29/4/19

1

ThegeneralcirculationoftheatmosphereSectionII:Theangular-momentumbudget

Eddies,theHadleyCell,andmonsooons

RecapoftheHeld&Hou modeloftheHadleycell

• Beginbyconsideringsimplestsolution:radiative-convectiveequilibrium.

• RCEsolutioncannotbemaintainedbecauseitviolatesHide’stheorem

EQ NPSP

RecapoftheHeld&Hou modeloftheHadleycell

• Consideraxisymmetriccirculationforsimplicity

• Assumeairrisesatequatorandmovespolewardattropopause,conservingangularmomentum

EQ POLE

𝑀 = constant

𝑀 = 𝑀#

RecapoftheHeld&Hou modeloftheHadleycell

• Withthermalwindbalance,thisprovidesastrongconstraintonthethermodynamicstructureoftheatmosphere:

𝑠% = 𝑠& −Ω)𝑅+)

2(𝑇% − 𝑇/)sin4 𝜙cos) 𝜙

1. Boundary-layerentropydistributionmustbeveryflatnearequator

2. AMconservingsolutioncannotcontinuetothepole

29/4/19

2

RecapoftheHeld&Hou modeloftheHadleycell

• EnergybalancewithintheHadleycellgivesapredictionofitsextent:

• Furtherpoleward,havetheRCEsolution

Hadleycirculationstrength

IntheHeld&Hou model,thestrengthofthecellisdeterminedbyenergyconservation:

1𝑅+ cos𝜙

𝜕𝐹# cos𝜙𝜕𝜙 = 𝑄 = −

𝑠% − 𝑠%=>#

𝜏

where𝐹# istheenergyfluxbythecell.

Hadleycirculationstrength

SupposetheHadleycellisthinlayersatthesurfaceandthetropopause

𝐹# = 𝜌𝑣𝛿𝑧ℎE − 𝜌𝑣𝛿𝑧ℎ%= ΨGHIΔℎ

EQ POLE

𝛿𝑧

𝛿𝑧

𝑣

−𝑣

WhatisΔℎ?

• Held&Hou consideradryatmosphere,ℎ isthedrystaticenergy.• Δℎ > 0 toensuregravitationalstability• Held&Hou takeΔℎ asaparameter

• Inamoistatmosphere,therelevantvariableismoiststaticenergy• moiststaticenergynon-monotonicwithheight• Size(andsign!)ofΔℎ dependentoncirculation• Willreturntothisquestionwhentalkingaboutenergybudget

29/4/19

3

Held&Hou modeloftheHadleycell

• TheHeld-Houmodelpredicts:• anoverturningcirculationinthetropics• astrongsharpsubtropicaljet• nooverturningfurtherpoleward

(ERA40 reanalysis 1980-2001)

Mean meridional streamfunction (1010 kg s-1)

Latitude

Sigm

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

NumericalcalculationsshowthattheHadleycellstrengthdependsonviscosity!

Whataboutthedescendingbranch?

EQ POLE

𝑀 = constant

𝑀 = 𝑀#

Airmustenterboundarylayerwithzerosurface

wind

29/4/19

4

Whataboutthedescendingbranch?

EQ POLE

𝑀 = constant

𝑀 = 𝑀#

Airmustenterboundarylayerwithzerosurface

wind

• Twoways:• dragslowsdownthedescendingbranch• Thedescendingbranchslopestowardtheequator

Slopingdescendingbranch

comparison to that of large-scale eddies. We confirm thisconclusion by a direct calculation of the momentum fluxconvergence associated with eddies of different wave-lengths below.Figure 5 shows the eddy momentum flux convergence

(EMFC), defined as

s521

r0

›r0u 0y0

›y2

1

r0

›r0u 0w0

›z, (6)

in the wide- and narrow-domain cases for DT5 40K.In the wide-domain simulation, large-scale eddies(.960-km wavelength) act to decelerate the flow in thedescending branch of the HC and accelerate it in themidlatitudes. While there is also some convergence ofmomentum directly over the equator associated with theweak superrotation noted previously, the simulatedlarge-scale EMFC pattern is broadly similar to that of

