Radiation Transfer March 2000 - ECMWF | Advancing … · Radiation Transfer 2 Meteorological Training Course Lecture Series ECMWF, 2002 qualitative point of view and to the introduction

Meteorological Training Course Lecture Series

ECMWF, 2002 1

Radiation TransferMar ch 2000

By Jean-Jacques Morcrette

European Centre for Medium-range Weather Forecasts, Shinfield Park, Reading Berkshire RG2 9AX, United Kingdom

Table of contents

1 . Introduction: An historical perspective

2 . The Earth’s radiative balance and its implications

2.1 The need for parametrization

2.2 Global mean considerations

2.3 Time and space variations of the solar zenith angle and their consequences

3 . The theory of radiation transfer

3.1 Terminology

3.2 Derivation of the monochromatic radiative transfer equation (RTE)

3.3 Basic laws

3.4 The spectral absorption by gases

3.5 The spectral scattering by particles

4 . Radiation schemes in use at ECMWF

4.1 The operational longwave scheme (as of March 2000)

4.2 The operational SW scheme

5 . Comparisons with observations

6 . Conclusions and perspectives

REFERENCES

Abstract

These notes aim at fulfilling two different and somewhat contradictory objectives:

1) to give a generalintroductionto theparametrizationof radiationtransfer(RT) in numericalweatherpredictionandclimatemodels.This might be useful to peoplewho wish to look at generalcirculation model (GCM) outputsand to understandqualitatively how radiationtransferin clearandcloudyatmospheresis linked to theotherphysicalprocesses,andhow it caninfluence the atmospheric motions. In that case, the internal behaviour of the radiation scheme might remain a black box.

2) to provide a reasonablycompletedescriptionof the RT parametrizationspresentlyusedin the ECMWF model to helpunderstandthespecificpropertiesof theECMWF forecasts.While theformertaskrequiresonly basicknowledgeof physics,the latter requires much more insight in how a RT scheme works.

The following pagessomehow attemptto tackle thesetwo tasks.After an introduction,the secondchapterdiscussestheradiationbudgetat thetop,andwithin theatmosphere,bothgloballyandmoreregionally in relationwith theothertermsof theatmosphericenergy balance.The third chapteris devoted to the derivation of the RT equationfrom an observational and

Radiation Transfer

2 Meteorological Training Course Lecture Series

ECMWF, 2002

qualitative point of view and to the introduction to the basic laws of physics necessaryto understandthe RT. Thesimplificationsthat canbe usedwhendealingwith RT in the Earth’s atmosphere,andthe variousapproximationsneededtomake theRT aproblemtractablefor anumericalmodelof theatmospherearealsopresentedin chapter3. ThemainaspectsofthevariousRT parametrizationspresentlyusedin theECMWFforecastingsystemarepresentedin chapter4, togetherwith thespecificationof the cloud optical properties.Examplesof validation of various aspectsof the parametrizationof cloud-radiationinteractionswith observedradiationfieldsarepresentedin chapter5. As a matterof conclusion,somecommentsonthe future of RT calculations, and on some other radiation-related hot topics are given in chapter 6.

1. INTRODUCTION : AN HISTORICAL PERSPECTIVE

Amongthevariousprocessesresponsiblefor thediabaticheatingof theatmosphere(convection,turbulence,large-

scalecondensation,heat,moistureandmomentumtransferat thesurface,radiationtransfer),whichhave to bede-

scribedby physicalparametrizationsin ageneralcirculationmodel(GCM), this lastonestandsbecauseits impact

is felt atall timesandoverthewhole3-D domain.Comparedto otherprocesses,radiationtransfer(RT) alsostands

ashaving a longhistoryof theoreticaldevelopments.Fromthestatisticalandquantummechanicsat theendof the

19thandbeginningof the20thcentury, Boltzmann,Stefan,Wien,PlanckandEinsteinmadepioneeringadvances

in thespectraldescriptionof theradiationemittedby ablackbody. Afterwards,spectroscopicstudiesof thegases

importantfor theradiativebudgetof theatmosphere,andthedevelopmentof variousapproximationsmadethecal-

culation of the radiation transfer in the atmosphere a tractable problem.

Thevery first suchapproximationis theseparationbetweenshortwave (SW) andlongwave (LW) schemesdueto

theobviouswavelengthdifferencebetweena black-bodyat Sun’s temperature(around5800K, with mostof its

energy below 4 microns,andtheenergy sourceoutsidetheatmosphere)andthatof theatmosphere(globallyaver-

agedequivalenttemperatureof about255K asseenfrom thetopof theatmosphere,with mostof its energy above

4 microns,andthesourcesbeingthesurface,theradiatively active gases(mostlyH2O, CO2, O3), andtheclouds

within the atmosphere).Chandrasekhar(1935,1958,1960) provided the first approximationsto the radiation

transferin a scatteringatmosphere.Elsasser(1938,1942),Goody(1952,1964),Curtis (1952,1956),Godson

(1953),Malkmus(1967)definedthefirst usablesimplificationsto thespectralrepresentationof theradiationtrans-

fer includingits dependenceon two key atmosphericparameters,pressureandtemperature,allowing atmospheric

heating/cooling rates to be computed (Rodgers and Walshaw, 1966).

To thepoint, that in themid-70s,it wassaid(WMO/GlobalAtmosphericResearchProgram,1975)thatradiation

transferwasa solvedproblem,providedbig enoughcomputerswereavailable.Theonly developmentsrequired

werefor fastparametrizations.A gooddescriptionof thevariousapproximationsusedin the80’s at thedifferent

stagesin thedevelopmentof computer-efficient radiationschemes(andof thenecessarytrade-offs betweenaccu-

racy and efficiency, two contradictory arguments) is given inStephens (1984) andFouquart (1987).

Unfortunately, theInterComparisonof RadiationCodesfor ClimateModels(ICRCCM) at theendof the80s(El-

lingsonet al., 1991;Fouquartet al., 1991,Baeret al., 1996)showedthat,if anoverall goodagreementin clear-

sky longwave computationsby line-by-linemodelsexisted,largediscrepancieswerefoundin theresultsof both

theLW parametrizedschemesandall typesof SW schemes.Hereafter, at leastfor clear-sky atmospheres,these

weretrackeddown to deficiencies,mainly in theimplementationof theapproximationsusedto dealwith thespec-

tral and vertical integrations.

SinceICRCCM,a numberof parametrizedRT schemeshave beendevelopedfrom line-by-line(LbLs) modelsso

thattheiraccuracy, at leastin clear-sky atmospheres,shouldbeveryclosefrom thatof LbLs (e.g.,Ramaswamyand

Freidenreich, 1992;EdwardsandSlingo, 1996;Mlawer et al., 1997,to namejust a few. As far ascloudsarecon-

cerned,thesituationis muchmorecomplex with anongoingdebatewhetheror notcloudsmightabsorbmorethan

whatcurrentparametrizationsaccountfor, with thesomewhatrelatedroleof thespatialinhomogeneitiesin the3-

dimensionaldistributionof watercondensateson theradiationfieldswithin andat theboundariesof acloud,with

Radiation Transfer


ECMWF, 2002 3

the role that theoverlappingof cloud layersplayson thevertical distribution of the radiative fluxesandheating

rates.Also thecomputationalenvironmentof ageneralcirculationmodelmightalsoimposesomeexternallimita-

tionsonthequalityof therepresentationof thecloud-radiationinteractions.In thefollowing sections,anoverview

of these different questions will be provided.

2. THE EARTH ’S RADIATIVE BALANCE AND ITS IMPLICA TIONS

2.1 The need for parametrization

Theneedfor a parametrizationof theradiationtransferin a large-scalenumericalmodelof theatmospherestems

for two reasons:

• first, like theotherphysicalprocesses(condensation,surfaceprocesses,turbulence,...) theradiation

transferactuallyoccursat spatialscalesmuchsmallerthanthe scales(e.g.,interactionwith cloud

droplets)explicitly resolved by the model in its treatmentof the adiabaticpart of the prognostic

equations;

• second,althoughthetheoryof radiationtransferhasbeenwell known for decades,thecomplexity

of thegoverningequationis suchthat it cannotbeusedstarightforwardly in a large-scalemodel.A

much more computationallyefficient schemehasto be designedto accountfor the effect of the

radiative processes.

2.2 Global mean considerations

Theimportanceof theradiationtransferprocessesfor theEarth’s atmospheresystemis obvious:Radiationis the

only way throughwhich theEarth-atmospheresystemcanexchangeenergy with therestof theuniverse.Theim-

portanceof a properrepresentationof theradiative processesin climatemodellingor weatherforecasting(aftera

few days) comes from this simple consideration;

A zero-dimensionalenergy balancemodeldescribestheglobalannualmeanequilibriumof theEarth-atmosphere

system as

where is the planetaryalbedo (the ratio betweenreflectedand incident solar energy at the top of the

atmosphere,ToA), is the“solar constant”(theflux of energy from theSunat themeanSun–Earthdistance),

is the outgoingterrestriallongwave radiation, is the radiometrictemperatureof the Earth,andσ is the

Stefan-Boltzmannconstant( W m-2 K-4 ). The factor 0.25 (= 1/4) arisesfrom the Earth

interceptingsolarradiationproportionallyto its cross-sectionandemitting terrrestrialradiationproportionallyto

its surface.Accordingto thelatestavailablesatellitemeasurements(ERBE,EarthRadiationBudgetExperiment,

BarkstromandSmith,1986) W m-2 and . Thus is 237W m-2 and

K. The discrepancy betweenthis value and the meanclimatological surface temperature( K) is

explained by the so-called “greenhouse” effect, which will be discussed later.

Fig.1 presentsthevariousradiativestreamswithin theatmosphereafterRamanathan(1987).Outof the343W m-

2 of solarenergy availableat ToA in the0.2 to 4 wavelengthrange,about30%is reflectedbackto theouter

spacewithout changeof wavelengthafter scatteringin the atmosphereand/orreflectionat the Earth’s surface.

Therethetruesolarimput to theatmosphereis about237W m-2. A largefractionof this reachesthesurfaceand

is absorbedby landmassesandoceans;only roughlyonequarterof this solarradiationis absorbedwithin theat-

mosphereandcreatesameanheatingof about0.6K/day. Theexactfractionsbeingabsorbedwithin cloudsandby

0.25 1 α–( )�

0

�σ � e

4= =

α �0� � e

σ 5.67 10 8–×=

�0 1372= α 0.295 0.010±=

� � e 255=

� s 288=

µm

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ECMWF, 2002

the “clear” atmospherearepresentlydebated(Cesset al., 1995;Ramanathanet al., 1995;PilewskieandValero,

1995;Stephens, 1996)with the role of aerosolsin clearsky atmosphereandof cloud inhomogeneitiesrecently

comingto theforefrontto explain thisexcessabsorptionnotproperlyaccountedby thecurrentgenerationof radi-

ationschemesin GCMs(e.g.,Cairnset al., 2000).Part of thesolarenergy input to thesurface(between155and

180W m-2 dependingon how muchis actuallyabsorbedwithin theatmosphere)is returnedto theatmosphereby

emissionof terrestriallongwaveradiationin the4 to 100 wavelengthrange,but thisemission(about63W m-

2, differencebetween390 W m-2, the longwave upward emissionof the surfaceand327 W m-2, the longwave

downwardemissionof theatmosphere)doesnot fully compensatefor thesolarflux into thesurface.Thedeficitof

about106W m-2 is compensatedby turbulenttransportof latentheat(for about90 W m-2) andsensibleheat(for

about16W m-2) from thesurfaceto theatmosphere.Theexistenceof aradiativebalanceatToA andof aradiative

imbalanceat thesurfaceimpliesthattheatmosphereitself is anetsourceof terrestriallongwaveradiationto com-

pensatefor thewarmingby thesolarheating,latentheatreleaseandsensibleheatflux. Thustheatmospherecools

through emssion of longwave radiation by about 1.6 K/day.

In theabove discussion,only thefiguresof theradiationbudgetat ToA areknown with somedegreeof accuracy

thanksto somedecadesof satellitemeasurements.Estimatesof thecomponentsof theenergy budgetat thesurface

aremoredifficult to obtaindueto thelackof globalcoverageby conventionalobservationsystemsandto thealrge

uncertainties in the ongoing tentative determination of these quantities from satellite measurements.

2.3 Time and space variations of the solar zenith angle and their consequences

Thesolarzenithangle is theanglebetweentheverticalatagivenpointontheEarthandtheSun’sdirection.Its

cosine, , which is therelevantparameterfor radiative computationscanbecomputedknowing thelatitude,the

longitude, the time of the year, and the time of the day. influences the radiation transfer in two ways:

• theamountof energy receivedat ToA is theproductof thesolarconstantby a factordependingon

thetimeof theyear( ) andby if is positiveandzerootherwise(seeFig. 2 );

• the atmospheric mass encountered by a solar beam is proportional to (for ).

Therefore,a diminishing meansnot only lessinput at ToA but alsomorescatteringandabsorptionwithin the

atmosphere,thereforeanevensmallersolarflux at thesurface.Thetime variationsof arethedaily cycle and

theyearlyaseasonalcycle,which inducecyclic temperaturevariationsandmany specificpatternsin theinstanta-

neousweather. But themostimportantfactorof influenceof radiationon theweatheris linkedto its specevaria-

tions.Dueto a betterinsolationof theequatorialbelt thanof thepolar regions,which is not compensatedby the

terrestriallongwaveoutput,theequatorialregion is warmerthanthepolarones.Thissituationhastwo majorcon-

sequences for the weather:

• thesamepressurelayersarethicker at theequatorthanat thepoles;sincethesealevel pressureis

observedto beuniformly distributeddueto friction in theplanetaryboundarylayer, andaccording

to Buys–Ballot’s law, thezonalmeanwind regimeis of zonalwesterlies,thereforetheatmosphere

precedes the Earth in its rotation;

• thereis a needto transportheatfrom the equatorialto the polar regions.This transportis partly

realizedby theoceans,partly by theatmosphere,But a simplezonalcirculationdoesnot allow the

atmosphere to fulfill this role and disturbances have to develop.

Theselasttwo pointsclearlyindicatewhatis requiredfrom aradiationschemein alarge-scalenumericalmodelof

theatmosphere:first a goodestimateof thepole–equatorradiative imbalance,secondanaccuratepartitionof the

poleward transport of heat between oceans and atmosphere.

In thetropics,themeanannualnetheatingof about60W m-2 is thesumof a320W m-2 heatingby absorptionof

solarradiationanda260W m-2 coolingby emissionof longwaveradiation.Thisnetheatingof only about20%of

µm

θ0

µ0

µ0

1.0000 0.0035± µ0 µ0

1 µ0⁄ µ0 0>

µ0

µ0

Radiation Transfer


ECMWF, 2002 5

theavailablesolarinput(theremaining80%heatthetropicalsurface)is thedriving forcefor thegeneralcirculation

of theatmosphereandoceans.Thereforea 10%systematicerror in thecolumnabsorbedsolarradiationor in the

longwave emissionmaypotentaillytranslateinto a factor5 larger(i.e.,50%)changein thepolewardtransportof

heat.

Also importantis thedipole-likedepositof radiativeenergy in theearth–atmospheresystem.As awhole,thetrop-

osphereissubjecttoanetradiativecooling(of about106W m-2, Ramanathan, 1987),whereasthesurfaceissubject

to anetradiativeheatingof thesameamplitude.Hereagaina10%systematicerrorin theamplitudeof thisdipole

will directly affect thedestabilizationof thetroposphere,i.e., thefundamentaldrive for thetroposphericconvec-

tion.

