Assignment#1
ATS761:Land-AtmosphereInteractions
DueTuesSept20
NOTE:thesetwoproblemsarenothard,butmaytakeyouquitealotoftimeifyou
haven’twrittencomputerprogramsinalongtime.Myowncomputerprogramsfor
theseproblemsareontheclasswebsite,andyoucanusetheseifyouwant.Itwill
helpfocusyourreadingandclasstimeifyoureadthroughtheassignmentand
startthinkingabouthowtosolvetheseproblems.I’veoutlinedprettymuchallyou
havetodobelow.
1. Writeasimplecomputerprogramtocalculatethediurnalcycleof
incomingsolarradiation(insolation)undercloudlessskiesforthe
followinglocationsanddates:
• FortCollins,COUSAonAugust21(P=855mb)
• FortCollins,COUSAonFebruary21(P=845mb)
• Mogadishu,SomaliaonMarch21(P=1010mb)
• Wellington,NewZealandonOctober21(P=1010mb)
Foreachlocationanddateabove,makeagraphofthesurface
insolationforeachhourofthedayfrom0=midnightto23=11PM(local
standardtime).
Here’show:
Intheabsenceofatmosphericscatteringandabsorption,thepowerofthe
Sun’sradiationincidentonahorizontalsurfaceis
S
(Bonaneqn4.4)
SH = C2 cos Z rv
whereSC=1367Wm-2isthe“solarconstant,”rvisthe“radiusvector”(the
ratiooftheactualdistancebetweentheSunandtheEarthtoitsannual
average),andcosZisthecosineofthesolarzenithangle.Whenthecosineof
thesolarzenithangleisnegative,theSunisbelowthehorizonandwecan
justsettheincomingsolarradiationtozero.
Thecosineofthesolarzenithangleis
(Bonaneqn4.1)
cos Z = sin φ sin δ + cos φ cos δ cos h where φ islatitude,disdeclination(latitudewheretheSunisdirectly
overheadatnoon),andhisthehourangle(h=0atnoon,h=p/2at6PM,h=
patmidnight,h=3p/2at6AM).
Tocalculatethehourangle,simplycompute
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Assignment#1
ATS761:Land-AtmosphereInteractions
h = −π +
DueTuesSept20
LST
2 π 24
whereLSTislocalsolartime.
Declination(latitudeofthesub-Solarpoint)isgivenby
δ = ε sin θ whereeisthetiltoftheEarth’saxis(currentlye=0.4091radians)andqis
theangular(seasonal)positionofEarthinit’sorbit.Theseasonalposition
angleq ismeasuredfromthevernalequinox(March21),sothat θ = π 2 at
theJunesolstice, θ = π attheautumnalequinox(Sept21),and θ = 3π 2 at
theDecembersolstice.
TheradiusvectordependsonboththepositionoftheEarthinitsorbitand
the“shape”oftheorbititselfaccordingto:
!
$
1
R
(1− e2 )
=# 0 &=
rv " RE % [1+ ecos(θ − ϖ )]
whereR0istheaverageSun-Earthdistance,REistheactualSun-Earth
distance,eistheeccentricityofEarth’sorbit(deviationfromaperfectcircle),
q istheseasonalpositionangle(seeabove),and ϖ istheanglebetweenthe
orbitalpositionoftheEarth’sperihelionandthevernalequinox.Currently,
e=0.0167and ϖ =1.7963.Seehttp://en.wikipedia.org/wiki/Insolationfora
derivationanddiscussion.
Thesolarbeamisalsoattenuatedbyatmosphericscatteringandabsorption.
Inaclearsky,incomingsolarradiationattheEarth’ssurfacecanbe
approximatedas:
(Bonaneqn4.6)
S = SH τ m whereSHistheincidentsolarradiationonahorizontalsurfaceabovethe
atmosphere(definedabove),tistheatmospherictransmissivity,andmisthe
airmassthroughwhichthesolarbeammustpass.Forthisproblem,assume
t=0.7.Theairmassis
1 P
(Bonaneqn4.7)
m=
cos Z PS
2
Assignment#1
ATS761:Land-AtmosphereInteractions
DueTuesSept20
wherecosZisthecosineofthesolarzenithangle(definedabove),Pis
atmosphericpressureandPS=1013.25mbistheatmosphericpressureat
sealevel.
2. Writeasimplecomputerprogramtocalculatetheaverage“daily”
variationofsurfacetemperatureontheEquatoroftheMoon!
Here’show:
TheMoonis“tidallylocked”totheEarthsothatthetimerequiredtorotate
onceonitsaxisisexactlythesameastheperiodofitsorbitaroundtheEarth
(thatis,thelengthoftheMoon’s“day”isthesameasthelengthofaLunar
“month:”29Earthdays,12hours,44minutes).
