere can diminish the effects of increasing carbon dioxide on the emitted flux at the top of the atmosphere. Using a Goody random model, calculate the mean transmissivity of water vapor for emission at the top of the atmosphere in the 25 cm−1 interval centered on 612.5 cm−1 . For this wavenumber∑ interval, the sum of the strengths of the water vapor lines is given by S =142.5 cm−1 (cm2 g−1 ) and the sum of the square roots of ∑ the √ strengths times the pressure-broadened half-widths is given by S�L = 1.22 cm−1 (cm2 g−1 )1/2 for a pressure of 1 atm. Assume that the mass mixing ratio for water vapor is given in terms of the pressure P by r(P) = rS (P∕PS )3 with rS = 0.008 kg H2 O/kg AIR the surface mixing ratio and PS = 1 atm the surface pressure. 4. CO2 is well-mixed throughout the atmosphere. Use the parameters below to calculate the mean transmissivities for the 725–800 cm−1 interval by (i) assuming that the mean transmissivity is the average of the transmissivities for the three subintervals and (ii) assuming that the mean transmissivity is that given by a single Goody random model for the entire 725–800 cm−1 interval. Perform the calculations for the path to the top of the atmosphere from the indicated pressure levels assuming pressure-broadened absorption lines. Assume that the cosine of the zenith angle is unity and that the concentration of CO2 is 390 ppm (parts per million). Compare the mean transmissivity obtained using the higher spectral resolution data with that obtained using the coarse resolution data.

Wavenumber interval (cm−1 ) 725–750 750–775 775–800 725–800

∑

S (cm−1 (g cm−2 )−1 ) 328 20.5 2.76 ?

∑√ S�L (cm−1 (g cm−2 )−1/2 ) 38.8 10.8 4.07 ?

183

184

5 Treatment of Molecular Absorption in the Atmosphere

Mean Transmissivities to Satellite Pressure level (hPa)

Single Goody model

Average of subinterval transmissivities

% difference

10 100 500 1000

References 1. Clough, S.A., Shepard, M.W., Mlawer,

2.

3.

4.

5.

6. 7.

8.

E.J., Delamere, J.S., Iacono, M.J., Cady-Pereira, K., Boukabara, S., and Brown, P.D. (2005) Atmospheric radiative transfer modeling: a summary of the AER codes. J. Quant. Spectrosc. Radiat. Transfer, 91, 233–244. Rothman, L.S., Gordon, I.E., Barbe, A. et al. (2009) The HITRAN 2008 molecular spectroscopic database. J. Quant. Spectrosc. Radiat. Transfer, 110, 533–572. Kotz, J.C., Treichel, P.M., and Townsend, J. (2012) Chemistry and Chemical Reactivity, Brooks/Cole, Cengage Learning, Belmont, CA. Goody, R.M. and Yung, Y.L. (1989) Atmospheric Radiation Theoretical Basis, Oxford Press. Ramanathan, V. (1975) Greenhouse effect due to chlorofluorocarbons: climatic implications. Science, 190, 50–52. Houghton, J.T. (2002) The Physics of the Atmosphere, Cambridge University Press. Malkmus, W. (1967) Random Lorentz band model with exponential-tailed S−1 line-intensity distribution function. J. Opt. Soc. Am., 57, 323. Rodgers, C.D. (1968) Some extensions and applications of the new random model for molecular band transmission. Q. J. R. Meteorolog. Soc., 94, 99–102.

9. Ambartzumian, V. (1936) The effect of

10.

11.

12.

13.

14.

15.

absorption lines on the radiative equilibrium of the outer layers of stars. Publ. Astron. Obs., Univ. Leningr., 6, 7–18. Lebedinsky, A. (1939) Radiative equilibrium in the Earth’s atmosphere. Proc. Leningr. Univ. Ser. Math., 3, 152–175. Wiscombe, W. and Evans, J. (1977) Exponential-sum fitting of radiative transmission functions. J. Comput. Phys., 24, 416–444. Domoto, G.A. (1974) Frequency integration for radiative transfer problems involving homogeneous non-gray gases: the inverse transmission function. J. Quant. Spectrosc. Radiat. Transfer, 14, 935. Lacis, A. and Oinas, V. (1991) A description of the correlated k-distribution method for modeling nongray gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres. J. Geophys. Res., 96, 9027–9063. Fu, Q. and Liou, K.-N. (1992) On the correlated k-distribution method for radiative transfer in nonhomogeneous atmospheres. J. Atmos. Sci., 49, 2139–2156. Lacis, A.A. and Hansen, J.E. (1974) A parameterization for the absorption of solar radiation in the earth’s atmosphere. J. Atmos. Sci., 31, 118–133.

185

6 Absorption of Solar Radiation by the Earth’s Atmosphere and Surface 6.1 Introduction

Lacis and Hansen [1] developed a parametric model for the absorption of sunlight by the Earth’s atmosphere and surface. A parametric model is a simplified description of a complex physical process. It achieves high accuracy by allowing parameters in the model to be adjusted so that the results obtained with the parametric model closely match those obtained with more detailed physical models. The model developed by Lacis and Hansen was designed to be a radiation module for numerical weather prediction and climate simulation. In such models, few computational resources can be devoted to the calculation of radiative heating rates and fluxes. The shortcuts adopted by Lacis and Hansen led to fast but relatively accurate estimates of the radiation field, thereby leaving most of the computational resources to the calculation of other factors governing weather and climate: winds, evaporation, condensation, precipitation, and so on. The Lacis and Hansen scheme has now been superseded by more complex, detailed, and more accurate methods [2–4]. Nonetheless, the simple scheme still provides a useful means for quickly calculating the scattering and absorption of sunlight in the Earth’s atmosphere and absorption by the Earth’s surface. In addition, as do the more complex radiation schemes, the Lacis and Hansen parameterization adheres to physical principles when calculating the combined effects of scattering and absorption by cloud droplets, ice crystals, and molecules. The parameterization differs from the most accurate calculations of solar heating in the numerical accuracies of the multiple scattering routines used in the calculations, in the details with which absorption by molecules is treated, and in the approximations used for scattering and absorption by aerosol particles, water droplets, and ice crystals. The spectral distributions of the Earth’s incident and reflected solar radiative fluxes were shown in Figure 1.8. The spectrum of the absorbed solar radiative flux is given by the incident minus the reflected radiative fluxes. The major absorption in the atmosphere is by water vapor and ozone. Oxygen and carbon dioxide, among other gases, also absorb sunlight but account for only 2–3% of the total absorption. In addition, the concentration of oxygen in the atmosphere is constant. While the concentration of CO2 in the atmosphere has increased by more Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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6 Absorption of Solar Radiation by the Earth’s Atmosphere and Surface

than a third since the latter half of the eighteenth century, the additional CO2 has had little effect on the absorbed sunlight.

6.2 Absorption of UV and Visible Sunlight by Ozone

Ozone has two prominent absorption features in the solar spectrum. It strongly absorbs UV radiation at wavelengths shorter than 340 nm. About 2% of the total incident sunlight is absorbed by ozone in the UV. Ozone weakly absorbs visible light between 450 and 750 nm, accounting for about 1% of the incident sunlight. In addition, almost all of the absorption occurs in the stratosphere where the concentration of ozone reaches its maximum. Because stratospheric ozone is above much of the atmosphere, absorption by ozone can be treated independently of scattering. Aside from occasional occurrences of polar stratospheric clouds and stratospheric aerosol layers that follow major volcanic eruptions, the scattering optical depth in the stratosphere is negligible. Sunlight is absorbed as it passes through the ozone layer. The sunlight that is transmitted by the layer is reflected by molecules, aerosols, and clouds in the lower atmosphere and by the Earth’s surface. The reflected light is subjected to absorption by ozone a second time as it passes through the layer on its way to space. Figure 6.1 shows the atmosphere divided into layers n = 1, 2, 3, to nS . Layer n = 1 is the top layer of the atmosphere and layer n = nS is the bottom layer. The layers are separated by levels � = 0, 1, 2, to nS . Level � = n is the bottom boundary and level � = n − 1 is the top boundary of layer n. As with optical depth, the counting begins at the top of the atmosphere with level � = 0. On the first pass of sunlight through the ozone in layer n, the amount of sunlight absorbed (W m−2 ) is given by Qn ABS FIRST PASS = �0 Q[A(Un ∕�0 ) − A(Un−1 ∕�0 )]

(6.1)

with �0 the cosine of the solar zenith angle and Q the solar radiative flux at the top of the atmosphere. In this chapter, Q is set equal to the solar constant, Q0 = 1360 W m−2 . For accurate calculations of absorption at a particular time, the value of Q has to account for the Earth’s elliptical orbit about the sun. In Equation 6.1 A(U) represents the fraction of sunlight absorbed for an absorber amount U. The absorber amount Un in Equation 6.1 represents the column amount of ozone above level � = n. The fraction of the incident sunlight absorbed above level � = n − 1 is subtracted from the fraction absorbed above level � = n. The remainder is the fraction that is absorbed by ozone in layer n. This strategy accounts for the non-exponential dependence of the average transmittance for finite spectral intervals, as discussed in Chapter 5, T(U1 + U2 ) ≥ T(U1 )T(U2 ). The sunlight that passes through the layer is reflected by molecules, aerosols, clouds, and the Earth’s surface. Let the albedo of this system be R′ . The albedo is

6.2

μ0Q

=0

Absorption of UV and Visible Sunlight by Ozone

θ0 n=1

=1 n=2 =2 n=3 =3

= nS – 1 n = nS

= nS Figure 6.1 Division of the atmosphere into layers. At the top of the atmosphere, level � = 0, the incident sunlight is �0 Q with Q the solar irradiance and �0 = cos θ0 with θ0 the solar zenith angle. Level � = n − 1 is the upper boundary and level � = n is the lower boundary of layer n. Sunlight is absorbed as it passes through the layers. For the absorption of near infrared radiation by water vapor as discussed in Section 6.3, upon reaching the surface the light is

reflected by the surface with albedo �S , the near infrared albedo. For the absorption of UV and visible radiation by ozone upon reaching the lower atmosphere–surface system, the light is reflected by the system with albedo R′ . Upon reflection, the sunlight is assumed to be isotropic. The reflected light passes upwards through the layers, being absorbed and scattered as it heads toward space.

calculated using the methods discussed in Chapter 4. For Rayleigh scattering, the molecules are nonabsorbing, and the single scattering albedo is � = 1. The scattering asymmetry is g = 0 and the Rayleigh optical depth at 0.55 μm is �0.55 = 0.09805. [5]. The Rayleigh optical depth is inversely proportional to the fourth power of the wavelength. It is given by ) ( 0.55 4 (6.2) �λ = �0.55 λ with λ the wavelength in microns. Upon reflection, the radiation is assumed to be isotropic. Lacis and Hansen used a diffusivity factor M to account for the propagation of diffuse light as it passes through the ozone layer on its way to space. The absorption for the layer n is thus given by Qn ABS = �0 Q{A(Un ∕�0 ) − A(Un−1 ∕�0 ) + R′ [A(UnS ∕�0 + M(UnS − Un−1 )) − A(UnS ∕�0 + M(UnS − Un ))]} (6.3)

187

6 Absorption of Solar Radiation by the Earth’s Atmosphere and Surface

1.50 UV σ03 (10–17 cm2)

188

Cross sections Model

1.25 1.00 0.75 0.50 0.25 0 0.20

0.25

0.30

0.35

Wavelength (μm) Figure 6.2 Molecular cross sections for the absorption of UV by ozone [6, 7] listed in Appendix E (•) and the normal distribution given by Equation 6.6 (solid curve). The units are 10−17 cm2 per molecule of O3 .

with the diffusivity factor M = 1.9 and UnS the column amount of ozone above the surface. In Equation 6.3 the sunlight absorbed by the ozone in a layer is the amount absorbed on the first pass through the layer plus the sunlight absorbed in passing through the entire atmosphere, being reflected by the lower atmosphere–surface system and passing back through the atmosphere to the top of the layer minus the amount absorbed in passing through the atmosphere, being reflected and passing back through the atmosphere to the bottom of the layer. Lacis and Hansen provided algebraic expressions for the absorption of UV and visible light by ozone. AO3 UV (U) =

1.082 U 0.0658 U + and (1 + 138.6 U)0.805 1 + (103.6 U)2

(6.4)

AO3 VIS (U) =

0.02118 U 1 + 0.042 U + 0.000323 U 2

(6.5)

with U the ozone amount in cm-STP and A(U) the fraction of the incident sunlight absorbed. Alternatively, the absorption can be approximately calculated from the absorption cross sections for ozone listed in Appendix E. Figure 6.2 shows the UV absorption cross sections for ozone as a function of wavelength. The absorption by ozone peaks between 0.20 and 0.30 μm, and is relatively weak at wavelengths less than 0.20 μm and between 0.30 and 0.45 μm where absorption at visible wavelengths begins to rise. The UV cross sections between 0.20 and 0.30 μm are approximately given by a normal distribution �O3 UV = �MAX O3 UV exp(−0.5[(λ − λ)∕�λ ]2 )

(6.6)

with �MAX O3 UV = 1.1 × 10−17 cm2 the maximum of the UV molecular cross sections, λ = 0.254 μm the wavelength at which the cross sections peak, and �λ = 0.018 μm the standard deviation of the distribution. While the normal distribution is an approximation, absorption by ozone in the UV is described as a continuum that varies slowly with wavelength with a relatively small amplitude structure superposed [8].

UV flux (Wm–2 μm–1)

6.2

1250

Absorption of UV and Visible Sunlight by Ozone

5000 K 5250 K 5500 K

1000 750 500 250 0 0.20

0.25

0.30

0.35

Wavelength (μm) Figure 6.3 Incident UV spectrum observed by the NASA SORCE satellite (solid line) and obtained for blackbodies at 5000 K (dotted line), 5200 K (dashed line), and 5500 K (dashdotted line). The radiances are for the top of the Earth’s atmosphere at 1 AU. The SORCE spectrum is from the SORCE SSI data for 14 July 2005.

