## 1. Introduction

That the meridional thermal contrast produced by radiative–convective equilibrium can produce a meridional circulation on a rapidly rotating planet like the earth is not at all obvious: a zonal wind in cyclostrophic balance is a solution of the equations of motions in the inviscid case. However, such a solution, with no meridional circulation, is untenable in the presence of friction.

The role of friction in symmetric models of the Hadley circulation is a contradictory one. On one hand, Ekman layers are the site at which the thermal wind balance is broken and a meridional circulation, down the pressure gradient, is allowed (Charney 1973). This frictionally driven flow results in a meridional cell, and theoretical ideas usually appeal to conservation of angular momentum in order to explain the cell properties and the associated surface zonal winds (Schneider 1977;Held and Hou 1980). Thus nonzero friction is required for the existence of the meridional flow, but angular momentum conservation is supposed not to be upset by friction.

The constraints imposed by quasi conservation of angular momentum are strong and lead to well-accepted scaling arguments for the extent of the meridional cell (Lindzen 1990). However, in order to make quantitative estimates beyond the scaling laws, the precise functional relationship between streamfunction and angular momentum must be determined. This is a difficult task in the present context because

- all the streamlines are closed and there is no “upstream region” where the dependence of the angular momentum on the streamfunction is set by external processes
- according to the numerical calculations presented in the published literature, all the streamlines pass through a viscous boundary layer forbidding application of the Prandtl–Batchelor theorem.

One hypothesis is that the meridional circulation acts as a vigorous blender so that angular momentum is approximately homogenized (Bretherton and Turner 1968;Fang and Tung 1996). While appealing, this scenario has not been deductively demonstrated, and its limits of validity (if any) have not been established.

In the following we examine a regime that attempts to bridge the gap between frictionally dominated linear dynamics and a quasi-inviscid, fully nonlinear flow. The meridional flow is assumed to be weak as in Charney’s linear model, but advection of angular momentum and potential temperature by the meridional circulation is included. Despite the weakness of the meridional flow, the streamlines are approximately coincident with the lines of constant angular momentum and potential temperature, except in thin Ekman layers. We show that, in all parameter ranges, the circulation homogenizes laterally the potential temperature within the bulk of the meridional circulation. Thus the thermal gradients imposed by the radiative–convective heating are greatly reduced. In the parameter range where the circulation becomes very vigorous, both in the meridional and zonal directions, the angular momentum is also homogenized.

## 2. Nondimensionalization and conservation laws

*a*is the radius of the sphere,

*φ*is the latitude, and

*y*

*a*

*φ,*

*c*

*y*

*φ*

*y*

^{2}/

*a*

^{2}

*ψ*is the meridional streamfunction defined by

_{y}

*ψ*

*w,*

_{z}

*ψ*

*cυ,*

*J*(

*A, B*) =

*A*

_{y}

*B*

_{z}−

*A*

_{z}

*B*

_{y}.

The right-hand side of (2.2) contains anisotropic viscosity. Apart from the anisotropy these dissipative terms have been constructed by analogy molecular processes. Thus, the horizontal viscosity *ν*_{H} mixes the rotation rate *ω* = Ω + *u*/(*ac*) so that there are no viscous stresses if the atmosphere is in solid-body rotation (e.g., Read 1986). If the viscosity is isotropic (i.e., if *ν*_{H} = *ν*_{V}), then the term proportional to *ν*_{H} in (2.2) will be much less than the term proportional to *ν*_{V} in the thin shell limit. This is why the formulations of Schneider (1977) and Held and Hou (1980, HH hereafter) do not include horizontal viscosity. However, our results suggest that if *ν*_{H} = 0 then (2.2) is ill-posed: singularities are not surprising since there is nothing that prevents *u* from becoming discontinuous in *y.* If *ν*_{H} ≠ 0 these discontinuities can be smoothed by lateral boundary layers.

*u*we will use the angular momentum

*M*

*ac*

^{2}

*uc*

*M*produces the more transparent conservation equation

*M*

_{t}

*J*

*ψ, M*

*ν*

_{V}

*M*

_{zz}

*ν*

_{H}

*c*

^{4}

*c*

^{−2}

*M*

_{y}

_{y}

*ν*

_{H}=

*ν*

_{V}except in lateral boundary layers. Again, the lateral friction tends to homogenize the zonal rotation rate,

*ψ*

_{zz}, rather than the zonal vorticity,

*c*

^{−2}

*ψ*

_{zz}.

*τ*is the relaxation time toward the radiative–convective equilibrium distribution

*θ*

_{E}(

*y, z*).

*ψ*

*M*

_{z}

*ψ*

_{zz}

*θ*

_{z}

*z*

*H.*

*M*to the local planetary value Ω

*ac*

^{2}.

*θ*

_{E}is of the form

*θ*

_{E}

*θ*

_{0}

_{V}

*z*

*H*

_{H}

_{E}

*y*

*u*is implicitly scaled with Ω

*a*:We have used a standard notation for the thermal Rossby number

*R,*the Ekman number

*E,*and the aspect ratio

*ϵ,*defined as

The viscous scaling in (2.11) for the streamfunction is imposed by the Ekman balance responsible for the meridional circulation: a meridional Ekman transport is generated to the left of the geostrophically balanced vertical shear. In turn the vertical shear of the zonal flow is in thermal wind balance.

*α*is the ratio of the viscous timescale across the depth of the model atmosphere to the relaxation time toward the radiative–convective equilibrium; Δ measures the prescribed stratification in the absence of circulation;

*γ*is the ratio of the spindown time due to the drag to the viscous timescale, and

*μ*measures the ratio of the horizontal to the vertical viscosity.