TheMoonhasnoatmosphere,andthe“tilt”ofitsaxisrelativetotheorbital
planeoftheEarth-MoonsystemaroundtheSunisnegligible.Thereforethe
calculationofinsolationattheLunarsurfaceismuchsimplerthantheone
youdidfortheEarthinproblem1above.AssumingthatthetiltoftheMoon’s
axisiszero,d=0.AccordingtoBonan’sequation4.1(above)then,atthe
Moon’sequatorthecosineofthesolarzenithangleisjustequaltothecosine
ofthehourangle.MuchsimplerthantheEarth!
OnEarththesurfaceenergybudgetis
ΔT
ρc
Δz = ( S ↓−S ↑+L ↓−L ↑) − H − λ E = G (Bonaneqn13.1)
Δt
Ontheleft-handside,risthedensityofthesoil(kgm-3),cistheheat
capacityofthesoil(Jm-3K-1),DTisthechangeinsoiltemperature(Kelvin)
overatimestep(Dt,seconds)foreachsoillayerwhosethicknessisDz
(meters).Ontheright-handsidethefourtermsinsidetheparenthesesare
shortwaveradiationdownandup,andlongwaveradiationdownandup.His
theturbulentfluxofsensibleheat,listhelatentheatofevaporationofwater,
Eistherateofevaporationofwaterfromthesurface,andGisthechangein
groundheatstorage.AlltermsinthisequationhavetheunitsofWm-2.
TheMoon’ssurfaceenergybudgetismuchsimpler!There’snowaterto
evaporateandnoturbulentairtocarrysensibleheat.There’salittlebitof
longwaveradiationemittedbytheEarththatheatstheMoon’ssurface,but
onthefarside(whichweneversee)thisiszerosowecanneglectthis.Allof
theupwardfluxofshortwaveradiationisjusttheincomingsolarradiation
timestheMoon’salbedo.Tocalculateupwardlongwaveemissionwecan
assumethesurfaceemitsasablackbody.SotheMoon’ssurfaceenergy
budgetis
3
Assignment#1
ATS761:Land-AtmosphereInteractions
ρc
ΔT
Δz = S ↓(1− albedo) − σ Ts 4 Δt
DueTuesSept20
Invisiblewavelengths,theMoonissurprisinglydark,withanalbedoofjust
7%(likeasphalt!).AcrossthewholeSolarspectrumit’salittlebrighter,so
pleaseusethebroadbandalbedoof11%.
Tocalculatethetemperature,dividetheLunar“soil”into15layers.Makethe
toplayerthinnest(0.02m=2cm),andthenmakeeachlayerunderneath1.5
timesasthickastheoneabove.Thiswillgetyou15lunarsoiltemperatures
downtoabout17.5metersdepth.
Thedownwardfluxofheatatthetopofeachlayerisproportionaltothe
differenceintemperaturebetweenadjacentlayers:
ΔT
F ↑= −k
(Bonaneqn9.1)
Δz
Theproportionalityconstantkiscalledthe“thermalconductivity.”The
negativesignistherebecausethelayersarecounteddownfromthesurface
buttheheatfluxisdefinedaspositiveupward,awayfromthesurface.
Changesinsoiltemperaturearedeterminedbythedifferenceintheheatflux
intoandoutofeachlayer.Thisiscalledthe“heatfluxdivergence:”
ΔT
ρc
Δz = −(Fz − Fz+Δz )
(Bonaneqn9.2)
Δt
There’saniceworkedexampleofthisonBonanpage133.Thebookalso
givestypicalvaluesofheatcapacityandthermalconductivityfordifferent
kindsofsoil.
TheLunar“soil”isactuallymadeupofmineralgrainsblastedtosmithereens
bybillionsofyearsofimpactbombardment.UnlikeEarthsoils,thepore
spaceisnotfilledwithwaterorair.It’sjustvacuum,sotheheatflowthrough
thelunarsoilismuchslowerthanonEarth,dependingonconductanceand
thermalradiationbetweenadjacentgrains!
AccordingtoColozza(1991),thevolumetricheatcapacityoflunarregolithis
about4.4millionJm-3K-1.Usethisvalueforrcintheequationabove.The
thermalconductivityoflunar“soil”isonlyabout0.025Wm-1K-1.Thisis
about100timeslessconductivethansoilsonEarth!
Tocalculatethechangesintemperaturewithdepthandtimeyou’llneedto
addsolarradiationtothesurface,subtractupwardlongwave,andstepthe
heatfluxequationsforwardintime.You’llfindthatifyouuseatimestep
4
Assignment#1
ATS761:Land-AtmosphereInteractions
DueTuesSept20
that’stoolong,thenumericalsolutionwillbecomeunstableandyou’llget
crazyvalues(forexamplenegativetemperaturesKelvin).IfindIcanget
awaywithatimestepofaboutanhour.
5