THE UN IV ER SIT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Meteorology and Oceanography Technical Report THE CALCULATION OF LONG-WAVE RADIATIVE TRANSFER IN PLANETARY ATMOSPHERES S?,.1ianJ 6ysoa' Edward xS.,E -er',roJe t Iirector ORA ProJect 07584 supported by, NATIONAL SCIENCE FOUNDATION GRANT NO. GP-4585 WASHINGTON, D. C administered througho: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR November 1967

This report was also a dissertation submitted by the first author in partial f,.7ulfillment of the requirements for the degree of Doctor of'Philosophy in The University of Michigan, 1.967.

ACKNOWLEDGMENTT The author wishes to express his thanks to all those who assisted him in this study. In particular, the guidance and help of Professor Edward S. Epstein, Chairman of the Doctoral Committee, is deeply appreciated. Professor Charles Young gave much assistance and participated in many helpful discussions with the author. The author is also grateful to the other members of the Doctoral Committee, Professors E. Wendell Hewson and Leslie M. Jones, for their interest and willingness to help. A number of helpful discussions on the form of the equations for the source function were held with Professor William R. Kuhn. Mr. J. W. Peterson kindly provided unpublished falling sphere temperature profiles to use in the cooling rate calculations. The interest and encouragment of other colleagues at the High Altitude Engineering Laboratory is gratefully acknowledged. The author is grateful for financial support from the National Science Foundation under Grant NO. NSF GP-4385. In addition to this work undertaken in the Department of Meteorology and Oceanography, much of the necessary background material was obtained at the High Altitude Engineering Laboratory, Department of Aerospace Engineering under contracts Nos. Cwb-11376 and Cwb-11106 with the Environmental Science Services Administration and contract No. JV'c-54(03) with the National Aeronautics and Space Administration. ii

TABLE OF CONTENTS Page IBST OF TABTLES LIST OF FIG:CES vi LIST OF SYi\BOLS vii ABSTRACT ix CEIAPTER 1i. ITI'IRODUCTION 2. ThE LT3-\ iERICAL EVALUATION OF THE EQUATIONS OF RADIATIVE TRANSFER 4 2.1 Equations of Radiative Transfer 4 2,2 The Evaluation of the Equations of Transfer 8 2.3 Atmiospheric Transmissivities 9 2.3.1 Homogeneous Path Absorption 10 2.3.2 Atmospheric Slant Path Absorption 20 2.4 Numerical, Procedures for Evaluating Radiative Flux 28 2.4.1 The Isothermal Layer Approximation 28 2.4 2 Polynomial Representation of the Source Function 32 2.5 Conclusions 41 35 APPLICATION TO THE EARTH'S MESOSPHERE AND MESOPAUSE 43 3.1. C.eneral Survey of Long-Wave Radiative Transfer in the Earth's Atmosphere 45 3.2 Equations for the Source Functions in the Upper Atmosphere 48 3.3 Calculation Details 57 3~4 Results of Cal.culations and Comparison with Previous Results 59 3.401 Cooling Rates for Model Atmospheres 59 3.4 2 Cooling Rates for Measured Temperatures Profiles 75 3.4.3 Comparison with Previous Calculations 80 355 S mmary and Conclusion 82 4. APPLICAT I.ON TO THE ATMOSPHERE OF MARS 84 41i Introduiction 84 4o2 Atmospheric Composition 8 iii

TABLE OF CONTENTS (Concluded) Pa.ge 4.3 Calculation Details and Results 86 4.4 Comparison with Previous Calculations 91 5. CONCLUSIONS 93 6. SUGGESTIONS FOR FUTURE WORK 96 APPENDIX:. THE POLYNOMIAL APPROXIMATIONS OF EXPONENTIAL INTEGRALS 98 BIBLIOGRAPHY 105 iv

LIST OF TABLES Table Page I. Band Intensities Used In Ca.lculating Rotational Line Intensities 60 II. Relative Abundance Of Isotopic Molecules 60 III. Cooling Rates In deg K/day For The Individua-l Bands, U.S. Standard Atmosphere (1962), ko = 10-5 Secs. 62 IV. Ratio J/B For Individual Bands, UoS. Standard Atmosphere (1962), X0 = 10-5 Secs. 63 V. Heating Rate Coefficients For The Martian Atmosphere 90 VI. Hea.ting Rates In The Martin Atmosphere For Typical Temperature Profiles. 91 VII. Coefficients For The Polynomial Approximation Of E4(x) 102 VIII. Coefficients For The Polynomial Approximation Of E3(x) 103 IX. Coefficients For The Polynomial Approximations Of E2(x) 104 V~~~~~~~~0

LIST OF FIGURES Figure Page 1. Diagram of the source function as a function of height. 30 2. Typical polynomial fits to vertical temperature structures. 35 3. Calculated cooling rates for the U.S. Standard Atmosphere (1962). 65 4. Calculated cooling rates for modified U.S. Standard Atmosphere (1962). 68 5. Calculated cooling ra.tes for ARDC-1959 atmosphere 7.1 6. Calculated cooling rates for CIRA-1961 atmosphere. 72 7. Calculated cooling rates for measured temperature profiles. 75 8. Model. of the Martian atmosphere. 87 vi

LIST OF SYMBOLS b subscript denoting boundary bi -line shape factor of ith line B(v,T) Planck black-body function c velocity of light in va.cuum E number density in vibrational quanta E value of E when vibra.tional energies have a. Boltzmann distribution En(x) exponential integral of order n e emission coefficient g subscript denoting ground gi statistical weight of ith energy level h Planck constant -3 -1 H flux divergence in quanta cm sec I(v,t,q) specific intensity J(v,T) source function k Boltzma.nn constant k absorption coefficient V M molecular weight N number density of absorbing molecules p pressure S total band intensity Si intensity of ith line vii

T tempera.ture u optical mass of absorbing ga.s CzD Doppler half-width CaL Lorentz half-width daN na.tural half-width y( T,) transmissivity zenith angle; radia.tive.lifetime 2 vibrational relaxation. time o0 value of x at 1 atmosphere pressure cosine of zenith a.ngle v frequency vo frequency of band center j0 a.zimuth angle T optical thickness viii

ABSTRACT A method is developed for calculating long-wave radiative transfer in a plane parallel atmosphere. Modern computing techniques are used to avoid approximations commonly used in existing methods. Two applications involving the:15t bands of C02 are given, the first to the earth's mesosphere and lower thermosphere, taking vibrational relaxation into account and the second to the Martian atmosphere, for use in a simple atmospheric circulation model.o Although the basic equations of radiative transfer in a plane parallel atmosphere are simple, their solution is a difficult task because of the complicated nature of molecular absorption spectra and'because planetary atmospheres are inhomogeneous. In this study atmospheric radiative fluxes and flux divergences are obtained by direct integration with respect to frequency across the absorption bands. The atmosphere is divided into horizontal layers which may be made arbitrarily thin, The direct integration method allows the Curtis-Godson approximation to be applied over successive thin layers, rather than thick atmospheric layers. Band models are not used, but instead the accurate contribution of all lines in the absorption band is ca:lculated at each frequency quadrature poidnt, Rather than using an approximation diffusivity factor, angular integration is performed exactly and expressed in terms of exponential integrals, Accurate, fast computing subroutines, using polynomial approximations, were developed for exponential integrals of the second, third, and fourth orders. Within each horizontal atmospheric layer the source function is assumed to'be linear i.n pressure,, This is shown to be more accurate than the assumption of isothermal layers and less problematic than the use of high order polynomials, To avoid numerical difficulties when the atmospheric layers are thin, flux divergences are calculated directly, rather than subtracting fluxes at adjacent levels. Long-wave radiative transfer in the earth's atmosphere between about 60 and 100 km is dominated by the 15bp C02'bands. The radiative transfer equations are solved separately to obtain source functions for each of the bands in this spectral region and hence obtain their contributions to the total cooling rate. Values of vibrational relaxation time in the range 2 x 10-6 to.2 x 10-5 sec at 1 atmosphere are used. Calculations are described for several model atmospheres In general small amounts of heating are obtained near the mesopause corresponding to discontinuities in temperature gradients which characterize most models, Vibrational relaxation affects the cooling rate above about 75 km, although the less abundant i.sotopic molecules depart from LoToEo near 60 km. Measured, as opposed to mean, temperature soundings also are used to calculate cooling rates. These show that the local temperature structure largely determines the radiative transfer and produces a predominance of cooling, rather than heating near the mesopauseo The mean cooling is significantly greater than the cooli.ng calculated from the mean profiles, suggestix

ing the presence of a mean heat sink for this part of the atmosphere, important for energy balance considerations. In the Martian atmosphere, the 15t CO2 bands must also dominate the long-wave radiative transfer. Flux divergences are calculated for a simple model atmosphere, assuming pure C02 and a surface pressure of 6 mb. Since the results are for use in a general circulation model, the cooling rates are expressed as a simple function of the temperature profile, so that rapid computations can be made. The cooling rates for several temperature profiles are presented.

CHAPTER 1 INTRODUCTION Although the atmosphere, oceans and land masses of the earth constitute an extremely complicated system for the transfer of energy, the long time-scales of tei-e-strial1 climatic changes imply that for the system viewed as a. whole there must be a close balance between the solar energy absorbed and the longswave radiant energy lost to space. The radiative and dynamic processes are not separate, but a.re related in a. complex manner. For example, the presence of clouds affects the amount a.nd distribution of sola.r energy received and modifies the out-going long-wave radiation. Some of the main features of the vertical atmospheric temperature profile can be accounted for by radiative processes, but dynamic processes e.g., convection must be introduced to account for other features. Simila.r processes occur in the atmospheres of other planets. In the Martian atmosphere, radiative processes are simpler; the absence of oceans on the surface, the virtual absence of water vapor and ozone in the atmosphere and the rarity of clouds provide a. much less complicated environment than that of the earth. On the other hand, Venus is permanently shrouded in clouds of an unknowncomposition, making the atmosphere the object of considerable speculation, and the problem of radiative transfer unsolvable at presentD In recent years, the development of meteorological satellites has renewed interest in atmospheric radiative processes. By making suitable 1

measurements of the outgoing radiation. fields it is possible to infer information on the vertical temperature structure of the a.tmosphere and on the distribution of water vapor and ozone A -po.lar orbiting sa.tellite ca~n provide globa.l coverage at regular:i.ntervals. In order -to extra-ct the maximum amount of in.forma.tion from the measurements9 a. detailed accurate knowiledge of atmospheric radiative R;processes is required, so that develo'sments in this field. have led to important theoretical advances Possible modification of the radiative regime'by air pollution is of considerable interest and importance, particularly that due to the steady increa.se in the amount of carbon dioxide in the atmosphere formed by the combustion of fossi.l fuels. The.long-term effect of this increase is difficult to ga.uge because of the complicated intera.ction of radia.tive and dynam.ic processes, but if the net result is to increase the su:rface tempera.ture of the earth by a.s little a.s.i'C it might'be sufficient to melt la.rge amo".: -..:' ai ice a-nd cause flooding over low-Lying regions of the ea.rth where much of the population is concentra.ted. The development of electron:ic computers has made possli.b.le the theoretica:l study of the general. circula.tion of the a.tmosphere by means o'f numerica.l mode.-ls and the development of numerica.l. weather prediction. In tihese models ra.diative tra:nsfer is a.pproximated in o7ni.-y a. rather crude manner Refinements in this area are essentia.l, pa.rticul.ar.ly foer long — range foreca.st.ingo When the ro.l.e which radia.tivre tran;sfer pla.ys i.n the general. circulation is underst'ood, prob.lem.s such as the inf.luence of the

5 carbon dioxide concentration in the atmosphere will be solvable. The ma.thematica.l equa.tions governing atmospheric long-wa.ve radia.tive transfer a.re compara.tively simple. However the structure of atmospheric absorption bands is exceedingly complicated and some a.spects such a.s the sha.pes of spectra.l absorption lines a.re not yet fully understood. The variations of temperature, pressure a.nd absorber concentration in the atmosphere combine to make the solution of the transfer equa.tion a. ta.sk of extreme difficulty. Many approximations were introduced before electronic digita.l computers were availab.le. It is now practicable to solve the equations more precisely. The purpose of this s'tudy is to develop a. numerical method of solution of the ra.dia.tive transfer euations with a.s few approximations as possible, using modern computing techniques. Direct integration with respect to frequency has already been applied to the ca.lculation of atmospheric slant path transmissivities. In this study the technique is extended to the calcula.tion of radiative fluxes and flux divergences a.nd is shown to be flexible and a.ccura.te, A critical discussion of many existing approximations is contained in Cha.pter 2, where the method of solution of the equations of radiative transfer is developed. The results are a.pplied to two atmospheric radiative prob.lems. Cha.pter 3 is devoted to an analysis of the role of the 15 t bands of carbon dioxide in the earth's upper atmosphere, including the deviation from loca.l thermodynamic equilibrium. Chapter 4 concludes the study with a. radia.tive transfer calculation for application to a. Martian atmospheric circulation model.

CHAPTER 2 THE NUMERICAL EVALUATION OF THE EQUATIONS OF RADIATIVE TRANSFER 2.1 EQUATIONS OF RADIATIVE TRANSFER In deriving the equations of radiative transfer the following a.ssumptions will be made. (a) The atmosphere is plane-parallel and horizontally homogeneous. (b) Atmospheric refraction ca.n be neglected. (c) Scattering effects at the wavelengths under consideration may be neglected. (d) The source function J(v,T) is isotropic. The differential equation may be written, a-s in Chandrasekhar 1960, P. 9, AS (vT,11) _ I(VTkL)-J(VT) (2.1.1) where I(v,7,4) is the specific intensity at frequency v and zenith angle 0 = cos Q Ad - is the optical thickness, defined by dT = kvdu u is the optical mass of the absorbing gas k, is the absorption coefficient e is the emission coefficient J(v,T), the source function, is the ratio ev/kv. For a gas in local thermodynamic equilibrium, the source function is equal to the Pla.nck bla.ck-body functiono The form of the source function 4

5 for radiating ga.s which is not in loca.l thermodynamic equilibrium will be discussed in a. later section. The usual convention will be followed here, that Tv is zero at the top of the atmosphere and increases with increasing pressure. A formal solution of Eq. (2.1.1) may be found in any text on radiative transfer e.g., Goody (i964) ~ -(-rb-F)/4' Tb -(t-T)/4 dt I(v,Tr,) I(v,Tb,p)e b + J(v,t)e( (2.1.2) where Tb is the value of,- at the boundary. For values of [ > 0 this boundary is the lower boundary of the atmosphere (the planetary surface). It is convenient to assume that I(v,Tb,k) is independent of k, although this assumption is not strictly correct. Its applicability depends on the angular distribution of specific intensity incident upon the surface, a.s well as its diffuse reflectance properties, which may be complicated functions of the incidence and reflection angles, and may vary also with frequency. Adequate experimental studies have not been made, either for the case where the lower boundary surfa.ce is a. cloud layer, or where it is the ground, consisting of soil, bare rock and vegetation. Except within the planetary boundary layer, this assumption is probably not important. It will also be convenient to assume that I(v,Tb) is given by the black body source function B(v,Tg) at frequency v and the temperature Tg of the surface. Frequently Tg will be taken equal to the temperature of the atmosphere bounding the surfaceo

6 For p < 0 the boundary is the top of the atmosphere, -and any radiation incident upon it will be neglected, i.e., I(v,0O,) = 0 for [I < 0 With these assumptions Eq. (2..1.2) may be rewritten I(v,T,y() = B(T,T )e + fg J(vt)e 4 ( ) (2.1.3a.a)'z -(t-T)/4 dt I(v,T.,.) = TO J(vt)e <- ( < ) (2.3b) The flux a.cross any horizontal surface at pressure pis defined by +1 2it F(p) f v f I(fp.) 4 de dp. dv V 4=-.1l Now the specific intensity Iv is independent of the azimuth angle ~, so that the equation may immediately be reduced to a double integra.l +1 F(p) = 2er V I I(v,T,4) p dp dv (2.1.14) To obtain the rate of loss of energy at any point, the flux divergence must be evaluated dF.+1 dI d-= 2rt f f. dp ddv (2.1.5) dp v -1 dp d Substitution of Eq. (2.1.1) enables the flux divergence to be written in the form dF du f1 = 2~r /,. 1 [I(v,T,p)-J(v,T)]k,,dp dv (2.1.6)

7 If the source function J(v,T) is known at every level in the atmosphere, the flux divergence ca.n be found, in theory at lea-st, by substituting the values of I(v,T,1) evaluated from Eqs. (2.1.3a.) and (2..1.3b) into Eq. (2.1.6) and integrating over t and v. In practice this procedure is very complicated, since kv is a. rapidly varying function of frequency, and is also pressure and temperature sensitive. Much effort has been devoted to the investigation of approximations which enable Eq. (2.1.6) to be evaluated while maintaining a. high degree of a.ccura.cy. Equation (2.1.6) gives the flux divergence at a. single level in the atmosphere. In a.tmospheric energy balance studies it is frequently important to know the rate- of loss of energy averaged over a. layer, rather than the flux divergence at a. point. The average value may be found'by evaluating Eq. (2..1.4) at the boundaries of the layer, and subtracting the fluxes. This finite difference method is ma.de difficult by the fact that at low pressures small differences in flux at adjacent levels can result in large cooling rates; it is essential to ensure that the noise introduced by the numerical processes used in the calculations does not mask the true flux difference. In spite of these difficulties the last method is the one employed in this study, with suitable precautions to ensure the suppression of such noise. However, Eq. (2-.1.6) provides a valuable physical insight into the radiative tra.nsfer process and will be referred to frequently.

