Lecture2Topics

•  PartI:Radia4ve-convec4veequilibrium(RCE);convec4vequasi-equilibrium(QCE)

•  PartII:Convec4vecri4calityandcolumnwatervaporprecipita4onsta4s4cs(Kathleen)

Ver4caltemperatureprofile

Whatdeterminesthisprofile?

NOAA

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.”

Cloudpopula4on

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: