Improved application of exponential sum fitting ... - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 104, NO. D6, PAGES 6291-6303, MARCH 27, 1999

Improved application of exponential sum fitting transmissions to inhomogeneous atmosphere Zhian Sun and Lawfie

Rikus

Bureau of Meteorology ResearchCentre, Bureau of Meteorology,Melbourne, Australia

Abstract.

An improvedmethodof applyingexponentialsum fitting transmissions

(ESFT) to inhomogeneous atmospheres (referredto as inhomogeneous ESFT or IESFT) is presented,and its use in a broadbandradiativetransfermodelis documented. Instead of using a scaling function to representthe pressureand temperature dependenceof the absorption coefficient we apply the ESFT to all pressuresand temperatures based on weight terms determined at reference conditions.The exponentialterms correspondingto all pressuresand temperatures form a look-up table whichcan be usedto interpolateto the corresponding value at the real pressureand temperature. The IESFT is performedusinga nonlinearleast squaresroutine with someconstraintsto ensurethat the ranking of the exponential terms is maintained. The method is tested againstline-by-linecalculations,and the results show that the use of IESFT in the two stream approximation of radiative transfer can reduce the cooling rate error by a factor of 2 relative to the use of a scaling function, particularly in the upper atmosphere. An application of the treatment of overlappinggaseousabsorptiondevelopedby Mlawer et al. [1997]to ESFT

is described.

The basic idea of this method

is to treat

the mixture

of two

absorbinggasesas a single gas by introducing a binary interpolation parameter

which reflectsthe relativeabundanceof the two designatedgases.Testsagainst line-by-line calculations show that this method is not only accurate but also computationally efficient. in eachspectralband, n monochromaticradiative transfer calculationsare performed,one for eachki term, and The method used in treatment of absorbingproper- the resultsare weightedwith the wi and summed.This ties of atmospheric constituents is crucial in radiative method has advantagesin that it can be employedin transfer computations. This is because both the ac- the monochromatic radiative transfer calculations such curacy and the speed of radiation calculations depend as two-stream approximations, which are simple and strongly on the way in which the absorption of solar efficient and in which the effect of scattering can be and infrared radiation by atmosphericgasesis calcueasilytaken into account.A difficulty ariseswhen this lated. The radiation schemes used in numerical weather method is appliedto inhomogeneous atmospheres.The prediction and climate models must be computationESFT can only be performed for homogeneouscondially fast yet accurate. To achieve these aims, one of tions. The scalingapproximationis normally used to the common methods, the exponential sum fitting of account for the variation of absorption coefficientwith 1.

Introduction

transmissions (ESFT) technique[ Wiscombe andEvans, pressureand temperature[Edwardsand Slingo,1996] 1977; $lingo and $chrecker,1982]has been usedfor a away from homogeneity.This approximationassumes long time. With the ESFT method the spectrally avthat the absorptioncoefficientcan be factorizedin the eraged gas transmissionsare expanded as a function of absorber amount u in terms of a sum of exponential

form

forms.

P) = 0(r,

(2)

so that the absorptioncoefficientat any pressureand temperaturecan be determinedby a productof the absorptioncoefficientat the referenceconditionand the Each exponentialterm ki actsas an equivalentabsorppressureand temperaturedependentscalingfunction4>tion coefficient. To calculate the irradiance for one gas This impliesthat the spectralabsorptioncoefficientsat different atmosphericlayers are fully correlated. This Copyright1999by the AmericanGeophysical Union. assumptionmay be appropriatefor a few specialcases

-

i=1

[seeGoodyand Yun9,1989]and not hold for general.

Paper number 1998JD200095.

Even for the specialcases,the accuracyof this assump-

0148-0227/99/1998JD200095509.00 6291

6292

SUN AND RIKUS.: IMPROVED

ESFT TO INHOMOGENEOUS

tion will dependon the form chosenfor the scalingfunction.

3.

ATMOSPHERE

Line-by-Line Calculations

To set up a spectral file required by the Edwards and Slingo code, a high spectral resolution radiation model to improve the performance of the ESFT in inhomomust be run to calculate the spectral transmittances for geneousatmospheres. In their approach the ESFT is each of the absorbinggases. These calculationswere performedfor all pressurelayers,successively increasing performed using the GENLN2 line-by-linecode [Edthe number of expansionterms as requiredto minimize wards, 1992], together with the HITRAN92 [Rothman the fitting error relative to the true transmissions.Thus et al.. 1992] molecular datebase; a spectralresolutionof the pressureand temperature dependenceof absorption 0.002 •um -• was used. These transmissions wereavercoefficientsis taken into accountdirectly. However,the aged across each spectral band, weighted by solar irranumber of expansionterms for a required accuracyis diance at the top of the atmosphere [Labs and Neckel, generally large for this method. 1970] for the solar spectrum and the Planck function at In this study we investigate a possibleway to imthe temperature at which the transmissions were calcuprovethe ESFT in its applicationto inhomogeneous atmosphere,particularly focusingon improvingthe accu- lated for the infrared spectrum. The results were then racy of irradianceand heating rate calculations.Section used to perform the investigationsfor improving the 2 briefly introducesthe radiation packageusedin this performance of the ESFT. In addition to the calculations of transmissions, study. Section 3 presentssome line-by-line calculations GENLN2 was also used to perform reference irradithat serveas a spectral basisfor the modificationworks ance and heating rate calculations. Although there and also provide a benchmark for verifications. The are some line-by-line long-wave irradiance and heatdetailed improvementsof the application of the ESFT ing rates available for comparison (e.g., line-by-linereto an inhomogeneousatmosphereare describedin secsults from the LBLRTM (Atmospheric Environment tions 4 through 6. Comparisonswith referenceresults Research [Clough et al., 1992]) and GFDL (Geophysare presentedin section 7, and the final discussionand ical Fluid Dynamics Laboratory, Princeton, New Jerconclusionare given in section8.

Armbrusterand Fischer [1996] proposeda method

sey[Ramaswamy andFreidenreich, 1991]),the line-byline references for the short-wave are not available. For

2. Edwards and Slingo New Radiation Package

this reason,the GENLN2 has been extendedto produceirradianceand heatingrate calculationsfor an absorbingatmosphere.The originalcodewasdesignedto

The new radiation package developed by Edwards calculate transmittance

and radiance and therefore the

and $1ingo[1996][hereinafterreferredto as ES96]is only extra work required for irradiance calculations is used to perform this study. This packageis very pow- to perform the angular integration. This is achieved erful in severalrespects. Its details have been described using a four-point first-momentGaussquadrature, as

by ES96, and hence only a brief descriptionis given here. The packagewas designedto suit both applications involving referencecalculationsand weather prediction and climate studies.Therefore the spectralresolution of the code is flexible. This is achievedby running a preprocessorto generatea spectral file containing the necessaryspectral information such as spectral band limit, absorbingspecies,and spectral properties of atmosphericconstituentsrequired by the radi-

suggestedby Cloughet al. [1992]. In addition, the variation of the Planck function within a layer was not taken into accountin the original GENLN2 radiance algorithm. The radiancewas calculatedusinga mean Planckfunctionevaluatedwith the path-weighted layer temperature. This has been modifiedby introducing the expressionproposedby Cloughet al.

ation code. This facility enables the user to have full control of the spectral resolution of the code and the

to replacethe meanPlanckfunction. In the aboveequa-

B(7-)- [•-3-(a7'-3-bv'2)Blev](1 4-a7'4-b7'2) -•

(3)

tion,B(7')istheeffective Planckfunction,• isthelayer

physicalprocesses involved. The gaseousabsorptionby meanPlanck function, •-isthelayeroptical depth,Blev different speciescan be treated using the exponential is the Planck functionevaluatedwith the layer upper sum fitting technique applied to either LOWTRAN7

(forupwelling radiance)or lower(fordownwelling radi-

[Kneizyset al., 1988] transmittancesor the HITRAN ance)boundarytemperature.The constantsa andbare [Rothmanet al., 1992]moleculardatebase.The radia- assignedvaluesof 0.193 and 0.013, respectively.Similar tion code is based on the two-stream equationsin both modificationsfor the GENLN2 codefor the long-wave the short-waveand the long-waveregionsof the spec- spectrumhave also been conductedindependentlyby trum. Radiative processessuch as gaseousabsorption, Brindley and Harries [1998]. They usedthe modified efRayleighscattering,absorptionand scatteringby cloud GENLN2 to studythe sensitivityof the greenhouse and aerosolparticles can be easily included or excluded fect to changesin atmosphericcomposition. dependingon the specificaim of study.

In the short-wavespectrum the calculationsare more

SUN AND RIKUS.'

IMPROVED

ESFT TO INHOMOGENEOUS

tion Codesin ClimateModels(ICRCCM) [e.g.,Luther et al., 1988] have been repeated using the modified

at Surface(in W m-2) Determined Using AER, GENLN2 for Three ICRCCM

GENLN2, and the results have been compared with those from other line-by-line models. Table 1 presents the upward irradiance at the top'of the atmosphere

Test Cases AER

GENLN2

and the downward

20 19

27

F½(TOA) F4 (surf) F • (TOA) F * (surf) F • (TOA) F* (surf)

333.55 268.82 319.40 333.03 282.19 345.68

6293

Several test casesfrom the Intercomparison of Radia-

Table 1. Upward Irradiance at Top of Atmosphere and Downward Irradiance

Case

ATMOSPHERE

332.53 270.01 320.04 334.74 282.81 346.37

irradiance

at the surface for three

ICRCCM test casesdeterminedusingthe GENLN2 and LBLRTM

models

for the midlatitude

summer

atmo-

sphere. The spectral regionfor this and the remaining

comparisons givenin this paper is 0-2200cm-t. Case 20 is for the water vapor line absorption,case19 for water vapor line plus continuum absorption, and case 27 for the absorptionby water vapor, carbon dioxide, and ozone. In general, the agreementbetween two models is very good. The irradiance difference at the top and

The midlatitude summer atmosphere is used in the calculations, and the spectral

regionis 0-2200cm-•.

the surface boundaries

between these two models is less

than2 W m-2. Theseresultsareverycloseto thoseobcomplicated due to the effects of scattering. If we do not considerthe effectsof scattering, however, the irradiance calculationsin the short-wave are greatly simplified as the upward and downward irradiance can be calculated separately like those in the long-wave spectrum: downward irradiance is just the direct solar beam traveling from the top of the atmosphereto the Earth's surface; upward irradiance for a nonzero albedo, nonspecularsurface is the isotropic upwelling diffuse beam. We adopted the formula presented by

tained by Brindley and Hatties [1998].Figure 1 shows the comparisonof long-waveheating rates for a midlatitude summer atmospheredue to absorptionby water

calculations. Again, we use the four-point first-moment Gauss quadrature to perform the angular integration for the upward transmission.

should be pointed out that the spectral line data used in the above line-by-line calculations are not from the same version of the HITRAN database, and this may partly explain some of the difference shown here.

vapor(Figurela) andozone(Figurelb)determinedusing GENLN2 with LBLRTM and GFDL models. It is seen that the heating rate due to water vapor absorption generatedusing GENLN2 is very closeto that from the LBLRTM

model. The maximum

two models is about 0.1 K d -•.

difference between

The maximum heat-

ing rate differencedue to ozoneabsorptiongeneratedby Ramaswamyand Freidenreich[1992]to perform these the GENLN2 and GFDL models is about 0.2 K d-t It

The Cloughwatervaporcontinuummodel[Cloughet al., 1989]is usedin GENLN2 andit hasbeenupdatedin this study with their modified versionknown as CKD2.2

[Han et al., 1997].

Ramaswamyand Freidenreich[1992] calculatedthe line-by-line solar heating rates due to water vapor line absorption using the modified GFDL model. In their

Table 2. Band Structure Mixture Speciesand Number of k Terms in Each Band 1020-100mbar 100-0.01mbar k (mix-gas)

k (random)

LW band, cm-• 0-400 400-560

H2Oc H20•

H2Oc H20•

8 10

8 10

560-800

H2Oc,CO2

CO2,O3

15

8 (H20) 9 (CO2)4 (03)

800-990

H20•

H20•

8

8

990-1070 1070-1200

H2Oc,O3 H2Oc

CO2, O3 H20•, 03

9 5

6 (H20) 3 (CO•) 7 (03) 6 (H20) 4 (03)

H20c

H20c

11

11

03 H20

03 H20

7 7

7 7

H•O• H•O•, CO•

CO2 H2Oc, CO•

11 13

12 (H•O) 5 (CO•) 16 (H•O) 12 (CO•)

1200-2200 SW band, •m 0.2-0.69 0.69-1.19

1.19-2.38 2.38-5.0

Note that the subscriptc of H•O refers to water vapor continuumand the number of k terms usedin the mix-gasand randomoverlappingschemesare listed in columns4 and 5, respectively. LW, long wave; SW, short wave.

6294

SUN AND RIKUS.'

IMPROVED

ESFT TO INHOMOGENEOUS

(o) H,O (0-2200 crn-' MLS) ....

0.01

i

....

i

....

i

....

i

ATMOSPHERE

(o) LBLRTM- GENLN2

....

i

'

,

' •

......

.

0.10 .

ß ß

0 ,

1.00 ß

LBLRTM

10.00

........

GENLN2

100.00 _

o .

,,,,i.... i,ii,]i•'1',,,iI,,•,

1000.00 0.0

0.5

1.0

1.5

2.0

2.5

,

,3.0 -0.4

-0.2

(b) O, (990-1200 crn-' MLS)

0.0

0.2

0.4

(b) GFDL- GENLN2

0.01

0.10

1.00

.

10.00

•

100.00

GFDL

1000.00 _

o

1

2

LW heoting rote (K doy-')

-0.4

-0.2

0.0

0.2

0.4

Heoting difference (K doy-')

Figure 1. Comparison of long-wave headtingrate for the midlatitude summer atmosphere determined using GENLN2, LBLRTM, and GFDL line-by-line models. The results of LBLRTM and GFDL are obtained from Carbon Dioxide Information Analysis Center.

