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Main Radiation in the Atmosphere - A Course in Theoretical Meteorology
Radiation in the Atmosphere - A Course in Theoretical MeteorologyLuther W. Brady, Luther W. Brady, Luther W. Brady, Hans-Peter Heilmann, Michael Molls, Carsten Nieder
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This page intentionally left blank RADIATION IN THE ATMOSPHERE A Course in Theoretical Meteorology This book presents the theory and applications of radiative transfer in the atmosphere. It is written for graduate students and researchers in the fields of meteorology and related sciences. The book begins with important basic definitions of the radiative transfer theory. It presents the hydrodynamic derivation of the radiative transfer equation and the principles of invariance. The authors examine in detail various quasi-exact solutions of the radiative transfer equation, such as the matrix operator method, the discrete ordinate method, and the Monte Carlo method. A thorough treatment of the radiative perturbation theory is given. The book also presents various two-stream methods for the approximate solution of the radiative transfer equation. The interaction of radiation with matter is discussed as well as the transmission in individual spectral lines and in bands of lines. It formulates the theory of gaseous absorption and analyzes the normal vibrations of linear and non-linear molecules. The book presents the Schrödinger equation and describes the computation of transition probabilities, before examining the mathematical formulation of spectral line intensities. A rigorous treatment of Mie scattering is given, including Rayleigh scattering as a special case, and the important efficiency factors for extinction, scattering and absorption are derived. Polarization effects are introduced with the help of Stokes parameters. The fundamentals of remote sensing applications of radiative transfer are presented. Problems of varying degrees of difficulty are included at the end of each chapter, so readers can further their understanding of the materials covered in the book. Wi l f o r d Z d u n kow s k i is Professor Emeritus at the Institute of Atmospheric Physics in the University of Mainz, Germany. T h o m a s Tr au t m a n n is Head of the Unit for Atmospheric Processors at the Remote Sensing Technology Institute, German Aerospace Center (DLR) Oberpfaffenhofen, Germany, and Professor for Meteorology at the University of Leipzig, Germany. A n d r e a s B o t t is a university professor for Theoretical Meteorology at the Meteorological Institute of the University of Bonn, Germany. Together, the authors have a wealth of experience in teaching atmospheric physics, including courses on atmospheric thermodynamics, atmospheric dynamics, cloud microphysics, atmospheric chemistry, and numerical modeling. Professors Zdunkowski and Bott co-wrote Dynamics of the Atmosphere and Thermodynamics of the Atmosphere (Cambridge University Press 2003 and 2004). RADIATION IN THE ATMOSPHERE A Course in Theoretical Meteorology W I L F O R D Z D U N KOW S K I Formerly University of Mainz, Germany THOMAS TRAUTMANN German Aerospace Center (DLR), Germany ANDREAS BOTT University of Bonn, Germany CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521871075 © W. Zdunkowski, T. Trautmann & A. Bott, 2007 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2007 ISBN-13 ISBN-10 978-0-511-27560-9 eBook (NetLibrary) 0-511-27560-9 eBook (NetLibrary) ISBN-13 ISBN-10 978-0-521-87107-5 hardback 0-521-87107-7 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. This book is dedicated to the memory of Professor Fritz Möller (1906–1983) and Professor J. Vern Hales (1917–1997). We have profited directly and indirectly from their lectures and research. Contents Preface page x 1 Introduction 1 1.1 The atmospheric radiation field 1 1.2 The mean global radiation budget of the Earth 3 1.3 Solar–terrestrial relations 8 1.4 Basic definitions of radiative quantities 16 1.5 The net radiative flux density vector 21 1.6 The interaction of radiation with matter 23 1.7 Problems 26 2 The radiative transfer equation 28 2.1 Eulerian derivation of the radiative transfer equation 28 2.2 The direct–diffuse splitting of the radiance field 34 2.3 The radiatively induced temperature change 36 2.4 The radiative transfer equation for a horizontally homogeneous atmosphere 37 2.5 Splitting of the radiance field into upwelling and downwelling radiation 46 2.6 The solution of the radiative transfer equation for a horizontally homogeneous atmosphere 51 2.7 Radiative flux densities and heating rates 57 2.8 Appendix 61 2.9 Problems 62 3 Principles of invariance 64 3.1 Definitions of the scattering and transmission functions 64 3.2 Diffuse reflection in a semi-infinite atmosphere 66 3.3 Chandrasekhar's four statements of the principles of invariance 70 3.4 The inclusion of surface reflection 75 vii viii 4 5 6 7 8 Contents 3.5 Diffuse reflection and transmission for isotropic scattering 3.6 Problems Quasi-exact solution methods for the radiative transfer equation 4.1 The matrix operator method 4.2 The successive order of scattering method 4.3 The discrete ordinate method 4.4 The spherical harmonics method 4.5 The finite difference method 4.6 The Monte Carlo method 4.7 Appendix 4.8 Problems Radiative perturbation theory 5.1 Adjoint formulation of the radiative transfer equation 5.2 Boundary conditions 5.3 Radiative effects 5.4 Perturbation theory for radiative effects 5.5 Appendix 5.6 Problems Two-stream methods for the solution of the radiative transfer equation 6.1 δ-scaling of the phase function 6.2 The two-stream radiative transfer equation 6.3 Different versions of two-stream methods 6.4 Analytical solution of the two-stream methods for a homogeneous layer 6.5 Approximate treatment of scattering in the infrared spectral region 6.6 Approximations for partial cloud cover 6.7 The classical emissivity approximation 6.8 Radiation charts 6.9 Radiative equilibrium 6.10 Problems Transmission in individual spectral lines and in bands of lines 7.1 The shape of single spectral lines 7.2 Band models 7.3 The fitting of transmission functions 7.4 Transmission in inhomogeneous atmospheres 7.5 Results 7.6 Appendix 7.7 Problems Absorption by gases 8.1 Introduction 78 80 82 82 92 95 104 110 118 128 131 133 133 137 143 147 153 157 158 158 161 166 172 176 179 183 190 195 201 204 205 216 237 255 261 265 271 276 276 Contents 8.2 8.3 8.4 8.5 8.6 9 10 11 12 Molecular vibrations Some basic principles from quantum mechanics Vibrations and rotations of molecules Matrix elements, selection rules and line intensities Influence of thermal distribution of quantum states on line intensities 8.7 Rotational energy levels of polyatomic molecules 8.8 Appendix 8.9 Problems Light scattering theory for spheres 9.1 Introduction 9.2 Maxwell's equations 9.3 Boundary conditions 9.4 The solution of the wave equation 9.5 Mie's scattering problem 9.6 Material characteristics and derived directional quantities 9.7 Selected results from Mie theory 9.8 Solar heating and infrared cooling rates in cloud layers 9.9 Problems Effects of polarization in radiative transfer 10.1 Description of elliptic, linear and circular polarization 10.2 The Stokes parameters 10.3 The scattering matrix 10.4 The vector form of the radiative transfer equation 10.5 Problems Remote sensing applications of radiative transfer 11.1 Introduction 11.2 Remote sensing based on short- and long-wave radiation 11.3 Inversion of the temperature profile 11.4 Radiative perturbation theory and ozone profile retrieval 11.5 Appendix 11.6 Problems Influence of clouds on the climate of the Earth 12.1 Cloud forcing 12.2 Cloud feedback in climate models 12.3 Problems Answers to problems List of frequently used symbols References Index ix 277 288 304 316 320 322 327 332 333 333 334 336 339 345 359 367 374 376 378 378 383 386 395 397 399 399 402 417 431 440 441 443 443 446 451 452 459 466 478 Preface Radiation in the Atmosphere is the third volume in the series A Course in Theoretical Meteorology. The first two volumes entitled Dynamics of the Atmosphere and Thermodynamics of the Atmosphere were first published in the years 2003 and 2004. The present textbook is written for graduate students and researchers in the field of meteorology and related sciences. Radiative transfer theory has reached a high point of development and is still a vastly expanding subject. Kourganoff (1952) in the postscript of his well-known book on radiative transfer speaks of the three olympians named completeness, up-to-date-ness and clarity. We have not made any attempt to be complete, but we have tried to be reasonably up-to-date, if this is possible at all with the many articles on radiative transfer appearing in various monthly journals. Moreover, we have tried very hard to present a coherent and consistent development of radiative transfer theory as it applies to the atmosphere. We have given principle allegiance to the olympian clarity and sincerely hope that we have succeeded. In the selection of topics we have resisted temptation to include various additional themes which traditionally belong to the fields of physical meteorology and physical climatology. Had we included these topics, our book, indeed, would be very bulky, and furthermore, we would not have been able to cover these subjects in the required depth. Neither have we made any attempt to include radiative transfer theory as it pertains to the ocean, a subject well treated by Thomas and Stamnes (1999) in their book Radiative Transfer in the Atmosphere and Ocean. As in the previous books of the series, we were guided by the principle to make the book as self-contained as possible. As far as space would permit, all but the most trivial mathematical steps have been included in the presentation of the theory to encourage students to follow the various developments. Nevertheless, here and there students may find it difficult to follow our reasoning. In this case, we encourage them not to get stuck with a particular detail but to continue with the x Preface xi subject. Additional details given later may clarify any questions. Moreover, on a second reading everything will become much clearer. We will now give a brief description of the various chapters and topics treated in this book. Chapter 1 gives the general introduction to the book. Various important definitions such as the radiance and the net flux density are given to describe the radiation field. The interaction of radiation with matter is briefly discussed by introducing the concepts of absorption and scattering. To get an overall view of the mean global radiation budget of the system Earth–atmosphere, it is shown that the incoming and outgoing energy at the top of the atmosphere are balanced. In Chapter 2 the hydrodynamic derivation of the radiative transfer equation (RTE) is worked out; this is in fact the budget equation for photons. The radiatively induced temperature change is formulated with the help of the first law of thermodynamics. Some basic formulas from spherical harmonics, which are needed to evaluate certain transfer integrals, are presented. Various special cases are discussed. Chapter 3 presents the principle of invariance which, loosely speaking, is a collection of common sense statements about the exact mathematical structure of the radiation field. At first glance the mathematical formalism looks much worse than it really is. A systematic study of the mathematical and physical principles of invariance it quite rewarding. Quasi-exact solutions of the RTE, such as the matrix operator method together with the doubling algorithm are presented in Chapter 4. Various other prominent solutions such as the successive order of scattering and the Monte Carlo methods are discussed in some detail. Chapter 5 presents the radiative perturbation theory. The concept of the adjoint formulation of the RTE is introduced, and it is shown that in the adjoint formulation certain radiative effects can be evaluated with much higher numerical efficiency than with the so-called forward mode methods. For many practical purposes in connection with numerical weather prediction it is sufficient to obtain fast approximate solutions of the RTE. These are known as two-stream methods and are treated in Chapter 6. Partial cloudiness is introduced in the solution scheme on the basis of two differing assumptions. The method allows fairly general situations to be handled. In Chapter 7, the theory of individual spectral lines and band models is treated in some detail. In those cases in which scattering effects can be ignored, formulas are worked out to describe the mean absorption of homogeneous atmospheric layers. A technique is introduced which makes it possible to replace the transmission through an inhomogeneous atmosphere by a nearly equivalent homogeneous layer. The theory of gaseous absorption is formulated in Chapter 8. The analysis of normal vibrations of linear and nonlinear molecules is introduced. The Schrödinger equation is presented and the computation of transition probabilities is described, xii Preface which finally leads to the mathematical formulation of spectral line intensities. Simple but instructive analytic solutions of Schrödinger's equation are obtained leading, for example, to the description of the vibration–rotation spectrum of diatomic molecules. Not only atmospheric gaseous absorbers influence the radiation field but also aerosol particles and cloud droplets. Chapter 9 gives a rigorous treatment of Mie scattering which includes Rayleigh scattering as a special case. The important efficiency factors for extinction, scattering and absorption are derived. The mathematical analysis requires the mathematical skill which the graduate student has acquired in various mathematics and physics courses. The effects of nonspherical particles are not treated in this book. So far polarization has not been included in the RTE, which is usually satisfactory for energy considerations but may not be sufficient for optical applications. To give a complete description of the radiation field the polarization effects are introduced in Chapter 10 with the help of the Stokes parameters. This finally leads to the most general vector formulation of the RTE in terms of the phase matrix while the phase function is sufficient if polarization may be ignored. Chapter 11 introduces remote sensing applications of radiative transfer. After the general description of some basic ideas, the RTE is presented in a form which is suitable to recover the atmospheric temperature profile by special inversion techniques. The chapter closes with a description of the way in which the atmospheric ozone profile can be retrieved using radiative perturbation theory. The book closes with Chapter 12 in which a simple and brief account of the influence of clouds on climate is given. The student will be exposed to concepts such as cloud forcing and cloud radiative feedback. Problems of various degrees of difficulty are included at the end of each chapter. Some of the included problems are almost trivial. They serve the purpose of making students familiar with new concepts and terminologies. Other problems are more demanding. Where necessary answers to problems are given at the end of the book. One of the problems that any author of a physical science textbook is confronted with, is the selection of proper symbols. Inspection of the book shows that many times the same symbol is used to label several quite different physical entities. It would be ideal to represent each physical quantity by a unique symbol which is not used again in some other context. Consider, for example, the letter k. For the Boltzmann constant we could have written kB , for Hooke's constant kH , for the wave number kw , and ks for the climate sensitivity constant. It would have been possible, in addition to using the Greek alphabet, to also employ the letters of another alphabet, e.g. Hebrew, to label physical quantities in order to obtain uniqueness in notation. Since usually confusion is unlikely, we have tried to use standard notation even if the same symbol is used more than once. For example, the Preface xiii climate sensitivity parameter k appears in Chapter 12, Hooke's constant in Chapter 8 and Boltzmann's constant in Chapter 1. The book concludes with a list of frequently used symbols and a list of constants. We would like to give recognition to the excellent textbooks Radiative Transfer by the late S. Chandrasekhar (1960), to Atmospheric Radiation by R. M. Goody (1964) and the updated version of this book by Goody and Yung (1989). These books have been an invaluable guidance to us in research and teaching. We would like to give special recognition to Dr W. G. Panhans for his splendid cooperation in organizing and conducting our exercise classes. Recognition is due to Dr Jochen Landgraf for discussions related to the perturbation theory and to ozone retrieval. Moreover, we will be indebted to Sebastian Otto for carrying out the transfer calculations presented in Section 7.5. We also wish to express our gratitude to many colleagues and graduate students for helpful comments while preparing the text. Last but not least we wish to thank our families for their patience and encouragement during the preparation of this book. It seems to be one of the unfortunate facts of life that no book as technical as this one can be written free of error. However, each author takes comfort in the thought that any errors appearing in this book are due to one of the other two. To remove such errors, we will be grateful for anyone pointing these out to us. 1 Introduction 1.1 The atmospheric radiation field The theory presented in this book applies to the lower 50 km of the Earth's atmosphere, that is to the troposphere and to the stratosphere. In this part of the atmosphere the so-called local thermodynamic equilibrium is observed. In general, the condition of thermodynamic equilibrium is described by the state of matter and radiation inside a constant temperature enclosure. The radiation inside the enclosure is known as black body radiation. The conditions describing thermodynamic equilibrium were first formulated by Kirchhoff (1882). He stated that within the enclosure the radiation field is: (1) isotropic and unpolarized; (2) independent of the nature and shape of the cavity walls; (3) dependent only on the temperature. The existence of local thermodynamic equilibrium in the atmosphere implies that a local temperature can be assigned everywhere. In this case the thermal radiation emitted by each atmospheric layer can be described by Planck's radiation law. This results in a relatively simple treatment of the thermal radiation transport in the lower sections of the atmosphere. Kirchhoff's and Planck's laws, fundamental in radiative transfer theory, will be described in the following chapters. While the derivation of Planck's law requires a detailed microscopic picture, Kirchhoff's law may be obtained by using purely thermodynamic arguments. The derivation of Kirchhoff's law is presented in numerous textbooks such as in Thermodynamics of the Atmosphere by Zdunkowski and Bott (2004).1 1 Whenever we make reference to this book, henceforth we simply refer to THD (2004). 1 2 Introduction The atmosphere, some sort of an open system, is not in thermodynamic equilibrium since the temperature and the radiation field vary in space and in time. Nevertheless, in the troposphere and within the stratosphere the emission of thermal radiation is still governed by Kirchhoff's law at the local temperature. The reason for this is that in these atmospheric regions the density of the air is sufficiently high so that the mean time between molecular collisions is much smaller than the mean lifetime of an excited state of a radiating molecule. Hence, equilibrium conditions exist between vibrational and rotational and the translational energy of the molecule. At levels higher than 50 km, the two time scales become comparable resulting in a sufficiently strong deviation from thermodynamic equilibrium so that Kirchhoff's law cannot be applied anymore. The breakdown of thermodynamic equilibrium in higher regions of the atmosphere also implies that Planck's law no longer adequately describes the thermal emission so that quantum theoretical arguments must be introduced to describe radiative transfer. Quantum theoretical considerations of this type will not be treated in this book. For a study of this situation we refer the reader to the textbook Atmospheric Radiation by Goody and Yung (1989). The units usually employed to measure the wavelength of radiation are the micrometer (µm) with 1µm = 10−6 m or the nanometer (nm) with 1 nm = 10−9 m and occasionally Ångströms (Å) where 1 Å = 10−10 m. The thermal radiation spectrum of the Sun, also called the solar radiation spectrum, stretches from roughly 0.2–3.5 µm where practically all the thermal energy of the solar radiation is located. It consists of ultraviolet radiation (< 0.4 µm), visible radiation (0.4–0.76 µm), and infrared radiation > 0.76 µm. The thermal radiation spectrum of the Earth ranges from about 3.5–100 µm so that for all practical purposes the solar and the terrestrial radiation spectrum are separated. As will be seen later, this feature is of great importance facilitating the calculation of atmospheric radiative transfer. Due to the positions of the spectral regions of the solar and the terrestrial radiation we speak of short-wave and long-wave radiation. The terrestrial radiation spectrum is also called the infrared radiation spectrum. Important applications of atmospheric radiative transfer are climate modeling and weather prediction which require the evaluation of a prognostic temperature equation. One important term in this equation, see e.g. Chapter 3 of THD (2004), is the divergence of the net radiative flux density whose evaluation is fairly involved, even for conditions of local thermodynamic equilibrium. Accurate numerical radiative transfer algorithms exist that can be used to evaluate the radiation part of the temperature prediction equation. In order to judiciously apply any such computer model, some detailed knowledge of radiative transfer is required. There are other areas of application of radiative transfer such as remote sensing. In the concluding chapter of this textbook we will present various examples. 1.2 The mean global radiation budget of the Earth 30 77 342 40 30 3 165 78 24 350 67 168 102 390 324 Fig. 1.1 The Earth's annual global mean energy budget, after Kiehl and Trenberth (1997), see also Houghton et al. (1996). Units are (W m−2 ). 1.2 The mean global radiation budget of the Earth Owing to the advanced satellite observational techniques now at our disposal, we are able to study with some confidence the Earth's annual mean global energy budget. Early meteorologists and climatologists have already understood the importance of this topic, but they did not have the observational basis to verify their results. A summary of pre-satellite investigations is given by Hunt et al. (1986). In the following we wish to briefly summarize the mean global radiation budget of the Earth according to Kiehl and Trenberth (1997). Here we have an instructive example showing in which way radiative transfer models can be applied to interpret observations. The evaluation of the radiation model requires vertical distributions of absorbing gases, clouds, temperature, and pressure. For the major absorbing gases, namely water vapor and ozone, numerous observational data must be handled and supplemented with model atmospheres. In order to calculate the important influence of CO2 on the infrared radiation budget, Kiehl and Trenberth specify a constant volume mixing ratio of about 350 ppmv. Moreover, it is necessary to employ distributions of the less important absorbing gases CH4 , N2 O, and of other trace gases. Using the best data presently available, they have provided the radiation budget as displayed in Figure 1.1. 