With the scaling (2.11), the Ekman number appears only in the zonal vorticity equation (2.13b), while the frictional and diffusive terms in the angular momentum and potential temperature equations are as large as the inertial terms. This is not surprising because in a symmetric model the meridional circulation is all generated inside viscous boundary layers.

In summary there are six nondimensional parameters, *R, E, α,* Δ, *γ, μ,* and one externally prescribed latitudinal distribution of temperature, Θ_{E}(*y*).

## 3. Asymptotic reduction in the small Rossby number limit

*R,*it is possible to obtain an expansion that reduces the system (2.13) to a simpler set of equations. There is a distinguished limit in which the parameters are ordered with respect to

*R*≪ 1. Specifically we take

*R*→ 0 with

*μ*

*R*

^{2}

*μ*

_{2}

*γ*

*R*

^{2}

*γ*

_{2}

*α*

*R*

^{2}

*α*

_{2}

*R*

^{−1}

*N*

^{2}

*α*and

*γ*should not be interpreted as a requirement that the vertical viscosity

*ν*

_{V}is large; rather, it is a statement about the slowness of the radiative processes,

*τ,*and of the surface drag. The importance of viscosity is measured by the Ekman number, which is assumed to be small.

Neglecting lateral dissipation, (3.2a) shows that the surface zonal wind, proportional to the difference between the surface angular momentum *M*(*y,* 0, *t*) and its planetary value, is given by the meridional transport of vertical shear. If the vertical shear *M*_{z} is in approximate thermal wind balance, then *M*_{z} = *O*(*R*), and the meridional flux of angular momentum *R*(*ψM*_{z}_{y} is *O*(*R*^{2}). The choice *γ* = *O*(*R*^{2}) on the right-hand side of (3.2a) guarantees a fair competition between advection, which makes angular momentum constant along streamlines, and bottom drag, which injects gradients in the surface angular momentum. Also with *μ* = *O*(*R*^{2}) and ∂_{t} = *O*(*R*) all the terms in (3.2a) have the same asymptotic order, namely, *O*(*R*^{2}).

Similarly, (3.2b) indicates that the lateral redistribution of vertically averaged equilibrium temperature is accomplished through the meridional transport of temperature that departs from the vertical average, that is, the term *R*(*ψθ*_{z}_{y}. If the imposed stratification Δ vanishes, then the stratification driven by the meridional circulation is *O*(*R*), and its meridional transport is *O*(*R*^{2}). The choice *α* = *O*(*R*^{2}) allows a fair competition between advective and radiative processes in establishing the temperature distribution. Given this scaling, Δ must be *O*(*R*^{−1}) in order to contribute to the vertical stratification as much as advection.

It is possible that other parameters orderings lead to interesting balances. The one selected here allows us to include many dynamical processes in the balance while being amenable to analytic exploration. Balances involving fewer processes are recovered on considering special limits of the final evolution equations.

Remarkably, as we will show in section 4, the distinguished limit (3.1), leads to quantitative agreement with the numerical solutions of (2.13) published in the literature, even when *α*_{2} and *γ*_{2} are large.

_{t}=

*R*∂

_{t1}

*M, θ,*and

*ψ*are expanded in powers of

*R*; that is,

### a. The O(R^{0}) balance

*O*(

*R*

^{0}) in the system (2.13) areAt this order both the top and bottom boundary conditions are

*M*

_{0z}=

*θ*

_{0z}= 0. Thus the solution of (3.4) is

*M*

_{0}

*y, t*

_{1}

*θ*

_{0}

*y, t*

_{1}

### b. The O(R) balance

*E*≪ 1, it is not so uniformly in height, and it is kept at this order. The solution of (3.6), which satisfies the no-flux and no-stress conditions at the top and bottom, isThe meridional circulation, characterized to a first approximation by

*ψ*

_{0}, depends on the yet unknown latitudinal temperature gradient Θ

_{y}, which is not necessarily clamped to that imposed by the radiative–convective equilibrium Θ

_{E}. In (3.7) we have introduced the variable

*ϕ,*which is defined to be the solution of the ordinary differential equation

*E*

^{2}

*c*

^{2}

*ϕ*

_{zzzz}

*y*

^{2}

_{y}

*ϕ*

*c*

^{4}

*ϕ*=

*ϕ*

_{zz}= 0 at

*z*= 0, 1. The explicit expression for

*ϕ*is somewhat complicated and is given in appendix A: it is a nonlinear version of Charney’s (1973) solution with an Ekman boundary layer thickness proportional to [−

*y*(

^{2})

_{y}/(

*cE*)

^{2}]

^{−1/4}, instead of (4

*y*

^{2}/

*E*

^{2})

^{−1/4}.