8 2.2 THE EVALUATION OF THE EQUATIONS OF TRANSFER To evaluate the flux F(p) at the pressure level p, the values of Iv are subsituted from Eqs. (2.1.3a.) and (2o1.3b) F(p) = 2tr fJ 1i[B(vTg)e~(Tg )/ + fg J(v,t) e -(t-)/ ]d +t f [fT JSvot~i~t e -(tT)/kt d)t.+ [ J(vdt) e -]d4 dv (2.2.1) At this point the formulation can be simplified by the introduction of the exponential integral En(x) defined by E 1 e-X/ n-l d n = 0,1,2..., x > O (2.2.2) En(x) foe The exponential integrals have a number of useful properties which are readily proved. In particular the following will be employed dxnx -Enl(x) n = 1,2..., x >0 (2.2.3a) En+l(x) =1 [eX-xEn(x) n = 0,1,., x > 0 (2.2 o3b) F(p) can then be expressed as F(p) = 2t JvB(v, Tg)E3(Tg-T) oJ(v,t )E2( -t)dt+fgJ(vtt)E2( t-T)dtdv The integral with respect to t may be transformed by integration by parts using rela.tion (2t2.3a.), yielding

9 F(p) = 2tfv( [B(v,mTg)_J(vTg) ]Ej(Tg-T) + E3(T)J(VO) + fTg dJ E3( 1t-T 1)dt)dv (2.2.4) 0 dt The expression for the flux divergence may be treated in a similar way. Substituting Eqs. (2.1.3a) and (2.1.3b) into Eq. (2.1.6), and integrating by parts dF = a2 d1 fv k (d gE2E( t-T) dt)-f T_ (-tt- ) tE(T)J(v,))dv (2.2.5) dp p v T d dt o 2 The last two equations involve three integrations, over frequency, over height, and over zenith angle (the latter being expressed as the exponential integral). Each of the integrations presents some difficulty. Before discussing techniques for performing the integrations, it is of interest to examine the problem of calculating atmospheric transmissivities, and the form of the absorption coefficient kv. 2.3 ATMOSPHERIC TRANSMISSIVITIES The calculation of transmissivities along an atmospheric slant path is a complicated problem. Part of the complexity arises because the temperature, pressure and concentration of the absorbing gas may vary along the path. Even in the absence of such variations, i.e., for homogeneous absorption paths, the difficulties are sufficiently great to have merited a large number of both experimental and theoretical investigations during the last twenty years. However, the comparative simplicity of the homogenous path absorption has led to non-homogeneous path absorption being ex

10 pressed in an approximate way in terms of an'equivalent' homogeneous path. Furthermore, homogeneous path theoretical calculations can easily be compared with experimental laboratory absorption measurements. Thus, it is of considerable interest to examine the simpler case in detail. 2.3.1 Homogeneous Path Absorption The results of experimental determinations of absorption are used in all methods of calculating transmissivities. The experimental values may be used directly if measurements have been made over a wide range of pressure, optical mass and temperature, and interpolated for the actual conditions required. However, the method has some serious limitations. Accurate absorption measurements are difficult to make at low pressure and for short paths where the transmissivity is near unity. Few data are available at temperatures below those normally encountered in the laboratory, such as are found in the earth's mesosphere, for example. In addition, very long paths, important for spectral regions where the absorption is weak, are impossible to simulate in the laboratory, frequently leading to extrapolation, rather than interpolation. A typical infrared absorption band consists of a large number of individual absorption lines, many of which may be resolved experimentally under suitable physical conditions. The frequency at the centers of these lines may be measured quite accurately, or calculated theoretically once the values of the relevant parameters have been experimentally determined (Hertzberg 1945). If the intensity of each line can be accurately measured,

11 and the line shape and half-width are adequately known, it is possible to calculate the transmissivity yv at any frequency v by the relation Y = exp(-T) (2.3.1) In general, TV will be equal to the sum of optical thicknesses of the different absorbing gases. Where there is only one absorbing gas TV = kv u (2.3.2) where kv is the absorption coefficient and u the optical mass of the absorption path. kv is the sum of the absorption coefficients of all the spectral lines: N kv iEl ikv (2.3.3) The absorption coefficient of the ith line, center vi is given by ikv = Si bi(vi,v) (2*3.4) where Si is the intensity of the ith line, and bi is a. line shape factor for the ith line, normalized so that fo bi(vi'v) dv = 1(2.35) There are two cla.ssical line shapes which may be regarded a.s limiting cases in the spectral region and under the physical conditions that are considered here. (a) Lorentz line shape, valid when the line broadening is due to

12 molecular collisions of the excited molecules, thus perturbing the energy level. In this case bi(vov) 1= _ (2.3.6).i is the Lorentz half-width (i.e., the half-width at half maximum). (b) Doppler line shape, caused by the Doppler shift due to the thermal motion of the molecules. bi(vo,v) = ko exp(-x2) (2.3.7) where k~ = 1 n2)1/2 V-VO \ 1/2 x = (n 2) CD and AD is the Doppler half-width Natural broadening has a half-width aN defined by where c is the velocity of light in vacuum and @ is the radiative life-time of the excited state In all cases the natural half-width is much smaller than the Doppler half

13 width and, except a.t extremely low pressures, much less than the Lorentz ha.lf-width and can be neglected. The Lorentz ha.lf-width GL is dependent on temperature and pressure; for most ga.ses the relation may be expressed by;aL - 0 ( -E~ S(2.3.8) where aL is the half-width at temperature T and pressure p and Co is the ha.lf-width at temperature To and pressure po. It should be noted that Co does not normally have the same va..lue for all lines in an absorption band. On the other hand, %9 is given by V 1/I kT:n2) CID M where k is the Boltzmann consta.nt and M is the molecular weight. Thus CD varies only slowly from line to line across an absorption band. Over a. wide range of atmospheric pressures it is found that both the Doppler and Lorentz broadening are important. The correct line shape for the mixed Doppler-Lorentz broadening is given by a convolution of the line sha.pes. k 00 -t2 b(vooV)f dt (2.3.9) b(vo )'A -, y2+ (xt)2 where y D

14 The resulting integral cannot be evaluated analytically, but a. number of numerical methods have been developed. The method adopted here is that of Young (.1965), with two modifications (Dra.yson and Young, 1966). (i) The range of Hermite-Gauss quadrature was extended to the region y 0, x > 70 (ii) 4-point Hermite-Gauss quadrature was used over this entire region. Although the Lorentz line shape gives a. good approximation when the broadening process is due to molecular collisions, important deviations have been observed for a number of molecules. For instance, Winters et al., (1964) have shown that beyond about 5 cm 1 from the line center the absorption coefficients for lines in the 4.3 11 CO2 band are less than those predicted by the Lorentz line shape. The deviations are dependent on the b-r.o..-ening gas, being less pronounced for nitrogenbroadened C02 than for pure CO2. Burch et al., (1965) have verified these results and made similar measurements in other CO2 absorption bands, where qualitatively the same phenomena. were observed, but with different magnitude of deviation. These absorption measurements have been made in regions free from strong lines, where the absorption is due to the wings of strong lines situated a. few wa.venumbers a.way. The absorption coefficient under these conditions changes only slowly with frequency, so that the instrumental response function does not appreciably distort the absorption.curve. It is more difficult to examine the shape near the line center, since

15 the best spectrometers have a resolution in the infrared comparable to the Lorentz half-width at about 1 atmosphere. Raising the pressure much higher than this causes the lines to overlap appreciably in most absorption spectra. However, it is not unreasonable to expect deviations closer to the line center. Indeed, Burch et al., (1965)have indirectly shown this to be true for C02 by comparing the absorption of the self-broadened and nitrogen-broadened gas. Burch et al., found that the line shapes of the two samples were different, implying that one at least is nonLorentzian. Investigations using a laser as an essentially monochromatic source may be able to yield information on line half-width and shape. In a recent report, Ray et al., (1966) have advanced the idea. that the non-Lorentzian effects may, in part at least, be due to the geometry of the absorption cell, an effect that would not be present in a. real atmospheric slant path. In this study the Lorentz line shape is assumed to be completely valid. At the present time the deviations are neither sufficiently understood on theoretical grounds, nor adequately measured experimentally, particularly at low concentrations such as the carbon dioxide mixing ratio in the earth's atmosphere, to warrant modification of the line shape. This is not a. limitation of the methods developed and described here, since any line shape can easily be incorporated into the calculations. In most cases the effect of deviations from the Lorentz line shape on the radiative cooling rates is probably small.

.16 It will now be assumed that the line shapes and half-widths are adequately known, and that the line intensities have also been precisely determined. Using Eq. (2.3.1), it is possible to calculate the transmissivity Yv, at any desired frequency v. In practice, however, it is usually desirable to calculate the average value of yv over some finite frequency interval Av. k - Av Av dv (2.3.10) The interval Av may contain a large number of absorption lines, and there may be many more lines outside the interval which contribute to the absorption within the interval. Thus yV may be a rapidly fluctuating function of v whose value is a complicated function dependent on the position, intensity and line shape of many individual absorption lines. To overcome some of these difficulties the concept of the band model was introduced. Band models approximate the actual distribution of line positions and intensities in such a way that a solution of integral (2.3.10) may be more easily obtained, the solution often being expressed in terms of analytic of semi-analytic functions. The simplest case, a single isolated line with the Lorentz shape, was solved by Ladenberg and Reiche (1913). The result can be extended to any number of lines, provided that they do not overlap. The Regular or Elsasser band (Elsasser, 1938) applies to an infinite array of Lorentzian lines of equal half-width and intensity, equally spaced. It simulates quite well the conditions encountered in the P and R branches of certain

17 bands e.g., the fundamental in the 15 k CO2 bands. This model has been generalized to include a. random superposition of different Elsa.sser bands in the Random Elsasser model. The Elsa.sser band and its derivatives are useful for some molecular absorption ba.nds, but for others such as water vapor and ozone there is little suggestion of regularity. The lack of regularity prompted the development of the Statistical. or Random model., suggested independently by Goody (1952) and Mayer (1947). The distribution and intensity of the lines is defined by probability functions which may be varied to suit the a.ctua.l distributions In addition to these important band models, numerous other models a.re scattered throughout the radiative tra.nsfer literature, most being,modifications or refinements of those described aboveo Band models have played a.n extremely important role in understanding radiative transfer processes in the a.tmosphere, transforming what would otherwise have been a problem of impossible complexity into one of manageable proportions. But it is important to realize that band models do have limitations and that alternative procedures are now available. The limitations include the following: (i) The spectra-1 resolution of the transmittances is finite: for the Elsasser model it is a. multiple of the line spa cings, while the sta.tistica.l model must use an averaging interval sufficiently large to ensure that the true distribution of the lines is adequately simulated by the statistical distribution. The finite resolution introduces further compli

cations for atmospheric slant paths. (ii) Many band models do not a.llow for accurate contributions of the wings of.lines lying outside the spectral. region under consideration. (iii) The actual distribution of lines can only be approximated by the band models. In some regions, eog., near Q-branches, it is extremely difficult, if not impossib.le, to accurately a.ccount for the complex distribution of lines. The development of the Qu.as i-Statistica..l or Qua.siRandom model] by Kaplan (1953) and Wyatt et al.. (.1962) ha.s done much to overcome these objectionso The resolution of the transmittances can be made arbit:arily small, the wings are accurately a.llowed for and the actual distribution of lines can be approached for sufficiently sma..l.l averaging interval. It shares one problem with the other models. (iv) When the models permit accurate calculation of the absorption due to a, real absorption band, the complexity and length of ca.lculation is considerable, even for the Lorentz line shape. For other line sha.pes including the mixed Doppler-Lorentz line shape, the complexities a.re even greater. Extensive tab.les using the Quasi-Random model ha.ve been prepared (Stull et al., 1963). In an effort to reduce computing time an averaging interval of a = cm'1 wa.s used.; this violates a basic requirement of the model, that should be sufficiently sma..l.l to allow the lines within each binterval to be assumed randomly distributed. In the 15 1P CO2 bands this violation resulted in the Q-branches being considerabLy enhanced, overestimating the total absorption appreciably (Drayson 1964).o

19 Modern computing techniques have made possible an alternative simpler approach, using direct numerical integration of Eq. (2.3.1). At each frequency quadrature point the contribution from all lines is evaluated accurately. The numerical accuracy of the calculations can be increased by making the quadrature intervals sufficiently small, especially in the neighborhood of line centers. Five significant figures are easily achieved. Computation time can be reduced by careful programing and the use of interpolative procedures for more distant lines (Drayson & Young 1966). This method of direct integration was first employed over a limited range in the 9.6 vt ozone bands by Hitchfeld and Houghton, (1961) and has since been employed by a number of authors in studies of several molecular absorption bands. The advantages of these methods are: (i) The resolution is not limited; the transmissivities may be weighted by an instrumental response function and compared with high resolution experimental spectra. (ii) There is considerably more flexibility in the calculations, especially when extended to non-homogeneous paths as in Section (2.3.2). (iii) The actual distribution of line positions and intensities is used and all contributions from the wings of distant lines are accurately included. Comparisons between theoretical calculations using direct integration techniques and experimental laboratory data do not always show good agreement. Indeed, the application of these techniques has shown that previously accepted values of band parameters are considerably in error. Part of the

20 discrepancy is due to experimental error, but this is probably much smaller than the error introduced in attempting to deduce the parameters from the measurements. Measurements on the same band by different experimenters produce widely differing estimates, often with mutually inconsistent error estimates, as in for example, Drayson and Young (1966). There is a clear need for accurate, well-designed experiments, coupled with a careful analysis to yield the desired band parameters. 2.3.2 Atmospheric Slant Path Absorption Calculations of absorption over atmospheric slant paths are more complicated than the case discussed above, because of variations in the pressure, temperature and absorber concentration. Eq. (2.3.2) must now be generalized to the form: Tv = J kv(2 the integral being taken along the absorption path. Variations in the three parameters affect the integral in the following ways: (a) The Lorentz half-width varies directly with pressure and inversely with the square root of temperature. It is also a. function of absorber concentration. (b) The Doppler half-width varies directly with the square root of temperature. (c) The line intensities are complicated functions of temperature. Typically the intensities of very strong lines decrease slowly with increasing

2.1 temperature. Lines belonging to wea.k bands increase in intensity very rapidly with increasing tempera.ture as much as an order of magnitude for a 25~K temperature rise, which has prompted the use of the term'hot band' to describe them, Generally, the practical problem of calculating slant path absorption ha.s been restricted to average values of the tra.nsmissivity -, defined in Eq. (2.3510). Of the several methods suggested, the one of most practical importance is the Curtis-Godson approximation (Curtis,.1952 and Godson 1953). This method seeks to find a mean pressure p, temperature T and a optical mass u such that the transmission is approximated by a. homogeneous absorption path with these three physical parameters. The expressions for p, T and u are required to satisfy the criterion that the limiting cases of weak line and strong line absorption shall be accurately represented. The expressions for U and T a.re simplest when the temperature is constant along the absorption path, and has been most thoroughly investigated to test its a.ccura.cy. Under these conditions Su P du udu u = du (2.3.12b) The errors in the approximation have been investigated for a. single isolated line for the case of a. uniformly mixed gas such as carbon dioxide in the earth's atmosphere, by Kaplan (1959) and extended to more general

22 pressure limits by Drayson (1966). The approximation was found to be quite accurate -for most a.pplications, although errors of up to about 6% can arise for certain combinations of pressure and line intensity. Simi.la.rly, Mayot and Vigroux (1965) found that the Curtis-Godson approximation was not a serious source of error in interpreting observations in the 9.6 p ozone bands. In ra.diative transfer calculations the computation of tra.nsmissivities is only an incidental procedure, and a.n approximation should be regarded a.s valid only if it produces accurate va.lues of flux divergence. This much more difficult task was attacked by Wa.lshaw and Rodgers (1963), who calculated the effect of the Curtis-Godson approximation on the accuracy of radiative heating-rate calculations. They considered radia.tive transfer in the earth's atmosphere in the 9.6 p. ozone bands, the 15 p. carbon dioxide bands and the rotational water vapor band. The Lorentz line shape only was used. Their conclusions were' (a.) For the 15 4 C02 bands the approximation is extremely accurate. (b) In the rotational water vapor bands, the result for the band a.s a whole is acceptable, although the errors are sometimes larger over sma.ll portions of the band. (c) The approximation is not valid in the 9.6 1. 03 bands. The failure was attributed to the fact that the ozone mixing ratio in the earth's atmosphere increases with decreasing pressures, a Condition for which Curtis (1952) found the approximation to be less aceurate.

