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

The General Circulationof the AtmosphereTapio SchneiderCalifornia Institute of Technology, Pasadena, California 91125; email: tapio@caltech.edu

Annu. Rev. Earth Planet. Sci.2006. 34:65588

The Annual Review ofEarth and Planetary Scienceis online atearth.annualreviews.org

doi: 10.1146/annurev.earth.34.031405.125144

Copyright c 2006 byAnnual Reviews. All rightsreserved

0084-6597/06/0530-0655$20.00

Key Words

Hadley circulation, macroturbulence, baroclinic eddies, jets,scaling laws

AbstractTheories of how Earths surface climate may change in the future, of how it mayhave been in the past, and of how it is related to climates of other planets must buildupon a theory of the general circulation of the atmosphere. The view of the atmo-spheric general circulation presented here focuses not on Earths general circulationas such but on a continuum of idealized circulations with axisymmetric ow statistics.Analyses of observational data for Earths atmosphere, simulations with idealized gen-eral circulation models, and theoretical considerations suggest how characteristics ofthe tropical Hadley circulation, of the extratropical circulation, and of atmosphericmacroturbulence may depend on parameters such as the planet radius and rotationrate and the strength of the differential heating at the surface.

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1. INTRODUCTION

In 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 denite 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

Sigm

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a c

Figure 1Temporal and zonal mean circulation statistics for January according to reanalysis data for theyears 19802001 (Kallberg et al. 2004). (a) Zonal wind (magenta) and potential temperature(light blue). Contour intervals are 4 m s1 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 ux streamfunction (magenta) and angular momentum (light blue). Contourintervals are 20 109 kg s1 for streamfunction and 0.1a2 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 ux convergence(eastward stress) due to mean circulations (light blue) and due to eddies (magenta), with eddiesdened as uctuations about the temporal and zonal mean. The vertical coordinate in (a) and(c) is = p/ps (pressure p normalized by surface pressure ps).

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General Trade-Winds have not been fully explained by any of those who have wroteon that Subject, Hadley wrote in 1735, for want of more particularly and distinctlyconsidering the Share the diurnal Motion of Earth has in the Production of them.Hadley recognized that Earths rotation is responsible for the direction of the meansurface winds and that a global circulation is necessary to maintain them.

According to Hadley, if one corrects minor shortcomings of his argumentation(Thomson 1892, Lorenz 1967), differential heating causes air to rise near the equatorand, by conservation of mass, to ow poleward aloft. Hadley envisaged a steady cir-culation symmetric about Earths rotation axis. Absent frictional forces, each parcelof poleward owing air in the axisymmetric circulation conserves its absolute angularmomentum about Earths axis, M = (a cos + u)a cos per unit mass in the ap-proximation of the atmosphere as a thin spherical shell, with radius a, angular velocityof planetary rotation , latitude , and eastward wind component u (Peixoto & Oort1992, Chapter 11). As an air parcel ows poleward, its distance to Earths axis a cos decreases, so that angular momentum conservation implies that the parcels eastwardwind component u increases.1 Differential cooling causes air to sink in higher lati-tudes, again under conservation of angular momentum of air parcels, and thus leadsto surface westerlies in higher latitudes. The surface winds are retarded by surfacedrag and are deected equatorward, to provide the return ow necessary to closethe circulation. Because of surface drag, the equatorward return ow does not con-serve angular momentum, but if one reverses the arguments for the poleward ow inthe upper branch of the circulation, it does, according to Hadley, give rise to surfaceeasterlies in lower latitudes, albeit weaker ones than angular momentum conservationwould imply. Because Earths spin angular momentum is nearly conserved (tidal andother external torques can be neglected in theories of the circulation of the lower at-mosphere), the eastward drag the surface westerlies exert on Earths surface in higherlatitudes must be balanced by the westward drag the surface easterlies exert on Earthssurface in lower latitudes for Earths rotation rate to remain constant. According tothis theory, a thermally direct meridional circulation in each hemisphere transportsangular momentum from low latitudes, where the drag on surface easterlies transfersangular momentum from the surface to the atmosphere, to high latitudes, where thedrag on surface westerlies transfers angular momentum from the atmosphere to thesurface. The circulation also transports heat from low to high latitudes.

In the nineteenth century, new observations showed that midlatitude surface west-erlies have, in the mean, a poleward component, instead of the equatorward compo-nent demanded by Hadleys theory. To account for the new observations, Thomson(1857, 1892) and Ferrel (1859) amended Hadleys theory by positing secondary, ther-mally indirect meridional circulation cells in midlatitudes. The indirect circulationcells were thought to be shallow, embedded under Hadleys hemispheric circulationcells, with, in todays terms, the Coriolis force on the poleward surface wind compo-nent approximately balancing the drag on the surface westerlies (Ekman balance).

1Hadley actually thought that absolute zonal velocity rather than absolute angular momentum was conserved,a conceptual but merely quantitative error in his qualitative argumentation (Thomson 1892, Lorenz 1967).

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General circulation:totality of mean large-scalemotions of atmosphere

Hadley cell: thermallydirect tropical circulationcell

Ferrel cell: thermallyindirect midlatitudecirculation cell

Macroturbulence: totalityof irregular motions oflarge-scale eddies,characterized by smallRossby number and smallaspect ratio

But the axisymmetric general circulation theories of Hadley and his successorsturned out to be untenable when Defant (1921), Jeffreys (1926), and others real-ized that large-scale eddiescyclones and anticyclones with length scales of order1000 kmare not perturbations of an idealized axisymmetric circulation but are es-sential, in particular, for the atmospheric transport of heat and angular momentum[see Lorenz (1967, 1983) for an historical overview]. The thermally direct meridionalcirculation named after Hadley extends from equatorial regions only to about 30

latitude in each hemisphere (Figure 1c), with slight seasonal shifts of the polewardboundaries of the Hadley cells. The thermally indirect midlatitude cells named afterFerrel cannot be understood as axisymmetric circulations. Angular momentum is notconserved along streamlines of the Ferrel cells. Horizontal eddy momentum uxesmake it possible that streamlines in the upper branches of the Ferrel cells cross angu-lar momentum contours (Figure 1c). The Coriolis force on the equatorward meanow in the upper branches of the Ferrel cells approximately balances a convergenceof horizontal eddy momentum uxes. Because horizontal eddy momentum uxes andmomentum advection by the mean ow are weak near the surface, the Coriolis forceson the mean meridional surface winds in the Ferrel and Hadley cells approximatelybalance the drag on the mean zonal surface winds, as implied in the axisymmetric the-ories of Ferrel and Thomson. Hence, the mean meridional surface wind is generallydirected poleward in regions of surface westerlies and equatorward in regions of sur-face easterlies. The angular momentum transfered from the atmosphere to the surfaceby the drag on midlatitude westerlies is transported into midlatitudes not primarilyby the mean meridional ow but by eddies, whose angular momentum transport fromhigh and low latitudes into midlatitudes implies eastward stresses on Earths surfacein midlatitudes and westward stresses in low and high latitudes (Figure 1d ). Thedivergence of eddy momentum uxes in high latitudes, concentrated in the uppertroposphere, and the drag on any high-latitude surface easterlies are approximatelybalanced by Coriolis forces on the mean meridional ow in weak, thermally directpolar circulation cells (not visible in Figure 1c). Instead of Hadleys hemisphericcirculation cells extending from equatorial regions to high latitudes, there are thusthree circulation cells in each hemisphere, at least two of which are due to eddyuxes.

A theory of the general circulation of the atmosphere must build upon a theory ofthe macroturbulence consisting of large-scale eddies. Eddies generated by baroclinicinstability generally transport angular momentum into the latitude zones in whichthey are generated (Held 1975, 2000, Simmons & Hoskins 1978, Ioannou & Lindzen1986). In a statistically steady state, the angular momentum ux into or out of anatmospheric column is balanced by surface drag, so surface westerlies appear in baro-clinic zones into which eddies transport angular momentum. The vertical structureof the winds and the strength of upper-level jets are connected to the surface windsthrough thermal-wind (or gradient-wind) balance, which relates the vertical shearof the zonal wind to meridional temperature (or potential temperature) gradients, asseen in Figure 1a. [See Holton (2004) for a discussion of thermal-wind and gradient-wind balance and other basic concepts of atmospheric dynamics.] The meridional

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temperature gradients, in turn, are determined by a balance of radiative processes,ocean circulations, and meridional uxes of latent and sensible heat, uxes that, inmidlatitudes, are primarily effected by eddies. Particularly in low latitudes, the meanmeridional circulations also contribute to the uxes of angular momentum, heat, andwater vapor, transporting, for example, angular momentum from equatorial regionsinto the subtropics, from where eddies transport it into midlatitudes (Figure 1d ).A minimal mean-eld theory of the general circulation of the atmosphere, then,must account quantitatively for the mutual dependence of eddy uxes and meanelds such as mean meridional temperature gradients. We do not have such atheory.