Absorption by ozone is strong between 0.20 and 0.30 μm. At 0.31 μm, where the absorption is relatively weak, the absorption cross section of an ozone molecule is 8.7 × 10−20 cm2 . For a typical summertime column amount of ozone at midlatitudes, ∼0.3 cm-STP, this cross section leads to an optical depth of 0.7. Even though relatively weak, the absorption is substantial. The large absorption cross sections and concentrations of ozone in the atmosphere make absorption by ozone in the UV a strongly nonlinear function of the ozone amount. In terms of the absorption cross sections, the fraction of sunlight absorbed by ozone is given by ∑ fi (1 − e−�i U ) (6.7) A(U) = i

with fi the fraction of the incident sunlight in the spectral interval associated with absorption cross section �i . Figure 6.3 shows the UV spectrum of the sun for 1 astronomical unit (1 AU). The observations are from the NASA Solar Radiation and Climate Experiment (SORCE) Solar Spectral Irradiance (SSI) data [9]. While a brightness temperature of 5783 K represents reasonably well the distribution of the incident sunlight for wavelengths greater than 0.6 μm [8], the temperatures that more closely fit the emission between 0.2 and 0.3 μm lie in the range of 5000–5500 K. The wavelength dependence of the incident sunlight exhibits considerable structure. This structure and the solar irradiance within this interval are stable from day to day and year to year [10]. As Lacis and Hansen noted, the UV absorption depends on the wavelength dependence of both the ozone absorption cross sections and the incident solar radiation. Figure 6.4 shows the UV absorption by ozone derived using Lacis and Hansen’s algebraic expression (6.4) and using the ozone absorption cross sections as given by Equation 6.6 with the SORCE UV spectrum in Equation 6.7. As expected on the basis of the optical depth estimated for the relatively small cross section at 0.31 μm, the absorption of UV by ozone is highly nonlinear. In fact, a doubling of O3 from 0.2 to 0.4 cm-STP increases the absorption by only about 20%. Agreement between the results of the two calculations is probably as good as can be

189

6 Absorption of Solar Radiation by the Earth’s Atmosphere and Surface

0.025 Fraction absorbed

190

0.020 0.015 0.010

Lacis and Hansen (1974) Model and SORCE UV

0.005 0 0.0

0.5

1.0

1.5

2.0

Ozone amount (cm–STP) Figure 6.4 Fraction of the total incident sunlight absorbed by ozone at UV wavelengths obtained using Equation 6.4 (solid curve) and using the absorption cross sections given by Equation 6.6 in Equation 6.7 with the SORCE UV spectrum shown in Figure 6.3.

expected given the simple wavelength dependence assumed for the absorption cross sections in Equation 6.6. In addition, the UV spectrum used by Lacis and Hansen was based on the observations and inferences of the incident solar spectrum made in the 1960s. These earlier spectra had UV fluxes that were significantly larger than current values between 0.25 and 0.30 μm where a sizeable fraction of absorption by ozone occurs [11, 12]. Since absorption of UV depends nonlinearly on the amount of ozone, subtracting even a small percentage of ozone causes a dramatic increase in the exposure of biological systems to radiation between 290 and 320 nm. Radiation at these wavelengths, referred to as UV-B, damages DNA molecules. Among the concerns is skin cancer in people with fair skin. Exposure to UV-B accumulates with time. Since ozone concentrations generally decrease from pole to equator, and since the annual average incident UV-B irradiance increases from pole to equator, per capita occurrence of skin cancer in fair-skinned people generally increases from high latitudes to low latitudes. Occurrence of other forms of cancer lacks this dependence on latitude [13]. From the latitudinal dependence for the rates of skin cancer, a 2% reduction in ozone has been associated with increases of 4–6% in the occurrence of skin cancer while a 10% reduction has been associated with increases of 50–90% [14]. Fortunately, as a result of the 1987 Montreal Protocol and the subsequent amendments, the manufacturing of chlorofluorocarbons has evidently stopped worldwide. In the mid-1970s, chlorofluorocarbons had been identified as a threat to the ozone layer [15]. Predictions indicated that the continued manufacturing of these compounds would cause substantial reductions in stratospheric ozone concentrations by the first decades of this century. The reductions would have caused substantial damage to the biosphere and, of course, substantial increases in the rates of skin cancer. By the time of the Protocol, chlorofluorocarbons had been shown to be the cause of the ozone hole that had abruptly appeared over Antarctica during the Austral Springs of the early 1980s. Owing to the cessation of production, the concentration of chlorine in the atmosphere is decreasing [16]. Chlorine atoms from the chlorofluorocarbons were the cause of the ozone destruction. By midcentury, the chlorine concentration is expected to

6.3

Fraction absorbed

0.04 0.03

Absorption of Sunlight by Water Vapor

Lacis and Hansen (1974) Model

0.02 0.01 0.00 0.0

0.5

1.0

1.5

2.0

Ozone amount (cm–STP) Figure 6.5 Fraction of the total incident sunlight absorbed by ozone at visible wavelengths as obtained using Equation 6.5 (solid line) and the cross sections [7] listed in Appendix E in Equation 6.7 with the spectral distribution of the incident sunlight given by a blackbody at 5783 K (dashed line).

return to the levels of the early 1980s when the ozone hole first appeared [17]. If the expected decrease in chlorine concentrations occurs, the ozone hole should disappear. Unlike absorption by ozone in the UV, absorption of visible sunlight by ozone is in the weak line limit. The largest absorption cross section at visible wavelengths is 5.1 × 10−21 cm2 , an order of magnitude smaller than the relatively small UV cross section at 0.31 μm. Since the absorption is weak, the average absorption is unaffected by absorption features between 450 and 750 nm. In addition, the incident sunlight varies relatively slowly with wavelength within this interval. As a result, agreement between Equations 6.7 and 6.5 is assured. This agreement is confirmed by the comparison shown in Figure 6.5.

6.3 Absorption of Sunlight by Water Vapor

Water vapor absorbs sunlight primarily in the near infrared. As a simplification, Lacis and Hansen assumed that the absorption by water vapor occurred at wavelengths longer than 0.9 μm. At these wavelengths, they assumed that Rayleigh scattering and scattering and absorption by aerosols were negligible. About 35% of the incident sunlight occupies wavelengths beyond 0.9 μm. For cloud-free conditions, the absorption by water vapor is treated in the same manner as absorption by ozone. In the case of water vapor, the underlying reflecting system is the surface. As they did for ozone, Lacis and Hansen fit the absorption by the near infrared bands of water vapor to an algebraic function. For a homogeneous path at constant temperature and pressure, the absorption is given by A(U ′ ) =

2.9U ′ (1 + 141.5 U ′ )0.635 + 5.295 U ′

(6.8)

with U ′ = UPEFF ∕P0 the pressure weighted absorber amount and U the amount of water vapor in g cm−2 at pressure P = PEFF with P0 = 1 atm. The pressure weighted

191

6 Absorption of Solar Radiation by the Earth’s Atmosphere and Surface

0.25 Fraction absorbed

192

0.20 0.15 0.10 0.05 0.00

0

2

4

6

8

10

12

Water vapor (g cm−2) Figure 6.6 Day–night average fraction of sunlight absorbed as a function of the column amount of water vapor. The cosine of the solar zenith angle is set at the global average value �0 = 0.5 to evaluate the optical path length through the water vapor.

amount is used in Equation 6.8 because for most atmospheric paths, absorption by water vapor is in the strong, pressure-broadened line limit. Figure 6.6 shows the fraction of sunlight absorbed by water vapor for different column amounts. The column amount of water vapor is given by U=

∫0

∞

�H2 O (z)dz =

∫0

∞

rH2 O (z)�AIR (z)dz =

∫0

PS

rH2 O (P)

dP g

(6.9)

with rH2 O (P) the mass mixing ratio of water vapor and PS the surface pressure. As was discussed in Chapter 1, water vapor is concentrated in the lower troposphere. For global average condition, the mass mixing ratio is approximately given by ( )3 P (6.10) rH2 O (P) = rS PS with rS the mixing ratio at the surface. Different values of rS are used to produce the different column amounts of water vapor shown in Figure 6.6. The global average column amount of water vapor is approximately 2 g cm−2 . It is obtained using Equation 6.10 in Equation 6.9 with a surface mixing ratio of 0.008 kg H2 O/kg air. The amount of sunlight absorbed in the layer n between levels � = n − 1 and � = n is given by Equation 6.3 with R′ , the reflectance of the lower atmospheresurface system at UV and visible wavelengths, replaced by the surface albedo for near infrared wavelengths �S . For water vapor, a diffusivity factor M = 1.66 is used. For atmospheric paths, the pressure weighted amount of water vapor for the path from pressure level P to space is given by ( )5 )3 P ( PS r P P dP P P rS (6.11) U′ = = S S ∫0 PS P0 g 5g PS P0 The pressure weighted amount was used in Equation 6.8 to derive the fraction of sunlight absorbed in Figure 6.6.

6.3

Altitude (km)

10 8

Absorption of Sunlight by Water Vapor

αs = 0 αs = 0.8

6 4 2 0 0.3

0.4 0.5 0.6 0.7 Heating rate (K/day)

0.8

Figure 6.7 Day–night average solar heating rates calculated using Equations 6.3 and 6.8 for the absorption of sunlight by water vapor. The surface albedo is set to 0 (solid line) and to 0.8 (dashed line). The calculations use the climatological profile of water

vapor mixing ratio given by Equation 6.10 with a column amount of 2 g cm−2 . The solar zenith angle is set to 60∘ to evaluate the optical path length through the water vapor.

Approximately 10–18% of the incident light is absorbed by water vapor. As this absorption occurs for wavelengths occupied by 35% of the incident sunlight, one-third to half of this near infrared radiation is absorbed by water vapor. The absorption is only weakly dependent on the amount of water vapor. Doubling the column amount of water vapor from 1 to 2 g cm−2 increases the absorption by approximately 20%. For larger column amounts, such as those found in the tropics, doubling leads to an even smaller percentage increase in absorption. Absorption by water vapor is saturated at the centers of the absorption lines. Any rise in absorption with increasing water vapor concentrations occurs primarily in the wings of the strong, pressure-broadened lines. Figure 6.7 illustrates the effect of the saturated centers of the absorption lines. Absorption within each atmospheric layer is calculated using Equations 6.3 and 6.8 with the incident solar radiation given by �0 Q = 340 W m−2 , suitable for a day–night global annual average. The cosine of the solar zenith angle is set at the average value for the daylight side of the Earth, �0 = 0.5, and is used to calculate the optical path through the water vapor. The absorption by the layer is converted to a heating rate as given by Equation 1.45b. In Figure 6.7 one of the calculations is performed with no surface reflection �S = 0 (solid line); a second is performed with a high surface albedo typical of fresh snow �S = 0.8 (dashed line). The reflecting surface enhances the absorption of sunlight but only in the layers nearest the surface. None of the reflected light reaches the upper levels of the troposphere. When clouds are present, both absorption and scattering occur within the cloud and the water vapor mean transmittance is approximated by an exponential sumfit model. For the exponential sum-fit, the fraction of sunlight absorbed is given by an expression analogous to Equation 6.7 but with the absorption cross sections for ozone replaced by the absorption coefficients and the ozone amount replaced by the pressure weighted water vapor amount. The values of the fractions of the

193

194

6 Absorption of Solar Radiation by the Earth’s Atmosphere and Surface

Table 6.1 Absorption coefficients for the near infrared bands of water vapor ki and the associated fractions of the incident solar radiation fi for the exponential sum-fit model (Equation 6.7) [1]. i

ki (cm2 g−1 )

1 2 3 4 5 6 7 8

4 × 10−5 0.002 0.035 0.377 1.95 9.4 44.6 190

fi 0.647 0.0698 0.1443 0.0584 0.0335 0.0225 0.0158 0.0087

incident sunlight fi and the associated absorption coefficients ki for water vapor are given in Table 6.1. Lacis and Hansen assumed that at wavelengths longer than 0.9 μm Rayleigh scattering and aerosol scattering and absorption are negligible compared with the absorption by water vapor. In Table 6.1 the absorption by water vapor is negligibly small for i = 1. This interval represents the visible and ultraviolet parts of the spectrum in which absorption by water vapor is negligible. It covers 65% of the incident sunlight. At visible wavelengths, the effects of the scattering and absorption are not negligible and can be included when performing the calculations associated with the absorption coefficient for i = 1. For Rayleigh scattering, a wavelength of 0.55 μm provides a suitable average optical depth. Figure 6.8 shows the solar heating rates for water vapor within atmospheric layers calculated using Equation 6.3 with Equation 6.8 (solid line) and Equation 6.7 (dashed line). Lacis and Hansen adjusted the fractions of the incident sunlight fi and the absorption coefficients ki so that Equations 6.7 and 6.8 would provide nearly the same fraction absorbed for any suitable atmospheric path, to within less than 5%. When calculating differences in fractions absorbed within single layers, however, any differences between Equations 6.7 and 6.8 are amplified, producing the level of disagreement shown in Figure 6.8. For simplicity, within clouds Lacis and Hansen also assumed that absorption in the near infrared by hydrometeors, cloud droplets, and ice crystals, is negligible compared with that due to water vapor. The hydrometeors have broad absorption bands that lie within the same wavelengths as those of the water vapor absorption bands. Lacis and Hansen recommended assuming that air within clouds be saturated. Saturated air enhances the absorption by water vapor within clouds and compensates slightly for neglecting the absorption by hydrometeors. For cloud layers, the optical depth due to water vapor is combined with the optical depth of the hydrometeors to obtain the total optical depth and single scattering albedo.

6.3

Absorption of Sunlight by Water Vapor

Altitude (km)

10 8 6 4 2

Sum–fit Algebraic

0 0.35

0.40

0.45

0.50

0.55

0.60

Heating rate (K/day) Figure 6.8 Day–night average solar heating rates due to absorption by water vapor as obtained using Equations 6.3 and 6.8 (solid line) and Equation 6.7 (dashed line). The surface albedo is set to 0. The

calculations use the global average conditions adopted for the absorption shown in Figure 6.7. The solid line is the same as that shown in Figure 6.7 for zero surface albedo.

The optical depth of a cloudy atmospheric layer is given by �i = �CLOUD + ki ΔU ′

(6.12)

with �CLOUD the optical depth of the hydrometeors and ΔU ′ the pressure weighted amount of water vapor in the cloud layer. The asymmetry parameter used inside the cloud is that of the hydrometeors. The single scattering albedo within the cloud is given by �CLOUD �i = (6.13) �CLOUD + ki ΔU ′ The total optical depth, single scattering albedo, and asymmetry parameter are used to evaluate the reflectance, transmittance, and absorptance for cloud layers. When clouds are present, the atmosphere is divided into layers as in Figure 6.1. In this example, the Eddington approximation is used to calculate reflectances and transmittances for each of the absorption coefficients ki in Table 6.1 and each layer that scatters light. Working from the top of the atmosphere downwards, the rules for adding layers are applied to determine the reflectances and transmittances at the bottom of each combination of layers. Starting at the top in Figure 6.1, layers 1 and 2 are combined and the reflectances and transmittances at the bottom of layer 2 for the combined layers are determined. Layer 3 is then added to the combined system composed of layers 1 and 2 and the reflectances and transmittances at the bottom of layer 3 for the combined layers are determined. This process is repeated until the surface layer nS in Figure 6.1 is included in the combination of layers. Once the surface is reached, rules for adding the surface are followed to evaluate the fluxes of sunlight incident on the surface and absorbed by the surface. Then, climbing back through the atmosphere, the adding rules are again used to calculate the reflectances at the top of each successive layer as it is added to the resulting multilayered surface system. In the process of moving upwards, the flux of sunlight incident on the top of each layer and the sunlight absorbed below the top level of the multilayered surface system are also calculated. For absorption coefficient

195

6 Absorption of Solar Radiation by the Earth’s Atmosphere and Surface

10 Altitude (km)

196

8 6

Cloud–free Overcast

4 2 0 0.0

0.2

0.4

0.6

0.8

Heating rate (K/day) Figure 6.9 Day–night average solar heating rates for absorption by water vapor in a cloudy atmosphere (dashed curve). A water cloud was placed between 1.2 and 1.6 km. The solid curve is the heating rate for cloud-free conditions and is shown for comparison. The cloud-free values are

the same as those shown by the dashed line in Figure 6.8 but with layer widths of 0.1 km instead of 0.5 km. The calculations use the global average conditions adopted for the absorption by water vapor shown in Figure 6.7.