23 To simplify their calculations Walshaw and Rodgers did not consider the complicated effect of variations of temperature on the transmissivity. It is interesting to examine the influence of such variations in the earth's atmosphere on a C02 line whose intensity increases rapidly with increasing temperature. At the stratopause, where the temperature is a maximum, it will be quite strong compared to the tropopause or mesopuase regions. Ca.rbon dioxide is fairly uniformly mixed up to at least 70 km, but since the transmissivity is a function of the product of the optical mass and the line intensities, it can be seen that the same effect on the transmissivity could be obtained by assuming that the line intensity remained constant and that the C02 concentration increases rapidly at the stratopause, i.e., a situation similar to that of ozone where the higher concentration is at the lower pressure, a circumstance which may lead to considerable error in the Curtis-Godson approximation. Moreover, because of the la.rge number of moderately weak tempera.ture-sensitive lines, they exert considerable cooling influence at the stratopause level and must be accurately calculated. The Curtis-Godson approximation is designed to give an estimate of the average transmission over a. finite frequency interval. At pressures where the Lorentz line shape is valid for homogeneous absorption paths, the Curtis-Godson approximation still gives a Lorentz line shape for the slant path, whereas the true line sha.pe has more absorption at the line center and less in the wings, tending to a common value in the far wings (Goody, 1964).

The methods developed in this paper are designed to be a.pplied to all gases of atmospheric interest, regardless of their distribution and temperature dependence of line intensity. Fortunately, the direct integration technique provides the flexibility to avoid some of the difficulties of the Curtis-Godson approximation, which are not easy to apply to band models or empirical data. The solution lies in dividing the atmosphere into horizontal layers over which variations in absorber concentrations and in temperature are small. Two alternatives are then possible: (a) The Curtis-Godson approximation may then be applied over each of these layers to obtain values of T, iU, T. This method wa.s used by Gates et al., (1963), and Drayson and Young (1966). (b) By assuming a value of T in each layer the absorption coefficient may be integrated with respect to pressure, giving a slightly more accurate solution than method a. For carbon dioxide, whose mixing ratio is assumed constant, the solution has been given by Drayson (1966). However, this variation is somewhat cumbersome and offers little advantage over the first method. To obtain the optical thickness for a path between two arbitrary points in the atmosphere, the sum of the optical thicknesses of the intervening layers is taken. It can easily be seen that this method is equivalent or nearly equivalent in case (b) to approximating the slant path by a series of homogeneous paths; the limiting case where the number of homogeneous paths is reduced to one is the Curtis-Godson approximation in its original form. The method of direct integration with respect to frequency

25 has previously been applied to the calculation of fluxes and flux divergences e.g., Hitchfeld and Houghton (1961), but the full potential of the method has not been exploited. In previous studies integration with respect to frequency wa.s used to compute average transmissivities. Here it is used to obtain fluxes and flux divergences directly. The expressions for flux (Eq. (2.2.4)) and flux divergence (Eq. (2.2.5)) do not explicitly contain the transmission function, but rather the exponential integral of the optical thickness. It will be recalled that the exponential integrals were obtained by integration of the transmission function with respect to angle and for this reason the exponential integral is frequently called the diffuse transmission function. One way to evaluate the integral is to use numerical quadrature. In this case the exponential integral En(T) is approximated by an expression of the form En(T) a.e i (2.5.13) N - Z ai /(biT) i=l where a.i and b. (i = 1,...,N) are constants. The advantage of this form is that the diffuse transmission function is expressed in terms of normal transmission functions, and since the source function J is usually a slowly varying function of frequency, and can be considered a.s effectively constant over a small frequency interval Av, the expressions for F(p) and dF/dp ma.y be readily integrated with respect to frequency and involve only

26 the average transmissivity y over the frequency interval Av. The most widely employed approximation for E3(T) is 1 E3(T) - 2 exp(-kT) (2.3.14) which is a special case of Eq. (2.3.13) with N = 1. The use of this form is discussed byC&ody (1964). Strictly speaking k is not a constant, but varies as T varies from 0 to oo. To maintain accuracy for small values of Ty k should be equal to 2.0, but this value is not a good approximation for larger values of T. Usually the diffusivity factor k is taken to be 1.66, a value Elsasser (1942) obtained for the strong line approximation of an Elsasser band. Suppose that an approximation of this form is used in Eq. (2.2.4) to evaluate the flux. The flux divergence is obtained by differentiating Eq. (2.2.5). From Eqs. (2.3.14) and (2.2.3a) we have E2(T) = dt - k/2 exp(-kT) As stated above, k = 2 must be used when T = 0 and this also gives a poor approximation for E2(T). In a later section an expression for F(p) which involves E4(T) will be derived. This leads to an expression E4(T) i_ exp(-kT) 4 2k Since E4(0) = 1/3, it is impossible simultaneously to obtain a good approximation for E2(T), E(T), E4(T), even at x = 0.

27 Despite these inaccuracies, the diffusivity factor of 1.66 has been almost universa.lly a.pplied. It is c.laimed that the errors in flux divergence are small, although it has not really been thoroughly tested under adverse conditions. Probably errors from one part of the absorption band are compensa.ted by cance.l.lation from. other parts of the band, reducing the inaccuracies to acceptable levels. In most physical problems a.n approximation as crude as Eqo (2.5..14) is not used. There are however two rea.sons for its use. (a) The exponential integral is not as easy to evaluate a.s the exponential function. This was particularly true before the development of electronic computers. (b) The more important reason is that evaluation of the exponential integral precludes the use of band models. The diffusivity fa.ctor a.pproximation used in conjunction with the Curtis-Godson approximation reduces the complex problem of diffuse slant path transmission to a. much simpler'equivalent' homogeneous path over a. finite frequency interval, where empirical data. or band model calculations may'be used. In the calculations made in -this study, direct integration with respect to frequency is employed, so that the reasoning in b does not apply. A number of methods of evaluating the exponential integrals were examined, but.none wa.s found suitable for use in the computer progra.ms of flux divergence because of their slowness or inaccuracy. A series of subroutines utsing polynomial approxima.tions was deve.loped and is fully described in the appendix. The avera.ge execution time of these subroutines was less

28 than that of the computing system exponential function subroutine. The development of fast subroutines is critical since a large proportion of - the execution time of the programs is spent in these subroutines. 2 4 NUMERICAL PROQEDURES FOR EVALUATING RADIATIVE FLUX For the purposes of this section it will be assumed that the problem of calculating the diffuse transmission function has been overcome and that the source function J(V,p) is known at every point in the atmosphere~ The flux F(p) given by Eqc (2.2a 4) can be evaluated numerically in a straightforwa.rd manner, using standard quadrature techniques. However, because of the approximate nature of quadrature evaluations, the calculated flux differs from the true flux by some sma.ll error e(p). The flux difference between two adjacent pressure levels may be found provided it is not of order c.(p). The quantity c(p) can be reduced by careful quadrature procedures, but will a.ways be present because of round-off and other inaccuracies introduced in computation, i.e., the well-known difficulties of numerical differentiation are encountered. 2-o40l The Isothermal Layer Approximation A frequently.used method of reducing the numerical noise is to a.ssume that the atmosphere is divided into isothermal layers or, more precisely, layers in which the source function is constant~ Equation (2~2.4) is no.long valid since its derivation depends on the continuity of the source function a.s a function of pressure. Instead Eq. (2.2ol) may be integrated to give

29 N =2it f g)E3N i J. [E2f IT -E3 ITv -T I) ( dv V 3(TN-T' ) j=lJ ij j-1 i = O,1,...N (24.4.I where the atmosphere has been divided into N layers and the subscript j refers to the value of the variables at the level j. The isothermal assumption is, of course, a gross distortion of the temperature profile expected in a planetary atmosphere and since radiative transfer processes tend to smooth temperature profiles, such discontinuities in the temperature field could not exist. The use of the isothermal assumption can be justified only if it gives realistic values of the average flux divergence in a layer. Consider, as an example, a part of an absorption band where the optical thickness is very large, i.e., the atmosphere becomes opaque within a very short distance. In this case the net flux is small and hence the true flux divergence is small also and independent of the temperature profile. This holds even if the temperature profile is similar to that in Fig. 1. If the isothermal slab approximation is used, the flux can easily be evaluated The assumption that the optical thickness is large implies _E1 i-T2 = j = o i # i Using this relation it is readily seen that Fi = S Iv (Ji+l-Ji)dv

3o0 LEVEL JI-2 JI-' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - i JJI HEIGHT I I 1+1 JI+2.. II2 TEMPERATURE Fig. 1. Diagram of the source function as a function of height. i.e., the flux at any level is proportional to Ji+l - JI. Thus I-2 I-1 I-2 FI O JI+l - JI > F I +2 -J > 0 (2.6.2) 1+1 1+2 I+> The heating at level i is proportional to AFi = Fi-Fi_1. Using these equations I= 0 AFI-1 AFJI JiJi-=1 I+1-JI > 0

31 The model correctly predicts no heating in level I-i, but produces a strong net heating in level I (the first layer in the isothermal region). The sign of the flux divergence in the I+lth depends on the relative sizes of JI, JI+l and JI+l By making layer I+1 small compared to layer I+2, the difference (JI+2-JI+1) can be made larger than (JI+l-Jj), producing a net heating in layer I+1. If however layer I+1 is small compared with layer I+2, the opposite effect can be produced and a net cooling is obtained. This means that the isothermal layer assumption has two serious flaws. (i) It tends to produce a greater amount of heating or cooling at the bottom of an isothermal layer in the atmosphere below which the temperature increases or decreases. This effect also occurs when the temperature gradient changes, not necessarily to an isothermal condition. (ii) The sign of the flux divergence can be altered depending on the thickness of the slabs chosen. It should be noted that even if the atmosphere is not optically thick in the region of the spectrum under consideration, these two errors still have a tendency to occur, although their effect is not so extreme. There are many regions of absorption spectra where the atmosphere is optically thick, at least near the line centers; an absorber in which this never happens has to be weak, and hence has to have little influence on the radiative transfer. Clearly the isothermal layer approximation must not be used to study radiative transfer in important regions where the atmospheric temperature reaches a. maximum or minimum (the tropopause, stratopause and mesopause),

32 and probably should not be used in other regions as well. A more realistic approach must be used, one which takes into account more accurately the structure of the atmosphere in the neighborhood of the point being investigated. 2.4.2 Polynomial Representation of the Source Function Since the simplest approximation, that of isothermal layers, leads to serious errors in calculated flux divergences, a more sophisticated technique must be used. Perhaps the most obvious method is to express the source function at points within the individual layers as a polynomial in pressure or height or some other vertical coordinate. The isothermal layer approximatfon is really a limiting case, in which the degree of the polynomial is zero. Suppose that the atmosphere has been divided into N horizontal planeparallel layers, with the i-th layer bounded by pressure levels Pi-l' and Pi (i = 1,2...,N) with 0 = Po < Pi <'' < Pi-l < Pi... < PN-1 < PN = Pg It will also be assumed that these levels have been chosen so that the atmosphere may, for the purpose of calculating atmospheric absorption, be approximated by homogeneous paths within the layers, as discussed in Section 2.3.2. For convenience, it will be assumed that B(v,Tg) = J(v,Tg). From Eq. (2.2.4) the expression for the net flux Fm at level m may be written.

33 N T.i Fm 2= I f (E3(Tm)J(v O) + f dE 3( ItTmI)dt}dv mi _l dt (m =0,1,...,N) (2.43) The assumption that the layers are homogeneous implies that within each layer t is a linear function of pressure, and that dJ/dt is a polynomial in p. Thus, each integral in the sum in Eq. (2.4.3) may be written T-. p n-l fl dJ E ( It-Tml)dt f i a pJE(b.+c.p) dp Ti-1 d 53( Pil =Ot 3 JJ 0 (2.4.4) where aj,bj,c. are constants within the level i, and the source function is expressed as a polynomial of degree n in p, within level i. Equation (2.4.4) can be evaluated using the relation (see appendix) x2 k ik. i xkE (x) dx = (k)' Xk-j n1 j j= ( k-j) 1 n+l+j 1 2 n+l+j (x (2.4.5) This method is very similar to that proposed by Curtis (1956), in which the source function is expressed in terms of a polynomial of degree 4, whose coefficients are chosen to give the source function exactly at five adjacent levels. This representation is used only in the neighborhood of the point at which the flux divergence is being calculated, a lower order polynomial being used elsewhere. The higher order polynomial wa.s used in an attempt to accurately include the contributions from nearby layers, which most influence the cooling rate in many parts of the atmosphere. This method was also used by Rodgers and Walshaw (1966).

