5. One-dimensional Radiative Transfer in
the Atmosphere
Ourzero-dimensionalmodelofearth’stemperature(Model4)predictedavalueforearth’s
surfacetemperatureofabout-20°C,whiletheobservedgloballyaveragedsurfacetemperatureis
roughly14°C.Thetemperatureestimatefromthezero-dmodelisalittlebitcold,butthatmodellacks
anatmosphere.Inthismodelwewillseehowaddinganatmosphereaffectsthesurfacetemperature.
Ourgoalistostudythetransferofradiationupanddownthroughtheatmosphere.
Onacleardaytheatmosphereisnearlytransparenttosolarradiation.Thesituationwith
respecttoterrestrialradiationismuchmorecomplicated.Somewavelengthspassrightthroughthe
atmospherewhileothersarecompletelyabsorbedwithinafewmeters.Thisisbecauseeachmolecular
speciesintheatmosphere,N2,O2,H2Oandabout100othermolecules,hasadistinctandinsanely
complicatedabsorptionspectrum,i.e.alistoffrequenciesthatitcanabsorbandothersthatitcannot.
Wearegoingtoignorethedetailsandimaginewehaveonlytwotypesofradiation:solar,alsoknownas
shortwave(SW),andterrestrial,alsoknownaslongwave(LW).Weimaginetheatmospheretoconsistof
anumberoflayers.Theshortwaveradiationpassesthrougheachlayer,butafraction𝑘ofthelongwave
radiationisabsorbed.Asafirsttry,wewillconstructa5layermodelwith𝑘 = 20%foreachlayer.Layer
1oftheatmosphereisclosesttothesurface,andlayer5isthetoplayer.Thelongwaveradiation
absorbedbyalayerwillwarmthatlayer.Inequilibrium,eachlayermustradiateawayasmuchenergyas
itabsorbs,otherwiseitwillgetwarmerorcoolerinviolationofitspostulatedequilibrium.
Considerthelongwaveradiationfirst.Let’susethefollowingnotationfortheLWfluxesbetween
layers:𝐿𝑊↑,(!!!)(!) and𝐿𝑊↓,(!!!)(!) ,wherethearrowsinthesubscriptstellwhethertheLWfluxis
upwardordownward,thefirstnumberinthesubscripttellswhichlayerthefluxisemittedfrom,and
thesecondnumbertellswhichlayertheLWfluxisdirectedtoward.Forexample,𝐿𝑊↑,!" istheupward
LWfluxfromlayer3tolayer4.Therearealotoftermstokeeptrackofinthismodel,solet’swork
throughanexamplewithnumbers.Supposewehave30Wm-2oflongwavedownwellingfromlayer5to
layer4(𝐿𝑊↓,!" ),and300Wm-2upwellingfromlayer3to4(𝐿𝑊↑,!" ).Then66Wm-2areabsorbedinlayer
4(20%of30plus20%of300).Halfofthe66Wm-2areradiateddownwardsandhalfupwards.So
𝐿𝑊↓,!" = 30 − 6 + 33 = 57Wm-2,and𝐿𝑊↑,!" = 300 – 60 + 33 = 273Wm-2.SeetheFigure5.1for
avisualrepresentationoftheflowoflongwaveradiationthroughalayeroftheatmosphere.
Nowconsiderthesun’sradiation.Theshortwaveradiationfromthesun,𝑆/4,passesthrough
theatmosphere.Afraction𝛼𝑆/4isreflectedatthesurfaceandescapestospace.Therestofthe
shortwaveradiation,(1 − 𝛼)𝑆/4,isabsorbedatthesurface.Thesurfacealsoabsorbsalldownwelling
longwaveradiationthatreachesit(𝑘 = 100% forthesurface).Theruleforthesurfaceboundary
conditionis:
𝐿𝑊↓,!" + (1 − 𝛼)𝑆/4 = 𝐿𝑊↑,!" .
where“layer0”isthesurface.
Therulesfortheone-dimensionalradiativetransfermodelaresummarizedbelow,andshown
visuallyinFigure5.1.
MiddleAtmosphere(layers1-4)
•
•
•
Foragivenlayer,saylayer4,thereisupwardlongwaveincidentonthelayerfrombelowand
downwardlongwaveradiationincidentfromabove.Afraction𝑘ofthisradiationisabsorbedby
thelayer:
(𝐿𝑊 absorbed by layer 4) = 𝑘 (𝐿𝑊↑,!" + 𝐿𝑊↓,!" ).
Afraction(1 − 𝑘)oftheincidentlongwaveradiationistransmittedthroughthelayer.
Thelayeremitsenoughradiationtobalancethetotalabsorbedradiation,andhalfofthe
emittedradiationisdirectedupwardsandhalfdownwards:
emitted 𝐿𝑊↑,!" = emitted 𝐿𝑊↓,!" = (𝐿𝑊 absorbed by layer 4)/2 = 𝑘 (𝐿𝑊↑,!" + 𝐿𝑊↓,!" )/2.
•
Thetotallongwaveradiationeitherupordownisequaltothetransmittedradiationplusthe
emittedradiation.Forexample:
𝐿𝑊↑,!" = 1 − 𝑘 𝐿𝑊↑,!" + 𝑘 (𝐿𝑊↑,!" + 𝐿𝑊↓,!" )/2.
total LW up
LW",54 = 273 Wm
2
transmitted LW up
(1
incident LW down
LW#,54 = 30 Wm
2
k)LW",43 = 240 Wm
2
emitted LW up
(kLW#,54 + kLW",43 )/2 = 33 Wm
2
absorbed
LW up
absorbed
LW down
kLW#,54 = 6 Wm
kLW",43 = 60 Wm
2
2
atmosphere layer 4
emitted LW down
(kLW#,54 + kLW",43 )/2 = 33 Wm
transmitted LW down
(1
k)LW#,54 = 24 Wm
2
incident LW up
2
LW",43 = 300 Wm
2
total LW down
LW#,43 = 57 Wm
2
Figure5.1Schematicdiagramshowingtherulesforenergybalanceinlayer4oftheatmosphere.
Surface
•
•
•
Solarradiationisabsorbedatthesurfaceonly.Thefluxofsolarradiationabsorbedbythe
surfaceis(1 − 𝛼)𝑆/4,where𝑆 = 1400Wm-2isthesolarconstantand𝛼 = 0.3isthealbedo.
Alldownwardlongwaveradiationthatreachesthesurfaceisabsorbed(𝑘 = 100%atthe
surface)
Theupwardlongwaveradiationemittedbythesurfaceisenoughtobalancethesolarradiation
anddownwardlongwaveradiationthatitabsorbs:
𝐿𝑊↑,!" = 1 − 𝛼 𝑆/4 + 𝐿𝑊↓,!" (Remember,“layer0”isthesurface.)
TopofAtmosphere(layer5)
•
Therulesarethesameatthetoplayeroftheatmosphereasforlayers1-4,exceptthatthereis
nodownwardlongwaveradiationfromspace:
𝐿𝑊↓, !"#$% ! = 0Wm-2.
Intherealworldthereisactuallyaverysmalldownwardfluxlongwaveradiationfromspace
thatisleftoverfromthebigbang,butwewillignoreitinthismodel.
Tocalculatethetablewebeginwithsomeinitialvalues,asshown.Thenapplytherulestoget
newvaluesanditerateuntilnothingchanges.BesuretoconfigureExcelforMANUALandITERATIVE
calculationasdescribedinmodel1.Figure5.2showsoneintuitivewaytoarrangethemodelinExcel.
Figure5.2Onewayofarrangingthemodelinaspreadsheet.Thelogicofthemodelisreflectedbythe
arrangementofthecells.
Box5.1|Satelliteobservationsoflongwaveradiation
Satellitesprovideaglobalviewofradiationatthetopoftheatmosphere.Measurementsofthefluxof
longwaveradiationatthetopoftheatmospherefromNASA’sCloudsandEarth’sRadiantEnergySystem(CERES)1
instrumentareshowninFigure5.3.Bothfiguresshowtheaverageof15yearsofmeasurementstakenfrom
March2000throughFebruary2015.