I review our present understanding of the general circulation of the atmosphere,focusing not on Earths atmosphere as such but on idealized atmospheres with ax-isymmetric boundary conditionswith incoming solar radiation only depending onlatitude and with a homogeneous surfaceatmospheres that have axisymmetric cir-culation statistics. Although the zonal-mean effects of statistically nonaxisymmetriccirculation features such as monsoons or stationary waves are signicant, a generalcirculation theory for idealized atmospheres with axisymmetric circulation statisticsis a prerequisite for any future, more complete, general circulation theory, whichmust be reducible to this canonical case. Present-day computational resources allowus to simulate such atmospheres with idealized general circulation models (GCMs)numerically, not only in the circulation regime comprising the seasonal cycle andrecent climatic variations of Earths atmosphere but over a broad continuum of possi-ble planetary circulations, of which circulations resembling that of present-day Earthrepresent a part. Investigations of such a circulation continuum are indispensable ifone wants to infer macroscopic laws governing atmospheric circulations and wantsto understand circulation changes that may have occurred in the past or may occurin the future and that go beyond those for which we have detailed observational data.Comparisons with the wealth of observational data for Earths atmosphere, togetherwith the much sparser data for other planetary circulations, can provide insurancesthat inferences drawn from simulations are empirically relevant. Unlike previousgenerations of atmospheric dynamicists, we are no longer limited by sparseness ofobservational data or insufcient computational resources to complete the projectof developing a quantitative theory of the general circulation at least of idealizedatmospheres.

I discuss the tropical Hadley circulation and the inuence macroturbulence mayhave on it in Section 2 and the extratropical circulation and aspects of atmosphericmacroturbulence in Section 3. Moist processes such as moist convection and latentheat release in phase changes of water are largely sidelined in the discussions, notbecause they would be unimportantthey generally are importantbut to draw at-tention to unresolved fundamental questions about the general circulation of dryatmospheres, questions whose resolution is a prerequisite for any future general cir-culation theory, moist or dry. I focus on the general circulation of the tropospherebecause it is primarily responsible for the maintenance and variability of the surfaceclimate.

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Hides theorem: in steadyaxisymmetric circulations,zonal wind cannot exceedangular momentumconserving zonal wind

2. THE TROPICAL HADLEY CIRCULATION

2.1. Nearly Inviscid Axisymmetric Circulations

In idealized atmospheres with axisymmetric boundary conditions, absent nonaxisym-metric perturbations, a circulation that is initially axisymmetric remains axisymmetric,albeit not necessarily steady. So we can consider theoretically and simulate numer-ically axisymmetric circulations in idealized atmospheres, postponing questions ofthe effects on these circulations of instabilities with respect to nonaxisymmetric per-turbations. We examine what Lorenz (1969) called the ideal Hadley circulation,essentially the circulation envisaged by Hadley.

A steady axisymmetric circulation in which angular momentum disperses diffu-sively, however weak the diffusion, cannot have an extremum of absolute angularmomentum away from boundaries, which constrains zonal winds to be weaker thanor at most equal to the zonal winds implied by an angular momentumconserving cir-culation (Hide 1969, Schneider 1977). If there were an angular momentum extremumaway from boundaries, there would exist a closed contour of constant angular momen-tum surrounding it in the meridional plane. The advective ux of angular momentumacross the contour would vanish because, in a steady state, the mass ux across the con-tour would vanish. But diffusion of angular momentumtypically, vertical diffusionrepresenting small-scale turbulent uxesimplies that there would be a downgradi-ent ux of angular momentum across the contour, which could not be balanced, so an-gular momentum extrema away from boundaries are impossible. Angular momentummust attain its maximum value in a region of easterlies at the surface, where the surfacedrag on the easterlies transfers angular momentum from the surface to the atmosphereimmediately above it, from where it can diffuse into the interior atmosphere. Theangular momentum per unit mass M = (a cos + u)a cos must therefore be ev-erywhere less than or equal to a2, and the zonal winds must be less than or equal tothe angular momentumconserving zonal wind uM = a sin2 / cos correspondingto M = a2, a result known as Hides theorem. A steady state with stronger eastwardwindsfor example, equatorial westerlies (superrotation) such as seen on the giantplanetscan only be maintained if diffusion of angular momentum is balanced byupgradient eddy uxes of angular momentum.

If the atmosphere were inviscid, radiative (or radiative-convective) equilibriumwithout a meridional circulation would be a steady axisymmetric solution of theequations of motion. But if angular momentum disperses diffusively, radiative equi-librium may not satisfy Hides theorem. Suppose that, above a planetary boundarylayer, angular momentum diffusion and frictional forces are weak, hydrostatic bal-ance holds, and meridional winds are much weaker than zonal winds, as is generallythe case in Earths atmosphere. The meridional momentum equation can then beapproximated by gradient-wind balance

f u + tan a

u2 1a

,

where f = 2 sin is the Coriolis parameter (vertical component of planetary vortic-ity), = gz the geopotential, and the meridional derivative is a derivative at constant

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Nearly inviscidaxisymmetric limit: limitof axisymmetric circulationsin which viscosityapproaches zero in freeatmosphere such thatangular momentum there ismaterially conserved

pressure. If we substitute Hides result u uM and make the small-angle approxima-tion 1 (adequate in the Tropics), we nd 22a23. In the Tropics,above the planetary boundary layer and independent of surface boundary conditions,geopotential can decrease at most quartically with latitude. Because the geopoten-tial at pressure p, according to the hydrostatic relation (p) = RT log(p/p0) foran ideal-gas atmosphere (gas constant R and reference surface pressure p0), is pro-portional to a vertical temperature average T between the surface and pressure p(Holton 2004, Chapter 1), vertically averaged temperatures in the Tropics can likewisedecrease at most quartically with latitude. If temperatures in radiative equilibrium de-crease more rapidly than quartically, and, in particular, if they have a nonzero gradientat the equator, a thermally direct meridional circulation must exist to transport heatand to reduce the rate of temperature decrease with latitude, such that Hides theo-rem is satised. For Earths atmosphere, in the annual mean, the latitude-dependenceof the radiative (or radiative-convective) equilibrium temperature vertically averagedover the troposphere can be approximated by TE h cos2 , with pole-to-equatortemperature difference h . In the small-angle approximation, Earths radiative equi-librium temperature in the troposphere thus decreases quadratically with latitude,so a thermally direct meridional circulation must exist if there is any diffusionof angular momentum. The circulation extends at least to the minimum latitudeM at which the equilibrium geopotential E = RTE log(p/p0) satises Hidestheorem, E 22a23M. This latitude is

M(z) (

gz2a2

h

T0

)1/2, (1)

where z = (RT0/g) log(p/p0) is the log-pressure height and T0 a reference tem-perature. The latitude bound M(z) can only be expected to be relevant at heightsz above the planetary boundary layer, where gradient-wind balance can be assumedto hold. Poleward of the latitude M, radiative equilibrium satises Hides theorem(cf. Held & Hou 1980). A pole-to-equator temperature difference in radiative equi-librium of h/T0 80 K/295 K, with Earths values for the other parameters, givesa minimum circulation extent at z = 15 km of M = 25, similar to the meridionalextent of Earths Hadley circulation (Figure 1c). However, Earths Hadley circula-tion above the planetary boundary layer does not exhibit the

z-dependence of

meridional extent on height that these considerations would suggest.Building on the work of Schneider (1977), Held & Hou (1980) offered a theory

of steady axisymmetric circulations in idealized atmospheres, in the nearly inviscidlimit in which angular momentum diffusion above a planetary boundary layer ap-proaches zero. Angular momentum diffusion above the planetary boundary layer isassumed to be nonzero, so that a meridional circulation exists, but is assumed to beso weak that the zonal wind in the upper branches of the circulation is approximatelythe angular momentumconserving wind uM. Held & Hou obtain the meridional ex-tent and strength of the circulation by requiring that the circulation is energeticallyclosed, with diabatic warming in the ascending branches balancing diabatic coolingin the descending branches, and by requiring that temperatures within the circulationmake a continuous transition to radiative equilibrium temperatures poleward of it.