The results of fitting such polynomials are shown in Fig. 2. The solid lines represent profiles typical of those encountered in standard or mean atmospheric profiles, while the dashed lines indicate the polynomial temperature distributions. Clearly the polynomials do not accurately represent the true profiles and may not give correct contributions to the cooling rate from adjacent layers. The polynomial frequently gives a. smoother temperature profile, particularly where the temperature gradient is discontinuous and this property is sometimes desirable; the important modifying effect of such smoothing will be shown in the next chapter. Since a new polynomial is used for each point at which the cooling rate is calculated, no two points have, in general, the same temperature profile associated with them. A further disadvantage of the higher order polynomials, at least for the method of calculation described here, is the difficulty in evaluating the integrals of the type in Eq. (2.4.5). Not only are they time consuming, but accuracy is difficult to maintain for some values of x; and x2 due to roundoff errors and inaccuracies in calculating the exponential integrals. The method applied here is to assume the source function is linear with pressure between adjacent pressure layers, and to simulate rapid temperature fluxuations by decreasing the thickness of the layers. The linear assumption enables Eq. (2.4.3) to be evaluated simply. N Aji Fm =2it fv(E5(Tm)J&,,O~) m iI ) E4( [Tm - i T-T (2.4.6)

(a) (b) TEMPERATURE TEMPERATURE / PRESSURE', PRESSURE 8 I I (C) (d) TEMPERATURE TEMPERATURE Fig. 2. Typical polynomial fits to vertical temperature structures. polynomial fitting temperature structure.

where AJi = Ji -Ji-1 and ATi = Ti-T_1 In many molecular absorption bands, the source function Ji at level i is a. slowly varying function of frequency, and may be considered constant over the band, or over some frequency subinterval of the band. For such a frequency interval Eq. (2.4.6) may be written N Fm- 2it[J(vO)[fvE3 (Tm)dv] + i] Z AJi[f A IE4( ITm —Til)-E4( IT T I ) I dv i.e., the flux may be expressed as a linear combination of the source functions at all points in the atmosphere (c.f., Curtis, 1956). It should be noted that the coefficients in the linear sum are functions of tempera.ture and strictly speaking should be used only for the atmospheric tempera.ture structure for which they are calculated. In some spectral regions the coefficients are only weakly temperature dependent and may be used to calculate fluxes for other temperature structures, provided they are not radically different from the original. As previously indicated, it is more difficult to compute flux divergences than fluxes, particularly in regions of the upper atmosphere where small flux differences cause large heating rates. For this reason

57 it is frequently advantageous to compute the divergence directly; Using Eq. (2.4.6) the flux difference at adjacent levels may be written AFm = Fm-Fm-l 2Jl, ([E((Tm) -E3( Tm_) ]JO m-l AJ. i=l+ E i [(E4(Tm-Ti)-E21(TiL-,T -(E(Tm-)-ET -T N AJi A [(E4(Ti -Tm)-E4(i-Tm) )-(E4(T i_-T -E4 (T i- ))]ldv i=m+l A-F. 4 i -l-in 4 m-1 (2.4.7) Suppose that ATi is small, for some i, and consider the errors which may arise in evaluating the terms in the sums in this equation. These terms involve differences of exponential integrals whose arguments differ by ATi. If AT. is of the same order as the accuracy with which E4 can be calculated, then the accuracy of calculating the difference will be relatively low. When the difference is divided by ATi the absolute error in calculating the term becomes large, and test calculations showed that these errors often led to meaningless results. Fortunately, it is not difficult to evaluate such terms with a much higher accuracy. The method is suggested by the mean value theorem [F(x+5x)-F(x)] = F'(x+%) for some 5 in the range 0 < ~ < bx

38 Consider the errors involved in approximating F' (x+:) by F' (x+2 tx). Expanding F(x) and F(x+bx) in a. Taylor series F(x+ox) = F(x+) + (3x-)F' (x+x) T)+ F (x+ ) + -(-) F'" (x3 X)+O(6x)4 F(x) = F(x) (5X)F' (xx) -+ 2 - ()F' ( x+x ) () Subtracting F(x+Fjx) - F(x) = 5xF'(x+6x) + 2(x)3F"'(x+ ox) + O(3x) Neglecting the terms of O(Fix)5 1 [(xt+x)-F(x) F' (x+56x) + 1 (x)2F,,, (x2 x) (2.4.8) 3x 42 -27 2 Applying this to the function E4(x) L [E4(x-' l +(x)2 E 1 -[E4(x)_E 4(x+ Sx)] = E4(x+-Sx) +- 3 E1(x+x) The values of bx for which the approximation E3(x + 2Sx) should be used depends on the accuracy to which E4(x) can be calculated. The subroutines developed in the appendix give an absolute accuracy of approximately 1 x 10O. However, since adjacent values of the a.rgume.nt tend to have similar errors, the absolute error of the difference [E4(x) - E4(x+6x)] is usually somewhat smaller. But for some x and x + 5x, particularly those that are computed from different polynomials, the error in the difference may be slightly more than 107 It will therefore be assumed that an error of 107 is a representative value. The errors from the two methods

39 will become equal when lo_? (,Ex)2 S2x )24 El(x) Except for very small values of x, El(x) is of order unity, so -2 5x - 10 Using the approximation E5(x + 1 5x) for Fx < 10-2, the order of the 2 error in computing!x [E4(x) - E4(x + 6x)] is 10-J, an acceptable value for the calculation of the flux divergences. Using this approximation, the terms in the first sum of Eq. (2.4.7) be come i [E (mT i-l/2) - E (Tm —i-1/2}) where Ti 1/2 is defined as ( Ti+i-1)}/2. Further error can arise when ATm is small, and can be treated in exactly the same way, reducing the term above to a still simpler form -AJi ATm E2(Tm 1/2 - Til1/2 Inspection of Eq. (2.4.7) shows that the contribution from layer i to the flux differences in level m is the same, apart from a difference in sign and the multiplying factor AJi/ATi, as the contribution from layer m to the flux differences in layer i. This duality is of great importance, since the differences of the exponentials need be evaluated only once for

each i and m, which halves the execution time of this important part of the computer program. It is also clear that the first term in Eq. (2.4.7) may, for small ATmY be approximated by E(Tm) - E3(Tm1) ATm E2(Tm) If the flux divergence at a point is required, the polynomial or linear approximation for the source function J may be substituted into Eq. (2.2.5), and integration with respect to t carried out in a similar manner. Assuming that the atmosphere has been divided into horizontal layers in which the source function is a. linear function of pressure, it is of interest to know how the flux divergence varies at points within these layers. Test calculations were made for layers as thin as 1 km. in the mesosphere, for the 15 4 CO2 bands. It was found that the absolute value of the flux divergence was greatest near the boundaries of the layers and lea.st near the center of the layers. The reason for this behavior is that at interior points the source function gradient is constant in the immediate neighborhood, whereas at the boundary it is usually discontinuous. The same problem undoubtably arises when a higher order polynomial is used; Fig. 2 shows that the source function gradient is not accurately reproduced. For example, in profile (b) the polynomial gives a local temperature maximum, which will tend to overestimate the cooling rate at that point. However, for temperature profiles which are smoother, the poly

nomia.l may be more realistic. Nevertheless, if the flux divergence is calculated at a point, rather than an integrated value over a, finite pressure layer, two important points must be taken into consideration. (i) The method of calculation may distort the local temperature field and lead to inaccuracies. (ii) The flux divergence at this point may not be representative.of the value at neighboring points, and cannot in general be used as the average value in a finite pressure layer. In the present study the thickness of layers was adjusted so that when a layer was halved and the flux differences recalculated the same values were obtained from the sum of the two sublayers, to within some acceptable error limits. The thickness of the layers depends on the a.ccura.cy required and on the characteristics of the spectral region under consideration. In the earth's mesosphere an accuracy of better than.1% can be obtained in the 15 4 CO2 bands with layers several km thick. The integrations with respect to frequency of equations such a.s Eq. (2.4.7) can be carried out in much the same way as those developed for calculating transmissivity (Drayson & Young 1966). The frequency interval of integration is divided into many subinterva.ls, whose length varies, being smallest near line centers and largest away from lines, and 4-point Gaussian quadrature is applied. over the intervals. 2.5 CONCLUSIONS In deriving the equations of radiative transfer and manipulating

42 them into a form suitable for numerical evaluation of flux divergences, a. concerted effort ha.s beet made to use a.s few approximations a.s possible. As a result several approximations, almost universally employed, have been avoided. This is not meant to imply that these approximations are valueless, but that under some circumstances they may lead to errors and must be used with caution. The more exact methods can provide a. means to check the approximations and define some conditions under which they can be employed. Flux divergences a.re obtained by direct integration with respect to frequency. The assumption that the source function is linea.r in pressure within the atmospheric layer is shown to have advantages over the use of isothermal layers and higher order polynomials, and also allows the vertical integration to be accomplished simply, without the use of a diffusivity approximation for angular integration. The disadvantage of the present method is that the formulation is rather complicated, and that quick simple solutions cannot be obtained. A considerable amount of computing time is required to solve any realistic problem. However, with computers of increasing speed, the time problem should steadily decrease. Apart from checking other methods, calculations of this type can be justified only if they result in considerable improvement over faster, simpler techniques. The next two chapters are devoted to two applications, with emphasis on comparison with other calculations.

CHAPTER 3 APPLICATION TO THE EARTH'S MESOSPHERE AND MESOPAUSE 3 1 GENERAL SURVEY OF LONG-WAVE RADIATIVE TRANSFER IN THE EARTH'S ATMOSPHERE In the earth' s atmosphere the important gases for long-wa.ve radiative transfer are minor constituents (with the possible exception of atomic oxygen in the thermosphere), comprising much less than 1% of its mass. Of these, the most influential a.re, (a) Carbon dioxide. CO2 is a linear symmetric molecule which has no-pure rotational spectrum. The important bands for infrared radiative transfer a.re a.s fol.lows. (i) The 15 t bands, consisting of the v2 fundamental and overtone bands. In addition to the normal molecule (12C 160 160), the isotopic species 13C 160 160 12c 160 180 12C 160 170 a.nd 13C 160 180 are abundant enough to exert some influence. The la-st three a.re asymmetric. (ii) The v3 fundamental at 4.3 u. Although the v3 band intensity is much greater than the total intensity of the bands near 15 t, it plays only a minor role in the earth's atmosphere because the thermal energy is low near 4.3 V at atmospheric temperatures. (b)i Water vapor is a nonlinear molecule with two important atmospheric absorption ba.nds.

44 (i) The pure rotational band, with lines distributed from the micro-wave region to 20.t. Between 10 and 20. the absorption is due partly to weak absorption lines in the interval and partly due to the wings of distant lines (ii) The v2 band near 6.3.a. Again the wings of lines in this band gives considerable absorption a.t least out to 10 p.o'Lines of isotopic molecules are again important. (c) Ozone is also an asymmetric molecule with complicated band structures. The two important absorption regions are: (i) The 9.6 p bands, which consist of two ba.nds, the vl and V3 fundamentals, centered at 1110 cm-l and.1043 cm-1 respectivelyo (ii) A weaker 14 p band, identified as the v2 band. For each molecule there a.re many more molecular absorption bands in the infrared but these are either too weak or occur at frequencies which are too high to influence the long-wave radiative transfer. Simila.rly, some of the even less abundant atmospheric constituents e.g., CH4, N20 have strong absorption bands, but their mixing ratio in the atmosphere is so low that they do not contribute significantly to atmospheric radia.tive transfer. To apply the methods developed in the previous chapter it is essential to know the frequency, intensity and Lorentz ha.lf-width of the absorption lineso As indicated eariler, a sound knowledge of the band structure is

45 often lacking, even for the common atmospheric absorbing molecules. The state of knowledge of these bands outlined above will now be examined. (a.) Carbon dioxide. The linearity of this molecule makes it comparitively easy to handle theoretically and the line positions and relative intensities of the individual bands can be readily calculated. (i) The 15 ~i bands. These important bands have been intensively studied, both theoretically and experimentally. Although the structure of the individual bands is simple, their overlapping makes it difficult to determine the relative intensities of the bands, and even the total band intensity. Knowledge of variation of Lorentz half-width with rotational quantum number is incomplete, particularly for air or nitrogen broadening. Despite these difficulties theoretical calculations show good agreement with laboratory spectra (Drayson et al., 1967) and atmospheric radiances (Chaney et al., 1967) (ii) The 4.3 ki bands. The somewhat simpler structure of these bands make this region easier to investigate, and satisfactory agreement between theoretical and experimental data exists (Gray, 1965). Larger errors can be tolerated in this spectral region, since the thermal energy is low at atmospheric temperatures. (bj) Water vapor is a non-linear molecule which has been investigated intensively in recent years.

(i) The pure rotational band. Line positions, intensities and Lorentz half-widths have been calculated by Benedict and Kaplan. Qualitative agreement with experimental data is good, although the amount of absorption is not always identical. The use of modified line shapes, in particular the Van Y1e~gknmWeisskopf line shape, may lead to improvement, particularly in the region between 10 and 20 i. (ii) The 6.3 [i band. This band has received comparatively little theoretical investigation and results are inconclusive at the present time. (c) The theoretical and experimental difficulties of determing water vapor absorption apply even more to ozone. The difficulties and dangers of generating the gas have detered most investigators, and for many years some of the bands were incorrectly identified. Quantitative measurements have been made by Walshaw (1957) and by McCa a & Shaw (1967). (i) The 9.6 V bands. Line positions and intensities in this region have been calculated by Kaplan (1956) and by Clough and Kneizys (1965); unfortunately neither gives agreement with experimental spectra. More work is required on this extremely difficult molecule. (ii) The 14 4 band has been measured quantitatively for the first time by McCaa and Shaw (1967). No theoretical work has been published.

Thus, at the present time the 15 k bands of carbon dioxide are best understood. The distribution of carbon dioxide with altitude has been measured frequently and the results show that the mixing ratio is approximately constant, except close to the surface. (Bolin and Keeling 1963). These. are small variations with latitude and nearness to land masses, as well as a slow increase with time. This presumably holds true up to the altitude where carbon dioxide begins to dissociate at about 100 km, although no measurements are available to support this supposition. In the stratosphere, mesosphere and mesopause regions, carbon dioxide is the principal absorber of lone-wave radiation; in the troposphere its relative role gradually decreases and is dwarfed by water vapor near the surface (M6ller, 1963). The distribution of water vapor is extremely variable and is not known with any certainty above the tropopause (Newell, 1967). Whatever its distribution, it is very important in the troposphere, particularly near the surface, and less important at higher altitudes. The influence of the 6.3 ct water vapor band is small compared to that of the rotational band (Kuhn, 1966). The situation as regards ozone is discouraging. The atmospheric absorption in the 14 4 band is weak and is masked by the strong absorption of the 15 Ct CO2 bands. The 9.6 4 atmospheric absorption is moderately strong and falls in an otherwise largely transparent region of the spectrum. Calculations by Plass (1956a) and Kuhn (1966) indicate that it provides a~n important cooling mechanism in the upper stratosphere,

falling off quite rapidly above and below this level. In the remaining part of this chapter the cooling due to the 15 t bands carbon dioxide in the mesosphere, mesopause and lower thermosphere will be investigated. It is clear from the preceding discussions that this is the dominant band at these altitudes, as well a.s being the one with the best understood structure. In addition the concentration of CO2 is better known than 03 or water vapor. Moreover, recent results (Kuhn 1966, Kondrat'yev et al1., 1966) have indicated that CO2 is responsible for some atmospheric heating at the mesopause level and the calculations to be described were designed to confirm or reject these results. A point of further interest lies in the evaluation of a source function in the mesopause a.nd lower thermosphere, where significant deviations from the Planck black body function occur. 3.2 EQUATIONS FOR THE SOURCE FUNCTIONS IN THE UPPER ATMOSPHERE For a gas that is in thermodynamic equilibrium it is well known that the source function is given by the Planck black-body function B(v,T) = 2hv3/c2[exp (hv/kT)-l] (2.2.1) where h is the Planck constant k is the Boltzmann constant c is the velocity of light in vacuum and

T is the absolute temperature In the atmosphere, where radiative heating or cooling is taking place, strict thermodynamic equilibrium is not mainta.ineda.nd the source function is not equal to the Pl.anck function. However, for much of the earth's atmosphere the deviation from the Planck function is slight, and the fluxes may be calculated from the equations of the previous cha.pter, assuming.equality between the two functions. Nea.r the mesopause, however, for the 15 Ci CO2 bands the differences become larger and the errors induced by assuming equa.lity'become serious (Curtis and Goody 1956). The explanation for the breakdown is as follows: for a C02 molecule in thermodynamic equilibrium the vibra.tional energy is distributed over the vibrationa.-l states according to a. Boltzmann distribution. In the non-equil.ibrium ca.se, the Boltzmann distribution is disturbed by the gain or loss of radiant energyo If molecular collisions a.re sufficiently frequent,the vibrational energy regains its Boltzmann distribution by loss or gain from translational energyo However, there is a,. possibility tha.t an excited molecule ma-y re-emit its energy before the- equilibrium process can be completed. These two mechanisms are chara.cterised by two time consta.nts, the former being the re.laxa.tion- time.. the latter the -radiative life-time 0,o The same is true for the -rotationa.l energy states. If X.< 0, a distribution close to the Boltzmann distribution is attained and the Pla.nck function becomes a. good a.pproxim.a.tiono Conversely if Q << X very littl.e collisiona.l excha.nge of energy wil.l occur a.nd the