Wecanunderstandthespatialvariationsinthefluxoflongwaveradiationusingintuitionfromour
modelsofone-dimensionalradiativetransferandblack-bodyradiation.Firstnotethat,withtheexceptionof
latitudesclosetotheequator,thelongwavefluxdecreasesasyoumovetowardsthepoles.Likeinourblackbodyradiationmodel,warmregionsemitmorelongwaveradiationthancoldregions.
Neartheequatorthingsaremorecomplicated.Thefluxoflongwaveradiationisrelativelysmallover
Indonesia,theAmazon,theCongoBasin,andanarrowstripspanningthePacificandAtlanticOceannearthe
equator,andlargeovertheSaharadesert,Australia,SouthernAfrica,andmostoftheoceansbetween10°-30°
latitude.Thesespatialvariationscanbeunderstoodbythinkingaboutcloudsandwatervaporintheatmosphere.
Cloudsareblackbodiesforlongwaveradiation,andwatervaporisastronggreenhousegas.Thus,wherever
therearehighcloudsandmoistair,theatmosphereisopaquetolongwaveradiation.Inourmodel,this
correspondstoalargevalueofk.Overregionswherethelongwavefluxtospaceissmall(Indonesia,theAmazon,
etc.),risingairanddeepcloudswithtopsataltitudesof10-12kmarecommon.Becausethecloudsareblack
bodies,andbecausetheirtopsarehighandcold,emittedlongwaveradiationtospaceissmall.Similarly,in
regionswherethelongwavefluxtospaceislarge(theSahara,Australia,etc.)itiscommontohavesinkingair
thatisdryandcloud-free.Intheseregionsthesurfaceiswarm,surfacelongwaveemissionisrelativelylarge,and
longwaveabsorptionbytheatmosphereisrelativelyweak.Thus,thefluxoflongwaveradiationatthetopofthe
atmosphereislarge.
Figure5.3Satelliteobservationsoftheupwardfluxoflongwaveradiationatthetopoftheatmosphereaveraged
fromMarch2000throughFebruary2015.a)showsamap,whileb)showstheaverageacrosscirclesofconstant
latitude.
Exercises
5.1Prepareaspreadsheettodothemodel3radiativetransfercalculation.Includearesetmacro.Use5
layers.Refertoexcelspreadsheetexcel3_1D_radiative_transfer.
5.2Ifyouset𝑘 = 0foreverylayeroftheatmosphere(butnotthesurface!)doyourecoverthevalues
fromthezero-dimensionalmodel(Model4)?Thisisagoodwaytocheckthatyourmodelisworking
properly.
5.3Togettemperatures,notethateachlayerisagraybodysothattheenergyradiatedbyalayer
𝐴𝐵𝑆/2(𝐴𝐵𝑆standsfortotalabsorbedlongwaveradiation)is𝑘times𝐴 + 𝐵(𝑇 − 273).Thisimplies
𝑇 = 273 + (
!"#
!!
− 𝐴)/𝐵[K]or(
!"#
!!
− 𝐴)/𝐵[°C].
Includethetemperatureofeachlayerinyourspreadsheet.
5.4ModifyyourmasterspreadsheettotreatvariousGreenhousescenarios.Howmuchmustyou
increasethelongwaveabsorptiontocausethesurfacetemperaturetochange1degree?
5.5Howmightyouincludecloudinessinthemodel?
5.6Areyouimpressedthatthemodelgivesasurfacetemperatureof14C?Youshouldn’tbe.Itunedthe
parameters𝑘untilIgotaplausibletemperature.
References
1.CERESScienceTeam,Hampton,VA,USA:NASAAtmosphericScienceDataCenter(ASDC),Accessed
August10,2016atdoi:10.5067/Terra+Aqua/CERES/EBAF-TOA_L3B.002.8
Solutions and hints
HintforthoseworkinginPythonorMatlab:Ifyouaredoingthecalculationinaloopthenyoushould
createvariablestosaveyourcomputedupward,downward,andabsorbedlongwavefluxesfromthe
previoustimestepandusetheseinyourcalculations.Otherwiseyoursolutionmaygrowwithout
bound.
5.1Thetrickypartisgettingthesurfaceboundaryconditionright.Atthesurfacetheincomingand
outgoingfluxesmustbalance.Thesurfacemustradiateenoughlongwaveradiationtogetridofthe
downwellingradiationandtheshortwaveradiationabsorbedthere.