If the rotation rate, proportional to *y* on the sphere, vanishes, then *ϕ,* the solution of (3.8), becomes very large, of order *E*^{−2}; that is, *ϕ* obeys the nonrotating balance. Because the actual meridional circulation, given in (3.7b), is proportional to the temperature gradient Θ_{y} as well as to *ϕ,* we expect a vigorous circulation unless the thermal gradient becomes small near the equator. Notice that the nonrotating behavior is also obtained if the angular momentum

*E*≪ 1, the streamfunction,

*ψ*

_{0}in (3.7b), is independent of height, outside the thin top and bottom boundary layers. Thus the zeroth-order angular momentum and temperature,

*J*

*ψ*

_{0}

*J*

*ψ*

_{0}

*y, t*

_{1}) andΘ(

*y, t*

_{1}) automatically satisfy the zero-order boundary conditions at the top and the bottom. Therefore, the angular momentum and the potential temperature do not need boundary layer corrections at

*z*= 0 and

*z*= 1. This makes the derivation of the evolution equations for

In this expansion the vertical stratification, *θ*_{1z} in (3.7c), is *O*(*R*) and is determined by two terms. The first one, proportional to *N*^{2}, is a familiar one: it arises from a balance between vertical diffusion and the stratification imposed by the radiative equilibrium. The second term on the right-hand side of (3.7c) is “dynamically driven” stratification: the shear of the meridional velocity tilts the isotherms from their vertical position until arrested by vertical diffusion. In this way differential advection by the meridional velocity turns *horizontal* temperature gradients into *vertical* temperature gradients. Because the strength of the meridional overturning is itself proportional to the latitudinal temperature gradient, the dynamically driven stratification depends on the *horizontal temperature gradient squared.* Thus the dynamically driven stratification is always stable.

In a similar way the zonal shear, *M*_{1z} in (3.7a), is generated by shear tilting the lines of constant

To obtain the *O*(*R*) angular momentum and temperature corrections, *M*_{1} and *θ*_{1}, (3.7a,c) must be integrated further in *z.* The additional boundary conditions necessary for the integration are arbitrary, and here they are chosen to be *M*_{1}|_{z=0} = 0 and ^{1}_{0}*θ*_{1} *dz* = 0. Thus *M*_{z} = *R*^{2}*γ*_{2}(*c*^{2}).

### c. The O(R^{2}) balance

*O*(

*R*

^{2}) angular momentum and potential temperature equations arewith boundary conditions

*θ*

_{2z}|

_{z=0,1}=

*M*

_{2z}|

_{z=1}= 0 and

*M*

_{2z}|

_{z=0}=

*γ*

_{2}(

*c*

^{2}). The equations in (3.10) have a“solvability condition,” which is obtained by integrating vertically over the whole depth:The parameter

*μ*

_{2}, which corresponds to horizontal viscosity, can be much less than one but it is necessary to retain this process in order to obtain solutions that are regular everywhere.

*ϕ*(

*z, y,*

*F*

^{2}

^{2}

_{y}

*c*

^{2}. Thus in (3.12a) there is a competition between homogenization of rotation rate, homogenization of angular momentum, and relaxation toward the planetary rotation rate.

The meridional advection of potential temperature produces an analogous term in (3.12b) proportional to the cube of the meridional temperature gradient, as well as a quasi-linear term where the diffusivity is proportional to the imposed stratification *N*^{2}: if the imposed stratification is unstable, the “diffusivity” of this term is negative. In (3.12b), all terms except the relaxation toward the radiative–convective equilibrium conspire to homogenize the vertically averaged potential temperature laterally.

The nonlinear terms in (3.12) all become singular when *yc*^{−2}(^{2})_{y} = *π*^{4}*E*^{2}: in appendix B it is shown that this value coincides with the threshold for symmetric instability in the original set of equations (2.13). Moreover, the diffusivity in the quasi-linear term of the temperature equation [the term *λFα*_{2}*N*^{2}Θ_{y} on the right-hand side of (3.12b)] becomes negative, even for positive prescribed stratification *N*^{2}. When the threshold of symmetric instability is passed, the validity of the evolution equations is lost: at the threshold of the instability, terms that were neglected become as large as those retained. The singularity signals a failure of our expansion. The regime where symmetric instability plays a role is certainly a very interesting one; however it will not be examined in this work.

The essential point emerging from (3.11) is that, in steady state, there is competition between advection, which tends to homogenize angular momentum, and surface drag, which injects horizontal gradients at the surface. Similarly, advection by the meridional cell homogenizes potential temperature laterally against the gradients imposed by the radiative equilibrium.

*ω*

_{r}, is

*O*(

*R*) and is given byUsing the expressions for

*M*

_{1}and

*ψ*

_{0}in (3.7a,b) the net relative rotation rate is given byAs long as the circulation is thermally and “angular momentum” direct,

*M*

_{1}is positive everywhere in the domain and the net relative rotation of the atmosphere,

*ω*

_{r}, is eastward.

## 4. Solutions with imposed stratification: *N*^{2} ≠ 0

Although (3.11) is considerably simpler than the two-dimensional system (2.13), its solutions depend on five parameters. (The original system has six parameters; our expansion has eliminated the Rossby number *R.*) Thus it is not possible to give a full description of the solution as a function of all parameters. We will therefore focus our attention on the range of parameters considered most relevant for the terrestrial circulation, that is, *μ*_{2}, *E* ≪ 1 and *α*_{2}, *γ*_{2} ≫ 1, and *N*^{2} ≲ 1. The expansion culminating in (3.11) was derived under the assumption that all the parameters appearing in the final evolution equations are of order one. However, it is still possible to consider limiting cases and then check the consistency of the approximations a posteriori.

_{E}= 4/3 −

*y*

^{2}, and consider a set of parameter values examined in that reference, except that we have a nonzero horizontal diffusivity,

*ν*

_{H}≠ 0:This choice is in the middle of the parameter range examined by HH (where 25 >

*ν*

_{V}> 0.5 m

^{2}s

^{−1}, with all the other parameters fixed). The value of the lateral friction was dictated by computational reasons: it was the smallest that we could use without requiring an excessive amount of resolution. The choice in (4.1) leads to the following set of nondimensional values:Figure 1 shows

*u,*and Θ obtained by solving the steady version of (3.11) numerically.

