J. Qunnr.Spectm~c.Radinr.Transfeer. Vol. 14,pp. 861-871.PergamonPress 1974.Printedin Great Britain. RADIATIVE TRANSFER DUE TO ATMOSPHERIC WATER VAPOR: GLOBAL CONSIDERATIONS OF THE EARTH’S ENERGY BALANCE* ROBERT D. CESS Department of Mechanics, State University of New York, Stony Brook, New York 11790, U.S.A. zyxwvutsrqponmlkjihgfedcbaZYXWVUT (Received 14 December 1973) Abstract-A simple analytical formulation is presented for describing radiative transfer due to atmospheric water vapor. The radiative model is then applied to a global energy balance for earth, and the net infrared flux to space is expressed in terms of the mean surface temperature and atmospheric lapse rate. Water vapor and clouds are assumed to be the only sources of infrared opacity. When compared with empirical information, and for a global mean surface temperature of 288 K, the radiative model indicates a cloud top altitude for a single effective cloud of 6.8 km. Alternatively, when applied to a more realistic three-cloud formulation, the model predicts a comparable value of 6.5 km for an average cloud top altitude. With respect to changes in mean surface temperature, again comparing with empirical results, a discussion relating to the model suggests that the cloud top altitude decreases with decreasing surface temperature, which results in the surface temperature being roughly twice as sensitive to changes in factors such as planetary al&do than for the conventional assumption of a fixed cloud top altitude. Implications of this are discussed with respect to possible albedo changes due to atmospheric particulate matter as well as cloudiness as a climate feedback mechanism. 1. INTRODUCTION THE MAIN source of infrared opacity within the earth’s lower atmosphere is water vapor, and numerous computational procedures have been presented for evaluating the atmospheric infrared radiative flux (e.g. Chapter 6 of GOODY,(‘) RODGERS,(~) MANABE and WETHERALD,(3) RASOOL and SCHNEIDER (4) and DOPPLICK(~)). It is not at all clear, however, that these various methods are mu&ally consistent. Furthermore, realistic radiativedynamical studies of the earth’s atmosphere require the incorporation of a large number of quite complicated phenomena, and for this reason it would be useful to reduce the radiative flux calculation to as simple a form as possible. The purpose of the present paper is threefold. The first objective is to reexamine the problem of calculating the infrared radiative flux within the earth’s atmosphere due to water vapor, with emphasis upon incorporating emissivity data which is based upon direct laboratory measurements. Secondly, the radiative flux formulation will be expressed in an extremely simple analytical form, employing a procedure which is quite analogous to that used by CESS and KHETAN’~) in describing radiation due to the pressure-induced opacity of hydrogen within the atmospheres of the major planets. Aside from computational ease, the advantage of a simple analytical formulation is that it facilitates physical insight into problems for which radiation is coupled with other processes. In this regard, the third objective of the paper is to consider a global mean *This work was supported by the National Science Foundation 861 through Grant No. K036988. ROBERTD. CESS 862 energy balance upon the earth’s atmosphere, with emphasis upon coupling mechanisms involving cloudiness and albedo changes. The importance of such mechanisms has been accentuated by recent concern over possible man-made modifications to the earth’s climate. 2. WATER VAPOR SCALE HEIGHT A prerequisite to investigating the water vapor opacity within the earth’s atmosphere involves specification of the water vapor scale height. This differs from the atmospheric scale height, since the water vapor mixing ratio is controlled by condensation, which in turn is governed by dynamical processes. As of now numerical models of the hydrological cycle have not correctly reproduced the distribution of water vapor within the earth’s atmosphere (e.g. HUNT(~)). Alternately, MANABE and WETHERALD(3) have employed an empirical expression for the vertical distribution of relative humidity (global average) within