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Modeling diabatic processes as Newtonian relaxation toward a radiative equilibriumtemperature with latitude-dependence TE h cos2 gives a meridional extentof the circulation that, in the small-angle approximation, scales with the minimumextent M(Ht) at the height Ht of the upper branches of the circulation near thetropopause, a height that is assumed to be constant. The strength of the circulation(extremum of mass ux streamfunction) scales as

|M| 0 a2 Ht

(g Ht2a2

h

T0

)3/2h

v, (2)

where 0 is a reference density, is the Newtonian relaxation timescale, and v is thegross stability, the effective potential temperature difference between the upper andlower branches of the circulation. Unlike the axisymmetric circulations envisaged byHadley or by Thomson and Ferrel, the ideal Hadley circulation found by Held &Hou, depending on values of parameters such as the planetary rotation rate and ra-dius a, does not necessarily span hemispheres but has limited meridional extent, withradiative equilibrium poleward of the circulation. The zonal surface wind is east-erly near the equator and westerly near the poleward boundaries of the circulation.Above the westerlies, there are shallow, thermally indirect circulation cells, similar tothose envisaged by Thomson and Ferrel, in which diffusion transfers angular momen-tum downward and Coriolis forces on the poleward near-surface ows balance dragon the surface westerlies. For parameters representative of Earths Hadley circula-tion, with h = 80 K, v = 70 K, T0 = 295 K, Ht = 15 km, 0 = 1 kg m3, and = 30 days, the circulation according to Held & Hous exact expressions (with-out making the small-angle approximation and taking nondimensional O(1) factorsinto account) extends to 29 latitude, similar to Earths Hadley circulation, and hasa strength of 10 109 kg s1, about an order of magnitude weaker than Earthsannual-mean Hadley circulation (cf. Figure 1c).

Held & Hou showed with numerical simulations that aspects such as the merid-ional extent and strength of steady axisymmetric circulations approach the predictionsof the nearly inviscid theory with decreasing momentum and potential temperaturediffusivities. Only vertical diffusion of momentum and potential temperature, rep-resenting vertical small-scale turbulent uxes, for example, in cumulus clouds, wasincluded in Held & Hous numerical model, as in other, similar models (e.g., Schneider& Lindzen 1977, Schneider 1977). The simulated circulations are unstable with re-spect to axisymmetric perturbations and become unsteady when diffusivities are lessthan a critical value, which prevented Held & Hou and others (e.g., Schneider 1977,Lindzen & Hou 1988, Fang & Tung 1999) from obtaining steady circulations in thenearly inviscid limit numerically. For the smallest diffusivities for which steady axisym-metric circulations have been obtained, the circulations (particularly their strength)still depend on diffusivities. Vertical diffusion is generally not negligible near thepoleward boundaries of the circulations, where meridional temperature gradientsand, with them, vertical zonal wind shears are greatest and meridional velocities aresmall (Held & Hou 1980). Diffusion also plays a role in middle and higher latitudes,which are not exactly in radiative equilibrium in the simulations but exhibit weak,diffusive meridional circulations.

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Nearly inviscid axisymmetric circulations represent one limit of atmospheric cir-culations, which, subsequent to the work of Held & Hou, has been studied exten-sively and has illuminated several aspects of tropical circulations. For example, Hou &Lindzen (1992) and Fang & Tung (1996) have studied how diabatic heating concen-trated in narrow regions representing intertropical convergence zones affects axisym-metric circulations; Satoh (1994) has studied how moist processes interact with andmodify axisymmetric circulations; Emanuel (1995) has derived a constraint on near-surface moist entropy gradients from the constraint on interior geopotential gradientsimplied by Hides theorem; Plumb & Hou (1992) have studied axisymmetric mon-soon circulations [see Plumb (2006) for a review]; and Pauluis (2004) has studied howdissipation in the planetary boundary layer interacts with axisymmetric circulationsin the free troposphere and affects the structure of cross-equatorial ows near the sur-face. However, the question is whether nearly inviscid axisymmetric theory accountsfor the dependence of Earths or an Earth-like Hadley circulation on external param-eters. It is clear that zonal winds within Earths Hadley circulation are signicantlyweaker than the angular momentumconserving wind, which would be, assumingzero zonal wind at the equator, 92 m s1 at 25 latitude (cf. Figure 1a). Figure 1dshows that Earths Hadley cells, especially the cross-equatorial winter cell, advect an-gular momentumthe nonlinear advection that is essential for angular momentumconservation in the upper branches of the circulation in the nearly inviscid axisymmet-ric limit. Yet this nonlinear advection is not efcient enough to homogenize angularmomentum along streamlines of the circulation above the planetary boundary layer(Figure 1c). In particular, the summer Hadley cell does not displace angular momen-tum contours strongly away from the vertical; streamlines cross angular momentumcontours throughout its upper branch. The seasonal cycle provides further clues aboutthe relevance of nearly inviscid axisymmetric theory for Earths Hadley circulation.

2.2. Seasonal Cycle

Lindzen & Hou (1988) showed that in the nearly inviscid limit, because of the nonlin-earity of the angular momentumconservation constraint, axisymmetric circulationsrespond nonlinearly to displacements of the latitude of maximum heating off theequator.

Figure 2a shows the mass uxes in the upper branches of nearly inviscid ax-isymmetric circulations for different latitudes 0 at which the radiative equilibriumtemperature toward which temperatures are relaxed is maximal. (Parameters are as inthe numerical example in the preceding subsection, but with h = 60 K, v = 30 K,and Ht = 12 km to be comparable with the simulations shown in Figure 3 below.)The poleward extent and strength of the cross-equatorial winter cell increase as|0| increases, whereas the strength of the summer cell decreases. Averaging thecirculations for 0 and 0 gives a circulation that is much stronger (by a factor of15 for 0 = 6) than the equinoctial circulation driven by relaxation toward the av-eraged radiative equilibrium, which corresponds to 0 = 0 (Figure 2b). Lindzen &Hou suggested that such a nonlinear amplication of the annually averaged re-sponse to seasonally varying thermal driving accounts for the discrepancy between

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a0

b

Latitude30

0

50

Figure 2Mass ux (a) and average (e.g., 0 = 6 with 0 = 6) mass ux (b) in upper branches ofnearly inviscid axisymmetric circulations with latitude of maximum radiative equilibriumtemperature 0 = 0 (light blue), 2 (magenta), 4 (orange), and 6 (blue). Positive valuesindicate northward ux. From Walker & Schneider (2005).

the strength of Earths annual-mean Hadley circulation and the strength of the nearlyinviscid axisymmetric circulation driven by relaxation toward an annual-mean radia-tive equilibrium.

Dima & Wallace (2003), however, examining the seasonal cycle of Earths Hadleycirculation, found no evidence for nonlinear amplication of the annual-mean cir-culation. They showed that a large fraction of Earths Hadley circulation can beattributed to an equinoctial component whose strength does not exhibit the dramaticequinox-solstice swings predicted by nearly inviscid axisymmetric theory. Seasonalvariations of the strength of Earths Hadley circulation are primarily due to a season-ally reversing solstitial component, which is associated with monsoons.

Indeed, nonlinear amplication of the circulation response to seasonally varyingthermal driving is already not evident in axisymmetric models with vertical diffusionof momentum and potential temperature, even if the diffusivities are close to thesmallest diffusivities for which steady circulations can be obtained.