5o atmosphere will act as pure scattering atmosphere. Between these two extremes both processes are important and some alternative form for the source function must be found. The problem is complicated because the source function depends on the radiation field, which in turn depends on the source function. Laboratory measurements have shown that the rotational relaxation time is approximately three orders of magnitude less than the vibrational relaxation time of the v2 CO2 ba.nd, so that levels where vibrational relaxation begins to become important a Boltzmann distribution over the rotational energy levels ma.y still be assumed. Rotational relaxation never becomes important in the earth's atmosphere, because the molecule is dissociated a.t the level around.120 km where the radiative -lifetime and the rota.tional relaxation time would become comparable (Curtis a.nd Goody,.1956). The first derivation of a.n equation for the source function in the eqrth's upper atmosphere for the.15 4 C02 ba,nds was made by Curtis and Goody (1956), in a, paper that has become a. classic in the fieldo They assumed a simple harmonic oscillator model in which the rotational energy.levels have a, Boltzmann distribution~ The equation of ra.diative transfer wa.s written in terms of Einstein transition probability coefficients and related to the number density of vibrational qua.nta,, E. The transfer of energy between the vibra.tional.. a nd translationa.l modes was ta.ken into account by use of the equation

51 dE = _- E) (2.2) dthe vbrtonl energes hve where E is the value of E when the vibrational energies have a Boltzmann distribution. The final form of the equation relating the flux divergence and the source function is dF = 4S (J _ B) (352.3) du X where S is the band intensity. Kuhn (1966) derived the source function in a rather different way, analogous to the two level atom problem in astrophysics. He determined the ratio of the populations of the O and 1 energy levels of the C02 molecule and showed further that the influence of a third level could be neglected for carbon dioxide in the earth's upper atmosphere. The derived equation is identical to that of Curtis and Goody, expressed in Eq. (3.2.3). An equation for the source function has also been obtained by Shved (1965a), using gas kinetic equations, and takes the form J B~ go H\exp(hvo/kT) l _ t N _J (3.2.4) where vo is the frequency of the band center, go and gl are the statistical weights of the lower and upper levels respectively, H is the flux divergence in quanta cm3 sec1 and

52 N is the number density of absorbing molecules. This expression apparently has a different form from Eq. (3.2.3) but can be changed using the approximation 2hvo2 B (v,T) 0nc exp(-hvo/kT) (3-52.5) This approximation is used in all three derivations and introduces a. maximum error of about 2o in B. Shved's Equation (3.2.3), becomes dF hvoH gl 1 c2 du - N gOX 2 (J-B) (3.2.6) The radiative lifetime Q is related to the total band intensity S by the following equation -1 go Vo2 - = 8t — 7 S (3.2.7) 91 c (Mitchell and Zermansky 1934, Penner 1959). Substituting into Eq, (3.2.3), Eq. (3.2.6) is obtained, i.e., the same equation for the source function is obtained i-n each of the three cases. It should be noted, however, that both Curtis and Goody, and Kuhn neglected to include the factor of the ratio of statistical weights in Eq. (3.2.7). For the C02 fundamental, gl = 2go, so that the final equations are in error by a factor of 2. This is equiva.lent to using vibrational relaxation times of twice the nominal values quoted in their papers. Whenreferring to their results this correction will be automatically applied. Although a factor of two a.ppea.rs to be an important quantitythe uncertainty of an order of magnitude in the relaxat'ion time X.prevents a. more accurat.e: determination of flux divergence,'as dis

cussed la.ter, The methods of deriving the equation for the source function employed by Shved and Kuhn are basically the sa.me, and are also contained in Kondrat'yev (1965. They do not depend on the simple ha.rmronic o-cil - l.ator model used by Curtis and Goody; the.latter represents a, simp.licica.tion of the energy states of the mol.ecule, but retains ma;ny essential fea.tures, so tha.t the equations obtained in ea.ch derivation are identicalO A ra.ther large number of assumptions and approximations in add.ition to those already mentioned have been made in the derivations of equatio-unsThe more important of these axre~ (i) It has been assumed that, in the regions of the atmosphere where deviations from Kirchoff' s law become important, the lines of the individual bands are non-overla.pping. This presents no difficulty since the lines in the earth' s mesosphere and above a.re very narrow, and no appreciable overlap occurs, even. in the Q-branches where the spa.cing is lea.st~ (ii) The source function J('v,T) has been approximated by J(v& T), where vo is the center of the band, i.e., it has been assumed independent of frequency within the confines of the band~ In particular this implies tha~t the Planck function is repla,ced'by a. frequency independent function over these frequencies. Fortunately, the Planck function varies slowly with frequency in the 15 4 region a~t mesospheric tempera.tures,. However, the effect; on the accuracy of the calculations is difficult to assesse It is better to look on this a.pproximat.ion as one which ma.kes the ca.lcula. tion possible a.t this time and accept it until a more complete solu+;tion ca.n be foundo

5 L. (iii) The equations apply to a. single band only, and then only to the fundamental This fact has some important implications~ (a) The equations cannot be applied to the weaker overtone bands, some of which are important in and above the mesosphereo (b) The equations are valid for the fundamentals of the molecules with the less common carbon and oxygen isotopes~ These molecules'play a, much more important role in radiative transfer than their relative abundance would suggest. The overtone bands of these isotopes are not significa.nto (c) Where local thermodynamic equilibrium breaks down there is no single source function applicable to all bands, but a different one must be derived for each individual band~ In general, the level at which deviations from the Planck bla.ckt-body function become significant will vary from band to band, and will be expanded upon later. Using Eq. (3.2l) and (2.1o6) the expression for the source function becomes g 2 2 +1 J B 1 h -g0 c' -l [I(v`r, ) 2 J(VT)]kV d~ dl go c (302~8) The dependence on the ratio </Q ha-s now reappeared since the absorption coefficient kv in the integrand is proportional to the band intensity S and therefore proportional to.1/0. This mea.ns that the source function for the ba.ad is determined solely by the rantio b/o annd the atmospheric structure parla~meters, a.lli o-ther quantcities being coanst~ant for -the'bando

55 To obtain the flux divergence the sourCe function is evaluated from Eqo (32.8) o In Section 2.o42 it was shown that the integral could be expressed as a. linea.r sum of source functions a.t all levels in the atmosphere At ea.ch level i the source function can be expressed a.s N Ji:B.i - j i J. J2. O1O2 N (3.2-9) This set of N linear simultaneous equations may be solved for the N unknow:ns Jl OOOJNo The solution presents no particular numerical difficulty, at lea.st in'the ca.ses encountered in the calculations made here. A double'baick substitution method wa,s employed, using a, library subroutine of the University of Michigan Computing Center. Another subroutine (a. GaussJorda.n reduction with a. complete pivota.l strategy) gave almost identical results o The flux divergence ma.y be evaluated from Eq (.3.2o6) or Eqo (21.L6), once the source functions have been obtained~ Using both equations provides a. good check on the accuracy of solution of the linea.r equa.tions. The hea.ting rates obtained from the two equations were very close, except a,t high pressures, where X is small~ Under these conditions J and B are almost identica.l i eo. local thermodynamic equilibrium exists, and the a.ccura.cy with which the numerical value of these functions ca.n be represented in the compute:r produces a. la.rge relative error in their differenceo It is'better to calculate flux diverge:nce using Eqo (2o1o6) a.t high pressures a For the reasons outline in Section 2o402 the finite difference for=

56 mulation of the flux divergence is preferredand Eqo (2.4.6) is used in place of Eqo (2.1o6)o The flux divergence in the ith layer is assumed to be that of the mid point where, because of the linearity of J and B, the source function and Planck function have a value equal to the avera.ge of the values a.t the bounda-rieso The system of linear Equations (3.2.11) is repla.ced by a. slightly modified system in the form Ji + Jill Bi+Bii j + PK 2 a j 2 /j (3 2.10) This number of unknowns is now N + 1, with only N equations to determine them. A further condition is that a.t the surfa.ce the source function is identical to the Planck function, i.e., N JN In the earth' s atmosphere no difficulty arises from this assumption beca.use almost a.ny reasonable value of JN leaves the cooling in the upper atmosphere unchanged. Otherwise the solution proceeds in exactly the same way o The equations for the source function ha.ve been derived on the a.ssumption tha.t the absorption lines in the mesosphere a.nd above are nonoverlapping. In the mesosphere Doppler broadening dominates the line s shape, where the Doppler half-width is approximately 6 x 104 cm-l1 The region of the 15 p CO2 bands near 668 cm-l has the highest line density,

57 containing the Q-branches of the 1i2ci602 fundamental and some overtones. The line spacing is approximately 1 x 102 cm. i so that non-overlapping is, a va.lid a.ssumptiono In the mesosphere much of the incident radiation field orgina.tes from the troposphere where overlapping is important; Nevertheless, the flux divergence at the upper level.s can be a.ccurately cormputed on the assumption that the lines a,re non-overlapping throughout the atmosphere. In Eq. (231o6) the expression for flux divergence contains the weighting factor..j in the integrand; a.way from the line centers kV is zero, but nea.r the.line center, where k, is la~rge, the absorption is complete well above the levels where overlapping is significant, unless the line is extremely weak a.nd ineffective in radiative transfero (CfO Shved: 1964)O 3-5 CALCULATION DETAI1 For ea-ch absorption'band in the mesosphere, the total. flux divergence wa.s expressed. a.s the sum of the flux divergences for the individua.l lineso In ea.ch of the P, Q and R branches of the bands, variations from line to line were found to'be sma.ll, so that ca.lculation of every third rotational level provided sufficient a.ccura-cy with a, considerable saving in computing time The most uncertain parameter is the vi'brational relaxation time X. Experimenta.lly it is difficu.lt to determiner especially for the tempera-tures encountered near the mesopause and theoretica.l estimations of P do not produce agreement with experimental data,.A comp.l.ete discussion, inl

58 eluding the possible catalytic effect of water vapor is given by Shved (1965a.) The value of X probably lies within the range 2 x 10 6 to 2 x 10-5 sec at 1 atmosphere and about 200OOK Since the relaxation time depends on the frequency of molecula.r collisions, \ is inversely proportional to pressure p. After the coo.ling rate calculations had been made, a. pa.per by -Read (1965) was brought to the a,uthor's attention. The vibrational relaxation time for C02CO02 collisions is approximately 6 x 10-6 sec a.t 1 atm and 280K. The values used here (between 2 x 10 a,nd 2 x 105 secs) bracket this result. The presence of small amounts of water vapor significantly reduces the relaxaition time. The values for C02-N2 collision is not given. The concentration of carbon dioxide was assumed to be consta~nt up to 120 km a.t.314 per cent by volume, but the calculated heating rates are not sensitive to sma.ll changes~ CO2 starts to dissociate a.t about 100 km (Curtis and Goody, 1956) so that radia.tive heating rates above this level a.re uncertain. The effect on heating rates.lower in the atmosphere is unimportant, as shown'by test calculationsO Beca~use the primary interest in this calculation was on the behavior near the mesopause, thin layers were used between 70 and 90 km (about 2 km) 9 a.nd rather thicker.layers elsewhere (5 km in the thermosphere and mesosphere). Layers of 5 km. thickness in the region of the mesopa.use were found to distort the cooling rates a.t these levels. The band parameters (line positions, intensities and Lorentz ha.lfwidth) used in the calculations are taken from Drayson and'Young (1966)

59 and Drayson et al., (1967). Table I lists all the bands which are important at atmospheric temperatures. The most common of the C02 molecules a.re given in Table II, with their abundance relative to 12C1602 The value of the Lorentz half-width used wa.s 0.08 cm-1 at 1 atm and 3000K. This value is somewhat uncertain, but at mesopheric pressures the lines are predominantly Doppler broadened, and errors in the Lorentz half-widths a.re not significant. The intensities of the weak bands in the 15 4 region are also uncertain, particularly those which are overlapped by strongly absorbing bands. However, their role in radiative transfer near the mesopause is small, largely 8 result of their rapid decrease in intensity with decreasing temperature. This point will be discussed more fully when the results of the calculations are presented. 3.4 RESULTS OF CALCULATIONS AND COMPARISON WITH PREVIOUS RESULTS 3.4.1 Cooling Rates for Model Atmospheres Radia.tive transfer calculations are frequently applied to standard or model atmospheres, which are important because they represent average temperature profiles and are not encumbered with fine structural details. Many radiative transfer models are not capable of representing such fine structure and this provides an additional reason for using the smooth profiles. The calculations for the U. S. Standard Atmosphere (1962) demonstrate many of the features common to all model atmospheres at the mesosphere

60 TABLE I BAND INTENSITIES USED IN CALCULATING ROTATIONAL LINE INTENSITIES (From Drayson et al., 1967) Level Band Center Intensity Lower Upper (12C 1602) (cm-l(atm cm)-1 at 3000K) 1 000:0 010:1 667.379 194 2 010:1 020:0 618.033 4.27 3 010:1 100:0 720.808 6.2 4 010: 1 020:2 667.750 15.0 5 020:0 030:1 647.054 1. 0 6 020:0 110:1 791.447 0.022 7 020:2 030:1 597.337 0.14 8 020:2 110:1 741.730 0.14 9 020:2 030:3 668.151 0.85 10 100:0 110:1 688.672 0.3 11 100:0 030:1 544.279 0.004 12 030:3 040:2 581.62 0.0042 13 030:3 120:2 757.47 0.0059 14 030:1 -120:2 828.284 0.00049 15 030:1 120:0 738.364 0.014 TABLE II RELATIVE ABUNDANCE OF ISOTOPIC MOLECULES (From Drayson and Young, 1966) Abundance Relative Band Center Molecule to 12C 1602 Of Fundamental o 12C 1 60 1.00 667. 4 1 13C 16 1.12 x 10-2 648. 5 2 12C 160 180 4.0 x 10'5 662.3 3 12C 160 170 8.0 x 10-4 664-7 4 13r 16A 18~ 4.5 x 10-5 645.6

61 and mesopause levels. The equations for the fundamental transition, described in the previous section, were applied to each of the isotopes separate using valued.of the vibrational relaxation time ko = 2 x 10 -. 4 x 10 6, 10 and 2 x 10- sec, where is the value of X at 1 atmosphere. It was found that only isotopes 0, 1, 2, and 3 (Table II) gave any contribution to radiative transfer in the upper atmosphere (greater than about 0.10K per day heating or cooling). The full results are too extensive to give here in detail; only representative values, will be given. The cooling rates for the four bands are shown in Table III for the value k0 = 1 x 10-5 sec. Table IV provides details of the deviation of the source functiorr from the black body function. For isotope 0 the devia.tion does not become apparent until about 80 km, while for isotope 3 the difference begins near the stratopause. It is interesting to note that at that level the radiative lifetime Q is still approximately two orders of magnitude greater than the relaxation time <, a condition generally unfavorable for vibrational relaxation. The reason can be explained by rewriting Eq. (3.2.6) in the form dF _ F du du gl 1 c2 dp du dp dp go X 2v02 du is proportional to the mixing ratio, which is very small for the less dp abundant isotopes. This implies that large differences between the source function and the Planck function are required before appreciable cooling Cain occur.