the earth’s troposphere, based upon a composite of experimental data, and this expression is = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG (0.77/0.98)(P/P, - 0.02), PJPwJar where P is atmospheric pressure, P, the surface pressure, and P, and P,,,,, the water vapor partial pressure and saturation pressure, respectively. For present purposes a sufficiently accurate representation within the troposphere is of the form P,IP,,,,t = 0.77(P/P,). (1) Equation (1) does not apply to the stratosphere, where the water vapor mixing ratio is constant, but as discussed in Section 6, within the limitations of the present model the stratosphere plays an insignificant role with regard to the overall opacity of the earth’s atmosphere. We now wish to rephrase equation (1) in terms of a water vapor scale height. and to accomplish this P,,,, satmay be expressed as a function of temperature T(K) by P w,,,,(atm) = (2.20 x 106)e-5385’T. (2) Equation (2) is simply the Clausius-Clapeyron equation with the constants fitting the equation to saturation vapor pressure data. Since P/P, = exp( -gz/RT) for a hydrostatic atmosphere, where g is the acceleration due to gravity, R is the gas constant for the atmosphere, then upon combining equations that P, = Pwse-z/Hw with P,, , the water vapor partial (1) and (2) to be pressure P,,(atm) at the earth’s surface, following = (1.69 x 106)e-s385’Ts, while the water vapor scale height, H,,,, is evaluated by (3) z is altitude, and (l)-(3), it follows (4) from equations (5) Radiative transfer due to atmospheric water vapor Furthermore, with l? = -dT/dz the troposphere, then denoting H, = Since the atmospheric the lapse rate, and taking r to be constant RTlg 1 + (5385RT/gT,) 863 within - scale height is H = RT/g, it follows that H/H,,, = 1 + 5385RT/gT,. (7) It is important to note that for this approximation H/H, is independent of local atmospheric temperature. With g = 981 cm set-’ and R = 0.287 x lo7 cm2 sece2 K-‘, together with r = 6.5K km-’ and T, = 288K, which are representative of global mean conditions, equation (7) yields H,JH = 0.22, which compares with the value HJH = 0.25* as employed for example by RASOOL and SCHNEIDER,(~) and SCHNEIDER.@) Consider now the application of the water vapor scale height to determining the effective broadening pressure P for the rotational lines of water vapor. The broadening pressure is the total atmospheric pressure, and application of the Curtis-Godson approximation (p. 238 of GOODY(‘)) yields p” = (l/P,,,) j-P dP,. Employing equations (3) and (4), P” is related to P by ‘= (1 +iIJH)” with p = 0.82P for H,IH (8) = 0.22. 3. WATER VAPOR EMISSIVITY Use of the gas emissivity in radiative flux calculations is well known, and in this section we present an analytic expression for the water vapor emissivity which greatly facilitates such flux calculations. Employing the strong-line limit, the water vapor emissivity, E, may be expressed in terms of a single variable as zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA E = c(P,PH,,,). A quantitative appraisal of the applicability of the strong-line limit is given by RODGERS.(~) Furthermore, the form of the strong-line parameter, P,P”H,, is consistent with the results of PENNER and VARANASI’~’for water vapor at elevated temperatures, as well as the band absorptance correlations for water vapor by EDWARDS and BALAKRISHNAN.(~‘) MANABE and WETHERALD, on the other hand, have employed P, H,,,(P)o.7 as the strong-line parameter. With the water vapor emissivity expressed as a function of P,pH,, then the emissivity for atmospheric applications may be determined from laboratory data taken at pressures other than those characteristic of the atmospheric applications. Laboratory measurements of the water vapor emissivity have been performed by HOTTEL and coworkers,(“J2) * This is equivalent to assuming that the water vapor mixing ratio varies as the fourth power of total pressure for a hydrostatic atmosphere. ROBERTD. CESS 864 SCHMITT, and EcKERT,(‘~) while these data have been expressed by HoTTEL(‘~,‘~) in terms of an emissivity chart for P” = I atm and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM P,,,/p - - f 0. Since P,,,/p 5 lo-’ within the earth’s atmosphere, then Hottel’s emissivity results, when expressed in terms of the strongline parameter, should be directly applicable to an atmospheric formulation. The water vapor emissivity, as taken from HOTTEL’S chart, (15*‘6) is illustrated in Fig. 1 for T = 300K, and this denotes a column emissivity. As discussed on pp. 197-198 of GOODY,(‘) the temperature dependence of the water vapor emissivity is quite small for the temperature range 220 K-300 K, which is consistent with the results of RODGERS.