The top row of Figure 3 shows simulations with an axisymmetric model, similarto the models used by Held & Hou (1980) and Lindzen & Hou (1988). The verticaldiffusivities for momentum and potential temperature (dry static energy, more pre-cisely) are chosen to be equal and to be close to the smallest diffusivity for which steadycirculations for an equilibrium temperature maximum at latitude 0 = 6 could beobtained [see Walker & Schneider (2005) for details]. Compared with heating cen-tered on the equator (0 = 0), the circulation with heating displaced off the equator(0 = 6) exhibits a signicantly strengthened winter cell and weakened summer cell(compare Figures 3a and b), although the changes are not as dramatic as predictedby nearly inviscid theory (cf. Figure 2a). However, the average of the circulationsfor 0 = 6 and 0 = 6 is not nonlinearly amplied but is even slightly weakerthan the circulation driven by relaxation to the averaged equilibrium temperature(compare Figures 3a and c and contrast them with Figure 2b).

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Sigm

aa

420.2

0.8

b123

11

c29

Latitude

Sigm

a

d 70

0 30

0.2

0.8

Latitude

e 108 47

30Latitude

f 68

0

Figure 3Eulerian mass ux streamfunctions (magenta) and angular momentum (light blue) inaxisymmetric (top row) and macroturbulent (bottom row) simulations. (a, d ) Maximum radiativeequilibrium temperature on equator (0 = 0). (b, e) Maximum radiative equilibriumtemperature north of equator (0 = 6). (c, f ) Average of circulations for 0 = 6 and0 = 6. Contour intervals are 15 109 kg s1 for streamfunction and 0.1a2 for angularmomentum. Streamfunction extrema above = 0.85 are identied. Based on Walker &Schneider (2005).

Local Rossby number:measures importance ofrelative vorticity comparedwith planetary vorticity, orof nonlinear momentumadvection compared withCoriolis force

The local Rossby number Ro = / f , with relative vorticity =(a cos )1(u cos ), is a nondimensional measure of the smallness of themeridional angular momentum gradient and hence is a measure of proximity tothe nearly inviscid limit because the meridional angular momentum gradient isproportional to the absolute vorticity, f + = (a2 cos )1 M. The nearlyinviscid limit corresponds to Ro 1 such that f + = f (1 Ro) 0 wherestreamlines are horizontal (in the upper branches of the circulations at latitudes atwhich vertical velocities vanish). The winter cell in the simulation with 0 = 6 isrelatively close to the nearly inviscid limit, with local Rossby numbers Ro 0.9 inthe upper branch indicating that angular momentum is nearly homogenized alongstreamlines (Figure 3b). The equinoctial (0 = 0) and summer cells, however,are further away from the nearly inviscid limit, with local Rossby numbers in theirupper branches of Ro 0.7 and Ro 0.5, respectively. Compared with advection,diffusion is relatively more important in the weaker circulation cells. Streamlinescross angular momentum contours near the poleward boundaries of the equinoctialcells and throughout the summer cell (Figure 3a,b; streamlines of the summer cellare not visible in the gure, but the streamfunction extremum is identied).

The different degrees to which winter, summer, and equinoctial cells approach thenearly inviscid limit, combined with the dependence of circulation strength on diffu-sivities even when diffusivities are relatively small, invalidate the predictions of nearlyinviscid theory for the relative strength of the cells in the axisymmetric simulations(Walker & Schneider 2005). To understand how axisymmetric circulations depend onexternal parameters, one needs to understand the dependence of the circulations ondissipative processes, in particular near the poleward boundaries of the circulations.

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Baroclinic instability:instability convertingavailable potential energy ofmean ow into kineticenergy of perturbation

Available potentialenergy: sum of potentialand internal energyavailable for conversion tokinetic energy in adiabaticredistribution of mass(Lorenz 1955)

2.3. Influence of Eddies

Axisymmetric circulations such as those shown in the top row of Figure 3 are baro-clinically unstable [see Pierrehumbert & Swanson (1995) for a review of baroclinicinstability theory]. Nonaxisymmetric perturbations of the circulations grow, rst aslinear baroclinic waves, which later break and lead to new, macroturbulent statisti-cally steady states in which eddies transport heat, mass, and angular momentum, aswell as, in Earths atmosphere and in more realistic models, water vapor and othertracers. Eddies transport angular momentum into midlatitudes; surface westerliesappear in latitude zones of upper-level angular momentum ux convergence. Ed-dies also transport heat poleward, reducing meridional temperature gradients, thevertical zonal wind shear, and the strength of the jets at the poleward boundaries ofthe axisymmetric circulation cells. New jet maxima may appear poleward of the for-mer axisymmetric circulation cells. (Chris Walker has provided an animation of thebaroclinic instability of an axisymmetric circulation and of the subsequent spin-up ofa macroturbulent circulation. Follow the Supplemental Material link in the onlineversion of this review or at http://www.annualreviews.org/.)

The bottom row of Figure 3 shows the temporal and zonal mean mass ux stream-function and angular momentum in the statistically steady states of the macroturbu-lent circulations that correspond to the axisymmetric circulations shown in the toprow of the gure. The Hadley cells in the macroturbulent simulations extend furtherpoleward than those in the axisymmetric simulations (cf. Held & Phillips 1990). (Forsuch comparisons, because the circulations in the axisymmetric simulations do nothave sharp poleward boundaries, but weak, diffusive meridional circulations extendinto midlatitudes, the latitudes at which absolute values of the mass ux streamfunc-tions drop to below, say, 10% of the maximum values should be taken to be themeridional boundaries of the axisymmetric circulations.) Streamlines in the upperbranches of the Hadley cells in the macroturbulent simulations cross angular mo-mentum contours, even in the winter cell (near its poleward boundary). In the upperbranches of the Hadley cells, the local Rossby numbers Ro = / f , now denedwith the temporal and zonal mean relative vorticity and measuring the importanceof nonlinear momentum advection by the mean ow relative to the Coriolis force,are reduced in comparison with the axisymmetric simulations. Where streamlines arehorizontal, the local Rossby numbers are Ro 0.3 in the equinoctial cells and sum-mer cell and Ro 0.5 in the winter cell. Near the poleward boundaries in the upperbranches of all cells, in the latitude zones of the descending branches, the local Rossbynumbers are Ro 0.1. Eddies strengthen the equinoctial Hadley cells (compareFigures 3a and d ). Eddies also strengthen the summer cell, but they weakenthe winter cell (compare Figures 3b and e). Unlike in nearly inviscid theory foraxisymmetric circulations, the mass ux in the Hadley cells in the macroturbu-lent simulations is of the same order of magnitude as that in Earths Hadley cells(cf. Schneider 1984, Becker et al. 1997, Kim & Lee 2001, Walker & Schneider2005).

As the local Rossby numbers suggest, nearly inviscid theory for axisymmet-ric circulations does not provide valid predictions of the response of the Hadley

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Had

ley

circ

ula

tion

exte

nt

1 2

30

=0.7

2e4e

=0.8=0.9

Ht h' '

Figure 4Meridional extent of Hadley circulations in simulations with idealized GCM. Shown areresults from simulations with different convective lapse rates d and terrestrial values forother parameters and from simulations with convective lapse rate 0.7d but with twice andfour times Earths rotation rate e . For each set of parameters, the gure shows a series ofsimulations obtained by varying the pole-to-equator surface temperature difference inradiative equilibrium. Adapted from Walker & Schneider (2006).

circulation to displacements of the latitude of maximum heating. If the latitude ofmaximum radiative equilibrium temperature is displaced from 0 = 0 to 0 = 6,the strength of the winter cell increases by a factor of 1.5 and the strength ofthe summer cell decreases by the same factor (compare Figures 3d and e)muchsmaller and more linear changes than predicted by nearly inviscid axisymmetrictheory (cf. Figure 2). The average of the circulations for 0 = 6 and 0 = 6is not nonlinearly amplied but is similar to the circulation driven by relaxationto the averaged equilibrium temperature (compare Figures 3d and f ). Inasmuchas they cause departures from angular momentum conservation along streamlinesand thus invalidate the predictions of nearly inviscid axisymmetric theory, large-scale eddies in macroturbulent simulations play a role similar to diffusion in ax-isymmetric simulations (Walker & Schneider 2005); however, their effects gen-erally cannot be modeled as diffusion of angular momentum and/or potentialtemperature.

