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ThegeneralcirculationoftheatmosphereSectionII:Theangular-momentumbudget

Theatmosphericangular-momentumbudget

Previously

Timemean:

�� = $% ∫ 𝐴𝑑𝑡%

* ,𝐴+ = 𝐴 − ��,-.-/= 0

Zonalmean:

𝐴 = $12 ∫ 𝐴𝑑𝜆12

* ,𝐴∗ = 𝐴 − [𝐴],-[.]-7

= 0

Decompositionofthecirculation

𝐴𝐵 = �� 𝐵9 + ��∗𝐵9∗ + 𝐴′𝐵′

Thezonal-meanmidlatitudecirculation

(NCEPreanalysis1979-2016)

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(ERA40 reanalysis 1980-2001)

Mean meridional streamfunction (1010 kg s-1)

LatitudeSigm

a

10

2

2

−8

−4

−60 −30 0 30 60

0.2

0.8

Latitude

Sigm

a 0.5

1

−60 −30 0 30 60

0.2

0.8

Contour interval 2

Contour interval 0.5

courtesyofPaulO’Gorman

ANRV273-EA34-21 ARI 17 April 2006 23:56

1. INTRODUCTIONIn the mean, zonal surface winds on Earth are easterly (westward) in low latitudes,westerly (eastward) in midlatitudes, and easterly or nearly vanishing in high latitudes.The strength of the mean zonal surface wind varies seasonally, but the pattern ofalternating easterlies and westerlies is present throughout the year, with slight seasonalshifts of the latitudes at which the mean zonal surface wind changes sign (Figure 1bshows January as an example). The mean meridional surface wind is weaker than themean zonal surface wind. It is directed poleward in regions of surface westerlies andequatorward in regions of surface easterlies. In boreal summer, the monsoons of theNorthern Hemisphere lead to a mean northward surface wind across the equator,which typically has a westerly component in monsoon regions.

That mean surface winds have definite directions has been exploited in centuriespast by navigators, who called winds with a prevalent direction trade winds, a term wenow use more restrictively to denote the tropical easterly winds. “The Causes of the

Sig

ma

300

350

270

280

32 440.2

0.8

40

Latitude

b

0˚ 50˚

0

8

Eas

twar

d s

tres

s (P

a)

Latitude

d

0˚ 50˚

0

0.2

Hadley cellsFerrel cells

Easterlies

WesterliesWesterliesEddies

Mean

a c

Figure 1Temporal and zonal mean circulation statistics for January according to reanalysis data for theyears 1980–2001 (Kallberg et al. 2004). (a) Zonal wind (magenta) and potential temperature(light blue). Contour intervals are 4 m s−1 for zonal wind and 10 K for potential temperature.The thick magenta line is the zero zonal wind contour. (b) Zonal wind at the surface.(c) Eulerian mass flux streamfunction (magenta) and angular momentum (light blue). Contourintervals are 20 × 109 kg s−1 for streamfunction and 0.1!a2 for angular momentum, withangular momentum decreasing monotonically from the equator to the poles. Negativestreamfunction values (dashed contours) correspond to clockwise rotation, positive values (solidcontours) to counterclockwise rotation. (d ) Vertically integrated momentum flux convergence(eastward stress) due to mean circulations (light blue) and due to eddies (magenta), with eddiesdefined as fluctuations about the temporal and zonal mean. The vertical coordinate in (a) and(c) is σ = p/ps (pressure p normalized by surface pressure ps).

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Kineticenergyperunitmass(m2 s-2)

(Peixoto and Oort, fig 7.22)

Kinetic energy (m s-2)

Transient

Stationary

Mean

Total

Zonal-meanmidlatitudecirculation

• Strongwesterliesaloft• Stronglatitudinaltemperaturegradients• Westerlywindsatthesurface• Thermallyindirectmeanoverturningcirculation• Strongeddykineticenergymaximum

Why?