Earth’s atmosphere during the equinoctial seasons [seeFig. 1 of Levine and Schneider (2011)].To provide a more quantitative comparison, we cal-

culate the mass-weighted mean of the large-scale hori-zontal EMFC between the latitudes of 108 and 258N andbetween 9 and 12 km in altitude (dashed black box inFig. 5a). This quantity has a value of21.4m s21 day21 inthe wide-domain simulation for DT5 40K. A similarmeasure based on pressure levels between 200 and300 hPa in the NCEP reanalysis is given in Fig. 2 ofCaballero (2008); for the Northern Hemisphere duringwinter, this is approximately 22.8m s21 day21 (sum-ming the contributions from transient and stationaryeddies). However, the Northern Hemisphere is stronglyaffected by stationary eddies, while in our zonallysymmetric simulations, stationary eddies are of littleimportance. Performing a similar calculation for theSouthern Hemisphere during spring gives a value

FIG. 5. EMFC (colors) and streamfunction (contours) inthe (a),(b) wide-domain and (c) narrow-domain simulationswith DT5 40K. EMFC associated with (a) large-scale eddies(wavelength . 960 km) and (b) small-scale eddies.

FIG. 4. Time-mean streamfunction C (black), zonal- and time-mean angular momentum M (red), and zonal- and time-meansaturation moist static energy h* (gray) in the (a) wide-domainsimulation and (b) narrow-domain simulation with DT5 40K. Thestreamfunction is defined by (7) and is shownwith contour intervalsof 500 kgm21 s21. The location of the streamfunction extremum ineach hemisphere is marked by a blue dot. Contours are plotted forangular momentum values corresponding to the angular momen-tumof the surface at latitudes of 08,658,6108, etc. Saturationmoiststatic energy is defined as h*5 cpT1F1Lyq*, where q* is thesaturation specific humidity; contours are given for values at whichh*/cp 5 280, 290, . . . , 450K.

JUNE 2016 S I NGH AND KUANG 2433

Singh&Kuang (2017)

high“viscosity”

low“viscosity”

angularmomentumcontours

streamfunction

Whatprocessesslowthedescendingbranch?

Whatprocessesslowthedescendingbranch?

Eddies!

TheHeld-Hou modelisaxisymmetricanddoesnotconsidertheeffectsofeddies

Let’sgobacktoourangularmomentumequationbutincludeeddyfluxes

29/4/19

5

Backtotheangularmomentumbudget

Considertheangularmomentumbudgetinpressurecoordinates.Assumingthepressurelevelweconsiderdoesnotintersectwiththesurfacewehave,

𝐷𝑀𝐷𝑡

= 𝑅+ cos𝜙 𝐹P

where𝑀 = 𝑅+ cos𝜙 Ω𝑅+ cos𝜙 + 𝑢 .

Aswehavedonebefore,wecanseparatemeanadvectionfromtheeddyforcing.Assumingasteadystate,thisgives,

[𝑣]𝑅+

𝜕[𝑀]𝜕𝜙

+ [𝜔]𝜕[𝑀]𝜕𝑝

= −𝛻 ⋅ [𝑢∗𝑀∗] + 𝑅+ cos𝜙 𝐹P

Thismaybewritten,

[𝑣]𝑅+

𝜕𝑀𝜕𝜙

+ [𝜔]𝜕𝑀𝜕𝑝

= 𝑅+ cos𝜙 −1

cos) 𝜙𝜕[𝑢∗𝑣∗] cos) 𝜙

𝜕𝜙−𝜕[𝑢∗𝜔∗]𝜕𝑝

+ 𝐹P

Eddy-meanflowinteractions

Considerapplyingthisequationintheuppertropospherenearthelatitudeofthestreamfunction maximum

Here[𝜔] ≈ 0,andsowemayneglectverticaladvectionbythemeanflow

Also,frictionisweak,sowemayneglect𝐹P

Eddy-meanflowinteractions

Thisallowsustowrite,

[𝑣]𝑅+) cos𝜙

𝜕[𝑀]𝜕𝜙 = −𝑆

where𝑆 istheeddymomentumfluxdivergence.