ki and associated weight fi with R′in the albedo of the multilayered surface system below the level � = n, R′in B the reflectance of the atmosphere for radiation incident on the bottom of the multilayered system above level � = n, and Tin′ the downward transmittance at the level, the downward solar radiative flux at the level is given by Q−in = �o Qfi Tin′ (1 − R′in R′in B )−1

(6.14)

The solar radiative flux absorbed below the level is given by Q−in − Q+in = �0 Qfi Tin′ (1 − R′in )(1 − R′in R′in B )−1

(6.15)

Moving upwards through the atmosphere, adding successive layers to the surface–atmosphere system, the top of the atmosphere is ultimately reached and the planetary albedo is obtained. Once the amounts of sunlight absorbed by the surface and each layer are obtained for all the absorption coefficients, the series is summed to determine the total amounts of sunlight absorbed. The sums give the solar irradiance absorbed at near infrared wavelengths by the surface, by the water vapor in each atmospheric layer, and by the Earth–atmosphere system. These procedures are followed to produce the water vapor solar heating rates shown in Figure 6.9. For these results, a low-level water cloud is placed between 1.2 and 1.6 km. The cloud is given an effective droplet radius of 12 μm and a droplet number concentration of 50 cm−3 , yielding a visible optical depth of approximately 18 and producing a visible reflectance of 0.67, values typical of marine stratus that completely cover regions of 50 km scale. For simplicity, absorption by ozone, Rayleigh scattering, aerosol scattering and absorption, and the surface albedo are all set to zero. Similar to reflection by the surface, reflection by the cloud enhances the absorption by water vapor above the cloud. Absorption is also enhanced in the upper layers of the cloud. The water droplets in the cloud scatter sunlight upwards, thereby

Problems

substantially reducing the sunlight absorbed by water vapor beneath the cloud and by the Earth’s surface. Since absorption in the centers of the pressure-broadened water vapor lines is nearly complete, low-level clouds have relatively little effect on the amount of sunlight absorbed by the atmosphere. In the example shown, without the cloud, water vapor in the atmosphere absorbs 14.1% of the incident sunlight. With the cloud present, it absorbs 12.2%, a 13% reduction. The sunlight that fails to make it through the cloud is absorbed either in the cloud or above the cloud on its way back to space. In the example, the water vapor mixing ratio profile (Equation 6.10) is used throughout the cloud. Lacis and Hansen recommended that the air inside clouds be saturated but no augmentation of water vapor is included for the heating rates shown in Figure 6.9. Low-level clouds have a tremendous impact on the amount of sunlight absorbed by the Earth’s surface and by the Earth–atmosphere system. Without the cloud, all the incident sunlight in this example is absorbed by the atmosphere and the surface, and the Earth’s albedo is zero. Water vapor absorbs 14% of the incident sunlight and the surface absorbs the remaining 86%. With the cloud, the fraction of sunlight absorbed by the Earth falls to 41%. Of the radiation absorbed by the atmosphere–surface system, the atmosphere absorbs slightly more than 12% leaving the surface with slightly more than 28%, a reduction of more than 65%. Most of the absorption by the surface occurs at visible wavelengths. Owing to the relative dryness of the upper troposphere and stratosphere, thick upper level clouds, unlike low-level clouds, reflect sunlight to space, shielding the Earth and atmosphere, thereby reducing the amount of sunlight absorbed not only by the surface but also by the atmosphere.

Problems

1. Use Equation 1.37 and the parameter values in Table 1.2 to determine the following: a. The total fraction of the incident sunlight absorbed by ozone at UV wavelengths in the tropics. In this calculation, use Equation 6.4 and a solar zenith angle of 60∘ . Use a spreadsheet or interactive computer software to obtain this fraction to two significant figures. b. Repeat (a) but determine the fraction of UV that is absorbed above the tropical tropopause. Assume that the tropical tropopause is at 18 km. c. Repeat (a) and (b) but use Equation 6.3 with R′ = 0.55, an approximate global average albedo for the lower atmosphere-surface system in the UV. By what percentage is the absorption by ozone increased when including the UV reflected by the lower atmosphere–surface system? 2. Ozone absorbs 2% of the total incident solar radiation by absorbing much of the incident UV between 200 and 340 nm. Use the short wavelength, large wavenumber approximation of the Planck function to determine the fraction of the incident UV absorbed within this spectral interval. Assume that the

197

198

6 Absorption of Solar Radiation by the Earth’s Atmosphere and Surface

sun radiates like a blackbody at 5300 K, an approximate average value for the incident UV spectrum shown in Figure 6.3. Adjust the incident solar flux at these wavelengths to 1 AU. 3. You are told that to avoid UV radiation, “Never sunbathe on a beach during the noontime hours in summer.” In subtropical high pressure systems along the western boundaries of the major oceans, however, there is always enough low-level marine stratocumulus to protect you from the UV. Or is there? The UV B dose rate peaks near 0.305 μm. At this wavelength the absorption cross section for an ozone molecule is 1.9 × 10−19 cm2 . a. Calculate the fraction of the solar UV B irradiance at 305 nm reaching the surface through a midlatitude, summertime amount of ozone, 0.3 cmSTP. Perform the calculation for noontime on the summer solstice at 35∘ latitude. Aside from the absorption by ozone, assume that the atmosphere is transparent at 305 nm. In other words, ignore the effects of Rayleigh scattering. b. What reduction in the UV B irradiance is achieved by avoiding exposure to direct sunlight for 2 h before and after noontime on the summer solstice? c. Assume that low-level marine clouds have droplets with radii r = 10 μm. The droplets have scattering cross sections equal to their extinction cross sections given by �EXT = 2πr2 . For the UV wavelengths, the asymmetry parameter for cloud droplets is g = 0.85. A typical droplet concentration for maritime clouds is n = 100 cm−3 . How thick would the clouds have to be at noontime to provide the same protection as staying out of the sun for the 4 h centered on noontime in (b)? Do not be deceived by the relatively small cloud thickness obtained in (c). Including the effects due to Rayleigh scattering at 0.305 μm and the surface albedo of beach sand, 0.09–0.12, amplifies the required depth of cloud by more than a factor of 3. In addition, note that the increase in path through the ozone layer caused by the change in the cosine of the solar zenith angle from noontime to 2 h before or after is also equivalent to more than a 10% increase in ozone amount, thereby providing substantial reductions in UV B dose rates. 4. Use the short wavelength, large wavenumber approximation of the Planck function to determine the fraction of incident solar radiation at wavelengths longer than 0.9 μm. Assume that the spectral distribution of the sun is that of a blackbody at 5783 K. 5. Estimate the Earth’s albedo at visible wavelengths as follows: a. Start by calculating the albedo for cloud-free regions, �CF . Take the optical depth for Rayleigh scattering to be �R = 0.1. Assume that there is also a haze with optical depth �A = 0.1, single scattering albedo �A = 0.95, and asymmetry parameter gA = 0.7. Let the haze layer be well-mixed with the Rayleigh atmosphere so that the optical depth, single scattering albedo,

Problems

and asymmetry parameter are given by � = �A + �R � =

g � � + gR �R �R �A �A + �R �R and g = A A A � ��

Set the average surface albedo �S = 0.1. Use the Eddington approximation to calculate the reflectances and transmittances of the cloud-free atmosphere and use the rules for adding the surface to calculate �CF . b. Set the cloud cover AC = 0.55. Assume that the Earth’s albedo is given by � = (1 − AC )�CF + AC �OVC with �OVC the albedo of regions overcast by cloud. Estimate the optical depth of the clouds so that the albedo of the Earth at visible wavelengths is � = 0.35. The albedo for the visible wavelengths is somewhat larger than the globally averaged annual mean albedo for the Earth, and that for the near infrared wavelengths is somewhat smaller. Obtain the albedo �OVC for the overcast regions by adding clouds to the mixed Rayleigh–aerosol atmosphere. Set the asymmetry parameter for clouds gC = 0.85. Assume that the cloud is well-mixed with the Rayleigh–aerosol atmosphere. Use the Eddington approximation to obtain the reflectances and transmittances for the overcast atmosphere above a reflecting surface. To obtain the optical depth of the cloud, first determine the required value of �OVC . Use a spreadsheet or interactive computer software to determine the cloud optical depth needed to obtain �OVC to two significant figures. c. If the droplet concentration is 100 cm−3 and the droplet radius is 10 μm, how thick (m) are the clouds on average? d. Are the properties of these average clouds typical of stratus or cumulus clouds? 6. A sizable component of global aerosols is associated with the burning of fossil fuels. Estimate the aerosol direct radiative forcing by repeating Problem 5, assuming that the human contribution affects only the visible portion of the reflected sunlight and that approximately 25% of the global aerosol particle number concentration is the result of human activity. 7. The increase in haze from the burning of fossil fuels is expected to increase the number of cloud droplets and reduce the average droplet size. The result is an increase in cloud albedo associated with the “indirect aerosol radiative forcing.” Estimate the aerosol indirect forcing for visible wavelengths as follows: a. Assume that the increase in cloud droplet number occurs with no change in cloud liquid water amount. In addition, assume that the scattering cross section of droplets is twice their geometric cross section. Show that when the liquid water concentration is held constant, the change in the cloud optical depth is proportional to n1∕3 with n the number of drops per unit volume.

199

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6 Absorption of Solar Radiation by the Earth’s Atmosphere and Surface

b. Show that in the Eddington approximation, the largest change in cloud reflectance for a given fractional change in cloud optical depth Δ�∕� occurs when the cloud reflectance is 0.5. c. Assume an asymmetry parameter g = 0.85, typical of water clouds, and calculate the optical depth for which the cloud reflectance is 0.5. d. For a cloud with a reflectance of 0.5, assume a typical droplet radius of 10 μm and calculate the change in radius that would accompany a 25% increase in the number of cloud droplets with constant liquid water amount. e. Calculate the change in cloud reflectance. f. If cloud cover fraction is approximately 0.55, what is the aerosol indirect radiative forcing? How does this forcing compare with the aerosol direct radiative forcing in Problem 6? How do these combined forcings compare with the current forcing by greenhouse gases, ∼2.5 W m−2 . 8. .a. Verify the equivalence of using Equations 6.8 and 6.7 for the fraction of sunlight absorbed by water vapor. For these calculations, use the exponential sum-fit absorption coefficients and weights listed in Table 6.1. Assume that the water vapor mixing ratio is given by Equation 6.10 and that the mixing ratio at the surface is rS = 0.008 kg H2 O/kg AIR. Calculate the pressure weighted absorber amount and the absorption of sunlight by water vapor above the pressure levels given in the table below. Use the global annual average incident sunlight, 340 W m−2 , and use a solar zenith angle of 60∘ for the optical paths through the water vapor. Assume that the albedo of the underlying surface is zero. A spreadsheet or interactive computer software will be needed to perform the calculations.

Pressure (hPa) Method 250 500 700 875 900 1013

U ′ (g cm−2 )

A(U ′ ∕�0 ) Equation 6.8 Equation 6.7

b. Calculate the solar heating rate (K/day) due to the absorption of sunlight by water vapor in the boundary layer (the atmosphere below 900 hPa). 9. Add a cloud between 900 and 875 hPa to the profile used in Problem 7 above. Assume that the cloud has a visible albedo of 0.5 and the phase function for water droplets has an asymmetry parameter g = 0.85. As was done in the example in the text, use the water vapor mixing ratio (Equation 6.10) to calculate the water vapor in the cloud.

References

a. What is the Earth’s albedo with the cloud present? b. What is the change in the fraction of sunlight absorbed by the atmosphere with the cloud present? c. What is the change in the fraction of sunlight absorbed by the surface with the cloud present? 10. Repeat Problem 9 but for a cloud between 700 and 250 hPa with a visible albedo of 0.9 and an asymmetry parameter g = 0.7, typical of ice crystals. Such clouds are referred to as deep convective clouds. Distribute the cloud in the layers so that the optical depth of the cloud in a layer is proportional to the pressure difference across the layer. References 1. Lacis, A.A. and Hansen, J.E. (1974)

2.

3.

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5.

6. 7.

A parameterization for the absorption of solar radiation in the earth’s atmosphere. J. Atmos. Sci., 31, 118–133. Fu, Q. and Liou, K.-N. (1992) On the correlated k-distribution method for radiative transfer in nonhomogeneous atmospheres. J. Atmos. Sci., 49, 2139–2156. Mlawer, E.J., Taubman, S.J., Brown, P.D., Iacono, M.J., and Clough, S.A. (1998) Radiative transfer for inhomogeneous atmospheres: RRTM, a validated correlated-k model for the longwave. J. Geophys. Res., 102, 16,663–16,682. Clough, S.A., Shepard, M.W., Mlawer, E.J., Delamere, J.S., Iacono, M.J., Cady-Pereira, K., Boukabara, S., and Brown, P.D. (2005) Atmospheric radiative transfer modeling: a summary of the AER codes. J. Quant. Spectrosc. Radiat. Transfer, 91, 233–244. Pendorf, R. (1957) Tables of the refractive index for standard air and the Rayleigh scattering coefficient for the spectral region between 0.2 and 20.0 μm and their application to atmospheric optics. J. Opt. Soc. Am., 47, 176–182. Houghton, J.T. (2002) The Physics of the Atmosphere, Cambridge University Press. Orphal, J. (2003) A critical review of the absorption cross-sections of O3 and NO2 in the ultraviolet and visible. J. Photochem. Photobiol., A: Chem, 157, 185–209.

8. Goody, R.M. and Yung, Y.L. (1989)

9.

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13. 14.

15.

Atmospheric Radiation Theoretical Basis, Oxford Press. SORCE. SORCE Data, http://lasp.colorado.edu/sorce/index.htm (accessed 22 January 2013). Woods, T.N., Chamberlin, P.C., Harder, J.W., Hock, R.A., Snow, M., Eparvier, F.G., Fontenla, J., McClintock, W.E., and Richard, E.C. (2008) Solar irradiance reference spectra (SIRS) for the 2008 Whole Heliosphere Interval (WHI). Geophys. Res. Lett., 36, L01101. doi: 10.1029/2008GL036373 Aversen, J.C., Griffin, R.N. Jr., and Pearson, B.D. Jr. (1969) Determination of extraterrestrial solar spectra irradiance from a research aircraft. Appl. Opt., 8, 2215–2232. Thuillier, G., Floyd, L., Woods, T.N., Cebula, R., Hilsenrath, E., Hersé, M., and Labs, D. (2004) in Solar Variability and Its Effects on Climate, Geophysical Monograph, Vol. 141 (eds J.M. Pap and P. Fox), American Geophysical Union, pp. 171–194. Turco, R.P. (2002) Earth Under Siege, 2nd edn, Oxford University Press. Graedel, T.E. and Crutzen, P.J. (1993) Atmospheric Change: An Earth System Perspective, W.H. Freeman and Company. Molina, M.J. and Rowland, F.S. (1974) Stratospheric sink for chlorofluorocarbons: chlorine atom catalyzed destruction of ozone. Nature, 249, 810–812.

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Berntsen, T., Betts, R., Fahey, D.W., Haywood, J., Lean, J., Lowe, D.C., Myhre, G., Nganga, J., Prinn, R., Raga, G., Schulz, M., and Vandorland, R. (2007) Changes in atmospheric constituents and radiative forcing, in Climate Change 2007: The Physical Basis. Contribution of Working Group I to the Fourth Assessment Report of

the Intergovernmental Panel on Climate Change (eds S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt, M. Tignor, and H.L. Miller), Cambridge University Press, Cambridge, New York. 17. Solomon, S. (1999) Stratospheric ozone depletion: a review of concepts and history. Rev. Geophys., 37, 275–316.