(‘) Furthermore, the modified emissivity (employing the temperature derivative of Planck’s function as the weighting function) results of MANABE and WETHERALD, based on the earlier work of MANABE and STRICKLER,“~ are also quite insensitive to temperature within the same temperature range. This relative invariance with temperature refers to a fixed absorber amount (gm cm-‘), and since H,,, N T from equation (6), then P, H,, as appears within the strong-line parameter, is not a pressure path length, but is instead proportional to the absorber amount. Thus Hottel’s emissivity results may be employed for temperatures other than 300 K providing H, corresponds to the same temperature as the emissivity information, i.e. 300 K, and from equation (6) HJkm) = 8.78 (9) 1 + ( 10241~~) ’ It is interesting to note that in terms of a fixed absorber amount, Hottel’s emissivity chart indicates a further invariance of the water vapor emissivity upon temperature for temperatures from 300 K to over 550 K. A large amount of information concerning the water vapor emissivity is available in the atmospheric literature. We do not attempt to summarize here all this information, but instead we consider only two of the more recent emissivity formulations. One of these is due to RODGERS(‘) and employs the same strong-line parameter as in the present study, so that a direct comparison with Hottel’s emissivities is possible, and this is shown in Fig. 1 (in Rodger’s nomenclature, the present emissivity corresponds to EJ. The second formulation is that employed by MANABE and WETHERALD. Since their strong-line parameter is P, H,,,(P)“‘7, a general comparison in terms of a single parameter is not possible. Such a comparison may be made, however, for conditions appropriate to the earth’s zyxwvutsrqponmlkjihg 0.6 I I - 0.6 - HOTTEL I I -----RODGERS - -- MANAEE 6 WETHERALD Iv 01 I I 102 IO P,FH,. I IO' olmPCm Fig. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1. Comparison of emissivity models for water vapor. Radiative transfer due to atmospheric water vapor 865 atmosphere. For this purpose we choose zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH T, = 288 K, H,,,/H = 0.22, and I’/P = 0.82, while, from equations (5) and (9) P,, = 0.0128 atm and H, = 1.93 km. It then readily follows that P,p”H,,, = 0.61 [P, Hw(~)o~7]1~05s, and thus Manabe and Wetherald’s emissivity values, as given in their Fig. 27, may be converted to the present strong-line parameter. Their results are also illustrated in Fig. 1, with their slab emissivities converted to column emissivities employing the diffusivity factor 1.66. The differences between the three emissivity formulations shown in Fig. 1 are not insignificant. Since Hottel’s chart is based upon direct laboratory measurements and seems to have stood well the test of time (e.g. discussions by PENNER and TIEN(‘~)), we deem it preferable to employ Hottel’s results. There is, in addition, theoretical justification for the dependence of Hottel’s emissivities upon the strong-line parameter. In the limit of nonoverlapping strong lines, the emissivity should vary as the square root of the strong-line parameter, and from Fig. 2 this is indeed the case, where the expression zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR 0.8 0.6 I I IO IO' x = P*Ptt*, Fig. 2. Comparison at12 cm of a simple approximation to Hottel’s emissivity results. is an empirical fit to Hottel’s curve for small values of P,PH,,,; i.e. the limit of nonoverlapping lines. This proper reduction to the square-root limit is consistent with the interpretation by PENNER and VARANASI(9) of Hottel’s emissivity values at elevated temperatures. Also illustrated in Fig. 2 is an empirical fit to Hottel’s curve given by E = 0.75 [l - exp( - 0.096JPxw)]. (10) The above expression is clearly a useful approximation for the range of values of the strong-line parameter which have been considered, and this range is characteristic of the earth’s troposphere. Equation (10) will be employed in the following section with regard to a simple analytical formulation for the radiative flux. 4. RADIATIVE FLUX FORMULATION There are a variety of ways, employing integration by parts, by which the radiative flux may be expressed in terms of emissivity, modified emissivity, and combinations thereof (see for example, p. 251 of GOODY,(‘) and RODGERS”)). For present purposes, since the 866 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ROBERT D. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA CESS water vapor emissivity is quite insensitive to temperature, we will follow the same procedure as employed by CESS and KHETAN(6) in formulating the radiative flux due to the pressure induced hydrogen opacity for the atmospheres of the major planets. Defining a dimensionless optical coordinate as [ = (3/2)(0.096,/P&,) (11) with pressure in atm and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA H, in cm, the formulation of the radiative flux due to water vapor, making use of equation (lo), becomes essentially identical to that of CESS and KHETAN.‘~’ Letting qR denote the net infrared flux measured in the downward direction, it then follows that qR = 0.757 _J,j’-7 e T4(5> 5d5 -T$(1/3 T4(t)ejt21.~2 + eJts2-)] (12) where g is the Stefan-Boltzmann constant, < is a dummy variable for [, and [,y denotes the value of < at the planet’s surface. This equation differs from that for the major planets (6) through inclusion of the last term, which accounts for surface emission. In addition, the arguments of the exponential functions are different, since the appropriate scaled quantity, (P, p), between levels is expressed as O.l44J<P, in accordance with previous discussion 5. GLOBAL B)H, = \:([2 - c2/ of the Curtis-Godson ENERGY (13) approximation. BALANCE To illustrate the preceding radiative flux formulation, we consider a global energy balance for earth, which amounts to relating the mean surface temperature to the outgoing infrared flux. For this purpose, we initially adopt the same cloud model as utilized by RASOOL and SCHNEIDER,(~)and SCHNEIDER.(*) This consists of a single cloud layer representing a global average of many cloud layers. The single cloud is assumed to be black in the infrared, the cloud top is located at the altitude z,, and the cloud covers a fraction of sky denoted by A,. The global mean surface temperature of the earth is roughly 288 K, as compared with the effective temperature of 253 K. The difference, 35 K, is the greenhouse effect due to the opacity of the earth’s atmosphere. Water vapor and clouds contribute most of this opacity, with the carbon dioxide contribution being quite small. The analysis of RASOOL and SCHNEIDER(~) for example, indicates that CO, contributes only about 1 K to the greenhouse effect. Correspondingly we shall neglect the opacity contribution due to CO,. Equation (12) is directly applicable to the clear portion of the atmosphere, while it applies to the region above the black clouds providing T, and [, are replaced by T, and c,, where the latter quantities refer to the cloud top. Letting F = -qR(0) denote the infrared flux emitted to space, it then follows that + 0.75 (T/Ts’J4ebr di + (Tc/Ts)4(l/3 -t edit) 1 A,. zyxwvutsrqponmlkjihgfedc (14) Radiative transfer due to atmospheric 867 water vapor Evaluation of the integrals within equation (14) requires description of the atmospheric temperature profile. Since H = RT/g, the tropospheric lapse rate, dT/dz = -I, may be rephrased as dT4/T4 = n de/c, where (15) and the temperature profile within the troposphere is in turn T”(i) = Ts”(Ws)“. (16) For the present it will be assumed that equation (16) is applicable throughout the entire atmosphere, although it is valid only for the troposphere. We will confirm this assumption in the following section by showing that inclusion of an isothermal stratosphere does not influence F. Typical values of [, are sufficiently large (I’,‘,2 6) such that the first integral within equation (14) may be evaluated from the asymptotic expression for the incomplete gamma function, and employing equation (16) scs(TITs) + 1) - e-‘*, zyxwvutsrqponmlkjihgfedcbaZYXWVUT (17) 4e-c d[ = [s-nr(n 0 where I(x) denotes the gamma function. Typical values of [,, on the other hand, are sufficiently small (5,s 1) that the second integral in equation (14) may easily be evaluated from m I0b’s)4e-idl = L-“j~o (_ jr(n l)j((,y+l+j + 1 +j) . (18) Equations (14), (17) and (18) thus completely describe the outgoing flux F for a specified surface temperature and cloud top altitude, with 5, evaluated from equation (11) together with equations (5), (8) and (9). The cloud top temperature is related to cloud top altitude by T, = T, - l-z,, while 5, = C,(Tc/TJ4’” from equation (16). For example taking I’ = 6.5 K km-‘, T, = 288 K, and z, = 5.5 km, then 5, = 6.47, n = 0.274, 5, = 0.932, and F/aTs4 = 0.656 - O.l37A,. 6. DISCUSSION For comparative purposes equation OF RESULTS (14) will be expressed F=c, -c2Ac, as (19) where c1 and cz depend upon T, and I, while cz is additionally a function of z,. In the following we take I = 6.5 K km-‘, and values for ci and cz for T, = 288 K, from both the present analysis and other sources, are listed in Table 1. Also included in Table 1 are results for which an isothermal (218 K) stratosphere has been included, and this accounts for the fact that H,,, = H within the stratosphere.(3) Recall that the analysis of the previous section ignored the stratosphere and assumed that equation (16) applied throughout the entire atmosphere. From Table 1, this is clearly a useful assumption and illustrates that the stratosphere opacity does not affect the overall energy balance of the planet. SCHNEIDER@)has employed the empirical formulation Of BUDYKO(“) as a standard against which a model atmosphere calculation may be compared, and we shall do the same. Budyko’s QSRT Vol. 14 No. 9-D 868 ROBERTD. CESS Table 1. Values of cr and cz for T, = 288 K cal cm-* min- Present analysis, z, = 5.5 km Present analysis, z, = 6.5 km Present analysis, z, = 6.8 km Present analysis, z, = 6.8 km (218 K stratosphere included) SCHNEIDER,(~) z, = 5.5 km BuDYKo,(~‘) empirical formulation, as based upon monthly c2, Cl, Reference 1 cal crnm2 min-’ 0.361 0,367 0,361 0,368 0.077 0.096 0.102 0.102 0.399 0.366 0.107 0.102 data from 260 meteorological stations, F = 0.3 19 + 0.00327’, - (0.068 + 0.00237’,)4,, may be expressed (20) with Fin cal cm-’ min-’ and T in degrees centigrade. From Table 1 we see that for T, = 288 K the present analysis is nearly identical to Budyko’s empirical expression for a cloud top altitude of 6.8 km. SCHNEIDER,@) on the other hand, comparing his c2 value with that of Budyko, suggests a cloud top altitude of 5.5 km. The difference between the present cloud top value of 6.8 km and that of 5.5 km suggested by Schneider is evidently due to the use of different water vapor opacity models. There appears to be justification for the present value, and for this purpose we consider the more realistic three cloud model of MANABE and WETHERALD (see their Table l), which assumes all three clouds layers to be black in the infrared, with the cloud top altitudes and fractions of cloud cover listed in Table 2. Also included in Table 2 are values of A,, i , which represents Table 2. Parameters for the three-cloud model of MANABEand WETHERALD z,, t , km 10 4.1 2.1 Ae. 1 A,.i 0.228 0.090 0.313 0.228 0.069 0.220 c,,,,calcm-*min-r zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON 0.171 0.053 0.033 the cloud cover fraction which is not overlapped by an upper cloud (or clouds), calculated from the A,,i values in accordance with MANABE and STRICKLER,(‘~) together with c2, i values which were calculated by treating each cloud layer individually by the single-cloud analysis of the previous section, i.e. evaluating c2 as a function of z, as in Table 1. It readily follows that the present analysis may be extended to multiple clouds by simply rewriting equation (19) as zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA F=c, -&i&i I From this we