Notealsothat

]^=_` abc d

e[f]ed

= 𝑓 + [𝜁].

EquationforStreamfunctionmaximum

Theprecedinganalysisleadstoaverysimpleequationforthepolewardflowintheuppertroposphere,

[𝑣](𝑓 + 𝜁 ) = 𝑆

absolutevorticityeddymomentumfluxdivergence

upper-troposphericzonal-meanmeridionalwind

29/4/19

6

• Twolimits:

1. Linearregime: 𝑓 ≫ [𝜁],[𝑣] isproportionaltotheeddymomentumfluxdivergence(asintheFerrelcell)

2. non-linearregime:𝜕d𝑀 ∝ 𝑓 + [𝜁] ≈ 0,angularmomentumconserved,Hadleycelldoesnotdependoneddyfluxes,Held-Hou modelisvalid

Constructequationforstreamfunction

[𝑣](𝑓 + 𝜁 ) = 𝑆

Integrateequationfromstreamfunction maximumtotropopausewithmassweighting

𝑓k 𝑣 1 − 𝑅𝑜𝑑𝑝𝑔

op

oq= k 𝑆

op

oq

𝑑𝑝𝑔

where𝑅𝑜 = −[𝜁]/𝑓 isameasureoftheRossbynumber

integrate

Constructequationforstreamfunction

𝑓k 𝑣 1 − 𝑅𝑜𝑑𝑝𝑔

op

oq= k 𝑆

𝑑𝑝𝑔

op

oq

Hence,

1 − 𝑅𝑜∗ Ψstu =𝑆𝑓

where𝑅𝑜∗ istheRossbynumberweightedbythepolewardmassfluxoftheHadleycell

integrate

• Twolimits:

1. Linearregime: 𝑓 ≫ 𝜁,ΨGHI isproportionaltotheeddymomentumfluxdivergence

2. non-linearregime:𝜕d𝑀 ∝ 𝑓 + 𝜁 ≈ 0,angularmomentumconserved,Hadleycelldoesnotdependoneddyfluxes,Held-Hou modelisvalid

Wheredoestheatmospherelie?

29/4/19

7

Walker&Schneider(2006):idealisedGCM

on the line !max " T . The other simulations that devi-ate from the line !max " T also show relatively largelocal Rossby numbers in the upper branches of the Ha-dley circulations. For example, the Hadley circulationsin the simulations with a dry-adiabatic convective lapserate (# " 1.0) have local Rossby numbers of about 0.5in their upper branches in the vicinity of the latitude ofthe streamfunction extrema. In these simulations, thethermal stratification within the Hadley circulations isnearly statically neutral, and the Hadley circulationsare up to an order of magnitude stronger than Earth’sannual-mean or equinox Hadley circulation.

The contribution of the vertical eddy momentum fluxu $%$

&to T is negligible in most simulations. An excep-

tion are the simulations with # " 1.0, in which the ver-tical momentum flux contribution, probably at least inpart owing to convection on the model’s grid scale,dominates T . (Poleward of the latitudes of the Hadleycirculation extrema at which T is evaluated, the hori-zontal eddy momentum flux divergence again domi-nates the vertical eddy momentum flux divergence inthe interior atmosphere in these simulations.)