4 Introduction The analysis employs a solar constant S0 = 1368 W m−2 . This is the solar radiation, integrated over the entire solar spectral region which is received by the Earth per unit surface perpendicular to the solar beam at the mean distance between the Earth and the Sun. Since the circular cross-section of the Earth is exposed to the parallel solar rays, each second our planet receives the energy amount π R 2 S0 where R is the radius of the Earth. On the other hand, the Earth emits infrared radiation from its entire surface 4π R 2 which is four times as large as the cross-section. Thus for energy budget considerations we must distribute the intercepted solar energy over the entire surface so that, on the average, the Earth's surface receives 1/4 of the solar constant. This amounts to a solar input of 342 W m−2 as shown in the figure. The measured solar radiation reflected to space from the Earth's surface–atmosphere system amounts to about 107 W m−2 . The ratio of the reflected to the incoming solar radiation is known as the global albedo which is close to 31%. Early pre-satellite estimates of the global albedo resulted in values ranging from 40–50%. With the help of radiation models and measurements it is found that cloud reflection and scattering by atmospheric molecules and aerosol particles contribute 77 W m−2 while ground reflection contributes 30 W m−2 . In order to have a balanced radiation budget at the top of the atmosphere, the net gain 342 − 107 = 235 W m−2 of the short-wave solar radiation must be balanced by emission of long-wave radiation to space. Indeed, this is verified by satellite measurements of the outgoing long-wave radiation. Let us now briefly consider the radiation budget at the surface of the Earth, which can be determined only with the help of radiation models since sufficiently dense surface measurements are not available. Assuming that the ground emits black body radiation at the temperature of 15 ◦ C, an amount of 390 W m−2 is lost by the ground. According to Figure 1.1 this energy loss is partly compensated by a short-wave gain of 168 W m−2 and by a long-wave gain of 324 W m−2 because of the thermal emission of the atmospheric greenhouse gases (H2 O, CO2 , O3 , CH4 , etc.) and clouds. Thus the total energy gain 168 + 324 = 492 W m−2 exceeds the long-wave loss of 390 W m−2 by 102 W m−2 . In order to have a balanced energy budget at the Earth's surface, other physical processes must be active since a continuous energy gain would result in an ever increasing temperature of the Earth's surface. From observations, Kiehl and Trenberth estimated a mean global precipitation rate of 2.69 mm day−1 enabling them to compute a surface energy loss due to evapotranspiration. Multiplying 2.69 mm day−1 by the density of water and by the latent heat of vaporization amounts to a latent heat flux density of 78 W m−2 . Thus the surface budget is still unbalanced by 24 W m−2 . Assigning a surface energy loss of −24 W m−2 resulting from sensible heat fluxes yields a balanced energy budget at the Earth's surface. The individual 1.2 The mean global radiation budget of the Earth 5 losses due to turbulent surface fluxes are uncertain within several percent since it is very difficult to accurately assess the global amount of precipitation which implies that the estimated sensible heat flux density is also quite uncertain. Only the sum of the turbulent surface flux densities is reasonably certain. Finally, we must study the budget of the atmosphere itself. Figure 1.1 reveals that the atmosphere gains 67 W m−2 by absorption of solar radiation, 102 W m−2 by turbulent surface fluxes, and additionally 350 W m−2 resulting from long-wave radiation emitted by the surface of the Earth and intercepted by atmospheric greenhouse gases and clouds. The total of 519 W m−2 must be re-emitted by the atmosphere. As shown in the figure, the atmospheric greenhouse gases and the clouds emit 165 + 30 = 195 W m−2 to space and 324 W m−2 as back-radiation to the surface of the Earth just balancing the atmospheric energy gain. We also see that from the 390 W m−2 emitted by the Earth's surface only 350 W m−2 are intercepted by the atmosphere. To account for the remaining 40 W m−2 we recognize that these escape more or less unimpeded to space in the so-called spectral window region as will be discussed later. By considering the budget in Figure 1.1, we observe that only the reflected solar radiation and the long-wave radiation emitted to space are actually verified by measurements. However, the remaining budget components should also be taken seriously since nowadays radiation models are quite accurate. Nevertheless, the output of the models cannot be any more accurate than the input data. In future days further refinements and improvements of the global energy budget can be expected. In order to calculate the global radiation budget, we must have some detailed information on the absorption behavior of atmospheric trace gases and the physical properties of aerosol and cloud particles. In a later chapter we will study the radiative characteristics of spherical particles by means of the solution of Maxwell's equations of electromagnetic theory. Here we will only qualitatively present the absorption spectrum of the most important greenhouse gases. Figure 1.2 combines some important information regarding the solar spectrum. The upper curve labeled TOA (top of the atmosphere) shows the extraterrestrial incoming solar radiation after Coulson (1975). For wavelengths exceeding 1.4 µm this curve coincides closely with a Planckian black body curve of 6000 K. The lower curve depicts the total solar radiation reaching the Earth's surface for a solar zenith angle θ0 = 60◦ . The calculations were carried out with sufficiently high spectral resolution using the so-called Moderate Resolution Atmospheric Radiance and Transmittance Model (MODTRAN; version 3.5; Anderson, 1996; Kneizys et al., 1996) program package. All relevant absorbing trace gases shown in the figure are included in the calculations. Not shown are the positions of the CO and CH4 absorption bands which are located in the solar spectrum and in the near infared spectral region of 6 Introduction TOA 2000 Irradiance (W m−2 µm−1) surface 1500 1000 500 0 0 1 2 Wavelength (µm) 3 4 Fig. 1.2 Incoming solar flux density at the top of the atmosphere (TOA) and at ground level. The solar zenith angle is θ0 = 60◦ , ground albedo Ag = 0. The spectral positions of major absorption bands of the trace gases are shown. thermal radiation. A tabulation of bands of these two trace gases is given, for example, in Goody (1964a). Since the radiation curve for ground level shows a high spectral variability, it was artificially smoothed for better display to a somewhat lower spectral resolution. Figure 1.3 depicts the spectral distribution of the upwelling thermal radiance as a function of the wave number (to be defined later) at a height of 60 km. For comparison purposes the Planck black body radiance curves for several temperatures are shown also. The maximum of the 300 K black body curve is located at roughly 600 cm−1 . The calculations were carried out with the same program package (MODTRAN) using a spectral resolution of 1 cm−1 . All relevant absorbing and emitting gases have been accounted for. The widths of the major infrared absorption bands (H2 O, CO2 , O3 ) are also shown in the figure. Kiehl and Trenberth (1997) produced similar curves for the solar and infrared radiative fluxes per unit surface. However, in addition to the absorption by gases shown in Figures 1.2 and 1.3, they also included the effects of clouds in their calculations by assuming an effective droplet radius of 10 µm and suitable liquid water contents. Moreover, assumptions were made about the spatial distributions of clouds. Their results indicate that water vapor is the most important 1.2 The mean global radiation budget of the Earth 7 Fig. 1.3 Upwelling infrared radiance at a height of 60 km for a clear sky midlatitude summer atmosphere. gas absorbing 38 W m−2 of solar radiation which is followed by O3 (15 W m−2 ) and O2 (2 W m−2 ) while the effect of CO2 may be ignored. Thus the greenhouse gases absorb 55 W m−2 . Figure 1.1, however, requires 67 W m−2 . The 12 W m−2 still missing must be attributed to partial cloudiness and to spectral overlap effects, i.e., cloud droplets and gases absorb at the same wavelength. Handling clouds in the radiative transfer problem is usually very difficult since in general water droplet size distributions are unknown. Finally, let us consider the gaseous absorption bands of the infrared spectrum. In the calculations of Kiehl and Trenberth (1997) analogous to Figure 1.3, the surface is assumed to emit black body radiation with a temperature of 15◦ C. The major absorbing gases are H2 O, O3 , and CO2 . Of course, the same distribution of absorbing gases and clouds as for solar radiation is assumed. Integration of the infrared curve at the top of the atmosphere over the entire spectral region yields 235 W m−2 as required by Figure 1.1. We conclude this section by considering a simple example to obtain the effective emission temperature of the system Earth's surface–atmosphere. As we have discussed above, the cross-section of the Earth intercepts the solar energy π R 2 S0 . Since the global albedo is 31%, the rate of absorption is 1368(1− 0.31) = 944 W m−2 . Assuming that the Earth emits black body radiation, we must apply the 8 Introduction April 4th a Ear th b Sun July 4th F1 January 3rd a F2 October 5th Fig. 1.4 Simplified elliptical geometry of the Earth's orbit. well-known Stefan–Boltzmann law so that the Earth's surface emits 4π R 2 σ T 4 where σ is the Stefan–Boltzmann constant. Assuming steady-state conditions, we have π R 2 × 944 W m−2 = 4π R 2 σ T 4 from which we obtain the temperature T = 254 K which resembles the effective emission temperature of our planet. 1.3 Solar–terrestrial relations To a high degree of accuracy the√Earth's orbit around the Sun can be described by an ellipse with eccentricity e = a 2 − b2 /a = 0.01673, where a and b are, respectively, the semi-major and semi-minor axis of the ellipse, see Figure 1.4. The Sun's position is located in one of the two elliptical foci (F1 , F2 ). For demonstration purposes, the figure exaggerates the eccentricity of the elliptical orbit. The perihelion, that is the shortest distance rmin between Sun and Earth, occurs around January 3rd, while the aphelion, that is the largest distance rmax between Sun and Earth, is registered around July 4th. These times are not constant, but they vary from year to year. Often the mean distance between the Earth and the Sun is approximated by a= rmin + rmax = 1.496 × 108 km 2 (1.1) The distances rmin and rmax are related to a and e via rmin =a(1 − e) = 1.471 × 108 km rmax =a(1 + e) = 1.521 × 108 km (1.2) 1.3 Solar–terrestrial relations 9 Beginning with January 1st, i.e. Julian day number 1 of the year, a normal year counts 365 days (for simplicity we will not take the occurrence of leap years into account). A particular day of the year is then labelled with its corresponding Julian day number J . We introduce the rotation angle of the Earth beginning with the 1st of January as = 2π (J − 1) 365 (1.3) where is expressed in radians. During the course of the year the angular distance Sun–Earth, the solar declination δ, and the so-called equation of time E T change in a more or less harmonic manner. In the following we will discuss simple expressions developed by Spencer (1971) which are accurate enough to evaluate the quantities (a/r )2 , δ, and E T , where r is the actual distance between Sun and Earth. The term (a/r )2 is given by a 2 = 1.000110 + 0.034221 cos + 0.001280 sin r + 0.000719 cos 2 + 0.000077 sin 2 (1.4) with a maximum error of approximately 10−4 . If S0 = 1368 W m−2 is the solar constant for the mean distance between Sun and Earth, the actual solar constant varies as a function of J a 2 (1.5) S0 (J ) = S0 r (J ) According to (1.4) the maximum change of S0 (J ) relative to S0 has an amplitude of approximately 3.3%. The solar declination δ is defined as the angle between the Earth's equatorial plane and the actual position of the Sun as seen from the center of the Earth. The Earth's rotational axis and the normal to the Earth's plane of the ecliptic make on average an angle of ε = 23◦ 27 , δ amounts to +23◦ 27 and −23◦ 27 at summer solstice (around June 21st) and winter solstice (around December 22nd), respectively. These relations are illustrated in Figure 1.5 and in the three-dimensional view of the Sun–Earth geometry of Figure 1.6. The equinox points are defined as the intersecting line (equinox line) between the Earth's plane of the ecliptic and the Sun's equatorial plane. A second line which is normal to the equinox line and which is located in the Earth's plane of the ecliptic intersects the Earth's orbit in the points W S (winter solstice) and SS (summer solstice). The perihelion P and the aphelion A, which both lie on the semi-major axis of the Earth's elliptical orbit, make an angle ψ = 11o 08 with the solstice line. 10 Introduction n N ε orbit Earth's Equ ato r Earth's ecliptic plane Fig. 1.5 Relation between the Earth's orbit, the normal vector n to the plane of the ecliptic, the Earth's rotational vector N and the angle of the ecliptic ε. A SS ψ s n' u S Eart ne la p l ria o t ua eq h's e clipt ic pl ane VE n N ε Earth AE Sun P WS Fig. 1.6 Schematical view of the Sun–Earth geometry. P, perihelion; V E, vernal equinox; SS, summer solstice; A, aphelion; AE, autumnal equinox; W S, winter solstice; ε, angle of the ecliptic; ψ, angle between the distances (SS, W S) and (A, P); N, vector along the rotational axis of the Earth; n, normal unit vector with respect to the Earth's plane of the ecliptic. It should be observed that the vector N is fixed in direction pointing to the polar star. At the solstices the vectors N, n and the line between the solstice points lie in the same plane so that δ = ± 23o 27 . At the equinox points (δ = 0o ) the line between the Earth and the Sun is at a right angle to the line (SS, W S). The solar declination δ is a function of the Julian day number J . It can be expressed as δ = 0.006918 − 0.399912 cos + 0.070257 sin − 0.006758 cos 2 + 0.000907 sin 2 (1.6) with δ expressed in radians. Due to Spencer (1971) this approximate formula has an error in δ less than 12 . Figure 1.7 depicts a plot of δ versus J . 