^{1}Despite the large nondimensional relaxation constant,

*α*

_{2}, the vertically averaged potential temperature Θ is much more homogeneous than the radiative–convective equilibrium Θ

_{E}in a region close to the equator that contains the bulk of “the Hadley circulation.” In the same region, however, there is no appreciable departure of the surface angular momentum

*c*

^{2}, so the angular momentum balance does not play a major role in the extent of the Hadley circulation for this parameter range [a result also found in the moist calculations of Satoh (1994)]. In Fig. 2 the first two terms in the expansion of the streamfunction, angular momentum, zonal velocity, and potential temperature fields are contoured in the meridional–vertical plane: for the last three fields the vertical structure appears at first order in the

*O*(

*R*) correction and it is given in (3.7a,c). The calculation of

*ψ*

_{1}in terms

The region of closed angular momentum contours coincides with the location of the breakdown of the approximation. Specifically in the upper equatorial region, for the choice of parameters in (4.2), the assumption that (*M*_{0}, *T*_{0}, *ψ*_{0}) is much greater than *R*(*M*_{1}, *T*_{1}, *ψ*_{1}) is violated. In the following we will explain the parametric restrictions necessary to avoid this breakdown.

### a. Steady solutions of (3.12) for α_{2}N^{2} ≫ 1, γ_{2} ≫ 1, E ≪ 1, μ_{2} ≪ 1, and N^{2} ≪ 1

*γ*

_{2}the surface angular momentum

*y*

^{2}

*O*

*γ*

^{−1}

_{2}

*O*(

*γ*

^{−1}

_{2}

*R*≪ 1, but with

*γ*=

*O*(1), rather than

*γ*=

*O*(

*R*

^{2}) as in (3.1). Based on this alternative expansion we conclude that (3.11) is valid even when

*γ*

_{2}≫ 1. Thus, encouraged by the results of section 7, we continue with the solution of (4.3) in the limit

*γ*

_{2}≫ 1. All the interesting information about the structure of the surface zonal wind is in the

*O*(

*γ*

^{−1}

_{2}

The strategy is to take advantage of the smallness of *E,* of *α*^{−1}_{2}*N*^{2} to simplify (3.12b) in nested equatorial regions of decreasing size and use matched asymptotics to construct a uniformly valid solution.

*α*

_{2}is very large there is an “outer” region, away from the equator, where Θ is clamped to the radiative–convective equilibrium Θ

_{E}. As the equator is approached, there is a region within which Θ departs from Θ

_{E}. In the limit of large

*α*

_{2}and

*N*

^{2}

*α*

_{2}, the term proportional to

^{3}

_{y}

*α*

_{2})For small

*N*

^{2}, the departures of Θ from the imposed equilibrium are small and confined to a small equatorial region of size

*δ*≡

*N*

*δ*it is useful to rescale

*y*=

*δξ.*As long as

*ξ*≫ (

*E*/

*δ*), we can neglect the term proportional to

*E*so that (4.4) reduces toIn the context of the approximation

*δ*≪ 1 and

*E*≪

*δ,*the boundary conditions that ensure matching with the outer radiative–convective solution areAlthough (4.5) can be solved exactly in terms of modified Bessel functions, it is more useful to look at the behavior of Θ for small

*ξ.*There is a series solution of the formwhere

*b*is an unknown constant that is determined by requiring that the solution in (4.7) matches the outer solution in the limit

*ξ*→ ∞ as required by (4.6a). A trick can be used to make this match: multiply (4.5) by a solution of the associated homogeneous problem,

*h*(

*ξ*), and integrate over

*ξ*from 0 to ∞. The homogeneous version of (4.5) has a solution that is regular at both

*ξ*= 0 and

*ξ*= ∞:

*h*

*ξ*

*ξ*

^{3/2}

*K*

_{3/4}

*π*

^{2}

*ξ*

^{2}

*h*(

*ξ*) and integrating by parts then leads towhich gives

*b*

*δ*

^{2}

*π*

^{−5/2}

^{2}

*δ*

^{2}

*y*=

*O*(

*E*). To this end, it is useful to introduce the boundary layer variable

*ŷ*≡

*y*/(

*π*

^{2}

*E*) and solve the rescaled form of (4.4),The first terms in an expansion in powers of

*E*

^{2}areNotice that the leading order term is the constant

*b,*determined through matching with the solution (4.7) as

*ξ*→ 0. In terms of the unscaled variable,

*y,*the uniformly valid approximation to Θ, valid for small latitudes, isFigure 3 compares the approximation for Θ given in (4.13) (dotted line) with the values obtained for Θ by solving (3.12) numerically (solid line). There is excellent agreement in the region

*y*=

*O*(

*δ*). Aside from the subdominant term of

*O*(

*E*

^{2}), the approximation (4.13) shows that the local behavior of the vertically averaged potential temperature gradient Θ

_{y}near the equator is proportional to

*y*

^{3}and not to

*y*as imposed by the radiative–convective equilibrium (shown in Fig. 3 as a dashed line); that is, the thermal gradient is substantially weakened in the Hadley cell region.