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7 Simplified Estimates of Emission 7.1 Introduction

As discussed in Chapters 4 and 5, evaluation of thermal emission in the Earth’s atmosphere requires detailed numerical integrations. There are integrations over the optical path to account for the variations with altitude of absorber concentrations, absorption coefficients, and the Planck function. Then in order to obtain the radiative flux, there are integrations over the angles of the emitted radiance and integrations over wavenumbers to include all spectral contributions. For accurate values of the emitted radiation, the numerical details cannot be avoided. Nonetheless, to illustrate the simplified approaches adopted in this book and gain insight into the factors that affect emitted radiation, absorption and emission by the 15 μm band of CO2 are estimated. The simplifications that have been discussed previously are suitable for calculating the absorption over most of this band. Absorption and emission are primarily by strong, pressure-broadened lines. Because carbon dioxide is well mixed in the atmosphere, the random model transmission for CO2 reverts to the idealized model used in Chapter 4 for the transmissivity and transmittance. The results of these relatively simple calculations lead to approximate estimates for the effect of increasing carbon dioxide on the greenhouse forcing of the Earth’s climate. They also lead to an estimate for the cooling of the stratosphere caused by increasing concentrations of carbon dioxide. Of course, these simple estimates also suffer from limitations. The chapter also exposes these limitations.

7.2 Emission in the 15 �m Band of CO2

In Chapter 4, the formal solution to the equation of radiative transfer gives the upward radiance at level z as 1

I + (z, �) = BS T(0, z, �) +

∫T(0,z,�)

dT(z′ , z, �)B(z′ )

(7.1)

with T(z′ , z, �) the transmissivity from z′ to z along the path with � the cosine of the zenith angle, B(z′ ) the Planck function evaluated at the level z′ , and BS the Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

204

7 Simplified Estimates of Emission

emission by the surface. To simplify notation, the wavenumber of the radiance is not indicated. The upward radiative flux is given by F + (z) = 2�

∫0

1

1

d� � I + (z, �) = �BS TF (0, z) +

∫TF (0,z)

dTF (z′ , z)�B(z′ )

(7.2)

with the transmittance given by TF (z′ , z) = 2

∫0

1

d� �T(z′ , z, �)

(7.3)

As discussed in Section 4.4, a diffusivity factor is used to approximate the transmittances with transmissivities. TF (z′ , z) ∼ T(z′ , z, �)

(7.4)

with � representing an “average” value of the cosine for zenith angles associated with diffuse radiation and M = 1∕� the diffusivity factor. In the limit of isolated strong, pressure-broadened lines, the diffusivity factor M = 16/9 = 1.78 (see Chapter 5, Problem 2). The diffusivity factor typically used for thermal emission is M = 1.66. The use of either 1.66 or 1.78 makes little difference in the emitted flux at the top of the atmosphere [1]. The total upward emission is the radiative flux integrated over all wavenumbers. Typically this integration is approximated by a sum over a finite set of wavenumber intervals. ∞ ∑ d�F�+ (z) ∼ F +i (z)Δ�i F + (z) = ∫0 i [ ] 1 ∑ �BSi T Fi (0, z) + = dT Fi (z′ , z)�Bi (z′ ) Δ�i (7.5) ∫T Fi (0,z) i with F +i (z) the average upward irradiance for spectral interval i, T Fi (z′ , z) the mean transmittance, �BSi the average irradiance emitted by the surface, and Δ�i the width of the interval. The average of the Planck function for the interval is given by �Bi (z) =

1 d��Bv (z) Δ�i ∫Δ�i

For random models the mean transmittance is given by ( ) W T F = exp − �

(7.6)

(7.7)

with � the average spacing and W the average equivalent width of the absorption lines in the spectral interval. For the Goody random model and pressurebroadened absorption lines the average equivalent width is given by �MU W= √ �MU 1+ ��L

(7.8)

7.2

Emission in the 15 μm Band of CO2

with � the average line strength for the interval and �L the associated pressurebroadened half-width. As discussed in Chapter parameters used in random ∑ 5, the∑ √ models are derived from compilations of Si and Si �Li , such as are found in Appendix D. Carbon dioxide is well-mixed in the atmosphere. The column amount above a particular pressure level P is given by U=r

P g

(7.9)

with r the mass mixing ratio for carbon dioxide. For the path to space, the effective pressure for the half-width �L is obtained using the Curtis–Godson approximation and is given by PEFF U′ = P0 U with ′

U =

∫0

P

dP′ P′ r(P′ ) = U g P0

(7.10) (

1 P 2 P0

) (7.11)

the pressure weighted amount. For carbon dioxide and the path from pressure level P to space, the effective pressure is half the pressure of the level, PEFF = P∕2. The half-width in Equation 7.8 is �L = �L0 (PEFF ∕P0 ) with �L0 the half-width at one atmosphere pressure P0 . In this simple estimate, the temperature dependence of the half-width and the line strength will be accounted for by choosing a midrange temperature appropriate for the atmospheric path. Two limits are considered for transmission by carbon dioxide. The first is the strong, pressure-broadened line limit. It is appropriate for the central portion of the 15 μm band. In this limit the absorption coefficient at the center of the lines is large so that, �MU ≫1 ��L

(7.12)

Consequently, with the values given for U and PEFF , for any path to space Equation 7.12 becomes 2�rMP0 ≫1 (7.13) ��L0 g This condition applies for all pressure levels. It depends only on the line parameters and the concentration of the gas. The results in Figure 7.1 illustrate the condition given in Equation 7.13. The spectral intervals in the central portion of the 15 μm band, wavenumbers 575–750 cm−1 , are in the strong, pressure-broadened line limit for the path to space. The second limit is the weak line limit. The weak line limit occurs when the absorption coefficients at the centers of the lines are small, �MU ≪1 ��L

(7.14)

205

7 Simplified Estimates of Emission

104 200 K 255 K

2

σMU/πα

10

100 10–2 10–4

(a) 10–6 104 102

σMU/δ

206

100 10–2 10–4 10–6 500

600

700

800

900

Wavenumber (cm–1)

(b)

Figure 7.1 Strong, pressure-broadened line limit, �MU∕��L ≫ 1, for all paths to space within the 15 μm band of CO2 (a), and weak line limit, �MU∕� ≪ 1, for the path from the surface to space (b). The number mixing ratio used in these calculations is 390 ppmv (parts per million by volume), a value close to that for 2010. The paths were calculated

using two temperatures, one typical of the midtroposphere, 255 K, and the other typical of the tropopause near the equator, 200 K. The dotted line separates spectral intervals into those for which the strong line limit applies and those in which the weak line limit applies.

Because line spacing is often much larger than the line half-width, the weak line limit implies that the mean transmittance is approximately given by �MU (7.15) � Figure 7.1 shows that within the 15 μm band of CO2 , there is a rapid transition from strong absorption lines at the center of the band to weak absorption lines in the wings of the band, wavenumbers less than 575 cm−1 and greater than 750 cm−1 . The weak line limit applies for paths from the surface to space that satisfy ∑ Sj rMPS 1 j ≪1 (7.16) Δ�i g TF = 1 −

7.2

Emission in the 15 μm Band of CO2

In these calculations the surface pressure is set to PS = P0 = 1 atm. Figure 7.1 shows results with the temperature of the atmosphere set at 255 K, a temperature typical of midtropospheric levels, and 200 K, a temperature typical of the equatorial tropopause. Consistent with the explanation in Chapter 5, as the temperature of a gas rises, the high energy rotational levels become populated and absorption increases in the wings of the band. For the evaluation of emission at the top of the atmosphere, the troposphere is given a constant lapse rate, and consistent with the approximations used in Chapter 4, the Planck function is assumed to depend linearly on the pressure. The stratosphere is taken to be isothermal with a temperature of 220 K. The altitude of the tropopause is the level at which the temperature given by a 288 K surface temperature and a 6.5 K/km lapse rate equals the stratospheric temperature. For those temperatures, the tropopause occurs at PT = 246 hPa. The tropospheric lapse rate and the stratospheric temperature are used to evaluate the Planck function but, as will be discussed later, a temperature of 255 K is used to determine the line strengths and pressure-broadened half-widths. The evaluation of the top of the atmosphere irradiances involves summing over the spectral intervals associated with the 15 μm band of carbon dioxide. The absorption coefficient varies with each pressure interval. Unlike the monochromatic illustrations in Chapter 4, multispectral calculations require a different treatment of the Planck function. Ensuring that for spectral interval i the emission as a function of pressure passes through that of a blackbody at the surface pressure and a blackbody at the tropopause pressure leads to ( Bi (P) = BSi +

BTi − BSi PT − PS

) (P − PS ) = ai + bi P

(7.17)

with BTi the average Planck function at tropopause temperature, the same as the temperature of the isothermal stratosphere, PS the surface pressure, and PT the tropopause pressure. Rewriting Equation 4.27a for the top of the atmosphere, the average irradiance is given by ) ( ( ′) PS P dP′ P F +i (0) = �BSi exp − S + �(ai + bi P′ ) exp − ∫PT Pi Pi Pi ( ′) PT ′ dP P + �BTi exp − ∫0 Pi Pi

(7.18)

The pressure Pi is determined by the absorption line strengths and half-widths and by the mass mixing ratio of carbon dioxide. It is the pressure at which the optical path length to space equals unity for the transmittance. For absorption lines in the strong, pressure-broadened line limit, √ Pi = Pi STRONG = �i

2P0 g ��i �L0i rM

(7.19a)

207

7 Simplified Estimates of Emission

108

200 K 255 K

106 Pi (hPa)

208

104 Surface Tropophere Stratosphere

102

100 500

600

700

800

900

Wavenumber (cm–1) Figure 7.2 Pressure at which the optical path length to space reaches unity for the transmittance. The concentration and temperatures are the same as those used for the results in Figure 7.1. The dotted

lines indicate the pressures associated with the surface and the tropopause. Pressures greater than the surface pressure indicate weak absorption.

For absorption lines in the weak line limit, Pi = Pi WEAK =

�i g �i rM

(7.19b)

Figure 7.2 shows Pi for the 15 μm band of carbon dioxide. As expected from the discussion on the radiance emitted from the top of the atmosphere in Chapter 4, absorption is strong in the center of the band and Pi is associated with tropospheric and stratospheric levels. In the wings of the band the absorption is weak and Pi is associated with pressures that are even greater than the surface pressure. As suggested by the results shown in Figure 7.1, the transition from strong to weak absorption occurs rapidly between 550 and 575 cm−1 and again between 750 and 775 cm−1 . While requiring somewhat more manipulation than was needed for the integrals in Chapter 4, the integrations in Equation 7.18 are readily performed using substitution of variables and integration by parts to give F +i (0) = �BTi + �bi Pi (exp(−PT ∕Pi ) − exp(−PS ∕Pi ))

(7.20)

which is identical in form to Equation 4.28a with P = 0 and �B� replaced by �bi . Figure 7.3 shows the average emitted radiance I +i (0, 1) at the top of the atmosphere with the cosine of the zenith angle equal to unity. This radiance would be observed by a satellite looking in the nadir direction. The radiance should be compared with the radiance shown in Figure 4.3 obtained with a line-by-line radiative transfer code. In Figure 7.3 the nadir radiance is obtained by setting the diffusivity factor M = 1.0 for the evaluation of the Pi in Equations 7.19a and b and dividing the irradiance by �.

7.3

Change in Emitted Flux due to Doubling of CO2

Radiance (mWm–2 sr–1/cm–1)

150 200 K 255 K

280 K 100 240 K 50

0 500

600

700

800

900

–1

Wavenumber (cm ) Figure 7.3 Nadir view of the average emitted radiances at the top of the atmosphere within the 15 μm band of CO2 . The concentration and temperatures used in the calculations are the same as those

used for the results in Figure 7.1. The dotted curves show the radiances emitted by blackbodies at 20 K intervals ranging from 220 to 280 K.

Figure 7.3 shows that absorption and emission by carbon dioxide are significant between 575 and 750 cm−1 . From the results shown in Figures 7.1 and 7.2 this emission occurs where carbon dioxide is, for the most part, in the strong, pressurebroadened line limit. On the wings of the 15 μm band the absorption is weak and carbon dioxide does little to alter the emission coming from the surface at 288 K.

7.3 Change in Emitted Flux due to Doubling of CO2

Figure 7.4 shows upward radiative fluxes at the top of the atmosphere, F +i (0) in Equation 7.20 multiplied by Δ�i = 25 cm−1 . These results were obtained using the tropospheric lapse rate and temperature of the isothermal stratosphere to evaluate the Planck function while using 255 K as the temperature to evaluate the line strengths and pressure-broadened half-widths. The results in the figure are for the preindustrial level of CO2 , 280 ppmv (solid line), and two times the preindustrial concentration, 560 ppmv (dashed-line). If current practices in the use of fossil fuels continue, as appears to be the case, this doubling could occur by midcentury [2]. The change in the net upward radiative flux at the top of the atmosphere is the radiative forcing caused by a doubling of CO2 . Figure 7.4 shows that with the doubling, emission will decrease, consistent with conclusions drawn in Chapters 1 and 4. According to these simplified calculations, the decrease in emission is negligible in the center of the band where the absorption is strong. Adding more carbon dioxide to the atmosphere will not alter the emission in the center of the 15 μm band. Those who deny the possibility of global warming sometimes argue that there is so

209

7 Simplified Estimates of Emission

12 10 Irradiance (Wm–2)

210

280 K

280 ppmv CO2 560 ppmv CO2 ΔF = –3.6 Wm–2

8 6

240 K

4 2 0 500

600

700

800

900

Wavenumber (cm–1) Figure 7.4 Top of the atmosphere emitted irradiance within 25 cm−1 intervals for the preindustrial concentration, 280 ppmv, and a doubling of CO2 . The doubling leads to a decrease in the emitted longwave flux of 3.6 W m−2 .

much carbon dioxide in the atmosphere that adding more will not make any difference. Based on these simplified calculations, they would appear to be correct, at least for changes near the center of the band. As will be shown shortly, however, it is the use of an isothermal stratosphere coupled with the strong absorption by CO2 that causes this lack of change. In the wings of the band, emission decreases. Although the absorption is still strong in the wings, it is not so strong that the stratosphere becomes opaque. At these wavelengths, emission from the troposphere is still capable of reaching the top of the atmosphere. The total decrease in emission is given by the sum of the differences between the dashed and solid lines, ∑ Δ F +i Δ�i = −3.6 W m−2 , approximately equal to the 4 W m−2 reduction in the longwave flux claimed in Chapter 1 for a doubling of CO2 . The near agreement of this simple estimate with more accurate estimates for the change in the longwave flux is partly luck. For one thing, the 4 W m−2 reduction in emission that is said to be the radiative forcing is, in fact, the net change in the longwave radiative flux at the tropopause, not at the top of the atmosphere. Furthermore, it is the change in the net longwave flux once the stratosphere has been allowed to achieve a new state of radiative equilibrium. Recall from Chapter 1 that the global average annual mean temperature profile of the atmosphere appears to be close to a state of radiative-convective equilibrium with the stratosphere in a state of radiative equilibrium. Consequently, the Earth’s troposphere–surface system must also maintain radiative equilibrium at the tropopause. The sunlight absorbed by the Earth’s troposphere-surface system must equal the net flux (upward–downward) of emitted radiation at the tropopause. The condition of radiative equilibrium at the tropopause led to the practice of calculating the radiative forcing for a given change in atmospheric composition, like a doubling of CO2 , by first allowing the stratosphere to reach radiative equilibrium while holding tropospheric temperatures fixed and then