The parameter dependence of the meridional extent and strength of Hadleycirculations in macroturbulent simulations also differs from that predicted bynearly inviscid axisymmetric theory. Figure 4 shows the extent of Hadley cir-culations simulated with an idealized GCM as function of the nondimensionalparameter H t

h = (Ht/H0)(h/T0) appearing in the nearly inviscid axisymmet-

ric scaling estimate M(Ht) (cf. Equation 1). The constant reference height fornondimensionalization is taken to be H0 = 12 km, and T0 is taken to be the mean

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surface temperature in radiative equilibrium.2 The idealized GCM is driven by New-tonian relaxation of temperatures toward radiative equilibrium temperatures that arehemispherically symmetric, maximal at the equator, and statically unstable in thelower troposphere. If an atmospheric column is less stable than a column with tem-perature lapse rate d , where is a rescaling parameter and d = 9.8 K km1 is thedry adiabatic lapse rate, a convection scheme relaxes temperatures toward a prolewith lapse rate d . The lapse rate in the interior of the simulated Hadley circu-lations is approximately equal to the convective lapse rate d that is, convectionis in quasi-equilibrium with its large-scale environment, and horizontal temper-ature gradients are weak in the interior of the Hadley circulations (Emanuel et al.1994). Except for the convection scheme mimicking the stabilization of the ther-mal stratication by moist convection, moist processes are not taken into account inthe GCM. Figure 4 shows series of simulations with convective lapse rates between0.7d and 0.9d and with planetary rotation rates twice and four times that of Earth.For each series of simulations, the pole-to-equator surface temperature difference inradiative equilibrium h was varied from 15 K to 360 K [see Schneider & Walker(2006) for a description of the simulations and Walker & Schneider (2006) for anal-yses of the Hadley circulations]. The Hadley circulation extent generally does notscale as (H t

h)

1/2 as nearly inviscid axisymmetric theory would suggest (cf. Equation1). [There are differences between the thermal driving in the GCM and in the modelof Held & Hou (1980), such as the presence of a convection scheme and a radiativerelaxation time that varies spatially near the surface; however, these differences arenot what is primarily responsible for the different scaling behavior of the Hadleycirculation in the simulations and in nearly inviscid axisymmetric theory (Walker& Schneider 2006).] The Hadley circulation extent increases relatively rapidly forsmaller H t

h and is nearly constant for larger H

t

h . The Hadley circulation extent

decreases with planetary rotation rate, but more slowly than the 1-dependencenearly inviscid axisymmetric theory would suggest. The Hadley circulation extentincreases with decreasing convective lapse rate, whereas nearly inviscid axisymmetrictheory suggests it would be independent of lapse rates (there would only be an indirectdependence via the dependence of the height Ht on lapse rate). The nonmonotonicdependence of Hadley circulation extent on H t

h for small H

t

h in the simulations

with increased planetary rotation rate is caused by the merging of subtropical andmidlatitude jets in circulation regimes with multiple jets in each hemisphere (Walker& Schneider 2006).

Figure 5 shows the strength of the Hadley circulations in the simulations, ex-pressed in terms of the potential temperature ux at the latitude of the extremalmeridional mass ux. According to nearly inviscid axisymmetric theory (Equation 2),the potential temperature ux should scale as vM (H t h)5/2. However, thepotential temperature ux exhibits two scaling regimes, one for small H t

h , in which

2The temperature T0 might also be taken to be the mean temperature or potential temperature of thetropical troposphere for a more direct comparison with Equation 1 or with Held & Hou (1980); however, inthe simulations, these temperatures scale approximately linearly with the (varying) mean surface temperaturein radiative equilibrium, so these other choices of T0 do not affect scaling laws (Walker & Schneider 2006).

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(Ht''h)5/2

Ht''h

Pote

ntia

l tem

pera

ture

flux

101 100103

102

101

100

Ht h' '

Figure 5Potential temperature ux (nondimensionalized) in Hadley circulations simulated withidealized GCM. Shown is the potential temperature ux at the latitude of extremal meridionalmass ux, vertically integrated over the atmosphere and nondimensionalized such that a ux ofone nondimensional unit, if multiplied by the specic heat at constant pressure, correspondsto a poleward energy ux of 4.8 PW. Simulations and plotting symbols as in Figure 4.Adapted from Walker & Schneider (2006).

the scaling of the ux depends on convective lapse rates, among other parameters,and one for large H t

h , in which the scaling of the ux is approximately independent

of convective lapse rates and is roughly linear (or of a weaker power) in H t h . The

potential temperature ux also does not exhibit the 3-dependence on planetaryrotation rate that nearly inviscid axisymmetric theory (Equation 2) would suggest[see Walker & Schneider (2006) for details and additional results].

One can rationalize some aspects of the parameter dependence of the merid-ional extent and strength of the simulated Hadley circulations by considering theinuence of large-scale eddies. The meridional extent of Hadley circulations maybe limited by baroclinic instability. If one takes as a heuristic stability criterion forthe Hadley circulation that a vertical average of the meridional temperature gra-dient must not exceed the critical temperature gradient for baroclinic instability inthe quasigeostrophic two-layer model, a1T c = (/ f )v with = a1 f , anangular momentumconserving circulation with u = uM for which the geopotentialsatises 22a23 in the small-angle approximation, with = gzT /T0,is baroclinically unstable poleward of the latitude (cf. Held 2000):

c (z) (

gz2a2

v

T0

)1/4. (3)

Using the stability criterion for the two-layer model is a heuristic because there is nocritical temperature gradient for baroclinic instability in a continuously stratied at-mosphere; however, this heuristic can be made more precise using the supercriticality

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concept discussed in Section 3.2, resulting in the same scaling behavior of the lat-itude c . If the meridional extent of the Hadley circulation is limited by baroclinicinstability near the latitude c (Ht), it does not depend on h but, unlike the latitudeM(Ht) at which radiative equilibrium satises Hides theorem, it increases with in-creasing gross stability v (or with decreasing lapse rate) near the poleward boundaryof the circulation. This parameter dependence is qualitatively consistent with thesimulations for larger H t

h shown in Figure 4. One might think that the Hadley

circulation extent is determined by the smaller of the two latitude scales c and M,so that the scale M is relevant for smaller meridional temperature contrasts and thescale c for larger meridional temperature contrasts (Held 2000). However, theselatitude scales do not account quantitatively for the parameter dependence of theHadley circulation extent (e.g., on planetary rotation rate) in the simulations, for theupper branches of the circulations are not close to the nearly inviscid axisymmetriclimit that underlies both scales (i.e., local Rossby numbers are signicantly less thanone).

A further indication of the inuence of eddies on the Hadley circulation is theparameter dependence of the circulation strength. The strength of the Hadley circu-lations in the simulations is related to the divergence of eddy momentum uxes. Inthe temporal and zonal mean, the zonal momentum balance (or, equivalently, the ab-solute angular momentum balance) in the upper atmosphere, away from the planetaryboundary layer, can be approximated by

( f + )v = f (1 Ro)v S,

where v is the meridional velocity, S is the eddy momentum ux divergence, andoverbars denote temporal and zonal means at constant pressure. We have neglectedthe vertical advection of zonal momentum, which is generally small in the Tropics ofEarths atmosphere and vanishes at the latitude of the meridional streamfunction ex-tremum, where mean vertical velocities vanish. Over a wide range of the simulationsincluded in Figure 5, the mean meridional velocity at the latitude of the stream-function extremum and with it the Hadley circulation strength scale as v f 1S,that is, to the extent that the circulation extent is approximately constant so that therelevant value of the Coriolis parameter is approximately constant, they scale withthe eddy momentum ux divergence S in the upper branches of the Hadley cells(Walker & Schneider 2006). The transition in scaling laws evident in Figure 5, froma scaling law that depends on convective lapse rates to a scaling law that does not,is a transition in scaling laws for eddy momentum uxes. Eddy momentum uxesundergo a transition in scaling laws from a regime in which the thermal straticationthroughout the atmosphere is set by radiation and convection to a regime in whichlarge-scale eddies inuence the thermal stratication in the extratropics and near thepoleward boundaries of the Hadley circulation (see Section 3.2). The scaling of theHadley circulation strength in the simulations is closer to the limit Ro 0, in whichthe Hadley circulation strength depends on the eddy momentum ux divergence,than to the nearly inviscid axisymmetric limit Ro 1, in which the Hadley circula-tion strength depends directly on the thermal driving. In the simulations, changes

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in Hadley circulation strength in response to changes, for example, in the pole-to-equator temperature gradient in radiative equilibrium, are mediated by changes ineddy momentum ux divergence. This underscores the importance of eddy momen-tum uxes, which cause departures from the nearly inviscid axisymmetric limit. Ifonly eddy heat uxes were important, they could be incorporated in nearly inviscidaxisymmetric theory by modifying the thermal driving of the axisymmetric circu-lations (cf. Schneider 1984). Eddy heat uxes depend nonlinearly on mean elds(in particular, mean potential temperature gradients) and thus on the mean circu-lation, so incorporating them in nearly inviscid axisymmetric theory would intro-duce an additional nonlinearity, but it would not invalidate the basic premises of thetheory.