The approximately linear relation between Hadleycirculation strength and eddy momentum flux diver-gence helps to explain the regime transition in scalingbehavior seen in the potential temperature flux (Fig. 4).Since over a wide range of flow parameters, the Hadleycirculation strength !max is approximately linearly re-lated to the eddy momentum flux divergence, the po-tential temperature flux '(!max is likewise directly re-lated to the eddy momentum flux divergence, with the

gross stability '( primarily controlled by the convectivelapse rate and, through its influence on the tropopauseheight, the tropical surface temperature. The gross sta-bility exhibits no transition in scaling behavior as afunction of H$t'$h, so the transition between the twoscaling regimes of the potential temperature flux is atransition in scaling regimes of the eddy momentumflux divergence. The transition between the two re-gimes occurs where the supercriticality of the extratro-pical thermal stratification and of baroclinic eddiesreaches saturation (Schneider and Walker 2006). In thelow H$t'$h regime, the extratropical thermal stratifica-tion is primarily maintained by convection. In the highH$t'$h regime, the extratropical thermal stratification ismodified by baroclinic eddies. As # is increased, thetransition between the regimes in which convection andeddies dominate the maintenance of the extratropicalthermal stratification occurs at smaller temperaturecontrasts 'h, and thus at smaller H$t'$h, such that the #" 0.9 and 1.0 simulations are all in the eddy-dominatedregime (Schneider and Walker 2006). Scaling laws ofbaroclinic eddy fluxes change at this regime transition,and they are imprinted on the scaling behavior of theHadley circulation. (We will discuss scaling laws ofbaroclinic eddy fluxes elsewhere.)

The approximately linear relation between Hadleycirculation strength and eddy momentum flux diver-gence implies that it is a coincidence that the scaling ofthe potential temperature flux in the low H$t'$h regimein some series of simulations in Fig. 4 is consistent withthe HH scaling '$(!$max ) (H$t'$h)5/2. This coincidence is

FIG. 6. Hadley circulation strength !max as a function of the quantity T , which is proportional toan integrated eddy momentum flux divergence (4) at the latitude of maximum absolute value of the Hadleycirculation streamfunction. The dashed line shows the identity !max " T . Simulations in (a) and (b) are the sameas in Figs. 4a,b.

DECEMBER 2006 W A L K E R A N D S C H N E I D E R 3341

Fig 6 live 4/C

Hadleycellstrength

Eddymomentumfluxdivergence 𝑆

It is often assumed that the Hadley circulation re-sponds directly to changes in thermal driving, in thesense that its changes are controlled by thermodynamicbalances. For that to be the case, the balance of angularmomentum about Earth’s spin axis, or the balance ofzonal momentum, would have to be degenerate in theupper troposphere, so that eddy momentum fluxes donot influence the Hadley circulation. In the upper tro-posphere, above the center of the Hadley cells wherevertical momentum advection by the mean meridionalcirculation can be neglected, the mean zonal momentumbalance may be used to quantify how strongly eddy mo-mentum fluxes influence the mean meridional flow:

f (1!Ro)y ’ Se. (1)

Here, Se is the eddy momentum flux divergence, f theplanetary vorticity (Coriolis parameter), and y the me-ridional velocity; overbars indicate a temporal and zonalmean. The local Rossby number Ro 5!z/f , with rela-tive vorticity z, is a nondimensional measure of the im-portance of nonlinear angular momentum advection bythe mean meridional circulation (Walker and Schneider2006; Schneider 2006). In the limit Ro / 1, the upperbranch of the Hadley circulation conserves angular mo-mentum and is unaffected by eddy momentum fluxes; itsstrength responds directly to changes in thermal driving.This is the limit considered in classical theories for theHadley circulation, which provide expressions for its widthand strength as a function of thermal and other parameters(Schneider 1977; Held and Hou 1980; Lindzen and Hou1988). In the limit Ro / 0, the strength of the Hadleycirculation is controlled by eddy momentum fluxes; itsstrength responds to changes in thermal driving only insofar

as they affect the eddy momentum fluxes (e.g., Dickinson1971). In between these limiting cases lie Hadley circula-tions with 0 , Ro , 1 in their upper branches, which re-spond to changes in thermal driving through changes inthermodynamic balances, in eddy momentum fluxes, andpossibly in Rossby numbers. There is no theory that cap-tures how the Hadley circulation responds to climatechanges in this intermediate range of Rossby numbers.

Figure 1 shows the meridional mass flux streamfunction,the eddy momentum flux divergence, and the local Rossbynumber for Earth’s atmosphere during equinox and in theannual mean. The quantities are similar during equinoxand in the annual mean, suggesting that the annual meanis not dominated by an average of solstitial circulations(cf. Lindzen and Hou 1988; see also Dima and Wallace2003; Walker and Schneider 2005). The eddy momentumflux divergence has a broad maximum in the upper tro-posphere centered near 218 latitude, extending deep intothe Hadley cells (color contours in Fig. 1). The Rossbynumber in the upper troposphere gradually decreasesfrom close to 1 at the equator to close to 0 in the subtropics,with values greater than 0.5 confined to a narrow bandwithin ; 48 latitude of the equator (gray shading in Fig. 1).Near the center of the Hadley cells (near 108 latitude),the Rossby number is between 0.35 and 0.45 during equi-nox and in the annual mean. This indicates that angularmomentum–conserving theories are inadequate for de-scribing Hadley cell dynamics, at least poleward of about108 latitude during equinox and in the annual mean. Un-derstanding how eddy momentum fluxes change with cli-mate therefore is integral to understanding how the Hadleycirculation responds to climate changes. This is borne outin observational data and models, which show that in-terannual variations in the strength of Hadley cells and

FIG. 1. Earth’s Hadley circulation (a) averaged over equinox seasons [March–May (MAM) and September–November (SON)] and (b) averaged annually. Black contours show the mass flux streamfunction, with dashed(negative) contours indicating clockwise motion and solid (positive) contours indicating counterclockwise motion.The contour interval is 25 Sv 5 25 3 109 kg s21. Arrows indicate the streamfunction extremum in each hemisphere,with the magnitude given in 109 kg s21. Colors indicate horizontal eddy momentum flux divergence div(u 9y9 cosf),with the overbar denoting a temporal and zonal mean and primes denoting deviations therefrom (i.e., eddy fieldsinclude stationary and transient eddies). The contour interval for the eddy momentum flux divergence is4 3 1026 m s22, with red tones for positive and blue tones for negative values; the darkest red tone corresponds tovalues greater than 2.4 3 1025 m s22. Gray shading indicates regions where jRoj. 0.5. Flow statistics are computedfrom reanalysis data for the years 1980–2001 provided by the European Centre for Medium-Range WeatherForecasts (Kallberg et al. 2004; Uppala et al. 2005).

770 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 68eddymomentumfluxdivergence

Ro>0.5

Inannualmean,Rossbynumber~0.2incentreofHadleycell

Levine&Schneider2010

HadleycellinSolsticial seasons

SofarwehaveonlytalkedaboutEquinoctialHadleycells.

ButSolsticial cellscancrosstheequator,thishasimplicationsfortheangularmomentumbudget

Forrestingatmosphere,angularmomentumdecreasespoleward

𝑀 = Ω𝑅+) cos) 𝜙≈ Ω𝑅+)(1 − 𝜙))

SPNP EQ

29/4/19

8

AngularmomentumconservingHadleycellmusthavetilteddescendingbranchwithsharp

discontinuity(jet)(𝑅𝑜 ≈ 1)

SPNP EQ

Orstreamlinesmustcrossangularmomentumcontours

(Ro<1)

SPNP EQ

Butwhenthecellcrossestheequator,streamlinescanoverturn!

Streamlinesonlycrossangularmomentumcontourswithinthe

boundarylayer

NP EQ

ObservedSolsticial Hadleycell

3.3. Mean state of the circulation

Latitude

Sigma

10

2

2

−8

−4

−60 −30 0 30 60

0.2

0.8

Latitude

Sigma

4

2

−20−4

−60 −30 0 30 60

0.2

0.8

Latitude

Sigma

24

−4

−2

−60 −30 0 30 60

0.2

0.8

Figure 3.13: Mean meridional streamfunction from ERA-40: annual mean (upper),NH winter (middle), NH summer (lower)

33

JJA

DJF

ERA40FigurecourtesyofPaulO’Gorman

29/4/19

9

LETTERS

250

500

750Pres

sure

(hPa

)

250

500

750

340

300

30° S 0Latitude

30° N

30° S 0Latitude

30° N

30° S 0Latitude

30° N 30° S 0Latitude

30° N

30° S 0Latitude

30° N

30° S 0Latitude

30° N

Pres

sure

(hPa

)

250

500

750Pres

sure

(hPa

)

250

500

750Pres

sure

(hPa

)

h (k

J kg

–1)

340

300h (k

J kg

–1)P (m

m d

–1)

0

12

a b

c d

e f P (mm

d–1)

0

12

Figure 2 Observed monsoon onset over Asia. Zonal- and temporal-mean circulation in the Asian monsoon sector at two 20-day periods before (left panels, Julian Day81–100) and after (right panels, Julian Day 161–180) monsoon onset. a,b, Streamfunction of meridional overturning circulation (black contours, contour interval50⇥109 kg s�1, with solid contours for anticlockwise rotation and dashed contours for clockwise rotation), angular momentum per unit mass (grey contours, contour interval⌦ a 2/15 with Earth’s rotation rate⌦ and radius a) and transient eddy momentum flux divergence div([u 0v 0]cos� ), with horizontal velocity vector v= (u, v ) (colourcontours, contour interval 0.6⇥10�5 m s�2 in a and 1.2⇥10�5 m s�2 in b, with red tones for positive and blue tones for negative values). Here, (·) denotes a temporal meanover the 20-day period and over all years of data, primes denote deviations from this mean and [·] denotes a zonal mean over the monsoon sector. c,d, Zonal wind (blackcontours, contour interval 6ms�1) and eddy momentum flux divergence (colour contours) as in a,b. e,f, Precipitation P (blue) and near-surface (850 hPa) MSE h (red). Exceptfor precipitation, all quantities are obtained from the ERA-40 reanalysis30 and are averaged over the years 1981–2000. In the latitude zones of the tropical overturningcirculation, the horizontal eddy momentum flux divergence shown in the figure is the dominant term balancing the Coriolis force on the mean meridional flow and the meanmeridional momentum advection in the zonal momentum budget; other terms, such as the stationary eddy momentum flux divergence and the zonal geopotential gradientacross the monsoon sector, are smaller.

primarily as a result of MSE advection (which dominates theMSE budget in the boundary layer). The transition betweenthe equinox and monsoon regimes in the simulation occursrapidly compared with variations in radiative heating. This ismanifest in the rapid intensification and relocation of the ITCZinto the summer subtropics (Fig. 1b) and the rapid changes insubtropical near-surface winds (Fig. 3). Overall, the transition inthe simulation resembles the onset of the Asian monsoon. A reversetransition occurs at the end of the warm season; this transitionoccurs more rapidly than in the Asian monsoon, suggesting thatprocesses not captured by our idealized GCM modulate the Asianmonsoon retreat.

For the feedback mechanisms discussed above to be able tomediate rapid transitions of the overturning circulation, the surfacethermal inertia must be suYciently low for the near-surface MSEto be able to adjust rapidly. Only then can circulation changesoccur rapidly because the near-surface MSE controls the locationof the ascending branch of the cross-equatorial cell (and henceof the main precipitation zone) and, by gradient-wind balance,the upper-level zonal wind, both of which must be able to adjust

0 90 180Julian day

270 360

–10

0

10

20

u (m

s–1

)

Figure 3 Rapid changes in near-surface zonal wind at 15�N. Seasonal cycle of

zonal- and pentad-mean near-surface zonal wind at 15� N from observations in theAsian monsoon sector (green) and from aquaplanet simulations with oceanmixed-layer thickness 1m (yellow) and 100m (red). The near-surface zonal wind isevaluated at 850 hPa in the observations and at � = 0.85 in the simulations, where� = p/ps is pressure p normalized by surface pressure ps.

nature geoscience VOL 1 AUGUST 2008 www.nature.com/naturegeoscience 517

AsiansectormeanforJulianday161-181

Bordoni &Schneider2008

eddy-momentumfluxdivergenceangular-momentumcontours

RapidonsetofSouthAsianmonsoonLETTERS

20

12

x

12

x

30° S

0

30° N

Latit

ude

Latit

ude

30° S

0

0 90 180Julian day

270 360

0 90 180Julian day

270 360

0 90 180Julian day

270 360

30° N

Latit

ude

30° S

0

30° N

a

b

c

x

Figure 1 Rapid shifts of precipitation zones. a, Seasonal cycle of zonal- andpentad-mean precipitation (colour contours, mmday�1) and sea-level airtemperature (SAT; grey contours) from observations in the Asian monsoon sector.b,c, The same from aquaplanet simulations with ocean mixed-layer thicknessof 1m (b) and 100m (c). SAT is evaluated at 1,000 hPa in the observations and atthe lowest model level in the simulations. The contour interval for precipitation is1mmday�1 in a and 2mmday�1 in b,c, with maxima identified by crosses. For SAT,the contour interval is 2 �C, with the solid grey line marking the 24 �C isoline.Precipitation rates in the simulations can exceed observed precipitation ratesbecause the lower boundary in the simulations is entirely water-covered. The thickdashed line in a shows the latitude at which the zonal-mean topography in the Asianmonsoon sector rises above 3 km. Arrows at the time axes in a,b indicate thecentres of the 20-day periods for which circulations are shown in Figs 2 and 4.Observed precipitation is from Global Precipitation Climatology Project29 data for theyears 1999–2005. In this and subsequent figures, zonal means for the Asianmonsoon sector are averages between 70� and 100� E. Results are robust tochanges in the averaging sector and do not change substantially if southeast Asia isincluded in the averages.

is controlled by eddy momentum fluxes and can respond tovariations in thermal driving only indirectly through changes ineddy momentum fluxes15. If Ro!1, the circulation approaches theangular-momentum-conserving limit, and its strength respondsdirectly to variations in thermal driving12. Local Rossby numbersin the upper troposphere in the Asian monsoon sector changeat monsoon onset, from Ro ⇠< 0.4 in the upper branch near thecentre of the southern cell pre-onset to Ro ⇠> 0.7 in the upperbranch near the centre of the cross-equatorial cell post-onset.Therefore, at monsoon onset, the southern cell transitions froman equinox regime, in which its strength is primarily controlled

by eddy momentum fluxes, to a monsoon regime, in which itsstrength is more directly controlled by the thermal driving. Areverse transition occurs at the end of the monsoon.

We have previously shown that when an overturning circulationundergoes such a regime transition, two dynamical feedbackmechanisms act, rendering the transition and accompanyingcirculation changes rapid even in the absence of surfaceinhomogeneities and an active hydrological cycle5. First,upper-level easterlies and, by gradient-wind balance, polewardtemperature and MSE gradients develop in the tropics in earlysummer as cross-equatorial flow develops (Fig. 2d,f). Because thelocal zonal wind determines the propagation characteristics ofthe energy-containing extratropical eddies16, upper-level easterliesshield the cross-equatorial cell from the eddies, which are primarilyconfined to regions of westerlies (Fig. 2d). This allows theoverturning cell to approach the angular-momentum-conservinglimit more closely, leading to strengthening of the cell andstrengthening and extension into the winter hemisphere of theupper-level easterlies5. Second, advection of cold (low MSE) airby the lower branch of the cross-equatorial cell pushes the MSEmaximum poleward (Fig. 2b,f), leading to poleward movementof the ascending branch because the main ascent region is near theMSE maximum9. The poleward movement of the ascending branchimplies strengthening of the cell and strengthening and extensioninto the winter hemisphere of the upper-level easterlies5,13.Together these mechanisms, discussed in ref. 5, render the regimetransitions rapid compared with the timescale of variations inradiative heating.