1.3 Solar–terrestrial relations 11 Solar declination ( ) 20 ˚ SS 10 0 VE −10 AE WS −20 0 50 100 150 200 250 300 350 Day of the year Fig. 1.7 Variation of the solar declination δ as a function of the Julian day J , see (1.6). V E, vernal equinox; SS, summer solstice; AE, autumnal equinox; W S, winter solstice. 1.3.1 The equation of time In the following we assume that the period of the rotation of the Earth around the North Pole is constant. The time interval between two successive passages of a fixed star as seen from the local meridian of an observer on the Earth's surface is called a sidereal day. Due to the fact that the Earth moves around the Sun in an elliptical orbit, the time interval between two successive passages of the Sun in the local meridian, i.e. the so-called solar day, is about 4 min longer than the length of the sidereal day. For a practical definition of time, one introduces the so-called mean solar day which is exactly divided into 24-h periods. Thus the local noon with respect to the local mean time (L M T ) is defined by the passage of a mean fictitious Sun as registered from the Earth observer's local meridian. Clearly, depending on the Julian day J the real Sun appears somewhat earlier or later in the local meridian than the fictitious Sun. The time difference between the noon of the true solar time (T ST ) and the noon of the local mean time (L M T ) is the so-called equation of time E T E T = T ST − L M T (1.7) Following the analysis of Spencer (1971), a functional fit expression can be derived for E T in the form ET = 1440 (0.000075 + 0.001868 cos − 0.032077 sin 2π − 0.014615 cos 2 − 0.040849 sin 2) (1.8) 12 Introduction 20 Equation of time 15 10 5 0 −5 −10 −15 −20 0 50 100 150 200 250 300 350 Day of the year Fig. 1.8 Variation of the equation of time E T (in minutes) during the course of the year as given by (1.8). where E T is expressed in minutes and 1440 is the number of minutes per day. The accuracy of this approximation is better than 35 s. The maximum time difference between T ST and L M T amounts to less than about ±15 min. Figure 1.8 depicts the variation of E T during the course of the year. Note that the irregularities of the Earth's orbit around the Sun lead to a complicated shape of the functional form of E T versus J . Universal time U T , or Greenwich mean time G M T , is defined as the L M T at Greenwich's (UK) meridian at 0◦ in longitude. Since 24 h cover an entire rotation of the Earth, L M T increases by exactly 1 h per 15◦ in eastern longitude, i.e. 4 min per degree of eastern longitude. Similarly, L M T decreases by 4 min if one moves by one degree of longitude in the western direction. For the true solar time we thus obtain the relation T ST = U T + 4λ + E T (1.9) where T ST , U T , and E T are given in minutes and the longitude λ is in units of degree (−180◦ < λ ≤ 180◦ ). The hour angle of the Sun H is defined as the angle between the local observer's meridian and the solar meridian, see Figure 1.9. If H is expressed in degrees longitude one obtains H = 15(12 − T ST ) (1.10) where T ST has to be inserted in hours. Note that H > 0 in the morning and H < 0 in the afternoon. 1.3 Solar–terrestrial relations 13 N Local vertical Observer δ H Solar meridian Local meridian Fig. 1.9 Relation between hour angle H , solar declination δ, the solar and the local meridian. The local standard time L ST is defined as the local mean time for a given meridian being a multiple of 15◦ away from the Greenwich meridian (0◦ ). Therefore, L ST and U T differ by an integral number of hours. For particular countries, differences of 30 and 45 min relative to the standard time meridians have been introduced for convenience. Note also that for locations with daylight saving time, the local mean time differs by 1 to 2 h relative to L ST . 1.3.2 Geographical coordinates and the solar position A particular point P on the Earth's surface is identified by the pair of geographical coordinates (λ, φ), where λ is the longitude and φ is the latitude. Note that φ is counted positive in the northern hemisphere and negative in the southern hemisphere. The coordinates of the Sun relative to P are defined by the solar zenith angle ϑ0 and the solar azimuth angle ϕ0 . If the Sun is at the zenith we have ϑ0 = 180◦ , and ϑ0 = 90◦ if it is at the horizon, see Figure 1.10. The solar height h is given by h = ϑ0 − π/2. The solar azimuth ϕ0 is defined as the angle between the solar vertical plane and a vertical plane of reference which is aligned with the north–south direction. Here, ϕ0 = 0◦ if the Sun is exactly over the southern direction and ϕ0 is counted positive in the eastward direction. Figure 1.11 depicts the apparent track of the Sun during the day. The position angles (ϑ0 , ϕ0 ) of the Sun are usually not measured directly and must be determined from other known angles. Utilizing the laws of spherical trigonometry it can be shown that the following relations are valid (a) (b) cos(π − ϑ0 ) = sin ϕ sin δ + cos ϕ cos δ cos H cos(π − ϑ0 ) sin ϕ − sin δ cos ϕ0 = sin(π − ϑ0 ) cos ϕ (1.11) 14 Introduction Zenith Sun vertical W N ϑ0 P E ϕ0 S Fig. 1.10 Coordinates defining the position of the Sun. Zenith W ϑ0 S ϕ0 Sunset N P Sunrise E Fig. 1.11 Apparent solar track during the course of the day. The dotted curve marks the projection of the solar path onto the horizontal plane. At solar noon at any latitude we have H = 0. In this case we obtain from (1.11a) (π − ϑ0 ) = ϕ − δ. At sunrise or sunset at any latitude ϑ0 = 90◦ and H = Dh . The term Dh is also called the half-day length since it is half the time interval between sunrise and sunset. Excepting the poles we find from (1.11a) cos Dh = − tan ϕ tan δ (1.12) At the equator on all days and at the equinoxes (δ = 0) at all latitudes (with ϕ = ±90◦ ) we find Dh = 90◦ or 6 h. The latitude of the polar night is found by setting in (1.12) Dh = 0 so that tan ϕ = − cot δ (with δ = 0) and ϕ(polar night) = 90◦ − |δ| in the winter hemisphere. The daily total solar radiation Q s incident on a horizontal surface at the top of the atmosphere is found by integrating the incoming solar radiation over the length 1.3 Solar–terrestrial relations n Circle of precession Sun F1 15 N Nutation C1 23° 27 ′ Ecliptic plane F2 C2 Equ ato rial pla ne Equatorial bulge Fig. 1.12 Precession and nutation of the Earth. of the day. Thus from (1.5) we find a 2 Dh Q s = S0 cos(π − ϑ0 )dt r (J ) −Dh (1.13) Since the angular velocity of the Earth can be written as = d H/dt = 2π day−1 we obtain from (1.11a) after some simple integration a 2 86400 (Dh sin ϕ sin δ + cos ϕ cos δ sin Dh ) J m−2 day−1 Q s = S0 r (J ) π (1.14) In the first term Dh must be expressed in radians. The expression (a/r (J ))2 never departs by more than about 3% from unity. Graphical representations of this formula are given in various texts, for example in Sellers (1965) where additional details may be found. 1.3.3 Long-term variations of the Earth's orbital parameters For completeness we briefly discuss the most important variations of the Earth's orbit around the Sun. The eccentricity e of the Earth's elliptical orbit varies irregularly between 0 and 0.05 with its current value e = 0.01673. The period of this oscillation is approximately 100 000 years. The Earth's rotational axis N precesses around the normal of the ecliptic plane n with an angle of 23◦ 27 . The reason for the precession of the Earth is that it is not an ideal sphere, but it has the shape of a geoid, that is, the poles are flattened and an equatorial bulge of about 21 km is observed, see Figure 1.12. First we investigate the influence of the Sun on the geoidal form of the Earth. At the center of the Earth the gravitational attraction of the Sun and the centrifugal force due to the revolving motion of the Earth around 16 Introduction the Sun are equal but opposite in sign. At the center of gravity C1 (left half of the geoid) the attractional force of the Sun is larger than the centrifugal force, which is due to the smaller distance of C1 to the Sun. At the center of gravity C2 (right half of the geoid) we observe the opposite situation, which is due to the larger distance of C2 to the Sun as compared to the center of the Earth; here the centrifugal force preponderates the attractional force of the Sun. Hence, at C1 the resultant force F1 is directed toward the Sun whereas at C2 the resultant force F2 is directed away from the Sun. Owing to the inclination of the ecliptic plane the forces F1 and F2 form a couple attempting to place the Earth's axis in the upright position. This results in the precession of the Earth's axis. The Moon, whose orbital plane nearly coincides with the orbital plane of the Earth, acts in the same way but even more effectively. Here, the small mass of the Moon in comparison with the mass of the Sun is overcompensated by the small distance between Moon and Earth. As a result of these forces, N revolves on the mantle of a cone as shown in Figure 1.12. The time for a full rotation around the circle of precession amounts to about 25 780 years. Apart from the Sun and the Moon the other planets of the solar system also exert an influence on the inclination of the ecliptic leading to changes in ε between 21◦ 55 and 24◦ 18 having a period of about 40 000 years. Finally, in addition to the precession, the Earth's rotational axis exhibits also a nodding motion. This effect is caused by the fact that the Moon's gravitational influence varies in time. This nutation leads to a small variation of the Earth's axis inclination and has a period of about 18.6 years. 