62 TAB.LE III COOLING RATES IN ~K/DAY FOR THE INDIVIDUAL BANDS, U.S. STANDARD ATMOSPHERE (1962), ko = 1 x 10-5 SEC Height Ioc;. 0 1 2 3 0 0 0 (km) Band 1 1 1 1 2 3 4 107.5 3.01.03.00.00.92.66 1.O1 102.5 3.80 ~0~3.01.00.60.43.89 97.5 4.67.03.00.00..18.09.45 92.5 5.03.01 -.02 -.01 -.05 -.11.04 89.0 3.49 -.03 -.05 -.12 -.17 -.26 87.0 3.25 -.05 -.07 -.02 -.13 -.17 -.28 85.0 2.48 -.08 -.10 -.02 -.13 -.18 -.30 83.0 1.5 3 -.13 -.14 -.03 -.14 -.18 -.33 81.0 -.16 -.20 -.18 -.04 -.15 -.19 -.37 79.0 -.45 -.23 -.21 -.05 -.14 -.19 -.35 77.0.30 -.17 -.19 -.04 -.09 -.16 -.21 75.0 *55 -.09 -.16 -.04 -.02 -.09.00 73.0.68.00 -.10 -.02.09.00.26 71.0.81.11 -.02.00.21.12.55 68.0.86.26.12.04.37.29.88 64.0 1.36.45.34.11.60.56 1.33 60.0 1.89.50.48.16.62.62 1.34 56.o 2.40.44.52.20.51.53 1.11 53.0 3.02.45.55.21.53. 55 1.19 49.5 3.23.39.47.21.45.46 1.07 44.0 2.85.31.50.15.34.34 33.85

TABLE IV RATIO J/B FOR INDIVIDUAL BANDS, U.S. STANDARD ATMOSPHERE (1962), o0 = lxlO5 SECS Height Iso. 0 1 2 3 0 0 0 (KM) Band 1 1 1 1 2 3 4 110 07.27.42.45.439.31 105.13.38.61.67.71.65.50 100.20.59.95 1.05.93.92 ~77 95.40.81 1.32 1.47 1.00 1.02.99 90.72 1.18 1.89 2.10 1.01 1.04 1.04 88.80 1.18 1.86 2.07 1.01 1.03 1.03 86.89 1.21 1.82 2.00 1.01 1.02 1.02 84 ~95 1.21 1.77 1.94 1.01 1.01 1.02 82.99 1.24 1.77 1.83 1.00 1.01 1.01 80 1.01 1.24 1.64 1.74 1.00 1.01 1.01 78 1.00 1.12 1.34 1 39 1.00 1.00 1.00 76 1.00 1.04 1.18 1.21 1.00 1.00 1.00 74 1.00 1.01 1.07 1.08 1.00 1.00 1.00 72 1.00.99 1.03 1.02 1.00.00 1.00 70 1.00.99.99.98 1.00 1.00 1.00 66 1.00.99.98.96 1.00 1.00 1.00 62 1.00.99.98.97 1.00 1.00 1.00 58 1.00 1.00.99.98 1.00 1.00 1.00 54 1.00 1.00.99.99 1.00 1.00 1.00 52 1.00 1.00 1.00.99 1.00 1.00 1.00 47 1.00 1.00 1.00 1.00 1.00 1.00 1.00

64 This result is important for the radiative transfer in upper levels of other planetary atmospheres. In Mars and Venus for example carbon dioxide is a major atmospheric constituent and vibrational relaxation of the fundamental will not take place until lower pressures a.re reached, possibly not until dissociation takes place, at least for the isotope 12C 160 The equations for vibrational relaxation have not been derived for overtone bands. Nevertheless the equations developed for the fundamental were applied to these bands. It was found that only the three strongest overtones were important for radiative transfer in the upper mesosphere and moesopause region (bands 2, 3 and 4) and these only for the molecule 12C 1602. For these bands the ratio of the statistical weights is unity. Furthermore, relaxation is not significant even at 80 km, as can be seen from Table IV. o This is due, in part at least, to the fact that the radiative lifetime of these weaker bands is at least 1 to 2 orders of magnitude larger than the fundamental. Because the bands are relatively weak, the magnitude of the heating and cooling rates is quite small. Unless the equations for vibrational relaxation are in a quite different form for'these bands, or the vibrational relaxation time X is much larger than for the fundamental, the method of calculation should introduce only a. small error. The total cooling for all bands for the U. S. Standard Atmosphere, (1962) is shown in Fig. 3, for values of ko = 2 x 10 6, 10 and 2 x 10 sec and are compared with the values of Kuhn (1966). For all values of

65 110 o=2 x 10-5 sec Xo 10-5 sec 100 90- 4h2XIO-6sec 80 -'t"C This Calculation 80 / E s, ----------.....Kuhn (1966) ~-70-:60 - lw uJ C5040 -4 -2 0 2 4 6 8 10 12 COOLING RATE (deg K/day) Fig. 3. Calculated cooling rates for the U.S. Standard Atmosphere (1962).

66 ko the present calculations show a. small a.mount of hea.ting nea.r the base of the mesopause, although its magnitude a.nd vertica.l extent are both significantly smaller then those of Kuhn. In the thermosphere the present results show much more cooling. This is a. consequence of the sepa,rate relaxation of the bands, since the weak overtone bands a,re not completely rela.xed, even a.t 110 kmo and have appreciable contribution to cooling at this level. Nea.r the stratopause the assumption of non-o-verla.pping.lines may begin to brea.k down and some of the weaker bands also provide an a.ppreciable source of cooling. The present calculations were not designed to be a.ccurate in this region of the atmosphere. There a.re a, number of important differences between the two methods of calcula.tiono Kuhn's calculations included the following features~ (a.) Isotherma,l layers were used (b) Equation (3.2.6) wa.s applied to the total flux divergence for all bands, i.e., the bands were not relaxed separately. (c) A constant temperature was used to calculate line intensities in the mesosphere. (d) The qua.si-random band model wa.s used. It ha.s already been shown that (a.) tends to produce greater heating at a. temperature minimum. Test calculations showed thart (c) exhibited the same tendency. Because the lines a.re isola.ted in the mesosphere the redistribution of the lines by the model does not change the absorption and (d) does not contribute errors directly.

67 Test ca.lculations were made to determine the effect of (b) The differences in cooling rates we:re greatest for the larger values of ko, the new calculation giving an additiona.] heating of about 0 4 deg k/day near 80 km for o = 2 x 1.0 qo'but only half this value for o 2 x 106 sec. In -the lower thermosphere the effect wa.s more pronounced, roughly having the cooling rates above 100 kmo Ca.lculated in this way, the coo.ling rates in the lower thermosphere are in good agreement with Kuhn, but the differences in the mesosphere and. at the mesopa.use a.re virtually unchanged One of the effects of atmospheric.long wave radiative transfer is to smooth discontinuities in temperature gradient, and tends to produce a. ma.ximum. of heating or cooling where such discontinuities exist. Ma.ny model atmospheres9 including the U. So Standa.rd Atmosphere (1.962), conta.in such features. A calculation was made for this atmosphere, with the temperature structure smoothed at the base of the mesopa.use, where hea.ting was previously obtained~ The new cooling rates are shown in Fig. 4. The heating has been very much reduced (about 0O6 deg K/day maximum) or entirely eliminated, depending on the value of k0o In view of the.ia,.:rge number of approximations made in the ca..culations, it is concluded that radiative transfer due to the L.5 g bands of carbon dioxide is not significantl.y different Lfrom equ.ili-briumn at the mresopause.level, for the smoothed a.tmosphere The effect of other temperature gradient discontinuities ca.n be seen at 90 and 60 kmr n.OLy one a.t.ospheric.layer was used for the stratopause,

TEMPERATURE (OK) 160 180 200 220 240 260 I I I I I10 L/t -L:'.E90- U. S. Stand a r d 100 D80 b.I Smoothed Prof ie 40 -2 0 2 4 6 8 10 COOLING RATE (deg/day) Fig. 4. Calculated cooling rates for modified U.S. Standard Atmosphere (1962).

69 but if a model. of greater:resoiution is u.sed for this region of the atmosphere, two maxima. of -ooling obtained, at the bottom and top of the isothermal.laye:r, a. result which was al so reported by Pla.ss (1.956b) A study of tahe r.oot.[iing,ra.tes for the individual'bands reveals that the weak overtone bands a.re in:ensitive to.i-cal changes in temperature gradient, and that the same is true of the less abundant isotopes. Only the fundamental, of 1-2C'160o showed a. large change from the cooling of the unsmoothed standard a,t+mosphere, although band.I of isotope I. and'band 4 of isot;ope C also show sma,.ller differenceso This result is to be expected since the bands with weak atmospheric absorption are almost transparent over r2 or 4 km layers in the upper atmosphere, and cooling does not depend on the temperature gradient in the neighborhood of the l.evel. under conside ration o Ca-cula-tions have a,.l.so been made for the ARRDC-.1959 and the CIRA-1961 model atmospheres. The ARDC-,.1959 (Minzner et a..L, 91 959) atmosphere has a, cold mesopa.use, typica.l of average conditions in summer a.t high latitudesO 3imi.La.r calcula~tions have been reported by Kondrat'yev et al., (1966) and these are of particular interest since separa-te results are given for the fundamentals of 122 162, and the sum of the contributions of the remaining isotopes~ Fu.Ll deta~i.ls of the met;hod of ca.lcul]ation are not given but presumably foll..ow the deve..opments of Shved (.1964, 1.965a. and 1965b), which feature (a) non-overlapping li. nes o (b) allowance for temperature dependence of line intensities~

70 (c) isothermal layers. (d) individual relaxation of bands. (e) line half-widths temperature independento Comparison of the cooling rates of this calculation with those of Kondrat'yev et. a!., are shown. in Fig. 5. The rates for the less abundant isotopes show agreement to within the error that the values ca.n be determined from the figure. For 12C 1602 the cooling rates are still quite close. both showing a heating of about 1 deg K/da.y near 80 km. In the present calculation this heating is localized nea.r the lower mesopause where the temperature gradient is discontinuous. Kondrat'yev et al,, shows heating sprea.d over a. layer twice as thick. As with Kuhn, the present calculation shows a. small but consista.ntly larger cooling rate throughout the mesosphere. In the lower thermosphere Kondra.t'yev et alo, have slightly smaller cooling rates. Unlike the other two models, the CIRA 1961 atmosphere (Ka-llma.nnBijl et a.l., 1961) ha.s a smooth profile, with slowly changing temperature gradients. This characteristic is reflected in the sum of the cooling rates due to the fundamentals (a.ll isotopes) (Fig. 6). Depending on the relaxation time k, a small a.mount of either heating or cooling is indicated at the mesopause. In contrast, Kondrat'yev et alo, show a.pproxima.tely 0.5 deg/day additiona.l heating at this level, with the heating region again of larger vertical extent. Although the differences between the present results and those of Kondrat'yev et a lo, are small, they are consistent. Possible reasons for

71 100 = 2 X I s e _-X I,/ ~ j2XIO-6sec 90- I. This Calculation -..... Kondrat'yev et al. (1965) 6 \ 570-2 0 2 4 6 8 10 COOLING RATE (deg K/day) Fig. 5. Calculated cooling rates for ARDC-1959 atmosphere.

72 I00 \~~~ Xo: 2 X I0- sec i s2~~~~~~~2XIO-6sec 90, /.~~.._C /5 / /~~~~ ---— ~~_ This Calculation 80/. 80E —- Kondrat'yev et al. (1965) H v \, \\ 700 I-& 0 w I CD 60 50-2 0 2 4 6 8 10 COOLING RATE (deg K/day) Fig. 6. Calculated cooling rates for CIRA-1961 atmosphere.

73 these a.re' (a.) the use of isothermal layers, already discussed, (b) the use of temperature independent line half-widths: this assumption has been discussed by Hummer and Rybicki (1966), who showed that variations in Doppler half -widths with temperature can be significa.nt in some situations 30402 Cooling Rates for Measured Temperatures Pr ofiles Although model atmospheres are of interest a.s objects of study in their own right, they are nevertheless different from rea.l a.tmospheres, in that the om.nipresent irregularities in the profiles have been removed. The study of the model atmospheres shows that, for the 12 C 1602 fundamental, loca.1 features la.rgely determined the cooling behavior nea.r the mesopa.use, so tha.t it is na.tural. to expect the fine structure of the atmosphere to be critical in determining cooling ra.teso Tempera.ture oscillations of up to 530-400C are not uncommon between 70 and 90 km. (Theon et a..l, 1967), and these oscillations frequently" exhibit a. wa.ve-like structure in the vertical The profiles used are from Jones and Peterson (1967) and Peterson (1967), measured by the falling sphere technique. Above the ma.ximum a.ltitude of -the soundings, the temperature was extrapolated to agree with the U.S. Standard Atmosphere (1..962) at 1.20 kmi; the structure at these levels is irrelevant to the cooling rate near the mesopause. LPike the ca~lculations for the model atmospheres, a resolution of 2 km

wa.s used between 70 and 90 km,, and approximately 5 km layers elsewhere. To reduce computing time the coefficients for the US.o Standard and Atmosphere (1962) were used to calculate cooling rate for each profile i.e., the temperatue dependence was neglected. The results showed that the fundamental of 12C 1602 is by fa.r the most important ba.nd, and the tempera, ture dependence for this band is small. Figure 7 show the temperature structures and calculated cooling rates for six soundings' for the two values of vibra.tional. relaxation time 2 x 10-6 sec and 2 x 05 sec. The contributions from a.ll isotopes and all bands are includedo Before commenting on the individual soundings, some general features common to all -the profiles a.re immediately noticeable. (a.) The smooth features of the cooling rates for model atmospheres a.re no longer appa.rento Instea.d, large variations in heating and cooling ra.tes a.re indicated, corresponding to the maxima, and minima. of the tempera.ture profiles or change in temperature gradiento (b) There appea.rs to be a. ma.rked tendency for greater overall cooling, particularly for the la.rger va~lue of vibrational relaxation time; for most, but not a.ll, the profiles the heating is restricted to rather thin layerso (c) In the mesosphere and mesopa.use region, the maximum cooling rate is frequently about the same or greater than the cooling rate at the much warmer straltopause o (d) For most profiles the cool.ing ra.te is the same for both values

75 TEMPERATURE (~K) 160 180 200 220 240 260 280 I i I i x| /2X10-5 sec. X2Xo0-6 sec. I00 90 80 w 70 I0, 60 50 -5 0 5 10 15 20 25 30 35 COOLING RATE (degdoay) (a) Wallops Island, 380N, 75~W, 1330 EST, 7 Aug, 1965. TEMPER ATURE (K) 0 0 20 240 260 20 110 12XI0-5 sec. - 2XI0-6 sec. 270 90 w60 4-5 0- 5 10 15 20 25 30 35 COOLING RATE (deg/doy) (b) Wallops Island, 38~N, 75~W, 2240 EST, 7 Aug, 1965. Fig. 7. Calculated cooling rates for measured temperature profiles.

76 TEMPERATURE (OK) 160 180 200 220 240 260 280 40 - 2XIo-5 sec. I7-00-6 sec. 10090 I 80 70e 0 5040 COOLING RATE (deg/day) / o ~~2XI0-5 sec. 1 2XI0-6 sec. I40 uJ w 060 50 40 -5 0 5 10 15 20 25 30 35 COOLING RATE (deg/day) (d) Wallops Island, 380N, 750W, 1810 EST, 19 Oct, 1965. Fig. 7. (Continued)

77 TEMPERATURE (~K) 160 180 200 220 240 260 280 I10x2XI0-5 sec. 2XIO-6 sec. I00 901 -80 -r-700 40 L 5 0 5 10 15 20 25 30 35 COOLING RATE (deg/day) (e) Waliops Island, 180N, 75W, 1114. EST, 25 Oct, 1965. TEMPERATURE (K) 160 180 200 220 240 260 280 1009050 40 -5 0 5 I0 15 20 25:0 35 COOLING RATE (deg/day) (f) Point Magu, 380N, 7l95W, 14.50 PST, 24 Oct, 1966.