7.3

Change in Emitted Flux due to Doubling of CO2

evaluating the solar and longwave fluxes at the tropopause [2]. Unlike the Earth–troposphere system, which responds relatively slowly to perturbations, the perturbed stratosphere achieves radiative equilibrium rapidly, within about 100 days. Unfortunately, there is no simple way using the assumptions adopted here to obtain the upward longwave flux at the tropopause. The evaluation of the upward flux at the tropopause requires the application of the Curtis–Godson approximation for optical paths from the surface to the tropopause. These paths introduce the pressure of the tropopause into the mean transmittances thereby foiling attempts to obtain relatively simple analytic solutions. The introduction of the tropopause pressure in the optical paths also forces the use of a single temperature for evaluating the line strengths and pressure-broadened half-widths. The evaluation of the upward irradiance at the tropopause requires numerical integration. By keeping the lapse rate constant for the entire atmosphere, insight is gained into the effects of using an isothermal stratosphere in this simple estimate. With a constant atmospheric lapse rate, the dependence of the Planck function on pressure requires that the temperature of the atmosphere falls to zero at the top of the atmosphere. As a result, the emission at the center of the 15 μm band is associated with extremely cold temperatures. For these calculations Equation 4.17 is used with the upward radiance at the top of the atmosphere multiplied by �, P = PS , B0 , and BS replaced by BSi , P0 by PS , and P� by Pi as given by Equations 7.19a and b. Figure 7.5 shows the resulting changes in the top of the atmosphere emitted irradiances. Differences with the irradiances shown in Figure 7.4 illustrate the role played by the isothermal stratosphere. With no isothermal stratosphere, a doubling of CO2 leads to a 5.7 W m−2 as opposed to a 3.6 W m−2 decrease in the emitted irradiance at the top of the atmosphere. Furthermore, the emission 12

Irradiance (Wm–1)

10

280 K

280 ppmv CO2 560 ppmv CO2 ΔF = –5.7 Wm–2

8 6

240 K

4 2 0 500

600

700

800

900

–1

Wavenumber (cm ) Figure 7.5 Top of the atmosphere emitted irradiances obtained for the preindustrial concentration of carbon dioxide, 280 ppmv, and double the preindustrial concentration for a model in which there is no stratosphere.

211

7 Simplified Estimates of Emission

decreases at all wavelengths, including those associated with the strongly absorbing center of the 15 μm band. The results in Figures 7.4 and 7.5 clearly illustrate that the forcing depends not only on the strength of the absorption but also on the temperature profile. The forcing, of course, also depends on the profile of the greenhouse gas concentration. The results in Figure 7.5 illustrate that if the greenhouse gas is confined to a region of the atmosphere in which the temperature falls with increasing altitude, then adding more gas will lead to a decrease in emission, regardless of the strength of the absorption. The results in Figure 7.4, on the other hand, suggest that if the absorption by the greenhouse gas is sufficiently strong and the gas is in a region in which the atmosphere is isothermal, then there will be no change in emission as the gas is added. Owing to the absorption of ultraviolet radiation by ozone, the temperature of the stratosphere increases with increasing altitude. As a result, at some wavelengths in the strongly absorbing center of the 15 μm band, increases in carbon dioxide lead to increases in emission. Figure 7.6 shows the changes in the emitted zenith radiance for a doubling of CO2 calculated using a line-by-line radiative transfer code. The changes in the radiance shown in the figure are normalized by multiplying the changes by �, which assuming an isotropic radiance converts the radiance to an irradiance, and dividing the result by the change in the emitted irradiance [3]. For the doubling of CO2 , the calculated change in the emitted 0.015 Normalized OLR change

212

0.010 0.005 0.000 –0.005 –0.010 500

600

700

800

900

Wavenumber (cm–1) Figure 7.6 Normalized spectrum of the change in the emitted outgoing longwave radiance due to a doubling CO2 calculated using a line-by-line radiative transfer model [4]. The calculations were performed for the 1976 U.S. Standard Atmosphere profiles of temperature and composition with the exception of the concentration for CO2 . For these calculations the concentration of CO2 was doubled from 280 to 560 ppmv.

The results are for cloud-free conditions. Changes in the radiance are normalized by first multiplying the radiance by � and then dividing them by the change in the emitted irradiance, which in this case was −2.78 W m−2 . Consequently, the changes with positive signs represent reductions in emitted radiance and changes with negative signs represent increases in emitted radiance.

7.4 Changes in Stratospheric Emission and Temperature Caused by a Doubling of CO2

longwave flux at the top of the atmosphere is −2.78 W m−2 . Positive changes in the figure represent reductions in the zenith radiance and negative changes represent increases in the radiance. Clearly, realistic representations of the temperature profile and atmospheric composition lead to a smaller reduction in the irradiance at the top of the atmosphere than is obtained with the simple estimate presented in Figure 7.4. Also, the increase in emission at the center of the 15 μm band of CO2 is clearly evident.

7.4 Changes in Stratospheric Emission and Temperature Caused by a Doubling of CO2

Figure 7.7 shows the emission by the stratosphere obtained using the simplified calculations for the preindustrial concentration of carbon dioxide and twice the preindustrial concentration. Because the stratosphere is isothermal in these calculations, the upward irradiance at the top of the atmosphere is the same as the downward irradiance at the tropopause. Also, in the calculations the stratosphere is isothermal at 220 K but the line parameters taken from Appendix D are those at 200 K. Since absorption in the wings of the band increases with increasing temperature, the calculated emission underestimates the emission for a stratosphere at 220 K. Using the line parameters at 255 K from Appendix D, increases the emission for the 280 ppmv concentration from 14.3 to 18.1 W m−2 , which overestimates the emission. The emission at 220 K is probably near the midpoint of these values, 16 W m−2 . For the 2010 concentration of CO2 , 390 ppmv, the emission is only slightly larger than that for the preindustrial concentration, 17 W m−2 equivalent

Irradiance (Wm–2)

5

280 ppmv CO2 560 ppmv CO2

4

14.3 Wm–2 16.1 Wm–2

3 2 1 0 500

600

700

800

900

Wavenumber (cm–1) Figure 7.7 Upward irradiance at the top of the atmosphere emitted by an isothermal stratosphere at 220 K for the preindustrial concentration of carbon dioxide and double the preindustrial concentration. For an

isothermal stratosphere the irradiance emitted downward at the tropopause is identical with the upward irradiance at the top of the atmosphere.

213

214

7 Simplified Estimates of Emission

to 5% of the incident sunlight. This estimate compares well with the 6% for the upward emission by the stratosphere given by Hartmann [5] as shown in Figure 1.7, which in addition to carbon dioxide, includes contributions from ozone and water vapor. For a doubling of CO2 , the downward irradiance at the tropopause increases by 1.8 W m−2 . Because the stratosphere maintains a state of radiative equilibrium and an increase in carbon dioxide leaves the absorbed sunlight largely unaffected, the increase in emission caused by a doubling of CO2 must be compensated by cooling. What temperature change is required to reduce the emission with increased CO2 to that obtained for the preindustrial concentration? The change in temperature is obtained by noting that the current emission is given by F ± = 2�BΔ� �F

(7.21)

with F ± the total emission by the stratosphere, a combination of the emitted flux upward at the top of the atmosphere and downward at the tropopause, BΔ� the Planck function integrated over the spectral intervals that contain the 15 μm band, and �F the average emittance of the stratosphere. For radiative equilibrium, the change in the emitted flux will be zero. Consequently, d� d2�BΔ� ΔT �F + 2�BΔ� F ΔU = 0 (7.22) dT dU The first term on the right hand side of Equation 7.22 is the change in the emitted irradiance owing to a change in stratospheric temperature. The second is the change in emitted irradiance owing to a change in carbon dioxide. The second term, according to Figure 7.7 is 3.6 W m−2 for a doubling of CO2 . One notes that by multiplying and dividing the first term in Equation 7.22 by B, the average of the Planck function, the expression for the change in the emitted irradiance can be rearranged to give ΔF ± =

d� d ln B ΔT + 2�BΔ� F ΔU = 0 (7.23) dT dU Substituting Equation 7.21 into Equation 7.23 and recognizing the second term in Equation 7.23 as the change in the emitted flux due to the change in carbon dioxide concentration ΔF ±U gives 2�BΔ� �F

ΔF ±U

(

d ln B dT

)−1

(7.24) F± The first factor on the right hand side of Equation 7.24 represents the fractional change in the longwave flux emitted by the stratosphere due to the change in carbon dioxide concentration. The logarithmic derivative of the Planck function for the 15 μm band of CO2 is approximated using the small wavelength, large wavenumber limit of the Planck function. It is evaluated for � = 667 cm−1 , the center of the band and is given by ΔT = −

d ln B hc� ∼ dT kT 2

(7.25)

7.5 Afterthoughts

with hc∕k = C2 = 1.44 cm K, the second Planck radiation constant. The change in temperature is thus given by ± 220 kT ΔF U 3.6 = −220 × × = −2.5 K. (7.26) hc� F ± 1.44 × 667 28.6 This temperature change is again a crude approximation. It ignores the variation in the changes with altitude. One suspects that the cooling of the stratosphere would be distributed so that the equilibrium temperature changes would range from being relatively small in the dense lower stratosphere to relatively large in the thin upper stratosphere. In addition, most of the downward emission by carbon dioxide into the troposphere occurs at wavelengths strongly absorbed by carbon dioxide, 625–700 cm−1 . Even though emission by the troposphere with doubled CO2 will be reduced, the reduction in emission at these wavelengths will be small. The downward emission by the strongly absorbing portions of the CO2 band will be compensated by absorption of the upward flux emitted by the troposphere at the same wavelengths. As a result, the stratosphere will cool largely by increasing emission to space as discussed in Section 4.4. In fact, detailed radiative transfer calculations suggest that “cooling to space” accounts for most of the longwave cooling rate particularly in the stratosphere [1, 6]. Nonetheless, the equilibrium temperature change remains a decrease of 2.5 K. For the cool to space approximation both the numerator and denominator in Equation 7.26 would be divided by two leaving the same value for the change in the equilibrium temperature. Though crude, this estimate for the change in the stratospheric temperature agrees favorably with the results obtained by Manabe and his coworkers [7, 8] with their radiative–convective models for altitudes less than 30 km, ∼10 hPa, which in turn agrees favorably with the results of current climate simulation models [2].

ΔT = −T

7.5 Afterthoughts

The method described in this chapter for exploring the effects of increasing carbon dioxide concentrations on the emission at the top of the atmosphere can be generalized to other applications involving atmospheric emission. Among the notable examples is the retrieval of upper level moisture based on emission in the water vapor 6.3 μm band [9]. Owing to its greenhouse effect and also the role it may play in the occurrence of cirrus and detraining cumulus, upper level moisture burdens are bound to prove critical to characterizing processes that affect the Earth’s water cycle and cloud feedbacks. Among other things, the method can be extended to the retrieval of pressures and thus altitudes of cloud layers based on emission in the window region and the short wavelength wings of the CO2 15 μm band [10]. Determining the processes involved in changes in cloud altitudes remains a challenge both observationally and theoretically. Although the approximate approaches offered in this book are instructive from the standpoint of learning the factors that affect emission by the Earth and the

215

216

7 Simplified Estimates of Emission

atmosphere and the scattering and absorption of sunlight by the atmosphere and surface, they are clearly limited. Numerical tools are essential for comparisons between observations and theory to reveal the shortcomings in both. Of course, although also instructive for learning, the straightjacket imposed by plane-parallel radiative transfer models will ultimately give way to approaches that include 3-D radiative transfer effects. Such effects are becoming increasingly noticeable in remote sensing problems involving clouds and reflected sunlight.

Problems

1. Follow the steps outlined below to estimate the flux of radiation emitted by the surface and lower atmosphere that is absorbed by the 9.6 μm band of ozone. First, based on the zenith radiances shown in Figure 4.3, estimate the spectral width over which emitted radiances are absorbed by ozone. Assume that the emission in the window region is that of a blackbody at a surface temperature of 288 K and that the atmosphere, aside from ozone, is transparent at these wavelengths. How does your estimate compare with that given by Hartmann [5] in Figure 1.7? 2. The water vapor rotation band overlaps the 15 μm band of CO2 . In the wings of the 15 μm band, low-level water vapor in the troposphere can diminish the effects of increasing carbon dioxide on the emitted flux at the top of the atmosphere. Using the Goody random model, calculate the mean transmittance of water vapor for emission at the top of the atmosphere in the 25 cm−1 interval centered on 737.5 cm−1 . For this wave number interval, the sum of the ∑ strengths of the water vapor lines is given by Sj = 31.84 cm−1 (cm2 g−1 ) and the sum of the square roots√of the strengths times the pressure-broadened ∑ half-widths is given by Sj �L0j = 3.621 cm−1 (cm2 g−1 )1∕2 with the halfwidths evaluated at a pressure of 1 atm [11]. Assume that the mass mixing ratio for water vapor is given in terms of the pressure P by ( )3 P r(P) = rS PS with rS = 0.008 kg H2 O/kg AIR the surface mixing ratio and PS = 1 atm the surface pressure. Use a diffusivity factor of M = 1.66 to obtain the transmittance from the surface to space. a. Show that for the path from the surface to space in this spectral interval water vapor is in the strong, pressure-broadened line limit. b. Show that the pressure associated with the level at which the optical path length to space is unity is given by √ 5g Pi = PS � ���L0 rS MPS

References

with

√ ���L0 �

=

2 ∑ Δ�i j

√ Sj �L0j

What is the value of Pi for this spectral interval? c. What is the mean transmittance for water vapor in this spectral interval? How would it affect the change in irradiance at the top of the atmosphere for a doubling of CO2 shown in Figure 7.4? 3. Prior to the ban on manufacturing, the rising concentration of chlorofluorocarbons (CFCs) in the atmosphere was recognized as potentially rivaling the radiative forcing due to increasing CO2 . The total of the line strengths for the strongest CFC absorber is 6.74 × 104 cm−1 (kg m−2 )−1 . Owing to the small concentration of the CFCs, all of the absorption lines in the bands are in the weak-line limit. In addition, the bands are in the atmospheric infrared window. For purposes of calculation assume that the absorption is at 11 μm and that the infrared window is transparent. Because of their long lifetimes, CFCs are well-mixed in the atmosphere. a. Assume that the Planck function in the window region may be approximated by ( ) P B(P) = BS PS with P the pressure, BS the Planck function evaluated at the surface air temperature, which in this case is the same as the surface temperature, and PS the surface pressure. Starting with Equation 7.2, derive an analytic expression for the radiative forcing due to increasing CFCs (see Chapter 4, Problem 3). b. Calculate the change in the emitted radiative flux at the top of the atmosphere for a 1 ppb (parts per billion) increase in the CFC with the strongest absorption bands. The molecular weight of the CFC with the strongest absorption is 4.17 times that of air.