1.4 Basic definitions of radiative quantities In this section we will present some basic definitions and the terminologies used in this book. The photon is considered to be an idealized infinitesimally small particle with zero rest mass carrying the energy e(ν) = hν (1.15) where h = 6.626196 × 10−34 J s is Planck's constant, and ν is the frequency of the electromagnetic radiation. Frequency units are Hertz (Hz) where 1 Hz is 1 cycle s−1 . Considering a single photon one may attribute to it a momentum p(ν) with magnitude p(ν) = |p(ν)| = e(ν) c (1.16) where c = 2.997925 × 108 m s−1 is the vacuum speed of light. Photons may travel in an arbitrary direction specified by the unit vector Ω. Therefore, the vectorial 1.4 Basic definitions of radiative quantities 17 z˜ z dA rdϑ ˜ ϑ x r ˜ y r sinϑ dϕ ˜ Ω r ϕ y˜ ˜ x Fig. 1.13 Definition of the local spherical (r̃ , ϑ, ϕ)-coordinate system and the direction Ω. notation of the photon's momentum can be expressed as e(ν) hν Ω= Ω (1.17) c c As soon as the photon interferes with matter, various types of interactions between the atoms of the material and the photon may occur. A single interaction may be an absorption or a scattering process. Between any two scattering interactions the photon is assumed to travel in a straight line with the speed of light c. We will also assume that during a scattering process the photon suffers no change in frequency. In this case one speaks of elastic scattering. In some situations inelastic scattering might be of importance where in addition to the change of flight direction a shift in the photon's frequency occurs. One important example for atmospheric applications is Raman scattering. Rayleigh scattering and Mie scattering to be discussed later are examples of elastic scattering processes. Inelastic scattering processes will not be investigated in this book. Six coordinates are required to unambiguously describe the photon at time t. These are the three coordinates of the position vector r, the magnitude of momentum p(ν) and two angles characterizing the direction of flight Ω. At a certain point in space a local system of Cartesian coordinates (x̃, ỹ, z̃) is introduced. At the origin of this system we define a spherical coordinate system r̃ , ϑ, ϕ, where r̃ is the radial distance from the origin located at r and (ϑ, ϕ) are the zenith and azimuthal angle, respectively, see Figure 1.13. In the latter system the direction Ω may be described p(ν) = 18 Introduction Table 1.1 Definition of special radiance fields Radiance field List of variables Stationary Isotropic Homogeneous Homogeneous and isotropic Iν Iν Iν Iν = = = = Iν (r, Ω) Iν (r, t) Iν (Ω, t) Iν (t) by the triple set of coordinates (r̃ = 1, ϑ, ϕ). The differential solid angle element d is defined by d = dA r̃ 2 (1.18) Here, d A = r̃ 2 sin ϑ dϑ dϕ is the differential area element on a sphere with radius r̃ , see Figure 1.13. Thus we obtain d = sin ϑ dϑ dϕ Integration over the unit sphere yields 2π π d = sin ϑ dϑ dϕ = 4π 0 0 2π 0 (1.19) 1 −1 dµ dϕ = 4π (1.20) where the abbreviation µ = cos ϑ has been introduced. The distribution function of photons f (ν, r, Ω, t) = f ν (r, Ω, t) is defined by2 Nν (r, Ω, t)dν = f ν (r, Ω, t) d V d dν (1.21) where Nν dν represents the number of photons at time t contained within the volume element d V centered at r, within the solid angle element d about the flight direction Ω, and within the frequency interval (ν, ν + dν). Therefore, f ν has units of (m−3 sr−1 Hz−1 ). In place of the photon distribution function f ν , in radiative transfer theory it is customary to use the radiance Iν (r, Ω, t) as defined by Iν (r, Ω, t) = chν f ν (r, Ω, t) (1.22) From this equation it is seen that the monochromatic radiance is expressed in units of (W m−2 sr−1 Hz−1 ). In the most general case the radiance field is time dependent, it varies in space, direction, and frequency. Table 1.1 briefly lists some special cases of Iν . 2 The dependence of radiative quantities on frequency is commonly denoted by the subscript ν. 1.4 Basic definitions of radiative quantities 19 Iν(r, Ω, t) z dΩ Ω θ n dσ r y x Fig. 1.14 Radiative energy streaming through the infinitesimal surface element dσ with surface normal n into the solid angle element d around the flight direction Ω of the photons. The physical meaning of the radiance can be illustrated with the help of the energy relation u ν (r, Ω, t)dν = Iν (r, Ω, t) cos θ d dσ dt dν (1.23) Thus u ν dν is the radiative energy contained within the frequency interval (ν, ν + dν) streaming during dt at r through the surface element dσ with unit surface normal n into the solid angle element d along Ω. The angle between Ω and the surface normal of dσ is denoted by θ , see Figure 1.14. Therefore, Iν dν is expressed in (W m−2 sr−1 ). The energy density û(r, t) of the radiation field, expressed in units of (J m−3 ), is obtained by integrating the term hν f ν over all directions and frequencies ∞ û(r, t) = 0 hν f ν (r, Ω, t) d 4π 1 dν = c ∞ 0 Iν (r, Ω, t) d dν (1.24) 4π Let us now consider the important case that the radiance is described by the Planck function Bν (W m−2 sr−1 Hz−1 ), which is also known as the spectral black body radiance. This special radiation field which is stationary, isotropic and homogeneous coexists with matter in perfect thermodynamic equilibrium at temperature T . The expression Bν dν = 2hν 3 hν/kT (e − 1)−1 dν c2 (1.25) 20 Introduction 25 4 Radiance (W m−2 sr −1 µm−1) 20 4: 3: 2: 1: 15 350 K 300 K 250 K 200 K 3 10 5 2 1 0 0 10 20 Wavelength (µm) 30 40 Fig. 1.15 Planckian black body curves for various temperatures. represents the energy (unpolarized radiation) emitted by a black unit surface area per unit time interval within a cone of solid angle 0 = 1 sr vertical to the emitting surface in the frequency range between ν and ν + dν. Figure 1.15 depicts four Planck curves as function of the wavelength for the temperatures 200, 250, 300 and 350 K. It is clearly seen that with decreasing temperature the maxima of the curves are shifted towards larger wavelengths. This phenomenon is also known as Wien's displacement law. The Planck curve of a black body with temperature 6000 K (the Sun) has its maximum around 0.5 µm while for a black body with T = 300 K (the Earth) the maximum is found at 10 µm. For a further discussion of Wien's displacement law see also Problem 1.1. The constant k = 1.380662 × 10−23 J K−1 appearing in (1.25) is known as the Boltzmann constant. The corresponding energy density follows from û = 1 c ∞ Bν d 0 4π dν = 4π c ∞ 0 2hν 3 hν/kT (e − 1)−1 dν c2 (1.26) The integral over frequency can be evaluated by substituting the new variable x = hν/kT and developing the exponential term (e x − 1)−1 into a power series, 1.5 The net radiative flux density vector 21 Ω θ iEnet,x dσ i x Fig. 1.16 Radiative energy streaming through the infinitesimal surface element dσ in x-direction. yielding 8π k 4 T 4 û = (hc)3 ∞ x 3 (e x − 1)−1 d x = 0 ∞ 48π (kT )4 1 3 (hc) n4 n=1 (1.27) Since ∞ π4 1 = n4 90 n=1 (1.28) the final result is 4 û = σ T 4 , c σ = 2π 5 k 4 = 5.67032 × 10−8 W m−2 K−4 15h 3 c2 (1.29) where σ is the Stefan–Boltzmann constant. Equation (1.29) can also be derived from purely thermodynamic arguments as shown, for example, in THD (2004). 1.5 The net radiative flux density vector Consider the special case that Ω is located in the (x, z)-plane and that the normal unit vector n of the surface element dσ points in the x-direction of a Cartesian coordinate system, that is n = i. According to (1.23) for the spectral differential radiative energy crossing dσ during dt we find the expression u ν (r, Ω, t)dν = Iν (r, Ω, t) cos θ d dσ dt = Iν (r, Ω, t) x d dν E net,x,ν (r, Ω, t)dν = dν (1.30) where x = Ω · i = cos θ = sin ϑ is the projection of Ω onto the x-axis, see Figures 1.16 and 1.17. Integrating this relation over the solid angle and over all 22 Introduction z Ω cosϑ ϑ y sin ϑ co sϕ ϕ x sin ϑ sinϕ Fig. 1.17 Cartesian components of the unit vector Ω pointing in the direction of photon travel. frequencies yields the radiative energy streaming within unit time through the surface element in the x-direction ∞ ∞ E net,x (r, t) = x Iν (r, Ω, t)d dν = x chν f ν (r, Ω, t)d dν 0 0 4π 4π (1.31) In the general case x will be a more complicated expression. If (E net,x , E net,y , E net,z ) are the three components of the net radiative flux density vector, then Enet is given by Enet (r, t) = ∞ 0 ΩIν (r, Ω, t)d dν = iE net,x (r, t) + jE net,y (r, t) + kE net,z (r, t) 4π (1.32) where E net,y (r, t) = 0 E net,z (r, t) = 0 ∞ ∞ y Iν (r, Ω, t)d dν 4π (1.33) z Iν (r, Ω, t)d dν 4π We will now derive an explicit form of Enet in Cartesian coordinates. From (1.32) follows the definition of the spectral net radiative flux density vector ΩIν (r, Ω, t)d (1.34) Enet,ν (r, t) = 4π 1.6 The interaction of radiation with matter 23 Thus the component of Enet,ν (r, t) in the arbitrary direction n is given by E net,n,ν (r, t) = Enet,ν (r, t) · n = Ω · nIν (r, Ω, t)d = cos(Ω, n)Iν (r, Ω, t)d 4π 4π (1.35) To find the Cartesian components ( x , y , z ), of the unit vector Ω = (1, ϑ, ϕ) we perform the scalar multiplication with the Cartesian unit vectors i, j and k. From Figure 1.17 we find immediately Ω·i= x = cos(Ω, i) = sin ϑ cos ϕ Ω·j= y = cos(Ω, j) = sin ϑ sin ϕ Ω·k= z = cos(Ω, k) = cos ϑ = µ (1.36) Thus, from (1.36) the Cartesian components of Enet,ν (r, t) are finally given as (a) E net,x,ν (r, t) = 2π 2π 0 (b) E net,y,ν (r, t) = 1 −1 1 0 −1 2π 1 0 −1 (c) E net,z,ν (r, t) = Iν (r, µ, ϕ, t) cos ϕ(1 − µ2 )1/2 dµ dϕ Iν (r, µ, ϕ, t) sin ϕ(1 − µ2 )1/2 dµ dϕ (1.37) Iν (r, µ, ϕ, t)µ dµ dϕ with (1 − µ2 )1/2 = sin ϑ and d = −dµdϕ. It is straightforward to show that for an isotropic radiation field Enet,ν = 0. For example, evaluating in (1.37a) for Iν = const the integral of the x-component yields 1 2π 2 1/2 cos ϕ (1 − µ ) dµ dϕ = =0 (1.38) xd −1 0 4π Similarly we obtain for the integrals of the y- and z-component = 0, =0 yd zd 4π (1.39) 4π 1.6 The interaction of radiation with matter 1.6.1 Absorption If a photon travels through space filled with matter, a certain absorption probability can be defined. For the mathematical description of this process the absorption coefficient kabs,ν (r, t) with units (m−1 ) is introduced. The dimensionless differential dτabs (r, t) = kabs,ν (r, t) ds (1.40) 24 Introduction is the so-called differential optical depth for absorption where ds is the geometrical distance travelled by the photon. Thus the differential dτabs (r, t) is a measure for the probability that the photon is absorbed along ds so that the photon disappears. It is important to realize that (1.40) is valid only for isotropic media. In general, for anisotropic media, the absorption coefficient not only depends on position, frequency and time but also on the direction Ω. For all practical purposes, the atmosphere can be considered an isotropic medium. Sometimes it is preferable to use the mass absorption coefficient κabs,ν (r, t) which is defined by the relation kabs,ν (r, t) = ρabs (r, t)κabs,ν (r, t) (1.41) where ρabs (r, t) is the density of the absorbing medium, and κabs (r, t) has units of (m2 kg−1 ). 1.6.2 Scattering In a similar manner the photon may suffer an elastic scattering process after having travelled a certain distance ds. The occurrence of a scattering process does not mean that the photon disappears at the location of scattering, instead of that it changes its flight direction from Ω to Ω. Let us denote the differential scattering coefficient by ksca,ν (r, Ω → Ω, t). In analogy to (1.40), the differential optical depth for scattering is defined as dτsca (r, Ω → Ω, t) = ksca,ν (r, Ω → Ω, t) d ds (1.42) This expression is a measure of the probability that a photon of frequency ν with initial direction Ω , in traveling the distance ds, is scattered into d having the new direction Ω. From (1.42) it is clear that ksca,ν (r, Ω → Ω, t) is expressed in units of (m−1 sr−1 ). It is noteworthy that the differential scattering coefficient agrees with the scattering function P̃(cos ) which will be introduced in a later chapter. The notation Ω → Ω has been chosen since in general the differential scattering probability depends explicitly on both directions Ω and Ω. However, for applications involving homogeneous spherical particles (e.g. cloud droplets) it is obvious that the scattering process depends only on the cosine of the scattering angle cos = Ω · Ω (1.43) This means that scattering is rotationally symmetric about the direction of incidence, see Figure 1.18. In case of randomly oriented inhomogeneous or non-spherical 1.6 The interaction of radiation with matter Backward scattering 25 Forward scattering Ω Θ Incident radiation Ω´ Fig. 1.18 Illustration of the rotationally symmetric scattering phase function. P(r, Ω → Ω) = const on the circle defined by all points with = const. particles (e.g. ice particles or aerosol particles) it is often assumed that the scattering angle may be defined analogously. If we integrate the differential scattering coefficient over all possible directions Ω, we obtain the ordinary scattering coefficient ksca,ν (r, t) ksca,ν (r, t) = ksca,ν (r, Ω → Ω, t)d (1.44) 4π This relation together with (1.42) can be used to define the scattering phase function or simply phase function Pν (r, Ω → Ω, t) ksca,ν (r, Ω → Ω, t) = 1 ksca,ν (r, t)Pν (r, Ω → Ω, t) 4π (1.45) The scattering phase function is a measure of the probability density distribution for a scattering process from the incident direction Ω into the direction Ω. The normalization of P is guaranteed since integrating (1.45) over the unit sphere, utilizing (1.44), yields 1 Pν (r, Ω → Ω, t)d = 1 (1.46) 4π 4π It is instructive to discuss a particularly simple form of scattering, namely an isotropic scattering process. In this case the phase function is simply given by Pν (r, Ω → Ω, t) = 1 (1.47) i.e. for each direction there is equal probability of scattering. The sum of absorption and scattering is called extinction. The extinction coefficient is defined by kext,ν (r, t) = kabs,ν (r, t) + ksca,ν (r, t) (1.48) 26 Introduction Another important optical parameter is the single scattering albedo ω0,ν (r, t) which is defined as the relative amount of scattering involved in the extinction process ω0,ν (r, t) = ksca,ν (r, t) kabs,ν (r, t) =1− kext,ν (r, t) kext,ν (r, t) (1.49) Of particular importance is the case ω0,ν (r, t) = 1 for which the scattering process is conservative. In a medium with conservative scattering no absorption of radiation occurs. Later it will be shown that a conservative plane-parallel medium is characterized by a vertically constant net radiative flux density. 1.6.3 Emission Emission is a process that generates photons within the medium. In the long-wave spectral region photons are emitted and absorbed by atmospheric trace gases such as water vapor, carbon dioxide, ozone, by cloud and aerosol particles, and by the Earth's surface. As already mentioned previously, in case of local thermodynamic equilibrium these emission processes can be described by the Planckian function. For the mathematical formulation of emission processes we introduce the socalled emission coefficient jν (r, t) for isotropic radiation sources. This coefficient defines the number of photons emitted per unit time and unit volume within the frequency interval (ν, ν + dν). The photons are contained in the solid angle element dΩ = Ω d ∂ Nν (r, t) = jν (r, t) d V d em ∂t dν (1.50) The emission coefficient is expressed in units of (m−3 s−1 sr−1 Hz−1 ). 1.7 Problems 1.1: With increasing temperature the maximum of the Planckian black body curve is shifted to shorter wavelengths. Observing that dν = −cdλ/λ2 , express (1.25) in terms of wavelength. (a) Differentiate Planck's law with respect to wavelength and estimate the wavelength λmax of maximum emission for a fixed temperature T . The resulting formula is known as Wien's displacement law. (b) Find λmax for the solar temperature T = 6000 K and for the terrestrial temperature T = 300 K. 1.7 Problems 27 1.2: Calculate for the two asymptotic situations (a) ν (b) ν 1: 1: Rayleigh–Jeans distribution Wien distribution the resulting simplified radiation laws of Planck. 1.3: Integrate Planck's formula (1.25) over all frequencies and directions to find the hemispheric flux density E b = σ T 4 . This is known as the Stefan– Boltzmann law. 1.4: A black horizontal receiving element (radiometer) of unit area is located directly below the center of a circular cloud at height z having the temperature Tc . The cloud radius is R. Find an expression for the flux density E incident on the receiving element in terms of the Stefan–Boltzmann law, z and R. Assume that the cloud is a black body radiator whose radiance is σ T 4 /π . Ignore any interactions of the radiation with the atmosphere. (a) Start your analysis using Lambert's law of photometry. (b) Rework the problem using equation (1.37c). 1.5: An idealized valley may be considered as the interior part of a spherical surface of radius a. The valley surface is assumed to radiate as a black body of temperature T . (a) Find an expression for the radiation received by a radiometer which is located at a distance z > a above the lowest part of the valley. Ignore any interaction of the radiation with the atmosphere. (b) Repeat the calculation with the radiometer located below the center of curvature, that is z < a. Hint: Use Lambert's law of photometry, see Problem 1.4. 1.6: A spherical emitter of radius a emits isotropically radiation into empty space. (a) Find the flux density Er = Er (r)er at a distance r ≥ a from the center of the sphere. er is a unit vector along the radius. (b) From Er obtain the power φ emitted by the sphere. (c) Find the energy density û(r). 1.7: For a monochromatic homogeneous plane parallel radiation field (solar radiation S0,ν ) find the energy density û ν and the net flux density Enet,ν . Ignore any interaction of the radiation with the atmosphere. 2 The radiative transfer equation 2.1 Eulerian derivation of the radiative transfer equation In the following section we will derive a budget equation for photons in a medium in which scattering, absorption and emission processes take place. The photon budget equation finally results in the so-called radiative transfer equation (RTE) which is a linear integro-differential equation for the radiance Iν (r, Ω, t). Let us consider a six-dimensional (6-D) volume element in (x, y, z, ϑ, ϕ, ν)-space with side lengths (x, y, z, ϑ, ϕ, ν). This volume element is assumed to be fixed in time t. According to (1.21), at point r the total number of photons Nν is given by Nν (r, Ω, t) = f ν (r, Ω, t)V ν (2.1) where V = xyz is the ordinary volume element in space. In order to simplify the notation, the dependence of different variables on (r, Ω, t) will henceforth be omitted except where confusion is likely to occur. The derivation of the photon budget equation requires the knowledge of the local time rate of change of the number of photons leaving and entering the 6-D volume element ∂ Nν ∂ fν = V ν ∂t ∂t This change consists of the following processes. ∂ Nν : Exchange of photons of the considered volume element with the ∂t exch exterior surrounding. ∂ Nν Absorption of photons with frequency ν and direction Ω. : ∂t abs ∂ Nν : Scattering of photons with frequency ν and direction Ω into all ∂t outsc other directions Ω (outscattering). 28 (2.2) 29 dz 2.1 Eulerian derivation of the radiative transfer equation ρv dx dz (ρv + ∂∂y (ρv)dy) dx dz dx dy Fig. 2.1 Mass flux entering the left face and leaving the right face of a cube with volume d x d y dz. ∂ Nν : ∂t insc ∂ Nν : ∂t em Scattering of photons with frequency ν and arbitrary direction Ω into the desired direction Ω (inscattering). Emission of photons with frequency ν in direction Ω. The individual contributions of these processes to the photon budget equation will now be discussed in detail. 2.1.1 The exchange of photons The exchange of photons can be treated in analogy to the continuity equation for the mass in fluid mechanics. We will consider a cube with side lengths (d x, dy, dz). The velocity of a fluid particle is given by v = ui + vj + wk, and ρ is the mass density of the medium. The volume d V is assumed to be fixed in space. The local time rate of change of the mass d M = ρ d x d y dz of the cube is given by adding all mass fluxes through its surface. Figure 2.1 depicts the mass fluxes entering and leaving the infinitesimal volume element d x d y dz through the vertical sides with area d x dz. Thus the net flux in y-direction is given by Fρ,y ∂ ∂ = − ρv + (ρv)dy d x dz + ρv d x dz = − (ρv)d x d y dz ∂y ∂y (2.3) In a similar manner we obtain the net fluxes Fρ,x and Fρ,z in the x- and z-direction. Adding up all three contributions and dividing by d x d y dz yields the well-known continuity equation ∂ρ = −∇ ·(ρv) ∂t (2.4) 30 The radiative transfer equation The concept of obtaining the continuity equation for the mass M will now be applied to the derivation of the photon budget equation. The velocity of a photon is given by v = cΩ = c( xi + yj + z k) (2.5) Analogously to (2.3) the net flux of photons in the y-direction through the 6-D volume element V ν can be expressed by means of Fy,ν = − ∂ c ∂y y fν V ν (2.6) Note that Fy,ν is expressed in units of (s−1 ). Adding up the contributions of the three directions yields the time rate of change for the number of photons due to the exchange with the surroundings ∂ ∂ ∂ ∂ Nν =− (2.7) ( x f ν ) + ( y f ν ) + ( z f ν ) cV ν ∂t exch ∂x ∂y ∂z The exchange term has been derived under the assumption that the medium's index of refraction is constant in space and time. This assumption is sufficient for most atmospheric applications. Otherwise, the photon path will be subject to refraction leading to photon trajectories which are curved in space. 2.1.2 The absorption of photons The absorption rate of photons within the 6-D volume element is given by the product of the photon number Nν and the probability that a photon will be absorbed during the time interval (t, t + dt). Dividing (1.40) by dt yields ds dτabs = kabs,ν = kabs,ν c (2.8) dt dt with c = ds/dt. Hence, for the total absorption rate of photons we obtain the expression ∂ Nν dτabs (2.9) = f ν kabs,ν cV ν = Nν abs ∂t dt 2.1.3 The scattering of photons Figure 2.2 illustrates the inscattering and outscattering process of photons. For inscattering the direction of the photons is indicated by solid arrows, outscattering is denoted by dashed arrows. It will be noticed that for the direction Ω inscattering represents a gain of photons whereas outscattering results in a reduction of the number of photons in this particular direction. 2.1 Eulerian derivation of the radiative transfer equation 31 dΩ′ ∆V dΩ′ Ω ∆Ω Fig. 2.2 Schematic view of the inscattering (solid arrows) and the outscattering (dashed arrows) processes. Analogously to (1.42) we may express the probability that at time t a photon is scattered from Ω → Ω by means of dτsca (Ω → Ω ) = ksca,ν (Ω → Ω )d ds (2.10) Dividing this equation by dt and multiplying by the number of photons Nν yields the time rate of change of photons resulting from the outscattering process Ω → Ω Nν d τsca (Ω → Ω ) = Nν ksca,ν (Ω → Ω )d dt c (2.11) The total loss of photons due to outscattering is obtained by integrating (2.11) over all directions Ω ∂ Nν = Nν c ksca,ν (Ω → Ω )d ∂t outsc 4π (2.12) c = f ν V ν ksca,ν Pν (Ω → Ω )d 4π 4π Here, use was made of (1.45) and (2.1). Utilizing the normalization condition for the phase function 1 Pν (Ω → Ω )d = 1 (2.13) 4π 4π 32 The radiative transfer equation finally gives ∂ Nν = f ν ksca,ν cV ν ∂t outsc (2.14) In a similar manner we may find the gain of photons for the direction Ω due to inscattering from all directions Ω . The number of photons moving in direction Ω , before inscattering takes place, is f ν (r, Ω , t) V d ν. In analogy to (2.11) we have d (2.15) Nν τsca (Ω → Ω) = f ν (Ω ) V d νksca,ν (Ω → Ω) c dt Integrating over all directions Ω , using equation (1.45), we find for the inscattering rate c ∂ Nν Pν (Ω → Ω) f ν (Ω )d (2.16) = ksca,ν V ν ∂t insc 4π 4π 2.1.4 The emission rate Finally, according to (1.50) the time rate of change of photons due to emission is given by ∂ Nν (2.17) = jν V ν ∂t em Now we have derived mathematical expressions for the five contributions for the photon budget equation as listed at the beginning of this section. 2.1.5 The budget equation of the photon distribution function The budget equation for the photon distribution function f ν is obtained by adding up the individual contributions. Considering absorption and outscattering processes as negative contributions, we may write ∂ Nν ∂ Nν ∂ Nν ∂ Nν ∂ Nν ∂ Nν − + (2.18) = − + ∂t ∂t exch ∂t abs ∂t outsc ∂t insc ∂t em Combination of (2.7), (2.9), (2.14), (2.16), and (2.17) gives ∂ fν ∂ ∂ ∂ = −c ( x f ν ) + ( y f ν ) + ( z f ν ) − c f ν kabs,ν − c f ν ksca,ν ∂t ∂x ∂y ∂z c + ksca,ν Pν (Ω → Ω) f ν (Ω )d + jν (2.19) 4π 4π where the common factor V ν has been cancelled out. 2.1 Eulerian derivation of the radiative transfer equation 33 Obviously, the unit vector Ω is divergence-free, that is ∇ ·Ω= ∂ y ∂ z ∂ x + + =0 ∂x ∂y ∂z (2.20) Therefore, the streaming term on the right-hand side of (2.19) may further be simplified yielding the final form of the photon budget equation for a nonstationary situation c ∂ fν = −cΩ · ∇ f ν − c f ν kext,ν + ksca,ν Pν (Ω → Ω) f ν (Ω )d + jν ∂t 4π 4π (2.21) Here, the extinction coefficient kext,ν as defined in (1.48) has been introduced. Since we consider the spatial change of the photon distribution function along ds in Ω-direction, ∇ f ν may be expressed in terms of ∇ fν = Ω d fν ds (2.22) so that Ω · ∇ fν = d fν = ds x ∂ fν + ∂x y ∂ fν + ∂y z ∂ fν ∂z (2.23) According to (1.36) the Cartesian components of Ω are given by x = Ω · i = sin ϑ cos ϕ, y= Ω · j = sin ϑ sin ϕ, z = Ω · k = cos ϑ (2.24) By introducing into (2.21) the radiance Iν as defined in (1.22), one obtains the general nonstationary form of the radiative transfer equation ksca,ν 1 ∂ Iν Pν (Ω → Ω)Iν (Ω )d + Jνe + Ω · ∇ Iν = − kext,ν Iν + (2.25) c ∂t 4π 4π Here, the source function for true emission Jνe (r, t) = hν jν (r, t) (2.26) has been introduced. This function has units of (W m−3 sr−1 Hz−1 ). Its relation to the Planck function will be described later. For most atmospheric applications the term 1/c(∂ I /∂t) in the RTE can be neglected in comparison to the remaining terms since the propagation speed c is very high. Thus (2.25) simplifies to ksca,ν Pν (Ω → Ω)Iν (Ω )d + Jνe (2.27) Ω · ∇ Iν = −kext,ν Iν + 4π 4π 34 The radiative transfer equation In the following we will assume that scattering takes place on spherical particles. Since according to (1.43) in this case the scattering process Ω → Ω