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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This textbook is a first-look at radiative transfer in planetary atmospheres with a particular focus on the Earth's atmosphere and climate. It covers the basics of the radiative transfer of sunlight, treating absorption and scattering, and the transfer of the thermal infrared. The examples included show how the solutions of the radiative transfer equation are used to evaluate changes in the Earth?s energy budget due to changes in atmospheric composition, how these changes lead to climate change, and also how remote sensing can be used to probe the thermal structure and composition of planetary atmospheres. The examples motivate students by leading them to a better understanding of and appreciation for the computer-generated numerical results.
Aimed at upper-division undergraduates and beginning graduate students in physics and atmospheric sciences, the book is designed to cover the essence of the material in a 10-week course, while the material in the optional sections will facilitate its use at the more leisurely pace and in-depth focus of a semester course.
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Seitenzahl: 431
Veröffentlichungsjahr: 2014
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Table of Contents
Cover
Series Page
Title Page
Copyright
Preface
Chapter 1: The Earth's Energy Budget and Climate Change
1.1 Introduction
1.2 Radiative Heating of the Atmosphere
1.3 Global Energy Budget
1.4 The Window-Gray Approximation and the Greenhouse Effect
1.5 Climate Sensitivity and Climate Feedbacks
1.6 Radiative Time Constant
1.7 Composition of the Earth's Atmosphere
1.8 Radiation and the Earth's Mean Temperature Profile
1.9 The Spatial Distribution of Radiative Heating and Circulation
1.10 Summary and Outlook
References
Chapter 2: Radiation and Its Sources
2.1 Light as an Electromagnetic Wave
2.2 Radiation from an Oscillating Dipole, Radiance, and Radiative Flux
2.3 Radiometry
2.4 Blackbody Radiation: Light as a Photon
2.5 Incident Sunlight
References
Chapter 3: Transfer of Radiation in the Earth's Atmosphere
3.1 Cross Sections
3.2 Scattering Cross Section and Scattering Phase Function
3.3 Elementary Principles of Light Scattering
3.4 Equation of Radiative Transfer
3.5 Radiative Transfer Equations for Solar and Terrestrial Radiation
References
Chapter 4: Solutions to the Equation of Radiative Transfer
4.1 Introduction
4.2 Formal Solution to the Equation of Radiative Transfer
4.3 Solution for Thermal Emission
4.4 Infrared Fluxes and Heating Rates
4.5 Formal Solution for Scattering and Absorption
4.6 Single Scattering Approximation
4.7 Fourier Decomposition of the Radiative Transfer Equation
4.8 The Legendre Series Representation and the Eddington Approximation
4.9 Adding Layers in the Eddington Approximation
4.10 Adding a Surface with a Nonzero Albedo in the Eddington Approximation
4.11 Clouds in the Thermal Infrared
4.12
Optional
Separation of Direct and Diffuse Radiances
4.13
Optional
Separating the Diffusely Scattered Light from the Direct Beam in the Eddington and Two-Stream Approximations
4.14
Optional
The
δ
-Eddington Approximation
4.15
Optional
The Discrete Ordinate Method and DISORT
4.16
Optional
Adding-Doubling Method
4.17
Optional
Monte Carlo Simulations
References
Chapter 5: Treatment of Molecular Absorption in the Atmosphere
5.1 Spectrally Averaged Transmissions
5.2 Molecular Absorption Spectra
5.3 Positions and Strengths of Absorption Lines within Vibration-Rotation Bands
5.4 Shapes of Absorption Lines
5.5 Doppler Broadening and the Voigt Line Shape
5.6 Average Absorptivity for a Single, Weak Absorption Line
5.7 Average Absorptivity for a Single, Strong, Pressure-Broadened Absorption Line
5.8 Treatment of Inhomogeneous Atmospheric Paths
5.9 Average Transmissivities for Bands of Nonoverlapping Absorption Lines
5.10 Approximate Treatments of Average Transmissivities for Overlapping Lines
5.11 Exponential Sum-Fit and Correlated
k
-Distribution Methods
5.12 Treatment of Overlapping Molecular Absorption Bands
References
Chapter 6: Absorption of Solar Radiation by the Earth's Atmosphere and Surface
6.1 Introduction
6.2 Absorption of UV and Visible Sunlight by Ozone
6.3 Absorption of Sunlight by Water Vapor
References
Chapter 7: Simplified Estimates of Emission
7.1 Introduction
7.2 Emission in the 15 µm Band of CO
2
7.3 Change in Emitted Flux due to Doubling of CO
2
7.4 Changes in Stratospheric Emission and Temperature Caused by a Doubling of CO
2
7.5 Afterthoughts
References
Appendix A: Useful Physical and Geophysical Constants
Appendix B: Solving Differential Equations
B.1 Simple Integration
B.2 Integration Factor
B.3 Second Order Differential Equations
Appendix C: Integrals of the Planck Function
Appendix D: Random Model Summations of Absorption Line Parameters for the Infrared Bands of Carbon Dioxide
References
Appendix E: Ultraviolet and Visible Absorption Cross Sections of Ozone
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Chapter 1: The Earth's Energy Budget and Climate Change
List of Illustrations
Figure 1.1
Figure 1.2
Figure 1.3
Figure 1.4
Figure 1.5
Figure 1.6
Figure 1.7
Figure 1.8
Figure 1.9
Figure 1.10
Figure 1.11
Figure 1.12
Figure 2.1
Figure 2.3
Figure 2.2
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8
Figure 2.9
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6
Figure 3.7
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10
Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12
Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4
Figure 6.5
Figure 6.6
Figure 6.7
Figure 6.8
Figure 6.9
Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4
Figure 7.5
Figure 7.6
Figure 7.7
List of Tables
Table 1.1
Table 1.2
Table 1.3
Table 1.4
Table 2.1
Table 2.2
Table 3.1
Table 3.2
Table 4.1
Table 4.2
Table 4.4
Table 4.3
Table 6.1
Table C.1
Wiley Series in Atmospheric Physics and Remote Sensing
Series Editor: Alexander Kokhanovsky
Wendisch, M. / Brenguier, J.-L. (eds.)
Airborne Measurements for Environmental Research
Methods and Instruments
2013
Coakley Jr., J. A. / Yang, P.
Atmospheric Radiation
A Primer with Illustrative Solutions
2014
Kokhanovsky, A. / Natraj, V.
Analytical Methods in Atmospheric Radiative Transfer
2014
North, G. R. / Kim, K.-Y.
Energy Balance Climate Models
2014
Davis, A. B. / Marshak, A.
Multi-dimensional Radiative Transfer
Theory, Observation, and Computation
2015
Minnis, P. et al.
Satellite Remote Sensing of Clouds
2015
Stamnes, K. / Stamnes, J. J.
Radiative Transfer in Coupled Environmental Systems
An Introduction to Forward and Inverse Modeling
2015
Zhang, Z. et al.
Polarimetric Remote Sensing
Aerosols and Clouds
2015
Huang, X. / Yang, P.
Radiative Transfer Processes in Weather and Climate Models
2016
Tomasi, C. / Fuzzi, S. / Kokhanovsky, A.
Atmospheric Aerosols
Life Cycles and Effects on Air Quality and Climate
2016
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
Spiesz Design, Neu-Ulm
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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 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 3-D 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.
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 3½ years that I spent transforming the notes into a book.
Corvallis, OregonJim Coakley
August 2013
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 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 Ping Yang
August 2013
References
1. Rodgers, C.D. and Walshaw, C.D. (1966) The computation of infra-red cooling rate in planetary atmospheres.
Q. J. R. Meteorolog. Soc.
,
92
, 67–92.
2. 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.
3. Goody, R.M. and Yung, Y.L. (1989)
Atmospheric Radiation Theoretical Basis
, Oxford University Press, New York.
4. Thomas, G.E. and Stamnes, K. (1999)
Radiative Transfer in the Atmosphere and Ocean
, Cambridge University Press, New York.
5. Liou, K.N. (2002)
Introduction to Atmospheric Radiation
, 2nd edn, Academic Press, New York.
6. Bohren, C.F. and Clothiaux, E.E. (2006)
Fundamentals of Atmospheric Radiation
, Wiley-VCH Verlag GmbH, Weinheim.
7. Wendisch, M. and Yang, P. (2012)
Theory of Atmospheric Radiative Transfer—A Comprehensive Introduction
, Wiley-VCH Verlag GmbH, Weinheim.
8. Mishchenko, M.I., Travis, L.D. and Lacis, A.A. (2006)
Multiple Scattering of Light by Particles–Radiative Transfer and Coherent Backscattering
, Cambridge University Press, New York.
1The 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 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
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 with the mass of air undergoing change and 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 with the pressure of the air and the incremental change in its volume. Combining the equation of state for an ideal gas with the conservation of energy gives
with = 1005 J kg−1 K−1 being the heat capacity of air held at constant pressure, , and = 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, ,
with often referred to as the heating rate and often expressed in units of Kelvin per day, being the rate at which energy is being added or removed from the air (J s−1 or W), and 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, . Consequently,
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 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 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, 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.
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, , 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 .
If the Earth were to absorb all sunlight incident on its surface, the rate of absorption would be with the Earth's radius. As shown in Figure 1.1, the area of the circle 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 . 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 = 680 W m−2. The fractional factor “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 = 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 with = 5.67 × 10−8 Wm−2 K−4 the Stefan–Boltzmann constant and 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.
The radiative equilibrium temperature for the Earth is given by
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
The Earth would be close to freezing.
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 . Figure 1.2 illustrates the energy exchanges for the Earth, the atmosphere, and the surface in the window-gray approximation.
Figure 1.2 Window-gray, radiative equilibrium model.
The window-gray, radiative equilibrium model gives rise to two equations with two unknowns, the temperature of the surface , and the temperature of the isothermal atmosphere . At the top of the atmosphere, radiative equilibrium for the Earth leads to
At the surface, radiative equilibrium of the surface leads to
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
Consequently, > . 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
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
For the Earth, = 288 K. With this surface temperature, Equation (1.11) gives the temperature of the atmosphere as = 242 K. Using = 288 K, = 0.3, and = 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, ε 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.
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 of radiation emitted, indicated here by a variable .
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
The term in the brackets in Equation (1.15) is the change in the net radiation budget of the Earth, . 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, , decreases and the Earth warms.
The equilibrium response to a radiative forcing is approximately given by a Taylor series expansion of the brackets in Equation (1.15):
with the albedo for the current equilibrium climate and 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, . The change in albedo due to a change in atmospheric composition, such as that brought about by a volcanic eruption, is given by . 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 . 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 . 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 . 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, 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
with representing the change in albedo caused by the buildup of the volcanic haze layer in the stratosphere and 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, . From Equation (1.17) the change in the equilibrium temperature is given by
Aside from the effects of clouds, the emitted radiation is expected to increase with increasing temperatures, . 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, . Numerical estimates indicate that . 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, = −4 W m−2 in Equation (1.17). Since the increase in carbon dioxide has almost no effect on the absorbed sunlight, . As a result, the change in the equilibrium temperature for an increase in carbon dioxide would be given by
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, . In addition, the emissivity of the atmosphere remains constant so that
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
with 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 = 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].
