lecture 2 topics - rutgers universityenvsci.rutgers.edu/~lintner/tropconv_lecture2.pdf ·...
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Lecture2Topics
• PartI:Radia4ve-convec4veequilibrium(RCE);convec4vequasi-equilibrium(QCE)
• PartII:Convec4vecri4calityandcolumnwatervaporprecipita4onsta4s4cs(Kathleen)
ThermalemissionandthePlanckfunc4on
€
R(λ,T) =2hc 2
λ5(ehc /λkBT −1)
h [Planck’s constant] = 6.625 x 10-34 J s-1
c [speed of light] = 3.0 x 108 m s-1
kB [Boltzmann’s constant] = 1.38 x 10-23 J K-1
The Planck function determines the amount of energy per unit time per area per wavelength radiated by an object at temperature T:
Sun
Simpleradia4vebalanceforaplanet(withnoatmosphere)
Integra4ngthePlanckfunc4onoverallwavelengthsandmul4plyingbythesurfaceareaofaplanetofradiusR:
ForaplanetaryalbedoawithasolarfluxdensityS0, the absorbedincomingsolarradia4onis:Atthermalequilibrium:
ForparametersappropriatetoEarth,thisyieldsaneffec4veemission(surface)temperatureof255K(-18°C),comparedtoanobservedsurfacetemperatureof288K(15°C).
F = 4πR2σT 4
Sabs = S0 (1− a)πR2
F = Sabs ⇒ Te = [S0 (1− a) / (4σ )]1/4
σ=5.67x10-8Wm-2K-4
Meantroposphericcomposi4onConstituent % by volume
of dry air N2 78.08
O2 20.95
CO2 0.033 ↑
Ar 0.934
Ne 1.82 x 10-3
He 5.24 x 10-4
CH4 2.0 x 10-4 ↑
Kr 1.14 x 10-4
N2O 5.0 x 10-5 ↑
H2 5.0 x 10-5
Xe 8.7 x 10-6
O3 4.0 x 10-6
H2O 0.0-4.0 [0.8]
F.W. Taylor, Elementary Climate Physics
Atmosphericabsorp4on
Inthevisiblepartoftheelectromagne4cspectrum,Earth’satmosphereiseffec4velytransparent(“solarwindow”).Atshorterandlongerwavelengths,ontheotherhand,theatmosphereistypicallystronglyabsorbing(exceptforthe“atmosphericwindow”).
Turco [2002]
Radia4vebalanceforasinglelayeratmosphere
Ts
Te = Ta
€
σTs4 =σTa
4 +σTe4 ⇒ Ts = 21/ 4Te
Surfacetemperature(~303K)isnowtoolarge…
IncomingsolarOutoingAtmospheric
longwave
OutoingSurfacelongwave
Perfectlyabsorbingatmosphere
Fullradia4vetransfercalcula4on
Pureradia4veequilibriumyieldsanunstableprofileinthetroposphere,althoughitisquitereasonableinthestratosphere.Thever4caltransportofheatviaconvec4oncounteractstheradia4vely-inducedinstabilityinthetroposphere.
Emanuel [2005]
Radia4veconvec4veequilibrium(RCE)
Simula4onsareperformedwitha1Dsinglecolumnmodelwithaconvec4onparameteriza4onandalbedotunedtogiveareasonablesoundingattheequator.Notethatmeansurfacetemperaturesarebelow0polewardof35degrees.RCEpredicts:heightofthetropopause,~constantrela4vehumiditywithwarming,precipita4onequalevapora4on.
Professor Adam Sobel
Convec4onaseventsThisprofilecontainsalargeamountofCAPE,sugges4ngthepossibilityofveryintenseconvec4on.Forconvec4ontoberealized,theCINneedstobeovercome.Weseethissitua4onoversummer4memidla4tudecon4nents.Butinthetropics(especiallyovertheoceans),theatmosphereisfrequentlyquiteclosetomoistadiaba4candneutralstability.
George Craig
Convec4onasasta4s4calequilibrium:convec4vequasi-equilibrium(CQE)
Becausethetropicalfreetroposphereisfrequentlyobservedtobeclosetomoistadiaba4c,itispositedthattheeffectofconvec4onismaintainingamoistadiaba4c(andneutrallystable)tropicalatmospheretroposphere.Theunderlyingno4onofthesta4s4calequilibriumviewofconvec4onisthatthesmall-scalecumulusconvec4oniscomparabletothelarge-scaleprocesseswithwhichitinteracts.Arakawa&Schubert[1974]firstappliedthisno4onasabasisforconvec4veclosure.Convec4veclosurereferstothenecessityincoarseresolu4onmodelsofrepresen4ngtheinherentlysubgrid-scaleconvec4veprocessintermsoftheresolvedgrid-scale[aswewilldiscussfurtherinLecture4].