(a) watervapor;(b) ozone.The right-handpanelsshowthe heatingdifferences.

calculations they employed a midlatitude summer at-

and four bandsin the short-wavefrom 0.2 to 5.0/•m. mospherefor three different solar zenith anglesand The band limits and the absorbinggasesconsidered in specified the surface albedo to be 0.80. We have re-

each band are listed in Table 2. The short-wave band

peated the samecalculationsusingGENLN2 with the limits are the same as those used in ES96. The detailed samespecificationsand comparedthe resultswith those descriptionsabout columns 2 to 5 in this table will be of Ramaswamy and Freidenreich. The differenceof the given in section 6. solar heating rates determinedby two modelsis about In the ES96 radiation codethe exponentialterms are 0.1K d-1 (notshownhere). scaledwith a scalingfunctionto accountfor the presIn the followingsectionswe have usedpublishedline- sureand temperaturedependence of the absorptioncoby-line resultsas the referencewhere possible.Other- efficient. The form of this scalingfunctionis a power wisewe usedthe resultsdeterminedfrom GENLN2, as law for the pressureand the quadraticfor the temperanoted in the text where appropriate. ture. It is seenfrom Figure3 in ES96 that the heating rate error in upper atmosphereis relativelylarge. This errorismostlikelydueto the useof the scalingfunction, 4. Inhomogeneous Exponential Sum asdiscussed in the introduction.To improvethe perforFitting Transmissions(IESFT) mance,we tried to fit a differentscalingfunctionto each We first use the preprocessorof the Edwards and term and found that the improvementcan be obtained Slingoradiation packageto generatethe spectralframe- in this way but only to a very limited degree.To overwork for the radiation codes. We chose seven bands in comethis deficiency,we replacedthe scalingfunction the long-wavespectrumrangingfromzeroto 2200cm-1 with a table describingthe pressureand temperature

SUN AND RIKUS.:

IMPROVED

ESFT TO INHOMOGENEOUS

ATMOSPHERE

6295

variation for each exponential term. The accuracy of this method in the correlated k-distribution approach

for the water vapor; i hPa and 250 K for the carbon dioxide and ozone. The numerical fitting of the expohas been clearly demonstratedby Fu and Lieu [1992] nential terms at pressuresand temperatures other than the referencecondition was performed with constraints and Mlawer et al. [1997]. Since the procedure of generating a look-up table on ki to ensurethat the appropriate ranking of ki terms for the ESFT is different from that for the correlated is maintained. This was achievedby using a Nonlinear k-distribution, a little bit of more details is described lea,stsquares (NLLS) routine(D. O'Brienpersonalcomhere. We first calculated gas transmissions using munication,1996). GENLN2 for 19 pressuresranging from 0.01 to 1020 hPa at approximately equal logarithmic intervals and 3 5. Treatment of Water Vapor

temperatures(200, 250, and 300 K) for eachpressure. The results were averaged acrosseach spectral band. The ESFT were first performed at a reference pressure and temperature to obtain weight terms wi using

the Wiscembeand Evans[1977]code. The ESFT were then generatedfor the pressuresand temperaturesother than the referencecondition using the fixed weightswi as determined at the reference pressure and temperature. The coefficientexponential terms ki thus obtained form a look-up table that can be used to interpolate each k term to the real pressureand temperature. This treatment is exactly the same as that used in the CKD

method,asdescribedby Fu andLieu [1992]andMlawer et al. [1997]. There are some issues that need to be addressed in the

numericalfitting of exponents. Firsfly, we found that accuracy in the transmissionfitting is not a sufficient conditionfor obtaining accurateirradiancesand heating rates. There are many casesin which the transmissions are fitted very accurately,but when the corresponding heatingrates are calculatedusinga two-streamapproximation model, the resultsare not very good. This problem arises from the fact that

the solution

of the nonlin-

ear least squaresfitting is not unique. There are many possiblesolutionsthat satisfy the required accuracyof the fit to the transmissionfunctions. These solutions, however, may not satisfy the physical requirementsfor monochromaticradiative transfer. The accuracyof the radiation calculations depends on the values of ki in each layer being consistentwith those in the other layers of the atmosphere. It is quite possiblethat the sum