Thesequestionsmayseemmainly relatedto climatestudies,especiallythoseperformedwith coupledocean–at-

mospheremodels.Howeverthepreviousargumentis alsorelevantto NWPmodelling.NWPmodelsareintegrated

with ananalysednon-interactive sea-surfacetemperature,which providestheright oceanicheattransport.In the

past,mostNWP(andclimate)modelswereusingcloudinessfixedand/orzonallyaveragedto gettheright answer

for thepole-equatorradiativegradient.Nowadays,NWPandclimateGCMshaveinteractivecloudinessandcloud-

radiationprocesses,asearlysatelliteobservationshaveshown thelongitudinalgradientsof radiativeheatingto be

of the same magnitude as the latitudinal gradients, due to the cloud distribution (Stephens and Webster, 1979).

Suchgradientsareillustratedin Figs.3 to 6 , which presentaveragedover thesummer1987(June–July–August)

thedifferentcomponentsof theradiationbudgetderivedfrom ERBEmeasurements.Fig. 3 displaysthetotal and

clear-sky absorbedSWradiationwhereasFig. 4 presentsthereflectedSWradiation.Theclear-sky fieldsobtained

by retainingthedataonly whencloudswerefound to beabsentfrom thefield of view of thesatellitearehighly

zonalover theoceans,with only thelandsurfacealbedodiffering with thevariouslandtypesintroducingmarked

departures.In comparison,the total fieldsclearlyshow the impactof thecloudinesswith higheralbedo(smaller

absorption)in theITCZ, theIndianmonsoonandover thestormtrack.Minimum albedo(maximumabsorption)is

foundover therelatively clear-sky subtropics,exceptover theextendedlow-level (stratocumulus)clouddeckson

the westernfacadesof thecontinents(California,Peru,Namibia).Fig. 5 presentsthetotalandclear-sky outgoing

longwave radiationfor thesamemonths.This field alsodisplaystheimpactof theclouds,this time mainly of the

high-level cloudinessusuallylinkedtoconvectionin theITCZ orovertheIndianmonsoonarea.Overtheequatorial

Pacific,theclear-sky OLR is lesszonalthanits SWcounterpartindicatingtheroleof moisteratmospheresoverthe

westpart thanover theeasternpartof thebasin.Fig. 6 presentstheshortwave andlongwave cloudforcing, i.e.,

thedifferencebetweenclear-sky andtotalfluxes.In theshortwave,cloudswith their reflectingeffectareresponsi-

ble for acoolingof theatmosphere-surfacesystem,throughadecreaseof theenergy availablefor heating.On the

opposite,in thelongwave,cloudscontributeto aheatingof theatmosphereby trappingtheradiationcomingfrom

thesurfaceandthe lower layersof theatmosphereandby emittingat thecold temperaturesrepresentative of the

higher clouds.

Cloud-radiationinteractionsare now thoughtto be of importancenot only for mesoscalephenomena(the sea

breezeis thebestbut not theonly exampleof intercationbetweenradiationandsurfacediscontinuitiesto create

local dynamicalcirculations),but also for synopticscalephenomenasuchas the onsetof the Indian monsoon

(Webster and Stephens, 1980).

Finally, somesystematicerrorsof aNWPmodelcanbetracedbackto adeficientradiationtransferparametrization

and at leastpartially correctedby an improved representationof the cloud-radiationinteractionsin the model

(Morcrette, 1990).

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ECMWF, 2002

Figure 1. The global energy balance for annual mean conditions. The top-of-the-atmosphere estimates of solar

insolation (343 W m-2), solar radiation reflected by the whole atmosphere-surface system (106 W m-2), and

outgoing longwave radiation (237 W m-2) are obtained from satellite data (Ramanathan, 1987). The other

quantites in the figure are obtained from various published mode and empirical estimates, and might still be

fraught with uncertainties. The quantities include: absorption of solar radiation at the surface (169 W m-2),

downwardlongwaveradiationat thesurfaceemittedby theatmosphere(327W m-2), upwardlongwaveradiation

emitted by the surface (390 W m-2), and the turbulent heat fluxes from the surface, the latent heat flux (90 W

m-2) and the sensible heat flux (16 W m-2).

��

Radiation Transfer


ECMWF, 2002 7

Figure 2. The daily variation of insolation at the top of the atmosphere as a function of latitude and day of the

year in units of cal cm-2 day-1 ( 1 cal cm-2 day-1 = 0.4844 W m-2) (afterPaltridge and Platt, 1976).

Radiation Transfer


ECMWF, 2002

Figure 3. Theabsorbedshortwave radiation(top)andtheclear-sky absorbedshortwave radiation(bottom)(W m-

2) derived from Earth Radiation Budget Experiment (ERBE) measurements for the summer 1987.

100� 100�150� 150�200� 200250 250�300�

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Radiation Transfer


ECMWF, 2002 9

Figure 4. As inFig. 3, but in terms of the reflected shortwave radiation at the top of the atmosphere (W m-2).

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Radiation Transfer


ECMWF, 2002

Figure 5. As inFig. 3, but for the outgoing longwave radiation at the top of the atmosphere (W m-2).

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Radiation Transfer


ECMWF, 2002 11

Figure 6. As inFig. 3, but for the shortwave (top) and longwave (bottom) radiative cloud “forcing” (W m-2).

3. THE THEORY OF RADIATION TRANSFER

3.1 Terminology

Generally, radiationis consideredto betheprocessof electromagneticwavespropagatingthroughamedium(nor-

mally a planetaryatmosphere).Someof theassociatedprocesseslike absorptionandthermalemissioncannotbe

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ECMWF, 2002

adequatelydescribedby classicaltheoryandrequiretheuseof quantummechanics,which treatsradiationasthe

propagationandinteractionof photons.Onecharacteristicpropertyof radiationis its wavelength,A classification

of radiation according to wavelength is presented inFig. 7 (from Liou, 1980).

Wavelength , frequency , andwavenumber arerelatedthroughtherelation , where is

thespeedof light ( m s-1 ). Thesensitivity of thehumaneyeis confinedto arathersmallinterval

from 0.4 to 0.7 (micrometres)known asthe visible region. However, mostof the atmosphericexchangeof

radiative energy occurs from the ultraviolet (from about 0.2 ) to the infrared (around 100 ).

In atmosphericstudies,two quantitiesaregenerallyused,theflux perunit areain W m-2 andthespecificintensity

or radiance, i.e., the radiative energy per time through unit area into a given solid angle, in W m-2 sr-1.

3.2 Derivation of the monochromatic radiative transfer equation (RTE)

Thefollowing discussionconcernsthetransferequationfor monochromaticradiationin its basicform for aplane-

parallel, horizontally homogeneous atmosphere (Chandrasekhar, 1960).

In GCM applications,theradiationfield is computedwithoutaccountingfor polarizationeffects,andassumingsta-

tionarity (noexplicit dependenceon time).Only thepointwhereradiationis considered,thedirectionof propaga-

tion, andthe frequency matterfor the problem.Fig. 8 presentsschematicallythe differentcontributionsto the

specificintensityatagivenpointPenclosedby aninfinitesimallysmallcyclindricalelementof length , of cross-

section , andof anorientationexpressedin termsof two angles,i.e., , with respectto the -axisand , the

anglebetweentheprojectionof thedirectionontothe planeandthe -axis itself. For a horizontallyhomo-

geneousatmosphericslice,thelocationof pointP is givenby its height aboveground(or any othersuitableco-

ordinate). Let us consider the various sources and sinks of radiative energy in this cyclindrical element:

3.2 (a) Extinction. Theradiance enteringthecylinder at oneendwill bepartially extinguished

within thevolume,proportionallyto theamountof matterencountered,thuscontributing a negative incrementof

radiative energy

where is the monochromaticextinction coefficient (units m-1) and is the solid angle differential

element.

3.2 (b) Scattering. Anothercontribution to a changein radiative energy in thevolumeis causedby thescat-

tering of radiation from any other direction into the direction of the considered beam, i.e.,

,

where is the monochromaticscatteringcoefficient, is the solid angledifferential elementfor the

originating beam, is the normalizedphasefunction, which representsthe probability for a

photon incoming from direction to be scatteredin direction . Since scatteredradiation may

originate from any direction, we have to integrate over all possible angle combinations of and .

.

λ ν ; λ < ν⁄ 1 ;⁄= = << 2.99792458=

µm

µm µm

d=d> θ ? φ@ –A @

?

Bν ? θ φ, ,( )

dC ν ext, ? θ φ, ,( ) βν ext,B

ν ? θ φ, ,( )d> d= dω–=

Dν ext, dω

dC ′ν scat, ? θ φ, ,( ) βν scat, E ν ? θ φ θ′ φ′,, , ,( ) 4π⁄{ }B

ν ? θ′ φ′, ,( )d> d= dω dω′=

βν scat, dω′

E ν ? θ φ θ′ φ′,, , ,( )θ′ φ′,( ) θ φ,( )

θ′ φ′

dC ν scat, ? θ φ, ,( ) βν scat, d> d= dωB

ν ? θ′ φ′, ,( ) E ν ? θ φ θ′ φ′,, , ,( ) 4π⁄{ }ω′∫ dω′=

Radiation Transfer


ECMWF, 2002 13

The direct (unscattered)solarbeamis generallyconsideredseparatelyandnot containedin the intensityof the

directionconsidered.Thusscatteringof thesolarbeaminto thedirectionof interestis aswell separatefrom the

scattering of diffuse radiation

,

where is thespecificintensityof incidentsolarradiation, and definethedirectionof theincidentsolar

radiationatToA, is thecosineof thesolarzenithangle,andif is theheightof thetopof theatmosphere,

is the optical thickness defined as

.

3.2 (c) Emission.Thelastcontributionto thechangein theradiativeenergy in thevolumeis thethermalemis-

sion:

,

where is the monochromatic absorption coefficient, and is the monochromatic Planck function.

3.2 (d) Total. Thetotalchangeof radiativeenergy in thecyclindercanbewrittenasthesumof theindividual

contributions

Replacingthe lengthdl of the cylindrical elementby the geometricalrelation where

, it follows from the expression of the optical thickness that

If wedefine astheratioof theabsorptionto theextinctioncoefficient andthesinglescat-

teringalbedo , wefinally obtainthemonochromaticRTE for aplaneparallel,hori-

zontally homogeneous atmosphere in the coordinate system given by ( as

In principle,theRTE derivedaboveallowsthecompleteproblemof theradiative transferin aNWPto besolvedif

thespecificintensities areknown for all modellayers,i.e., includingthesurface,all directionsandall wave-

numbersof thespectrum.But in thispresentform, theRTE is muchtoocomplicatedto beusedassuchin aNWP

or climateGCM. We will now considervarioussimplicationslinkedeitherto thebasiclaws of physicsor to the

very specific circumstances prevailing in the Earth’s atmosphere.

dC 0ν scat, ? θ φ, ,( ) βν scat, E ν ? θ φ θ0φ0, , ,( ) 4π⁄{ } F ν

0 δν

µ0-----–

d> d= dωexp=

F ν0 θ0 φ0

µ0 ? ToA

δν

δν βν ext, d?GG

ToA

∫=

dC ν emis, ? θ φ, ,( ) βν abs,D

ν �H?( )( ) d> d= dω=

βν abs,D

ν

dC ν ? θ φ, ,( ) dC ν ext, dC ν scat, dC ν scat,0 dC ν emis,+ + + d

Bν ? ω φ, ,( ) d> d= dω= =

d= d? θcos⁄ d? µ⁄= =

µ θcos= dδν βν ext, d?–=

κν κν βν abs, βν ext,⁄=

ω 1 κν– βν scat, βν ext,⁄= =

δν µ, θ φ,cos=( )

µdB

ν

dδν-----------

Bν δν µ φ, ,( ) 1 κν–( ) dφ′

Bν δν µ′ φ′,,( ) E ν δν µ φ µ′ φ′,, , ,( ) 4π⁄{ } dµ′

1–

+ 1

∫0

2π

∫–=

1 κν–( )F ν0 δν

µ0-----–

E ν δν θ φ θ0 φ0, , , ,( ) 4π⁄{ }exp–

κνD

ν δν( ).–

Bν

Radiation Transfer


ECMWF, 2002

3.3 Basic laws

3.3 (a) Planck’s law. Theenergy of anatomicoscillatoris quantized,asis any changeof energy state.The

changeof the energy stateby onequantumnumbercorrespondsto an amountof energy (eitherradiatedor ab-

sorbed)of where is thefrequency of theatomicoscillatorand J s

is Planck’sconstant.For a largesampleof atomicoscillatorswherethedistributionof energy levelsfollowsBoltz-

mannstatistics,onecanderivePlanck’s functionfor theemissionof radiantenergy by ablackbody(seedetailsin

Liou, 1980)

,

where is thewavelengthof emission, is Boltzmannconstant, J K-1, is thevelocity

of light in a vacuum, and is the absolute temperature of the black body (in K).

3.3 (b) Wien’s law. Thedependenceof on for varioustemperaturesis shown in Fig. 9 . Thedotted

line in thediagrammarksthewavelengthof themaximumintensity. It increaseswith thetemperatureof theblack

body and its variation follows Wien’s displacement law. Extreme values of the Planck function are defined by

.

With and , anddefining and , the previous condition

can be rewritten as

.

After somemanipulation(see,e.g.,Liou, 1980),we obtaina transcendentalequation ,which

can be shown to give provided that .

This translates into .

Wien’sdisplacementlaw considerablysimplifiestheradiationtransferproblemin theEarth’satmosphere.Atmos-

phericandsurfacetemperaturesaretypically in the200–300K rangeandthemaximumemissionoccursin the10

to 15 wavelengthrange.Ontheotherhand,theSun,theEarth’sexternalsourceof radiativeenergy emitsmost

of its energy atwavelengthsaroundabout0.5 correspondingto anequivalentblackbodyatabout6000K. At

themeanEarth–Sundistanceof km, themonochromaticradiantintensityreceivedfrom theSunover a

wide rangeof thespectrumis muchlessthantheemissionby theEarth’s atmospheresystemat equivalentwave-

lengths.Thecross-overpointisapproximatelyat3.5 . Forwavelengthslowerthan3.5 , theterrestrialemis-

sionis negligible andsois theenergy receivedfrom theSunat ToA for wavelengthslargerthan3.5 . For this

reason,whendesigningaradiationschemefor applicationto theEarth’satmosphere,onemayseparatethetransfer

problemin two parts,theshortwaveandlongwaveradiativetransfer, by dealingindependentlywith thesourcesand

sinks that are important for each part of the spectrum.

3.3 (c) Stefan-Boltzmann law. Thetotal radiantintensityof a blackbodyfollows from a spectralintegration

of the Planck function over all wavelengths, i.e.,

∆F I ν= ν < λ⁄= I 6.6260755 10 34–×=

Dλ �( ) 2IJ< 2

λ5------------

IJ<λ KL�-----------

exp 1–

W m 2– µm 1–=

λ K K 1.380658 10 23–×= <�

Dλ �( ) λ

dD

λmax

dλ---------------- 0=

< 1 2IJ< 2= < 2 IJ<MK⁄= @ < 2 λ �( )⁄= λ5 < 25 @ 5 � 5( )⁄=

dd@------

< 1 � 5@ 5

< 2@( ) 1–exp( )

------------------------------------ln

0=

5 5 @–( )exp– @=@ 4.9651= 5 @( ) 1<exp

λmax � max 2897 µm K=

µm

µm

1.5 108×

µm µm

µm

Radiation Transfer


ECMWF, 2002 15

.

Using the same substitutions as when deriving Wien’s displacement law, we get

.

The integral term has the value , so we finally obtain Stefan–Boltzmann law

,

where W m-2 K-4 is Stefan’s constant.The total emittedflux givenby this equationalready

impliesthattheangularintegrationover thetotal radiantintensityis performedassuminganisotropicemissionby

the black body (which is verified), leading to the factor .

3.3 (d) Kirchhoff ’s law. Theemissivity of amediumis theratioof theemittedenergy to thePlanckfunc-

tion at thetemperatureof themediumfor agivenwavelength . If theabsorptivity is definedastheamountof

energy absorbedat thatwavelengthdividedby theappropriatePlanckfunction,Kirchhoff ’s law statesthatthetwo

quantites are equal, i.e.,

.