Temperatureandzonal-windarerelatedbythethermalwindrelation

Considerthemeridionalprimitiveequationinpressurecoordinates:

𝐷𝑣𝐷𝑡

= −2Ω sin𝜙 𝑢 −𝑢1

𝑅Etan𝜙 −

1𝑅E𝜕Φ𝜕𝜙

+ 𝐹M

Outsidethedeeptropics,thedominantbalanceisgeostrophic,

2Ω sin𝜙 𝑢 = −1𝑅E𝜕Φ𝜕𝜙

Bytakingtheverticalderivativeandusinghydrostaticbalance,wehave,

2Ω sin𝜙𝜕𝑢𝜕𝑝

=1𝑅E𝜕𝛼𝜕𝜙

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Temperatureandzonal-windarerelatedbythethermalwindrelation

Usingtheidealgaslaw,wehave,

2Ωsin𝜙𝜕𝑢𝜕𝑝 =

𝑅𝑝𝑅E

𝜕𝑇𝜕𝜙

(Sinceinpressurecoordinates-Q-M= 0)

Inmorefamiliarnotation:

𝜕𝑢𝜕𝑝 =

𝑅𝑝𝑓𝜕𝑇𝜕𝑦

Temperatureandzonal-windarerelatedbythethermalwindrelation

𝜕𝑢𝜕𝑝

=𝑅𝑝𝑓𝜕𝑇𝜕𝑦

• Strongestuppertroposphericwindsoccurintheregionofstrongesttropospherictemperaturegradients.

• Thisdoesnotexplainthesurfacewind

• Nordoesitexplainthelocalisationofjets

Zonal-meanmidlatitudecirculation

• Strongwesterliesaloft• Stronglatitudinaltemperaturegradients• Westerlywindsatthesurface• Thermallyindirectmeanoverturningcirculation• Strongeddykineticenergymaximum

Why?

thermalwind

balanceAllthesepropertiesarerelated

Considerableinsightmaybegainedbyconsideringthebudgetofangularmomentumintheatmosphere

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Angularmomentum

• Angularmomentumisdefinedasthecrossproductofthemomentumandamomentarm.

• Thatis,angularmomentumperunitmassisdefined

𝑴 = 𝒓×𝒖𝒊

• Whenactedonbyaforce(perunitmass)𝑭 evolvesaccordingto

𝐷𝑴𝐷𝑡

= 𝒓×𝑭 = 𝝉

AngularmomentumabouttheEarth’saxis

TakethemomentarmfromthecentreoftheEarth

𝑴 = 𝒓×(𝛀×𝒓 + 𝒖)

Weareinterestedinthecomponentofangularmomentuminthedirectionoftherotationvector:

𝑀 = Ω𝑟1 cos1 𝜙 + 𝑢𝑟 cos𝜙

𝛀|𝒓c| = 𝑟 cos𝜙

𝒓c

Thinshellapproximation

Weapplythethinshellapproximationsothatwecanreplace𝑟 inthedefinitionofangularmomentumbytheEarth’sradius:

𝑀 = Ω𝑅E1 cos1 𝜙 + 𝑢𝑅E cos𝜙

planetaryangularmomentum

relativeangularmomentum

Conservationofangularmomentum

Recalltheprimitiveequationforzonalvelocityinpressurecoordinates:

𝐷𝑢𝐷𝑡 = 2Ωsin𝜙 𝑣 +

𝑢𝑣𝑅Etan𝜙 −

1𝑅E cos𝜙

𝜕Φ𝜕𝜆 + 𝐹7

Canexpressthisas:

𝐷𝑀𝐷𝑡 = −

𝜕Φ𝜕𝜆 + 𝑅E cos𝜙𝐹7

Intheabsenceofpressuregradientandfrictionalforces,wehave,

𝐷𝑀𝐷𝑡 = 0.

Angular-momentumisconservedalongparcelpaths!

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Relativeangularmomentum

Dividetheangularmomentumintoaplanetaryandrelativecomponent:

𝑀Q = Ω𝑅E1 cos1 𝜙 𝑀e = 𝑢𝑅E cos𝜙

Notethat:fgh

f/= 𝑅E cos𝜙 (𝑓𝑣)

𝐷𝑀e

𝐷𝑡= −

𝜕Φ𝜕𝜆

+ 𝑅E cos𝜙 (𝑓𝑣 + 𝐹7)

Relativeangularmomentumbudget

Wecanalsowritethisequationinfluxform(easyinpressurecoordinates!):

𝜕𝑀e𝜕𝑡

+ 𝛻 ⋅ (𝒖𝑀e) = −𝜕Φ𝜕𝜆

+ 𝑅E cos𝜙(𝑓𝑣 +𝐹7).

Considerfirsttheuppertroposphere,wherefrictionissmall.Takingthezonalandtimemeanwehave

𝛻 ⋅ ([𝒖𝑀e]) = 𝑅E cos𝜙𝑓[ 𝑣]

Theaboveequationmaybewritten,

1𝑅E cos1 𝜙

𝜕 cos1 𝜙 [𝑢𝑣]𝜕𝜙

+𝜕[𝑢𝜔]𝜕𝑝

= 𝑓[𝑣]

Thisisanequationforthesteady-statemeridionalflowintheuppertroposphere!

Decomposingthisintomeanandeddy,wehave,

𝑓 𝑣

=1

𝑅E cos𝜙𝜕 cos1 𝜙 𝑢 𝑣

𝜕𝜙 +𝜕 𝑢 𝜔𝜕𝑝

+1

𝑅E cos1 𝜙𝜕 cos1 𝜙 [𝑢′𝑣′]

𝜕𝜙 +𝜕[𝑢+𝜔′]𝜕𝑝

+1

𝑅E cos1 𝜙𝜕 cos1 𝜙 [𝑢∗𝑣∗]

𝜕𝜙 +𝜕[𝑢∗𝜔∗]

𝜕𝑝

mean

transienteddy

stationaryeddy

Usingthetime- andzonal-meancontinuityequation,andnotingthattheverticaladvectionofrelativeangularmomentumbythemeanflowisweak,wehave,

𝑓 + [𝜁] 𝑣 =1

𝑅E cos1 𝜙𝜕 cos1 𝜙 𝑢+𝑣+

𝜕𝜙+𝜕 𝑢+𝜔+

𝜕𝑝+

1𝑅E cos1 𝜙

𝜕 cos1 𝜙 𝑢∗𝑣∗

𝜕𝜙+𝜕 𝑢∗𝜔∗

𝜕𝑝

Inthemidlatitudes, 𝑓 ≫ 𝜁 (thisisequivalenttoassumingQGscaling),andwecanthereforewrite,

𝑓 𝑣 ≈ −𝑆

WhereSisthedivergenceofzonal(angular)momentumbyeddies.

Meridionalflowdependsoneddymomentumfluxconvergence– notdirectlyonenergytransportrequirements!

Theseeddyfluxesaredominatedbythehorizontalcomponent.

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Northwardfluxofmomentum(m2 s-2)Northward flux of momentum (m2s-2)

Peixoto and Oort, Fig 11.7

Transient

Stationary

Mean

Totaltotal

transienteddy

stationaryeddy

mean

Fluxofmomentumbythecirculation

• Momentumfluxisroughlyantisymmetricbetweenhemispheres• Magnitudeofmomentumfluxpeaksatmidlatitudes• Primarilyaresultoftransienteddies(somestationaryeddiesinNorthernHemisphere)• Impliesconvergenceofmomentumatroughly50degreesineachhemisphere• Bythebalanceequation,thisimpliespolewardflowinthemidlatitudeuppertroposphere!

Schematicofmomentumfluxes

POLEEQUATOR 30deg 60deg

convergence

momflux

divergence

uppertroposphericflow

Eddyfluxes&theFerrelcell

• Theconvergenceof(angular)momentumintothemidlatitudes drivespolewards flowintheuppertropospherecorrespondingtotheupperbranchoftheFerrelcell

• Bymassconservation,werequirethedepthintegratedmeridionalflowtobezero

p 𝑣𝑑𝑝Qq

*= 0

• Wheredoesthisreturnflowoccur?

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EddyfluxesandtheFerrelcell

Multiplybydensityandintegratethebalancevertically

∫ 𝑣 rQs

Qq* ≈ −∫ t

uQq*

rQs.

Butitisclearfromtheobservationalestimatethat,atmidlatitudes t

u> 0,evenintheverticalintegral.

Whatarewemissing?

Friction!

Atlowlevels,thefrictionaltermbecomesimportantandthisallowsthecirculationtoclose

Thustheupper-troposphericeddy-momentumfluxesdeterminethemeridionalflowaloft,andthisdetermineshowthelowlevelfrcitionalflowmustbehave

This“downwardcontrol”principlewasdevelopedfirstinthestratosphericliteraturetoexplaintheBrewer-Dobsoncirculation(Haynesetal.,1991)

Whatdoesthismeanforthesurfacewinds?

Backtotherelativeangularmomentumbudget𝜕𝑀e

𝜕𝑡 + 𝛻 ⋅ (𝒖𝑀e) = −𝜕Φ𝜕𝜆 + 𝑅E cos𝜙(𝑓𝑣 +𝐹7).

Nowwemustconsiderthefrictionalterm.Notethatwecanwritethefrictionalforceasthedivergenceofthestresstensor

𝜌𝑭 = −𝛁𝐳 ⋅ 𝝉

Thezonalcomponentofthisis

𝜌𝐹7 = −1

𝑅Ecos𝜙𝜕𝜏77𝜕𝜆 +

𝜕𝜏7M cos𝜙𝜕𝜙 −

𝜕𝜏7{𝜕𝑧 ≈ −

𝜕𝜏7{𝜕𝑧 = 𝑔

𝜕𝜏7{𝜕𝑝

Relativeangularmomentumbudget

Nowintegratetherelativeangularmomentumequationequationverticallyfromthesurfacetothetopoftheatmosphereandtakethezonal- andtime-mean.

Thelefthandsidebecomes,

1𝑅E cos𝜙

𝜕[∫ 𝑣𝑀eQq* 𝑑𝑝/𝑔 ]cos𝜙

𝜕𝜙

Foreaseofnotation,wecanwritethisas,

p1

𝑅E cos𝜙𝜕[𝑣𝑀e ]cos𝜙

𝜕𝜙

Q�

*

dp𝑔

forsome𝑝* greaterthanthesurfacepressure,whereweset𝒖 = 𝟎 atlevels𝑝� <𝑝 belowthesurface.

Weconsidertheright-handsidetermbyterm

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TheterminvolvingtheCoriolisacceleration,

p 𝑅E cos𝜙𝑓 𝑣Q�

*

𝑑pg= 0

bymassconservation

Thesurfacefrictiontermisgivenby,

𝑅E cos𝜙 p𝜕𝜏7{𝜕𝑝

𝑑𝑝Q�

*= 𝑅E cos𝜙 [𝜏�],

where𝜏� isthefrictionalstressatthesurface

Considerthezonalintegralofthegeopotentialgradientatagivenpressure𝑝 = 𝑝$,

p𝜕Φ𝜕𝜆

𝑑𝜆12

*

Thisiszeroif𝑝$ < 𝑝� atalllongitudes

longitude

pressure

𝑝$

𝑝�

𝜆$ 𝜆10 2𝜋

Otherwise,wemustsplittheintegral

p𝜕Φ𝜕𝜆

𝑑𝜆12

*= p

𝜕Φ𝜕𝜆

𝑑𝜆7�

*+ p

𝜕Φ𝜕𝜆

𝑑𝜆12

7�

= Φ 𝜆$ − Φ 0 + Φ 2𝜋 − Φ 𝜆1

= Φ 𝜆$ − Φ 𝜆1

= Φ� − Φ�

Ifthereismorethanonemountain,wemustsplittheintegral.

longitude

pressure

𝑝$

𝑝�

𝜆$ 𝜆10 2𝜋

Peixoto &Oort Fig.11.5

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Sowecanwritethepressuregradienttermasasumofpressuredifferencesacrossmountainsthatintersectagivenheight,

12𝜋𝑇

p p p𝜕Φ𝜕𝜆

Qq

*𝑑𝑧

12

*

�

*=

12𝜋

p �Φ�� − Φ�

��

�

Qq

*

𝑑𝑝𝑔

ThesepressuredifferencesrepresentaformdragontheatmospherebythesurfaceoftheEarth.

WecanalsouseLeibnizintegralruletoexpressthisformdragpurelyintermsofsurfacequantities

Thepressuregradienttermmaybeexpressedas,

12𝜋𝑇

p p p𝜕Φ𝜕𝜆

Qq

*𝑑𝑝/𝑔

12

*

�

*𝑑𝑡𝑑𝜆

WemayuseLeibnizintegralruletoremovethelongitudinalderivativefromthefirstintegral.Leibnizrulestates

𝑑𝑑𝑥

p 𝑓 𝑥, 𝑡 𝑑𝑡 =� �

*𝑓 𝑥, 𝑏 𝑥 𝑏+ 𝑥 + p

𝑑𝑓𝑑𝑥

𝑑𝑡� �

*

Applyingthistothepressuregradient,wehave,

12𝜋𝑇

p p p𝜕Φ𝜕𝜆

Qq

*𝑑𝑝/𝑔

12

*

�

*𝑑𝜆𝑑𝑡 =

12𝜋𝑇

p p𝜕𝜕𝜆p ΦQq

*

𝑑𝑝𝑔−𝜕𝑝�𝜕𝜆

Φ�

𝑔

12

*

�

*𝑑𝜆𝑑𝑡

Andtherefore,

12𝜋𝑇

p p p𝜕Φ𝜕𝜆

Qq

*𝑑𝑝

12

*𝑑𝜆𝑑𝑡

�

*= −

𝜕𝑝�𝜕𝜆

𝑧�

Puttingthisalltogether,wehave,

∫ $�� ��� M

-[�g� ]��� M-M

Q�*

���= −𝑅E cos 𝜙 𝑝�

-{q-7

+𝑅E cos 𝜙 [𝜏�]

Thisequationstatesthattheconvergenceofangularmomentumintoalatitudebandmustbebalancedbyfrictionaltorquesatthesurfaceandmountaintorques.

Inparticular,weexpectthefrictionalforcetoopposethesurfacezonalwindspeed.Themountaintorquetermisnotassimple,butitgenerallyalsoactsaadragonthesurfacezonalwind.

Neglectingthemountaintorqueswemaywrite,

p1

cos1 𝜙𝜕 cos1 𝜙 [𝑢𝑣]

𝜕𝜙

Q�

*𝑑𝑝/𝑔 = [𝜏�]

Inregionsofzonalmomentumfluxconvergence,thesurfacezonalwindsarewesterly,inregionsofzonalmomentumfluxdivergence,thesurfacezonalwindsareeasterly.

Schematicofmomentumfluxes

POLEEQUATOR 30S 60S

convergence

momflux

divergence

uppertroposphericflow

North

surfacewinds

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Atmosphericangularmomentumcycle

• Theprecedinganalysispaintsapictureinwhichangularmomentumisconvergedintotheuppertroposphereinmidlatitudes,whereitistransporteddownwardsandremovedatthesurfacebyfriction/mountaintorques(westerlywinds)• Thisangularmomentumisprimarilydrawnfromsubtropicallatitudes,whereitisprovidedbyfrictionwiththeEarth’ssurface(easterlywinds)• Notethatthisangular-momentumtransportisupgradient!