Continuum Absorption In these studies

and as described

in ES96

the wa-

ter vapor continuum due to foreign-broadeningand to self-broadeninghave been treated separately. The contribution due to the foreign-broadeningcontinuum absorptionhas beentaken into accountby multiplying the foreign continuumtransmissionswith the water vapor line transmissionsand then integrating acrossa band. Therefore the effect of the foreign continuum absorption is included in the treatment of the water vapor line absorption,and the spectral dependenceof the foreign continuum absorption is better resolved. The treatment of the self-broadeningcontinuum is slightly different from ES96. In ES96 the self-continuum absorption was treated as a grey process,and the self-continuumtransmissionswere fitted in terms of an exponential function. The exponential coefficientwas scaled using the same form of scaling function as used for the water vapor lines, i.e., a power law for the pressureand a quadratic for the temperature. Although the scalingapproximation is appropriate for the water vapor continuum absorption, we found that it may be more logical to scale the self-continuumabsorption coefficientin terms of the water vapor partial pressurebecauseit is proportional to water vapor partial pressure. The self-continuum transmissionfor five vapor partial pressuresand three temperatures were calculated usingthe CKD2.2 model. The averagedtransmissionswere then fitted in terms of an exponential function with an exponent scaled by a scalingfunction

from equation(1) is closeto the true valueof the averaged transmissionfor a given layer, but the exponential ki terms do not represent the absorption coefficientsin each subinterval for that layer, which will introduce errors to the heating rate calculations. Secondly,the heating rates in the stratosphere are sensitive

to small

deviations

of the transmission

from

unity for small amounts of absorber,so even if the trans-

missionis fitted to a good accuracy,the heating rates may easily showlarger errors. The last point is that the choice of the reference condition is important in determining the accuracy of the heating rate; in particular, the referencepressureneedsto be chosenat the pressure level where maximum heating rates occur, as suggested by Harshvardhanet al. [1987]. In our calculations the referencepressureof 500 hPa and the reference temperature of 250 K were specified

f(E T) - a

'

Eref

--

Tref

(4)

where T and E are temperature and water vapor partial pressure,a and m are constantcoefficientsdetermined by the numericalfitting. The referencetemperatureand

watervaporpartialpressure arespecified Tref = 250K and Eref = 45 hPa.

6.

Overlapping Absorption by Mixture

Gases

Overlappingabsorptionby a mixture of gasesis a commonproblemin radiative transfercalculations.The common solution is to assumethat the absorption spec-

tral linesfrom two gasesare randomlydistributedover

6296

SUN AND

RIKUS.-

IMPROVED

ESFT

TO

a spectral band. Under such a condition the multiplication property of the monochromatic transmissionholds for band-averagedtransmissionof the gasmixture; that is, the mean transmissionsof the mixture can be calculated by

-

(5) i=1

INHOMOGENEOUS

ATMOSPHERE

tion (6) for fixedvaluesof r/and f. Sinceu2 is altered proportionallyto u• in the transmissioncalculations, the results are equivalent to those calculated for absorberamountsof a singlegas and can be fitted with

Y(gl,g2)--

?J.)i e--k'(uldFfu2) = ?J.)i e(--k*ttmix) , (8) i=1

i=1

j=l

Umix-- Ul -•-fu2.

where ux and u2 are the absorber amounts of the two

(9)

gases.Althoughby usingthispropertyonecansolvethe Defined in this way, Umixcan be thought of as a mixed problem of overlappinggaseousabsorption,the num- absorberamount. The value of f defined by equation ber of monochromatic calculations involved would be (6) hasclearphysicalconnection to the valueof r/. In n•nu. The treatment of overlappinggaseousabsorp- our case, however,the value of f must be tuned to obtion in this way is obviously computationally expen- tain appropriatefitting resultsin termsof equation(8). sive. On the basisof Ritter and Geleyn[1992],Edwards This is becausethe mixing ratio for one gas can be sev[1996]proposeda methodcalledequivalentextinction eral ordersof magnitude greater than for another, and to reduce

the number

of monochromatic

calculations

required in the treatment of overlappinggaseousabsorption. This method is computationally ef•cient; the number of monochromatic calculationsrequired for two overlappingabsorptiongasesis n• + an2, where a _• 1. Here we present a different approachto overlapping gaseousabsorption,which is, in principle, an application to the ESFT of a method proposed by Mlawer et

thereforef mustbe large (or small)enoughto ensure that the secondterm of equation(9) is the sameorder of magnitude as the first term. If f is chosenas the ratio of S2 to S1 as in the Mlawer et al. work, then the secondterm may not influence the fit, and the results may depend entirely on the first term alone. We

i õ• 2 ß.- • 1 whichare the same chose 9 r/ valuesof 0, õ• as used by Mlawer et al. in the lower-altitude regime.