AkioArakawa
“The2010VilhelmBjerknesMedalisawardedtoAkioArakawainrecogniNonofhispioneeringandfundamentalcontribuNonstophysicallybaseddiscreNsaNontechniquesinatmosphereandoceanmodelsandtorepresenta*onsofconvec*vecloudsinatmospheremodels,andforhisconNnuingworkonbridgingthegapbetweentheresolvedandunresolvedscalesinatmosphericgeneralcirculaNonmodelling.”
Cloudsubensemblesandworkfunc4onsConsideraquan4tyλthatdenotesasubensembleofconvec4veclouds,characterizedbyacommonajributesuchastheal4tudeofthecloudtops.IfMλandBλrepresent,respec4vely,thever4calmassfluxandbuoyancyoftheλsubensemble,the4merateofchangeofenergyconsumed(conversionofpoten4alenergytokine4c)bytheλsubensembleis:
Here,thelimitsofintegra4onarethecloudbaseandtop,Aλisthecloudworkfunc=on,andthesubscriptcdenotesaconvec4ve-(cloud-)scaleprocess.Ontheotherhand,therateatwhichenergyisgenerated,eitherbythelarge-scaleorthroughinterac4onsamongsubensembles,is:
€
−dAλ
dt$
% &
'
( ) C
= MλBλdzzbλ
ztλ
∫
€
dAλ
dt#
$ %
&
' ( LS
= Fλ − (1−δλ + λ ) M + λ Jλ + λ zb + λ
zt + λ
∫ dz
Here,FλistheexplicitLSrateofenergyproduc4onandJλλʹaccountsforinterac4onsbetweentheλandλʹsubensembles. Arakawa & Schubert [1974]
Interac4ontermsInfluenceofver4calmassfluxinλʹsubensemblecloudsonλclouds(top)andλcloudsonλʹclouds(bojom).
Influenceofcloud-topdetrainmentofλʹsubensemblecloudsonλclouds.Notefortheinterac4onasshown,theeffectofλcloudsonλʹcloudsiszero,sincethedetrainmentintheformeroccursabovethelajer.Decrease of cloud work function
Increase of cloud work function
Arakawa & Schubert [1974]
Applica4onofQCECQEimpliesthatthetotal4merateofchangeofthecloudworkfunc4oniszero,implyingtherateofLSenergyproduc4onisequaltoitsconsump4onattheconvec4vescale,i.e.,
dAλdt
=dAλdt
⎛
⎝⎜
⎞
⎠⎟C
+dAλdt
⎛
⎝⎜
⎞
⎠⎟LS
= 0
⇒dAλdt
⎛
⎝⎜
⎞
⎠⎟LS
= −dAλdt
⎛
⎝⎜
⎞
⎠⎟C
€
Kλ # λ = (1−δλ # λ )Jλ # λ + δλ # λ Bλ
Thatis:
€
Fλ = M # λ Kλ # λ zb # λ
zt # λ
∫ dz
wheretheintegra4onkernelKλλʹisdefinedas:
GivenFλandacloudmodelthatdeterminesBλ,Mλcanbedetermined:thisisthebasisofArakawa&Schubert[1974]typeconvec4onschemesusedinmodels.However,thisdetermina4onisnontrivial,asitdependsonthefeaturesofthecloudmodelsandhowtheytreatcloudmicrophysics.
Observational support for CQE Es4matedtendenciesofAfromobserva4onsintheMarshallIslands[Yanai1973]suggestlijle4mevaria4onofworkfunc4on,consistentwiththeCQEhypothesis.CQEisonenframedwithCAPEreplacingtheworkfunc4on.Fortropicalcondi4ons,netsurfacefluxandcolumnradia4vecoolinggenerate~4000Jkg-1day-1,whileCAPEvaluesinthetropicalatmospherearetypicallybelow1000Jkg-1.