References 1. Rodgers, C.D. and Walshaw, C.D. (1966)

The computation of infrared cooling rate in planetary atmospheres. Q. J. R. Meteorol. Soc., 92, 67–92. 2. Forester, P., Ramaswamy, V., Artaxo, P., Berntsen, T., Betts, R., Fahey, D.W., Haywood, J., Lean, J., Lowe, D.C., Myhre, G., Nganga, J., Prinn, R., Raga, G., Schulz, M., and Vandorland, R. (2007) Changes in atmospheric constituents and radiative forcing, in In

Climate Change 2007: The Physical Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (eds S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt, M. Tignor, and H.L. Miller), Cambridge University Press, Cambridge, New York. 3. Huang, Y., Leroy, S., Gero, P.J., Dykema, J., and Anderson, J. (2010) Separation

217

218

7 Simplified Estimates of Emission

4.

5.

6.

7.

8. Manabe, S. and Wetherald, R.T. (1967) of longwave climate feedbacks from Thermal equilibrium of the atmosphere spectral observations. J. Geophys. with a given distribution of relative Res., 115, D07104. doi:10.1029/ humidity. J. Atmos. Sci., 24, 241–259. 2009JD012766 9. Soden, B.J. and Bretherton, F.P. (1993) Clough, S.A., Shepard, M.W., Mlawer, Upper tropospheric relative humidity E.J., Delamere, J.S., Iacono, M.J., from GOES 6.7 μm channel: method and Cady-Pereira, K., Boukabara, S., and climatology for July 1987. J. Geophys. Brown, P.D. (2005) Atmospheric radiaRes., 98, 16,669–16,688. tive transfer modeling: a summary of the 10. Kahn, B.H., Fishbein, E., Nasiri, S.L., AER codes. J. Quant. Spectrosc. Radiat. Eldering, A., Fetzer, E.J., Garay, M.J., Transfer, 91, 233–244. Hartmann, D.L. (1994) Global Physical and Lee, S.-Y. (2007) The radiative Climatology, Academic Press, San Diego, consistency of atmospheric infrared CA. sounder and moderate resolution imagFels, S.B. and Schwarzkopf, M.D. (1975) ing spectroradiometer cloud retrievals. The simplified exchange approxiJ. Geophys. Res., 112, D09201. doi: mation: a new method for radiative 10.1029/2006JD007486 transfer calculations. J. Atmos. Sci., 32, 11. Houghton, J.T. (2002) The Physics of the 1475–1488. Atmosphere, Cambridge University Press, Manabe, S. and Strickler, R.F. (1964) Cambridge. Thermal equilibrium of the atmosphere with convective adjustment. J. Atmos. Sci., 21, 361–385.

219

Appendix A Useful Physical and Geophysical Constants Table A.1 Physical and geophysical constants used in the book. The symbols are those most commonly used for the constant in the book. Earth Average radius of Earth’s orbit, 1 AUa) Current eccentricity of Earth’s orbit Current obliquity of Earth’s orbit Current time of perihelionb) Average radius of the Earth Earth’s acceleration due to gravity Sun Angle subtended by solar disk at 1 AU Solar constant Earth’s atmosphere Gas constant for dry air Global average surface pressure Global annual average surface temperature Mass of dry air Specific heat at constant pressure for dry air Latent heat of vaporization for water at 0 ∘ C Heat capacity of liquid water Physical constants Avogadro’s number Boltzmann’s constant Universal gas constant Planck’s constant Plank’s first radiation constantc) Plank’s second radiation constant

Symbol

RE g

1.496 × 1011 m 0.01675 23.45∘ 3 January 6.371 × 106 m 9.81 m s−2

Q0

0.533∘ 1361 W m−2

R, RAIR PS TS mAIR CP

k R* h C1 C2

287 J K−1 kg−1 1013.25 hPa 288 K 28.97 g mol−1 1005 J K−1 kg−1 2.5 × 106 J kg−1 4218 J K−1 kg−1 6.022 × 1023 mol−1 1.38 × 10−23 J K−1 8.314 J K−1 mol−1 6.626 × 10−34 J s −5 1.19 × 10 mW m−2 sr−1 (cm−1 )−4 1.44 K(cm−1 )−1 (continued overleaf )

Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

220

Appendix A Useful Physical and Geophysical Constants

Table A.1 (Continued) Earth Speed of light in vacuum Standard temperature Standard pressure Stefan-Boltzmann constant Volume of ideal gas at STP

Symbol c T0 P0 �

2.998 × 108 m/s 273.15 K 1013.25 hPa 5.67 × 10−8 W m−2 K−4 0.224 m3 mol−1

a) AU is one Astronomical Unit, the average distance between the Earth and sun. b) Perihelion as of 2010. c) In this book Planck’s first radiation constant is given by C1 = 2hc2 . The units are those most commonly used for radiances emitted by the Earth’s surface and atmosphere.

221

Appendix B Solving Differential Equations There are three types of differential equations encountered in this introductory treatment of atmospheric radiation. The equations and methods for solving them are given in the following text. Interestingly, these methods were developed by scientists who faced problems that led to these particular equations. In some cases the methods were developed a couple of centuries ago. The practical means that scientists adopted for solving the equations remain the preferred methods of solution. B.1 Simple Integration

The simplest equation is df (x) =A (B.1) dx with A a constant. Such an equation is encountered, for example, for the mean radiance under conditions of radiative equilibrium for which the net radiative flux is constant and the radiative heating is zero, for example, Equations 4.71a and b. The solution is obtained by integration. f (x) = Ax + B

(B.2)

with B a constant that is determined through the application of a boundary condition. The application of a boundary condition ensures that the value of f (x) is realistic at a particular value of x. For example, f (x1 ) = Ax1 + B might be an observed value from which B is obtained. B.2 Integration Factor

The second type of equation is a first order differential equation with constant coefficient. It is solved using an “integration factor.” The equation is df (x) + Af (x) = h(x) dx

(B.3)

Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

222

Appendix B Solving Differential Equations

with A a constant and h(x) any function of x. The solution is obtained by seeking another function g(x), the integration factor. Multiplying through the equation by g(x) converts the left side of the equation into a derivative of the product g(x)f (x). This product is equal to the integral of g(x)h(x) over x. The integration factor that accomplishes this outcome is determined as follows. First, multiplying Equation B.3 by g(x) gives g(x)

df (x) + Ag(x)f (x) = g(x)h(x) dx

(B.4)

Next, the “chain rule” for derivatives is applied. The chain rule is given by d[g(x)f (x)] df (x) dg(x) = g(x) + f (x) dx dx dx

(B.5)

If g(x)

dg(x) df (x) df (x) + f (x) = g(x) + Ag(x)f (x) dx dx dx

(B.6)

for all x, then dg(x) = Ag(x) dx

(B.7)

The integration factor that converts the left side of Equation B.3 into a derivative of g(x)f (x) is given by g(x) = BeAx

(B.8)

with B a constant. The value of B is immaterial. Any value of B satisfies Equation B.7. As a result, setting B = 1 has no effect on the solution of Equation B.3. With g(x) determined, Equations B.4–B.6 lead to d[g(x)f (x)] = g(x)h(x) dx x1

∫xo

x1

d[g(x)f (x)] =

∫x0

dx′ g(x′ )h(x′ ) x1

g(x1 )f (x1 ) − g(x0 )f (x0 ) =

∫x0

dx′ g(x′ )h(x′ )

and x1

g(x0 )f (x0 ) + f (x1 ) =

∫x0

dx′ g(x′ )h(x′ )

g(x1 )

(B.9)

In the case of the time dependent solution for the perturbation of the atmospheric temperature in the radiative equilibrium, window-gray model as given by Equation 1.30, the initial perturbation temperature was set to zero at the start of the perturbation, corresponding to f (x0 ) = 0 for x0 = 0 in Equation B.9.

B.3 Second Order Differential Equations

The solution given here is known as the particular solution. In satisfying the boundary conditions so as to make the solution of Equation B.3 physically reasonable, the addition of a solution for the homogeneous equation may be required. The homogeneous equation is given by df (x) + Af (x) = 0 dx

(B.10)

In this case, the solution is the same as that for the integration factor given in Equation B.8. f (x) = Ce−Ax

(B.11)

with C a constant. Since equations such as Equation B.3 are linear, any solution of the form (Equation B.11) can be added to the particular solution (Equation B.9) and the sum satisfies Equation B.3. The addition of a solution for the homogeneous equation to the particular solution is used in Section 4.12 for the Eddington approximation of radiative transfer in which the diffuse and direct radiative fluxes are separated.

B.3 Second Order Differential Equations

The third and final equation encountered is a second order differential equation with constant coefficient. The equation is d2 f (x) − �f (x) = 0 d2 x

(B.12)

with � a constant. This type of equation arises in the Eddington approximation for radiative transfer, for example, Equation 4.57a. The solution is f (x) = Ae�x + Be−�x

(B.13)

with A and B √ constants. Inserting Equation B.13 into Equation B.12 gives � 2 = � so that � = ± �. Since the equation is second order, there are two constants, A and B, which are determined through the application of boundary conditions. Known values of f (x) or df (x)∕dx are used for specific values of x to determine A and B. For example, if f (x1 ) = Ae�x1 + Be−�x1

(B.14a)

f (x2 ) = Ae�x2 + Be−�x2

(B.14b)

and

223

224

Appendix B Solving Differential Equations

the two equations can be solved algebraically for A and B. If instead the physical reasoning leads to a value for a derivative as given by df (x) | = �Ae�x1 − �Be−�x1 (B.15) dx x1 then Equation B.15 could serve as one of the algebraic equations used to obtain A and B.

225

Appendix C Integrals of the Planck Function Table C.1 provides integrals of the Planck function. In Section 2.4 the Planck function is given by 2hc2 v3 Bv (T) = ( hcv ) e kT − 1

(C.1)

The integral of the Planck function over all wavenumbers leads to the Stefan–Boltzmann law for thermal emission by a blackbody as given by �

∫0

∞

d�Bv (T) = �

∫0

∞

2hc2 v3 d� ( hcv ) = �T 4 e kT − 1

(C.2)

Using the change of variable x = hc�∕kT in the integrals of Equation C.2 leads to ∫0

∞

dxB(x) =

∫0

∞

dx

(ex

�4 x3 = − 1) 15

(C.3)

The fraction of the integral contributed by wavenumbers less than a given value associated with x is given by x

�(x) =

15 x3 dx x 4 � ∫0 (e − 1)

(C.4)

Table C.1 provides values of �(x). For the radiative flux emitted by a blackbody for a particular spectral interval �1 ≤ � ≤ �2 associated with x1 ≤ x ≤ x2 , the fraction of the flux integrated over all wavenumbers is given by ] [ x2 x1 x2 15 x3 x3 x3 15 = �(x2 ) − �(x1 ) = − dx dx dx (ex − 1) � 4 ∫0 (ex − 1) � 4 ∫x1 (ex − 1) ∫0 (C.5) Multiplying the fraction in Equation C.5 by �T 4 provides the irradiance between the two wavenumbers. Dividing the irradiance by � gives the integrated radiance for the spectral interval. Dividing the integrated radiance by the width of the spectral interval gives the average radiance.

Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

226

Appendix C Integrals of the Planck Function

Table C.1 Fractions of the total integral of the Planck function as given by �(x) in Equation C.4. x

�(x)

x

�(x)

x

�(x)

x

�(x)

x

�(x)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6

0.00005 0.00038 0.0012 0.0028 0.0053 0.0088 0.0134 0.0192 0.0263 0.0346 0.0442 0.0551 0.0671 0.0804 0.0948 0.1102 0.1267 0.1440 0.1622 0.1811 0.2007 0.2209 0.2416 0.2626 0.2840 0.3056

2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2

0.3274 0.3493 0.3712 0.3930 0.4147 0.4363 0.4576 0.4786 0.4994 0.5197 0.5397 0.5593 0.5784 0.5970 0.6151 0.6328 0.6499 0.6664 0.6825 0.6980 0.7129 0.7273 0.7412 0.7545 0.7673 0.7796

5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

0.7913 0.8026 0.8133 0.8236 0.8334 0.8427 0.8517 0.8601 0.8682 0.8758 0.8831 0.8900 0.8966 0.9028 0.9086 0.9142 0.9194 0.9244 0.9291 0.9335 0.9377 0.9416 0.9453 0.9488 0.9521 0.9552

7.9 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.4

0.9581 0.9608 0.9634 0.9658 0.9680 0.9701 0.9721 0.9740 0.9757 0.9774 0.9789 0.9803 0.9817 0.9829 0.9841 0.9852 0.9862 0.9872 0.9881 0.9889 0.9897 0.9904 0.9911 0.9917 0.9923 0.9928

10.5 10.6 10.7 10.8 10.9 11.0 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13.0

0.9934 0.9938 0.9943 0.9947 0.9951 0.9954 0.9957 0.9961 0.9963 0.9966 0.9968 0.9971 0.9973 0.9975 0.9977 0.9978 0.9980 0.9981 0.9983 0.9984 0.9985 0.9986 0.9987 0.9988 0.9989 0.9990

227

Appendix D Random Model Summations of Absorption Line Parameters for the Infrared Bands of Carbon Dioxide Table D.1 Sums of line strengths, S =

∑

Sj , and square roots of the product of line strengths ∑√ and pressure-broadened half-widths, R = SJ �L0J for carbon dioxide and the indicated wavenumber intervals. The parameters are given for three different temperatures, T. The pressure-broadened half-widths are for the given temperatures and 1 atm pressure. The units of S are cm−1 cm2 g−1 and those of R are cm−1 (cm2 g−1 atm−1 )1/2 . The line parameters are from the HITRAN2008 molecular spectroscopic database [1].