al. [1997]for the correlatedk-distributiontechnique. The ESFT is performed for each r/, and the results are Application to the ESFT requiresan approachslightly stored for 19 pressuresand 3 temperatures. To calcudifferent from that of Mlawer et al. The distinction late optical depth for a mixture of two gasesin broadbandmodel,onefirstcalculatesr/from equation(6) and between the two methods is given below. In the Mlawer et al. work, the overlappingabsorbing then obtains the effective absorption coefficientby linspeciesare addressedby including an additional inter- ear interpolation from the stored ki values at reference polation variable, the binary speciesparameter r/, in the r/. Multiplying this effective absorption coefficientby parameter space associatedwith the stored ki values. t/mix givesthe optical depth for the mixture of gases. This variable is defined as To generate the coefficient table for the mixture of two gases,we have to repeat the same line-by-linecal,•'1t•l = (6) culationsfor eachr/, and this is extremely expensive.To •1- S•u•+ S2u2 u•+ fu2' avoid such a large amount of calculations,we perform where S1 and S2 are the respective integrated line line-by-line calculationsfor each speciesindividually for strengthsof the two speciesin the spectral band and f a reasonablerange of absorber amount and store the is the ratio of $2 to $1. Parameter r/representsthe rel- mean transmissions across the G ENLN2 wide-mesh inative radiative importance of the two absorbinggases. tervalswhichare specifiedto 1 cm-•. We then usethe For each reference pressure and temperature, values of multiplication property of the transmission to obtain ki in the relevant bands are stored for atmosphereswith the transmissionsfor the mixture of two speciesat the valuesof r/spacedappropriately for linear interpolation, wide-meshinterval. The band-averagedtransmissions which is used by the model to compute the ki values for for mixed gasesare obtained by integrationof the transmissions at the wide-mesh interval across the spectral the specifiedatmosphericcondition. In our applications the variable r/ is also used to band. In this procedure we have avoided the applicatreat the overlappingabsorbingspeciesin the relevant tion of the multiplication property to the broadband, spectral bands, but the values of f are different from which is inevitable in the random overlap approach. To those used in the method

of Mlawer

et al.

The

band-

calculate

the transmissions

of the mixture

at the wide-

averagedtransmissionsfor two mixed gasesare calcu- mesh intervals, one speciesis chosenas the key species. lated in terms of the momJchromatictransmissionequa- The absorber amount for the second speciesis calcution

T•2(u•, u2)- •

lated usingequation(6), basedon the amountof the key species.The transmissioncorrespondingto the ab-

•,exp[-(kxvu•+ k2vu2)]dl,' (7)

for a reasonablerange of absorberamountsul and u2, where the variation of u• and uu is constrained by equa-

sorber amount for the secondspeciesis then determined by linear interpolation from the precalculatedvalues. The method describedabove can be applied to a spectral band in whichonly two gasescontributeto absorp-

SUN AND RIKUS.'

IMPROVED

ESFT

TO INHOMOGENEOUS

tion. In principle, however, this method may be extended to the casein which more than two gasesabsorb in a band. In this paper we only consider three major absorbinggases' water vapor, carbon dioxide, and ozone. These specieshaveoverlappingabsorptionin the spectral intervals of 560-800, 990-1070, and 1070-1200

7.1.

ATMOSPHERE

6297

Scaling for Water Vapor Self-Continuum

We first show an improvement of long-wave heating rate due to the use of the new scaling function for the water vapor self-continuum absorption. In this comparison the ES96 scaling function has been applied to the water vapor line absorption in order to separate cm-1 Followingthe techniqueof Mlaweret al. the the difference caused by the use of different continuum key speciesin each band are designatedfor two sepscalingschemes.The calculationswere performedusing arate altitude regimes based on the fact that the imthe midlatitudesummeratmosphere(for the ICRCCM portant absorption by water vapor occurs in the tropocase19), and the resultsare shownin Figure2. In this sphere,while the absorptionsby ozoneand carbondioxfigure the referenceresult, determinedusing GENLN2, ide have important impacts on heating rates above the is plotted in solid curve; the result from the original tropopause[Cloughand Iacono,1995].In eachregime, ES96 scalingscheme,which is referred to as scale-C1is only two speciesare considered.The transition between plotted as a dotted curve; the result from the current the lowerand the upper altitude regimesis chosenas the scheme, referred to as scale-C2, is plotted as a dashed 100 hPa level; theseregimesare listed in Table 2. Using curve. A linear pressurescale is used to emphasizethe this method to treat the overlappinggaseousabsorption effect of the water vapor continuum absorption in the is not only more accurate, as will be seen in section 7, lower atmosphere. It is seenthat the new scalingfuncbut also computationally more efficient becausewe estion (scale-C2)for the water vaporcontinuumabsorpsentially treat two absorbingspeciesas one. The extra tion leads to a better representation of the radiative cost in the interpolation with respectto parameter r/is heating in the low troposphere.The maximum heating small. We have found that this idea can be extended rate error shownin the right-hand panel is reducedfrom to the short-wave spectral region. The details for the 0.18 to 0.10 K d -1. short-wave spectrum are also listed in Table 2; columns 4 and 5 of Table 2 show the number of k terms required 7.2. Improvement Due to IESFT in each spectral band in both the current schemeand Next we show the improvement due to IESFT relathe random overlappingscheme. tive to the scalingfunction for the casesof spectralline absorption. Figure 3a showsthe results for the water 7. Comparison With Line-by-Line vapor plus continuum absorption. In this and the folModel lowing comparisonsthe scale-C2has been used for the In this section we will verify the modified schemes calculation of the water vapor self-continuum absorpdescribed in section 6 by a nurnber of comparisonsof tion. We use the term "scaling"to representthe results the heating rate determined using the ES96 code with determined using the ES96 scalingfunction, and the term "IESFT" denotes those using the current method. that using the line-by-line model. GENLN2

H•O+CONT(0-2200 cm-' MLS)

I

0L,

-

ES96

)...... 2--L....'1 ?

200

400

,

•

,

600

ß

ß

10hPa and lessthan 0.5 K d-• for p < 10 hPa. The resultfrom

icrccm case 25 (trop.) 0.01

calculations

codes, only gaseousabsorptionsare included for consistency with the line-by-line calculations. The results are shownin Figure 8. The heating rate error from the

-

ES96

'

0.10

1.00

10.00

100.00

i i i i i i i i i | .......

1000.00

o

5

lO

LWheatingrate (K day-t)

15

-2

-1

"_ ß , •1:,, ! .......

0

I , , , , ....

1

2

Heatingerror (K day-')

Figure7. Long-wave heating rateforICRCCIVl case 25(tropical atmosphere). Theoverlapping gaseous absorption duetowatervapor, carbon dioxide, andozone occurs in thespectral bands 3, 5, and6 andaretreatedwiththetwomethods, mix-gas andrandom overlap. Right-hand panel showsthe heating rate errors. ,

6302

SUN AND

RIKUS.-

IMPROVED

ESFT

TO INHOMOGENEOUS

SW heating in band4 .........

0.01

i .........

i .........

ATMOSPHERE

GENLN2

I .........

i ........

-r-

'

i

-

ES96

.......

.o

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f

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,

,

,

-,,,

, ,

7,

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2

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sw heoting rote (K

-

,

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,

I

....

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Heoting error (K cloy-')

Figure 8. Solar heatingrate in the near-infraredspectralband due to the overlappingabsorption by water vapor and carbon dioxide. The calculationsassrunea solar zenith angle of 30ø and

a surfacealbedoof 0.8. The solarconstantusedis 1360W m-2. The water vaporcontinuum absorption is included in this band.

good accuracythe number of k terms usedin the mixgas schemeis also much lessthan that usedin the random scheme,as seenin Table 2. In the long-waveband 3, for example, the number of k terms required by the mix-gasschemeis 15, whereasthat required by the random overlapping schemeis 288. These comparisonsclearly indicate that the achievement of the mix-gas method is of interest in two respects. The first is its efficiency; the number of monochromatic calculations required by the mix-gas scheme is almost 1 order of magnitude smaller than that required by the random overlap scheme. It is also smaller than that required by the fast ESFT scheme

sorptionbandsof overlappinggaseshas beentaken into accountin a more appropriatefashionthan assuming random overlap acrossthe whole broad and. It is seen from Figure 7 and 8 that the improvementof the heat-

ingratein theupperatmosphere (p < 10hPa) issignificant, indicatingthat the new methodhas a particular advantagein improvingheatingrate calculationsin the upper atmosphere.

8.

Summary

An improvedapplicationof the ESFT method to inhomogeneousatmosphereshas been describedin this [Ritter and Geleyn,1992]and the equivalentextinction paper. The main aspectsof the problemwe have adscheme[Edwards,1996].The secondis its accuracy.In dressed in thisstudyare the useof the scalingapproxour comparisonsthe accuracyof the mix-gas schemeis imation and treatmentof the overlapping gaseous abbetter than the random overlap scheme. The reason sorption.Althoughthe scalingapproximationis a comfor this is due to the fact that the randran overlapping monwayto dealwith gaseous absorptionin inhomogeschemeis basedon the assumptionthat the absorption neousatmospheres and has been usedfor a longtime, spectral lines from two gasesare randomly distributed the errorof radiativeheatingarisingfromthismethodis over a spectral band, •nd thereforethe mean transmis- relativelylargein the upperatmosphere (p < 10hPa). sionsof the mixture, averagedover the spectral band, We solvethis problemby replacingthe scalingfunction are determined by the product of the mean transmis- with an interpolationof the k exponentialterms from sions of the two gasesseparately. The multiplication a precalculated look-uptable. This table is generated property of transmissionsis appropriate over narrow by the method called IESFT, as describedin the text. bands[Goodyand Young,1989]but is lessjustifiedover This approach canreducethe heatingrate errorarising broad bands. As can be seenfrom Figure 6, the heating from the more commonlyusedscalingapproximation rate error from the random overlap schemefor the spec- by a factor of 2.