depends on the cosine of the scattering angle cos = Ω · Ω only, henceforth the term Ω → Ω will be replaced by Ω · Ω or by cos . Utilizing in (2.27) the definition of the single scattering albedo as given in (1.49), we obtain the standard form of the RTE for a three-dimensional medium ω0,ν − Ω · ∇ Iν = Iν − kext,ν 4π 1 Pν (Ω · Ω)Iν (Ω )d − 4π 1 kext,ν Jνe (2.28) The derivation of the RTE is based on arguments of radiation hydrodynamics as presented by Pomraning (1973) where many additional and interesting details may be found. The RTE can also be derived on the basis of geometric reasoning in the manner described by Chandrasekhar (1960). In passing we would like to remark that the RTE is part of the atmospheric predictive system. With changing composition of the atmospheric constituents the radiation parameters are also changing so that radiance continues to be a function of time. 2.2 The direct–diffuse splitting of the radiance field The total solar radiation field is defined as the sum of the direct solar beam and the diffuse solar radiation. Usually one writes the RTE (2.28) in a different form by splitting Iν (r, Ω) into the unscattered direct light Sν (r) and the diffuse light Id,ν (r, Ω) Iν = Id,ν + Sν δ(Ω − Ω0 ) (2.29) where δ is the Dirac δ-function and Ω0 is the direction of the solar radiation. While Id,ν is expressed in (W m−2 sr−1 Hz−1 ), the units of the parallel solar radiation are (W m−2 Hz−1 ). In order to have a consistent set of units, the Dirac δ-function δ(Ω − Ω0 ) must refer to the unit solid angle, i.e. (sr−1 ). According to (2.23) we may write Ω · ∇ Iν = d Iν d Id,ν d Sν = + δ(Ω − Ω0 ) ds ds ds (2.30) The attenuation of the direct Sun beam Sν along its way from the top of the atmosphere (s = 0) to the location s at r follows from Beer's law d Sν = −kext,ν (s)Sν , ds Ω = Ω0 (2.31) 2.2 The direct–diffuse splitting of the radiance field 35 Using the boundary condition Sν (s = 0) = S0,ν , where S0,ν is the solar radiation at the top of the atmosphere, the solution of (2.31) is s Sν (s) = S0,ν exp − kext,ν (s )ds , (2.32) Ω = Ω0 0 S0,ν is also called the solar constant or the extraterrestrial solar flux. The exponential function occurring in (2.32) is also known as the transmission function since it determines the fraction of radiation which is transmitted through the path s. Obviously, the transmission function is bounded by 0 (no transmission) and 1 (total transmission). The integral expression on the right-hand side of (2.28) contains the contributions due to multiple scattering processes. Introducing into this term the definition (2.29) yields ksca,ν ksca,ν Pν (Ω · Ω)Iν (Ω )d = Pν (Ω · Ω)Id,ν (Ω )d 4π 4π 4π 4π ksca,ν + Pν (Ω · Ω)Sν δ(Ω − Ω0 )d 4π 4π (2.33) The last expression in this equation represents the direct solar radiation scattered from Ω0 into the direction Ω. Hence, we may also write Pν (Ω · Ω)Sν δ(Ω − Ω0 )d = Pν (Ω0 · Ω)Sν (2.34) 4π We are now ready to obtain the RTE for the diffuse radiation. Substituting (2.29), (2.31), and (2.33) together with (2.34) into (2.28) gives Ω · ∇ Id,ν − kext,ν Sν δ(Ω − Ω0 ) = Jνe −kext,ν Id,ν − kext,ν Sν δ(Ω − Ω0 ) ksca,ν Pν (Ω · Ω)Id,ν (Ω )d + 4π 4π ksca,ν + (2.35) Pν (Ω0 · Ω)Sν 4π It can been seen that the two terms involving the direct solar light cancel. Thus, after introducing the direct–diffuse splitting (2.29), the final version of the RTE for the stationary diffuse radiation field of a three-dimensional medium reads Ω · ∇ Id,ν = −kext,ν Id,ν + ksca,ν 4π 4π Pν (Ω · Ω)Id,ν (Ω )d + ksca,ν Pν (Ω0 · Ω)Sν + Jνe 4π (2.36) The third term on the right-hand side describes the generation of diffuse light from the unscattered part of the direct sunlight. 36 The radiative transfer equation 2.3 The radiatively induced temperature change According to (1.34) the net flux density vector is obtained by taking the first moment of the radiance with respect to Ω ΩIν d (2.37) Enet,ν = 4π The total net flux density vector follows from an integration of this equation over the entire electromagnetic spectrum ∞ ∞ Enet,ν dν = ΩIν d dν (2.38) Enet = 0 0 4π Since ∇ · Ω = 0 the RTE for the total (diffuse plus direct) radiation field (2.28) can be written as ksca,ν Pν (Ω · Ω)Iν (Ω )d + Jνe (2.39) ∇ ·(ΩIν ) + kext,ν Iν = 4π 4π Employing the integral operator 4π d . . . to the above equation we find in view of (2.37) ksca,ν Iν d = Pν (Ω · Ω)Iν (Ω )d d + 4π Jνe ∇ ·Enet,ν + kext,ν 4π 4π 4π 4π (2.40) since the source function Jν was assumed to be isotropic. Applying the normalization condition for the scattering phase function 1 Pν (Ω · Ω)d = 1 (2.41) 4π 4π to the integral on the right-hand side, we obtain for the divergence of the net flux density vector ∇ ·Enet,ν = ksca,ν Iν d − kext,ν Iν d + 4π Jνe 4π 4π (2.42) e = −kabs,ν Iν d + 4π Jν 4π If the medium does not contain any true interior sources (Jνe = 0), then the vector of the net flux density is given by = −k Iν d < 0 (2.43) ∇ ·Enet,ν abs,ν 4π Whenever ∇ · Enet,ν = 0 the medium is said to be in radiative equilibrium for the frequency ν. The concept of radiative equilibrium will be discussed in more detail in a later chapter. 2.4 The RTE for a horizontally homogeneous atmosphere 37 In the absence of frictional effects the first law of thermodynamics can be written as dv dh 1 dp 1 de +p = − = − ∇ · (Jq + Enet ) dt dt dt ρ dt ρ (2.44) see e.g. Chapter 3 of THD (2004). Here, v = 1/ρ is the specific volume of the air with density ρ. The quantities e, h, p and Jq stand for the specific internal energy, the specific enthalpy, the total pressure, and the heat flux, respectively. Here we are interested only in the contribution of radiative processes to the atmospheric temperature change. For an isobaric process (d p = 0) enthalpy and temperature changes are related by 1 dT dh = cp = − ∇ · Enet dt dt ρ (2.45) where cp is the specific heat at constant pressure. The local time rate of change of the temperature caused by radiative processes alone is then given by 1 1 ∂ T e =− ∇ · Enet,ν = − Iν d + 4π Jν −kabs,ν (2.46) ∂t rad,ν ρcp ρcp 4π From this equation the following conclusions are drawn. (i) The first term on the right-hand side describes the absorption of photons. Since this term is never negative it causes local warming. (ii) The second term on the right-hand side describing the emission of photons is never positive, thus resulting in local cooling. (iii) In the absence of absorption and emission no radiatively induced temperature changes take place. 2.4 The radiative transfer equation for a horizontally homogeneous atmosphere The simplest geometry for a scattering and absorbing medium is the so-called plane– parallel approximation, where in the horizontal direction the medium stretches to infinity. In such a homogeneous plane–parallel slab all optical properties are independent of the horizontal position. Moreover, the incident radiation, including the parallel solar beam, is assumed to be independent of the horizontal coordinates along the upper and lower boundaries of the atmosphere. In many cases the plane– parallel assumption represents a good approximation to a planetary atmosphere. It is important to note that the plane–parallel approximation to the RTE is best whenever the vertical variations of all radiative quantities dominate over the horizontal variability which is often the case. Two specific examples for which the plane–parallel theory is inadequate are: (i) radiative transfer in finite clouds located 38 The radiative transfer equation S0 s=0 z µ=1 dz µ> 0 ϑ s ϑ µ=0 µ <0 z ϑ ds y ϑ0 Reference level ϕ0 x µ = −1 Fig. 2.3 Illustration of the plane–parallel geometry. Upwelling radiation: µ > 0. Downwelling radiation: µ < 0. The zenith (µ = 1), nadir (µ = −1), and horizontal (µ = 0) directions are indicated. Solar radiation is incident from (ϑ0 , ϕ0 ). over a heterogeneously reflecting ground; and (ii) radiative transfer in a spherical atmosphere for solar positions near or below the horizon. In the following we derive the RTE for a plane–parallel medium. For ease of notation the subscript ν will henceforth be omitted from all radiative quantities. We start with the RTE for the diffuse radiation in the form (2.36) (omitting for simplicity the subscript d) and using Ω · ∇ I = d I /ds, ω0 1 d I (s) P(Ω · Ω)I (s, Ω )d = −I (s) + kext ds 4π 4π ω0 1 e + P(Ω·Ω0 )S(s) + J (s) (2.47) 4π kext For a black body the emission source function is given by Kirchhoff's law J e (s) = kabs B(s) with kabs = 1 − ω0 kext (2.48) where B(s) = B(T (s)) is the Planck function which depends on frequency and local temperature. The form of (2.48) will be motivated in the Appendix to this chapter. In a plane–parallel medium the only spatial variable is the altitude z. The direction of the radiation is defined by the angle ϑ with respect to the z-axis and by the azimuth angle ϕ counted from an arbitrary origin. As illustrated in Figure 2.3, the path element ds is related to dz by ds = dz µ with µ = cos ϑ (2.49) 2.4 The RTE for a horizontally homogeneous atmosphere 39 From the figure it is also seen that for upward directed radiance µ > 0 while for downward directed radiance µ < 0. At an arbitrary reference level the horizontal direction is characterized by µ = 0. Radiation propagating in the upward and downward vertical direction is specified by µ = 1 and µ = −1, respectively. However, in order to simplify the notation, for the downward directed radiance with ϑ > 90◦ the cosine of the zenith angle will henceforth be denoted by −µ so that for arbitrary directions of the radiance µ is positive definite. The direct solar radiation is expressed by s ∞ 1 S(s) = S0 exp − kext (s )ds = S0 exp − kext (z )dz µ0 z 0 (2.50) where according to (2.49) and due to the fact that θ0 > 90◦ , the path increment ds has been replaced by −dz /µ0 . The integration of the direct beam starts at the upper boundary s = 0 of the medium and proceeds along a straight path of length s. The optical depth τ is defined as the integral of the extinction coefficient over height along a path perpendicular to the horizontal plane, i.e. ∞ τ= kext (z )dz or dτ = −kext dz (2.51) z Since according to (2.49) and (2.51) kext ds = kext dz/µ = −dτ/µ we obtain for the solar radiation τ S(τ ) = S0 exp − µ0 with µ0 > 0 (2.52) With these results one finds for the plane–parallel approximation of the RTE the integro-differential equation µ d ω0 2π 1 P(cos )I (τ, µ , ϕ )dµ dϕ I (τ, µ, ϕ) = I (τ, µ, ϕ) − dτ 4π 0 −1 ω0 τ − P(cos 0 )S0 exp − − (1 − ω0 )B(τ ) 4π µ0 (2.53) We repeat that the cosine of the scattering angle is given by cos = Ω · Ω, which is a function of the four angles ϑ , ϕ , ϑ, ϕ. An expression for cos will be derived later. 40 The radiative transfer equation The scattering phase function P(cos ) is assumed to be rotationally symmetric along the direction of the incident light, an assumption that is exact for spherical scattering particles. The scattering of light by homogeneous spheres is rigorously described by the so-called Mie theory which will be thoroughly discussed in a later chapter. The scattering properties of a single particle are solely a function of two physical parameters; the complex index of refraction N = n + iκ,1 and the so-called Mie size parameter x= 2πr λ (2.54) where r is the radius of the scattering sphere and λ is the wavelength of the incident electromagnetic wave. The schematic illustration of the rotationally symmetric form of P(cos ) shown in Figure 1.18 indicates a strong forward peak of the scattering phase function. This is typical of cloud droplets and spherical aerosol particles when they are illuminated by solar radiation. Having specified the size parameter and the index of refraction, the Mie theory provides a table of numbers of P versus . In order to give a mathematical description of P, the scattering phase function will be expanded as an infinite series of Legendre polynomials Pl (cos ) P(cos ) = ∞ pl Pl (cos ) (2.55) l=0 For practical purposes the phase function will be truncated after a finite number of terms. The Legendre polynomials for argument x, with −1 ≤ x ≤ 1, are defined by Pl (x) = 1 dl 2 x −1 2l l! d x l l (2.56) It will be readily seen that the first three polynomials are given by P0 (x) = 1, P1 (x) = x, P2 (x) = 3 2 1 x − 2 2 (2.57) The Legendre polynomials are a special case of the more general associated Legendre polynomials Plm (x) as defined by Plm (x) = 1 m (1 − x 2 )m/2 d l+m 2 l 2 m/2 d (x − 1) = (1 − x ) Pl (x) 2l l! d x l+m dxm n and κ are, respectively, the real and the imaginary part of the index of