We have theories for Hadley circulations in the nearly inviscid axisymmetric limitRo 1, in which nonlinear angular momentum advection by the mean ow in theupper branches of the circulations is important, and the Hadley circulation strengthresponds directly to changes in thermal driving (e.g., Schneider 1977, Held & Hou1980, Lindzen & Hou 1988, Satoh 1994, Fang & Tung 1996, 1999). We also havetheories for Hadley circulations in the limit Ro 0, in which nonlinear angularmomentum advection by the mean ow in the upper branches of the circulationsis unimportant, and the Hadley circulation strength responds to changes in ther-mal driving via changes in the thermal structure and in eddy momentum uxes (e.g.Dickinson 1971, Schneider & Lindzen 1976, 1977, Fang & Tung 1994). To accountboth for the displacement of angular momentum contours away from the vertical,for example, in Earths winter Hadley cell (Figure 1c), and for the lack of a stronglynonlinear response of the Hadley circulation strength to seasonal variations in ther-mal driving (Dima & Wallace 2003), we need a theory of Hadley circulations withlocal Rossby numbers between the limiting cases of zero and one. One part of thetheory must account for the structure of the Hadley circulation given the structureof eddy uxes. Another part of the theory must account for the structure of eddyuxes given the Hadley circulation and other mean elds. The rst part of the theorymay be based on the framework of the weak-temperature gradient approximation fortropical dynamics, which allows one to derive properties of Hadley circulations withintermediate local Rossby numbers (Sobel et al. 2001, Polvani & Sobel 2002). Thesecond part of the theory must be based on a theory of atmospheric macroturbulence,aspects of which are discussed in the next section. We need such a theory to accountfor the response of Earths Hadley circulation not only to seasonally varying thermaldriving but also, for example, to increases in greenhouse gas concentrations (e.g., Rind1998, Dai et al. 2001) and to changes in tropical temperatures such as are associatedwith El Nino (e.g., Hou 1993, 1998, Chang 1995, Robinson 2002). A theory linkingthe Hadley circulation and eddy uxes in idealized dry atmospheres with axisymmet-ric circulation statistics is a prerequisite for more complete theories that account forstatistically nonaxisymmetric circulation features and moist processes, such as mon-soon circulations, cloud-radiative feedbacks, or inuences of the circulation on theconvective stability of the atmosphere through its effect on the distribution of watervapor.

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Isentropic coordinates:coordinate system withentropy or potentialtemperature as verticalcoordinate

3. THE EXTRATROPICAL CIRCULATION

3.1. Isentropic Mass Circulation and Eddy Fluxes

In the extratropics, the bulk of the transport of heat, mass, angular momentum, andof tracers such as water vapor is effected by large-scale eddies. If one disregards moistconvection and latent heat release in large-scale phase changes of waterprocesseswhose role in extratropical dynamics is poorly understoodlarge-scale eddies areonly weakly inuenced by diabatic processes because radiative damping timescalesare a factor of 520 greater than eddy turnover timescales. As already recognizedby Shaw (1930, Chapter 6), it is therefore convenient to analyze the extratropicalcirculation in isentropic coordinates, in which air parcel displacements in large-scaleeddies approximately follow horizontal coordinate surface.

In isentropic coordinates, one can generalize many relations known from quasi-geostrophic theory without making assumptions such as that variations of the Coriolisparameter or uctuations of the static stability about a xed reference value be smallassumptions central to quasigeostrophic theory that are likely not adequate in studiesof the general circulation. With the development of the theory of baroclinic instability(Charney 1947, Eady 1949) and the rst numerical demonstration of how large-scaleeddies affect the general circulation (Phillips 1956), to name two examples, quasi-geostrophic theory has led to some of the most fundamental insights in atmosphericdynamics in the twentieth century. In the 1980s, the realization that relations fromquasigeostrophic theory can be generalized in isentropic coordinates without makingrestrictive quasigeostrophic assumptions, largely because quasigeostrophic dynamicscan be viewed as representing dynamics on isentropes (Charney & Stern 1962), ledto renewed interest in analyzing the extratropical circulation and especially eddy-mean ow interactions in isentropic coordinates (e.g., Andrews 1983, Hoskins et al.1985, Tung 1986, Johnson 1989, Hoskins 1991). The dynamics of the circulationnear the lower boundary, which is a uctuating boundary in isentropic coordinatesbecause the surface potential temperature uctuates, and their correspondence toquasigeostrophic dynamics have recently become clearer (Held & Schneider 1999,Koh & Plumb 2004, Schneider 2005). We are now in a position to analyze the ex-tratropical circulation in isentropic coordinates, relate it to eddy uxes of quantitiesthat are materially conserved in adiabatic and inviscid air parcel displacements, andaddress questions not amenable to quasigeostrophic theory, such as the question ofwhat controls the extratropical static stability, a xed parameter in quasigeostrophictheory.

The mean mass ux along and across isentropes forms hemispheric meridional cir-culations (Figure 6). Upward and downward uxes across isentropes imply diabaticwarming and cooling. The divergence of cross-isentropic uxes is balanced by theconvergence of along-isentropic uxes, and vice versa. To the extent that air parceldisplacements in large-scale eddies approximately follow isentropes, the isentropicmass ux streamfunction is approximately the Lagrangian mass ux streamfunction.Isentropic mass ux streamlines are approximately mean Lagrangian trajectories ofair masses. The meridional mass ux along isentropes represented by the isentropic

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Latitude

Pote

ntia

l tem

pera

ture

(K)

250

500

750

20

60

20

220

120

50 500250

300

350

Hadley cells

Figure 6Isentropic mass ux streamfunction for January (temporal and zonal mean according to thesame reanalysis data as in Figure 1). Negative streamfunction values (dashed magenta contours)correspond to clockwise rotation, positive values (solid magenta contours) to counterclockwiserotation. The contour interval is 20 109 kg s1. The solid blue line is the median surfacepotential temperature. The dotted blue lines are mean pressure contours (250, 500, and750 hPa).

streamfunction transports entropy: The meridional entropy ux is the mass ux mul-tiplied by specic entropy, a constant along isentropes. The tropospheric meridionalux of (dry) entropy (or potential temperature) at any given latitude is the tropo-spheric extremum of the isentropic mass ux streamfunction multiplied by the grossstability, the effective specic entropy (or potential temperature) difference betweenthe poleward upper branch and the equatorward lower branch of the circulation. It isevident that the circulation shown in Figure 6 transports entropy poleward. Like thecirculation envisaged by Hadley, it is thermally direct and hemispheric, but unlikeHadleys circulation, at least outside the Tropics, it cannot be understood as an ax-isymmetric circulation with an angular momentumconserving upper branch. Thatthe isentropic mass circulation in the extratropics is primarily an aggregate of eddymotions is obvious when one compares it with the Eulerian mean mass circulation(compare Figure 6 with Figure 1c): The direction of the isentropic mass ux inmidlatitudes is opposite to that of the Eulerian mean mass ux in the Ferrel cells.This also implies that, in the extratropics, the eddy entropy ux dominates the meanentropy ux associated with the Ferrel cells.

