JULY 1969 B. H. A R M S T R O N G 741The Radiative Diffusivity Factor for the Random Malkmus Band ~B. I-I. AR~ISTRONGIBM Scientific Center, Palo Alto, Calif. (Manuscript received 5 March 1969)ABSTRACT The diffusivity factor r that converts an intensity transmission function to a flux transmission function iscalculated for the Lorentz line shape as a function of the optical parameters for the new molecular bandmodel proposed by Malkmus. The results are compared with those for Goody's random exponential model.An estimate of the error in the transmission function due to the use of the usual r= 1.66 value can be obtainedfrom our results, and some typical errors are calculated.1. Introduction In a previous paper (Armstrong, 1968a), calculationsof an exact radiative diffusivity factor were presentedfor a number of molecular band models and line shapesof interest in atmospheric radiation studies. Theseresults can be used to accurately account for the integration of the radiative flux over zenith angle, or, inother words, to convert an intensity transmissionfunction to a flux transmission function$f (u) -- ~ -u). (1)In this definition, u is the mass of absorbing gas per unitarea, Tr and Tf the intensity and flux transmissionfunctions, respectively, the bar denotes an average overa selected frequency interval, and r is the diffusivityfactor. The practical magnitude of this factor wasshown a number of years ago by Elsasser (1942) to be~ 1.66. As has been demonstrated (Armstrong, 1968a),the exact numerical result which can be obtained as afunction of the optical parameters involved does notusually deviate much from Elsasser's value. It is usually assumed that the errors involved in theprecise choice of diffusivity factor are sufficiently smallthat an accurate treatment is not required. Hitschfeldand Houghton (1961) and Walshaw and Rodgers(1963) have made comparisons for a constant r= 1.66with more exact results (such as by numerical oranalytic integration over the zenith angle) that showedthe accuracy achieved with r--1.66 typically to be ofthe order of 1-2%. However, these results are forindividual comparisons and cannot be trusted in general.We have, e.g., carried out computations on the CO2levels as given in Table 2 by Walshaw and Rodgers thatindicate considerably larger errors for the separateupper and lower integrals that contribute to the heatingrate at a given level. The errors for these particularlevels, however, show a systematic cancellation thatreduces the net error to a lower figure. It appears likely,as more elaborate calculations involving more levelsand/or different formulations are performed, that someof this cancellation may cease to occur, and more accurate treatment of the zenith-angle integral may berequired. We also note the suggestion of Rodgers (1968)that use of the new Malkmus model with improvementsin the Curtis-Godson approximation~ may yield a moreaccurate accounting of the ozone heating rate than hasheretofore been available. A substantial improvementin this heating rate may permit the error arising fromthe constant diffusivity factor approximation to becomecompetitive with the remaining errors. Therefore, wehave calculated the exact diffusivity factor (as a function of the optical parameters) for this new model tosupplement the author's (1968a) results pertaining tothe older models.2. Results for the Malkmus model The new model consists of the line-strength distribution function (Malkmus, 1967; Rodgers, 1968) P(S)=S-~ exp[--S/So~, (2)where S is the integrated line strength and So a constantof the distribution. For a detailed discussion of theconstant of proportionality in Eq. (2) (which has herebeen arbitrarily taken as unity), and other effectsassociated with the cutoff at small S, the reader isreferred to Malkmus' original paper (1967) and to ~ More recent results on the Curtis-Godson approximation(Armstrong, 1968b) afford simpler and more accurate improvements to this approximation than those suggested by Rodgers(1968). He suggested a two-path version of the random model anda four-parameter approximation to the pressure integral in theoptical depth. The use of second-order Gaussian quadrature asrecommended in Armstrong (1968b) has the form of Rodger'stwo-path approximation, while simultaneously offering a significant improvement in optical depth over the Curtis-Godsonapproximation.742 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLU~E 261.9 Fro. 1. The diffusivity factor r for a random Malkmus band ofLorentz lines as a function of the optical parameterfor various values of the width-to-spacing ratio R. The dashedline is a comparison curve for the random exponential modelfor R=0.1.Rodgers (1968). The quantities that we consider hereare independent of the constant of proportionality ifthe range of S is taken to be infinite. The equivalent width Wu that the new model yieldsfor a regular array of non-overlapping Lorentz lines is W~ =--~(lq-4q)~-- 1-], (3) 2 ~where R------~ra/b~ qm , a is the line halfwidth, and /~ 7rotthe line spacing. Since W/ti=A~ where A~= 1--5~ is theaverage absorption (Goody, 1964) over the interval 