Arakawa & Schubert [1974]
Cri4cismsorlimitsofCQECausality:Theformula4onofCQEisonenunderstoodintermsofthelarge-scaleforcingconvec4on(thermodynamicanalogytoasystemrespondingtoitsenvironment),butwhatabouttheoppositedirec4on,namelyconvec4onforcingthelarge-scale?Lackof(evidencefor)scalesepara=on:Temporally,theconvec4ve4mescaleshouldbeshortcomparedto4mescaleofevolu4onofthelarge-scale.Also,thesta4s4calviewsuggeststhatconvec4veelementsshouldbesufficientlysmallcomparedtothedomainsize,sothatalargenumbercanexistwithinthedomain.Applicabilitytoland/midla=tudes:Thepresenceofastrongboundarylayercycle(withconvec4onhappeningpreferen4allyinlateanernoon)suggestsatriggerisrequiredtorealizecondi4onalinstability.Transi=ontostrongconvec=on:Theindica4onofaclearonsetthreshold(inmoisture,asaproxyforinstability)challengestheequilibriumview.
Interpreta4onsandalterna4vestoCQE*
*FollowingheretherecentreviewbyYano&Plant[2012].Schema4cslightlyembellished.
/WTG
QTCM
Interpreta4ons
Alterna4ves
Rela4ngABLθeandcumulusconvec4on
ConsidertheCAPEdefinedas:.Usingthedefini4onofspecificvolumeandtransformingtheintegraltooneoverpressure[throughhydrosta4cequilibrium]gives:
Wecanrelateconstant-pressureperturba4onsinparcelspecificvolumetoperturba4onsinentropyviaMaxwell’srela4ons.Thus,
∂α∂t
!
"#
$
%&p
= limt→0
δα( )pδt
= limt→0
∂α∂s*!
"#
$
%&p
δs*δt
=∂T∂ p!
"#
$
%&s*
∂s*∂t
Ontheotherhand,fromthehydrosta4crela4onship,expressedintermsofgeopoten4al:
CAPE = gρ−1( ʹρ − ρ)dzzLFC
zEL∫
CAPE = (α −α')dppEL
pLFC
∫UnderCQE,wehave:
∂CAPE∂t
= (∂α∂t
−∂α'∂t)dp ≅ 0
pEL
pLFC
∫
s*= cP lnTT0
!
"#
$
%&− Rd ln
pp0
!
"#
$
%&+
LcqsT
∂α'∂t
⎛
⎝⎜
⎞
⎠⎟p
=∂∂t
−∂Φ∂ p
⎛
⎝⎜
⎞
⎠⎟
Emanuel et al. [1994]
ABLθeandcumulusconvec4on(cont’d)
Thus:
Wehaveassumedsatura4onentropy[orcplnθe]isisconservedfollowingascentwithintheconvec4ngcloudandcanthereforebedescribedbyitsABLvalue.Theaboverela4onship,namelythattemporalfluctua4onsinthe“thickness”oftheCQEconvec4nglayerarepropor4onaltoABLfluctua4onsinmoistentropy/equivalentpoten4altemperature,isafundamentalresult.Itimpliesthataposi4veperturba4oninABLθeproducesaposi4veperturba4oninthicknessabovetheABL(note:ΦEL-ΦLFC>0).Thus,toalargeextentintheTropics,predic4ngtheresponseofABLθetodisturbancesdeterminestheimpactofsuchdisturbancesonconvec4on.
And:
∂CAPE∂t
=∂T∂ p!
"#
$
%&s*
∂s*∂t
+∂∂t
∂Φ∂ p!
"#
$
%&
(
)*
+
,-dp ≅ 0
pEL
pLFC
∫
€
∂ cp lnθe( )b∂t
TLFC −TEL( ) +∂∂t
ΦLFC −ΦEL( ) ≈ 0
Emanuel et al. [1994]
Emanuel et al. [1994]
€
lnθe( )b eq ≅1
w0 + we + Md
×
w0 ln(θe )s + we ln(θe )e + Md ln(θe )d +zb
˙ Q rad
cpT
%
& '
(
) *
Equilibrium subcloud θe increases with: 1. Increasing SST [I.e., higher θes ] 2. Increasing surface windspeed [I.e., higher w0] 3. Increasing above subcloud layer equivalent potential temperature [I.e., higher θee] 4. Increasing downdraft equivalent potential temperature [I.e., higher θed]
ABLθeandcumulusconvec4on(cont’d)Therela4onshipderivedonthepreviousslideallowsustocasttheques4onofcontrolsonconvec4onintermsofcontrolsonθe: