FEBRUARY 1992 RITTER AND GELEYN 303A Comprehensive Radiation Scheme for Numerical Weather Prediction Models with Potential Applications in Climate Simulations BODO RITTERDeutscher Wetterdienst, Offenbach/Main, Germany JEAN-FRANCOIS GELEYNDirection de la Meteorologie Nationale, Paris, France(Manuscript received 23 January 1991, in final form 25 June 1991) ABSTRACT A comprehensive scheme for the parameterization of radiative transfer in numerical weather prediction(NWP) models has been developed. The scheme is based on the solution of the 6-two-stream version of theradiative transfer equation incorporating the effects of scattering, absorption, and emission by cloud droplets,aerosols, and gases in each part of the spectrum. An extremely flexible treatment of clouds is obtained by allowing partial cloud cover in any model layer andrelating the cloud optical properties to the cloud liquid water content. The latter quantity may either be aprognostic or diagnostic variable of the host model or specified a priori depending on cloud type, height, orsimilar criteria. The treatment of overlapping cloud layers is based on realistic assumptions, but any differentapproach requires only minor modifications of the code. The scheme has been tested extensively in the framework of the intercomparison of radiation codes in climatemodels (ICRCCM, WMO 1984, 1990). Radiative fluxes and heating rates, calculated in a few milliseconds ofCPU time with our scheme, are in very good agreement with reference calculations, which may require severalthousand CPU seconds for the same purpose. First experiments, using our parameterization scheme within the framework of a global weather forecastmodel, give promising results. Subject to the results of further experimentation, our code will be part of theparameterization schemes used in the operational weather prediction models of the DWD (Deutscher Wetterdienst). However, the generality of the scheme, particularly the flexibility of the code, extends its scope to otherapplications, such as climate simulations. In the long run, one of the decisive advantages of the method described here lies in the fact that the cost ofcomputations varies only linearly with the number of atmospheric model levels, unlike the quadratic behaviorof the so-called "emissivity-type" methods.1. Introduction The aim of radiative transfer parameterization inatmospheric models, as discussed by Stephens (1984),is to provide an accurate and fast method to calculateradiative fluxes and heating rates in the course of themodel integration. Accuracy and computational efficiency are, at least in principle, competing objectives,so that radiative transfer schemes for the aforementioned applications will necessarily have to compromisebetween these two criteria. The required degree of accuracy varies with the particular application of the host Corresponding author address: Dr. Bodo Ritter, Deutscher Wetterdienst, Frankfurter Str. 135, D-6050 Offenbach am Main, Germany.atmospheric model. For a very short-range forecast ofthe midtropospheric geopotential height field, a simpleNewtonian cooling, which avoids a systematic warmingof the atmosphere, may be an adequate parameterization. However, state-of-the-art NWP models with acomprehensive physics package, as well as climatemodels, require radiative transfer schemes that describeas accurately as possible the interaction of radiativetransfer processes with the other components of thehost model. In NWP models the interaction betweenradiation and clouds will have a strong impact on thesurface energy balance and the subsequent evolutionof the atmospheric profiles in the lower troposphere,among other equally important effects. As the modifiedatmospheric structure affects in turn the generation anddissipation of clouds (e.g., radiation fog), a feedbackwith a strong impact on quantities like the diurnal cycleof the near-surface temperature field may be estabc 1992 American Meteorological Society304 MONTHLY WEATHER REVIEW VOLUME 120lished. In order to be able to simulate such effects, atleast in principle, a radiative transfer scheme for a stateof-the-art NWP model must be both accurate andcomprehensive, in the sense that it can respond appropriately to any relevant input (e.g., partial cloudcover in any model layer; cloud liquid water content)provided by the host model. Similar arguments holdfor climate simulations. Studies in which climatologicalcloud fields were used as input to the radiative-transferscheme (e.g., Manabe and Wetherald 1975) lack themodulating effect of temporal variations in cloudinesson the climatic response of the model. As discussed byvarious authors (e.g., Slingo 1989), it is not only thechange in cloud amount and distribution that may affect the climatic response of the model, and therebythe conclusions drawn from the results of model integrations. Since cloud optical properties are stronglydependent on the cloud liquid water content and therespective drop-size distribution (Stephens 1979), arealistic simulation of the interaction between cloudsand radiative transfer in climate simulations requiresa parameterization scheme that is both flexible andaccurate enough to take those effects into account.. Whether the desired degree of accuracy can beachieved, bearing in mind the computational constraints, uncertainties in spectroscopic properties of theoptical constituents of the atmosphere, and the deficiencies of the host models with regard to the simulation of atmospheric properties like cloud cover andliquid water content, is open to question. However, itfollows from these arguments that there is at presentno obvious reason to design basically different radiative-transfer schemes depending on the application inNWP or climate models. The desire for flexibility of the parameterizationscheme extends beyond the capability to consider allrelevant optical constituents and corresponding radiative processes. It should indeed be possible to changeother model aspects, such as the vertical resolution,without major modifications to the radiative transferscheme. For example, assuming independence of eitherthe local partial cloud-cover effects (i.e., random cloudcloud overlap) or the local cloud optical depths fromall other layers causes dependencies on the vertical discretization, a drawback avoided by our method. Linearity with regard to vertical model resolution isalso an important aspect when computational costsare considered, particularly because a considerably increased number of model layers is one of the mostlikely future developments in NWP. The radiative transfer scheme presented in this paperwas designed with the aforementioned arguments inmind. In the next section we will discuss the underlyingtheoretical concept, followed by a description of opticalproperties of the constituents considered. Finally, results from validation and preliminary forecast experiments are presented in section 4.2. Basic theory The basic formulation of the ~-two-stream equationsfollows the concepts discussed in the review paper byZdunkowski et al. (1980), where it is shown 'that thevarious two-stream methods described in the literaturecan be cast in a standard form and differ only by theiirparticular choice of fundamental optical parameters.In the following subsections we will briefly describe thebasic equations and solution methods used in. the formulation of our radiative transfer model.a. List of symbols In order to abbreviate the description of the equations we define those symbols that will be used fi:equently in the following sections, using a notation similar to Zdunkowski et al. ( 1982):F.i,gU~(~0)As(t~o), AsEsBBsT, Tsdiffuse upward, downward flux;blackbody differential fluxes, corre sponding to F~, F2;parallel solar flux with respect to a hori zontal plane; solar constant; cosine of solar zenith angle; optical thickness; optical thickness of gray constituents only;optical thickness of a gas, considering absorption coefficient i;single-scattering albedo;asymmetry factor of the phase function;fraction of radiation contained in the dif fraction peak of the phase function; diffusivity factor; mean fractional backscattering coefficient for diffuse light;mean fractional upward-scattering coef ficient for primary scattered solar ra diation;earth's surface albedo for parallel, re spectively diffuse solar radiation; earth's surface emissivity; blackbody radiation; blackbody radiation at the temperature of the earth's surface;temperature, earth's surface temperature; andatmospheric pressure.b. The basic flux equations for the b-two-stream method Zdunkowski et al. (1982) demonstrate that the fluxequations for a b-two-stream method can be formallywritten to describe either the radiative transfer ofmonochromatic solar or infrared radiation in a scatFEI~RU^RY 1992 R1TTER AND GELEYN 305tering and absorbing atmosphere. For reference, werecall the basic set of equations:dF~db--= atFl - a2F2- a3JdF2d*-- = ot2F! - atF2 + a4J -(1 -&f)~- (2.1)db uowith the following definitions:a~ = U 1 - ~[1 -/~0(1 -f)] ) solar/infrared;a2 = U/~o&( 1 - f) solar/infrared; I /~(tto)&( 1 -f) solara3=[U(l_~)=al_a2 infrared; = I[1 - ~(t~o)]b(1 -f) solarOr4 t a3 infrared; I S/t~o solar (2.2) J = [ ~rB infrared.In contrast to Zdunkowski et al. (1985) we include thethe factor ( 1 - f) as normalization also in the corresponding coefficient for diffuse radiation, thus leadingto a consistent notation for the coefficients ak, as previously defined. The backscattered fractions for paralleland diffuse radiation are derived from the phase function of the scattering particles (cfi Wiscombe andGrams 1976). Development of the phase function inLegendre polynomials following Joseph et al. (1976 )and subsequent angular integration leads to (Zdunkowski et al. 1985):4+g/~o = -- (2.3a)8(1 +g) 1 3 g~(/~o) = 2 4 1 + g/~o. (2.3b) For reasons of computational efficiency, we do notsolve the set of differential equations (2.1) that represents the explicit formulation of the b-two-streammethod. Instead, we use the equivalent implicit formulation. Defining:b'= (1 -&f)~ (2.4a)&,= &(1 -f) 1 -&f ' (2.4b)one obtains from Eq. (2.1): rift 'db' = odlFi - a[F2 - ot[J dF2 = a~F~ - ot'~F2 + a~J dg' dS db'with:a'~ = U[1 - $'(1 - ~0)] solar/infrared;a h = U[~o&' solar/infrared; I &'~(t~o) , solara~ = [ U( 1 - b ) infrared; &'[1 -/~(~)] solara~= a~ infrared.(2.5)(2.6)c. The solution of the differential fiux equations For the solution of (2.5) we subdivide the atmosphere into layers of constant optical properties as discussed by Geleyn and Hollingsworth (1979). In orderto solve the radiative transfer problem in the thermalpart, we introduce the blackbody differential fluxes,defined by Pl ~ ~i'B -- F1 and P2 = ~rB - F2. (2.7)For solar wavelengths, that is, ~rB --- 0, we use the sameconvention to keep a unified notation. If we now assume a linear relation between the blackbody radiationand the optical depth, we can integrate (2.5) analytically for each layer. We obtain fluxes directed outwardat the layer boundaries as linear functions of the corresponding fluxes directed inward of this layer. Thelinear coefficients, which are analytical functions of theoptical properties of the layer (i.e., of ~ and a~, k = I,4), are defined in the Appendix. Thus, we have:withF~j+lt =' (~2 ~4 t~5 ?2,j F~,~ / g3 fi~ fi4 F~,~+~/ s~Q~ -- ~(1~+~ - 1~) ~1solarfi~ = infrared; = I-a2 solar [ a6 infrared;solarinfrared;(2.8)306 MONTHLY WEATHER REVIEW VOLUME 120 & = an solar/infrared; = l-a3 solar a3 t a6 infrared; ~ = as solar/infrared. (2.9) If we use Nlayers to subdivide the atmosphere, (2.8)represents a system of 3N + 3 linear equations thatcan be solved by an elimination-backsubstitution algorithm, if we specify appropriate boundary conditions.At the upper boundary, the top of the atmosphere, wehave: & =/~0S0 and F2,~=F2,~=O solar (2.10a) ff2,~ = ~rBi i.e., F2,~ = 0 infrared. (2.10b)At the lower boundary (i.e.,j = N + 1 ) the conditionsare:P~,N+~ ---- --F~,N+~ = --As(IZO)SN+~ -- AsF2,N+~solar(2.11a)~I,N+I = ( I -- ES)t~2,N+i, i.e.,Ft,N+~ = (1 - Es)F2,iv+i + Es~rBs infrared. (2.11b) Note that in our model we do not consider a temperature discontinuity between the air in the lowestmodel layer and the surface of the earth. However, theinclusion of such a discontinuity, as at all model levelinterfaces, would be straightforward as indicated byZdunkowski et al. (1982).d. Treatment of partial cloudiness Partial cloudiness can be treated by the method presented in the last section, if we characterize each layerby two sets of optical properties and fluxes, one for thecloudy and one for the cloud-free part. This treatmentof cloudiness, originally introduced by Geleyn andHollingsworth (1979), requires the specification of ageometrical relation between the cloudy and cloud-freefractions of any layer and its adjacent layers. Havingdefined such a relation, one obtains the incoming fluxesfor the cloudy and cloud-free parts of that layer ascomposites of the outgoing fluxes of the adjacent layers. To define such a geometrical relation in-our model,we assume that clouds in adjacent model layers havemaximum overlap. However, clouds, separated bycloud-free layers are independent of each other. Thisapproach is supported by Tian and Curry (1989), whoinvestigated the errors introduced in the calculated totalcloud coverage for various overlap assumptions. Usingdata from three-dimensional nephanalysis, they conclude that the maximum overlap assumption is fullyjustified if there is no clear interstice between adjacentcloud layers, and that the random overlap assumptionis a realistic approach for all other cases. The commonassumption in many radiation schemes of randomoverlap for all cloud layers (e.g., Ramanathan et aL1983; Morcrette and Fouquart 1986) will, accordingto the results of Tian and Curry, generally overestimalethe total cloud cover. Furthermore, the resulting totalcloud cover in such an approach will depend on thevertical resolution of the model, unlike in our case. The importance of the overlap assumptions for thecalculated radiative fluxes has been illustrated byHarshvardhan et al. ( 1987 ). They show that the choiceof maximum or random overlap has a significant itnpact on the vertical distribution of therma~ coolingrates, and thus may cause a different subsequent evolution of the cloud field itselfi For details of the treatment of partial cloudiness inour scheme, we refer the interested reader to the description by Geleyn and Hollingsworth (1979).e. Spectral integration The previously described method is valid in principleonly for monochromatic problems. To obtain fluxesfor a whole spectrum, one should specify ve~3~ detaileddiscretized spectral distributions for the source functionand the optical properties, solve (2.8) for each of theso-defined small intervals, and sum the individual results. A direct solution of(2.8) with simply a spectrallyaveraged input would lead to erroneous results, owingto the strong nonlinearity of the coefficients aj withrespect to the coefficients a~. However, for numerical weather prediction, constraints on the computational resources entirely prohibit the use of a large number of intervals for a bruteforce solution to the problem. Thus, it is an absolutenecessity to use fairly wide intervals for the solution ofthe 6-two-stream equations, but with impact valueschosen to minimize the nonlinearity errors. Atmospheric properties involved in radiative-transfer processes vary in a number of ways (Stephens1984). For instance, Planck function and opticalproperties of cloud droplets and aerosols change relatively slowly with wavelength, and can therefore beparameterized over fairly wide spectral intervals without unacceptable errors (see section 3). Unfortunately,this is not the case for gaseous absorption, whiclh hasan extreme spectral dependence, making it necessaryto use the concept of band models (Goody 1952;Malkmus 1967 ). However, the application of such band modds requires the a priori knowledge of the amount of absorbing gas encountered by the photons, and this effective pathlength is not solely a function of the amountof gas contained in the model layers: scattering byclouds and aerosols will elongate the effective path,whereas absorption by the same constituents and eventhe gases themselves will shorten it.lZ'EBRUARY 1992 RITTER AND GELEYN 307 For the solar domain, methods exist to approximately determine the effective pathlength (e.g., Geleynand Hollingsworth 1979). In the thermal part of thespectrum, additional complications arise from the factthat the atmosphere contains internal radiative sources;thus, the concepts used in effective pathlength methodsare invalidated. This problem is commonly bypassedthrough the assumption that scattering of thermalfluxes can be neglected by comparison with absorption/emission by clouds and gases; this is the basis for theso-called emissivity-type methods used in the majorityof radiative transfer parameterization schemes for atmospheric modeling (e.g., Ramanathan et al. 1983;Harshvardhan et al. 1987; Morcrette 1990). However,the validity of the underlying assumption is at leastquestionable, since, as shown for instance by Stephens(1980), cases can be found where only an (unacceptable) effective emissivity greater than 1.0 could lead tothe correct solution. This effect is caused by the reflection of photons originating from the warmer lower troposphere and the earth's surface, a process that cannotbe accounted for in the simple emissivity approach.More generally, this approach neglects the fact that thereal effective emissivity of a cloud will be a complexfunction of not only optical properties of the clouditself but also its atmospheric environment (see section 4). The simultaneous and consistent treatment of scattering and molecular absorption is possible, if we employ the k-distribution method for describing the absorption properties of gases in wide spectral intervals(Wiscombe and Evans 1977; Stephens 1984; Hansenet at. 1983). The simplest practical approach of thismethod is to calculate the average gaseous transmissionfunction for a wide spectral interval, using a narrowband model, and then to fit this function of the absorberamount by a series of exponentials (Wiscombe andEvans 1977): 1 rrX(u) ~ ~ wie-Iqu. (2.12) i=l This version of the method is generally called exponential sum-fitting technique (hereafter ESFT). Thecoefficients ki can be considered as a set of pseudomonochromatic gaseous absorption coefficients of thegas in the spectral band, representative of the ensembleof all monochromatic absorption coefficients. Theweights wE are constrained to sum to one to ensureenergy conservation. Pseudomonochromatic solutionsof (2.8) in the manner described in subsection 2c canbe obtained for a combination of gray effects and eachone of the terms of the series, the obtained fluxes beingsubsequently accumulated with weights wi; that is, 1 F ~-. ~ wiF(ho + hi). (2.13) The increase in computational costs caused by thenecessity of solving the h-two-stream system once foreach value of k~ is compensated by the facts that fairlywide spectral intervals may be used, gaseous absorptionand cloud-droplet scattering by cloud droplets can beconsidered simultaneously, and "saturation effects" ofthe gaseous absorption are automatically taken intoaccount without any need to recompute the transmission functions at all model levels. The trade-offbetweenaccuracy and computational costs depends, of course,only on the number I of coefficients used to describethe spectrally averaged transmission function. For thechoices in Our model, see Table 1. However, if more than one gas has important absorption bands in a particular spectral interval, thecomputational costs increase dramatically. Multiplicative properties of transmission functions require thatthe total flux is evaluated as a weighted sum of resultsfrom all possible permutations of gaseous absorptioncoefficients; that is, for two gases we would have: F~ ~ ~ w/,w0F(/~0+ ~, + ~/2). (2.14) i~=1 i2=1Thus, the computational burden increases as the product of the number of coefficients of the different series. Since the absorption bands of gases are quite wellseparated in the solar domain, one could neglect gasesof minor importance in each broad interval and useonly (2.13 ). However, as discussed by Chou (1990),such an approach would cause a systematic underestimation of the solar absorption in the atmosphere withpotential consequences for climate simulations. We canavoid both the "explosion" of computational costs andthe aforementioned underestimation, using a conceptintroduced by Zdunkowski et al. (1982). We first solvethe radiative transfer equation in the absence of gaseousabsorption to obtain gray fluxes F- (here F- stands forupward, downward, or parallel solar fluxes). Subsequently, we solve the transfer problem including onlyone gas in addition to the gray constituents (Rayleigh,aerosols, and clouds) in order to compute an effectivegaseous transmission for each model level and type offlux; that is, r~ FO, (2.15)where F~ represents, for example, the flux calculatedwith inclusion of gas number one according to (2.13).We can finally approximate the solution in the presenceof all gases and of the gray constituents by M F~ 1-I ~mxF-, (2.16)where M is the number of considered gases. The mainadvantage here is that the computational costs increase308 MONTHLY WEATHER REVIEW VOLUME 120now like the sum rather than like the product of the 0number of coefficients. For the current version of ourmodel the gain is a factor of about 3. ~00The accuracy of the second method, fast ESFT(hereafter FESFT), is demonstrated in Fig. I in a case 200where all gases are considered as well as Rayleigh effectsand the presence of a cloud. In this example, as in all a00investigated cases, the difference between the twomethods remains extremely small. Nevertheless, we ~- 400have coded our scheme with maximum flexibility, giving the possibility, interval by interval, of choosing either FESFT or ESFT. ~ 500-.In the thermal domain, overlapping ofspectrai bandsis a more serious problem, complicated by the fact that 600~simultaneous absorption and emission preclude the useof an effective transmission as introduced by (2.15 ). 