refraction. (2.58) 2.4 The RTE for a horizontally homogeneous atmosphere 41 They have the following properties Plm=0 (x) = Pl (x), (a) Plm (x) = 0, m > l, 1 2 (l + m)! (b) Pnm (x)Plm (x)d x = δnl 2l + 1 (l − m)! −1 Plm (−x) = (−1)l+m Plm (x) (2.59) Equation (2.59b) expresses the orthogonality relation of the associated Legendre polynomials on the interval [−1, 1]. In order to obtain an analytical expression for the phase function P we need to evaluate the expansion coefficients pl occurring in (2.55). This is done by multiplying both sides of (2.55) by Pn (cos ) and then integrating over the unit sphere ∞ 1 1 P(cos )Pn (cos )d = pl Pl (cos )Pn (cos )d (2.60) 4π 4π 4π 4π l=0 2π In this equation the integration 4π . . . d may be replaced by 0 dϕ 1 −1 . . . d cos . Application of the orthogonality relations for Pl , which are given by (2.59b) if we set m = 0, finally yields pl = 2l + 1 2 1 −1 Pl (cos )P(cos )d cos (2.61) In addition to specifying P with the help of the Mie theory, mathematical models or experimentally determined values of the phase function may be employed. Due to the normalization of P, i.e. 2π 1 1 1 P(cos )d = P(cos )d cos dϕ 4π 4π 4π 0 −1 (2.62) 1 1 = P(cos )d cos = 1 2 −1 the first expansion coefficient p0 is always given by 1 1 p0 = P(cos )d cos = 1 2 −1 (2.63) Thus the coefficient p0 expresses conservation of energy in case of pure scattering. In the following we require an explicit expression for the cosine of the scattering angle . From Figure 1.16 and equation (1.36) one easily obtains for Ω · Ω = cos the expression cos = (i cos ϕ sin ϑ + j sin ϕ sin ϑ + k cos ϑ ) · (i cos ϕ sin ϑ + j sin ϕ sin ϑ + k cos ϑ) (2.64) 42 The radiative transfer equation With µ = cos ϑ, µ = cos ϑ we find cos = µµ + (1 − µ2 )1/2 (1 − µ )1/2 cos(ϕ − ϕ ) 2 (2.65) This relation allows us to express the scattering angle with respect to the angles ϑ , ϕ and ϑ, ϕ. From (2.53) it may be seen that the angles ϑ, ϕ are fixed while ϑ , ϕ vary over the unit sphere. To continue the mathematical development we also need to state the addition theorem for the associated Legendre polynomials which is given by Pn (cos ) = Pn (µ)Pn (µ ) + 2 n (n − m)! (n + m)! m=1 Pnm (µ)Pnm (µ ) cos m(ϕ − ϕ ) (2.66) (cf. Jackson, 1975). This theorem will now be used to separate the angles of the scattering phase function. Substituting (2.66) into (2.55) gives ∞ l (l − m)! pl Pl (µ)Pl (µ ) + 2 Plm (µ)Plm (µ ) cos m(ϕ − ϕ ) P(cos ) = (l + m)! l=0 m=1 = ∞ l (2 − δ0m ) plm Plm (µ)Plm (µ ) cos m(ϕ − ϕ ) (2.67) l=0 m=0 with plm = pl (l − m)! (l + m)! Since Plm (x) = 0 for m > l, P(cos ) can be reformulated as P(cos ) = ∞ ∞ (2 − δ0m ) plm Plm (µ)Plm (µ ) cos m(ϕ − ϕ ) l=0 m=0 = ∞ (2 − δ0m ) m=0 ∞ plm Plm (µ)Plm (µ ) cos m(ϕ − ϕ ) (2.68) l=m = P(µ, ϕ, µ , ϕ ) It can now be seen that in the scattering phase function the (µ, µ )-dependence has been completely separated from the (ϕ, ϕ )-dependence. In order to remove the azimuthal dependence of the radiance field we assume that the radiance can be expressed by means of a Fourier expansion for an even function of the azimuth angle ϕ I (τ, µ, ϕ) = ∞ m=0 (2 − δ0m )I m (τ, µ) cos m(ϕ − ϕ0 ) (2.69) 2.4 The RTE for a horizontally homogeneous atmosphere 43 where ϕ0 is the azimuth angle of the direct solar beam. It is customary to orient the Cartesian (x, y, z)-coordinate system in such a way that ϕ0 = 0, see Figure 2.3, so that I (τ, µ, ϕ) = ∞ (2 − δ0m )I m (τ, µ) cos mϕ (2.70) m=0 The expansion (2.70) may be used whenever the phase function is expressed as a series of Legendre polynomials. Substituting (2.68) and (2.70) into the RTE (2.53) yields µ ∞ (2 − δ0m ) m=0 = ∞ d m I (τ, µ) cos mϕ dτ (2 − δ0m )I m (τ, µ) cos mϕ m=0 ∞ ∞ ω0 τ − (2 − δ0m ) plm Plm (µ)Plm (−µ0 ) cos mϕ S0 exp − 4π µ0 m=0 l=m ω0 − 4π × 0 ∞ 2π 1 ∞ −1 m=0 (2 − δ0m ) ∞ plm Plm (µ)Plm (µ ) cos m(ϕ − ϕ ) l=m (2 − δ0i )I i (τ, µ ) cos iϕ dµ dϕ − (1 − ω0 ) i=0 ∞ δ0m B(τ ) m=0 (2.71) The above equation can be simplified by employing the orthogonality relations of the trigonometric functions 2π cos m(ϕ − ϕ ) cos lϕ dϕ = (1 + δ0l )δlm π cos lϕ (2.72) 0 An expression for I m (τ, µ) can be isolated from (2.71) by carrying out the following 2π two steps: (i) Multiply (2.71) by cos kϕ; (ii) Carry out the operation 0 . . . dϕ. As a consequence, the following relations are needed 2π cos mϕ cos kϕ dϕ = (1 + δ0k )δmk π 0 2π (2.73) cos kϕ dϕ = (1 + δ0k )δ0k π 0 0 2π cos m(ϕ − ϕ ) cos kϕ dϕ = (1 + δ0k )δmk π cos kϕ 44 The radiative transfer equation From (2.71) it is seen that for the multiple scattering term an integration over ϕ remains to be done for which we obtain the result 2π cos iϕ cos kϕ dϕ = (1 + δ0i )δik π (2.74) 0 Using the above intermediate steps we find from (2.71) µ ∞ (2 − δ0m ) m=0 = ∞ d m I (τ, µ)(1 + δ0k )δmk π dτ (2 − δ0m )(1 + δ0k )δmk π I m (τ, µ) m=0 − ∞ ∞ ω0 τ (2 − δ0m ) plm Plm (µ)Plm (−µ0 )(1 + δ0k )δmk π S0 exp − 4π µ0 m=0 l=m − ω0 4π × ∞ 1 ∞ −1 m=0 (2 − δ0m ) ∞ plm Plm (µ)Plm (µ )(1 + δ0k )δmk π l=m (2 − δ0i )I (τ, µ )(1 + δ0i )δik π dµ− (1 − ω0 ) i i=0 ∞ δ0m B(τ )(1 + δ0k )δ0k π m=0 (2.75) We recognize immediately that, with the exception of m = k, all terms vanish. Therefore, after separation of the azimuthal dependence, for the m-th Fourier mode of the radiance the RTE is given by (a) µ (b) µ ∞ d m ω0 1 I (τ, µ) = I m (τ, µ) − p m P m (µ)Plm (µ )I m (τ, µ )dµ dτ 2 −1 l=m l l ∞ ω0 τ S0 exp − − p m P m (µ)Plm (−µ0 ) − (1 − ω0 )B(τ )δ0m 4π µ0 l=m l l d m I (τ, µ) = I m (τ, µ) − J (τ, µ) dτ (2.76) (2.76b) is the standard form of the RTE for plane–parallel atmospheres. The source function J (τ, µ) summarizes the terms describing multiple scattering, primary scattering of solar radiation and thermal emissions. For the special case m = 0 the same result is obtained by performing an azimuthal average of (2.53). Consider the averaging process for each individual term separately. 2.4 The RTE for a horizontally homogeneous atmosphere 45 (1) Averaging the intensity I (τ, µ, ϕ) as defined by (2.70) yields 1 2π 2π ∞ 1 I (τ, µ, ϕ)dϕ = (2 − δ0m ) I m (τ, µ) cos mϕ dϕ 2π m=0 0 2π 0 =I m=0 (2.77) (τ, µ) = I (τ, µ) Only m = 0 contributes to the azimuthally averaged radiation field since the integral of the cosine function over a complete cycle vanishes. (2) Since the Planck function is isotropic the thermal emission term retains its form. (3) Analogously to (1) the primary scattering term can be averaged according to the expansion (2.68) 1 2π 2π ∞ P(cos 0 )dϕ = 0 pl Pl (µ)Pl (−µ0 ) = P(µ, −µ0 ) (2.78) l=0 (4) For the multiple scattering term we proceed as in (2.77) and (2.78) obtaining ω0 8π 2 = = 2π 0 ω0 4π ω0 2 1 −1 2π I (τ, µ , ϕ ) −1 2π 0 1 P(cos )dϕ dµ dϕ 0 1 −1 I (τ, µ , ϕ )P(µ, µ )dµ dϕ (2.79) I (τ, µ ) P(µ, µ )dµ The azimuthally integrated form of the phase function P(µ, µ ) is given by 2π 1 P(cos )dϕ P(µ, µ ) = 2π 0 2π ∞ 1 P(µ, ϕ, µ , ϕ )dϕ = pl Pl (µ)Pl (µ ) = 2π 0 l=0 (2.80) Application of steps (1)–(4) results in the azimuthally integrated form of the RTE d ω0 1 µ I (τ, µ) = I (τ, µ) − P(µ, µ )I (τ, µ )dµ dτ 2 −1 ω0 τ − S0 exp − P(µ, −µ0 ) − (1 − ω0 )B(τ ) 4π µ0 (2.81) Comparison of (2.81) with (2.76) reveals that for m = 0 both equations are identical. The separation of the ϕ-dependence of the radiance field results in an infinite system of first-order ordinary differential equations for the Fourier modes I m (τ, µ). 46 The radiative transfer equation It is convenient to introduce into (2.76) the following abbreviations (a) R m (µ, µ ) = ∞ plm Plm (µ)Plm (µ ) l=m (b) ω0 1 m R (µ, µ )I m (τ, µ )dµ (2.82) J (τ, µ) = 2 −1 ω0 τ + S0 exp − R m (µ, −µ0 ) + (1 − ω0 )B(τ )δ0m 4π µ0 m With these definitions the RTE reads in short-hand notation µ d m I (τ, µ) = I m (τ, µ) − J m (τ, µ), dτ m = 0, 1, . . . , (2.83) For practical applications the infinite series (2.70) must be truncated after a finite number of terms, , as indicated above. 2.5 Splitting of the radiance field into upwelling and downwelling radiation It is common practice to distinguish between the upwelling (µ > 0) and downwelling (µ < 0) radiation, see Figure 2.3. For the radiance field I m and the source term J m the following quantities are introduced2 I+m (τ, µ) = I m (τ, µ > 0), I−m (τ, µ) = I m (τ, µ < 0) J+m (τ, µ) = J m (τ, µ > 0), J−m (τ, µ) = J m (τ, µ < 0) Now the integral 1 and 0 . . . dµ 1 1 −1 . . . dµ in (2.82b) will be split into the two parts R (µ, µ )I (τ, µ )dµ = m m −1 0 m −1 0 1 R m (µ, µ )I m (τ, µ )dµ 0 1 = 0 m . . . dµ R m (µ, µ )I m (τ, µ )dµ R (µ, −µ )I (τ, −µ )dµ + m 0 −1 1 0 1 = R (µ, µ )I (τ, µ )dµ + m (2.84) R m (µ, −µ )I−m (τ, µ ) + R m (µ, µ )I+m (τ, µ ) dµ (2.85) With the above convention µ and µ are always positive quantities. As a consequence of this, the system (2.83) can be treated as two separate differential equations, one governing the upwelling part and the other one the downwelling part of the radiation 2 Some authors define the zenith angle ϑ in the opposite way, that is µ > 0 for downwelling radiation and µ < 0 for upwelling radiation. 2.5 Splitting of the radiance field into up- and downwelling radiation 47 field d m I (τ, µ) = I+m (τ, µ) − J+m (τ, µ) dτ + d µ I−m (τ, µ) = −I−m (τ, µ) + J−m (τ, µ) dτ m = 0, 1, . . . , , µ>0 µ (2.86) In a similar manner the source terms of (2.82b) can be split to give (a) (b) J+m (τ, µ) = J−m (τ, µ) ω0 2 1 I+m (τ, µ )R m (µ, µ ) + I−m (τ, µ )R m (µ, −µ ) dµ ω0 τ S0 exp − + R m (µ, −µ0 ) + (1 − ω0 )B(τ )δ0m 4π µ0 0 ω0 = 2 1 I+m (τ, µ )R m (−µ, µ ) + I−m (τ, µ )R m (−µ, −µ ) dµ ω0 τ + S0 exp − R m (−µ, −µ0 ) + (1 − ω0 )B(τ )δ0m 4π µ0 (2.87) 0 with µ > 0 and µ > 0. The above expressions demonstrate that (2.86) is a system of 2( + 1) coupled integro-differential equations. The source terms J+m and J−m contain integrals over µ which may be approximated by means of quadrature formulas. Best suited for the interval [−1, 1] are the Gauss–Legendre quadrature formulas, which are obtained by neglecting in the expression 1 r f (µ)dµ = wi f (µi ) + Rr −1 i=1 (2.88) 22r +1 (r !)4 (2r ) with Rr = f (ξ ), −1<ξ <1 (2r + 1) [(2r )!]3 the error term Rr (see, e.g. Abramowitz and Stegun, 1972). The wi are the weights and the µi are the zeros or roots of the Legendre polynomials Pr (µ). The weights are given by r 2 wi = > 0, wi = 2 (2.89) 2 (1 − µi2 ) Pr (µi ) i=1 where Pr is the derivative of Pr . The last expression follows immediately from (2.88) by setting there f (µ) = 1. It is convenient to divide the interval [−1, 1] according to the zeros of evenorder Legendre polynomials, which is achieved by choosing r = 2s. For such even divisions we have 2s different weights and zeros. By denoting them as (wk , µk , 48 The radiative transfer equation k = −s, . . . , −1, 1, . . . , s) the following relations hold wi = w−i , µi = −µ−i , i = 1, . . . , s (2.90) Now we split the integral of (2.88) into two parts yielding 1 0 1 1 f (µ)dµ = f (µ)dµ + f (µ)dµ = f (−µ)dµ + −1 −1 0 0 1 f (µ)dµ 0 (2.91) For the sum on the right-hand side of (2.88) we may write s wi f (µi ) = i=−s = −1 wi f (µi ) + s wi f (µi ) i=−s s i=1 s i=1 i=1 wi f (−µi ) + wi f (µi ) (2.92) where use was made of (2.90). Hence, the notation is equivalent to the summation from i = −s to i = s thereby omitting the term i = 0. By comparing (2.91) with (2.92) we obtain the Gauss–Legendre quadrature formulas of order 2s in the form 0 1 f (−µ)dµ ≈ s wi f (−µi ), 1 0 i=1 f (µ)dµ ≈ s wi f (µi ) (2.93) i=1 For ease of notation (2.93) will henceforth be called the Gaussian quadrature formulas. It is important to note that the µi , i = 1, . . . , s are the positive zeros of the Legendre polynomials P2s (µ). Furthermore, from (2.89) and (2.90) we see that the weights fulfill the requirement s wi = 1 (2.94) i=1 We conclude that a particular Gaussian quadrature formula is defined by the set of quantities (wi , µi , i = 1, . . . , s). The weights and the nodes may be found in tabulated form (see e.g. Abramowitz and Stegun, 1972) or may be found using efficient numerical algorithms (Press et al., 1992). Thus the nodes define the directions of the radiances for which the system of equations (2.86) will be discretized. In order to express the directional dependence of the radiance it is convenient to introduce a compact matrix–vector notation. One defines two vectors for the upwelling and downwelling radiation field by means of m m I+ (τ, µ1 ) I− (τ, µ1 ) .. .. Im (2.95) Im , . . + (τ ) = − (τ ) = I+m (τ, µs ) I−m (τ, µs ) 2.5 Splitting of the radiance field into up- and downwelling radiation 49 In a similar manner vectors for the sources of the up- and downwelling primary scat