The mass ux along isentropes in the extratropics is associated with eddy uxes ofpotential vorticity and of surface potential temperature, the two principal materiallyconserved quantities in adiabatic and inviscid air parcel displacements in the extrat-ropics (Hoskins et al. 1985). The isentropic mass ux streamfunction is proportionalto the vertical (i.e., cross-isentropic) integral of the meridional mass ux along isen-tropes, which is related to eddy uxes of potential vorticity and of surface potential

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temperature by3

ib

vd

ib

v P

Pd 0 vgs s

s + ME . (4)

The meridional mass ux along isentropes (left-hand side) is associated with an eddyux of potential vorticity P along isentropes (rst term on right-hand side), an eddyux of surface potential temperature s (second term), and an Ekman mass ux MEdue to zonal frictional forces (third term). The potential temperature b () is a nom-inal lower boundary of the integration, smaller than or equal to the lowest surfacepotential temperature that occurs at the latitude under consideration; the poten-tial temperature i () is some potential temperature greater than the highest surfacepotential temperature that occurs at the latitude under consideration. The bal-ance equation (4) is a statement of zonal momentum or potential vorticity balancein the temporal and zonal mean in isentropic coordinates. It holds for circulationswith small Rossby numbers, such as in the extratropics of Earths atmosphere, andgeneralizes similar expressions known from quasigeostrophic theory; see Schneider(2005) for a derivation, comparisons with quasigeostrophic theory, and a discussionof the conventions on which the balance equation depends. [Tung (1986) and Held &Schneider (1999) give precursors (incomplete in the treatment of the intersections ofisentropes with the surface) and Koh & Plumb (2004) offer an essentially equivalentvariant of the balance equation (4).]

The signicance of the balance equation (4) lies in the fact that it relates theisentropic mass circulation and thus entropy uxes to eddy uxes of two scalars thatare materially conserved in adiabatic and inviscid air parcel displacements. A closuretheory for the eddy uxes of surface potential temperature and of potential vorticityalong isentropes must be one central part of any general circulation theory for theextratropics. If the dependence of these eddy uxes and thus of entropy uxes onmean elds is known, and if eddies are only weakly inuenced by diabatic processesso that correlations between eddy uctuations and diabatic heating can be neglectedin mean-eld equations, the thermal structure of the atmosphere is determined byenergy-balance requirements, and the vertical zonal wind shear is determined bythermal-wind balance. If the upper boundary of the integration in the balance equa-tion (4) is taken to be the top of the atmosphere (i ), the left-hand side vanishesby mass conservation (cf. Andrews et al. 1987, Chapter 3.8). Upon integration overan atmospheric column, the two eddy-ux terms on the right-hand side balance theEkman mass ux owing to zonal frictional forces. Therefore, to the extent that zonalfrictional forces can be modeled as boundary-layer turbulent dissipation acting onmean zonal winds, the vertical integral of the eddy-ux terms determines the mean

3Overbars denote temporal and zonal means: ()s along the surface; () along isentropes if the argumentdepends on a vertical coordinate; and () = ( )/ along isentropes weighted by the isentropic density = g1pH( s ), where p is pressure, is potential temperature, and H() is the Heaviside stepfunction. Primes denote uctuations about the respective means. The subscript s marks surface quantities,0 is the mean isentropic density at the mean surface potential temperature s , and vg is the geostrophicmeridional velocity.

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zonal surface wind [see Green (1970) and Held & Hoskins (1985) for analogousrelations in quasigeostrophic theory]. A theory of the general circulation of dry at-mospheres can thus be constructed from a closure theory for the eddy uxes of surfacepotential temperature and of potential vorticity along isentropes. Developing a clo-sure theory for eddy uxes of two scalars that are conserved in adiabatic and inviscidair parcel displacements appears to be easier than developing a closure theory, forexample, for eddy entropy uxes in height-coordinates (vector-valued uxes) or foreddy momentum uxes (uxes of a quantity that is not materially conserved in adia-batic and inviscid displacements). Yet we do not have such a closure theory. We haveelements of it, and we have some qualitative understanding of the eddy uxes.

Qualitatively, the direction of the eddy uxes can be understood by assuming thateddies mix potential vorticity and surface potential temperature downgradient (Held& Schneider 1999, Schneider 2005). The isentropic meridional gradient of poten-tial vorticity is generally positive in the interior atmosphere, that is, on isentropesthat, at any given latitude, typically do not intersect the surface. In the interior at-mosphere, downgradient mixing leads to southward eddy uxes of potential vorticityalong isentropes, which, by balance equation (4), are associated with poleward massuxes. The equatorward uxes necessary to close the isentropic mass circulation oc-cur in the surface layer, that is, on isentropes that, at any given latitude, sometimesintersect the surface ( Juckes et al. 1994). In the surface layer, the isentropic merid-ional gradient of potential vorticity is generally negativeit is related to the surfacepotential temperature gradient, similar to the potential vorticity gradient in the lowerlayer of a quasigeostrophic two-layer model (Schneider 2005)so that downgradi-ent mixing leads to northward eddy uxes of potential vorticity along isentropes. Bybalance equation (4), the northward eddy uxes of potential vorticity are associatedwith equatorward mass uxes. Similarly, downgradient mixing implies poleward eddyuxes of surface potential temperature and so, by balance equation (4), also equator-ward mass uxes. Much of the near-surface equatorward mass ux occurs at potentialtemperatures less than the median surface potential temperature (Figure 6), that is,in cold air outbreaks. Figure 7 illustrates the isentropic mass circulation and eddyuxes in the troposphere. The Ekman mass ux owing to zonal frictional forces (themean mass ux in the lower branch of the Ferrel cell) is poleward in the region ofsurface westerlies and hence has opposite direction of the mass uxes associated withthe near-surface eddy uxes.

3.2. Eddy Fluxes and the Extratropical Thermal Stratification

A semiempirical quantitative result of how eddy uxes constrain the thermal stratica-tion of the extratropical troposphere can be obtained under the assumption that eddiesmix potential vorticity and surface potential temperature diffusively, with an eddy dif-fusivity that has no essential vertical structure in the altitude range over which eddieseffect signicant entropy uxes, or mass uxes along isentropes (Schneider & Walker2006). The assumption that the eddy diffusivity has no essential vertical structure isjustiable in the limit of strongly baroclinically unstable circulations. In this limit,one would expect the inverse cascade of eddy energy from the scales of baroclinic

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Latitude

Pote

ntia

l tem

pera

ture

(K)

0 50

300

350 Tropopause

v P *

vgsss

Mass flux

Figure 7Sketch of isentropic mass circulation and eddy uxes (based on reanalysis data for theNorthern Hemisphere annual mean). Solid magenta lines are contours of the isentropic massux streamfunction. Wavy lines indicate eddy uxes. The solid blue line is the median surfacepotential temperature (approximately equal to the mean surface potential temperature); thedotted blue lines demarcate the surface layer of isentropes within which, at any given latitude,the instantaneous surface potential temperature lies 90% of the time.

Inverse energy cascade:turbulent cascade ofbarotropic eddy kineticenergy from smaller tolarger scales

Barotropic kinetic energy:kinetic energy of verticallyaveraged ow

instability to larger horizontal and vertical scales to lead to barotropization of theenergy-containing eddies, which dominate the advection of potential vorticity andsurface potential temperature uctuations (e.g., Charney 1971, Salmon 1980, 1982,Held & Larichev 1996, Smith & Vallis 2002). For scaling estimates, the assumptionthat the eddy diffusivity has no essential vertical structure may also be justiable inweakly baroclinically unstable circulations, provided that, as appears to be the casein Earths atmosphere, the streamfunction amplitudes of linear or weakly nonlinearbaroclinic waves vary only weakly in the altitude range over which the waves effectsignicant entropy uxes (cf. Simmons & Hoskins 1976, 1977, Held 1978, Valdesand Hoskins 1988).