further define an equivalent single-cloud aititude and Wetherald’s three-cloud model by noting that an equivalent may be defined by corresponding to Manabe single-cloud parameter c2 (21) Radiative transfer due to atmospheric water vapor 869 since xi A,, i represents the fraction of the cloud cover for the three-cloud model. From to an equivalent Table 2, cp = 0.097 cal cmm2 min- ‘, and from Table 1 this corresponds single-cloud altitude of z, N 6.5 km. The present analysis thus predicts a single-cloud altitude of 6.8 km when compared with Budyko’s empirical results and an equivalent cloud altitude of 6.5 km employing Manabe and Wetherald’s three-cloud model. We therefore conclude that the single-cloud altitude of 6.8 km appears to be reasonable. A partial comparison of the present radiation model with that employed by Manabe and Wetherald may be made by noting that c1 represents the ‘clear sky’ outgoing flux. From Fig. 10 of MANABE and WETHERALD, c1 N 0.37 cal cm-’ min-‘, which is in excellent agreement with present results (Table 1). This is somewhat surprising, however, in view of the different emissivity models which have been employed (Fig. 1). Again following SCHNEIDER,@’ a second test of the accuracy of the radiation model is to compare the dependence of F upon surface temperature. Employing Budyko’s empirical expression, we have from equation (20) that (22) E = 0 0032 - 0 0023A aT, ’ . =’ while holding the cloud top altitude constant and varying T,in the present analysis gives = 0.0028 + O.O006A,. (23) The first term is in reasonable agreement with Budyko’s equation (22), whereas the difference in the second terms is significant and cannot be reduced by merely changing z,. Equation (23), however, appears to be consistent with other model atmosphere calculations. SCHNEIDER”) gives aFIaT, = 0.0033* for A, = 0.5, which is in good agreement with equation (23), while from Fig. 10 of MANABE and WETHERALD(3J it is found that aFIaT, increases slightly with increasing cloud cover, and this again is consistent with equation (23). All three analyses, i.e. equation (23), Schneider, and Manabe and Wetherald, have one feature in common, which is that the cloud top altitudes are held constant while T, is varied. This suggests that perhaps an alternate cloud model should be considered. A particularly simple possibility is one in which the cloud top temperature remains constant as T, is varied, which would correspond to a decrease in cloud altitude for decreasing T,. With T,= 288K and z, = 6.8 km, then T,= 243.8 K and, holding the cloud top temperature constant at this value, the present analysis yields zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ = 0.0028 - 0%-)025A,. (24) Equation (24) is in much better agreement with Budyko’s equation (22) than is equation (23). This suggests that, at least to a first approximation, an appropriate single-cloud model is one in which the cloud top temperature is held constant when the surface temperature is varied. l Schneider indicates agreement with Budyko’s result, stating that iJF/aT, = 04031 from equation (20). This is an error, however, since equation (20) yields aF/aT,= 04021 for A, = 0.5. 870 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ROBERT D zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML . Cess 7. CONCLUDING REMARKS In this section we briefly and qualitatively describe some implications of the present results to considerations of global climate changes. For this purpose it will be convenient to consider the sensitivity of surface temperature upon the effective temperature zyxwvutsrqponmlkjihgfed T,, where F = aTe4. If the water vapor content of the atmosphere is held constant, by describing for example a fixed distribution of absolute rather than relative humidity,‘3) then in the absence of clouds the opacity of the atmosphere will be independent of surface temperature, and it may easily be shown that aT,/ aT, = 1.11. As discussed by MANABE and WETHERALD, however, it is more realistic to employ a fixed distribution of relative humidity, and this results in a self-amplification effect of water vapor, since a decrease in surface temperature produces a decrease in water