T (K) Interval (cm−1 )

S

200 R

S

255 R

S

310 R

500–525 525–550 550–575 575–600 600–625 625–650 650–675 675–700 700–725 725–750 750–775 775–800 800–825 825–850 850–875 875–900 900–925 925–950 950–975 975–1000 1000–1025 1025–1050

6.44E−03 1.22E−01 1.82E−01 2.75E+01 3.24E+02 5.92E+03 7.93E+04 2.39E+04 7.24E+02 1.06E+02 2.60E+00 3.54E−01 1.08E−01 2.40E−03 5.65E−04 1.25E−03 6.25E−03 9.24E−02 1.62E−01 5.36E−02 3.46E−03 1.23E−01

1.41E−01 7.11E−01 1.23E+00 1.23E+01 4.19E+01 1.61E+02 6.44E+02 2.16E+02 5.62E+01 1.74E+01 3.12E+00 1.25E+00 4.07E−01 1.20E−01 6.01E−02 6.40E−02 1.62E−01 4.09E−01 5.08E−01 2.65E−01 8.50E−02 4.21E−01

8.87E−02 1.07E+00 2.17E+00 1.17E+02 9.16E+02 8.63E+03 8.23E+04 2.69E+04 2.06E+03 3.28E+02 2.05E+01 2.76E+00 8.78E−01 5.27E−02 1.33E−02 2.03E−02 1.12E−01 9.52E−01 1.27E+00 6.09E−01 4.99E−02 1.15E+00

6.03E−01 2.33E+00 4.67E+00 3.10E+01 7.97E+01 2.32E+02 7.71E+02 2.86E+02 1.11E+02 3.88E+01 1.08E+01 4.07E+00 1.31E+00 5.93E−01 3.27E−01 3.31E−01 7.41E−01 1.51E+00 1.49E+00 9.55E−01 3.33E−01 1.49E+00

4.91E−01 4.38E+00 1.14E+01 3.10E+02 1.81E+03 1.14E+04 8.57E+04 2.95E+04 4.13E+03 7.00E+02 7.90E+01 1.10E+01 3.39E+00 3.89E−01 1.07E−01 1.61E−01 8.06E−01 4.33E+00 4.71E+00 2.93E+00 3.42E−01 4.92E+00

1.57E+00 5.28E+00 1.16E+01 5.96E+01 1.28E+02 3.17E+02 9.26E+02 3.72E+02 1.81E+02 6.99E+01 2.52E+01 9.43E+00 2.88E+00 1.68E+00 9.98E−01 1.02E+00 2.09E+00 3.58E+00 3.05E+00 2.21E+00 8.90E−01 3.49E+00

(continued overleaf )

Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

228

Appendix D Random Model Summations of Absorption Line Parameters

Table D.1 (Continued)

T (K) Interval (cm−1 )

S

200 R

S

255 R

S

310 R

1050–1075 1075–1100 1100–1125 1200–1225 1225–1250 1250–1275 1275–1300 1300–1325 1325–1350 1350–1375 1375–1400 1400–1425 1800–1825 1825–1850 1850–1875 1875–1900 1900–1925 1925–1950 1950–1975 1975–2000 2000–2025 2025–2050 2050–2075 2075–2100 2100–2125 2125–2150 2150–2175 2175–2200 2200–2225 2225–2250 2250–2275 2275–2300 2300–2325 2325–2350 2350–2375 2375–2400 2400–2425 2425–2450 2450–2475 2475–2500 2500–2525

3.15E−01 1.66E−01 3.18E−05 1.63E−03 1.35E−01 2.29E−01 7.75E−02 1.02E−03 6.70E−02 2.52E−01 1.78E−01 4.26E−03 7.08E−05 1.00E−03 2.39E−02 1.68E−01 3.99E+00 2.76E+00 1.97E−01 1.22E−02 2.88E−01 2.96E+00 3.02E+01 3.13E+01 6.19E−01 3.45E−01 9.49E−03 4.55E−03 3.35E−01 1.40E+02 4.01E+03 7.06E+03 7.81E+04 5.08E+05 6.06E+05 7.52E+03 1.87E−02 2.12E−02 6.15E−03 8.28E−02 9.13E−02

8.23E−01 6.07E−01 8.16E−03 7.27E−02 6.73E−01 9.92E−01 4.77E−01 4.25E−02 4.17E−01 1.09E+00 8.85E−01 1.43E−01 9.43E−03 3.78E−02 2.86E−01 1.11E+00 3.06E+00 2.62E+00 5.10E−01 2.19E−01 7.70E−01 3.45E+00 7.73E+00 1.08E+01 1.76E+00 1.09E+00 2.65E−01 1.72E−01 1.59E+00 2.08E+01 9.27E+01 1.63E+02 4.59E+02 1.06E+03 1.01E+03 6.20E+01 1.50E−01 2.11E−01 1.83E−01 5.04E−01 5.31E−01

2.16E+00 1.56E+00 1.98E−03 5.88E−03 1.61E−01 2.15E−01 1.02E−01 1.97E−03 9.16E−02 2.39E−01 2.07E−01 1.29E−02 9.89E−04 8.45E−03 7.46E−02 4.81E−01 5.01E+00 2.89E+00 3.51E−01 4.98E−02 4.01E−01 6.17E+00 3.44E+01 3.47E+01 1.92E+00 1.06E+00 7.27E−02 7.17E−02 3.59E+00 3.74E+02 4.64E+03 8.67E+03 1.38E+05 5.18E+05 6.43E+05 2.05E+04 1.43E−01 1.52E−01 3.35E−02 8.82E−02 9.16E−02

2.43E+00 2.20E+00 6.63E−02 1.42E−01 8.13E−01 1.02E+00 5.60E−01 6.33E−02 5.04E−01 1.14E+00 1.03E+00 2.59E−01 3.67E−02 1.16E−01 5.59E−01 1.83E+00 3.88E+00 2.93E+00 8.05E−01 4.37E−01 1.13E+00 4.71E+00 9.53E+00 1.38E+01 3.68E+00 2.34E+00 8.39E−01 8.25E−01 5.87E+00 4.12E+01 1.27E+02 2.59E+02 7.21E+02 1.23E+03 1.16E+03 1.15E+02 4.23E−01 5.72E−01 4.10E−01 5.79E−01 5.33E−01

7.46E+00 6.87E+00 3.11E−02 1.33E−02 1.83E−01 2.05E−01 1.22E−01 3.52E−03 1.13E−01 2.31E−01 2.31E−01 2.68E−02 5.42E−03 3.40E−02 1.71E−01 1.08E+00 5.92E+00 3.03E+00 5.38E−01 1.31E−01 5.64E−01 1.05E+01 3.83E+01 3.93E+01 4.21E+00 2.35E+00 2.97E−01 5.06E−01 1.85E+01 7.45E+02 5.26E+03 1.43E+04 2.05E+05 5.27E+05 6.75E+05 3.94E+04 5.25E−01 5.39E−01 1.43E−01 9.64E−02 9.03E−02

5.08E+00 5.23E+00 2.65E−01 2.18E−01 9.22E−01 1.06E+00 6.22E−01 8.51E−02 5.73E−01 1.19E+00 1.15E+00 3.80E−01 8.77E−02 2.39E−01 9.16E−01 2.72E+00 4.68E+00 3.28E+00 1.15E+00 7.06E−01 1.58E+00 6.16E+00 1.17E+01 1.74E+01 6.32E+00 4.14E+00 1.86E+00 2.48E+00 1.47E+01 6.98E+01 1.85E+02 4.22E+02 1.03E+03 1.45E+03 1.32E+03 1.75E+02 8.30E−01 1.11E+00 7.72E−01 6.75E−01 5.29E−01

Reference

Table D.1 (Continued)

T (K) Interval (cm−1 )

S

200 R

S

255 R

S

310 R

2525–2550 2550–2575 2575–2600 2600–2625 2625–2650 2650–2675 2700–2725 2725–2750 2750–2775 2775–2800 3000–3100 3100–3200 3200–3300 3300–3400 3600–3700 3700–3800 3800–3900 3900–4000 4000–4100 4400–4500 4500–4600 4600–4700 4700–4800 4800–4900 4900–5000 5000–5100 5100–5200 5200–5300 5300–5400 5600–5700 5700–5800 5800–5900 5900–6000

9.36E−03 3.93E−04 5.11E−02 1.58E−01 9.93E−02 5.57E−03 2.41E−05 1.57E−02 2.15E−02 6.05E−03 7.99E−05 7.18E−02 4.45E−02 9.75E−01 1.39E+04 1.57E+04 4.90E−02 7.60E−03 1.01E−02 1.07E−03 8.82E−03 2.22E−01 1.64E+00 1.07E+02 4.45E+02 7.75E+01 7.60E+01 6.73E−02 5.57E−01 3.78E−04 4.41E−05 4.60E−03 9.33E−03

1.66E−01 1.79E−02 3.35E−01 7.76E−01 6.30E−01 1.11E−01 1.79E−03 1.84E−01 2.36E−01 1.10E−01 6.42E−03 5.30E−01 4.23E−01 2.44E+00 2.80E+02 2.34E+02 6.12E−01 2.18E−01 1.93E−01 3.37E−02 2.03E−01 1.75E+00 5.58E+00 2.67E+01 4.86E+01 1.70E+01 1.49E+01 6.04E−01 1.52E+00 2.94E−02 4.89E−03 1.33E−01 3.11E−01

1.50E−02 7.76E−04 6.80E−02 1.48E−01 1.18E−01 6.80E−03 7.43E−05 1.76E−02 1.99E−02 8.27E−03 5.02E−04 9.31E−02 8.45E−02 1.30E+00 1.54E+04 1.64E+04 2.49E−01 1.55E−02 3.10E−02 1.36E−03 1.35E−02 2.54E−01 3.53E+00 1.17E+02 4.78E+02 9.38E+01 8.56E+01 1.12E−01 5.89E−01 2.84E−03 4.58E−04 4.72E−03 1.67E−02

2.07E−01 2.67E−02 4.20E−01 8.12E−01 7.49E−01 1.32E−01 3.14E−03 2.02E−01 2.26E−01 1.31E−01 1.67E−02 7.00E−01 6.54E−01 3.34E+00 3.38E+02 2.93E+02 1.24E+00 3.38E−01 3.36E−01 3.86E−02 2.72E−01 2.10E+00 9.16E+00 3.21E+01 6.04E+01 2.17E+01 2.00E+01 1.00E+00 1.70E+00 8.23E−02 1.62E−02 1.36E−01 4.33E−01

2.10E−02 1.53E−03 8.27E−02 1.42E−01 1.36E−01 8.35E−03 1.49E−04 1.89E−02 1.85E−02 1.01E−02 1.64E−03 1.20E−01 1.45E−01 1.69E+00 1.68E+04 1.75E+04 8.79E−01 3.52E−02 7.05E−02 1.60E−03 1.99E−02 2.97E−01 7.19E+00 1.28E+02 5.14E+02 1.12E+02 9.74E+01 1.83E−01 6.20E−01 1.04E−02 2.04E−03 4.71E−03 3.37E−02

2.42E−01 3.64E−02 4.91E−01 8.44E−01 8.52E−01 1.50E−01 4.45E−03 2.14E−01 2.17E−01 1.47E−01 3.08E−02 8.76E−01 9.14E−01 4.35E+00 4.11E+02 3.65E+02 2.27E+00 5.01E−01 5.06E−01 4.19E−02 3.39E−01 2.48E+00 1.41E+01 3.79E+01 7.49E+01 2.71E+01 2.62E+01 1.44E+00 1.88E+00 1.60E−01 3.46E−02 1.36E−01 5.94E−01

Reference 1. Rothman, L.S., Gordon, L.E., Barbe,

A., et al., (2009) The HITRAN 2008 molecular spectroscopic database. J.

Quant. Spectrosc. Radiat. Transfer, 110, 233–244.

229

231

Appendix E Ultraviolet and Visible Absorption Cross Sections of Ozone Table E.1 Absorption cross-sections � at ultraviolet and visible wavelengths � for an ozone molecule at a temperature of 273 K. The tabulated cross sections were taken from Houghton [1] for wavelengths less than 230 nm and interpolated from a compilation by Orphal [2] for wavelengths greater than 230 nm.

� (nm)

� (cm2 )

200 210 220 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315

3.1540e−19 5.7160e−19 1.7850e−18 4.5402e−18 6.3253e−18 8.2837e−18 9.8091e−18 1.1142e−17 1.1449e−17 1.0670e−17 9.6011e−18 7.9403e−18 5.8245e−18 3.9711e−18 2.4419e−18 1.3902e−18 7.5886e−19 3.8053e−19 1.9143e−19 9.5910e−20 4.7492e−20

� (nm) 320 320 325 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410

� (cm2 ) 2.8969e−20 2.8969e−20 1.6465e−20 1.6465e−20 4.0623e−21 2.1269e−21 1.4333e−21 6.0016e−22 2.1597e−22 8.0739e−23 6.1943e−23 2.9826e−23 8.6870e−24 7.4008e−24 5.0366e−24 4.9339e−24 6.8539e−24 1.0865e−23 1.1825e−23 1.5995e−23 2.8581e−23

� (nm) 415 420 425 430 435 440 445 450 475 500 525 550 575 600 625 650 675 700 725 750

� (cm2 ) 2.9155e−23 3.7333e−23 7.0091e−23 6.3741e−23 8.6232e−23 1.3599e−22 1.6409e−22 1.8953e−22 4.6177e−22 1.2122e−21 2.2134e−21 3.4262e−21 4.9167e−21 5.2562e−21 3.9109e−21 2.5680e−21 1.5713e−21 8.9957e−22 5.4672e−22 4.3830e−22

References 1. Houghton, J.T. (2002) The Physics of the

Atmosphere, Cambridge University Press. 2. Orphal, J. (2003) A critical review of the absorption cross-sections of O3 and

NO2 in the ultraviolet and visible. J. Photochem. Photobiol. A, 157, 185–209.

Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

233

Index

a absorptance 90, 106, 118 absorption – atmosphere 30, 186, 190, 191, 193, 197 – band strength, molecular 170 – coefficient 78, 90 – cross section 65, 69, 77 – earth 4, 26, 30, 31, 33, 35, 197 – inhomogeneous path 174, 176, 178 – line strength 163 – overlapping bands, molecular 182 – overlapping lines 171 – particle 121, 136 – radiative transfer equation 79, 81 – spectra, molecular 28 – strong, pressure-broadened line limit 166, 171, 193, 205, 206 – surface 30, 197 – weak line limit 164, 166, 168, 170, 172, 175, 191, 206 absorptivity 6, 90 actinic flux 115 adding layers – adding-doubling method 138 – Eddington approximation 121 adding-doubling method 138 adiabatic lapse rate 22 aerosol 18 – continental, composition 76 – direct radiative forcing 199 – extinction cross section, example 69 – indirect radiative forcing 200 – infrared scattering and absorption 80 – optical depth, example 69 – reflectance, �-Eddington approximation 133, 136 – reflectance, Eddington approximation 120, 133, 136

– reflectivity, single scattering approximation 136 – scattering phase function, example 75 – upward scattering fraction, example 107 – volcanic layer 11, 37 albedo – Bond 118 – �-Eddington approximation 120 – earth 4, 26, 27, 32, 33, 35, 197, 198 – Eddington approximation 120 – planar, planetary 106 – single scattering 79 – spherical 118 – zonal 35 angular momentum 155 – quantum number, component fixed in space 158 – quantum number, total 155, 158 – units, Planck’s constant 53, 155 anharmonic oscillation 158 astronomical unit (AU) 3, 219 asymmetric top molecule 157 atmospheric composition 14 average radiance – angular 115 – partly cloudy sky, infrared 126 – wavelength, band average 150 – wavelength, blackbody 56, 225 average transmission – apparent breakdown of exponential law 150, 159, 165 – correlated k 178, 180, 182 – exponential sum-fit 177, 182 – inhomogeneous path 166 – nonoverlapping lines 169 – overlapping lines 171 – overlapping molecular bands 182 – random model, Goody 176

Atmospheric Radiation: A Primer with Illustrative Solution, First Edition. James A. Coakley Jr and Ping Yang. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