Peixoto &Oort

Totalangularmomentumoftheatmosphere

𝜕𝜌𝑀e

𝜕𝑡 + 𝛻 ⋅ (𝜌𝒖𝑀e) = −𝜕𝑝𝜕𝜆 + 𝜌𝑅E cos𝜙(𝑓𝑣 +𝐹7).

Integrateoverwholeatmosphere,

𝑑𝑑𝑡� 𝜌𝑀e𝑑𝑉

 ¡¢£= −p

𝜕𝑧�𝜕𝜆 𝑝�

�

- ¡¢£𝑑𝑆 +p 𝜏�

�

- ¡¢£𝑑𝑆.

AtmosphereexchangesangularmomentumwiththeEarththroughformdragandfriction.Reflectedinchangestothelengthofday.

changeinangularmomentum formdrag friction

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Summary

• Themeanmeridionalcirculationintheextratropicsisdeterminedbythedistributionofeddymomentumfluxesintheuppertroposphere

• Thesurfacewindsaredeterminedbytheverticallyintegratededdyfluxconvergenceordivergenceintoagivenlatitudeband

• Thusinaquasi-geostrophicatmosphere,barotropicjetsarecanonlybemaintainedinthepresenceofananisotropicdistributionofeddies

• Whatdeterminesthedistributionofeddy-momentumfluxesinEarth’satmosphere?

Epilogue:BarotropicvsBaroclinic

Thesepiecesofusefuljargoncomefromthevorticityequation:

𝐷𝜻𝐷𝑡

= 𝜻 ⋅ 𝛁𝐮 − 𝜻 𝛁 ⋅ 𝐮 +1𝜌¦𝛻𝜌×𝛻𝑝 +

1𝜌𝛻×𝑭

Abarotropicfluidhasnobaroclinic productionofvorticity.

Thisrequiresthatisobarsandisopycnalsareparallel

Onewayofachievingthisisifpressureisafunctionofdensityonly

tilting stretching baroclinic friction

Epilogue:BarotropicvsBaroclinic

Foranidealgas,

𝛻𝜌×𝛻𝑝 =𝜌𝑝𝛻𝑝×𝛻𝑝 −

𝑝𝑇𝛻𝑇×𝛻𝑝

Thus,anidealgasisbarotropicifisothermsareparalleltoisobars.

Thatis,iftherearenohorizontaltemperaturegradients(whenworkinginpressurecoordinates)

Epilogue:BarotropicvsBaroclinic

Now,forafluidthatobeysquasi-geostrophicscaling,theflowwillapproximatelysatisfythethermalwindrelation,

𝜕𝒗𝜕𝑝 =

𝑅𝑝𝑓 𝒛

©×𝛁𝑇

where𝒗 isthehorizontalwind.

Sohorizontaltemperaturegradientsareassociatedwithverticalgradientsinthewind.

Abarotropicatmospherethereforehasdepth-independentflow

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Epilogue:BarotropicvsBaroclinic

Moregenerally,wethinkofthedepth-independentcomponentoftheflowasthe“barotropiccomponent”.

Thisoriginatesinanalyticmodelsoftheatmospherebasedonafinitenumberofmodes.Thegravestmodeisthebarotropicmode,thenextmodeisthe“1stbaroclinic mode”etc.

barotropic

1stbaroclinic

2ndbaroclinic

3rdbaroclinic

Epilogue:BarotropicvsBaroclinic

Sometimesyouwillheartheterm“equivalentbarotropic”

Thismeansthattheshearisalwaysinthesamedirectionasthewind(Thewinddoesnotturnwithheight)