vapor content of the atmosphere, which in turn results in a decreasing greenhouse effect with the result that WJ3T, > 1.11. Let us now compare this amplification effect for two different models, assuming average cloudiness A, = 0.5. Considering the cloud top altitude to be fixed, as has been conventional in previous studies, then from equation (23) of the present analysis = zc=6.8 On the other hand, employing constant, equation 1.64. km (24). for which the cloud top temperature = is held 3.18. This illustrates that the cloud model with fixed cloud top temperature, which appears reasonable in view of previous discussion, produces nearly twice the amplification upon surface temperature than does the model employing fixed cloud top altitude. This surface temperature amplification is extremely important with regard to the problem of possible global climate changes due to increased particulate matter within the atmosphere. Preliminary studies of this have been conducted by RASOOL and SCHNEIDER,(~) and YAMAMOTOand TANAKA.“‘) The essential point is that increased particulate matter could alter the earth’s albedo, thus changing T,. The subsequent change in surface temperature would thus depend upon the above mentioned amplification due both to water vapor and clouds. A second mechanism for global climate changes involves the possibility of increased cloud cover. This has been discussed by SCHNEIDER,“’ and for a fixed cloud altitude he shows that increasing A, from 0.50 to 0.58 reduces the surface temperature by 2 K. Budyko, however, as quoted by Schneider, suggests that the effect of increased cloud cover might be offset by a corresponding increase in cloud altitude. But this is contrary to the suggestion of the previous section that, within the context of a single cloud layer, z, decreases with decreasing T, . In fact the present analysis indicates an amplification of Schneider’s calculated reduction in surface temperature from 2 K to 4 K. We do not imply that a positive amplification necessarily exists, but rather that the realities of global cloudiness probably defy simple explanation. Indeed, as pointed out by SCHNEIDER,@)a global-mean energy balance is probably insufficient for estimating average climate changes due to varying cloud cover. Quite clearly cloudiness as a coupling mechanism needs much additional study, and the present investigation is intended merely as a guide in this direction. Since the water vapor radiation model formulated in this paper combines both accuracy and simplicity, the results of Sections 4 and 5 should prove useful in such future endeavors. Radiative transfer due to atmospheric water vapor 871 REFERENCES 1. R. M. GOODY, Atmospheric Radiation. Oxford University Press, London (1964). 2. C. D. RODGERS,Q. J. Roy. meteor. Sot. 93,43 (1967). 3. S. MANABEand R. T. WETHERALD, J. atmos. Sci. 24,241 (1967). 4. S. I. RA%XL and S. H. SCHNEIDER, Science 173, 138 (1971). 5. T. G. DOPPLICK,J. atmos. Sci. 29, 1278 (1972). 6. R. D. CESSand S. KHETAN,JQSRT 13, 995 (1973). 7. B. G. HUNT, Tellus 25, 337 (1973). 8. S. H. SCHNEIDER, J. atmos. Sci. 29, 1413 (1972). 9. S. S. PENNERand P. VARANASI,JQSRTS, 391 (1965). 10. D. K. EDWARDSand A. BALAKRISHNAN, Inc. J. Heat Mass Transfer 16,25 (1973). II. H. C. HOTTELand R. B. EGBERT,Trans. Am. Inst. them. Engrs 38, 531 (1942). 12. H. C. HOSTELand H. G. MANGELSDORF, Trans. Am. Inst. them. Engrs 31, 517 (1935). 13. E. SCHMIDT,Forsch. Geb. Inghles. 3, 57 (1932). 14. E. R. G. ECKERT,VDZ Forschungsheft 387, 1 (1937). 15. H. C. HOTTEL,Heat Trunsmission (Edited by W. H. MCADAMS),Chapter IV, p. 85. McGraw-Hill, New York, (1954). 16. H. C. HOTTELand A. F. SAROFIM, Radiative Transfer, p. 232. McGraw-Hill, New York (1967). 17. S. MANABEand R. F. STRICKLER, J. atmos. Sci. 21, 361 (1964). 18. S. S. PENNER,Handbook of Heat Transfer (Edited by W. M. ROHSENOW and J. P. HARTNETT),Section 15-D. McGraw-Hill, New York (1973). 19. C. L. TIEN, Advances in Heat Transfer, Vol. 5. Academic Press, New York (1968). 20. E. M. SPARROW and R. D. CESS,Radiation Heat Transfer, Brooks/Cole, Belmont, California (1966). 21. M. I. BUDYKO,TeIlus 21, 611 (1969). 22. G. YAMAMOTOand M. TANAKA,J. afmos. Sri. 29, 1405 (1972),