5,the definition (1) can be written ~+~-~ a~=~C(~+4rq)~-~3. (4)The cosine of the zenith angle has been denoted by ~and appears in the formula to modify the mass ofabsorbing gas, which becomes u/~ for a slant ray. Theintegration in Eq. (4) can be performed wi~ theresult that / (q+~) (1+4q)~ ~(~+4q)~+lnl~ (5)However, the primary interest in the new model arisesin its use in a random array of lines, for which theaverage transmission function ~na is given by ~n~= exp(-W/b) where W is given by Eq. (3). Applicationof the definition of r [-Eq. (1)-] to this transmissionfunction yields the equation2 L\ y-? J / R= exp -~-[(1 q-4r~)--1~1 , (6)which determines the diffusivity factor r for a randomMalkmus baud. Eq. (6) can be readily seen to yieldback Eq. (4) in the limit R --~ 0. (It can also be verifiedthat Eq. (4) yields r= 2 and r= 16/9 in the limits q --~ 0and q -~ m, respectively.) Eq. (6) has been solved numerically by methodsalready described (Armstrong, 1968a) on an IBM S/360Model 50 computer and the results for r as a function ofq and R are shown in Fig. 1. A comparison with Fig. 5of Armstrong (1968a) shows the results for this modelto be very similar to those for the random exponentialband. In fact, if the arguments in Eq. (6) are expandedto neglect terms of order unity compared to ~, oneobtains 1 r~qR~ --log~4E~ ( ~)-~ , (7)which is identical to the result obtained in the previouspaper for the same limit in the case of the random exponential band. This is to be expected since the equivalent widths have the same strong- and weak-line limitsand are closely related to each other as has been pointedout by Rodgers (1968). A graph of Eq. (~) is given inArmstrong (1968a). The dashed line in Fig. 1 shows for comparison thevalue of r for the random exponential band for a widthto-spacing ratio R=0.1. The asymptotic agreementmentioned above is quite evident. The curves for theMalkmus model are smoother, with less variation in theintermediate q region (they both, of course, have thesame weak-line limit as q -~ 0).3. Transmission function error due to use of constant r Although considerable detailed calculation is requiredto obtain the accuracy of the heating rate when aConstant value of r is employed, the error in the transmission function itself can be estimated quite readily.The error 5~' that will appear in the average flux transmission function due to an incorrect choice of r given by ~ ~ O ~ (rU)3r' (8) Orwhere 5r is the error in r. For a random model forwhich ~ has the characteristic exponential form givenJuLY 1969 B. H. A R M S T R O N G 743above, the fractional error in ~'~, e---bT/T, becomes,approximately, 1 OW ~ =- ~r, (9) ~arwhere W is the equivalent width for the particularrandom model. In the case of the Malkmus model,Eq. (9) yields Su/~ '] -I' ' (10) ( l +4rSu/,m)~ ~To obtain the order of magnitude of the error involvedin the use of r= 1.66 for the ozone transmission function,we approximate a single main level for the ozone concentration from the work of Walshaw and Rodgers(1963). We take this to contain a total ozone amount of--~5X 10-4 gm cm-2 between pressures of, say, 200 and5 rob. Thus, an approximate Curtis~Godson line width&~a(Pi+JP2)/2 is estimated as 0.03 cm-~, wherea=0.28 cm-~, as given by Rodgers (1968). Using thestrong contribution to Rodgers' two-interval fit to theabsorption of the 9.6u band, we find Su/a~l.1,q--=Su/~ra~ 1.2, and R--:~ra/~0.094. For these valuesof q and R, Fig. 1 yields r,-~l.71 for the Malkmus band.Hence, $r~ 0.05 for an estimate of the error in using theusual 1.66 value. Inserting the above values into Eq.(10) yields ~n~u~l.9%. The error equation for therandom exponential band is quite similar to Eq. (10).For large optical depth, Eq. (10) can be written as a(q)~e~,u= 2-~-~lr, (11)which indicates that fairly large errors are possible inthis case. For example, if R=0.1, ~=2000 (which onemight obtain, e.g., for a CO= layer), the error given byEq. (11) is ~9%.REFERENCESArmstrong, B. H., 1968a: Theory of the diffusivity factor for atmospheric radiation, f . Quant. Spectros. Radiative Transfer, 8, 1577-1599. , 1968b: Analysis of the Curtis-Godson approximation and radiation transmission through inhomogeneous atmospheres. J. Atmos. Sci., 25, 312-322.F, lsasser, W. M., 1942: Heat Transfer by Infrared Radiation in the Atmosphere. Harvard Meteorological Studies No. 6, Harvard Univ. Press, 107 pp.Goody~ R. M., 1964: Atmospheric Radiation I, Theoretical Basis. Oxford University Press, 437 pp.Ititschfeld, W., and J. T. Houghton, 1961: Radiative transfer in the lower stratosphere due to the 9.6 micron band of ozone. Quart. J. Roy. Meteor. Soc., 87, p. 562.Malkmus, W., 1967: Random Lorentz band model with expo nential-tailed S-~ line-intensity distribution function. J. Opt. Soc. Amer., 57, 323-329.Rodgers, C. D., 1968: Some extensions and applications of the new random model for molecular band transmission. Quart. J. Roy. Meteor. Soc., 94, 99-102.Walshaw, C. D., and C. 1). Rodgers, 1963: The effect of the Curtis-Godson approximation on the accuracy of radiative heating-rate calculations. Quart. J. Roy. Meteor. Soc., 89 122-130.