6. Budyko’s model
Inthemodelsthatwehavestudiedsofar,wehaveexploredhowradiationcontributestothe
energy budget of earth. In Model 3 we studied incoming solar radiation, in Model 4 we studied the
balance between solar radiation and terrestrial radiation on an earth-like planet with no atmosphere,
andinModel5wesawhowradiationistransferredupanddownintheatmosphere.However,inthe
realworld,heatisalsotransportedtowardsthepolesbytheatmosphereandtheocean.Ourgoalinthis
modelistoaccountforbothradiationandpolewardheattransportbytheoceanandatmosphere,and
predictdifferenttemperaturesatdifferentlatitudes.Wewillalsoseehowearth’ssurfacetemperature
respondstofluctuationsinthesolarconstant.
The single dimension of the model is latitude. This model is still a box model, but instead of
havingtheboxescorrespondtolayersintheatmosphere,wenowconsidertheboxestoliesidebyside
in the horizontal corresponding to different latitudes. Take boxes, or cells, of width 10o centered on
latitudes5o,15o,…,85o,foratotalof9cells.Wewilltreatonlythenorthernhemisphere.Foreachcell
wewanttodeterminethetemperature.Todoso,wekeeptrackofthefluxesofheatintoandoutof
eachcell.Considerthe𝑛thcell.Ithasthefollowingheatfluxes,whicharealsoshownschematicallyin
Figure6.2:
1. NetShortwaveRadiation
Foragivenlatitude,thenetfluxofshortwave,orsolar,radiationisgivenby
𝐹!" = 𝑆(1 − 𝛼)
where𝑆istheannual-averagefluxofshortwaveradiationreachingthesurfaceand𝛼isthealbedoof
thesurface.For𝑆,referbacktoModel3,andchooseavalueforeachofthe9boxes.Rememberthat,in
Model3,wecomputedthedownwardfluxofshortwaveradiationatthetopoftheatmosphere.About
onethirdofthisradiationisreflectedbycloudsbeforereachingtheearth’ssurface,sobesuretoscale
thevaluesthatyougetfromModel3accordingly.ForthosewhodidnotcompleteModel3,thereare
valuesof𝑆fromthatmodelthatyoucanuse,andsatelliteobservationsforcomparison,inBox6.1.
Foralbedo,weuseasimplefunctionoftemperature:
𝛼=
0.3
0.65
𝑇 ≥ 0°𝐶
𝑇 < 0°𝐶
Ifthesurfaceiscolderthan0°Cthenitisassumedtobeice-coveredandbright.Ontheotherhand,ifthe
surfaceiswarmerthan0°Cthenitisassumedtobeocean-orland-coveredanddark.
Wewillalsowanttoconsiderwarmerandcoolerclimates.Onewaytodothisistomultiply𝑆at
all latitudes by the same scale factor 𝑠𝑓. A value 𝑠𝑓 > 1 will produce a warmer climate and 𝑠𝑓 < 1 a
coolerclimate.
Box6.1|ObservationsofDownwellingShortwaveRadiationattheSurface
Figure 6.1 shows an estimate of the downward flux of shortwave radiation at the surface of earth, 𝑆,
fromsatelliteobservations.These estimatesare made by taking satellite measurementsofradiationand cloud
propertiesatthetopoftheatmosphereandfeedingthemintoaradiativetransfermodelthatissimilarto,but
more sophisticated than, our model in Model 5. Figure 6.1b also shows an estimate of 𝑆 from our Model 3,
DistributionofSolarRadiation.Thisestimatewascomputedasfollows:
1. Thesolarzenithangle𝛾,ortheanglebetweenthesunandthelocalverticaldirection,wascomputedfor
differentlatitudes,timesofdayandtimesofyearfollowingtheprocedureofModel3.
2. Usingthesevalues,𝑆wascomputedas
𝑆 = max (0, 𝑆! cos(𝛾) (1 − 𝛼!"#$% ))
where𝑆! = 1400W/m2isthesolarconstant,orthedownwardfluxofshortwaveradiationatthetopof
theatmospherewhenthesunisdirectlyoverhead,andthefactor𝛼!"#$% = 0.3isincludedtoaccountfor
thefactthataboutonethirdofthedownwardfluxofshortwaveradiationatthetopoftheatmosphere
isreflectedbycloudsbeforereachingthesurface.𝑆issettozeroatnight.