tral bands560-800cm-• is muchlargerthan that for For overlappinggaseousabsorptionwe investigated the spectralbands990-1070cm-•; this may reflectthe a technique suggested by Mlawer et al. [1997]. The effect of the bandwidth on the accuracy of the random overlap assumption. In the mix-gas approach,the mean transmissions of the mixture are calculated at i cm -• intervals

and therefore

the correlation

between

the ab-

principleof this techniqueis to treat the mixture of two gasesas one, and therefore the number of the monochro-

matic calculations is greatlyreduced.The comparison with line-by-linemodelshowsthat the heatingrate er-

SUN

AND

RIKUS.:

IMPROVED

ESFT

TO

INHOMOGENEOUS

ATMOSPHERE

6303

ror from the random overlap scheme is larger in the

Goody, R. M. and Y. L. Yung, Atmospheric Radiation: Theoretical Basi::, end ed., 519 pp., Oxford Univ. Press, New York, 1989. cating that the multiplication property may not be apHan, Y., J. A. Shaw, J. H. Churnside, P. D. Brown, and propriate to apply to the broadband. The method used S. A. Clough, Infrared spectral radiance measurementsin in this study has avoided the application of the multithe tropical Pacific atmosphere, J. Geophys. Res., 102,

long-wave band3 (wider)than in band6 (narrow),indi-

4353-4356, 1997. plication property to the calculationsof the broadband transmissions,which is inevitable in the random overlap Harshvardhan, R. Davies, D. A. Randall, and T. G. Corsetti, A fast radiation parameterization for atmospheric circuapproach. The accuracytherefore is better than that of lation models, J. Geophys. Res., 92, 1009-1016, 1987. the random overlap scheme. The idea is not new, but Kneizys, F. X., E. P. Shettle, L. W. Abreu, J. H. Chetwynd, this is the first time it has been applied in the context G. P. Anderson, W. O. Gallery, J. E. A. Selby, and S. A. of ESFT.

Clough, User's guide [o LOWTRAN7, AFGL Tech. Rep. AFGL-TR-88-0177, Air Force Geophys. Lab., Bedford,

Mass., 1988. Acknowledgments. The authors gratefully acknowledge the generoussupport of J. Edwards and A. Slingo for Labs, D., and H. Neckel, Transformation of the absolute providingthe radiationpackagecontainingthe preprocessor solar radiation data into the international practical temperature scale of 1968, Solar Phys., 15, 79-87, 1970. and the radiation code which have been used for much of Luther, F. M., R. G. Ellingson, Y. Fouquart, S. Fels, N. A. the work presentedin this paper. D. Edwardsis thanked Scott, and W. J. Wiscombe, Intercomparison of radiation for allowingus to use his GENLN2 line-by-linecode. We

codesin climatemodels(ICRCCM): Longwaveclear-sky results-A workshop summary, Bull. Am. Meteorol. Soc., some very useful discussions.Two reviewersmade many 69, 40-48, 1988. valuablecommentsand were greatly appreciated.The lineby-line resultsof GFDL and LBLRTM wereobtainedfrom Mlawer, E. J., S. J. Taubman, P. D. Brown, M. J. Iacono, and S. A. Clough, Radiative transfer for inhomogeneous the CarbonDioxideInformationAnalysisCenter(CDIAC).

are gratefulto D. O'Brienfor providingthe NLLS codeand

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Edwards, J. M., and A. Slingo, Studies with a flexible new radiation code. I, Choosinga configurationfor a largescalemodel, Q. J. R. Meteorol. Soc., 122, 689-719, 1996. Fu, Q., and K. N. Liou, On the correlated k-distribution method for radiative transfer in nonhomogeneousatmospheres,J. Atmos. Sci., 49, 2139-2156, 1992.

properties of stratiform water clouds, Q. J. R. Meteorol. Soc., 108, 407-426, 1982. Stamens, K., and S.-C. Tsay, Optimum spectral resolution for computing atmospheric heating and photodissociation rates, Planet. Space. Sci., 38, 807-820, 1990.

Wiscombe, W. J., and J. E. Evans, Exponential sum fitting of radiative transmissionfunctions, Jo Cornput. Phys., 416-444, 1977.

L. Rikus and Z. Sun (correspondingauthor), BMRC, GPO Box 1289K,Melbourne,VIC. 3001, Australia. (e-mail: [email protected]) (ReceivedMarch 9, 1998; revisedNovember12, 1998; acceptedNovember16, 1998.)