78 of vibrational relaxation time below 75 km., although small differences occur down to 70 km,. for some of the soundings. The first three profiles (Figs~ 7 (a,), (b), and (c)) are from consecutive soundings taken within a. 24 hour period in August 1965 a.t Wallops Island Ea.ch shows some oscilla.tory temperature structure in the mesopau.se region. The maximum cooling ra.te between 90 and 70 km for the first sounding is about 7 deg K/day for 2 x 10 seco The radiative tra.nsfer due to the 15 J bands of CO2 is incapable of expla,ining by itself the reduction by a.'bout 200 K of the tempera.ture ma.ximum a.t 88 km during the 10 hour interval between the first and second soundingso The oscillations increase during the night and are more marked in the- third sounding. Although the carbon dioxide plays a. substantia.l role in the transfer of energy at these levels, it is clear that even more important dynamic processes are controllingthe temperature profile0 Soundings (d) and (e), ta.ken in October 1965 at Wall.ops Isla.nd, ha.ve similar temperature gradient discontinuities in the mesosphere near 70 km which provide la.rge cooling a.t this level.. The mesopause is particularly well defined with the temperature rising ra~pidly above the temperature minimum. This chara.cteristic explains the substantial heating observed a.round 80 km. for'both soundings. Figure 7 (f) shows a. rema.rka~b.le sounding from. Point Ma~gu on 24 October 1966 A tem.pera.ture perturbation of amplitude almost 50~K wa.s obta.ined at 85 kmo Smaller perturbations a.re common at this level, but this amplitude is appa.rently one of the largest recorded~ As expected, this perturba

79 tion produces a large cooling rate, of about 55 deg K/day for ko = 2 x 10-6 sec, and 17 deg K/day for o 2 x lO15 seCo A heating of comparable magnitude might have been expected in the region of temperature minimum immediately above and below the warm layer, but its va~lue is only about one seventh that of the maximum cooling rate~ Also shown is the cooling rate obtained in the absence of the temperature perturbation, with near equilibrium conditions at the mesopauseo The calculations for the measured profiles show that the cooling implied by mean atmospheres is not the same a.s the mean cooling derived from individua.l soundings. This result is significa.nt for a.n understanding of the energy balance in the mesopause region of the atmosphere; radiative trainsfer by carbon dioxide may be able to provide a. mean energy sink, pa.rticularly for situations in which waves are being propagated vertically. The effect of temperature oscillations on cooling rates ha.s been discussed by Goody (1964) a.nd by Sasa.mori and London (1966) for homogeneous atmospheres. In the mesosphere the atmosphere is not homogeneous, or even approximately so, and the departure from.loca.l thermodynamic equilibrium makes it impossible to apply their methodso The calculations made in this section used a vertical resolution of 2 km between 70 a.nd 90 kmo The method could be applied to soundings of higher vertical resolution if reliable sounding data become available. The profiles avai.lable are not suitable for this applica.tion, since they contain a. considerable amount of high frequency noiseo

80 3~4~3 Comparison with Previous Calculations A full comparison of the methods and results of the calculations of Kondrat'yev (1965) and Kuhn (1966) has already'been made in the previous sectionso Young (1964) attempted calculations that were essentially similar to those of Kuhn, but was unable to obtain cooling rates because of numerica.l difficultieso At the time that Gurtis and Goody (1956) made their calculations fa.st computers were not so readily available. They chose an iterative procedure to solve the radiative transfer equations using a.s an initia~l guess for the flux divergence the'cooling to space' (ioe. the a.mount of energy lost directly to space), a. concept that has appea.red a number of times in radiative transfer literature. In the mesopause region it has been shown that for strong ba.nds the cooling is dependent on the local temperature structure, while for weak bands heating, rather than cooling, takes place. Lower in the atmosphere where absorption is stronger, *the loca.l structure may dominate a.ll but the weakest bands, so that'cooling to space' cannot be expected to provide a. reasonable approximation to the cooling rate. Moreover, iterative procedures for solving the tra.nsfer equations have been shown to converge extremely slowly and to depend on the initial guess, a.nd may even diverge (Kuhn, 1966)o Curtis and Goody do not make it clea-r what wea.k lines, if any, were included or if the less a.bundant isotopes were considered~ These lines have been shown to be important~ Murgatroyd and Goody (1958) also made calculations for themesosphere, citing Curtis a.nd Goody for the method of computationo Because of the uncertanties in

81 the actual parameters and method used, no direct comparison can be made between their results and the present calculations. The calculations of Prabhaka.ra, and Hoga.n (1966) contain two serious errors. The first lies in the attempt to simulate the absorption in given spectral regions by lines of constant intensity and frequency spacing, using transmission tables of Stull et al., (1963) to deduce some of the parameters. The transmission tables contain the contributions from lines of many different intensities, so that the simple model cannot be expected to represent the transmissivity accurately, especially when extrapolated to low pressures. A further consequence of this treatment is that all bands are considered together, instead of treating the relaxation separately for each band. The second error lies in the method of solution of the radiative transfer equation. The energy absorbed by a horizontal layer in the atmosphere is 2i 1, fd I(v,f,T)[l-7(vI ) doi dv where, is the tra.nsmissivity of the layer. This equation is correct. However, Prabhakara. and Hogan applied it using the average values of I and y over finite frequency intervals, i.e., they integrated over frequency assuming that the integral of the product I(v) [1-y(4)] is equal to product of the integrals of I(v) and [l-y(v)]. In fact, I(v) is closely correlated with y(v). When the absorption is large the radiation originates from a region near the layer under consideration. Conversely, when

82 the absorption is small, the radiation originates from deep in the atmosphere. The method used by Prabha.kara, and Hogan is the finite difference form of Eq.(2.1.6). Their assumption is equivalent to replacing the integral f (I -J ) kv dv by fV k, dv * v(I -J ) dv S / ( I-JV) dv where S is the total band intensity. This reasoning is false. The same method was also applied to the Martian atmosphere (Prabhakara and Hogan, 1965). ~.5 SUMMARY AND CONCLUSION The most important results of the investigation of radiative cooling due to carbon dioxide in the mesosphere and mesopause are: (a) The equation linking the source function to the flux divergence should contain a. degeneracy factor. (b) Although some radiative heating near the mesopause is typical, its magnitude is small and is usually correlated with either a temperature minimum or an abrupt change in temperature gradient. (c) Oscillations in the temperature structure near the mesopause dominate the cooling rate, resulting in a. considerable increase in cooling at this level. Most of this cooling is provided by the 12C 1602 fundamental, with the other bands playing a minor role. When the profile is smooth, the remaining bands (fundamentals of the isotopes and overtone

83 bands of 12C 1 02) -have a, role approximately equal to that of the ~*12C 1602 fundamentalo (d) Vibra.tional relaxation begins to influence the cooling ra.te above about 75 kmo Uncertainties in the vibrationa.l relaxation time lea.d to lalrge uncertainties in the cooling ra-te above this level. The mesosphere is not in radiative equilibrium (Leovy 1964a. and 1964b)0 This ha.s a.lso been demonst:rated by the series of soundings at Wallops Island However, there can be little doubt tha.t radiative cooling is a.n important factor in determining the atmospheric termperature profile in the mesosphere a.nd mesopause and tha.t it must be accurately accounted for in any model of-this part of the atmosphere. Mean temperature profiles cannot be used to derive mea~n cooling rateso Wa.ve-like temperature oscillations in the vertica..19 which are lacking in simple models, a.re essential in the determination of mean cooling rates

CHAPTER. 4 APPLICATION TO THE ATMOSPHERE OF MARS 4.1 IETRODUCTION In the last decade significant advances have been made in observations of the surface and atmosphere of Ma.rs. These have come from ground ba-sed spectra.l observations, including the remarkable high resolution interferometric observations of Connes a.nd Connes, 1966, and also from a, planetary f.ly-by missiono Plans are now being made for the soft.landing of scientific pa.ckages on the surface, and for manned exploration a.t a, la.ter da.te. It would be of significant help to the spacecra.ft designers to know the range of meteoro-.logical conditions such as temperature, pressure, wind velocity to be expected at the surfa.ce Meteorologists a.re also interested in the Martian atmosphereo Its structure a.ppears to be simpler tha~n that of the.ea.rth' s, and therefore interesting for genera.l circulation studieso The virtua..l absence of clouds and of wa.ter vapor from the atmosphere and oceans from the surfa.ce is in marked contrast to conditions on the ea.rth. Long wave ra.diative tra.nsfer, in pa.rticu.lar tha.t due to ca~rbon dioxide, is of great importance. The calculations described in this chapter are designed to be used in conjunction with a. simple a.tmospheric circuilation model. An essential requirement is tha~t the flux divergence must be simple a.nd quick to eva.lua.te, since it has to be ca.lculated many timeso 84v

85 402 ATMOSPHERIC COMPOSITION It has been known for a number of years that the Martian atmosphere contains considerable amounts of carbon dioxide (eogo, Grandjean and Goody, 1955), deduced from the absorption spectra of reflected solar radiationo Beginning with Kaplan et alo, (1964) a series of more precise measurements wa.s ma.de of the carbon dioxide content and the surface pressure. The Mariner IV occulation experiment has also provide valuable information on the surface pressure and composition of the Martia.n atmosphere (Kilore et al.,.1965)o Although the exact amount of carbon dioxide is still somewhat uncertain, the amount in a. vertica.l column is believed to be in the range 68 + 26 m atm., (a. partial pressure of 5 ~ 2 mb at the surface) and the surface pressure between 5 and.13 mb (Belton and Hunten, 1966). Thus, carbon dioxide is a. major atmospheric constituent and the possibility that the atmosphere consists almost entirely of carbon dioxide is not inconsistent with the.spectroscopic observations, nor with the occultation observations of Ki.lore et al. The amount of water vapor in the atmosphere is small. Kap.lan et a.1, (1964) deduced 14 + 7 4 precipitable water from their observations but, as in the earth's atmosphere, the amount may be variable. Kaplan et al., also fa.iled to detect any molecular oxygen in the atmosphere, and set a.n upper limit of 70 cm a.tm. This implies that the amount of ozone in the atmosphere is strictly limited, a.nd is consistent with the detection of carbon monoxide in Martian spectra, approximately three orders of magnitude

86 less abundant than carbon dioxide (Ka.plan, 1967). Ka.plan wa.s also able to deduce that the relative abundance of the isotopes 12C, 13C, 160, 170, 18 O is approximately the same a.s in the ea.rth's atmosphere. Although there are still uncertainties in the exa~ct composition of the Martian atmosphere, the important constituents, particularly those tha.t affect long wave radiative transfer, are known quite wello In any case, a. circulation model is probably not sensitive to small uncertainties, and new calculations can easily be undertaken when more a.ccurate'data, become available. In the radiative transfer calculations described here, a. pure CO2 atmosphere was chosen, with a, surfa~ce pressure of 6mb. The influence of wa.ter vapor was neglected; Goody and Belton (1967) have shown its influence to be small0 Carbon monoxide has no vibra.tionrotation band in the thermal infrared region, but a. pure rotation band with lines between 100 and 600 4 has been detected 9 Thus, the radiative role of CO is small a~nd was a.lso neglected. 453 CALCULATION DETAILS AND RESULTS The details of the bands of ca~rbon dioxide in the 15 [i region have already been given in the previous chapter, and the abundance of the isotopes wa~s.left unchangedo The only major change is in the Lorentz ha.lf-width of the lines, which are now self-broadened. Burch et a.lo, (1962) have shown that the self broadened ha.lf-width is about 1o3 times the nitrogen broadened value, i.e., about 00.105 cm at 1 atmosphere and 300 Ko This is the value used in these calculationso Madden (1961) has

87 shown that there are variations in ha.lf-width with rotational quantum number in the 15 1 region, and these should be taken into account. However, the correct value is not known for all lines, and probably does not have a large effect, especially when compared to other uncertainties in the model. Mixed Doppler-Lorentz broadening was used throughout the Martian atmosphere. The atmospheric model used for the circulation calculations is quite simple. The atmospheric temperature profile is defined by the temperature at three presure levels. pg/3, 2Pg/3 and pg, where pg is the surface pressure. At pressures lower than p /3 the atmosphere is isothermal. At higher pressures the temperature is given by a quadratic in logarithm of pressure, through the three pressure levels. LEVEL PRESSURE LAYER TEMPERATURE (mb.) ( K) 0 0 170 1 2. 170 2 3 190 3 4 210 4 5 6 220 5 6 230 Fig. 8. Model of Martian atmosphere, showing temperatures and pressures used in radiative transfer calculations.

88 For the radiative transfer calculation, two further levels at pg/2 and 5pg/6 were inserted, and the calculation methods described in Chapter 2 were applied to this simple atmosphereo From Eqo (2~4~7), the flux difference between adjacent levels may be written in the form Fm v il ami Ji m = (431) In order to arrive at this form of the equation the source function (identical to the Planck function in these calculations) has to be assumed frequency independent over finite frequency intervalso It wa.s found that using the average value of Ji in 10 cm1l intervals was satisfactory. This should be contrasted to the mesospheric calculations in Chapter 3, where the source function was a.ssumed constant over a. whole absorption band, approximately 100 cm wideo In each of the 10 cm-1 intervals the source function Ji was expanded in the form Ji(V,T) do i + dli T + diT (432) The coefficients d d and d2 were obtained by requiring the quadratic expression to be exact for three temperatures, Ti and Ti + 30K. In this wa.y awn error of less than 1% can be obtained over the ra.nge Ti~ 50~Ko The T.'s used are given in Fig. 80 Substituting Eqo (4~352) into Eqo (4o351), AFm may be expressed in the form

89 AFm = a,+m l (bmi Ti + Cmi Ti2) m = 1,2,... 5 (4 3~3) One difficulty remains: the coefficients a mi in Eq. (4.3.1) are functions of temperature, because of the dependence of line intensity and half-widths on temperature, and should therefore only be used for the temperature profile for which they are calculated. The temperature dependence of ami could be allowed for in a simple way by expanding the coefficient as a linear function of Tloo.,T5, but the amount of computation involved would be very large since the integration over the whole band would have to be made for many temperature profileso In view of the many assumptions made in the atmospheric circulation madel'.,the temperature dependence of the coefficients wa.s neglected. In theory, the dependence is easy to account for but practical limitations on computing time make it impossible at the present time. The values of the coefficients are shown in Table V. The a.ccura.cy of solution using Eq. (4o3 3) was compared with that of Eqo (4.3o1) for an extreme profile, assuming the atmosphere to be 50K colder, than the mean profile at all levels. The largest error was about 0.5 deg K/day in the cooling rate. These results are shown by way of example. It is clearly possible to repeat the computations a.s necessary, for any desired model atmosphere. Table VI shows the cooling rates derived from several representative temperature profiles.

TABLE V HEATING RATE COEFFICIENTS FOR THE MARTIAN ATMOSPHERE.A 1) -. 3 1658 E +0 2 B(1, 1)..B (1,5).65578E+00 -.96416E-01 -.22109E-01 -.90128E —02 -.16149E-01 C(1, 1)...C f (1,5) -.26919E-02.37790E-03.84839E-04.34486E-04.61785E-04 A(2) -.11207E+02 B(2, 1)...B (2,5).20483E+00.26009E+00 -.19079E+00 -.36926E-01 -;.49174E-01 C(2,1)..oC(2,5) -.84228E-03 -. 10202E-02.73195E-03.14135E-03.18819E-03 A (3) -. 19648E+02 B (3, 1).B (3,5) -.22993 +0O0.512514E+00.33239E+D0 -..22086E+00 - 11244E+00 C(3, 1)..C (3,5).994419E-03 -.200BOE-02 -.12735E-02.84507E-03. 43014E-03 A(4) -. 19465E+02 o B (4, 1)..B (4,5) -. 45554E-01 -. 23934E+003.54998E+00.38205E+00 -.37851E+00 C(4, 1)..C(4,5). 18731E-03. 93777E-03 -.21044E-02 -..14587E-02.14504E-02 A(5) - 14975E+02 B(5, 1)...B(5,5) -.23383E-01 -.42701E-01 -.26976E+00 54628E+O.000OOOOOE+00 C(5, 1)..C (5,5).96087E-04.16736E-03.10332E-02 -.20834E-02.0000OE+00

91 TABLE VI HEATING RATES IN THE MARTIAN ATMOSPHERE FOR TYPICAL TEMPERATURE PROFILES Temperature at level I (~K) or heating rate in layer I (~K/Day) I 1 2 3 4 5 Temperature 170 190 210 220 230 Heating -4.3 1.4 -3 7 -7.6 -10.0 Temperature 170 190 210 220 250 Heating -4.0 2. -1.8 -1.8 -13 -10.0 Temperature 170 190 210 220 210 Heating -4.5 0.8 -5.3 -12.8 -10.0 Temperature 170 190 210 210 200 Heating -4.7 0.2 -7.3 -12.5 -6.5 Temperature 190 210 230 240 250 Heating -8.8 0. -6.7 -11.4 -13.8 Temperature 170 180 190 210 230 Heating -5.0 -0.3 0.7 -0.4 -9.6 The form of the solution in Eq. (4.3.3) is simple and quick to evaluate, and therefore meets the requirements for use in the circulation model. 4.4 COMPARISON WITH PREVIOUS CALCULATIONS Calculations of radiative transfer in the Martian atmosphere have been made by Ohring and Marino (1966) for several different combinations of surface pressure and atmospheric composition, although not for pressures as low a.s 6 mb. Their long-wave radiative transfer considered only the 1~5 4 bands of carbon dioxide, using a modified form of Elsasser' s (1960) radiation tables. It is not clear exactly what allowance was

92 made for temperature effectso Their results showed stra.tospheric equilibrium temperatures some.15 to 20~K lower than those of Pra.bha.kara, and Hogan (1965)o This may be a, consequence of the error in computing the absorbed radiation in the latter paper, as discussed in section.3403. The results of Ohring a~nd Ma.rino cannot be directly applied to a, circulation model, although they could probably be modified to do so. However, calculations by Leovy (Ijeovy,.1966 and Leovy a~nd Mintz, 1966) were made specifically for this purpose~ The transmissivities were calculated by the same method as Prabha.kara, a,nd Hogan (1965), which introduces an unknown error. However, the manner of calculating transmissivities is not essential to the method. The circulation model used by Leovy and Mintz assumes a, constant lapse rate in the troposphere (except for a, surface boundary la.yer), making the temperature dependence of the transmission functions ea.sier to incorporate. A similar method could probably be used for the calculations described here, although it may be'better to use the consta.nt coefficients initially to test the circulation model, and to modify them a.fterwards when it becomes appa.rent precise.ly wha.t kind of temperature variations ca.n be expected.