*O*(

*γ*

^{−1}

_{2}

*O*(

*R*) baroclinic correction to the total angular momentum. Using (3.7a,b), (4.13), and the first term of the series expansion (A.4) we find that near the equatorIn the asymptotic limit

*γ*

_{2}→ ∞, the baroclinic term in (4.14) (i.e., the third term in the square bracket) is much larger than the departures of

*M*in (4.14) also allows one to determine the constraints on the Rossby number

*R*in order for the perturbation expansion (3.3) to be uniformly valid. The region where the breakdown first occurs is at the top of the equatorial region, and the requirement thatis satisfied if

*Rbπ*

^{2}

*N*

^{2}

*b*= −0.172

*δ*

^{2}and

*δ*

^{2}=

*N*we find

*R*

*N*

*π*

^{2}

*N.*

*RM*

_{1}breaks down in a small region. However, there is a range in the Rossby number–stratification parameter space where the constraint (4.17) is satisfied. For example, with the value of

*N*

^{2}used here, (4.17) is satisfied for

*R*⩽ 0.06, that is, half the value used in HH. Moreover, (4.17) is much less restrictive than the constraint

*R*≪ 80

*E*

^{2}found by Fang and Tung (1994) in the purely linear case [originally proposed by Charney (1973)].

Because in the range of large surface drag the angular momentum is, to leading order, equal to the planetary value, the solution for the streamfunction *ψ*_{0} is equivalent to Charney’s as long as the temperature gradient used is appropriately modified from the prescribed radiative–convective equilibrium.

The essential point is that, when *N*^{2} is small but *α*_{2}*N*^{2} and *γ*_{2} are large, Θ is significantly different from Θ_{E} in a region of thickness *N**R*Δ_{V}/Δ_{H})^{1/4}. The effect of the Hadley circulation is to homogenize the horizontal temperature gradient compared to the radiative–convective equilibrium prescription, and to reduce the vertically averaged equatorial potential temperature by an amount proportional to *N* = (*R*Δ_{V}/Δ_{H})^{1/2}. Notice that in this limit in which the relaxation time *τ* is short compared to the viscous time *H*^{2}/*ν*_{V}, the extent of the Hadley circulation is independent of both these timescales.

In dimensional units, the size of the Hadley circulation is then given by (*a*^{2}*gH*Δ_{V}/Ω^{2})^{1/4}, which corresponds to the equatorial Rossby radius of deformation, a scaling also suggested in Schneider and Lindzen (1976) and Schneider (1977).

The equatorial homogenization of the vertically averaged potential temperature is achieved through the meridional advection of the *baroclinic component* of potential temperature, which acts as an effective *lateral* diffusion with a diffusivity proportional to the vertical stratification: this is the term (*λFα*_{2}*N*^{2}Θ_{y})_{y} in (3.12b). However, unlike the linear model analyzed by Schneider and Lindzen (1976), the vertical stratification is not clamped to the radiative–convective equilibrium but is determined through a diffusive balance. This is quite apparent in Fig. 4, where we compare the departures from the vertically averaged potential temperature prescribed by the radiative–convective equilibrium to the vertically varying potential temperature obtained in our expansion, evaluated at the equator, that is, *Rθ*_{1} obtained from (3.7c). The two fields differ in two thick diffusive layers next to the two boundaries. Despite the apparently diffusive and viscous nature of the balances analyzed here the final amplitude and scale of the temperature field are independent of viscosity.

## 5. Solutions with no imposed stratification: *N*^{2} = 0

A model of the symmetric circulation should consider the meridional motion that develops in response to the distribution of temperature imposed by the radiative–convective equilibrium. The main role of nonhydrostatic convection is to erase the unstable stratification built by the radiative equilibrium alone: we thus expect that convective processes would leave the atmosphere in a state of neutral vertical stratification. However, because moisture is an essential element of tropical convection, potential temperature will follow a moist adiabatic profile, which corresponds to stably stratified potential temperature. In the context of a conceptual dry model, it is reasonable to analyze the consequences of “dry convection” and assume that the radiative–convective equilibrium leaves the atmosphere in a neutrally stratified state; that is, *N*^{2} = 0. The solution in the previous section depended crucially on the assumption that *N*^{2} ≠ 0. In this section we examine the qualitative changes contingent upon taking *N*^{2} = 0.

Figure 5 shows *U,* and Θ resulting from a numerical solution of (3.11) for the same parameter values given in (4.1) and (4.2) except that now *N*^{2} = 0. Despite the large value of the surface drag, there is now an equatorial region where the surface angular momentum (shown in Fig. 5a) departs from the planetary value and is homogenized. The vertically averaged potential temperature is homogenized in a region of similar extent (Fig. 5c). This leads to easterly surface winds that are maximum at the equator (Fig. 5b) and westerlies that are stronger than those obtained if *N*^{2} ≠ 0 (cf. Fig. 1b).

Inside the “cavity” of homogenized ^{2}_{y}*ϕ* in (3.7c)] the homogenized equatorial cavity is unstratified, as illustrated in Fig. 6d.

### a. Steady solutions of (3.12) for α_{2} = γ_{2} ≫ 1, E ≪ 1, μ_{2} ≪ 1, and N^{2} = 0

*E*and

*μ*

_{2}can be neglected and we have the approximate balanceIn the special case when

*α*

_{2}=

*γ*

_{2}, this formidable-looking system can be solved, at least when Θ

_{E}= 4/3 −

*y*

^{2}, by noticing that the solution must satisfy

*α*

_{2}=

*γ*

_{2}is unchanged when

*α*

_{2}≈

*γ*

_{2}. Then

*γ*

_{2}limit, we anticipate that the departures of

*δ*

^{2}

*m*(

*ξ*), where

*δ*≡ (

*γ*

_{2})

^{−1/4}is the characteristic size of the Hadley cell and

*ξ*≡

*y*/

*δ.*Then in the limit of small

*δ,*(5.2) becomeswhich has the same form as (4.5). Using the method outlined in section 4 we find that the local behavior of

*m*isThe local solution (5.4) shows that the meridional circulation reduces the gradient of the angular momentum

*fourth*power of

*y*=

*δξ*rather than the variation imposed by solid-body rotation

*y*

^{2}. Because of the kinematic constraint (3.13), the gradients induced by the solid-body rotation are expelled outside the region of meridional circulation, giving rise to surface westerlies (cf. Fig. 5b).

*E*≫

*μ*

_{2}, the assumption that

*E*

^{2}is much less than

*y*(

^{2})

_{y}in (3.12), with

*Eδ*

*δ*≫

*E*so that the viscous sublayer is smaller than the characteristic size of the Hadley cell,

*δ.*In this viscous region we adopt the rescalingThe rescaled form of (3.12) then readsso the dominant balance isIt is now clear that close to the equator

*m*

_{1}≈ 3(

*π*

^{10}

*a*

_{0}

*η*

^{4})

^{1/3}/8, and a singularity at the equator appears in the

*second derivative*of the angular momentum and temperature. In terms of the nondimensional latitude

*y,*this corresponds to

*a*

_{0}

*γ*

^{−1/2}

_{2}

*π*

^{10}

*a*

_{0}

^{1/3}

*E*

^{4/3}

*γ*

^{1/6}

_{2}

*y*

^{4/3}

*μ*

_{2}, in a very small sublayer (poorly resolved in the numerical calculation shown in Figs. 4 and 5). Naturally, the nonanalytic behavior of the solution is cause of a failure of the ordering of the perturbation expansion:

*Rψ*

_{1}becomes larger than

*ψ*

_{0},

*R*

^{2}Θ

_{2}becomes larger than

*R*Θ

_{1}, and so forth as the equator is approached. This singularity in high derivatives (here the second derivative of the relevant dynamical fields) is reminiscent of the difficulties encountered in critical layers and signals the local importance of neglected processes, for example, lateral friction or time dependence. A detailed analysis of this equatorial “critical layer” would require examination of the full system (2.13) and it is beyond the scope of this presentation. We conjecture that, given the weakness of the singularity, the approximate behavior of the surface angular momentum and vertically averaged potential temperature given in (5.4) and (5.7) is essentially correct.

In summary, we find that if the radiative–convective equilibrium does not impose explicit stratification, the downgradient mixing due to the meridional circulation reduces the gradient of the surface angular momentum as well as the gradient of the vertically averaged potential temperature. Because of the “equal-area rule” for the surface angular momentum (3.13) the equatorial value of ^{2}

Homogenization of *θ* and *M* fields: recall that the corrections to the vertically homogenous fields, _{1} and *θ*_{1} given in (3.7), are proportional to the *lateral* gradients of

*γ*

^{−1/4}

_{2}

*R*

^{2}

*ν*

_{V}

*CH*

^{1/4}

*R*

*ν*

_{V}/

*CH*

Thus, at least in the limit examined in this section, the size of the Hadley cell depends on the bottom drag *ν*_{V}/*CH.*

## 6. Solutions with weak surface drag, *γ*_{2} = *O*(1)

*γ*

_{2}is not large, the region of homogenized angular momentum occupies a considerable portion of the spherical shell and the surface winds are strong. Figure 7 shows

*U,*and Θ obtained by solving the steady version of (3.11) numerically using the parameter values given in (4.2) except that

*γ*

_{2}

*ϕ,*solution of (3.8) approaches the nonrotating limit, which implies a very vigorous meridional circulation. However, this strong meridional flow equilibrates to a lower value by mixing Θ despite the fast temperature relaxation time (here

*α*

_{2}= 509). The circulation that results from this balance, represented by the leading order streamfunction

*ψ*

_{0}in Fig. 8, is indeed of

*O*(1) and not of

*O*(

*E*

^{−2}) as in the nonrotating regime.

The range of parameters described in this section is not easily amenable to analytical exploration, but it appears to approach the ideal situation envisaged in HH:inside the meridional overturning cells angular momentum is conserved (here it is almost homogeneous), while outside it is given by the planetary value. This is best illustrated in Fig. 9a, where we show the vertical vorticity at the surface, −∂_{y}_{y}*y,* that is, the planetary gradient. The region of zero vertical vorticity coincides with the meridional circulation (cf. Fig. 8). The potential temperature exhibits a behavior analogous to the angular momentum: inside the meridional cell potential temperature gradients are very small, while outside Θ hugs the radiative–convective equilibrium Θ_{E}. The contrast between the two behaviors is most apparent if the potential temperature gradient, shown in Fig. 9b, is examined. Thick internal boundary layers connect the equatorial region characterized by weak gradients with the high latitudes where ∂_{y}Θ ≈ ∂_{y}Θ_{E}.

However, while the emphasis in HH is on the *baroclinic* component of the angular momentum, here the small surface drag allows a large *barotropic,* depth-independent part of the angular momentum to dominate. Numerical calculations varying the parameter *γ*_{2} in a range of *O*(1), not presented here, show that the scale of the circulation is dictated by the value of the surface drag *γ*_{2}. Notice also that the surface wind at the equator is negative and large. In this sense, the regime shown in Figs. 7 and 8 is akin to the unstratified case, described above, even though stratification is present. As in the unstratified case, weak singularities in the high derivatives of *μ*_{2}.