700However, even though we could not find any consistenttheoretical explanation for that fact, numerical exper- 8ooimentation shows that (2.15 ) and (2.16) also providesatisfactory approximate results when applied to the 00onet thermal flux. As a typical example, we present results of ESFT and FESFT calculations in Fig. 2 for the t00osame atmospheric situation as in Fig. 1. The errors inheating rates introduced through the approximationare slightly larger than in the solar case, so that the two~.ESFT........ FESFT -'7 -'6 -'5 -'4 -'3 -'2 -'lFIG. 2. Same as Fig. I for thermal radiation.'0'100'200-300-400500600700'800900 000 100' C -100 200' '200 300' ESFT '300 ........ FESFT 400' -400 500- -500 ~ ooo- -soo 7Off "700 800- -800 1000' -1000 F~G. 1. Comparison of solar heating rates in a midlatitude summeratmosphere using the exponential sum-fitting technique in its o~ina~(ESFT) and its approximate fast (FESFT) version. A cloud with 10g m-2 liquid water content is located between the 1000- and 2000-mheights. Solar zenith angle is 30- and a surface albedo of 0.20 isassumed.curves can just he distinguished. The approximationwill typically cause a reduction of the OLR in the orderof a few watts per square meter, depending on the actualatmospheric conditions. The impact on the net thermalsurface flux is generally less than I W m-2. Overall,this implies a slight reduction of the total atmosphericdivergence of the thermal flux, but this is evenly spreadthroughout the atmosphere, the errors in the heatingrates of individual model layers being very small. Mostof the deviations occur, as expected, in the intervalcontaining both the 15-~zm CO2 band and the wing ofthe water vapor rotation band (cf. section 3), so thatremaining discrepancies could be very much reducedby using the FESFT in all spectral intervals except forthis one. However, for the purpose of numericalweather prediction, we prefer to employ the app:roximate technique to treat gaseous overlap in all intervals,thus minimizing the computational costs in an operational environment. We have not yet addressed the problem of the pressure and temperature dependency of gaseous transmission functions. The so-called Curtis-Godson approximation, well fitted to band-model calculations,cannot be introduced as such in the k-distributionmethods. The scaling approximation (cf. Chou andArking 1980), requiring the separation between aspectrally dependent factor and a pressure-temperature-dependent one in the absorption coefficient, is ofcourse better suited to the latter. Equation (2.11 ) wouldthen read:FEBRUARY 1992 RITTER AND GELEYN 309 , [7X(u,p, T) ~ ~ wiexp -ki . (2.17) i~ 1The temperature and pressure correction factors a and o~ ~_~/3 for a given gas and a given broad spectral interval ~are tuned to get the best fit to reference calculations. ~ g0However, as demonstrated by Ackerman et al. (1976), ..this oversimplification of the temperature and pressure '~ c~correction will lead to erroneous results. The same au- 71thors suggest the use of a matrix of k values, a proposal ~ 0.tending toward the full use of lookup tables for the k ~= 71coefficients, the weights remaining the same at all con- ~ditions (for obvious energy conservation reasons). In- ~'stead of this cumbersome refurbishing of the scalingapproximation, we decided to use the following procedure, suggested by Y. Fouquart (personal commu- Cnication ):- First, compute a set of wi and ki at reference temperature and pressure values chosen after Chou andArking (1980) to be representative of the most relevantatmospheric conditions for the gas and spectral interval ~considered. ~ ~)- Second, vary the atmospheric pressure in the ref- ~ toerence calculations in order to obtain a least-squares ~ e~fit of pressure-dependency exponents ai, that is, ;2 tO ' ~ ffi ~X(u,p, T) ~ ~ wiexp - . (2.18) ~ i= ! ~Pref ] ] ~ to 1 - Third, repeat step two, but this time for temper- ~ature, thus leading to exponents/~i: ~' 0 I1) ' ~ (5;1 -x(u,p, T) ~. ~ wiexp -ki u . (2.19) 0 i=1 -0.5The transmission function at any pressure and temperature configuration will finally be approximated by [ / p \~/ T \~ ] [ )[ ) l' (2.20)As proof of the validity of this spectrally independentscaling, it is interesting to note that the numerical fittingproduces pressure-scaling coefficients a~ that are closeto I for weak absorption coefficients ki. This is in goodagreement with the theoretical expectation for absorption far from the center of Lorentz lines. Similarly, forstrong absorption, the pressure exponents becomesmall, or in some cases even negative. This differentialbehavior suppresses the aforementioned difficulties associated with the use of a single scaling parameter. Thegeneral correlation between the strength of the absorption coefficient and the corresponding pressure-scalingcoefficient is well illustrated in Fig. 3a. The temperature-scaling exponents exhibit a similar coherent structure (Fig. 3b). According to Chou and10-t 10 0 10 10 ~ 10absorption coefficient Pa-~10-~ 10 0 10 t 10 ~ 10absorption coefficient Pa-t FiG. 3. (a) Pressure-scaling coefficients in the water vapor rotationband as function of the strength of the absorption coefficients. Thewidth of the bars indicates visually the relative contribution by linesof that particular strength to the average absorption in the spectralinterval considered. (b) Same as (a) for the temperature-scaling coefficients.Arking, absorption is more sensitive to temperature inband wings than close to the band centers; furthermore,they demonstrate that, at least for water vapor, commonly used correction factors (Stephens 1984) takinginto account only the half-width effect are even wrongin sign, this effect being dominated by that of the lineintensifies. In the example illustrated, only the verystrongest absorption is associated with a temperatureexponent close to -0.5, as expected from theory, whileall other exponents are positive, and thus in goodagreement with the findings of Chou and Arking. The accuracy of the overall approximation is dem310 MONTHLY WEATHER REVIEW VOLUME 1201.00.8O'~ 0.6'~o.40.2T ref : 282.iT : 230.,p ref : 861.p : 304.1.0 K K 'k\ hPa "\ hPa 'k '1 data fit with variable scaling.... [? ........ , 10" 10- '101 10z 103 absorber amount Pa0.80'~ 0.6~=o.4p ret : 608. hPap : 203. hPa0.0 ........, 0.0 ......... 10'~ l(f~datafit with variable scaling i0q I0- 10~ 10~absorber amount Pa FIG. 4. (a) Comparison of band-model transmission function for H20 in the interval 12.5-20.0 ~m with fitted transmission functionsusing constant temperature and pressure scaling or absorption coefficient dependent scaling for deviations from the reference :atmosphericconditions. (b) Same as (a), but for the absorber CO2.onstrated by Fig. 4a. The full line corresponds to thewater vapor transmission function in the wing of thethermal rotation band as calculated directly at thespecified temperature and pressure values, as will beindicated in subsection 3e. The dashed line, representing the full approximation after the three coefficient-fitting steps, is very close to it, an excellent resultif one considers the large differences between the reference temperature and pressure and the actual conditions. The constant scaling, using the values suggestedby Stephens (1984), leads to large deviations betweenthe actual curve and the fitted one. This is mainly dueto the aforementioned problem with the temperaturedel~ndency in band wings. Figure 4b provides a similarillustration for the '15-~tm CO2 band and the differences,albeit Smaller, could be of some importance in the context of the CO2 climate problem.3. Optical properties and spectral intervals As already mentioned in section 2, radiative transferparameterizati0n in a numerical forecast model is severely constrained 'by the necessity of minimizing theuse of computational resources. The cost effectivenessof a parameterization scheme is subject to many influences. First, there is the solution method for the radiative transfer equation and the approximation associatedwith it (e.g., solution of two-stream formulation). Second, there is the number of intervals used to resolvethe spectrum. In a typical parameterization scheme fornumerical forecasting and/or climate simulations thisnumber is well below ten.' Further economy is achievedin many models by considering only those optical constituents of the atmosphere that are important in a particular spectral domain: However, the question of importance is very much' a question of the problem considered. For example, solar absorption by 02 and CO2is definitely small compared to absorption by H20 or03, but it may have a nonnegligible effect in climatestudies, as discussed by Chou (1990), Kiehl and 'Yamanouchi (1985), and Kiehl et al. (1985). In orderto constrain the applicability of our model as little aspossible, we have therefore adopted the approach toinclude, at least in principle, each optically relevantconstituent in each spectral interval for which there isevidence in the basic spectroscopic data. Naturally, wewill omit some constituents in spectral intervals wlheretheir impact is well below the noise level for the operational version of the scheme in order to reduce; thecomputational costs. Finally, the scheme should beimplemented within the numerical forecast model using efficient coding techniques, adopted to .moderucomputer technology. In its basic version, 'the schemedescribed in this paper requires approximately 3 ras ofCPU time on a CRAY-YMP per call and grid pointof the European Centre for Medium Range WeatherForecasts (ECMWF) 19-level model. In summary, we have sought what we think is thebest compromise among generality, flexibility, efficiency, and accuracy. For this reason we will refer toour scheme in subsequent sections as GRAALS (general radiative algorithm adapted to linear-type solutions). In the following subsections the spectral 'intervals ofthe scheme will be described, together with some detailsabout the derivation and sources of optical propertiesof atmospheric constituents considered.a. Spectral intervals and principal optical constituents Table I describes the spectral intervals chosen forour model and the most important optical constituentsFEBRUARY 1992 RITTER AND GELEYNTABLE 1. Spectral intervals and major optical constituents considered in the radiation scheme.311 Solar ThermalInterval number I 2 3 4 5 6 7 8Limits (~m) 1.53-4.64 0.70-J.53 0.25-0.70Gaseous H20, CO2 H20, CO2, 03, I-[20absorption, CH4, N:O 02 02No. of k~ for (7, 6, 0) (7, 3, 0) (3, 2, 5)H:O, CO:and 03Dropletscattering yes yes yesabsorption yes yes yesRayleigh scattering yes yes yesAeorsolscattering yes yes yesabsorption yes yes yes20.0-104.5 12.5-20.0 8.33-9.01 9.01-10.31 4.64-8.33 10.31-12.5H20H:O, CO:, H:O, CO2, H20, 03, H:O, CHn, N20 N:O CO2, q20 N20, CO2(7, 0, 0) (7, 7, 0) (4, 3, 0) (3, 3, 5) (7, 4, 0)yes yes yes yes yesyes yes yes yes yesno no no no noyes yes yes yes yesyes yes yes yes yesconsidered. Even though we consider this division ofthe spectrum appropriate for our purposes, any othercombination of intervals may be realized easily, sinceexisting software enables us to produce the requiredoptical properties for any part of the spectrum with aminimum of effort. Some arguments for our choice ofintervals will be given in the following subsections.b. Cloud optical properties A successful parameterization of radiative transferin numerical weather prediction and climate simulations requires a realistic treatment of cloud effects, inparticular that of the so-called cloud radiative forcing(CRF); that is, the impact of clouds on actual radiativefluxes at the top of the atmosphere (see Charlock andRamanathan 1985) and that of the related cloud radiative feedback mechanism. In the real atmosphere,cloud optical properties are a function of the clouddrop-size distribution and of the considered wavelength. In models, assuming spherical droplet shapes,Mie theory can provide a suitable input for spectralaveraging of basic cloud optical properties. Nevertheless, one must realize that this is the case only becauseof the moderate variations of these properties insidethe considered broad spectral intervals. However, current modeling cannot provide any reliable information on cloud-size distributions; furthermore, Mie calculations are too expensive for suchmodels. Thus, we chose to use the cloud optical properties provided by Stephens (1979) as a basis for ourparameterization. In this work eight different clouddroplet distributions, covering a wide range of cloudtypes and corresponding liquid water contents, areconsidered and processed for a spectral resolutionmeeting our needs. Therefore, we expect to reasonablysimulate real atmospheric conditions when using themethod described next. A common feature of the extinction by droplets ofall cloud types is the fact that for wavelengths smallerthan 0.7 ~m the single-scattering albedo is virtuallyidentical to 1; that is, no absorption takes place at thesewavelengths. As the same wavelength presents an important boundary for the solar albedo of vegetated surfaces, it is a natural choice as one of our intervalboundaries (Table 1 ). For the remainder of the solarspectrum Slingo (1989) recommends the use of at leastthree more spectral intervals to resolve the importantcharacteristics of droplet absorption and scattering. Hisrecommendation is based on calculations with ascheme that uses linear, weighted averaging in the determination of the optical properties of wide intervals,that is, (Slingo and Schrecker 1982): fx kxSoxdXkx = (3.1) fx SoxdXwith Sox as the solar irradiance at the top of the atmosphere and kx as the droplet absorption or scatteringcoefficient at wavelength X. If we disregard the trivial case of constant opticalproperties, extinction coefficients derived this way willlead to the right results only in the limit of opticallythin clouds. For thicker clouds the application of( 3.1 )will cause a mishandling of saturation effects, that is,strong extinction in one part of the spectrum influencing adjacent spectral regions contained in the samewide interval. In order to alleviate this problem, wedecided to derive spectrally averaged optical propertiesfrom the condition:Okx w,[~,x- exp(-kx/X~z)l2 = 0, (3.2)312 MONTHLY WEATHER REVIEW VOLUME 12:0where the spectrally averaged transmission through apathlength An Z in the cloud is calculated from thespectrally resolved extinction coefficients as: fx exp( -kx A, z ) Sox dX ~,x = (3.3) fS0xdXA dominance of trivial cases is avoided by an appropriate choice of the weights w, in (3.2). Thus, the errordue to the use of spectrally averaged absorption andscattering coefficients is minimized over a wide rangeof cloud pathlengths. However, we are aware that wehave displaced the main physics problem to the choiceof the optimal weighting factors w,. Depending onwhich aspect (transmission, reflection; thin clouds,thick clouds) the emphasis is put, results may varysubstantially for the other ones. Using this approach,we decided to accept the residual error linked with thechoice of three intervals. For the asymmetry factor, which varies only slowlywith wavelength, we use the same averaging procedureas Slingo and Schrecker (1982), that is: ~x gxkxSCat Sox dX ~x = (3.4) fx kt~scat Sox dX For intervals in the thermal part of the spectrum,the Planck function at 255.8 K is used instead of thesolar irradiance (Labs and Neckel 1970; Neckel andLabs 1984) as weighting function in (3.3) and (3.4). At this point we still have not related the cloud optical properties to variables provided by the forecastmodel. Without going so far as to diagnose arbitrarilydistinct model cloud types, the strong impact of thedrop-size distribution on the optical properties shouldbe taken into account. For this purpose Slingo andSchrecker (1982) propose the following approximativerelations between the specific liquid water content p [wand effective radius re of the droplet distribution andthe optical properties of the cloud: t2 = c3 + c4re g = c5 + core; (3.5)where Az is the layer thickness and c~-c6 are constantsderived from a least-squares fit of (3.5) to the spectrallyaveraged optical properties of the eight cloud types.Since we do not expect the forecast model to be,ableto explicitly provide the effective radius of the dropsize distribution, we will assume that this quantity isitself related to the specific cloud liquid water contentand approximate it by re = c? + CspLw. (3.6)The validity of these approximations, based on. implicitdependencies in Stephens' data, is demonstrated bythe example of Fig. 5. The vertical columns representthe spectrally integrated original data and the horizonlalbars correspond to the fit according to (3.5) in combination with (3.6). The agreement is sat!isfactory,bearing in mind the wide variation in cloud liquid wamrcontent and effective radius in the original data andthe crude methods used in weather forecast models toparameterize cloud cover and cloud liquid water content. An even better agreement between data and fi~ isalso achieved for the asymmetry factor and the singlescattering albedo. Both preceding remarks apply to allspectral intervals, of course. Even though we cannotexpect the model to provide accurate values for theliquid water content in the cloud, either as a prognosticor diagnostic variable, the above parameterization allows the simulation of the basic relation between cloudoptical properties and their integrated and specific liquid water content. In the context of climate simnlations, this approach will enable us to incorporate threeaspects of cloud radiative feedback mechanisms simultaneously. Time-dependent cloud cover, prognosedor diagnosed layer liquid water, and local specific liquidwater content will influence layers' transmissivities andreflectivities, respectively, via geometrical effects, optical depth determination, and, more importantly150'140'130'1~0'110100' 90'approximation 1.00 0.220.140.2 0.~0.501-ST 2 SC 1 ST 1 AS SC 2 NS CU CB FIG. 5. Approximation of spectrally averaged extinction coefScientsfor different cloud types derived from data of Stephens (1979) forthe solar spectral interval 0.70-1.53 ~m. Above the vertical bars,which represent the original extinction coefficients, the specific liquidwater content of each cloud type is indicated in grams per cubicmeter (g m-3).FEBRUARY 1992 RITTER AND GELEYN 313(Twomey et al. 1984), choice of single-scattering alhero. The approach we have chosen still lacks the incorporation of a specific treatment for ice clouds. However,we feel that the presently available information aboutthe optical properties of ice clouds is not stable enoughto warrant its implementation in operational weatherforecasting models.c. Aerosol optical properties The parameterization scheme considers the effect ofscattering and absorption by aerosols on the radiativetransfer in all spectral intervals. Optical properties offive aerosol types (continental, maritime, urban, volcanic, and background stratospheric) are derived forthe spectral intervals described in Table I from basicdata provided in the Appendix of the World ClimateResearch Program report No. 55. For further detailsconcerning the radiative impact of the aerosol distribution and the method of implementation of aerosolclimatologies in the forecast model, the interestedreader is referred to Tanre et al. (1984).d. Rayleigh scattering Data for the optical depth due to molecular scatteringare taken from the Handbook of Geophysics and SpaceEnvironments ( 1965 ) and spectrally averaged over thesolar intervals of the scheme using I f~(1 + bR~)-~SoxdX = (3.7) 1 + fSo~dXThe form of(3.7) ensures that for pure molecular scattering the solar surface flux is unaffected by the use ofwide spectral intervals.e. Gaseous optical properties We compute the optical properties of the gases considered in GRAALS (H20, CO2, 03, CO, CH4, N20,and 02) from the 1982 edition of the AFGL spectroscopic database (Rothman et al. 1983). Since the codeis mainly intended for NWP applications, we did notinclude other trace gases, such as halocarbons, despitetheir potential importance for climate studies (Kagannet al. 1983), but such an implementation would bestraightforward, should the need arise. For 840 narrow spectral intervals covering the spectral range from 0.245 to 104.5 t~m, we use the AFGLdata to generate appropriate input parameters for acalculation of gaseous transmission functions at varioustemperatures and pressures using a band model. In addition to the standard band-model parameters (meanline intensity and half-width), we compute a pressuretype continuum term to account for the fact that thearbitrary choice of interval boundaries will otherwiseconfine the impact of the lines to the interval containingthe line center. Thus, the equivalent width for themodified Malkmus model reads: W(u, Ur) --~ 5 U ~0 ~rr] -- 1 -- &Ur, (3.8)where u (Ur) is the (pressure- and temperature-reduced)absorber amount. The mean line intensity ~ and themean half-width &o [at standard temperature andpressure (STP)] are derived from the spectroscopicAFGL data. The contributions of all lines in the spectrum to the continuum term ~c in interval k are definedby: ~ = ~ (Sigik - Si~ik), (3.9)where the Kronecker symbol $~k expresses the fact thatthe total intensity of lines centered in the consideredinterval is already included in the mean line intensityof the classical equivalent width. The weights gik dependon the assumed line shape and represent the part ofany line contained in the considered interval. Sinceobservational evidence precludes the use of a Lorentzline shape far from the line center, we assume the following modified Lorentzian line shape: 1 1 --/~SL(V-- V0)2+ Ot2 1 - 1 -/Ss~(V - Vo)2 + a2//~L' (3.10)where the tSsL parameter, describing the sub-Lorentzianshape, is determined empirically. If we relate/~SL to themean line half-width am, that is, ~SL ~ -- (3.11) b'a choice of b = 200 cm-~ provides good agreementwith measured values of the pressure-broadened continuum (Butch and Gryvnak 1979) in the windowspectral regions (for comparison, the water vapor amis approximately 0.07 cm-~ at STP). It should be notedthat the form of (3.10) ensures that the integral properties of the line shape are preserved, since the subLorentzian behavior far from the line center is compensated by a super-Lorentzian shape in the vicinityof it. In a similar fashion we obtain e-type continuumabsorption coefficients for water vapor in good agreement with empirical values (Burch and Gryvnak 1979;Roberts et al. 1976) by reapplying (3.9) with the assumption that the half-width for self-broadening is anempirically determined multiple of the correspondingforeign pressure-broadened value. As a best compro314 MONTHLY WEATHER REVIEW VOLUME 12,0mise between somewhat diverging constraints (for different parts of the spectrum) we chose a value of 50for the corresponding multiplying factor. The inclusion of a continuum term in the equivalentwidth of the Malkmus (or Goody) model has severaladvantages over the standard procedure of using theclassical band model and adding published continuumabsorption coefficients in the region of the atmosphericwindows. As illustrated in Fig. 6, the widths of thespectral intervals used in the narrow-band model havea substantial impact on the spectrally averaged transmission function. This is commonly attributed to 'thefact that for too large intervals, the basic assumptionof the Malkmus or Goody band models (random distribution of line center positions) is invalidated (Morcrette and Fouquart 1985; Tjemkes and Duynkerke1988). To justify the choice of a particular intervalwidth, comparisons with line-by-line calculations areoften put forward. However, the agreement betweenband-model results and line-by-line model results doesnot necessarily imply that the "right" interval widthwas chosen. In the classical narrow-band model, theeffective spectral range of impact of any absorptionline will be restricted by the interval boundaries, thuseliminating the effect of strong lines on adjacent intervals and reducing the total absorptivity of the considered spectral region. In the extreme case, it could meanthat the agreement between the line-by-line model andthe "tuned" narrow-band model is fortuitous, the linecutoffintroduced implicitly through the narrow intervalwidth being comparable to the explicit cutoff used inthe line-by-line model. This point is supported by the0.6 "~ x ~\ 40 intervals, inel.eontinuui~~O.g'. ............... glO intervals, inel.eontinuum g40 intervals, exel.eontinuum .............. glO intervals, exel.eontinuu~0.0 10-~ 10-~ 10- 10~ 10~ 10a absorber amount Pa'~ 0.6'~0.4' no. 6. Comparison of band-model transmission functions for thewing of the water vapor rotation band (12.5-20.0 ~m) for differentinterval widths in the narrow-band model. Solid line: 840 intervals(i.e., z~v = 5 cm-~ in this spectral region) for the total spectrum considered, including continuum effects; short dashed line: 210 intervals(i.e., Av = 20 cm-~ ), including continuum effects; long dashed line:840 intervals, excluding continuum effects; chain dashed line: 210intervals; excluding continuum effects. Curves are valid at standardtemperature and pressure.results of Tjemkes and Duynkerke, who get ~:he bestagreement when the narrow-band interval width isclose to the cutoff chosen for the reference line-by-linecalculations. Since there is a wide range of cutoffs usedin line-by-line models, there could as well be a widerange of justifiable interval widths. The introduction of a pressure-type continuum inthe formulation of the band model alleviates the abow~mentioned problems. As demonstrated by Fig. 6, themodified model exhibits a strongly reduced i~npact ofthe interval width. Of course, the quantitative effect ofthe continuum term will depend on the chosen intervalwidth, the relative position of the interval with regardto strong absorption bands, and the value of the lineshape factor/~sr. In the vicinity of strong absorptionbands, where sub-Lorentzian effects are small, the vabaeof/~sr is of minor importance. For spectral regions thraway from major absorption bands (e.g., spectral windows) the choice of the interval width has very littleinfluence on the wide-band transmission function. Forsuch regions the absence or presence of a continu~.mterm is far more important and its magnitude in ourmethod will be mainly controlled by the value of ~st.(chosen to obtain agreement with measured continuumabsorptions). Thus, the inclusion of the continurtmterms avoids to a large extent the arbitrariness associated with the choice of the interval widths, as well asthe need to merge gaseous optical properties obtainedfrom various sources (i.e., narrow-band model parameters from spectroscopic data and continuum coefficients in some parts of the spectrum from publishedvalues). The narrow-band transmission functions calculatedusing (3.8) are averaged over the spectral intervals ofthe radiation scheme using the appropriate weights forthe energy distribution. The obtained wide-bandtransmission functions are subsequently approximatedby a series of decaying exponentials as discussed insubsection 2e. This is done separately for H20 and 03. However,for gases with constant atmospheric mixing ratios wecreate a "composite" gas by combining their spectroscopic properties as a sum of the individual propertiesof CO2, CO, 02, CH4, and N~O, using the actual relative atmospheric concentration of each gas with re~rdto CO2 as weights in the summation. Figure 7 illustratesthe absorption bands of the so-created composite CO2;the contributions by the individual components areindicated in the drawing. In this way, we reduce thecomputational costs in those intervals of our modelwhere more than one of the gases contained in thecomposite has absorption bands. This compensates forthe disadvantage of having to recalculate band-modelparameters, transmission functions, and correspondingCOe absorption exponential sum-fitting coefficients,when we want to consider modified relative concentrations of the gases that belong to the composite. AFEBRUARY 1992 RITTER AND GELEYN 315t0~-'10t10-"30.3 10- 3 I0~ 30 tO~ wave length ~ FIG. 7. Spectral distribution of absorption bands included in the composite CO2. Importantcontributions of the individual components are indicated in the drawing, i.e., I = pure CO2, 2= CH4, 3 = N20, and 4 = O2.modification of the code to account for each gas separately would be straightforward, should the need arise,for instance, in order to consider time-dependent concentrations of each gas in very long-range model integrations.4. Results In this part of the paper we will present results ofexperiments conducted to verify the radiation schemeand to study its impact on results of a global forecastmodel.a. Single-column experiments The verification of a radiative-transfer parameterization scheme is hampered by a number ofdifticulties.First, there is a lack of reliable flux and heating rateobservations suitable for comparison with model resuits. This point has been addressed frequently in thepast (cf. WMO 1984), but it is unlikely that the socalled "complete" radiation experiment will becomeavailable in the near future. In the absence of adequate observations, it is common practice to validate highly approximative radiation schemes like the one presented in this paper againstthe results of highly sophisticated reference schemes.In the context of the ICRCCM project (WMO 1984,1990) such reference schemes have been applied to anumber of well-defined standard atmospheric ~itdations. To validate the.thermal part of o, ur radiationscheme in clear-sky situations we use results of'the lineby-line model of the Laboratoire de Meteorologie Dynamique (LMD, Scott and Chedin 1981) and of theMax Planck Institut fuer Meteorologie Hamburg (MPI,Hollweg 1989). As an example,' Fig. 8 demonstratesthe generally excellent agreement between our schemeand the reference models. Illustrated are the heatingrates obtained for a clear-sky midlatitude summer atmosphere, when all gases are included. Both the resultsof the MPI model and of our scheme include the effectsof the minor trace gases (N20, CH4, CO, and O2),whereas the LMD model considers only H20, CO2,and 03. However, this discrepancy in the underlyingatmospheric composition is not responsible for the deviations between the two line-by-line models. A comparison of MPI results for the same case with and without the minor gases indicates that their contributionto the curves in Fig. 8 is small. It is interesting to notethat the magnitude of deviations of our results fromeither of the reference models is very similar to thedifferences between the two line-by-line results. Similardiscrepancies between line-by-line models have beendocumented in WMO (1984). This indicates that uncertainties in spectroscopic properties (e.g., water vapore-type absorption) contribute probably as much to thetotal error as the approximations involved in thespeeding up of our type of parameterization. Unfortunately, a comprehensive set of referenceCalculations is not yet available for the solar spectraldomain and not at all for cloudy conditions. However,results from a second part of ICRCCM calculations(WMO 1990), which mainly include calculationswith narrow-band and broadband radiative transferschemes, indicate that results of our model for solarcases and cloudy conditions are well within the limits316 MONTHLY WEATHER REVIEW VOLUME 120 , GRAALS~ LMI-o-100-200-300-400~00700~00900'1000 -'4 -'~ -'~ -'1 o K/d FIO. $. Heatin rates in a dear-sky midlatitude summer atmospheredue to thermal radiation. Comparison between line-by-line models(LMD, MPI ) and GRAAL$.of uncertainty. One reason for a disturbingly largespread among the results of various models is shownin Fig. 9, which illustrates the spectrally integrated solartransmission function for water vapor as used in different radiative transfer parameterizations. For largewater vapor amounts the maximum absolute differencein absorptivity is close to 5%, that is, approximately25% of the total absorption. This discrepancy will ofcourse be reflected in large differences in solar heatingrates, as discussed by Wang ( 1976),. Kratz and Cess(1985), and WMO (1990). However, some of the differences can be related to properties of the individualparameterization schemes. The low total absorptivityfor long pathlengths obtained with the parameterizationof Chou and Arking ( 1981 ) may be related to the factthat their scheme omits the water vapor bands at 0.72,0.81, and 6.3 ~m. As already pointed out by McDonald(1960) and illustrated by Davies et al. (1976), the weakbands at 0.72 and 0.81 nm contribute bonsiderably tothe total absorption at large pathlengths since they coincide with high values of the solar energy curve. Kratzand Cess estimate that the bands neglected by Chouand Arking contribute as much as 0.2 K day-1 to thesolar heating rates in the lower troposphere. For thelow absorptivity computed from LOWTRAN data using the parameterization according to Fouquart andBonnel (1980), no obvious explanation could be found(cfi Kratz and Cess 1985). For large absorber amounts the total absorption calculated with our scheme will exceed the values of Ladsand Hansen (1974) by approximately one percent. Asdiscussed by various authors (Kiehl and Ramanathan1982; Kratz and Cess 1985; Morcrette and Fouquart1985), excessive absorption could be caused by toowide spectral intervals in the narrow-band model calculations, and an interval width smaller than 10 cm-1has therefore been recommended. The latter conditionis violated in the narrow-band model that we: employto calculate the spectrally averaged transmission functions. The typical width we use in this part of the spectrum is approximately 20 cm-L However, as mentioned in subsection 3e, to what extent the recommendation of 10 cm-~ is a consequence of an implicit linecutoff tuning to fit line-by-line models that use a similarly explicit cutoff is open to discussion. This questioncan only be finally answered if reliable wide-band measurements of the solar gaseous transmission functionfor a wide range of atmospheric conditions and absorber amounts become available. In a comparisonwith measured transmission functions for selectedbands in the thermal spectral domain, Hunt and Mattingly (1976) claim agreement within the experimentalerror with band-model calculations using an intervalwidth of 50 cm-L A minor contribution to the largerabsorption in our model may also be caused by thefact that we do not confine our calculations to specificabsorption bands of limited width. We include all linescontained in the spectroscopic data, so that even veryweak bands in the region from 0.60 to 0.70 #m aretaken into account. Such bands, as well as the continuum effects discussed in subsection 3e, could be relevant for very long pathlengths, which may occur dueto multiple scattering inside clouds. The relevance ofcontinuum effects in the solar part of the spectrum hasalso been a major topic in a recent paper by Stephens absorber amount g c~n-~ v.5. 7.s ~ 5. 0 2.5' .5 I0- 10' 10~ 10~ absorber a~ount Pa FIG. 9. Compad~n of water vapor abso~tion o~ sol~ radiationat standard temperature and pressure as used in different radiativetransfer schemes.FEBaU^RY 1992 RITTER AND GELEYN 317and Tsay (1990), who discuss possible explanationsfor the so-called anomalous absorption of solar radiation in clouds. For the other major gaseous absorber in the solardomain, ozone, the agreement between variousschemes is much better. The solar ozone absorptioncurve used in GRAALS is based on the studies of Vigroux (1985) and Inn and Tanaka (1953) and closelyfits those of two other radiation schemes (Fig. 10). In order to illustrate the importance of scattering forradiative transfer processes in the thermal part of thespectrum, we performed two sets of transfer calculations with our scheme in cloudy conditions, as suggested for the second part of the ICRCCM project. Forthe first set of calculations, we derive the appropriatecloud optical properties from the data of Stephens(1979), including scattering coefficients for thermalintervals. For the second set of calculations, we consideronly absorption and emission by cloud droplets, bysetting the scattering optical depth to zero. An exampleof the results is illustrated in Fig. 11. For the sameatmospheric situation as in Fig. 2, substantial differences to the complete calculation are caused if scattering is neglected. Errors in the cooling rates insidethe cloud may still be acceptable, but the net surfaceflux is wrong by more than 20% (Table 2a), since reflection of the upward flux in the cloud layer contributes considerably to the downward flux, an effect notaccounted for when scattering is neglected. For comparison, we include the results of the calculation withthe FESFT version of GRAALS, including scattering,in the same figure. Obviously, the choice of ESFT orFESFT causes much smaller deviations than the neglectof scattering. If we run the same case assuming a different drop-size distribution in the cloud layer (i.e., thecloud type Cb instead of Scl of Stephens 1979), theresults change dramatically as a consequence of thestrong impact that the drop-size distribution has on absorber amount em NTP 0 lO'~ 0'~ .... t10-' .... 1'0-~ .0 i_~'" _-~0,/~... ......../ ............. Llc l S/ItANSEN I17.5~/15.0]~0.03 5.0 10-a10 absorber amount Pa ~G. 10. Same as Fig. 9, but for ozone.20.0'17.5'15.0'12.5'10.0'7.5'5.0'2.5100200'300~ 400500'soo GRAALS/E GRAALS/E no scat........ GRAALS/F'0'100200300'400500'600 700' ~ -700 soo ....... ~ .......................... l'so0 ,ooo~ ....... -7 -6 -5 -4 -3 -2 - 1 0 K/d FiG. l 1. Comparison of heating rates when scattering by clouddroplets is included (thick solid line) or excluded (thin solid line) inthe calculations. The dotted line represents the results obtained withthe FESFT version of GRAALS, including scattering. Atmosphericconditions are for midlatitude summer with a cloud between I and2 km above ground. Cloud optical properties are derived from Stephens (1979) for his cloud model Scl, scaled to a specific liquidwater content of 0.01 g m2.absorption and scattering optical efficiencies. For thislow cloud case, the dependence of the heating ratesand fluxes on the choice of method (ESFT with orwithout scattering, or FESFT with scattering) remainssmall; but the cooling rate and the downward flux atthe surface change dramatically compared to the firstcase, even though we assume the same total cloudwater content (Fig. 12 and Table 2b). Note the changeof scale for the abscissa between the two figures. Suchlarge differences are a strong argument for a link between the cloud optical properties and the specific liquid water content inside the cloud as formulated insubsection 3b. In standard emissivity-type schemes,which use only the integrated cloud liquid water content to determine an effective cloud emissivity, thiskind of behavior could not be simulated. For a cloud positioned higher up in the atmosphere,the impact of scattering on the downward flux at thesurface is reduced since reflected photons will be partlyabsorbed in the lower-tropospheric layers. However,reflection still contributes more than 5 W m-2 to thedownward flux at the surface and influences the outgoing longwave radiation at the top of the atmosphereconsiderably. In the case considered (Fig. 13), neglect318 MONTHLY WEATHER REVIEW VOLUME 120 TABLE 2a. Thermal radiative fluxes and total atmospheric flux divergence calculated for a midlatitude summer atmosphere in the presence ofa stratiform cloud (Stephens 1979, model Sc 1) with liquid water content of 10 g m2 in a 1-km-thic.k layer. GRAALS / EGRAALS E GRAALS / F no scatteringF, (top of atmosphere) -269.4 -266.8 (-268.9) -277.7 Cloud topF2 (surface) - 19.8 - 19.6 (-19.7) - 24.6 at 2 km heigJatAF (arm) -249.7 -247.3 (-249.2) -253.1F~ (TOA) -142.2 -138.0 (-141.7) -158.4 Cloud topF2 (surface) - 61.3 - 60.9 (-61.2) - 66.8 at 13 km heightAF(atm) - 80.9 - 77.1 (-80.5) - 91.5of scattering increases the outgoing longwave radiation(OLR) by 16.2 W m-2 and increases the lower-tropospheric cooling rates, since the cloud's opacity is reduced by the omission of the scattering optical depth.The error in the cooling rates inside the cloud due tothe FESFT approximation is comparable to the errorinduced by the neglect of scattering, because the FESFTexhibits a small reduction of the net flux at the heightof the cloud base. However, this reduction is spreadevenly in the vertical, so that the lower4roposphericcooling rates are almost unaffected by the approximation. It should be noted that the calculation of fluxesand heating rates was performed at the original resolution of the ICRCCM input data. Therefore, the heating above the cloud layer in Fig. 13 is merely an artifactof the plot software used for the illustration. 0100'200.300400'500.600. GRAALS/E GRAALS/E no scat........ C~An~S/~0'100200'300'400500'600 700- '700 800 ..... ~ ..................... 800 1000' -1000 -'4 -'3 -'2 -'l 0 ~/dFiG. 12. Same as Fig. 11, but cloud optical propertie~ from Stephensmodel Cb scaled to the same liquid water content as in Fig. l 1. The commonly used emissivity concept is not exactlyequivalent to setting the scattering optical depth to zero,as was done in our experiments. The effective emittanceof a cloud is considered as a parameterization of theoverall effect of the cloud, including the combined effectof droplet absorption and scattering as well as absorption by trace gases (cf. Stephens 1984). However, aswe have illustrated above, this complex phenomenoncannot be simulated adequately by a simple relationbetween cloud' liquid water path and effective emittance. In Tables 2a and 2b we have also included theresults obtained when we apply the FESFT version ofthe code to all spectral intervals but the one with the15-ttm CO: band. In this case (see values in brackets)the errors vanish almost completely.b. Global results The radiation scheme presented in this paper wasimplemented in cycle 34 of the ECMWF global fore:castmodel (Simmons and Jarraud 1984) and in the DWDversion of this model. The latter is part of the newforecasting system of DWD, which consists of nestedhigh-resolution limited-area models and the globalmodel. The flexibility of our scheme will enable us tosee the same basic code in all parts of the forecastingsystem, thus avoiding inconsistencies between the individual models, which might cause undesirable problems. Within the basic code of GRAALS we maychoose approximations (e.g., number of spectral intervals, choice of ESFT or FESFT, neglect of gases ofminor importance) according to the computationalconstraints and the accuracy requirement and application of the host model. The validation of a radiation scheme within a modelis an order of magnitude more difficult than in standalone mode. The resulting radiative fluxes and heatingrates are a complicated product of the performance ofindividual model components, the most importantbeing the cloud generation scheme and the radiationscheme itself. In the ECMWF global model the cloudscheme is based on a modified version of the diagnosticformulation of Slingo (1987). Clouds are diagnosedfrom basic-model variables and the parameterizedFEBRUARY 1992 RITTER AND GELEYN 319 TABLE 2b. Same as Table 2a but with Stephens optical properties for a Cb cloud. GRAALS / E GRAALS / E GRAALS ! F no scatteringF, (top of atmosphere) -279.3 -275.0 (-278.3) -282.5 Cloud topF2 (surface) - 50.9 - 50.5 (-50.8) - 53.5 at 2 km height~F (atm) -228.4 -224.6 (-227.5) -229.0convective activity. On global and regional scales thediagnosed cloud cover can be validated against climatological data and satellite estimates. However, thereis no adequate observational data for the verificationof the modeled cloud liquid water content. In the current operational model of ECMWF, the cloud liquidwater content is assumed to be one percent of the saturation specific humidity in the cloud layer. The concept of diagnosing the cloud liquid watercontent as a percentage of the saturation specific humidity is an oversimplification of the atmosphericreality, but there is little observational evidence to provide guidance for a more realistic approach. In theECMWF operational model a value of 1% was chosen,as it gave good agreement between satellite observationsand model-calculated OLR and planetary albedo.However, preliminary experiments indicate that thenew radiation scheme may require a somewhat highervalue for the same purpose. No attempt was made toverify the liquid water content as an independent ~00'400500'600700'800'90~10O0'_~l~d_~_~. '~ '-illllli~ ~ GRAALS/E /[i ORAALS/E no scat l________ GRAALS/F :4 ' -'~ ' ' ~ ; ~ ~ K/dFIG. 13. Same as Fig. 11, but cloud located between 12 and 13 km above ground. -0 -100 -200 -300 '400 '500 -600 -700 '800 '900 'i000l0 quantity. The latter point raises the question of whether one should not bypass the tuning of the cloud liquid water content and specify instead the cloud optical properties directly as a means to get the overall model results in agreement with observations. Even though the latter procedure is commonly used in climate models (Ramanathan et al. 1983 ), several arguments support the indirect approach. First, a direct specifi cation of optical properties based on observations of the present climate will limit the cloud radiative feed back to variations in cloud cover and distribution. The important link between cloud optical and microphys ical properties would be lost. Second, an increasing number of forecast models include cloud liquid water content explicitly as a prognostic model variable. A radiative transfer scheme with predefined optical properties will not be able to use this information, and thus will create an unnecessary inconsistency between the various model components. As a preliminary step to validate the overall perfor mance of our scheme, the global model was integrated for 10 days starting from an operational global analysis of 21 April 1991. The 10-day mean globally averaged OLR of 236.1 W m-2 predicted with the new scheme- is in good agreement with recently published satellite observations that provide a monthly mean value for April 1985 of 234.5 W m-= (Harrison et al. 1990). For the same month Kiehl and Ramanathan (1990) pro vide a mean global planetary albedo of 0.295, which is somewhat smaller than the value of 0.305 obtained as a result of our 10-day integration. Thus, the obser vations imply a small net heating of the earth-atmo sphere system caused by an imbalance in the radiative fluxes of more than 6 W m-= at the top of the atmo sphere. This has to be compared to an imbalance in the results of the model integration of approximately -4 W m-2. Since the global planetary albedo depends strongly on the correlative distribution of insolation and cloud cover, the difference between observations and model simulation may well be a consequence of errors in the modeled cloud field. Furthermore, a dis crepancy of this magnitude could also be explained as the combined effect of measurment uncertainties (e.g., sampling errors) and incompatibility of observation and simulation period (e.g., different years, different time ranges). The impact of clouds on the radiative budget and the distribution of radiative heating rates in the earth320 MONTHLY WEATHER REVIEW VOLUME 120atmosphere system has been the subject of many studiesin recent publications. Cess and Potter (1987) demonstrated that even on a global scale, different modelsof the atmosphere produce a widely varying radiativeresponse to clouds in the atmosphere. The models investigated produce global mean values of the so-calledcloud radiative forcing ranging from -34 to + 1 W m-2for the earth-atmosphere system. A similar disagreement occurs if the net effect is split into its individualcomponents, with a solar CRF between -45 and -74W m-2 and a thermal CRF of 23 to 55 W m-2. Someguidance about the true value of the cloud effect onradiative fluxes at the top of the atmosphere can beobtained from satellite estimates. But values of the netCRF ranging from -17 to -35.5 W m-2 for differentsatellite systems indicate considerable disagreementbetween the various observations (Arking 1990). For this reason we consider the preliminary resultsobtained with our model for this quantity mainly as acrude test of whether the model's overall performancewill provide a reasonable CRF. An agreement with observed values will of course be a necessary but not sufficient part of the model validation. As demonstratedby Cess and Potter (1987) for the Oregon State University-Lawrence Livermore National Laboratory(OSU-LLNL) model, errors in the cloud scheme caneasily compensate errors in clear-sky and cloudy fluxesto produce a misleading agreement with satellite estimates of the CRF. In our 10-day integration of theglobal model, starting from initial data for 21 April1991, the total global cloud cover of 52% resulted in aglobal mean solar CRF of -43.9 W m-2 and a corresponding thermal value of 21.5 W m-2 so that the netCRF at the top of the atmosphere amounts to -22.4W m-2. Comparison of these values with recently published monthly mean values for April 1985 (Harrisonet al. 1990) indicates very good agreement with regardto the solar CRF, where the observed value is -45.1W m-2. A comparison of the modeled zonal-mean distribution (Fig. 14b) with the corresponding illustrationof Harrison et al. demonstrates that this good agreement is also evident in the latitudinal structure of thesolar CRF. Small absolute values of the forcing in thepolar region of the Northern Hemisphere, caused by aconvolution of high background reflectivity (i.e., snowcovered surfaces) and little solar insolation, are adjacentto a very pronounced minimum in the region of themidlatitude storm tracks. In this area, the abundanceof clouds combines with a fairly high amount of solarinsolation for the period considered, causing a substantial impact of cloud on the amount of solar energyavailable to the earth-atmosphere system. A considerably smaller impact is present in the subtropical regions, where the large insolation encounters far fewerclouds. Even though a distinct minimum of the solarCRF in the tropics is also present in the observations,the simulation obviously exhibits a considerable exaggeration of this feature. This discrepancy cannot be1.0[~3~ global mean value :0.520.60.40.2060 30 0 -30 -61) -90 latitude 100 solar global mean 75~ .............. thermal solar :-43.9 W/m~ ~ net thermal: 21.5 Wltm~ 501 net :-23.4 W/m~ -25-75-100 b 40 40 --90 latitude50 solar .............. thermal 1~ netglobal meansolar : 1.0 W/methermal: 3.9 W/menet : 5.0 ~/m~ 60 30 0 -30 --60 -90 latitude F3G. 14. Zonal-mean distribution of (a) total cloud, cover, (b)cloud radiative forcing at the top of the atmosphere, and (c) for theatmosphere averaged over a 10-day integration of the ECMWF ~lobalmodel employing GRAALS. Initial date: 21 April 1991.explained completely by uncertainties in the observations (e.g., scene identification errors) or the fact thatthe model integration period is not identical to the period of the observations. However, this kind of' discrepancy could be easily explained by errors in themodeled cloud field, which in this region of high insolation would project strongly onto the solar CRF. Inparticular, this discrepancy may well be related to thesame causes that lead to the slight overestimation ofthe global planetary albedo, discussed earlier. Themagnitude of the minimum solar CRF in the regionFEBRUARY 1992 RITTER AND GELEYN 321of the Southern Hemisphere storm tracks is reducedin our integration compared to the observations as aconsequence of the fact that the integration period isvalid for the latter part of the month considered. Thus,the insolation available for interaction with the cloudsof this area is reduced compared to the monthly meanvalue. The discrepancy between model results and observations is much larger for the thermal CRF, which isunderestimated by approximately 10 W m-2 in theglobal mean, mainly as a consequence of too low valuesfor the maxima in the tropics and the midlatitudes inthe Northern Hemisphere (Fig. 14b). A nonnegligiblepart of this discrepancy may be caused by the differentalgorithms used for the determination of the CRF inthe model and from satellite observations. Whereas inthe model clear-sky fluxes are calculated for each gridpoint by assuming zero cloud cover, the Earth Radiation Budget Experiment (ERBE) algorithm uses fluxesfrom grid points designated as cloud free in the sceneidentification algorithm for the same purpose. Thisimplies that in the model, clear-sky fluxes for gridpoints with a substantial cloud cover will still be affectedby the typically moist structure of the underlying atmosphere, whereas ERBE clear-sky fluxes will be dominated by relatively dry conditions. A quantitative assessment of this effect was presented by Arking (1990).Assuming that cloud-free pixels are typically 25% drierthan the adjacent cloudy areas, he interpreted the results of corresponding differences in clear-sky flux calculations as an indication that a significant part of 10W m-2 difference in the global values of the CRF between different satellite estimates might be explainedby the way clear-sky fluxes are obtained in the ERBEalgorithm. However, if we believe that the lack ofstructure in our simulation of the thermal CRF is atrue discrepancy with reality, there is little reason toassume that this is caused by a lack of clouds in theareas of interest. Neither the cloud distribution (cf. Fig.14a) nor the solar CRF indicates that the discrepanciesmight be caused by a lack of clouds in areas of largeobserved thermal CRF values. More likely, the cloudspresent in the model integration, particularly those inthe upper troposphere, have too little impact on theOLR. The inefficiency of modeled upper-level cloudswas discussed also in a paper by Kiehl and Ramanathan(1990), describing experiments with the NCAR Community Climate Model that exhibited similar discrepancies. In our integration this inefficiency is probablycaused by the linear relation between the layer saturation humidity and the assumed cloud liquid watercontent in the model. Consequently, high clouds atlow temperatures will systematically be associated withvery small liquid water contents and correspondingoptical depths. As an alternative to .the presently employed linear formulation, the cloud liquid water content could be based on an approach used in the FrenchNWP models. Following Betts and Harshvardhan(1987) and Somerville and Remer (1984), the cloudliquid water content is estimated from: = 1 Oqsat I = constant ql p* O( 1/p) oe p* = 8 x 106 Pa (tunable constant), (4.1)with qt being proportional to the change of liquid wateralong a given vertical path following a saturated adiabat. The choice of the 1/p vertical coordinate (different from Betts and Harshvardhan) was done in orderto keep qt as independent ofp as possible, even at lowpressures. However, as this paper is confined to the presentationof the radiation scheme itself, no attempt is made toalleviate the problems inherent in the cloud generationscheme. In agreement with the observations, the net CRFcalculated with our model is negative for the majorityof the globe. However, due to the underestimation ofthe thermal CRF, the difference between the simulatedand the observed global mean values is more than 8W m-2 and the near cancellation of solar and thermaleffects, observed in the tropical region, is not simulatedat all. The major part of the CRF affects the earth's surface.However, as illustrated in Fig. 14c, clouds also have asubstantial direct impact on the radiative heating ofthe atmosphere. This impact is most pronounced atlow latitudes where clouds cause both a solar and aninfrared contribution to the atmospheric warming,which is on the order of 10 W m-2 atm-~. The solarcontribution results from both a direct absorption ofincoming solar radiation by cloud droplets and the enhanced absorption by gases as the effective pathlengthsare elongated through multiple scattering. In the infrared part of the spectrum, the loss of radiative energyto space experienced by lower tropospheric layers isreduced mainly by clouds located at higher atmosphericlevels. Even though we intend to use the radiation schemepresented in this paper mainly for applications in numerical weather prediction, it could also be used in thecontext of climate studies. As an example, we presentan estimate of the initial radiative forcing resulting froman increase of the atmospheric concentration of CO2.This forcing is the necessary prerequisite for climatesimulations concerning the problem of a possible climatic change related to the generation of increased atmospheric CO2 levels through human activities. Chouand Peng (1983) provide estimates of the initial radiative forcing at the surface and the top of the atmosphereresulting from a doubling of the atmospheric concentration of CO2. Using our scheme for the same purposewill provide answers to two questions. First, does ourmodel agree with the results of Chou and Peng withinthe expected uncertainties, and second, will the choice322MONTHLY WEATHER REVIEWVOLUME 120 of ESFT or FESFT have a significant impact on prob lems of this kind? The thick solid lines in Fig. 15 il lustrate the zonal-mean distribution of the change in the radiative fluxes obtained in our model, if we double the amount of our composite CO2 for the interval con mining the 15-~tm CO2 band and use the ESFT method to treat the overlapping with other gases. Similarly, the thin solid lines represent the results, when the FESFT is used instead. The difference between the two sets of curves is very small, except for a part of the antarctic region where CO2 forcing at the top of the atmosphere becomes slightly positive when the ESFT is used. This positive forcing is due to a strong temperature inversion in the lower model layers with associated clouds, which causes the effective radiative temperature of the earth atmosphere system to be larger than the corresponding surface temperature. This complex situation cannot be handled well enough by the approximations associated with the FESFT to give the correct CO2 forcing, but absolute values of fluxes and atmospheric heating rates are in good agreement with the ESFT results. The global mean forcing at the top of the atmosphere is -2.43 W m-2 (-2.63 W m-2 with FESFT) and at the surface- 1.01 W m-2 (0.96 W m-~ with FESFT). The overall agreement with the results of Chou and Peng (dashed lines) is satisfactory. Both schemes exhibit a minimum of the CO2 surface forcing at low latitudes, which is due to the dominant role of water vapor e-type ab sorption in these areas of high atmospheric humidity. In contrast, the forcing in the OLR decreases toward the poles as a consequence of the lower temperatures at high latitudes. Weak local extreme values in our results are caused by corresponding extremes in the modeled cloud cover. The maximum impact on the OLR found in the subtropical regions is more pro nounced with our model. This could be related to a 5.0' Surface~ 2.5~ -5 01 ..... 60 30 0 -30 -60 -90 latitude FIG. 15. Zonal-mean distribution of the initial radiative forcingdue to a doubling of the atmospheric CO2 content from 330 to 660ppm for the spectral region 12.5-20.0 #m, solid lines: ESFT versionof GRAALS, dashed lines: FESFT version, dotted: Chou and Peng(1983).less cloudy and drier atmosphere, since we use theECMWF analysis from 19 October 1989 as initial data,whereas Chou and Peng based their calculations on aclimatological dataset. In addition, the following re.asons may explain the generally higher sensitivity in ourmodel, which is most obvious in the forcing at the topof the atmosphere. First, the doubling of CO2 in ourmodel affects a larger spectral region (12.5-20.0 u~n)compared to Chou and Peng's model ( 12.5-18.5 tzrn).The additional part of the spectrum exhibits only weakCO2 absorption, but as pointed out by Chou and Peng,unsaturated band wings are very sensitive to an increasein the CO2 concentration, therefore contributing significantly to the total CO2 forcing. A small part of ourenhanced sensitivity may be due to the fact that theabsorptive properties of our composite CO2 take intoaccount the 17-ttm N20 band, even though this band- is overlapped by considerably stronger proper CO2lines. The assumption that almost all clouds are black,which is implied in the scheme of Chou and Peng (Penget al. 1982), may also contribute to the differences,since black clouds will of course reduce the sensitivityof cloudy areas to an increase in atmospheric CO2.This argument is supported by the fact that we (:anreduce the discrepancies between our results and thoseof Chou and Peng by assuming a higher liquid watercontent, and thereby increase the optical depth ofclouds in our model. In contrast to the generally acknowledged importance of the proper representation of radiative processesin climate models, it is still widely believed that radiative transfer parameterization in NWP is of lesser importance and its impact is confined to the latter partof medium- or long-range forecasts. As a demonstralionof the potential impact of a change of radiative transferschemes on the results of a forecast model, we willaddress the subject of model performance in terms of.standard forecast skill scores. A limited set of completeforecast experiments can, of course, serve only as ademonstration of the model's sensitivity to a replacement of one operational radiation scheme (Morcrette1990) by another. In order to illustrate this sensitivity,we present the time evolution of the troposphericanomaly correlation of geopotential height in theNorthern Hemisphere for one 10-day experiment inFig. 16. For this particular forecast, the integration using GRAALS scores considerably better for most ofthe forecast period than the one where the operationalradiation scheme is employed. Even though we presentonly the results of the forecast showing the greatestsensitivity to a replacement of the radiation scheme,GRAALS has not had a detrimental impact on forecastskill in any of the six experiments performed so far.However, further experimentation is required to confirm these preliminary results and to allow a thoroughassessment of all aspects of the interaction of GRAALSwith the remainder of the forecast model.FEBRUARY 1992 RITTER AND GELEYN 323IO08060 /,0 0 0............ ' KR$. ~ '"....~.. i , "7,, , , ,'"""' 1 2 3 /~ 5 6 7 8 9 10 forecast deys FIG. 16. Time evolution of the tropospheric (1000-200 hPa)anomaly correlation ofgeopotential height averaged over the NorthernHemisphere (20-N-83-N) for two forecasts with the cycle 34 versionof the ECMWF global model. The forecast with the operationalECMWF radiation scheme is represented by the dotted line (A15)and the thick solid line (KRS) represents the experiment with theGRAALS scheme. The thin solid line indicates the performance ofa persistence forecast. Initial date: 15 September 1990.5. Final remarks and conclusions The use of two-stream methods allows a consistentsolution of the radiative transfer problem includingabsorption and scattering by all relevant constituentsin each part of the spectrum. In the thermal domain,commonly applied emissivity-type schemes are basedon the questionable assumption that scattering processes can be neglected or at least be implicitly treatedby the concept of effective cloud emittances. Anotherdisadvantage of emissivity schemes is the nonlineardependence of computational costs on the verticalmodel resolution. The computational cost of the twostream method is linear in the number of layers, a significant factor in light of the likely future developmentof numerical weather forecast models toward higherresolution. At the present resolution of the ECMWFforecast model (i.e., 19 layers), the scheme presentedrequires approximately the same computation time and60% of the memory requirements of the ECMWF operational radiation scheme (Morcrette 1990). Acknowledgments. The authors would like to thankN. A. Scott and A. Chedin as well as H.-D. Hollwegfor the provision of reference results from line-by-linecalculations. We also benefited from Y. Fouquart'sstimulating suggestion concerning the treatment oftemperature and pressure effects in k-distributionmethods. Finally, we wish to express our gratitude toour colleagues at the DWD and the DMN who contributed to this collaborative effort. In addition, thecontributions by the anonymous reviewers are gratefully acknowledged. APPENDIX Analytical Expressions of the ai Coefficients The layer coefficients ai, i = 1, 5 are defined as follows:wherea~ = exp(- A~'/Ho)a2 = --a472 -- a571a~ + 72a~a3 = --as'Y2 -- a23'la~ + Tia4 = E( 1 - M2)( 1 - E2M2)-~a5 = M( 1 - E2)( 1 - E2M2)-~ I -an+a5a6 = E = exp(-~Aj6') a5 - ~0(a;-~ + ~[a;) ~1 = 1 ~ ~2~02 ~2 = 1 -- e2~02The resonance case, that is, ~o = 1, can be avoidedby a sm~l displacement A~o. REFERENCESAckerman, T. P., K.-N. Liou, and C. B. Leovy, 1976: Infrared ra diative transfer in polluted atmospheres. J. ,4ppl. Meteor., 15, 28-35.Arking, A., 1989: The radiative effects of clouds and their impact on climate. Report to the IAMAP International Radiation Com mission, WCRP-52, 39 pp.Betts, A. K., and Harshvardhan, 1987: Thermodynamic constraint on the cloud liquid water feedback in climate models. J. Geophys. Res., 92, 8483-8485.Burch, D. E., and D. A. Gryvnak, 1979: Continuum absorption by H20 vapor in the infrared and millimeter regions. 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