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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
Meteorological Training Course Lecture Series
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
Radiation Transfer
4 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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).
Radiation Transfer
6 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
8 Meteorological Training Course Lecture Series
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�
300
erbe_nfov ASW Area Mean: 0.258E+03ERBE Narrow FOV on (2.5 deg)^2 grid for 198706-08
198706-08�
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60°W 30°W
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25 - 50 50 - 75 75 - 100 100 - 125 125 - 150 150 - 175 175 - 200 200 - 225 225 - 250 250 - 275 275 - 300 300 - 325
325 - 350 350 - 375 375 - 400 400 - 417.18
200� 200�250� 250�300� 300�350� 350�400�
400�
erbe_nfov CSASW Area Mean: 0.317E+03ERBE Narrow FOV on (2.5 deg)^2 grid for 198706-08
198706-08
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75 - 100 100 - 125 125 - 150 150 - 175 175 - 200 200 - 225 225 - 250 250 - 275 275 - 300 300 - 325 325 - 350 350 - 375
375 - 400 400 - 419.7
Radiation Transfer
Meteorological Training Course Lecture Series
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).
90�130�170�
erbe_nfov REF Area Mean: 0.100E+03ERBE Narrow FOV on (2.5 deg)^2 grid for 198706-08
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18.765 - 25 25 - 40 40 - 55 55 - 70 70 - 85 85 - 100 100 - 115 115 - 130 130 - 145 145 - 160 160 - 175 175 - 190
190 - 205 205 - 220
Radiation Transfer
10 Meteorological Training Course Lecture Series
ECMWF, 2002
Figure 5. As inFig. 3, but for the outgoing longwave radiation at the top of the atmosphere (W m-2).
-270 -270!-270"
-270#-270$
-240%-240&
-240'-240( -240)
-210*-210+
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198706-08�
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-329.74 - -315 -315 - -300 -300 - -285 -285 - -270 -270 - -255 -255 - -240 -240 - -225 -225 - -210 -210 - -195 -195 - -180 -180 - -165 -165 - -150
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-2900 -2901-2902
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198706-08
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-330 - -320 -320 - -310 -310 - -300 -300 - -290 -290 - -280 -280 - -270 -270 - -260 -260 - -250 -250 - -240 -240 - -230 -230 - -220 -220 - -210
-210 - -200 -200 - -190 -190 - -180 -180 - -170 -170 - -160 -160 - -158.03
Radiation Transfer
Meteorological Training Course Lecture Series
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
erbe_nfov SWCF Area Mean: -0.477E+02ERBE Narrow FOV on (2.5 deg)^2 grid for 198706-08
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0 - 10:
10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 - 100 100 - 100.67
Radiation Transfer
12 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
14 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
16 Meteorological Training Course Lecture Series
ECMWF, 2002
Figure 7. The electromagnetic spectrum
Radiation Transfer
Meteorological Training Course Lecture Series
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
18 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
20 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
22 Meteorological Training Course Lecture Series
ECMWF, 2002
Figure 13. As inFig. 11, but for CO2.
Figure 14. As inFig. 12, but for O3.
Radiation Transfer
Meteorological Training Course Lecture Series
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
24 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
26 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
28 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
30 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
32 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
34 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
36 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
38 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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 3µe a–( ) 4⁄=¿
cl r Á cl r
δ+ δR δa 1 ϖa a2–( )+=
 +  a
1 Â a+---------------
δa
δR δa+( )----------------------=
ϖ+ δR
δR δa+-----------------ϖR
δa
δR δa+-----------------
ϖa 1 Â a2–( )
1 ϖa a2–
--------------------------+=
Radiation Transfer
40 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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  c ϖc
δc Ï LWP
δc
3 Ï LWP
2Ë e-----------------=
Ë e Ë e
µ µ
 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–=
 i Ù i Û i Ë e+=
Ë e µ
Radiation Transfer
42 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
44 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
46 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
48 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
50 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
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
0.001996 - 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
52 Meteorological Training Course Lecture Series
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
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
0.001013 - 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
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
0.000019 - 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
Meteorological Training Course Lecture Series
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
54 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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
�
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
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
�
RandomMaximumMax−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
RandomMaximumMax−Ran
Radiation Transfer
56 Meteorological Training Course Lecture Series
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
Meteorological Training Course Lecture Series
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%
Y DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...