A blackbody is definedby its ability to absorball the incomingradiationat a givenwavelengthandthereforeto

emit isotropicallyaccordingto Planckfunction; then for a black body . In contrast,grey bodies

absorb and emit only partially with values of and being less than unity.

TheEarth’satmosphereis farfrom beingamediumwith auniformtemperaturedistribution.However, theassump-

tion of athermodynamicequilibrium( ) is still valid in alocalsense,sinceat leastfor thelowerpartof the

atmosphere(approximatelybelow 40km), collisionbetweenmoleculesis aprocessefficientenoughto maintaina

local thermodynamic balance.

D �( )D

λ �( ) dλ0

∞

∫=

D �( ) 2K 4 � 4

I 3 < 2----------------

@ 3

@( ) 1–( )exp------------------------------- d@

0

∞

∫=

@ 4 15⁄

�πD �( ) σ � 4 W m 2–= =

σ 5.6697 10 8–×=

π

ελλ > λ

ελ > λ=

ελ > λ 1= =

ελ > λ

ελ > λ=

Radiation Transfer


ECMWF, 2002

Figure 7. The electromagnetic spectrum

Radiation Transfer


ECMWF, 2002 17

Figure 8.Processes contributing to the radiative transfer in a cylindrical volume

Figure 9. The dependence of the Planck function on wavelength and temperature. Wavelengths of maximum

emssion are connected by a dotted line

Radiation Transfer


ECMWF, 2002

In Subsection3.2, thederivationof theRTE andits formalsolutionwerecarriedoutmonochromatically. It is now

necessaryto considerthespectralvariationsof thevariousparameters(scatteringandextinctioncoefficients,single

scatteringalbedoandphasefunction).By contrastwith theemissionof a blackbodywhich displayssmoothvar-

iationswith temperatureandwavelength,theabsorption,emissionandscatteringby particlesandgasesgive rise

to highly variable spectra of parameters.

3.4 The spectral absorption by gases

Any moving particlehaskineticenergy asaresultof its motionin space.Thisenergy is equalto andis not

quantized.Howevermoleculeshaveotherradiativeenergy types,whichcanbedescribedfrom amechanicalmodel

of themolecules.Theabsorptionof radiativeenergy by gasesis aninteractionprocessbetweenmoleculesandpho-

tonsandthusobey quantummechanicslaws.Absorptionandemissiontake placewhentheatomsandmolecules

undergotransitionsfrom oneenergy stateto another. In contrastto “grey” absorberssuchassolidparticlesandliq-

uids,which absorbradiationfairly uniformly with respectto wavelengths(a consequenceof thedensepackingof

moleculesandatomsin these),theabsorptionandemissionby atmosphericgasesarehighly selectivewith regards

to wavelength, due to the selection rules that govern the transitions.

Thegaseousabsorptionspectrumcanbecategorizedin threepartsaccordingto thetotal (radiative) energy of the

molecules:

Themoleculemayrotateor revolveaboutanaxisthroughits centreof gravity. Therelatively low rotationalenergy

of themoleculesis associatedwith wavelengthsin thefarinfrared(i.e., mm).In thisspectralregionabsorp-

tion lines are well separated and are related to the transformation from one discrete rotational state into another.

Theatomsof themoleculeareboundedby certainforces,but theindividualatomscanvibrateabouttheirequilib-

rium positionrelative to eachother. A combinationof rotationalandvibrationalenergy transformationscausesab-

sorptionlines in the1 to 20 µm region. Thecoexistenceof both typesleadsto very complex structures(e.g.,the

thevibration-rotationbandof H2O around6.3 ) asanenormousnumberof linesareinvolved,whichpartially

overlap each other.

Towardsshorterwavelengths,thethird form of energy transformation,thechangeof theenergy level of electrons,

occursandthischangein theelectroniclevelsof energy increasesfurtherthecomplexity of thespectrum(e.g.,O2

bands in the ultraviolet).

3.4 (a) Line width. Eachspectralline correspondsexactly to oneform of energy transformationandshould

in principlebediscontinuousdueto thediscreteenergy levels.However, dueto Heisenberg’suncertaintyprinciple

(naturalbroadening),thecollision betweenmolecules(Lorentzor pressurebroadening),andDopplereffects(re-

sultingfrom thethermalvelocityof atomsandmolecules),absorbedandemittedemissionis notstrictly monochro-

matic,but is ratherassociatedwith spectrallinesof finite width. Whereasthenaturalbroadeningaffectsthe line

width only marginally, theothertwo have a markedimpacton theshapeof thespectrallinesandthereforeon the

absorption process itself.

TheDopplerbroadeningoccursdueto thethermalagitationwithin thegas.For amoleculeof mass radiatingat

frequency , with a velocity component in theline of sight following a Maxwell–Boltzmannprobabilitydis-

tribution

,

the absorption coefficient of such a Doppler broadened line is

KL� 2⁄

λ 20>

µm

Nν0 ;

O ;( ) d; N2π KP�---------------

0.5 N ; 2

2KP�-----------– exp= d;

Radiation Transfer


ECMWF, 2002 19

,

with .

Thepressurebroadeningis dueto collisionsbetweenthemolecules,whichmodify theirenergy levels.Theresult-

ing absorption coefficient is

,

with making it proportional to the frequency of collisions.

Fig. 10 compares typical Lorentz and Doppler line shapes.

3.4 (b) Line intensity. Theintensityof a line varieswith temperature,dueto thevariationwith of thesta-

tistical population of the energy levels of a molecule.

,

where is theenergy of thelower stateof thetransition, is anexponentthevalueof which dependson the

shapeof themolecule(1 for CO2, 3/2 for H2O, 5/2 for O3) and is thereferencetemperatureat which theline

intensities are known (and compiled, usually 296 K).

An exampleof thefinescalesinvolvedis givenin Fig. 11 takenfrom Fisher(1985).Thefigureshows the15

bandof CO2 atvaryingspectralresolution.Evenonawavenumberinterval assmallas0.01cm-1 (the15 band

coversroughly450to 800cm-1), theabsorptionvariesover thefull rangefrom total absorptionto total transmis-

sion.It is this extremespectraldependency of gaseousabsorptionthatmakesanexplicit integrationof themono-

chromaticRTE andsubsequentspectralintegrationof fluxesquitealengthy task,evenonthefastsuper-computers,

andrendersit impossiblefor operationalusein aNWPmodel.Verydetailedmodelsof theradiationtransferexist,

whichperformsthespectralintegrationoverwavenumberintervalsthewidth of whichis typically smallerthanthe

half-widthof anindividualline.Theseso-called“line-by-line” calculationscanserveasreferencefor moreapprox-

imatemethods.At presentthenumberof known spectrallinesfor gasesexisting in theEarth’satmosphereis of the

order of 300,000 (Rothman et al., 1998).

Figs.12 to 14 show thespectraldependency of thegaseousabsorptioncoefficient for thethreemainatmospheric

absorbersH2O, CO2 andO3 over thelongwavepartof thespectrum.Thevaluesarederivedfrom anevaluationof

spectroscopicdataover 5 cm-1 intervalsbetween0 and1110cm-1, 10 cm-1 between1110and2200cm-1, and20

cm-1 beyond2200cm-1, with thenarrow-bandmodelof MorcretteandFouquart(1985). Similarly, Figs.15 to 17

show the temperaturedependency of the absorption coefficients for the same absorbersin terms of

and , with correspondingto ,

the absorption coefficient at 250 K.

Fig. 18 shows thespectraldependency of thegaseousabsorptioncoefficient for thethreemainabsorbersover the

wholerangeof wavenumbersrelevantfor atmosphericenergy budget.To put theimportanceof of theabsorption

linesfor theatmospherictransferin perspective, thenormalizedPlanckfunctionfor 6000K (solaremission)and

255K (terrestrialemission)areplottedunderneath.It is obvious thatH2O is themostimportantabsorberin the

terrestrialspectrumleaving atransparentregiononly around10 . Thesocalled9.6 bandof ozonepartially

K D ν( )�

D

> D π( )0.5--------------------

ν ν0–( )> D

-------------------2

–

exp=

> D ν0 <⁄ 2KP� N⁄( )0.5=

K L ν( )�

αL

π ν ν0–( )2 αL2+[ ]

-------------------------------------------=

αL αL0

OQO0⁄ �SRT�⁄( )0.5=

�

� �0

� 0�------

em F K---- 1�---- 1� 0------–

–

exp=

F em� 0

µm

µm

log10 K 300 K( ) K 250 K( )⁄[ ] log10 K 200 K( ) K 250 K( )⁄[ ] 0 log10 1( )= K 250 K( )

µm µm

Radiation Transfer


ECMWF, 2002

fills this gap.However, astheamountof O3 in theatmosphereis not very largeandis concentratedin thestrato-

sphere,the atmosphereis fairly transparent(with respectto gaseousabsorption)in this region of the spectrum

calledtheatmosphericwindow. Absorptioneffect of tracegasessuchasmethane(CH4), nitrousoxide(N2O) and

chlorofluorocarbons(CFC-11andCFC-12)aremainly felt in thisspectralregiondueto thesmallbackgroundab-

sorption.

Towardslongerwavelengthsthis window is boundedby CO2 absorption(the15 band)andtheH2O rotation

band.Towardsshorterwavelengthsbut still in theterrestrialradiation,wefind thevibration-rotationof H2O around

6.3 . At very shortwavelengths(in theultraviolet region below 0.25 ), ozoneabsorbsalmostcompletely

thesolarradiationpenetratingtheatmosphere.However, in thesocalledvisible (0.4–0.7 ) andnear-infrared

(0.7–4 ) regionsof thesolarspectrum,solarradiationis weakenedonly by H2O andmoremodestlyby CO2

and other trace gases (NO2, O2, CH4, N2O, not shown in Fig. 18)

Figure 10. Transmission in the 15µm band of CO2 at various spectral resolutions (afterFisher, 1985)

µm

µm µm

µm

µm

Radiation Transfer


ECMWF, 2002 21

Figure 11. Lorentz and Doppler line shapes for similar intensity and line width (afterLiou, 1980)

Figure 12.Theabsorptioncoefficient for H2O at250K asa functionof thewavenumber. The

abscissa is given by the fraction of the black body function between 0 and (in cm-1) divided by the integral of

between 0 and 2500 cm-1.

K δ⁄�JU

∆ν⁄∑=

νDν 250 K( )

Radiation Transfer


ECMWF, 2002

Figure 13. As inFig. 11, but for CO2.

Figure 14. As inFig. 12, but for O3.

Radiation Transfer


ECMWF, 2002 23

Figure 15. Effect of the temperature of the absorption coefficient for H2O. The upper curve is for 300 K, the

lower one for 200 K.

Figure 16. As inFig. 14, but for CO2.

Radiation Transfer


ECMWF, 2002

Figure 17. As inFig. 14, but for O3.

Figure 18. Absorption coefficients of H2O, O3 and uniformly mixed gases (indexed as CO2) at different

wavelengths and the normalized Planck functions at 6000 K and 255 K (bottom panel).

Radiation Transfer


ECMWF, 2002 25

3.5 The spectral scattering by particles

A primaryelectromagneticwave encounteringa particlewill excite a secondarywave originatingat theparticle’s

location.Suchprocesscanoccurat all wavelengthscoveringthewholeelectromagneticspectrum.Thescattering

efficiency will dependonthesize,thegeometricalshapeandontherealpartof its complex refractiveindex, where-

astheabsorptionefficiency will essentiallydependson theimaginarypartof therefractive index. Particlesin the

atmospherearegenerallyat suchdistancesfrom eachotherthattheradiationoriginatingfrom individualparticles

is incoherent(i.e.,nointerferencebetwwenscatteredradiationfrom differentparticles).Thereforethefield of scat-

teredradiationdueto anensembleof particlesis thesumof thefieldsfrom theindividualscatteringprocesses.At-

mosphericconstituentscanbeseparatedinto differentbroadcategoriesaccordingto their size:Moleculeshave a

typical radiusof 10-4 µm, thesocalledaerosolsrangefrom 0.01to 10 , cloudparticlesfrom 5 to 200 , rain

drops and hail particles up to 10-2 m.

Therelativeintensityof thescatteringpatterndependsontheso-calledMie parameter where is the

radius of the particle (assumed spherical) andis the wavelength.

3.5 (a) Rayleigh scattering. Thesizeof air moleculesis smallcomparedto thewavelengthof radiationin the

Earth’satmosphere.Soin asimplemodelonecanassumethatmoleculesinteractingwith anelectromagneticwave

startto vibrateandtherebyradiateas linearoscillators.By consideringsphericalair moleculesstatisticallydistrib-

uted in a volume, Lord Rayleigh derived (1871) the phase function of such molecules

Rayleigh scattering being conservative ( ), the integral over the phase function is unity, i.e.,

.

It is completelysymmetric( ) (seeFig. 19 ), andthescatteringcoefficient, i.e., theprobabilityof a photon

beingscatteredin a volumeis for Rayleighscatteringproportionalto thedensityof air andinverselyproportional

to thefourth power of thewavelength.Thevisible consequenceof this spectraldependency is thebluenessof the

sky, causedby thepreferentialscatteringof shorterwavelengthsin a clearatmosphere.Therapiddecreaseof the

Rayleighscatteringcoefficient with wavelengthsimplifiesthetransferproblemfor theEarth’s atmosphere,since

Rayleigh scattering can be completely neglected for the terrestrial part of the spectrum.

3.5 (b) Mie scattering. Thesizeof aerosolsandor clouddropletsis comparableto thewavelengthsat which

theradiativeenergy is dominantin theEarth’satmosphere.Rayleighscatteringis notapplicableanymore.Photons

encounteringaparticleoff suchsizewill excitesecondarywavesin variouspartsof theparticles.Thesewavesare

coherentandthereforeinterferencecausespartialextinctionin somedirectionsandenhancementin others.For this

reason,scatteringby aerosolsandcloudparticlesis far from beingisotropic,with apronouncedpreferencefor the

forwarddirection.Undertheassumptionof sphericalparticleshape,Mie theorycanbeusedto derive thephase

function(seeSubsection3.2(b)) for a givensizedistribution of particles.Unfortunately, thereis no analyticalso-

lution andtheresultis obtainedin theform of aninfinite seriesof Besselfunctions.A normalizedphasefunction

for a standard aerosol model (afterQuenzel, 1985) is shown in Fig. 20.

For detailedcalculationsinvolving radiances(e.g.,in remotesensingapplications)the phasefunction is usually

developed into Legendre polynomials with up to several hundred terms to get the true angular representation

µm µm

α 2π V λ⁄= Vλ

OR

34--- 1 cos2θ+( )=

ω 1=

Oθ( ) dθ

4π∫ 1=

W 0=

Radiation Transfer


ECMWF, 2002

.

When dealing with the impact of scattererson fluxes, only the very few terms necessaryfor an adequate

descriptionof thehemisphericallyintegratedfluxesarerequiredandonecanusesomeanalyticformulasuchas

Henyey–Greenstein function

where is the asymmetry factor, the first moment of the expansion

.

When , all theenergy is backscattered, all energy appearsin the forwarddirection,and

correspondsto anequipartitionbetweentheforwardandbackwardspacesdefinedby aplaneperpendicularto the

incoming direction

In large-scaleatmosphericmodels,a scatteringmedium(cloudor aerosols)is usuallycharacterizedby its single

scatteringalbedo(seeSubsection3.2 (d)) andits phasefunction(seeSubsection3.2 (b)) andits opticalthickness

(seeSubsection3.2 (b)). For very largedropletsthe laws of geometricopticsapplymakingthecalculationsvery

simple. In theEarth’s atmospherea largevarietyof dropletsizescanbefounddependingvery muchon thetype

of cloudthey areembeddedin. Stephens(1979)quotesradii rangingfrom 2.25 for stratuscloudsto 7.5

for stratocumulus.Hanet al. (1994)derivedradii for watercloudsfrom satellitemeasurementsbetween6 and15

. Most radiativetransferschemesin usein GCMsreuiqreonly thevolumeextinctioncoefficient (relatedto the

optical thickness),thesinglescatteringalbedoandtheforwardscatteredfractionof radiation(or asymmetryfac-

tor). All these quantities as given byStephens (1979) are plotted inFigs. 21 to 24 for various cloud types.