A constraint on the thermal stratication of the extratropical troposphere fol-lows from the balance equation (4) upon substitution of diffusive eddy ux closuresv P

Dy P| and vgs ss Dy s , with eddy diffusivity D and with latitude

coordinate y = a. Suppose that, at each latitude, there is a well-dened potentialtemperature e up to which signicant eddy entropy uxes extend and that above thatpotential temperature the atmosphere can be assumed to be in radiative or radiative-convective equilibrium. With the potential temperature e as the upper boundary ofthe integration, the left-hand side of the balance equation (4) then vanishes by massconservation. If one ignores the Ekman mass ux (the mean mass ux in the Ferrelcell), substitutes the diffusive eddy ux closures, and uses the assumption that theeddy diffusivity has no essential vertical structure, the eddy diffusivity drops out ofthe balance equation (4), and one obtains an estimate of the pressure pe = p(e ) up

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Supercriticality:nondimensional measure ofslope of isentropes or ofvertical extent of eddyentropy uxes

to which signicant eddy entropy uxes extend as a function of mean-eld quantities(Schneider & Walker 2006). [Alternatively, and potentially more generalizable to thecirculations of the oceans and of planets without a solid surface, one can assume thatthe geostrophic component of the isentropic mass circulation closes at potential tem-perature e ; see Schneider (2006).] If convective entropy uxes extend higher thanentropy uxes associated with large-scale eddies, the pressure pe will be higher thanthe pressure pt at the tropopausethe upper boundary of the atmospheric layerwithin which the bulk of the entropy the atmosphere receives at the surface is redis-tributed (Schneider 2004). If entropy uxes associated with large-scale eddies extendas high or higher than convective entropy uxes, large-scale eddies will modify orset the tropopause height, so that the pressure pt at the tropopause will be similar tothe pressure pe . Combining the constraint pt pe on tropopause pressure with theestimate of the pressure pe obtained from the balance equation (4), one nds that thesupercriticality

Sc = f

y s

v, (5)

with bulk stability v = 2p s ( ps pt), cannot signicantly exceed one: Sc 1(Schneider & Walker 2006). If the tropopause height is somewhat greater than thescale height, as is the case in Earths atmosphere (apparently by coincidence), thebulk stability is approximately equal to the potential temperature difference betweentropopause and surface, v t s , similar to the gross stability that appearedin the discussion of the Hadley circulation. The supercriticality is a nondimensionalmeasure of the vertical extent of eddy entropy uxes relative to the tropopause height,Sc ( ps pe )/( ps pt), or a nondimensional measure of the slope of near-surfaceisentropes.

Figure 8 shows bulk stabilities v and scaled surface potential temperature gradi-ents ( f/)y s in the macroturbulent idealized GCM simulations already discussedin the context of the Hadley circulation in Section 2.3. Since the supercriticality con-straint Sc 1 is based on diffusive eddy ux closures, it can only be expected tohold on scales greater than eddy length scales, so Figure 8 shows averages over ex-tratropical baroclinic zones. For small scaled surface potential temperature gradients( f/)y s and for sufciently small convective lapse rates d , bulk stabilities vare approximately independent of the scaled surface potential temperature gradients.In this regime, eddy entropy uxes are weak, the thermal stratication and tropopauseheight are set by radiation and convection, and Sc < 1. When scaled surface potentialtemperature gradients ( f/)y s approach the value at which Sc 1 for a radiative-convective thermal stratication, bulk stabilities v increase in proportion to thescaled surface potential temperature gradients, such that the supercriticality does notsignicantly exceed one. In this regime, eddy entropy uxes stabilize the thermalstratication and modify the tropopause height. The smaller the bulk stability for aradiative-convective thermal stratication (i.e., the larger the convective lapse rated ), the smaller is the scaled surface potential temperature gradient ( f/)y s atwhich the line Sc 1 is reached. For sufciently large surface potential temperaturegradients, all simulations condense onto the line Sc 1. The simulations span a wide

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Earth

10 50 100

10

50

100

v (K

)

(K)

Sc < 1

Sc > 1

=0.7

2e4e

=0.8=0.9

Figure 8Extratropical bulk stability v and scaled surface potential temperature gradient ( f/)y sin idealized GCM simulations. Displayed quantities are averages over extratropical barocliniczones, for the same simulations as in Figures 4 and 5. The corresponding quantities forEarths atmosphere (from reanalysis data for the Northern Hemisphere annual mean) areshown for comparison. The dashed line represents supercriticality Sc = 1, with Sc < 1 aboveit and Sc > 1 below it. Based on results of Schneider & Walker (2006).

range of atmospheric circulations, including Earth-like circulations and circulationswith multiple jets and regions of surface westerlies in each hemispherefeatures suchas those seen on the giant planets (Figure 9a and b).

The supercriticality structurally resembles quasigeostrophic instability measures(e.g., Pedlosky 1970, 1979, Held & Larichev 1996). For example, a necessary con-dition for baroclinic instability in the quasigeostrophic two-layer model is that acounterpart of the supercriticality exceeds one. (We used this condition in the discus-sion of the meridional extent of the Hadley circulation in Section 2.3.) For Earth-likeatmospheres, in which the bulk stability is approximately equal to the potential tem-perature difference between tropopause and surface and in which baroclinic zonesare centered in midlatitudes such that | f/| = |a tan | a , a supercriticality Sc 1implies t s |ay s |, which means that the lowest isentrope that crosses thetropopause in high latitudes grazes the surface in low latitudes, as is the case inEarths atmosphere and in Earth-like atmospheres in the idealized GCM simulations(cf. Figures 1a and 9a). That Earths atmosphere is close to a state with supercritical-ity one has led to hypotheses that it may be in a baroclinically adjusted state, that is,a state approximately neutral with respect to baroclinic instability (e.g., Stone 1972,1978, Lindzen & Farrell 1980, Held 1982, Lindzen 1993, Stone & Nemet 1996).Although structurally similar to quasigeostrophic baroclinic adjustment constraints,the supercriticality constraint Sc 1 neither presupposes nor implies, however, thatthe atmosphere is neutral or stable with respect to baroclinic instability.

The supercriticality constraint Sc 1 on the thermal structure of the extratropicaltroposphere constrains the changes in tropopause height and thermal stratication

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Sigm

a

300270

0.2

0.8

Latitude

Sigm

a

330 270

50 0 50

0.2

0.8

36

24

a

b

Figure 9Mean zonal wind (magenta), potential temperature (light blue), tropopause (orange), and eddylength scales (green) in two idealized GCM simulations. As in Figure 1, the contour intervalsare 10 K for potential temperature and 4 m s1 for zonal wind. The meridional extent of thewavy green lines indicates eddy length scales (the lines are centered at the latitude ofmaximum eddy kinetic energy). (a) Simulation with planet radius and rotation rate of Earth.(b) Simulation with planet radius of Earth and rotation rate four times that of Earth. Adaptedfrom Schneider & Walker (2006).

that are possible as the climate changes. See Schneider (2006) for a review of howthe tropopause height and the thermal stratication are related and can be inuencedby eddy dynamics such as those discussed here and by other processes such as moistconvection. Evidence that changes in Earths extratropical tropopause height andthermal stratication in recent decades scale in a manner consistent with Sc 1 isgiven by Schneider (2004).

3.3. Eddy Energies, Eddy Length Scales, and SeparationScales between Jets

The supercriticality constraint Sc 1 also constrains characteristics of atmosphericmacroturbulence. In quasigeostrophic models, circulations with Sc 1 are possibleand have been simulated (e.g., Panetta 1993, Lapeyre & Held 2003). Quasigeostrophicmacroturbulence with Sc 1 is characterized by strong nonlinear eddy-eddy inter-actions, which mediate an inverse cascade of barotropic eddy kinetic energy from thescale of the linearly most unstable baroclinic waves to larger scales (Held & Larichev1996). Because the thermal stratication is taken to be xed in quasigeostrophic the-ory, eddies in quasigeostrophic models cannot stabilize the thermal stratication or

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Baroclinic kinetic energy:kinetic energy ofuctuations about verticallyaveraged ow

Eddy

APE

(MJ m

-2 )

Baroclinic eddy kinetic energy (MJ m-2)10-4 10-2 100

10-4

10-2

100Earth

10-1 100

10-1

100

Figure 10Eddy available potentialenergy (EAPE) versusbaroclinic eddy kineticenergy (EKEbc). Thedashed lines represent thelinear relationEAPE = 2.25 EKEbc.Simulations and plottingsymbols as in Figure 8.Based on results ofSchneider & Walker (2006).

modify counterparts of the tropopause height. In contrast, for more realistic atmo-spheric macroturbulence, the result that surface potential temperature gradients andthe extratropical thermal stratication adjust such that Sc 1 suggests, by analogyto the quasigeostrophic counterparts of the supercriticality, that nonlinear eddy-eddyinteractions and the concomitant inverse energy cascade are inhibited.

Perhaps the clearest indication that nonlinear eddy-eddy interactions and an in-verse energy cascade beyond the scale of the linearly most unstable baroclinic wavesare inhibited comes from eddy energies in the idealized GCM simulations. Because itis nonlinear eddy-eddy interactions that distort the linearly most unstable baroclinicwaves in macroturbulent ows, scales and structures of the energy-containing e