AREA=S.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%
Y DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...
AREA=EUROPE 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
Radiation Transfer
58 Meteorological Training Course Lecture Series
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/...
AREA=N.HEM TIME=12 MEAN OVER 12 CASES
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
0.05 DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...
AREA=S.HEM TIME=12 MEAN OVER 12 CASES
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 * 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
0.18 DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...
AREA=TROPICS TIME=12 MEAN OVER 12 CASES
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 15:51:13 1999 Verify SCOCOM * 1 ERROR(S) FOUND *
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/...
AREA=N.HEM TIME=12 MEAN OVER 12 CASES
MEAN ERROR FORECAST
500 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 15:51:13 1999 Verify SCOCOM * 1 ERROR(S) FOUND *
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 DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...
AREA=S.HEM TIME=12 MEAN OVER 12 CASES
MEAN ERROR FORECAST
500 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 15:51:13 1999 Verify SCOCOM * 1 ERROR(S) FOUND *
0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10
Forecast Dayi-0.1
-0.05
0
0.05
0.1
0.15 DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...
AREA=TROPICS TIME=12 MEAN OVER 12 CASES
MEAN ERROR FORECAST
500 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 15:51:22 1999 Verify SCOCOM * 1 ERROR(S) FOUND *
Radiation Transfer
Meteorological Training Course Lecture Series
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
0.1 DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...
AREA=N.HEM TIME=12 MEAN OVER 12 CASES
MEAN ERROR FORECAST
200 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 15:51:22 1999 Verify SCOCOM * 1 ERROR(S) FOUND *
0_ 1 2a 3b 4c 5d 6e 7f 8g 9h 10
Forecast Dayi-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1 DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...
AREA=S.HEM TIME=12 MEAN OVER 12 CASES
MEAN ERROR FORECAST
200 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 15:51:23 1999 Verify SCOCOM * 1 ERROR(S) FOUND *
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 DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...
AREA=TROPICS TIME=12 MEAN OVER 12 CASESj MEAN ERROR FORECAST
200 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 15:51:32 1999 Verify SCOCOM * 1 ERROR(S) FOUND *
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
1.4 DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...
AREA=N.HEM TIME=12 MEAN OVER 12 CASES
MEAN ERROR FORECAST
50 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 15:51:32 1999 Verify SCOCOM * 1 ERROR(S) FOUND *
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
1 DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...
AREA=S.HEM TIME=12 MEAN OVER 12 CASES
MEAN ERROR FORECAST
50 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 15:51:32 1999 Verify SCOCOM * 1 ERROR(S) FOUND *
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
1 DATE1=19970915/... DATE2=19970915/... DATE3=19970915/... DATE4=19970915/...
AREA=TROPICS TIME=12 MEAN OVER 12 CASES
MEAN ERROR FORECAST
50 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 15:51:32 1999 Verify SCOCOM * 1 ERROR(S) FOUND *
Radiation Transfer
60 Meteorological Training Course Lecture Series
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
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Time-Space Interpolation: REF�
Radiation Transfer
Meteorological Training Course Lecture Series
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.
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Radiation Transfer
62 Meteorological Training Course Lecture Series
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Figure 44. As inFig. 43, but for the mean error in temperature at 850, 500, 200 and 50 hPa.
-0.6
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NeuroFluxEC-OPE
Radiation Transfer
Meteorological Training Course Lecture Series
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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Radiation Transfer
64 Meteorological Training Course Lecture Series
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Figure 46. As inFig. 45, but for water clouds with optical properties fromFouquart (1987).
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0¨
5©
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Optical Thickness
0.0
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