3. 𝑆wasaveragedoverthecourseofoneyear.𝑆valuesareshowninTable6.1forthosewhowishtouse
themwithoutcompletingModel3.
Figure 6.1 Estimate of the downward flux of shortwave radiation at the surface of earth from satellite
observations.Averagesaretakenbetween2000-2015.(a)showsamapofannual-averagevalues,and(b)shows
annual-, December- and June-averages averaged over circles on constant latitude. (b) Also shows an estimate
basedoncalculationsfromModel3,DistributionofSolarRadiation.DatafromCERES2.
Latitude(°N):
𝑆(W/m2)
5
299
15
290
25
274
35
251
45
221
55
187
65
153
75
132
Table6.1Estimateof𝑆fromthecalculationsinModel3,DistributionofSolarRadiation.
85
123
2. NetLongwaveRadiation
TheoutgoinglongwaveradiationcomesfromthelinearapproximationoftheStefan-Boltzmann
law, 𝐴 + 𝐵𝑇, with 𝑇 expressed in units of [°C]. A fraction, 𝛽, of the outgoing longwave radiation is
absorbedbytheatmosphereandre-radiatedbacktothesurface.Thenetfluxoflongwaveradiationis
therefore:
𝐹!" = (𝛽 − 1)(𝐴 + 𝐵𝑇).
Wewilluseavalueof𝛽 = 2/3.Youmaywanttoverifyorchangethisvaluebasedonresults
fromourone-dimensionalradiativetransfermodelfromModel5.
3. Fluxesofheatfromadjacentcells
Ifneighboringcellshavedifferenttemperatures,heatwillflowfromthewarmertothecooler
cell. This heat transport is done by the ocean and the atmosphere. Following Budyko’s lead1, we will
assumethattheheatfluxisproportionaltothetemperaturedifferencebetweenadjacentcellsandthe
lengthoftheboundarybetweenthetwocells.Thereisadetailthatmustbefacedhere.Considertwo
adjacentcells,say𝑛 − 1and𝑛,centeredonlatitudes𝜃!!! and𝜃! .Thelengthoftheboundarybetween
thecellsisproportionaltocos((𝜃!!! + 𝜃! )/2).Theareaofthecell𝑛isproportionaltocos(𝜃! ).Sothe
heat flux per unit area of cell 𝑛 − 1 must have a factor cos((𝜃!!! + 𝜃! )/2) in the numerator and
cos(𝜃!!! ) in the denominator. The same flux per unit area of cell 𝑛 will have a factor cos(𝜃! ) in the
denominator.Thebottomlineisthatthe𝑛thcellhas:
•
fluxinfromthelowerlatitudes:𝐹!" = 𝑘
•
fluxouttohigherlatitudes:𝐹!"# = −𝑘
(!!!! !!! ) !"#((!!!! !!! )/!)
!"#(!! )
(!! !!!!! ) !"#((!! !!!!! )/!)
!"#(!! )
where𝑘isaconstant.Thecellnearesttotheequatorwillonlyhaveheatflowingouttohigherlatitudes,
andthecellnearesttothepolewillonlyhaveheatflowinginfromlowerlatitudes.
Net SW
FSW = S15 (1
Net LW
FLW = (
↵15 )
Heat Transport In
Fin /
(T5
=) Fin = k
5°
L10 / cos(10o )
(T5
Heat Transport Out
T15 )L10
A15
Fout /
T15 ) cos(10o )
cos(15o )
10°
1)(A + BT15 )
=) Fout =
15°
(T15
k
T25 )L20
A15
(T15
20°
o
A15 / cos(15 )
T25 ) cos(20o )
cos(15o )
25°
L20 / cos(20o )
Figure 6.2 Schematic diagram of heat flux terms in the model for the cell centered at 15° latitude. 𝐿
standsforthelengthoftheboundarybetweentwocells,and𝐴standsfortheareaofthecell.Subscripts
correspondtolatitudes.“∝”means“proportionalto.”
Set up your model with cells centered on latitudes of 5°, 15°, … 85°. Start with an initial
temperature at each cell and assume a value of 𝑘 (a reasonable starting value could be 𝑘 = 10). The
proportionalityconstant𝑘determineshoweffectivelytheatmosphereandoceantransportheatfrom
lowlatitudestohighlatitudes.Then,followthesesteps:
1. Nowthatyouhavetemperaturesforallcells,compute𝐹!" ,𝐹!" ,and𝐹!"# foreachcell.
2. Compute the net heat flux into each cell, 𝐹!"# , by adding the longwave, shortwave, and
transportterms:
𝐹!"# = 𝐹!" + 𝐹!" + 𝐹!" + 𝐹!"# 3. Adjustthetemperatureofeachcellbasedonthenetheatflux:
𝑇 𝑡 + 1 = 𝑇 𝑡 + 𝑐𝐹!"# 𝑇 𝑡 + 1 istheupdatedtemperatureatthenexttimestep,𝑡 + 1,and𝑇(𝑡)isthetemperature
atthecurrenttimestep,𝑡.Theheatcapacity,𝑐,determineshowquicklyacellwillwarmwhen
there is a net flux of heat into the cell. Because we are only concerned about equilibrium
solutionsinthismodel,where𝐹!"# = 0 Wm!! ,wewillset𝑐 = 0.02andnotworryaboutit.
4. Repeat steps 1-3 until 𝐹!"# ≈ 0 W/m2 in every cell. If you find that your model produces
solutionswhere𝐹!"# growstoinfinity,tryusingasmallervalueof𝑘.
Box6.2|ObservationsofSeaSurfaceTemperature
Figure 6.3 Observed sea surface temperature averaged between 1982-2013 for comparison with Budyko’s
model.Observationswerecollectedfromsatellites,shipsandbuoys.DatafromNOAAOISST3.
Exercises
6.1Torunthemodelrequiresthatwespecifyvaluesforthemodelparameters.Makealistofmodel
parameters, their values, and the range of uncertainty. It is almost impossible to resist the
temptationtochooseparametersthatwillmakethemodelresultslookgood.
Supposethetransportcoefficient,𝑘, is zero.Thenthenineequationsareuncoupled.Infact,they
arenine,non-interacting,zero-Dmodels(seeModel4).Thisisn’tveryinterestingfromageophysical
point of view, but it is a nice test for internal consistency. Set 𝑘 = 0 and run to equilibrium.
Comparethezero-DresultswiththefullBudykomodel.
6.2 Find a way to run the model while slowly decreasing 𝑆. By slowly decreasing, we mean that 𝑆
decreases slowly enough that the state is always nearly in equilibrium. For example, you could
define𝑆! = (𝑠𝑓)𝑆
!!!
!
where𝑁isalarge,positiveinteger.Whatwillhappen?
6.3 Nowlet𝑆varybyincrementingthescalefactor𝑠𝑓.Findparametervaluesthatsupportatemperate
climate:warmatlowlatitudesandcoldathighlatitudes.Doesthetemperateclimateresemblethe
observationsshowninBox6.2?
6.4 InBudyko’s1969paper1,hefoundthatthepresentmodelclimatewasquitesensitivetovariations
insolarradiation.Ifhereducedthesolarconstantby2%,themodeliceedgemovedsouthplunging
earth into a totally ice covered climate, or “snowball earth.” Does your model behave this way?
Whatisitaboutthemodelthatmakesitsosensitive?
6.5 Can the model planet recover from a transition to a snowball state? Use your model to find out.
Supposethetransitionfromthepresenttemperatestatetothesnowballstateoccursatacertain
scale factor, 𝑠𝑓!"# . How much must you increase 𝑠𝑓 beyond 𝑠𝑓!"# to cause a transition from
snowballtotemperateclimate?
References:
1.Budyko,Mikhail,1969:“TheeffectofsolarradiationvariationsontheclimateoftheEarth”Tellus,21,
pp.611-619.
2.CERESScienceTeam,Hampton,VA,USA:NASAAtmosphericScienceDataCenter(ASDC),Accessed
August10,2016atdoi:10.5067/Terra+Aqua/CERES/EBAF-TOA_L3B.002.8(dataavailableonlineat:
https://ceres.larc.nasa.gov/order_data.php)
3. Reynolds, R. W., T. M. Smith, C. Liu, D. B. Chelton, K. S. Casey, and M. G. Schlax, 2007:“Daily highresolution-blended analyses for sea surface temperature”Journal of Climate,20, 5473–
5496,doi:10.1175/JCLI-D-14-00293.1 (data available online at: https://www.ncdc.noaa.gov/oisst/dataaccess)