Themostcomplex calculationof thephasefunctionis required for thetreatmentof ice crystals.Theapplication

of Mie theoryis notpossiblesincetheassumptionof asphericalshapeis certainlyinvalidatedin thiscase.Icecrys-

tals take all sortsof shapesrangingfrom thin needlesto complex combinationsof hollow prismsforming snow-

flakes.If aspecificshapeis assumedfor theicecrystals,geometricalopticsmaybeusedfor particlesthatarelarge

comparedto thewavelength(TakanoandLiou, 1989;EbertandCurry, 1992;Fu andLiou, 1993;Fu et al., 1999,

1988,1999).However, theresultsdependvery muchon theorientationof thecrystalsrelative to thedirectionof

propagationof theradiation.At present,it is almostimpossiblefor a GCM-typeradiationschemeto take into ac-

countsuchdetailedandcomplicateddependenciesof theopticalproperties,speciallyastheGCM is still far from

providing the reuiqred information necessary to initiate such calculations.

In thecruderform of parametrizationcurrentlyused,theicewaterloadingof thecloudsis prognosedby thecloud

schemeandthusprovidesan interactive optical thickness.The effective dimensionof the particlesis diagnosed

from temperatureand/orthetypeof processfrom which thecloudis originating(large-scalecondensationor con-

vection), but the single scattering albedo and asymmetry factor are specified.

E ν δ µ µ′, ,( ) ω0 2= 1+( )ψ X O X µ( )O X µ′( )∑=

E ν δ µ µ′, ,( ) 1 W 2–

1 W 2 2W µµ′+ +--------------------------------------=

W

W 12--- E δ µ µ′, ,( )µ dµ

1–

+ 1

∫=

W 1–= W 1= W 0=

µm µm

µm

Radiation Transfer


ECMWF, 2002 27

Figure 19. Rayleigh phase function for a molecule located at the origin of the polar coordinate system. The

probability of scattering in a certain direction is proportional to the length of the arrow for the outgoing beam.

Figure 20. Normalized phase function in polar coordinates with logarithmic axis for a standard aerosol model.

Note that the scattering is strongly dominated by the forward peak.

Radiation Transfer


ECMWF, 2002

Figure 21. The droplet distribution of eight cloud models (afterStephens, 1979) [top left]

Figure 22. Wavelength dependency of the total extinction coefficient [top right]

Figure 23. Wavelength dependency of the single scattering albedo [bottom left]

Figure 24. Wavelength dependency of the asymmetry factor [bottom right]

Radiation Transfer


ECMWF, 2002 29

4. RADIATION SCHEMES IN USE AT ECMWF

4.1 The operational longwave scheme (as of March 2000)

ThelongwaveradiationisanupgradefromtheoriginalschemedefinedbyMorcretteetal. (1986),Morcrette(1990,

1991, 1993).

Assuming a non-scattering atmosphere in local thermodynamic equilibrium, is given by

(1)

where is themonochromaticradianceat wavenumber at level , propagatingin a direction (the

anglethat this direction makes with the vertical), where and is the monochromatic

transmissionthrougha layerwhoselimits areat and seenunderthesameangle , with . The

subscript ‘surf’ refers to the earth’s surface.

After separatingtheupwardanddownwardcomponents(indicatedby superscripts+ and–, respectively), andin-

tegratingby parts,we obtaintheradiationtransferequationasit is actuallyestimatedin thelongwave partof the

radiation code

(2)

where,takingbenefitof theisotropicnatureof thelongwaveradiation,theradiance of (5.3)hasbeenreplaced

by thePlanckfunction in unitsof flux, (here,andelsewhere, is assumedto alwaysincludes

the factor). is thesurfacetemperature, thatof theair justabovethesurface, is thetemperature

at pressure-level , that at the top of the atmosphericmodel. The transmission is evaluatedas the

radiancetransmissionin adirection to theverticalsuchthat is thediffusivity factor(Elsasser, 1942).

Theintegralsin (2)areevaluatednumerically, afterdiscretizationovertheverticalgrid,consideringtheatmosphere

asapile of homogeneouslayers.As thecoolingrateis stronglydependenton local conditionsof temperatureand

pressure,andenergy is mainlyexchangedwith thelayersadjacentto thelevel wherefluxesarecalculated,thecon-

tribution of thedistantlayersis simply computedusinga trapezoidalrule integration,but thecontribution of the

adjacent layers is evaluated with a two-point Gaussian quadrature, thus at the level,

(3)

Y

Y µ µ ν Z ν E surf µ,( ) [ ν E surf E µ, ,( ) Z ν E ′ µ,( ) [ νd

E ′ E surf=

0

∫+

d

0

∞

∫d

1–

1

∫=

\ν ] µ,( ) ν E θ

µ θcos= ^ ν ]_] ′ ;,( )E E ′ θ ` θsec=

Yν+ E( )

Dν � surf( )

Dν � 0+( )–[ ] [ ν E surf E ; V,( )

Dν � E( )( ) [ ν EaE ′; V,( )

Dνdb ′ b surf=

b∫+ +=

Yν

– E( )D

ν � ∞( )D

ν � top( )–[ ] [ ν E 0; V,( )D

ν � E( )( ) [ ν E ′ E ; V,( )D

νdb ′ b=

0

∫+ +=

Z νDν �( ) W m 2– D

νπ � surf � 0+ � E( )

E � top [ νθ V θsec=

c-th

[ ν EaE ′; V,( )D

ν =db ′ b surf=

b U∫

dD

νX 1=

2

∑ =( ) deXf[ ν E U E X ; V,( ) 12---

Dν g( ) [ ν E U Eeh ; V,( ) [ ν E U Eeh 1– ; V,( )+[ ]dh 1=

U2–

∑+

Radiation Transfer


ECMWF, 2002

where is thepressurecorrespondingto theGaussianrootand is theGaussianweight. and

are the Planck function gradientscalculatedbetweentwo interfaces,and betweenmid-layer and interface,

respectively.

The longwave spectrum is divided into six spectral regions.

1) 0 – 350 & 1450 – 1880

2) 500 – 800

3) 800 – 970 & 1110 – 1250

4) 970 – 1110

5) 350 – 500

6) 1250 – 1450 & 1880 – 2820

correspondingto the centresof the rotationandvibration-rotationbandsof H2O, the 15 bandof CO2, the

atmosphericwindow, the 9.6 bandof O3, the 25 “window” region, and the wings of the vibration-

rotation band of H2O, respectively.

Integrationof (2) overwavenumber within the spectralregiongivestheupwardanddownwardfluxesas

(4)

(5)

The formulationaccountsfor the different temperaturedependenciesinvolved in atmosphericflux calculations,

namelythaton , thetemperatureat thelevel wherefluxesarecalculated,andthaton , thetemperaturethat

governs the transmissionthrough the temperaturedependenceof the intensity and half-widths of the lines

absorbing in the concerned spectral region.

Thebandtransmissivitiesarenon-isothermalaccountingfor thetemperaturedependencethatarisesfrom thewav-

enumberintegrationof the productof the monochromaticabsorptionandthe Planckfunction.Two normalized

bandtransmissivitiesareusedfor eachabsorberin agivenspectralregion:thefirstonefor calculatingthefirst right-

hand-side.termin (2), involving theboundaries;it correspondsto theweightedaverageof thetransmissionfunc-

tion by the Planck function

E X deX dD

ν g( ) dD

ν =( )

cm 1– cm 1–

cm 1–

cm 1– cm 1–

cm 1–

cm 1–

cm 1– cm 1–

µm

µm µm

ν K th–

YSi + E( )D i � surf( )

D i � 0+( )–{ } [kj i Vml E surf E,( ) �on E surf E,( ),{ }D i � b( )+=

+ [ dBi Vpl EqE ',( ) �on, EaE ',( ){ }

D idb ′ b surf=

b∫

YSi – E( )D i � 0( )

D i � ∞( )–{ } [rj i Vpl E 0,( ) �sn E 0,( ),{ }D i � b( )–=

[ dBi Vpl E ' E,( ) �on E ' E,( ),{ }

D idb ′ b=

0

∫–

� b �on

Radiation Transfer


ECMWF, 2002 31

(6)

thesecondonefor calculatingtheintegral termin (2) is theweightedaverageof thetransmissionfunctionby the

derivative of the Planck function

(7)

where is the pressure weighted amount of absorber.

In thescheme,theactualdependenceon is carriedout explicitly in thePlanckfunctionsintegratedover the

spectralregions.Althoughnormalizedrelativeto or , thetransmissivitiesstill dependon ,

boththroughWien’s displacementof themaximumof thePlanckfunctionwith temperatureandthroughthetem-

peraturedependenceof theabsorptioncoefficients.For computationalefficiency, thetransmissivitieshavebeende-

veloped into Pade approximants

(8)

where is aneffective amountof absorberwhich incorporatesthediffusivity factor ,

the weighting of the absorberamountby pressure , and the temperaturedependenceof the absorption

coefficients.

The function takes the form

(9)

The temperaturedependencedue to Wien’s law is incorporatedalthoughthere is no explicit variation of the

coefficients and with temperature.Thesecoefficientshave beencomputedfor temperaturesbetween187.5

and312.5K with a 12.5K step,andtransmissivities correspondingto thereferencetemperaturetheclosestto the

pressure weighted temperature are actually used in the scheme.

Theeffectonabsorptionof theDopplerbroadeningof thelines(importantonly for pressurelower than10hPa) is

includedsimplyusingthepressurecorrectionmethodproposedby Fels(1979)andincorporatedby Giorgettaand

Morcrette (1995).

Theincorporationof theeffectsof cloudson thelongwave fluxesfollows thetreatmentdiscussedby Washington

andWilliamson (1977).Whatever thestateof thecloudinessof theatmosphere,theschemestartsby calculating

[kjHl E � b �on, ,( )

Dν � b( ) [ ν l E �sn,( ) νd

ν1

ν2

∫

Dν � b( ) νd

ν1

ν2

∫

------------------------------------------------------------=

[ djHl E � b �on, ,( )

D � b( )d �d⁄{ } [ ν l E �on,( ) νdν1

ν2

∫

D � b( )d �d⁄{ } νdν1

ν2

∫

-------------------------------------------------------------------------------=

l E� b D � b( ) d

D � b( )/dT �on

[tl E �vu,( )< c l ef f

U/2U

0=

2

∑w h l ef f

h /2h 0=

2

∑--------------------------=

l ef f VTl E( )Ψ �onxl E,( )= Vl E

Ψ �on , l E( )

Ψ �snyl E,( ) >ql E( ) �sn 250–( ) zxl E( ) �on 250–( )2+[ ]exp=

< U w h�on

Radiation Transfer


ECMWF, 2002

thefluxescorrespondingto aclear-sky atmosphereandstoresthetermsof theenergy exchangebetweenthediffer-

entlevels(theintegralsin (5.4)Let and betheupwardanddownwardclear-sky fluxes.Forany cloud

layeractuallypresentin theatmosphere,theschemethenevaluatesthefluxesassumingauniqueovercastcloudof

emissivity unity. Let and theupwardanddownwardfluxeswhensucha cloudis presentin the

layerof theatmosphere.Downwardfluxesabove thecloud,andupwardfluxesbelow thecloud,areassumed

to be given by the clear-sky values

(10)

Upwardfluxesabove thecloud( for ) anddownwardfluxesbelow it ( for )

can be expressedwith expressionssimilar to (3) provided the boundaryterms are now replacedby terms

corresponding to possible temperature discontinuities between the cloud and the surrounding air

(11)

where is now thetotal Planckfunction(integratedover thewholelongwave spectrum)at level , and

and are the longwave fluxes at the upperand lower boundariesof the cloud. Termsunderthe integrals

correspondto exchangeof energy betweenlayersin clear-sky atmosphereandhave alreadybeencomputedin the

first stepof thecalculations.Thisstepis repeatedfor all cloudylayers.Thefluxesfor theactualatmosphere(with

semi-transparent,fractional and/ormulti-layeredclouds) are derived from a linear combinationof the fluxes

calculatedin previousstepswith somecloudoverlapassumptionin thecaseof cloudspresentin several layers.

Let betheindex of thelayercontainingthehighestcloud, thefractionalcloudcover in layer , with

for theupwardflux at thesurface,andwith and to have the right

boundary condition for downward fluxes above the highest cloud.

Themaximum-randomoverlapassumptionis operationallyusedin theECMWF model,andthecloudyupward

and downward fluxes are obtained as

(12)

In caseof semi-transparentclouds,the fractionalcloudinessenteringthecalculationsis aneffective cloudcover

equalto the productof the emissivity due to the condensedwaterand the gasesin the layer by the horizontal

coverage of the cloud layer, with the emissivity, , related to the condensed water amount by

Y0+ c( ) Y

0– c( )

YS{ + c( ) YS{ – c

( )| th

YS{ + c( ) Y

0+ c( )= for

c |≤YS{ – c

( ) Y0– c( )= for

c |>

YS{ + K( ) } ~ 1+≥ YS{ – K( ) K |>

YS{+ K( ) Ycld+ D | 1+( )–{ } [ E i E { 1+ ; V,( )

D K( ) [ E i E ′; V,( )D

db ′ b { 1–=

b i∫+ +=

YS{ – K( ) Ycld– D |( )–{ } [ E i E { ; V,( )

D K( ) [ E i E ′; V,( )D

db ′ b i=

b {∫+ +=

D c( )

c Ycld�Y

cld–

� �cldc)( )

c�

cld 0( ) 1=�

cld�

1+( ) 1=Y��

1+– Y

0–=

Y + Y –

Y + c( ) Y0+ c( )= for

c1=

Y – c( )�

cldc

1–( ) Y U 1–+ c

( )�

cld| 0=

c2–

∑ |( ) YS{+ c( ) 1�

cld =( )–{ }X {1+=

U1–

∏+= for 2c �

1+≤ ≤

Y + c( )�

cld

�( ) Ye�+ c

( )�

cld|( ) YS{+ c( ) 1

�cld =( )–{ }X {

1+=

�∏{

0=

�1–

∑+= forc �

2+≥

εcld

Radiation Transfer


ECMWF, 2002 33

(13)

where is thecondensedwatermassabsorptioncoefficient (in ) following SmithandShi (1992)for

water clouds andEbert and Curry (1992) for ice clouds.

4.2 The operational SW scheme

The rate of atmospheric heating by absorption and scattering of shortwave radiation is

(14)

where is the net total shortwave flux (the subscript SW will be omitted in the remainder of this section).

(15)

is thediffuseradianceat wavenumber , in a directiongivenby theazimuthangle, , andthezenithangle, ,

with . In (15), we assumea planeparallelatmosphere,andtheverticalcoordinateis theopticaldepth

, a convenient variable when the energy source is outside the medium

(16)

is theextinction coefficient, equalto thesumof thescatteringcoefficient of theaerosol(or cloud

particleabsorptioncoefficient ) andthepurelymolecularabsorptioncoefficient . Thediffuseradiance

is governed by the radiation transfer equation

(17)

is the incidentsolarirradiancein thedirection , is thesinglescatteringalbedo( )

and is the scatteringphasefunction which definesthe probability that radiationcomingfrom

direction ( ) is scatteredin direction ( ). The shortwave part of the scheme,originally developedby��r�and Bonnel (1980) solves the radiation transferequationand integratesthe fluxes over the whole

shortwave spectrumbetween0.2and4 . Upwardanddownwardfluxesareobtainedfrom thereflectanceand

transmittancesof thelayers,andthephoton-path-distributionmethodallowsto separatetheparametrizationof the

scattering processes from that of the molecular absorption.

Solarradiationis attenuatedby absorbinggases,mainlywatervapour, uniformly mixedgases(oxygen,carbondi-

oxide,methane,nitrousoxide) andozone,andscatteredby molecules(Rayleighscattering),aerosolsandcloud

particles.Sincescatteringandmolecularabsorptionoccursimultaneously, theexactamountof absorberalongthe

photonpathlengthis unknown, andbandmodelsof thetransmissionfunctioncannotbeuseddirectly asin long-

wave radiationtransfer(seeSubsection4.1). Theapproachof thephotonpathdistribution methodis to calculate

εcld 1 K abs l LWP–( )exp–=

K abs m2kg

1–

�∂ [∂-------W< b-----

YSW∂

E∂--------------=

YSW

Y δ( ) ν φ µ Z ν δ µ φ, ,( ) µd-1

+1

∫

d0

2π

∫d0

∞

∫=

ν φ θµ θcos=

δ

δ E( ) βextν E ′( ) E ′db

0

∫=

βextν E( ) βν

sca

βνabs K ν

Bν

µd Z ν δ µ φ, ,( )

dδ------------------------------ Z ν δ µ φ, ,( )

ϖν δ( )4

--------------O

ν δ µ φ µ0 φ0, , , ,( ) � ν0exp -δ µ �⁄( )–=

ϖν δ( )4

-------------- φ'd Φν δ µ φ µ' φ', , , ,( ) Z ν δ µ' φ', ,( ) µ'd-1

+1

∫

0

2π

∫–

� ν0 µ0 θ0cos= ϖν = βν

sca K ν⁄Φ δ µ φ µ' φ', , , ,( )

µ' φ', µ φ,

µm

Radiation Transfer


ECMWF, 2002

theprobability thataphotoncontributing to theflux in theconservativecase(i.e.,noabsorption,

, ) hasencounteredanabsorberamountbetween and .With thisdistribution, theradi-

ative flux at wavenumber is related to by

(18)

andtheflux averagedover thespectralinterval canthenbecalculatedwith thehelpof any bandmodelof the

transmission function

(19)

To find the distribution function , the scatteringproblem is solved first, by any method,for a set of

arbitrarily fixed absorptioncoefficients , thus giving a set of simulatedfluxes . An inverseLaplace

transformis thenperformedon (4.19) (��r�

, 1974).The main advantageof the methodis that the actual

distribution is smooth enoughthat (4.19) gives accurateresults even if itself is not known

accurately. In fact, need not be calculated explicitly as the spectrally integrated fluxes are

where and .

Theatmosphericabsorptionin thewatervapourbandsis generallystrong,andtheschemedeterminesaneffective

absorber amount between and derived from

(20)

where is an absorptioncoefficient chosento approximatethe spectrallyaveragedtransmissionof the clear

sky atmosphere

(21)

where is the total amountof absorberin a vertical columnand . Oncethe effective absorber

amountsof and uniformly mixed gasesare found, the transmissionfunctionsare computedusing Pade

approximants

(22)

Π l( )d l Ycons

ων 1= K ν 0= l l d l+

ν Ycons

Yν

Ycons Π l( ) K ν l–( )exp ld

0

∞

∫=

∆ν[ ∆ν

Y 1∆ν-------

Yν νd

∆ν∫

Ycons Π l( ) [ ∆ν l( ) νd

0

∞

∫= =

Π l( )K 1

YSi1

Π l( ) Π l( )Π l( )

Y Ycons[ ∆ν l⟨ ⟩( ) in the limiting case of weak absorption=

Y Ycons[ ∆ν l 1/2⟨ ⟩( ) in the limiting case of strong absorption=

l⟨ ⟩ Π l( ) l�ld0

∞∫= l 1/2⟨ ⟩ Π l( ) l 1/2 ld

0

∞∫=

l e l⟨ ⟩ l 1/2⟨ ⟩

l eY�i

e

� Ycons( )/ K eln=

K e

K e1l tot µ0⁄

------------------- [ ∆ν l tot/µ0( )( )ln=

l tot µ0 θ0cos=

H2O

[ ∆ν l( )> U l

U1–U

0=

�∑

z h l h 1–

h 0=

�∑-----------------------------=

Radiation Transfer


ECMWF, 2002 35

Absorptionby ozoneis also taken into account,but sinceozoneis locatedat low pressurelevels for which

molecularscatteringis smallandMie scatteringis negligible, interactionsbetweenscatteringprocessesandozone

absorption are neglected. Transmission through ozone is computed using(22) where the amount of ozone is

is thediffusivity factor(seeSubsection4.1), and is themagnificationfactor(� ��k�r�r� 1967)used

instead of to account for the sphericity of the atmosphere at very small solar elevations

(23)

To performthe spectralintegration, it is convenientto discretizethe solarspectralinterval into subintervals in

which the surface reflectancecan be consideredas constant.Since the main causeof the important spectral

variationof thesurfacealbedois thesharpincreasein the reflectivity of thevegetationin thenearinfrared,and

sincewatervapourdoesnot absorbbelow 0.68 , theshortwave schemeconsiderstwo spectralintervals,one

for thevisible (0.2–0.68 ), onefor thenearinfrared(0.68–4.0 ) partsof thesolarspectrum.Thiscut-off at

0.68 alsomakestheschememorecomputationallyefficient, in asmuchastheinteractionsbetweengaseous

absorption(by watervapouranduniformly mixed gases)andscatteringprocessesareaccountedfor only in the

near-infrared interval.

4.2 (a) Vertical integration. Consideringanatmospherewhereafraction (asseenfrom thesurfaceor

thetop of theatmosphere)is coveredby clouds(thefraction dependson which cloud-overlapassumption

is assumedfor thecalculations),thefinal fluxesaregivenasa weightedaverageof thefluxesin theclearsky and

in the cloudy fractions of the column

wherethe subscripts‘ ’ and ‘ ’ refer to the clear-sky and cloudy fractionsof the layer, respectively. In

contrastto theschemeof � �r� �r¡£¢ andHollingsworth(1979),thefluxesarenot obtainedthroughthesolutionof a

systemof linear equationsin a matrix form. Rather, assumingan atmospheredivided into homogeneouslayers,

the upward and downward fluxes at a given layer interface are given by

(24)

where and arethe reflectanceat the top andthe transmittanceat the bottomof the th layer.

Computationsof ’s start at the surfaceand work upward, whereasthoseof ’s start at the top of the

atmosphere and work downward. and account for the presence of cloud in the layer

(25)

l O3

l O3

d ¤ l O3db0∫= for the downward transmission of the direct solar beam

l O3

u V l O3l O3

d E surf( )+dbs

0

∫= for the upward transmission of the diffuse radiation

V 1.66= ¤V

¤ 35/ µ02 1+=

µm

µm µm

µm

� totcld� tot

cld

Y – g( )� tot

cldY

cld– g( ) 1

� totcld–( ) Y cl r

–+=

cl r cld

g

Y – g( ) Y0 ¥ bot K( )i h=

�∏=

Y + g( ) Y – g( ) ¦ top g 1–( )=

¦ top g( ) ¥ bot g( ) g¦ top ¥ bot¦ top ¥ bot

¦ top

�cld ¦ cld 1

�cld–( ) ¦ clr+=

¥ bot

�cld ¥ cld 1

�cld–( ) ¥ clr+=

Radiation Transfer


ECMWF, 2002

where is the cloud fractional coverage of the layer within the cloudy fraction of the column.

4.2 (b) Cloudy fraction of the layer. and arethereflectanceatthetopandtransmittanceatthebot-

tomof thecloudyfractionof thelayercalculatedwith theDelta-Eddingtonapproximation.Given , , and ,

the optical thicknessesfor the cloud, the aerosoland the molecularabsorptionof the gases,respectively, and

( ), and and thecloudandaerosolasymmetryfactors, and arecalculatedasfunctions

of the total optical thickness of the layer

(26)

of the total single scattering albedo

(27)

of the total asymmetry factor

(28)

of thereflectance of theunderlyingmedium(surfaceor layersbelow the th interface),andof thecosineof

an effective solar zenith angle which accountsfor the decreaseof the direct solar beam and the

corresponding increase of the diffuse part of the downward radiation by the upper scattering layers

(29)

with the effective total cloudiness over level

(30)

and

(31)

, and aretheoptical thickness,singlescatteringalbedoandasymmetryfactorof thecloud in

the th layer, and is thediffusivity factor. Theschemefollows theEddingtonapproximationfirst proposedby§�¨ �r�©�ª� �and Weinman(1970), then modified by « ��k�r¬ ¨ et al. (1976) to accountmore accuratelyfor the large

fraction of radiation directly transmittedin the forward scatteringpeak in caseof highly asymmetricphase

functions.Eddington’s approximationassumesthat, in a scatteringmediumof optical thickness , of single

scattering albedo , and of asymmetry factor , the radiance entering(15) can be written as

(32)

In thatcase,whenthephasefunctionis expandedasa seriesof associatedLegendrefunctions,all termsof order

greater than one vanish when(15) is integrated over and . The phase function is therefore given by

�cld

� totcld

¦ tcdy ¥ bcdy

δc δa δg

= K e ® c ® a ¦ tcdy ¥ bcdy

δ δc δa δg+ +=

ϖ* δc δa+

δc δa δg+ +----------------------------=

® * δc

δc δa+---------------- ® c

δa

δc δa+----------------ga+=

¦ _ gµeff g( )

µeff g( ) 1�

cldeff g( )–( )/µ V � cld

eff g( )+[ ]-1

=

�cldeff g( ) g

�cldeff g( ) 1 1

�cldc

( ) F c( )–( )U h 1+=

�∏–=

F c( ) 1

1 ϖcc

( ) ® cc

( )2–( )δcc

( )µ

-------------------------------------------------------–exp–=

δc

( ) ϖc

( ) ® ¯ c( )c V

δ*

ω ® ZZ δ µ,( ) Z 0 δ( ) µ Z 1 δ( )+=

µ φ

Radiation Transfer


ECMWF, 2002 37

where is the angle between incident and scattered radiances. The integral in(15) thus becomes

(33)

where

is the asymmetry factor.

Using(33) in (15) after integrating over and dividing by , we get

(34)

We obtain a pair of equations for and by integrating(34) over

(35)

For the cloudy layer assumed non-conservative ( ), the solutions to(4.35), for , are

(36)

where

Thetwo boundaryconditionsallow to solve thesystemfor and ; thedownwarddirecteddiffuseflux at the

top of the atmosphere is zero, i.e.,

OΘ( ) 1 β1 Θ( )µ+=

Θ

φ′O

µ φ µ' φ', , ,( ) Z µ' φ',( ) µ'd-1

+1

∫ w

0

2π

∫ 4π Z 0 π Z 1+( )=

® β1

3-----

12---O

Θ( )µ µd-1

+1

∫= =

µ 2π

µ ddδ------ Z 0 µ Z 1+( ) Z 0 µ Z 1+( )– ϖ Z 0 ® µ Z 1+( )+

+1/4ϖ Y 0 δ/µ0–( ) 1 3 ® µ0µ+( )exp

=

Z 0 Z 1 µ

d Z 0

dδ--------- 3 1 ϖ–( ) Z 0–

34---ϖ Y 0 δ/µ0–( )exp+=

d Z 1

dδ--------- = – (1 –ϖ ® ) Z 1

34---ϖ® µ0

Y0 δ/µ0–( )exp+

ϖ 1< 0 δ δ*≤ ≤

Z 0 δ( ) ° 1 ± δ–( ) ° 2+ +± δ( ) α δ/µ0–( )exp–expexp=

Z 1 δ( )O ° 1 ± δ–( ) ° 2 +± δ( ) β δ/µ0–( )exp–exp–exp{ }=

± = 3 1 ϖ–( ) 1 ϖ®–( ){ }1 2⁄

O= 3 1 ϖ–( ) 1 ϖ®–( )⁄{ }1 2⁄

α = 3ϖ Y 0µ0 1 3® 1 ϖ–( )+{ } 4 1 ± 2µ20–( ){ }⁄

β = 3ϖ Y 0µ0 1 3® 1 ϖ–( )µ20+{ } 4 1 ± 2µ2

0–( )( )⁄

° 1 ° 2

Y – 0( ) Z 0 0( ) 23--- Z 1 0( )+ 0= =

Radiation Transfer


ECMWF, 2002

which translates into

(37)

Theupwarddirectedflux at thebottomof the layer is equalto theproductof thedownwarddirecteddiffuseand

directfluxesandthecorrespondingdiffuseanddirectreflectance( and , respectively) of theunderlyingme-

dium

which translates into

(38)

In theDelta-Eddingtonapproximation,thephasefunctionis approximatedby aDiracdeltafunctionforward-scat-

ter peak and a two-term expansion of the phase function

where is the fractionalscatteringinto the forward peakand the asymmetryfactorof the truncatedphase

function. As shown byJoseph et al. (1976), these parameters are

(39)

The solution of the Eddington’s equationsremainsthe sameprovided that the total optical thickness,single

scattering albedo and asymmetry factor entering(36) to (38) take their transformed values

(40)

Practically, theopticalthickness,singlescatteringalbedo,asymmetryfactorandsolarzenithangleentering(4.36)

to (4.38) are , , and defined in(39) and(40).

4.2 (c) Clear-sky fraction of the layers. In the clear-sky part of the atmosphere,the shortwave scheme

accountsfor scatteringandabsorptionby moleculesandaerosols.Thefollowing calculationsarepracticallydone

1 2O

/3+( ) ° 1 1 2O

/3–( ) ° 2+ α 2β/3+=

¦ d ¦ _

Y + δ*( )= Z 0 δ*( ) 23--- Z 1 δ*( )–

¦ _ Z 0 δ*( ) 23--- Z 1 δ*( )+

¦ dµ0

Y0 δ* /µ0–( )exp+=

1 ¦ _– 2– 1 ¦ _+( )O

3⁄{ } ° 1 ±– δ*( )exp

+ 1 ¦ _ +– 2 1 ¦ _+( )O

3⁄{ } ° 2 ±+ δ*( )

= 1 ¦ _–( )α 2 1 ¦ _+( )β 3⁄– ¦ dµ0Y

0+{ } δ* /µ0–( )exp

P θ( ) 2² 1 µ–( ) 1 ²–( ) 1 3 ® ′µ+( )+=

² ® ′

² ® 2=

® ′ ® ® 1+( )⁄=

δ′* 1 ϖ²+( )δ*=

ω′ 1 ²–( )ϖ1 ϖ²–

----------------------=

δ* ϖ* ® * µeff

Radiation Transfer


ECMWF, 2002 39

twice, once for the clear-sky fraction ( ) of the atmosphericcolumn with equal to , simply

modified for the effect of Rayleighand aerosolscattering,the secondtime for the clear-sky fraction of each

individual layer within the fraction of the atmospheric column containing clouds, with equal to .

As theopticalthicknessfor bothRayleighandaerosolscatteringis small, and , thereflectance

at thetopandtransmittanceat thebottomof the th layercanbecalculatedusingrespectively afirst andasecond-

order expansionof the analyticalsolutionsof the two-streamequationssimilar to that of ³ ��´�� r¡ and Chylek

(1975).For Rayleighscattering,theoptical thickness,singlescatteringalbedoandasymmetryfactorarerespec-

tively , , and , so that

(41)

The optical thickness of an atmospheric layer is simply

(42)

where is the Rayleighoptical thicknessof the whole atmosphereparametrizedas a function of the solar

zenith angle (µ�¶r·r¸º¹£»�¼e½£· et al., 1983)

For aerosolscatteringandabsorption,theopticalthickness,singlescatteringalbedoandasymmetryfactorarere-

spectively , , with and , so that

(43)

(44)

where is the backscattering factor.