Using an idealized general circulation model (GCM) with anactive hydrological cycle and with a homogeneous lower boundary,we demonstrate that these feedback mechanisms can mediaterapid transitions of an overturning circulation with circulation andprecipitation changes resembling those in the Asian monsoon. Wesimulate seasonal cycles by prescribing variations of mean dailyinsolation at the top of the atmosphere in an idealized GCM17

in which the lower boundary is a mixed-layer (slab) ocean ofconstant depth, with a prescribed time-independent meridionalheat transport estimated to match the observed annual-meanocean heat transport in low latitudes18,19. We show results fromtwo simulations, one with a thin (1 m) and one with a thick(100 m) mixed layer, respectively implying low and high surfacethermal inertia.

Like the overturning circulation in the Asian monsoon sector,the overturning circulation in the simulation with a thin mixedlayer undergoes a rapid transition from an equinox to a monsoonregime. In the equinox regime, upper-level westerlies and strongeddy momentum flux divergence extend into both circulation cells(here, Hadley cells because the simulated climate is statisticallyaxisymmetric), leading to deviations from angular momentumconservation in the upper branches (Fig. 4a,c). Local Rossbynumbers satisfy Ro ⇠< 0.3–0.4 in the upper branches and show thatthe circulation strength is primarily controlled by eddy momentumfluxes. In the monsoon regime, the streamfunction maximum islocated near the equator, in a region of upper-level easterlies, whereit is shielded from extratropical eddies and the eddy momentumflux divergence is small (Fig. 4b,d). Rossby numbers are poorlydefined in the upper branch of the cross-equatorial cell close toits centre near the equator, but that the flow in the ascendingand upper branches is closer to angular-momentum-conservingis manifest from the near-coincidence of angular momentumcontours and streamlines there. Similar to what is seen duringthe Asian monsoon, westerly winds prevail near the surface inthe summer subtropics. Strong easterlies prevail at upper levelsthroughout the tropics (Fig. 4d); by gradient-wind balance, thenear-surface temperature and MSE gradients are reversed (Fig. 4f),

516 nature geoscience VOL 1 AUGUST 2008 www.nature.com/naturegeoscience

PrecipitationinAsianmonsoonsector

Bordoni &Schneider2008

ObservedSolsticial Hadleycell

Characterisedby:

• Dominanceofcrossequatorialcell• overturningofangular-momentumcontours• closertoangular-momentumconservationthanequinoctialcell• rapidonsetofmonsoon(atleastinAsiansector,butseeDima&Wallace,2005)

Lindzen &Hou solution

Lindzen&Hou extendedtheHeld&Hou modeltoincludeanoff-equatorialmaximumintheRCEsubcloud entropy:

𝑠%=># = 𝑠& − 𝛿𝑠& sin𝜙 − sin𝜙& )

Again,assumeangularmomentumconservationinupperbranch,butallowrisingbranchtobeofftheequator:

Uppertroposphericwindvelocityisthengivenby,

𝑢E =Ω𝑅+cos𝜙

cos) 𝜙v − cos) 𝜙 )

29/4/19

10

Lindzen &Hou solution

Lindzen-Housolutionstrengthensnonlinearlyas𝜙& isdisplacedoffthe

equator

Eddy-meanflowinteractionandmonsoons

• TheprecedingresultspaintapictureofanequinoctialHadleycellthatisstronglyinfluencedbyeddiesandaSolsticial Hadleycellthatisclosertotheangular-momentumconservinglimit

• Intheequinoctialcase,thewindsuppertroposphericarewesterlyinthecentreoftheHadleycells,allowingeddiestopropagateintotheHadleycellinfluencethecirculation

• IntheSolsticial case,thewindsareEasterlybetweentherisingbranchandtheequator,thisisinpenetrable toRossbywaves,potentiallyshieldingthecirculationfromeddies

• Bordoni &SchneiderarguethismayaccountfortherapidonsetoftheSouthAsianmonsoon

Summary

• AxisymmetricHadleycelltheorypredictsanumberoffeaturesofthetropicalcirculation

• However,equinoctialcirculationknowntobestronglyaffectedbyeddies

• Solsticial Hadleycellclosertoangular-momentumconservinglimit

• Quasi-equilibriumtheoryhighlightsimportanceofboundarylayerentropyratherthansurfacetemperature