234

Index

average transmission (contd.) – random model, Malkmus 179 – single, pressure-broadened, strong line limit 164 – weak line limit 163 azimuthal component, radiance and phase function 112 azimuthally averaged – phase function 113 – radiance 112

b Babinet’s principle. See diffraction ballistic bolometer 47 Beer-Bouguer-Lambert law 68, 126, 159 blackbody – radiating temperature, cloud 62 – radiating temperature, zonal average 35 – radiation, Planck function 52 – radiative equilibrium temperature 5 – spectral irradiances, examples 54, 61 – Stefan-Boltzmann law 5 – sun, as a blackbody 26, 53–55 – Wien’s displacement law 51, 55, 62 Boltzmann distribution 51, 154, 164 Bond albedo 118

c carbon dioxide, CO2 – absorption cross section, example 69, 81 – absorption of sunlight 185 – concentration 14 – column amount, path to space 205 – concentration 18 – concentration increase, fossil fuels 10, 209 – concentration variations, natural 10 – doubling 209 – effective pressure, path to space 205 – electric dipole 19, 154 – emitted radiance, 15 μm 81, 209, 212 – mass mixing ratio 18 – pressure weighted amount 205 – radiative forcing 11, 209, 210 – random model parameters 175, 205, 227 – stratospheric emission 26, 214 – stratospheric temperature change 213 – strong, pressure-broadened line limit 205 – surface temperature change 12 – vibration-rotation band, 15 μm 29, 154, 156 – weak line limit 207 Chandrasekhar, S. 111, 135 chlorofluorocarbons, CFCs 14, 170, 190, 217 climate feedbacks 8, 31, 33, 38

climate sensitivity 8 – greenhouse forcing and greenhouse effect 8 – uncertainty 31 – window-gray model 12 cloud – absorption, infrared 125 – absorption, near infrared 194 – effective fraction 126 – emission by overcast regions 125 – extinction cross section 69 – greenhouse forcing 30 – hydrometeor 66 – longwave radiative forcing 29 – optical depth, example 69 – scattering phase function, example 75 – shortwave radiative forcing 31 – single scattering albedo, infrared 125 – single scattering albedo, visible light 120 Cloud and Earth’s Radiant Energy System, CERES – blackbodies 50 – Earth’s radiation budget 32 – radiometer 47 – zonal average radiative fluxes 35 cm-STP 18 column amount 17 – carbon dioxide 205 – ozone 17 – water vapor 192 Community Climate System Model (CCSM) 33, 34 conservation of energy – adiabatic process 23 – first law of thermodynamics 2 conservative scattering 119, 122 cool to space approximation 101, 215 cosine series 111, 112 critical lapse rate 24 cross section 65 – absorption 77 – scattering and differential scattering 70 – extinction 66 Curtis-Godson approximation 169, 182, 205, 206

d daylight hours 60, 61 declination of the sun 59 degeneracy 158 �-Eddington approximation 130 – scaled optical properties 132 – scattering phase function 130

Index

– transmittance, total, diffuse, and direct beam 132 �-M method 137, 139 dielectric media 41, 76 diffraction 74, 125 diffuse reflector 124 diffusion length 119 diffusivity factor 95 – diffusely scattered light 188, 192 – infrared radiation 95, 204 – strong, pressure-broadened line limit 183 – weak line limit 183 dipole, electric – oscillating 42, 154, 155 – permanent 153 Dirac delta function 102, 130, 159, 162 direct solar beam – attenuated by the atmosphere 102, 126 – top of the atmosphere 58, 102, 127 discrete ordinate method and DISORT 119, 132, 135 division of atmosphere into layers 186, 195 doppler broadening 162 doubling layers 139

e Earth’s energy budget 3, 26, 30 Earth’s orbit 61, 219 Eddington approximation 114, 195, 224 – adding layers 121 – adding surface albedo 123 – clouds, infrared 124 – reflectance and transmittance, examples 120 Einstein, A. 53, 154 elastic scattering 65 electric field vector 43, 72, 154 electromagnetic waves 41, 52, 72, 111, 138 electronic transition 152 emission level and emitting layer 91, 93, 98 emissivity 6, 90, 92, 125 equation of state, ideal gas 3, 16, 161 equivalent width 164 – random band model 174, 176 – strong, pressure-broadened line limit 165, 171 – weak line limit 164, 170 evaporation and evapotranspiration 24, 30 exponential extinction. See Beer-Bouguer-Lambert law exponential integral function 95 exponential sum-fit transmission model 177, 182

extinction coefficient and cross section extinction of species 121

69

f first law of thermodynamics 2 flat plate radiometer 63 formal solution, radiative transfer equation. See radiative transfer equation Fourier decomposition 112

g gas constant 3, 16, 219 global annual mean radiative imbalance 32 Goody random band model 176 Goody, R.M. 173 gray body 6, 30 greenhouse effect 6 greenhouse forcing 8, 30, 31 greenhouse gas 1, 31, 93, 99, 212

29,

h harmonic oscillator 152, 158 haze. See aerosol heat capacity 219 – air 3 – Earth 9 – surface 13 – water 38 heating rate – cloud-free solar due to water vapor 193, 195, 200 – cool to space approximation 101, 215 – first law of thermodynamics 3 – infrared 25, 93, 98 – overcast solar due to water vapor 197, 201 – radiative 21, 25, 100, 116, 178 – radiative–convective equilibrium 25 – solar 25 Henyey-Greenstein phase function 82, 108, 132, 144 hot bands 158 hour angle 58 hours of daylight 60, 61 hydrometeor 66, 194 hydrostatic balance 22, 23, 90

i ice crystals 26, 66, 69, 76, 124, 194 ideal gas law 3, 16, 161 impedance of free space 43 incomplete gamma function 91 index of refraction 72 infrared radiation 1

235

236

Index

infrared radiation (contd.) – diffusivity factor 95, 204 – emission level and emitting layer 91, 93 – heating rate. See heating rate – wavelength range, Earth 1, 41, 56 infrared spectra – CO2 15μm band 156 – H2 O rotation band 157 – N2 O 8.6 μm band 153 – surface 28 – top of atmosphere 28, 94, 212 infrared window, Earth’s atmosphere 28, 30, 49, 54, 56, 62, 100, 143, 145, 217 inhomogeneous atmospheric path 124, 137, 139, 166, 174, 176, 178 integration factor 13, 87, 221 internal energy 2, 152 inverse laplace transform 178 irradiance 46, 53, 58, 90 isotropic phase function 74, 75, 135, 136 isotropic radiation 43, 46, 51–53, 212

Mie theory 77 moment of inertia, rigid rotor 155 Monte Carlo simulation 140 Montreal protocol 190 Mt. Pinatubo eruption 11, 37

n near infrared – absorption of sunlight by clouds 194 – absorption of sunlight by water vapor 26, 27, 191 – wavelengths 27 net radiative flux 20, 21, 26, 30, 34, 35, 115 – longwave 98, 101, 210, 213, 214 – shortwave 196 nitrous oxide, N2 O – absorption spectra, 8.6 μm band 155, 158 – concentration and residence time 14 – electric dipole moment 18, 153 – greenhouse forcing with CH4 31 nonlocal thermodynamic equilibirium 154 nuclear winter 120

k k-distribution method 178, 182 Kirchoff ’s radiation law 6 Kronecker delta symbol 111

l Lambertian reflector 124 Laplace transform 178 lapse rate 24 latent heat 26, 219 Legendre polynomials 75, 114, 130, 135 Liou, K-N. 136 local thermodynamic equilibrium 79, 86, 88, 154 longwave radiation. See infrared radiation Lorentz half-width 162 Lorentz line shape factor 161 Lorentz, H.A. 159, 165 Lorenz–Mie theory. See Mie theory

m magnetic field vector 42 Malkmus random band model 179 mass mixing ratio 16 Maxwell, J.C. 41 – distribution 162 – equations 43, 72, 75 methane, CH4 – concentration 10, 14 – electric dipole moment 19 – greenhouse forcing with N2 O 31 – residence time 14

o obliquity, Earth’s orbit 59, 219 ocean mixed layer 9 optical depth 78 – examples 69 – path to space 91 – scaled, �-Eddington approximation 132 orthonormal reslationships – cosines 111 – Legendre polynomials 113 oscillating dipole 19, 154 – polarization of the radiation 43 – power radiated 47 – radiation pattern 42 ozone hole 191 ozone, O3 – 9.6 μm vibration-rotation band 29, 100, 216 – absorption cross sections, UV 189 – absorption cross sections 81, 231 – absorption of UV 27, 186, 190, 212 – absorption of UV-B 191, 198 – absorption of visible light 186, 191 – concentration 14, 18, 190 – electric dipole moment 18, 153 – residence time 14

p P branch 153, 156 paleoclimate

Index

– temperature 4, 5, 34, 35 – temperature profile, Earth 23 – tropopause 25, 211 – window-gray model 7, 19 radiative flux, See also net radiative flux 20, 46 – blackbody. See blackbody – incident solar, Earth 27, 30, 35, 58 – longwave downward, surface 26, 30 – longwave emitted, top of atmosphere 26, 30, 33, 35, 208, 210 – shortwave downward, surface 27, 106, 118, 124, 132 – shortwave reflected, top of atmosphere 26, 27, 30, 106, 117 radiative flux density 46 radiative forcing 9, 210 – aerosol direct 199 – aerosol indirect 200 – chlorofluorocarbons, CFCs 217 – cloud, longwave 29 – cloud, shortwave 31 – CO2 , doubling 11 – volcanic, Mt. Pinatubo 11, 37 radiative heating rate. See heating rate radiative time constant 13 q radiative transfer equation 77 Q branch 156 – adding-doubling method 138 quantum mechanics – �-Eddington – degeneracy 158 approximation 132 – electric dipole transition rules 155 – diffuse radiances 126 – electronic transitions 19, 152 – discrete ordinate method, DISORT 135 – molecular energy levels 152 – Eddington approximation 114, 116, 128 – photon. See photon – formal solution 86 – rotational energy 156 – formal solution, absorption and emission – rotational transitions 156 88 – vibrational energy 152 – formal solution, scattering and absorption – vibrational transitions 153, 158 102 – Monte Carlo simulation 140 r – single scattering approximation 103 R branch 153, 156 – spherical harmonic decomposition 111, radiance 46, 52, 58 135 – diffuse 104, 126 – sunlight and infrared radiation, Earth 80 – direct beam 102, 104, 126 radiative–convective equilibrium 24 – downward 86 Ramanathan, V. 170 – surface downward, emitted 28 Rayleigh scattering 18, 31, 80, 194 – surface downward, transmitted 105 – top of atmosphere, emitted 28, 94, 209, 212 – extinction cross section 69 – optical depth 69, 187 – upward 88 – phase function 74, 82 radiating temperature. See blackbody and – reflectance, �-Eddington approximation emission level 133 radiative equilibrium – reflectance, Eddington approximation 120, – Earth 3, 26 135 – radiative forcing 9 – reflectivity 109 – stratosphere 26, 214 – changes in atmospheric composition 10 – extinction of species 121 parametric models 185 perihelion, Earth’s orbit 60, 219 permeability and permittivity – dielectric media 72 – free space 43 phase function. See scattering phase function photochemical reactions 16, 18, 57, 115 photon 50, 54, 119, 140, 153 planar albedo. See albedo, planar Planck function. See blackbody, radiation, Planck function Planck’s radiation constants 53, 219 polarization 43, 51, 72, 76, 124 precipitable water, See also column amount, water vapor pressure weighted amount 169 – CO2 , path to space 205 – water vapor, path to space 192 pressure-broadened line – equivalent width, strong line limit 165 – half-width 162 – shape factor 161

237

238

Index

Rayleigh scattering (contd.) – upward scattering fraction 106 Rayliegh–Jeans approximation 51 recursion relation, Legendre polynomials 113 reflectance 90, 117, 122 – examples, Eddington and �-Eddington approximations 133 – examples, Eddington approximation 120 reflectivity 90, 105, 136, 138, 143 remote sensing – aerosol optical depth 143 – surface temperature and atmospheric temperature profile 93 residence time – aerosols 18 – trace gases 14 right ascension, sun 59 rigid rotor molecule 153 rotation band 29, 154, 157 rotation constant 155 rotational energy 153 – degeneracy 158 – rigid rotor 155 rotational partition function 158 rotational transition 154 rotational wavenumber 155

s saturated absorption, line center – CO2 15 μm band 205 – strong, pressure-broadened line limit 165 – water vapor, near infrared 193 – water vapor, rotation band 216 scale height – atmosphere 15 – ozone 17 – water vapor 15 scattering coefficient 78 scattering cross section 68 scattering phase function 71 – asymmetry parameter 76 – cumulative distribution 82, 141 – �-Eddington approximation 130 – Eddington approximation 114 – examples 75 – Henyey-Greenstein 76 – Mie theory 75 – upward scattering fraction 106 Schuster and Schwarzshild, two-stream approximation 128 shortwave radiation. See solar radiation similarity principle 132 single scattering albedo 79 size parameter 72

skin cancer 190 solar constant 3, 53, 219 solar radiation – absorption, atmosphere 26, 30, 197 – absorption, ozone, UV 27, 186, 190, 198, 212 – absorption, ozone, visible light 27, 186, 191 – absorption, surface 26, 30, 197 – absorption, water vapor, near infrared 197 – effective radiating temperature 27, 53, 55, 189 – incident, Earth 4, 26, 27, 30, 34, 35, 58 – reflected, Earth 4, 26, 27, 30, 35, 197 – wavelengths 1, 41, 55 Solar Radiation and Climate Experiment, SORCE 4, 32, 189 solar zenith angle 5, 58, 79 solid angle 44, 46, 50, 63, 66 source function 86 – absorption and emission 86 – scattering and absorption 86 spectrally averaged transmission. See average transmission speed of light 41, 43, 72, 219 spherical albedo 118, 120 Stamnes, K. 137 Stephan–Boltzmann law 5, 53, 225 stimulated emission 154 strong, pressure-broadened line limit 164 subsolar point 58 surface – absorption, longwave 7, 26, 30 – absorption, shortwave 26, 30 – albedo 5, 36, 123, 193, 198 – downward shortwave flux, Earth 27 – energy budget, Earth 26, 30 – energy budget, window-gray model 7, 13, 37 surface temperature – global average annual mean, Earth 6 – radiative–convective model, cloud-free 23 – role in climate feedbacks 8 – sensitivity to radiative forcing and atmospheric composition 8 – window-gray model 7 symmetric top molecule 157

t terrestrial radiation. See infrared radiation thermal inertia 9 thermal infrared 54 thermal radiation. See Infrared radiation top of atmosphere 3

Index

transmissivity, See also average transmission 68, 81, 89, 90, 105, 138, 139, 150 transmittance 90, 95, 106, 118, 119, 122, 125 – examples, Eddington and �-Eddington approximations 133 – examples, Eddington approximation 120 tropopause 16, 25, 210 two-stream approximation 128

u U.S. Standard Atmosphere 25 ultraviolet catastrophe 51 ultraviolet radiation, UV 54 – abosrption by ozone 17, 27, 186, 190 – absorption cross sections, ozone 189, 231 – UV-B 191 UV incident solar radiative flux 189

v van de Hulst, J.C. 132 vibrational state 152 vibration-rotation band 156 – CO2 15 μm band 29, 156 – H2 O 6.3 μm band 29 – N2 O 8.6 μm band 153 – O3 9.6 μm band 29 – random model parameters, CO2 175, 205, 227

visible light – absorption by ozone 27, 191 – absorption cross sections, ozone 231 – extinction cross section, example 69 – optical depth, example 26, 69 – Rayleigh scattering, optical depth 187 – top of atmosphere 55, 197, 198 – wavelengths 1 Voigt line shape 162 volcanic haze. See aerosol, volcanic layer volume mixing ratio 18

w wavenumber 28, 41 weak line limit 163 Wien’s displacement law 51, 55, 62 window-gray approximation – single-layer, radiative equilibrium model 6, 36 – two-layer, radiative equilibrium model 19, 38

z zonal mean radiation budget

35

239