Practically, and arecomputedusing(41) andthe combinedeffect of aerosolandRayleighscattering

comesfrom usingmodifiedparameterscorrespondingto theadditionof thetwo scattererswith provision for the

highly asymmetricaerosolphasefunctionthroughDelta-approximationof theforwardscatteringpeak(asin (39)

and(40))

(45)

1 ¾ totcld– µ µ0

¾ totcld µ µe¿

cl r À 1–( ) Á cl r À( )

À

δR ϖR 1= Â R 0=

¿R =

δR

2µ δR+-------------------

Á R =2µ

2µ δR+( )------------------------

δR

δR δR* Ã À( ) Ã À 1–( )–{ } Ã

surf⁄=

δ*R

δa ϖa 1 ϖa 1«– Â a

den 1 1 ϖa– back µe( )ϖa+{ } δa µe⁄( )+=

1 ϖa–( ) 1 ϖa– 2back µe( )ϖa+{ } δ2a µ2

a⁄( )+

¿ µe( )back µe( )ϖaδa( ) µa⁄

den-------------------------------------------------=

Á µ Ä( ) 1 den⁄=

back µe( ) 2 3µeÂ a–( ) 4⁄=¿

cl r Á cl r

δ+ δR δa 1 ϖaÂ a2–( )+=

Â + Â a

1 Â a+---------------

δa

δR δa+( )----------------------=

ϖ+ δR

δR δa+-----------------ϖR

δa

δR δa+-----------------

ϖa 1 Â a2–( )

1 ϖaÂ a2–

--------------------------+=

Radiation Transfer


ECMWF, 2002

As for their cloudy counterparts, and must accountfor the multiple reflectionsdue to the layers

underneath

(46)

and is the reflectance of the underlying medium and is the diffusivity factor.

SinceinteractionsbetweenmolecularabsorptionandRayleighandaerosolscatteringarenegligible, theradiative

fluxesin a clear-sky atmospherearesimply thosecalculatedfrom (24) and(41) attenuatedby thegaseoustrans-

missions(22).

4.2 (d) Multiple reflections between layers. To dealproperlywith themultiplereflectionsbetweenthesurface

andthecloudlayers,it shouldbenecessaryto separatethecontributionof eachindividual reflectingsurfaceto the

layer reflectanceandtransmittancesin asmuchaseachsuchsurfacegivesrise to a particulardistribution of ab-

sorberamount.In caseof anatmosphereincluding cloudlayers,thereflectedlight abovethehighestcloudcon-

sistsof photonsdirectly reflectedby thehighestcloudwithout interactionwith theunderlyingatmosphere,andof

photonsthathave passedthroughthis cloudlayerandundergoneat leastonereflectionon theunderlyingatmos-

phere. In fact,(17) should be written

(47)

where and aretheconservative fluxesandthedistributionsof absorberamountcorrespondingto the

different reflecting surfaces.Å�Æ�Ç�È�Ç »�ÉrÊ andBonnel(1980)haveshown thataverygoodapproximationto thisproblemis obtainedby evaluating

thereflectanceandtransmittanceof eachlayer(using(34)and(40)) assumingsuccessively anon-reflectingunder-

lying medium( ), thena reflectingunderlyingmedium( ). First calculationsprovide thecontribu-

tion to reflectanceandtransmittanceof thosephotonsinteractingonly with the layer into consideration,whereas

the second ones give the contribution of the photons with interactions also outside the layer itself.

Fromthosetwo setsof layerreflectanceandtransmittances( ) and( ) respectively, effectiveab-

sorberamountsto beappliedto computingthe transmissionfunctionsfor upwardanddownwardfluxesarethen

derived using(18) and starting from the surface and working the formulas upward

(48)

where and are the layer reflectanceand transmittancecorrespondingto a conservative scattering

medium.

Finally the upward and downward fluxes are obtained as

¿cl r Á cl r

¿cl r

¿ µe( ) ¿– Á µÄ( ) 1

¿ * ¿––( )⁄+=

¿–

¿–

¿t À 1–( )= Ë

Ì

Í Ícl Î l Ï( ) Ð ∆ν Ï( ) νd

0

∞

∫Ñ0=

Ò∑=

Ícl Î l Ï( )

¿– 0=

¿– 0≠

Á t0 Á b0, ¿t ≠ Á b ≠

Óe0– ln Á b0 Á bc⁄( ) Ô e⁄=

Ó –e ≠ ln Á b ≠ Á bc⁄( ) Ô e⁄=

Ó +e0 ln

¿t0¿

tc⁄( ) Ô e⁄=Ó +

e ≠ ln¿

t ≠¿

tc⁄( ) Ô e⁄=

¿tc Á bc

Radiation Transfer


ECMWF, 2002 41

4.2 (e) Cloud shortwave optical properties. As seenin (36), the cloud radiative propertiesdependon three

different parameters: the optical thickness, the asymmetry factor , and the single scattering albedo.

PresentlythecloudopticalpropertiesarederivedfromÅ�Æ�Ç�È�Ç »�ÉrÊ (1987)for thewaterclouds,andÕLÖ�¶rÉrÊ andCurry

(1992) for the ice clouds

is related to the cloud liquid water amount by

where is the meaneffective radius of the size distribution of the cloud water droplets.Presently is

parametrizedasalinearfunctionof heightfrom 10 m at thesurfaceto 45 m at thetopof theatmosphere,in an

empirical attemptat dealing with the variation of water cloud type with height. Smaller water dropletsare

observed in low-level stratiform clouds whereas larger droplets are found in mid-level cumuliform water clouds.

In thetwo spectralintervalsof theshortwave radiationscheme, is fixedto 0.865and0.910,respectively, and

is given as a function of followingÅ�Æ�Ç�È�Ç »�ÉrÊ (1987)

(49)

Thesecloud shortwave radiative parametershave beenfitted to in situ measurementsof stratocumulusclouds

(Bonnel et al., 1983).

For the optical properties of ice clouds, we have

(50)

wherethecoefficientshavebeenderivedfrom ÕPÖ�¶rÉrÊ andCurry (1992)for thetwo intervalsof theshortwave radi-

ation scheme, and is fixed at 40 m.

5. COMPARISONS WITH OBSERVATIONS

As for therepresentationof clouds,thereis animperativeneedto performat leastsomeof thevalidationonshort-

time andlimited spacescales,beforethedrift in theGCM climatemakescomparisonwith observationsvery dif-

ficult to interpret.Relative to otherphysicalprocesses,theexistenceof referencemodels(LbLs) thathavebeenor

canbevalidatedagainsthighly spectrallydetailedmeasurements(suchasthoseof spectrometerandinterferometer,

presentonARM sites,or embarkedonNimbus-3)offersaveryusefultool for validation.However, althoughcur-

rentlydonefor LW, thisapproachis notreallyusedfor SW, becauseof therelativelackof highly spectrallydetailed

measurementsin theSW comparedto LW, andthereforeof carefully validatedLbL modelsin theSW (Ramas-

Í + À( ) Í0¿

t0 Ð ∆ν Ï +e0( ) ¿

t ≠¿

t0–( ) Ð ∆ν Ï +e ≠( )+{ }=

Í – À( ) Í0 Á b0 Ð ∆ν Ï +

e0( ) Á b ≠ Á b0–( ) Ð ∆ν Ï –e ≠( )+{ }=

δc Â c ϖc

δc Ï LWP

δc

3 Ï LWP

2Ë e-----------------=

Ë e Ë e

µ µ

Â c

ωc δc

ϖc1 0.9999 5 10 4–× 0.5δc–( )exp–=

ϖc2 0.9988 2.5– 10 3–× exp 0.05δc–( )=

δci IWP × i Ø i Ë e⁄+( )=

ϖi Ù i Ú i Ë e–=

Â i Ù i Û i Ë e+=

Ë e µ

Radiation Transfer


ECMWF, 2002

wamy and Freidenreich, 1992).

Theapproachdiscussedabovecouldalsobeused,albeitwith moredifficulties,for cloudyatmospheres.However,

for years,mostof thevalidationeffort for radiationtransferin cloudyatmosphereshasdealtwith comparisonof

radiationfluxesat TOA with satellite-derivedfluxes(Nimbus-7EarthRadiationBudget,EarthRadiationBudget

Experiment).More often thannot, this wasoften donewithout consideringat the sametime a validationof the

cloudcharacteristicssothattweakingthecloudproperties(particularly, therelationshipsgiving thecloudfraction

andcloudwaterloadingin diagnosticcloudschemes)allowedaneasyagreementwith TOA fluxes,speciallyover

monthlymeantime-scales.With theavailability of carefullycalibratedmeasurementsof theradiationfluxesat the

surface(BaselineStationRadiationNetwork (Ohmuraetal., 1998),GlobalEnergy BalanceArchive,NOAA-ARL

SURFaceRADiationnetwork), or asis thecasewith theARM program,of radiationfluxesandrelatively detailed

informationon thestateof theoverlyingatmosphereandsomeinformationon thecloudstructure,systematicver-

ificationof theradiationfieldsproducedby theECMWFmodelatvariousstagesin theforecastscannow becarried

out.

In thefollowing, we presentoneexampleof suchvalidationfor a sitewheregoodquality surfaceradiationmeas-

urementsareavailablewith a high frequency (suchsitesareavailableaspartof theBSRN,SURFRADor ARM

projects).For a weatherforecastsystem,it is oftenpreferableto usemeasurementsnot too remotein time,asthe

analysisandforecastsystemundergoesregularimprovementovertheyears.In thisrespect,theNOAA-ARL SUR-

FaceRADiation network offers real-timeavailability of surfaceradiationmeasurementsover 6 sitesin the U.S.

Figure5.1 comparesthedownwardandupwardLW radiationover GoodwinCreek,Mississippi,over theperiod

17Novemberto 15December1997,whenanew physicalparametrisationpackagewasbeingtested.Fig.26 shows

thecorrespondingdownwardSWradiation.As canbeseenfrom theLW comparisons,themodelis rathersuccess-

ful at producingtheobservedvariability in temperatureat thesurface(asseenfrom theupwardLW radiationin

Fig. 25 , bottompanel)andin temperature,humidity, andcloudfractionin thelowestlayersof theatmosphere(as

seenfrom thedownwardLW radiationin Fig. 25 , toppanel).On theotherhand,it appearslesssuccessfulathan-

dling theamountof condensedwaterin clouds,with toosmallopticalthicknesstranslatingin too largedownward

SW radiationat thesurface(Fig. 26 ). Next stagein this typeof comparisonsis to make surethat thefluxesare

computedfor theproperverticalprofilesof radiatively importantquantities.Albeit obvious,thisstatementis very

difficult to beput in practice,asacomprehensiveknowledgeof all theatmosphericprofilesis rarelyattained(and

attainable).TheARM programis thebesttry in thisdirection,with, over theSouthGreatPlainssitein Oklahoma,

longtime-seriesof surfaceradiationflux measurements,radiosoundings,synopticobservations,verticallyintegrat-

edwatervapourandcloudwatermeasurementsfrom amicrowaveradiometer, andverticalprofilesof temperature,

humidity, andcloudwaterfrom theinversionof interferometerandcloudradarmeasurements.Figure5.3presents

thetotalcolumnwatervapourandcloudliquid waterin December1997asobservedby theMicrowaveRadiometer

andproducedby theECMWFmodelin aseriesof 31operationalforecastsall startingat12UTC. Betweenthe13

and17December, thesky is essentiallyclear. Fig.28 comparesthesurfacedownwardLW radiationobserveddur-

ing these5 daysandcomputationsby differentradiationschemesfrom theforecastfieldsof temperatureandhu-

midity. A differenceof 5 to 8 W m-2 is systematicallyfound betweenthe pre-December’97ECMWF radiation

scheme(old_EC)andtheRRTM scheme.However, eventhisstate-of-the-artradiationschemeunderestimatesthe

observedflux by 10–15W m-2, showing that,eventhis thought-to-be-simplecomparisonexerciseis not straight-

forward.It is very likely thatthemodellow-level temperatureandhumidity profilesareat theorigin of thesedis-

crepancies.

Anothervalidationexerciseinvolving themodelradiationschemeis theverificationof thecloudsignatureonTOA

radiances.Fig.29 illustratessuchcomparisonfor theERICAstorm:thelongwavewindow channelbrightnesstem-

perature/radianceis simulatedfrom themodelcloud, temperatureandhumidity fieldsusingthemethodologyof

Morcrette (1991a).

Radiation Transfer


ECMWF, 2002 43

Figure 25.Downward(top)andupwardLW radiationfrom theECMWF6-hourfirst-guessforecastscomparedto

the SURFRAD measurements in Goodwin Creek, Mississippi over the period 971117 00 UTC to 971215 00

UTC.

0Ü

120Ý 240Þ 360ß 480à 600Ü

720ÝTime since 971117 00GMT

150

200

250

300

350

400

Rad

iatio

n W

/m2

á

Goodwin Creek, MississippiâSurface Down LW: SURFRAD, ECMWF First Guess

Obs.ãOldãNewä

0å

120æ 240ç 360è 480é 600å

720æTime since 971117 00GMT

250

300

350

400

450

Rad

iatio

n W

/m2

Goodwin Creek, MississippiêSurface up LW: SURFRAD, ECMWF First Guessë

Obs.ìOldìNewí

Radiation Transfer


ECMWF, 2002

Figure 26. As inFig. 25, but for the downward SW radiation at the surface averaged over 24-hour periods.

0î 120ï 240ð 360ñ 480ò 600î 720ïTime since 971117 00GMTó

0

20

40

60

80

100

120

140

160

180

200R

adia

tion

W/m

2

ô

Goodwin Creek, MississippiõSurface Down SW: SURFRAD, ECMWF First Guessö

Obs.OldNew

Radiation Transfer


ECMWF, 2002 45

Figure 27. The total column water vapour (top) and total column cloud liquid water over the ARM-SGP site in

December 1997 from a series of 31 forecasts all starting at 12 UTC. Points represent hourly averaged values of

TCWV and TCCLW derived from Microwave Radiometer measurements.

0÷ 120ø 240ù 360ú 480û 600÷ 720øHours since 97120100ü

0

10

20

30

TC

WV

(kg

/m2)

ý

SGP − Total column water vapourECMWF vs ARM−MWR − December 1997

ObsMod 1x2

0÷ 120ø 240ù 360ú 480û 600÷ 720øHours since 97120100ü

−50

0

50

100

150

200

250

300

350

400

450

500

550

600

650

700

LWP

(g/

m2)

ý

SGP − Liquid water pathþECMWF vs ARM−MWR − December 1997

ObsMod 1x2

Radiation Transfer


ECMWF, 2002

Figure 28.ThesurfacedownwardLW radiationduring5 clear-sky daysover theARM-SGPsite.Theuppercurve

is the SIRS observation, the lower curves are computed by different radiation schemes from and fields

operationally produced from forecasts between 12 and 35 hours, all starting at 12 UTC.

288ò 312ÿ 336� 360ñ 384ò 408îTime since 971201 00 GMT (hours)ÿ

190

200

210

220

230

240

250

260

270

280

290

Dow

nwar

d LW

Rad

iatio

n (

W/m

2)

�

ARM−SGP December 1997õICRCCM−type comparison of Surface Downward LW Flux

old_ECnew_ECRRTM_1aRRTM_3aEC_ZHobs_sirs

� �

Radiation Transfer


ECMWF, 2002 47

Figure 29.TheGOESlongwavewindow channelbrightnesstemperatureover theERICA storm.Toppanelis the

ISCCP-DXimagefor 18UTC 4 January1989,bottomis theradiancesimulatedfrom theECMWFTL639model

after a 18 hour forecast starting at 00 UTC 4 January 1989.

Radiation Transfer


ECMWF, 2002

6. CONCLUSIONS AND PERSPECTIVES

Recentstudieshavefocussedat thedirectand/orindirectradiative impactof aerosolsin climatesimulations(Jous-

saume, 1990;Genthon, 1992;BoucherandLohmann, 1995;Chyleket al., 1995;Mitchell et al., 1995;Lohmann

andFeichter, 1997;Tegenetal., 1997;Timmrecketal., 1997;Cusacketal., 1998;LeTreutetal., 1998;Schulzet

al., 1998).Sincethepioneeringwork of Tanreetal. (1984),theECMWFmodelhasincludedanannualmean,but

geographicallydistributedclimatologyof aerosols.Thegeographicaldistributionof themaritime,continental,ur-

bananddesertaerosolsaregivenin Figs.30 and31, whereastheverticaldistribution is illustratedin Fig. 32. As

discussedin Cusacketal. (1998),thepresenceof theseaerosolsin theECMWFmodelis certainlyoneof therea-

sonswhy thesurfacedownwardSW radiationis lessoverestimatedin theECMWF modelthanin otherleading

climateGCMs(Garratt, 1994;Garratt etal., 1998).However, amonthlyspecificationof theaerosoldistributions

or better a prognosed aerosol loading is a possible future development for the ECMWF model.

Recentstudiesby WangandRossow(1995,1998)andStubenrauchet al. (1997)have focussedat thepotentially

largeimpactontheatmosphericcirculationlinkedto uncertaintiesin theverticaldistributionof cloudsandrelated

radiativeheating/coolingrates.Similar resultswerealsoobtainedwith theECMWFmodel(MorcretteandJakob,

2000).Fig. 33 presentsthecloudoverlapsthattheoperationalradiationschemecanhandle.One-dimensionalcal-

culationsareperformedfor differentcloudconfigurationsoftenencounteredin three-dimensionalsimulations.The

first onecorrespondsto a typical convective systemandincludesa convective tower of fractionalcover 0.15from

820to 290hPa toppedby stratiformanvil of cover0.4and70hPa thick. Thesecondcaseincludesthreelayersof

stratiformcloudof cover 0.3between890and820hPa (low-level), 640and545hPa (middle-level), and290and

220hPa(high-level), respectively. Theprofilesof thecloudlongwave,shortwaveandnetradiativeforcingfor these

twocasesarepresentedin Fig.34, left andrightpanelsrespectively. Thecloudforcingtermissimplythedifference

betweentheheatingobtainedfor thecloudyatmosphereminustheclear-sky heating.ThedifferentCOAs aretreat-

edexactly in theLW partandonly approximatelyin theSWpartof theradiationscheme.Therefore,theLW MRN

andMAX resultsareessentiallythesamefor case1,andthesameholdsfor theLW MRN andRAN resultsfor case

2. In theSW, theagreementontheSWforcingprofilesis alsoverygood.In thefirst case(seeFig. 34, left panels),

theanvil producesastrongLW coolingwhereasarelativeheatingis foundin therestof theconvectivetowerbelow.

Comparedto MRN/MAX, theRAN LW shows a smallercoolingpeakin theanvil, smallerheatingbelow within

thetower, andlargerwarmingin theclearPBL.All thesefeaturesareconsistentwith thelargereffectivecloudiness

thattheRAN assumptionproducesbetweenany two cloudylayers.This largereffectivecloudiness(i) preventsthe

radiationfrom theloweratmosphericlayersfrom escapingto space,thustheheatingin thePBL, (ii) decreasesthe

possibleradiativeexchangeswithin thetower, thusthesmallerradiativeheating,and(iii) decreasestheupwardLW

flux atthebaseof theanvil, hencethesmallerLW divergencein theanvil. Similarly, astrongerSWheatingis linked

in RAN to thelargercloudfractionavailableto screenthedownwardradiationandto themorediffusecharacterof

this radiationin thecloudy fractionof thecolumn.Consequentlya smalleramountof SW radiationis available

below thecloudbaseandthis translatesinto a relative cooling.Thesmallerheatingof theanvil in RAN is linked

to thesmallerSW flux divergencethrougha largerupwardradiationdueto the increasedreflectionon the larger

effective amount of lower level clouds.

In thesecondcase(seeFig.34, rightpanels),eachof thethreestratiformcloudlayersproducesaLW coolinginside

andaLW heatingimmediatelyunderneath.DifferencesbetweenMAX andMRN/RAN resultsareconsistentwith

thesmallertotaleffectivecloudinessgivenbyMAX. Thecoolingin thelow-level cloudis reducedin MAX because

themiddle-andhigh-level cloudsprevent the radiationat the top of the low-level cloud from escapingto space.

Similarly, thecoolingin thehigh-level cloudis increasedin MRN/RAN becausethelow- andmiddle-level clouds

prevent the warmerradiationfrom the lower layersto reachthe baseof the high-level cloud.The smallertotal

cloudinessin MAX alsoexplainsthesmallerLW heatingbelow thelowestcloud.In theSW, thedifferencein heat-

ing is very small within the clouds.The slightly larger relative cooling immediatelybelow the cloudsin MAX

Radiation Transfer


ECMWF, 2002 49

comesfrom thelargerfractionof radiationbeingdirectlytransmitted,as,in MRN/RAN, theradiationin thecloudy

fractionof thecolumnis morediffuseandthereforemorelikely to beabsorbed.ThesmallerSW heatingof the

uppercloudwith MRN/RAN is againlinkedto theincreasedreflection.Theseone-dimensionalcomputationsshow

the validity of the treatmentof the differentCOAs in the ECMWF radiationscheme.They alsoshow the larger

impact of a change of COA on the LW than on the SW radiative profiles.

Fig.35 presentsthetotalcloudiness(TCC)averagedover thelastthreemonths(DJF)of theEPRsimulationswith

theMRN, MAX andRAN COAs. Not surprisingly, thesetof MAX simulationsdisplaytheminimumtotal cloud

coverat0.609,followedby theMRN at0.639,whereastheRAN simulationshaveamuchlargertotalcloudcover

at 0.714.Theincreasefrom MRN to RAN (respectively, thedecreasefrom MRN to MAX) is apparentover most

of theglobe,but is morepronouncedoverthesub-tropics,particularlythoseof theSouthernhemisphere.Are these

changessimply reflectingthedifferentgeometricaldistribution of thecloudelementson thevertical or arethey

linkedto actualchangesin thevolumeof thecloudelements?A simplecheckwhetherthecloudvolumeis actually

modifiedis obtainedby diagnosingthe total cloudcover for all setsof simulationsandapplyingtheoperational

maximum-randomoverlapassumptionto thevariouscloudelements.Fig. 36 presentstheMRN-equivalentTCC

for theEPRexperiments.Themainresultis thatmostof thechangein TCCis notreflectedin theMRN-equivalent

TCC,showing thata largefractionof thesignalis dueto thegeometricaloverlappingin theradiationscheme,and

not to a real significant change in the cloud volume.

Radiationtransferparametrisationis generallythemostexpensive in computertime amongthephysicalparame-

trisationsusedin a GCM. In theECMWF operationalforecastmodel,full radiationcomputationsareperformed

only every 3 hoursandon a reducedgrid (1 point out of 4 alongthe longitudedirection).Evenso,radiationac-

countsfor about15%tof thetotalcostof themodel.Althoughsuchspatialandtemporalsamplingof theradiation

forcing doesnot appeardetrimentalon theshorttime scalesencompassedin theoperationalanalysesof meteoro-

logical observationsandthe 10-dayforecastsmadewith the high resolution(TL319) ECMWF weatherforecast

model(Fig. 37 for theanomalycorrelationof thegeopotentialat 500hPa,Figs.38 and39 for themeanerrorin

temperatureat 850,500,200and50 hPa), theimpactbecomessystematicand(possibly)detrimentalon seasonal

time-scalesand/orat lower resolution(Morcrette, 2000).For example,Figures6.11and6.12presentthe zonal

meancloudinessandtemperature,with theoperationalconfiguration(S4T3h= spatialsampling1 pointoutof 4 in

thetropics,full radiationcalculationonly every3 hours),andthedifferencewith configurationsfor whichradiation

is eithercalledat every grid point (S1)and/orfor every time step(T1). Whentheradiationcalculationsaresyn-

chronouswith the restof thephysics(andof themodel),thestability in the ITCZ is decreased,with a resultant

(small) increasein high-level clouds(Fig. 40 ), but a morespatiallyextensive decreasein stratospherictempera-

tures (Fig. 41).

New algorithmsarebeingtestedbasedon theneuralnetwork approachof Cheruyet al. (1996)andChevallieret

al. (2000a,2000b),or on the linearizedapproachproposedby ChouandNeelin(1996),bothof which introduce

majorsavingsin computertime.They would thereforeallow for morefrequentand/ornonspatiallysampledradi-

ationcalculations.For example,Fig. 42 presentthebiasandstandarddeviationof theLW coolingratescomputed

by a neuralnetwork methodrelative to theoperationalLW scheme,whereas.Figs.43 and44 displaystandard

objectivescores,theanomalycorrelationof thegeopotentialat500hPa(Fig.43) andthemeanerrorin temperature

atvariouslevelsin theatmosphere(Fig. 44 ). Suchresultsindicatethattheneuralnetwork approachis potentially

a viable replacement. for traditional LW parametrizations.

Otherareasof developmentconcerntherole of thecloudinhomo/heterogeneityon themodelhorizontalsub-grid

scale,andthatof theverticaloverlappingof cloudlayers.Onthefirst point,(notequivalentto a full accountof the

three-dimensionalcloud-inducedradiative effects,but goingaway from theplane-parallelapproachusedover the

last25years),Tiedtke(1996)proposedasimpleparametrisationof thiseffect,while Barker(1996,1998)modified

a SW schemesimilar to theECMWF oneandincorporatedanapproximatetreatmentof the impactof the liquid

Radiation Transfer


ECMWF, 2002

waterinhomogeneitiesvia agamma-distributionapproach.Figs.45 and46 presentthedifferencesin transmissiv-

ity, reflectivity andabsorptivity for somestandardliquid andice watercloudsin thetwo spectralintervalsof the

SW scheme.The impactis importantover mostof the rangeof optical thicknessandfor all quantities.Suchan

“horizontal” effect, to be accountedfor alsoin the LW part of the spectrumis likely to changethe effect of the

cloudson theradiative distribution within thewholeatmosphericcolumn.In view of thelargeeffectsof thesub-

grid scaledistribution of thecondensedwateron theradiative fluxes,it is likely that,in thecomingyears,GCMs

will incorporateanequationto at leastdiagnosethis sub-gridscalevariability andaccountfor its effect on thera-

diation fields.

Up to now, thevalidationof thelarge-scaleradiative fieldsproducedby a climateor a weatherforecastGCM has

mainly consistedof checkson thetotal cloudcover, andrelatedtop of theatmosphereandsurfacelongwave and

shortwave radiationfields.Thesefields(for example,Darnell et al., 1992;Guptaet al., 1993;LaszloandPinker,

1993;Li andLeighton, 1993;Rossow, 1993;Zhanget al., 1995)areprovidedby dedicatedsatelliteobservations

suchasERB (Stoweet al., 1989),ERBE(Harrison et al., 1988),ScaRaB(Kandelet al., 1998),CERES(Wielicki

etal., 1996)and/oroperationalsatelliteobservationsaspartof ISCCP(Rossowetal., 1987).Althoughveryuseful,

theseessentiallytwo-dimensional(3-D if timehistoryis considered)validationeffortsareonly afirst steptowards

whatwould really berequiredto ascertaintheadequacy of therepresentationof thecloud-radiationinteractions:

the4-dimensionaldistribution of cloudvolumeandcloudwaterloadingtogetherwith relevant4-dimensionalra-

diation parameters: a real challenge for the future!

Radiation Transfer


ECMWF, 2002 51

Figure 30. The annual mean climatological distribution of maritime and continental aerosols in the ECMWF

operational system. The relative weight has to be multiplied by the optical thickness, i.e., 0.05 for maritime, and

0.2 for continental aerosols.

0.20.

2

0.4 0.4

0.40.4

0.60.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

0.8�

Maritime

Relative WeightClimatological Aerosols in the ECMWF Forecast System

60°S60°S

30°S 30°S

0°0°

30°N 30°N

60°N60°N

30°E

30°E 60°E

60°E 90°E

90°E 120°E

120°E� 150°E�

150°E 180°

180° 150°W�

150°W 120°W

120°W� 90°W

90°W 60°W

60°W 30°W

30°W

0.1 - 0.2�

0.2 - 0.3�

0.3 - 0.4�

0.4 - 0.5�

0.5 - 0.6�

0.6 - 0.7�

0.7 - 0.8�

0.8 - 0.9�

0.9 - 1�

0.2

�0.2

0.20.4

0.4

0.4

0.6�

0.6

0.8

0.8

Continental


60°S60°S

30°S 30°S

0°0°

30°N 30°N

60°N60°N

30°E

30°E 60°E

60°E 90°E

90°E 120°E

120°E� 150°E�

150°E 180°

180° 150°W�

150°W 120°W

120°W� 90°W

90°W 60°W

60°W 30°W

30°W

0.001996 - 0.1 0.1 - 0.2 0.2 - 0.3 0.3 - 0.4 0.4 - 0.5 0.5 - 0.6 0.6 - 0.7 0.7 - 0.8 0.8 - 0.9 0.9 - 1�

Radiation Transfer


ECMWF, 2002

Figure 31. As inFig. 30, but for the urban and desert-type aerosols. Relevant optical thickness is 0.1 for urban,

and 1.9 for desert aerosols.

0.2

0.2

0.2

0.2�

0.40.6

Urban


60°S60°S

30°S 30°S

0°0°

30°N 30°N

60°N60°N

30°E

30°E 60°E

60°E 90°E

90°E 120°E

120°E� 150°E�

150°E 180°

180° 150°W�

150°W 120°W

120°W� 90°W

90°W 60°W

60°W 30°W

30°W

0.001013 - 0.1 0.1 - 0.2 0.2 - 0.3 0.3 - 0.4 0.4 - 0.5 0.5 - 0.6 0.6 - 0.7 0.7 - 0.8 0.8 - 0.9 0.9 - 1�

0.2

0.2

0.4

0.6

Desert


60°S60°S

30°S 30°S

0°0°

30°N 30°N

60°N60°N

30°E

30°E 60°E

60°E 90°E

90°E 120°E

120°E� 150°E�

150°E 180°

180° 150°W�

150°W 120°W

120°W� 90°W

90°W 60°W

60°W 30°W

30°W

0.000019 - 0.1 0.1 - 0.2 0.2 - 0.3 0.3 - 0.4 0.4 - 0.5 0.5 - 0.6 0.6 - 0.7 0.7 - 0.8 0.8 - 0.9 0.9 - 1�

Radiation Transfer


ECMWF, 2002 53

Figure 32. The vertical distribution of aerosols. Urban aerosols follow type 1 distribution, all other tropospheric

aerosols follow type 2.

0.0 0.2 0.4 0.6 0.8 1.0Relative Weight

1.0

0.8

0.6

0.4

0.2

0.0

Sig

ma

Leve

l

�

Relative Vertical Distribution of Aerosol Types�

Tropos. Type 1Tropos. Type 2Stratos.

Radiation Transfer


ECMWF, 2002

Figure 33. Schematics of the various cloud overlap assumptions used in this study. The clouds are shown as

rectangular blocks filling the vertical extent of a layer. The total cloud fraction from the top of the atmosphere

down to agivenlevel is givenby theline on theright delineatingthecloudsandtheshadedareabelow them.The

cloud overlap configurations are, from top to bottom, the maximum-random, maximum, and random cloud

overlap.

0.0 0.2 0.4 0.6 0.8 1.0Cloud Cover

0.0

0.2

0.4

0.6

0.8

1.0

1−S

igm

a

Maximum−Random Overlap

0.0 0.2 0.4 0.6 0.8 1.0Cloud Cover

0.0

0.2

0.4

0.6

0.8

1.0

1−S

igm

a

Maximum Overlap�

0.0 0.2 0.4 0.6 0.8 1.0Cloud Cover

0.0

0.2

0.4

0.6

0.8

1.0

1−S

igm

a

Random Overlap

Radiation Transfer


ECMWF, 2002 55

Figure 34. The longwave (top), shortwave (middle), and net cloud radiative forcing profiles for an atmosphere

with a anvil-topped convective tower (left panels) and three independent stratiform cloud layers (right panels).

−3� −2� −1 0�

1� 2�Cloud LW Radiative Forcing K/day

1.0

0.8

0.6

0.4

0.2

0.0

Sig

ma

Pre

ssur

e

�

Convective Tower with Cirrus Anvil�

RandomMaximumMax−Ran

−3.5� −2.5� −1.5 −0.5�

0.5 1.5Cloud LW Radiative Forcing K/day�

1.0

0.8

0.6

0.4

0.2

0.0

Sig

ma

Pre

ssur

e

Three Independent Stratiform Cloud Layers�RandomMaximumMax−Ran

−1 0�

1� 2� 3� 4�

Cloud SW Radiative Forcing K/day

1.0

0.8

0.6

0.4

0.2

0.0

Sig

ma

Pre

ssur

e

�


−1 0�

1� 2� 3� 4�

Cloud SW Radiative Forcing K/day

1.0

0.8

0.6

0.4

0.2

0.0

Sig

ma

Pre

ssur

eRandomMaximumMax−Ran

−1 0�

1� 2� 3� 4�

Cloud Net Radiative Forcing K/day

1.0

0.8

0.6

0.4

0.2

0.0

Sig

ma

Pre

ssur

e

�


−1 0�

1� 2� 3� 4�

Cloud Net Radiative Forcing K/day

1.0

0.8

0.6

0.4

0.2

0.0

Sig

ma

Pre

ssur

e


Radiation Transfer


ECMWF, 2002

Figure 35.Thetotalcloudiness(TCC)averagedoverthelastthreemonths(DJF)of theRAD simulationswith the

maximum-random (MRN), maximum (MAX) and random (RAN) cloud overlap assumption (COA).

Figure 36.Themaximum-random-equivalenttotal cloudinessof thesimulationswith theMRN, MAX andRAN

COAs

45� �

45�45�

65�65

65! 65"

65#65

8585$

EPR_22 MAX Global Mean: 0.609E+02

MAX

5.1540 - 15 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 75 - 85 85 - 95

95 - 96.318

65%

85&85'85(

85)

85*85

85+

EPR_33 RAN Global Mean: 0.714E+02

RAN

7.4617 - 15,

15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 75 - 85 85 - 95

95 - 100

45-65.65

/ 0

651 652

853 4

85585

6857

EPR_11 MRN Global Mean: 0.639E+02

MRN

6.9888 - 158

15 - 25 25 - 35 35 - 45 45 - 559

55 - 65 65 - 75 75 - 85:

85 - 95

95 - 98.833;

Total Cloud Cover % Between 3024- 720 HoursImpact of Cloud Overlap Assumption: EPR

45<

65

65

85=85>

85

EPR_22 MAX Global Mean: 0.638E+02

MAX

5 - 15 15 - 25 25 - 35 35 - 45 45 - 55?

55 - 65 65 - 75 75 - 85 85 - 95

95 - 99.366

65@ A

65B

6565C

85D E85F

85G

85

EPR_33 RAN Global Mean: 0.645E+02

RAN

5 - 15 15 - 25 25 - 35H

35 - 45 45 - 55 55 - 65 65 - 75 75 - 85 85 - 95

95 - 99.540

45I

65

65

85J 85K L

85M

EPR_11 MRN Global Mean: 0.641E+02

MRN

5 - 15 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 75 - 85 85 - 95

95 - 99.703

Total Cloud Cover seen as MRN % Between 3024- 720 HoursImpact of Cloud Overlap Assumption: EPR

Radiation Transfer


ECMWF, 2002 57

Figure 37. The anomaly correlation of the geopotential at 500 hPa for the various time-space configurations for

theNorthern(toppanels)andSouthern(middlepanels)hemispheres,andthetropicalregion(bottompanels).Full

line is S4T3h, long dash line is S4T1, dotted line is S1T3h, and dot-dash line is S1T1.

0N 1O 2P 3Q 4R 5S 6T 7U 8V 9W 10

Forecast DayX0

10

20

30

40

50

60

70

80

90

100%

Y DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...

AREA=N.HEM TIME=12 MEAN OVER 12 CASES

ANOMALY CORRELATION FORECAST

500 hPa GEOPOTENTIALZFORECAST VERIFICATION TL159_18r6_S4T3h

[TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1\

MAGICS 6.0 styx - pam Wed Apr 7 15:07:14 1999 Verify SCOCOM

0N 1O 2P 3Q 4R 5S 6T 7U 8V 9W 10

Forecast DayX0

10

20

30

40

50

60

70

80

90

100%


AREA=S.HEM TIME=12 MEAN OVER 12 CASES


500 hPa GEOPOTENTIALZFORECAST VERIFICATION

]TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h[TL159_18r6_S1T1

^


0N 1O 2P 3Q 4R 5S 6T 7U 8V 9W 10

Forecast DayX0

10

20

30

40

50

60

70

80

90

100%


AREA=EUROPE TIME=12 MEAN OVER 12 CASES


500 hPa GEOPOTENTIALZFORECAST VERIFICATION TL159_18r6_S4T3h

[TL159_18r6_S4T1

\TL159_18r6_S1T3h

TL159_18r6_S1T1


Radiation Transfer


ECMWF, 2002

Figure 38.Themeanerrorof thetemperatureat850and500hPafor thevarioustime-spaceconfigurationsfor the

Northern and Southern hemispheres, and the tropical region. Panels are read from left to right, and top to bottom.

0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10

Forecast Dayi-0.15

-0.1

-0.05

0

0.05

0.1 DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...


MEAN ERROR FORECAST

850 hPa TEMPERATURE/DRY BULB TEMPERATURE

FORECAST VERIFICATION

TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1

MAGICS 6.0 styx - pam Thu Apr 22 15:51:04 1999 Verify SCOCOM

0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10

Forecast Dayi-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0



MEAN ERROR FORECAST



TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1

MAGICS 6.0 styx - pam Thu Apr 22 15:51:04 1999 Verify SCOCOM * 1 ERROR(S) FOUND *

0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10

Forecast Dayi0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16


AREA=TROPICS TIME=12 MEAN OVER 12 CASES

MEAN ERROR FORECAST



TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1


0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10

Forecast Dayi-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0 DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...


MEAN ERROR FORECAST



TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1


0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10

Forecast Dayi-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1



MEAN ERROR FORECAST



TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1


0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10

Forecast Dayi-0.1

-0.05

0

0.05

0.1



MEAN ERROR FORECAST



TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1


Radiation Transfer


ECMWF, 2002 59

Figure 39. As inFig. 38, but for the mean error of the temperature at 200 and 50 hPa.

0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10

Forecast Dayi-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0



MEAN ERROR FORECAST



TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1


0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10

Forecast Dayi-0.5

-0.4

-0.3

-0.2

-0.1

0



MEAN ERROR FORECAST



TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1


0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10

Forecast Dayi-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


AREA=TROPICS TIME=12 MEAN OVER 12 CASESj MEAN ERROR FORECAST



TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1


0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10

Forecast Dayi-0.2

0

0.2

0.4

0.6

0.8

1

1.2



MEAN ERROR FORECAST



TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1


0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10

Forecast Dayi0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9



MEAN ERROR FORECAST



TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1


0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10

Forecast Dayi-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9



MEAN ERROR FORECAST



TL159_18r6_S4T3h

TL159_18r6_S4T1

TL159_18r6_S1T3h

TL159_18r6_S1T1


Radiation Transfer


ECMWF, 2002

Figure 40. The zonal mean distribution of the cloud fraction averaged over the last 3 months of TL95 L31

simulations (in percent). Top left panel is the operational configuration (S4T3h), top right is the difference S4T1-

S4T3h, bottom left is the difference S1T3h-S4T3h bottom right is the difference S1T1-S4T3h.

Figure 41. As inFig. 40, but for the zonal mean temperature (in degrees Kelvin).

-0.5

-0.5 -0.5

-0.5

0.5

0.5

0.5

0.5

S1T3hk

90. 75. 60. 45. 30. 15. 0.l -15. -30.m -45.n -60. -75. -90.oLatitudep 996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

Pre

ssur

e Le

vel

(hP

a)

q

996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

-4.5 - -3.7457 -3.7457 - -3.5 -3.5 - -2.5 -2.5 - -1.5 -1.5 - -0.5 0.5 - 1.5 1.5 - 2.5 2.5 - 3.5 3.5 - 4.5

4.5 - 5.5 5.5 - 6.5 6.5 - 7.5 7.5 - 7.7446

-0.5 -0.5

-0.5

0.5

0.5

S1T1k

90. 75. 60. 45. 30. 15. 0.l -15. -30. -45. -60. -75. -90.Latitudep 996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

Pre

ssur

e Le

vel

(hP

a)

q

996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

-5.5 - -5.4460 -5.4460 - -4.5 -4.5 - -3.5 -3.5 - -2.5 -2.5 - -1.5 -1.5 - -0.5 0.5 - 1.5 1.5 - 2.5 2.5 - 3.5

3.5 - 4.5 4.5 - 5.5 5.5 - 5.8971

5r

5s

5t5

u15

15v

15w 15x25

y z

35{ |S4T3h

k 90. 75. 60. 45. 30. 15. 0.l -15. -30.m -45.n -60. -75. -90.o

Latitudep 996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

Pre

ssur

e Le

vel

(hP

a)

q

996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

5 - 10 10 - 15 15 - 20 20 - 25 25 - 30 30 - 35 35 - 40 40 - 45 45 - 50

50 - 55 55 - 60

-0.5

-0.5

0.5

0.5

0.5

0.5

S4T1k

90. 75. 60. 45. 30. 15. 0.l -15. -30. -45. -60. -75. -90.Latitudep 996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

Pre

ssur

e Le

vel

(hP

a)

q

996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

-4.5 - -3.7050 -3.7050 - -3.5 -3.5 - -2.5 -2.5 - -1.5 -1.5 - -0.5 0.5 - 1.5 1.5 - 2.5 2.5 - 3.5 3.5 - 4.5

4.5 - 5.5 5.5 - 6.5 6.5 - 7.5 7.5 - 8.5}

8.5 - 8.5191

8706-8708 Cloud Fraction (%) Zonal M Diff. to OPE~ ECMWF TL95 L31: 4/3h, 4/1, 1/3h, 1/1

Time-Space Interpolation: REF�

-0.1

-0.1�-0.1�

0.1

0.1

0.1

0.1

0.1

0.5

0.5

S1T3hk

90. 75. 60. 45. 30. 15. 0.l -15. -30.m -45.n -60. -75. -90.oLatitudep 996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

Pre

ssur

e Le

vel

(hP

a)

q

996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

-0.9 - -0.887457 -0.887457 - -0.7 -0.7 - -0.5 -0.5 - -0.3 -0.3 - -0.1 0.1 - 0.3 0.3 - 0.5 0.5 - 0.7 0.7 - 0.9

0.9 - 1.1 1.1 - 1.3 1.3 - 1.4489

-1.3-0.9� -0.5�

-0.1

-0.1

0.1

0.1

0.1

0.1

0.5

0.5

S1T1k

90. 75. 60. 45. 30. 15. 0.l -15. -30. -45. -60. -75. -90.Latitudep 996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

Pre

ssur

e Le

vel

(hP

a)

q

996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

-2.5 - -2.3646 -2.3646 - -2.3 -2.3 - -2.1 -2.1 - -1.9 -1.9 - -1.7 -1.7 - -1.5 -1.5 - -1.3 -1.3 - -1.1 -1.1 - -0.9

-0.9 - -0.7 -0.7 - -0.5 -0.5 - -0.3 -0.3 - -0.1 0.1 - 0.3 0.3 - 0.5 0.5 - 0.7 0.7 - 0.762493

200�210�

220� �

220�230�230�

240

�

240�250�

250�260

�260�

270�

270�280

� �290� �S4T3h

k 90. 75. 60. 45. 30. 15. 0.l -15. -30.m -45.n -60. -75. -90.o

Latitudep 996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

Pre

ssur

e Le

vel

(hP

a)

q

996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

174.78 - 175 175 - 180 180 - 185 185 - 190 190 - 195 195 - 200 200 - 205 205 - 210 210 - 215

215 - 220 220 - 225 225 - 230 230 - 235 235 - 240 240 - 245 245 - 250 250 - 255 255 - 260

260 - 265 265 - 270�

270 - 275�

275 - 280�

280 - 285�

285 - 290�

290 - 295�

295 - 300�

300 - 300.77

-2.1�-1.3-0.9�

-0.5� -0.1�

-0.1

-0.1

� -0.1�

0.10.1

0.5

S4T1k

90. 75. 60. 45. 30. 15. 0.l -15. -30. -45. -60. -75. -90.Latitudep 996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

Pre

ssur

e Le

vel

(hP

a)

q

996.

959.

891.

807.

717.

626.

538.

453.

374.

302.

237.

181.

132.

90.

50.

10.

-2.5 - -2.3577 -2.3577 - -2.3 -2.3 - -2.1 -2.1 - -1.9 -1.9 - -1.7 -1.7 - -1.5 -1.5 - -1.3 -1.3 - -1.1 -1.1 - -0.9

-0.9 - -0.7 -0.7 - -0.5 -0.5 - -0.3 -0.3 - -0.1 0.1 - 0.3 0.3 - 0.5 0.5 - 0.7 0.7 - 0.9 0.9 - 1.0868

8706-8708 Temperature (K) Zonal M Difference to OPE~ ECMWF TL95 L31: 4/3h, 4/1, 1/3h, 1/1

Time-Space Interpolation: REF�

Radiation Transfer


ECMWF, 2002 61

Figure 42. The bias and standard deviation of the longwave cooling rates computed with the neural network

approach with respect to those computed with the operational longwave scheme, for an ensemble of profiles

sampled out of the first days of the months during one year, with the TL319 L31 operational model.

Figure 43.Theanomalycorrelationof thegeopotentialat500hPafor asetof experimentsstartingon the15thof

the month for one year. EC-OPE uses the operational LW radiation scheme, Neuro-Flux the neural network

version of it.

5

10

15

20

25

30

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Cooling rates (K/d)

Bias

polarmid-latitude

tropical

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Cooling rates (K/d)

Standard deviation

polarmid-latitude

tropical

10

20

30

40

50

60

70

80

90

100

0 2 4 6 8 10

Ano

mal

y co

rrel

atio

n (%

)

�

Forecast day

500hPa geopotential (m)

NeuroFluxEC-OPE

Radiation Transfer


ECMWF, 2002

Figure 44. As inFig. 43, but for the mean error in temperature at 850, 500, 200 and 50 hPa.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 2 4 6 8 10

Tem

pera

ture

(K

)

�

Forecast day

850hPa

NeuroFluxEC-OPE

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 2 4 6 8 10

Tem

pera

ture

(K

)

Forecast day

500hPa

NeuroFluxEC-OPE

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 2 4 6 8 10

Tem

pera

ture

(K

)

¡

Forecast day

200hPa

NeuroFluxEC-OPE

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10

Tem

pera

ture

(K

)

¡

Forecast day

50hPa

NeuroFluxEC-OPE

Radiation Transfer


ECMWF, 2002 63

Figure 45. The transmissivity, reflectivity and absorptivity for ice clouds with optical properties fromEbert and

Curry (1992), computed withBarker’s (1996) IPA, the Delta-Eddington approximation and a Delta-Eddington

approximation accounting forTiedtke’s (1996) inhomogeneity factor.

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Figure 46. As inFig. 45, but for water clouds with optical properties fromFouquart (1987).

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Comparison IPA vs. Plane−Parallel / Y.Fouquart Water CloudªSpectral Interval: 0.25 − 0.68 CSZA=1.0 Sigma= tau/sqrt(1.0) Fmt=0.7«

Re_IPATr_IPA

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