CHAPTER 5 CONCLUSIONS The main result of this study is the development of a. method of computing flux divergences in a plane parallel atmosphere by direct integration with respect to frequency over molecular absorption ba.nds. Direct integration methods, which have already been used to calculate atmospheric slant path transmissivities, offer advantages over previous methods: 1. The Curtis-Godson approximation is applied over successive thin atmospheric layers, instead of thick layers. 2. Band models are completely a.voided and the actual position, intensities and half-widths of the lines a.re used, at lea.st to within the a.ccuracy that they are known. The calculation of flux divergences by direct integration shares these two advantages, but also makes possible other features: 3. A diffusivity factor is not employed. Instead, exponential integrals are used and efficient polynomial approximations for E2(x), E3(x) and E3(x) have been developed. 4. A source function which is linear in pressure in thin horizontal layers is used. It is shown to be more accurate tha.n using isothermal layers or higher order polynomials to approximate the source function. 5. Flux differences rather than fluxes are calculated, avoiding numerical difficulties when the differences are small, This is important at low pressures where small differences in flux can provide large amounts 93

94 of cooling. The atmospheric layers can be made arbitarily thin to a.cco-modate fine structure in the vertical temperature profile. 6. The method is versatile and can be used for any absorber or mixture of a:bsorbing ga.ses~ The disadvantages of the method a.re~ lo It requires a~n intimate knowledge of atmospheric absorption ba.nds, although this is true to a! large extent of a.ny method~ 2. It requires comparatively large amounts of computing time. This factor can be expected to decrease in the future as fa.ster electronic computers are developed, and computing becomes less expensive. Two applications of the method a.re given. The first is to the earth's atmosphere between about 60 and 100 km, where radiative cooling due to the 15 p carbon dioxide bands is investigatedo Vibrationa.l relaxation is taken into account to derive source functions for the individual bands. Cooling rates a.re obtained for'both model atmospheres and a.ctua.l soundings. The results show that cooling predominates near the mesopause, but that slight heating frequently occurs at the temperature minimum or where discontinuities in the temperature gradient exist~ Local features of the vertical temperature structure dominate the cocling rate Vibra.tiona.l relaxation influences the cooling rate above 70 to 75 km where the rates arre strongly dependent on the vibrationa.l relaxation time. The radiative cool.ing rates calculated from mean profiles are significantly smaller tha.n the mean of cooling rates obtained from measured temperature soundings,

suggesting a, possible mean energy sink for the upper mesosphere and lower thermosphere. The second application is to the Ma.rtian atmosphere, where radiative transfer calculations are performed for incorporation in a, simple circula,tion model. A pure carbon dioxide atmosphere and a, surface pressure of 6 mb are used to illustrate the method.

CHAPTER 6.. SUGGESTIONS FOR.FUTURE WORK The techniques developed in this paper can be applied to a. wide variety of atmospheric radiative problems9 The speed of the available computer was a. definite Limit ation on the scope of applications described here, but future computing developments will make more extensive computae tions poss ible The lack of detailed knowledge of the absorption bands of water vapor (653 [A and pure rotational bands) and of ozone (996 p bands) prep vents the accurate calculation of cooling rates in other spectral regions for the earth's atmosphere~ Both experimental and theoretical investigations of these absorption bands are being made and positive results can be expected within a. few years. This will make possible the application of the present method to other spectral regions, as well as to the rest of the atmosphereo In the region of the earth's atmosphere where vibrationa.l relaxation is important, accurate values of relaxation time are needed. An equation for the source function for the carbon dioxide overtone bands in the 15 p. region is required. Because of the influence of local temperature structure features, more calculations using measured profiles are required to determine mean cooling rates nea.r the mesopaluse, and to obtain an understanding of the role of the 15 pC02 bands in the energy balance of the upper atmosphere9 96

97 As more details of the composition.of the Martian atmosphere become available, it will be possible:to modify radiative transfer calculations. The problem of vibrational relaxation in the upper atmosphere is also of considerable interest, but cooling rates may again be dominated by local temperature features, which are unknown at the present time. It would be of interest to apply the present method of calculation to obtain radiative equilibrium temperature profiles~ Although the methods described in this study were developed specifically to examine long-wa,ve radiative transfer, they ca.n be modified and applied to the absorption of solar radiation in the near infrared; An understanding of the combined effect of all radiative- transfer processes in the upper atmosphere would constitute a. significant advance in atmospheric physics.

APPENDIX.. THE POLYNOMIAL APPROXIMATION OF EXPONENTIAL INTEGRALS In radiative transfer problems a.ngular integration of the tra.nsfer equations frequently produces te-rms involving exponential integrals. The nth order exponential integral En(x) is defined by En(x) I e Xt/tn dt E (X) may be eva.luated in a. number of ways, including series and asymptotic expansions,- or less,precisely using polyn.om.ial or rationa.l approximations. (Abranmowitza -and Steguin 1964). A recurrence relation may be used to obtain higher order integrals from E1(x). T4e rational approximations are quite suitable for use in a computer subroutine, but they involve the evaluation of the exponential function or natura.l logarithm, or both (depending on the value of x), a. fa.ctor which considerably increases computing time. Furthermore El(x) has a singularity at the origin, whereas the higher order integrals a.re well behaved. In addition, va.lues obtained by the recurrence relation for x > 5 are subject to considerable error. Thus there are advantages in computing the higher order integrals directly, rather than from the first order integral. The approach adopted was to approximate the integrals by polynomials of intervals on the posi'tive x9axis, using a.east; squares technique. 98

99 Suppose a polynomial of degree m m pm(X) i a=i X is used in the interval (xl,x2). The condition to be satisfied is that x2 2 fxl [pm(X) - E(x) dx is a. minimum. In practice, pm.(x) is expressed in terms of Legendre polynomials Pi(x) which are orthogonal on (x1,x2) m pm(x) iZO A. P.(x) i=O The condition then becomes x X2 Jx Pi(x) [Pm(x) En(x) ] dx = 0 i = O,...,m or x2 Ai Ci =fXl Pi(X) En(x) dx i Om (A-1) AX~Pi xI En~x~dx i = O,..o,m (A-i) where 2 C. = f 2 (P.(x)) dx The integral in Eq.-A.-l can be expressed a.s a: linear sumn of integrals of the form dj = 2x En(x) dx j = O,.o. i J X~~

100 By successive integration by pa.rts, or by induction on j, it can be shown that d. ( J ) [x j -kE+l+k( xi)- En++k(2 ] dj = k'~O (j k), This method can be used for values of xl greater than about.o0. The values of El(xl) and El(x2) were obtained fromr published. tables and the recurrence relation employed for the higher order integrals. Double precision arithmetic was used. For small values of xl and x2 the numerical value of the individual terms in the series expansion are many orders of magnitude greater than the sum, so that even double precision does not produce sufficient a.ccura.cy. The integrals were evaluated for x1 and x2 between 0o0 and 1. 0 by Gaussia.n quadrature. The values of the exponential integrals at the quadrature points were accurately obtained from the series expansion of El(x) (which converges rapidly on this interval), followed by the recurrence relation. Radiative tra nsfer applications generally require a. high absolute accuracy for the exponential integrals, rather than a high relative accuracy. This implies that for larger values of x, the number of significant figures of accura.cy may be quite low. The aim of investigation wa.s to produce a.n efficient method to calculate the exponential integrals to an absolute accuracy of about 1 x 10'7. For E4(x) this can be readily achieved, although it requires the interval (0o,) to be broken into rather a large number of subintervals. Details are given in Table VII. E3(x)

101 requires slightly more subintervals but is otherwise similar. Table VIII lists the intervals a.nd coefficients. Near x = 0 E2(x) presents some problems, since its slope is not finite at x = 0, a characteristic which make approximation by a polynomia.l in this region very difficult. In the interval (0,0.005). function wa.s expanded: E2(x) 1 + 1 + n x - 1 x + n x] where y is Euler's constant. For x >.005 a. polynomial expansion was employed (see Table IX). The resulting computer subroutines, although not mathematically elegant, provide a rapid means of evaluating E2, E3 and E4. With the exception of the region near x = 0 for E2, the average time for evaluation is about 0.24 m sec for the IBM 7090, which is almost twice the average speed of the system subroutine for the exponential function.

TABLE VII COEFFICIENTS FOR THE POLYNOMIAL APPROXIMATION OF E4(x) Xi X2 N A O) At(1) A(2) A(3) A(4) A(5) A( ).00.0 2 3. 33333321E+00 -.49991784E+00.48723621E+O0 -.4,0065675F 0O *.02.10 4.33333145E+00 -.49976349E+OQO.48623636E+00 -.:54877654E+00.67427701E+0Q.10.20 4.33329039E+00 -.49820931E+00.46314345E+00 -.38876234E+00.23942219E+00.20.40 4.333,00485Ee00 -.4930687,2E+00.42766140E+00 -.27717782E E00.10435406E+00.40.70 5.33253226E+00 -.48847193E+00 1.41209314E+00 -.26007762E+00.11597315E+00 -.26C919O5E-01.70 1.00 4.32618509E+00 - 44764367E+00.30568652E+00 -..11924852E+O0.21083163,E-C1 1.00 1.50 5.323 69356E+00 -.44028710E+00.30183790E+00 -. 12713362 E+00.3 1436526E-01 -.348479 74E-C2 1.50 2.50 6.31416286E+00 -640980641E+0O.26366584E+00 -.'10465057E+0O.26191553E-C1 -.38024567E-C2.244183t7 E-03 2.50 4.00 6.26476052E+00 -.29454781E+00.-14995841E+00 -.43917471E-Ol.76716851E-02 -.74718008E-03.31347324E-04 4.00 6.00 6.16347779E+00 -.14351490E+00.54917252E-01 -.11618356E-01.14 22i296E-02 -.94971 422E-O4.26e86468E-0 I6.00 8.00 5.45805090E-01 -.27338178E-01.66093-63E-02 -.82506626E-03 a 511796823F-C4 - 13152 07E-C5 R) 8.00 10.00 4.61485633E-02 -.24149872E-02.35942825E-03 -.23983739E-04.60455421E-06 10.00 12.00 3 43135189E-03 -.10'716388E-03.89272471E-05 -.24915099E-06 12.00 14.00 2.16129888E-04 -.23012068E-05.82365827E-07

TABLE VIII COEFFICIENTS FOR THE POLYNOMIAL APPROXIMATIONS OF E3(x) Xl X2 N A(0) A(l) A(2) A(3) A(4) A(5) A(6).00.01 5.5O00000E+00 -.99927661 E+00.35030680E+01 -.14451565E+03.91145339E+04 -.27510454E+06.01.05 5.49998486E+00 -.99367759E+00.24543231E+01 -.18666429E+02.16957034E+03 -.7823345 E+C3.05.10 5.49987973E+00 -.98250148E+00.19531425E+01 -.68473836E+01.24141827E+C2 -.43633200E+C2 -.10.20 5.49950246E+00 -.96447504E+00.16000413E+01 -.33014884E+01.55086080E+01 -.53301442E+01.20.40 5.49797289E+00 -.92834808E+00.12496552E+01 -.15556700E+01.14440030E+C1 -.65387457E+C0.40.70 5.49291603E+00 -.86641115E+00.93952389E+00 -.76213922E+ 00.40763092E+00 -.10229208E+C0.70 1.00 5.48203747E+00 -.78785009E+00.70970913E+00 -.42183647E+00.1 271436E+00C -.25C82541E-C 1.00 1.50 5.46228186E+00 -o69286540E+00.52553268E+00 -.24177842E+00.(:395S788E-C1 -.74386331E-C2 1.50 2.50 6.43684018E+00 -.61053170E+00.42011457E+00 -.17665828E+00.46264005E-C1 -.69 480434E-02.45746718E-03 2.50 4.00 6.34656643E+00 -.39789156E+00.20831318E+00 -.62459659E-01.11121848E-01 -.10999885E-C2.46719667E-04 4.00 6.00 6.20000240E+00 -.17767995E+00.68695208E-01 -.14662192E-01.1086938E-C2 -.12154629E-C3.34603193 E-05 6.00 8.00 5.52636089E-01 -.31518925E-01.77010713E-02 -.95616821E-03.60148400E-04 -.15299014E-05 8.00 10.00 4. 68132871E-02 -.26793015E-02.39918259E-03 -.266660540E-04.67255760E-C6 10.00 12.00 3.46744948E-03 -.11619393E-03.96840454E-05 -.2703841 E-06 12.00 14.00 2 17225127E-04 -.24581282E-05.88003757E-07 ~~ ~ ~ ~

TABLE IX COEFFICIENTS FOR TUE POLYNOMIAL APPROXIMATION OF E2(x) Xl X2 N A(O) A 1) A(2) A(33 A(4) A(5) ( 5).00,5.0 1 5.99863854E+00 -.64849862E+01.28699864E+03 -.202302 11 E+05.92987275E+C6 -6 18864517E+C8.01.02 5.99719845E+00 -.57667626E+01.14015630E+03 -.48733755E+04.11095948E+06 -.11211459E+C7.02.03 5.99512047E+O0 -.52178244E+01.81215706E+02 -.16637561E+04.22536892E+05 -.13650376E+C6.03.05 5.99227478E+00 -.47562499E+01.50897054E+02 -.65548040E+O3. 5 5339225E+C4 -.2087'15 3 5E+05.05.08 5.98743435E+00 -.42707709E+01.31161615E+02 -.249118857E+03.13013803E+C4 -.30413637E+04.0 8.14 5.97890829E+O0 -.37515615E+01.18314077E+02 -.87710226E+02.27097516E+03 -.37351891F+C3 14.215" 6.96 617789E89 0 EA - 0.32935617E'+01'.116'33582E+02 9.3855427 55E+2.9O5i739E+02 -. 12506467f403.75047'64E+02. 2 5 i~i 1.40:1 6.94'232903F+QOd 0 ~~~-.,2 76'02 169E-0 1;.66185588E+1 -.813205705E14-O2.18236745E+C2 -.14695374E+C2.5i12657OE0.40.70 6.910OO-254-OEE-00 -.23189262E+01.40972991 1E+101 -.5512366'9E OiJ.5C584481E+O1 -.27377738E+O01.6 312535f+0.70 1.00" 5.83257-7151E+00 -.16'82,4257E+01.18822436E+O1 -.13316793E+01.54534O9(E+00 -.975555:65E-01 1.00' 1.50", 6.7896;4619E-0Ob -.147142:04E+01.14733369E+Ol -.951335+.3:167846E+CO 1.50 2.50 6.6768079W7E+0O -.10419146E+01.78456730E+0O -.35571203E+ZYO.9d812997E-C1 -.155218UE-O1.10573839E-02 2.50 4.00- 6.48261635-E+00 -.57616-474,E+00.31201874E+00 -.96201227E —QJ. 0.1751.8586E-1i -. 17638563E-02.759E8859E-04 4.00 6.00 6.2531862OEL00 -.228163453E+00.89305974E-01l -.19262687E-01.23975807E-02 -.16236047E-03.46529556E-05 6.00 8.00 5 -,.61520229E-01 -.36981421E-01.90654130E-02 -. 11 287050-02. 71105 86E-04 -. 18139326E-C5 8.00'' 10.00' 4.76228Q,02E-02'-.30017878E-02.44776459E-03 -.29936337E-04..75588066E-C6 10.00- 12.00'3. 509636 09E-03 —'e.12675641E- 03.10569960E-04 -.29525857E-06 12.00 14002.18470210E 04''-k-.26366002E -05 -.94418581E-07

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