## 7. Asymptotic reduction in the limit of large surface drag

In the previous sections we have seen that the qualitative structure of the solutions depends crucially on the surface drag *γ*_{2}. It is important to understand how the asymptotic reduction changes if one makes alternative scaling assumptions about bottom drag. Recall that in (3.1) we wrote *γ* = *R*^{2}*γ*_{2} and held *γ*_{2} fixed as *R* → 0. This ensured that the velocity at both *O*(1) and *O*(*R*) satisfied a free-slip condition on the ground. Now we will consider an alternative scaling in which *γ* is held fixed as *R* → 0. In this case the velocities satisfy a mixed boundary condition at the ground at all orders. We will show that the evolution equations hold their validity even in this limit. To do this we quickly rederive the *O*(*R*^{2}) balance and solvability condition assuming that *γ* = *O*(1). In the limit of *R* ≪ 1, and regardless of the size of *γ,* we still expand all fields in powers of *R* as in (3.3).

*O*(

*R*

^{0}) in the system (2.13) are still given by (3.4); however, now the boundary condition at

*z*= 0 is

*M*

_{0z}=

*γ*(

*M*

_{0}−

*c*

^{2}). Thus, the solution to (3.4) is

*M*

_{0}

*c*

^{2}

*θ*

_{0}

*y, t*

_{1}

*γ*to

*O*(1) from

*O*(

*R*

^{2}) (as was the case in section 3) leads to the demotion of the angular momentum from its former prognostic prominence. Now

*M*

_{0}is clamped to the planetary value

*c*

^{2}and only the vertically averaged temperature remains as a “master variable.”

*O*(

*R*) balance is as in (3.6) and (3.7), that is,The boundary condition at

*z*= 0,

*M*

_{1z}=

*γM*

_{1}, together with (7.2a) implies that both

*M*

_{1}and

*M*

_{1z}vanish at the bottom, since

*ψ*

_{0}|

_{z=0}= 0. The vertical structure of the meridional streamfunction

*ϕ*satisfies

*E*

^{2}

*ϕ*

_{zzzz}

*y*

*c*

^{2}

_{y}

*ϕ*

*c*

^{2}

*ϕ*=

*ϕ*

_{zz}= 0 at

*z*= 1, and

*ϕ*=

*ϕ*

_{zz}−

*γϕ*

_{z}= 0 at

*z*= 0. It will be apparent shortly that the different bottom boundary condition for

*ϕ*is the main point of departure between the evolution equations given in this section and those given in section 3. However, in the limit of small Ekman number

*E,*the difference between the solution of (7.3) and the solution of (3.8) is confined to a thin bottom boundary layer of thickness

*E*

In anticipation of the *O*(*R*^{2}) balance, we compare in Fig. 10 the vertical average of *ϕ*^{2} for *γ* = 0, that is, the assumption made in section 3, and for *γ* = 10. When the Ekman number is small, which is the limit of interest here, the two integrals differ by a small amount.

*O*(

*R*

^{2}) angular momentum and potential temperature equations are vertically integrated, namely,The evolution equation for the vertically averaged potential temperature Θ is formally the same as that found in section 3c, except for the already noted difference in

*ϕ*and, most importantly, for the fact that Θ is determined independently of

*M*

_{2}. Indeed, (7.3) shows that

*ϕ*is independent of

*M*

_{2}because the surface angular momentum is, to a first approximation, clamped to the planetary value. In this sense, the potential temperature is a master variable: it is determined independently of the unknown contribution to the surface angular momentum

*M*

_{2}.

On the other hand, the solvability condition (7.4a) determines the surface value of *M*_{2} in terms of Θ. In this sense the angular momentum is a “slave variable” (as is *ψ*): its distribution in space and time is dictated by Θ.

*γ*

_{2}≫ 1 in the “master equation” (3.11a), as it is appropriate in the range

*γ*=

*O*(1) and

*R*≪ 1. In that limit, as shown in section 4, (3.11a) can be solved perturbatively; that is,

*c*

^{2}

*γ*

^{−1}

_{2}

*m*

_{1}

*O*

*γ*

^{−2}

_{2}

*O*(1) we find that

*m*

_{1}satisfiesUsing (7.5) and recalling that

*γ*

_{2}≡

*R*

^{−2}

*γ*and that

*M*(

*y,*0,

*t*

_{1}), (7.6) can be rewritten asNow it is clear that (7.4a) is equivalent to (7.7) up to

*O*(

*R*

^{2}), because

*R*

^{−2}

*M*

*y,*

*t*

_{1}

*c*

^{2}

*M*

_{2}

*y,*

*t*

_{1}

In summary, we find that the choice *γ* = *O*(*R*^{2}) in the small Rossby number limit leads to an evolution equation for the surface angular momentum, that is, (3.11a), which is more general than that obtained for the choice *γ* = *O*(1). Specifically, (7.4a) is a limiting case of (3.11a) when *γ*_{2} ≫ 1.

## 8. Summary and conclusions

We have tried to understand the contradictory role of friction in the axially symmetric circulation on a rapidly rotating planet, driven by meridional thermal gradients. Recognizing that meridional motions rely on friction leads us to postulate that the strength of the meridional overturn is directly proportional to the explicit viscosity *ν*_{V}, as is the case for Charney’s linear model. Despite this conjecture, it is still possible to consider the limit of small dissipation by considering the Ekman number to be small. Our formulation is consistent with the notion of weakly dissipated and weakly forced flows where the streamlines are aligned with the quasi-conserved quantities, here the angular momentum and the potential temperature, everywhere except in thin boundary layers.

The compromise between a weak circulation and a quasi-inviscid, fully nonlinear balance is possible in the limit of small Rossby number *R* as long as a distinguished limit relating the nondimensional parameters of the problem to *R* is considered. This contrived range of parameters is not always within the reach of atmospheric values, but it allows the explicit calculation of the advective fluxes in terms of two fundamental variables: the surface angular momentum and the vertically averaged potential temperature. One result is that the meridional circulation induces a stably stratified vertical temperature gradient, proportional to the *square* of the horizontal temperature gradient, by tilting the latitudinal gradients generated by the differential solar heating. Thus, even if the radiative–convective equilibrium has no vertical stratification, the large-scale dynamics can supply one. The same tilting mechanism turns horizontal gradients of angular momentum, provided by the earth’s rotation and by relative rotation, into vertical angular momentum gradients, that is, zonal wind shears. The constraint that this dynamically induced vertical shear is in cyclostrophic balance determines the meridional circulation in the interior.

The overturning of these dynamically generated vertical gradients in turn changes the *horizontal* gradients of potential temperature and angular momentum from the values prescribed by the solar heating and the planetary rotation respectively. Specifically, the meridional flow acts to diffuse laterally and downgradient angular momentum and potential temperature. This is a reassuring result: lateral homogenization of potential temperature near the equator is an observed feature, while angular momentum homogenization has often been advocated (e.g., Bretherton and Turner 1968) and has been also obtained in some numerical solutions (e.g., Satoh 1994).

While there is quantitative agreement of our solutions with some of the numerical computations presented in HH, the calculations in HH with very small viscosity differ from those found here. Specifically, the calculations published in HH show a qualitative change as *ν*_{V} → 0. In Fig. 11 we plot the ratio of the dimensional streamfunction maxima to the vertical viscosity as a function of *ν*_{V} for the sequence shown in HH (their Fig. 4). The analysis presented in section 4 predicts that *ψ*_{0}, marked by circles in Fig. 11, is approximately independent of *ν*_{V}. If the *O*(*R*) correction is added to the streamfunction (diamonds) a weak dependence upon *ν*_{V} emerges, but the qualitative structure of the streamfunction is as shown in Fig. 2.

The numerical results of HH, marked by squares in Fig. 11, show that, for *ν*_{V} ≥ 2.5, the streamfunction maxima scale approximately linearly with *ν*_{V}, as predicted by our reduction. For *ν*_{V} < 2.5, *ψ*_{max}/*ν*_{V} rapidly increases as viscosity decreases. This sudden increase is accompanied by a qualitative change in the streamfunction structure: the position of the streamfunction maximum in HH moves from the bottom of the vertical domain to the top (cf. Fig. 4 in HH). We conjecture that, in this range of small viscosity, a symmetric mode, which is only very weakly damped, can be nonlinearly excited, leading the system through a bifurcation that is not captured by our reduction.

In the range of parameters where our results agree with the direct computations of HH, the potential temperature is laterally homogenized within the cells, and the advection of the stable stratification imposed by the radiative–convective equilibrium is essential for the determination of the scale of the meridional cell, while the angular momentum balance plays a marginal role. Thus, the Hadley circulation expands linearly with the equatorial baroclinic radius of deformation, in agreement with Schneider and Lindzen (1976) and Schneider (1977). This scaling result departs from the conclusions drawn by HH.

When stratification is absent or the surface drag is much weaker than that used by HH, both angular momentum and potential temperature are homogenized inside the Hadley cells. This regime approaches the HH ideal of the axially symmetric meridional circulation, where conservation of angular momentum is important. In this regime, surface easterlies are found at the equator, even for a heating distribution that centered at the equator.

## Acknowledgments

I am especially indebted to Piero Malguzzi for illuminating discussions and for providing me with his Newton–Keller subroutines. Numerous conversations with Bill Young are gratefully acknowledged. Ed Schneider and an anonymous referee are thanked for their constructive criticism. Funding for this research is provided by the Department of Energy through its CHAMMP Program (DOE DEFG03 93ER61690).

## REFERENCES

Bretherton, F. P., and J. S. Turner, 1968: On the mixing of angular momentum in a stirred rotating fluid.

*J. Fluid Mech.,***32,**449–464.Charney, J. G., 1973: Planetary fluid dynamics.

*Dynamic Meterology,*P. Morel, Ed., Reidel, 97–351.Fang, M., and K. K. Tung, 1994: Solution to the Charney problem of viscous symmetric circulation.

*J. Atmos. Sci.,***51,**1261–1272.——, and ——, 1996: A simple model of nonlinear Hadley circulation with an ITCZ: Analytic and numerical solutions.

*J. Atmos. Sci.,***53,**1241–1261.Held, I. M., and A. Y. Hou, 1980: Nonlinear axially symmetric circulation in a nearly inviscid atmosphere.

*J. Atmos. Sci.,***37,**515–533.Lindzen, R. S., 1990:

*Dynamics in Atmospheric Physics.*Cambridge University Press, 310 pp.——, and H. Y. Hou, 1988: Hadley circulation for zonally averaged heating centered off the equator.

*J. Atmos. Sci.,***45,**2416–2427.Read, P. L., 1986: Super-rotation and diffusion of axial angular momentum: II. A review of quasi-axisymmetric models of planetary atmospheres.

*Quart. J. Roy. Meteor. Soc.,***112,**253–272.Satoh, M., 1994: Hadley circulations in radiative convective equilibrium in an axially symmetric atmosphere.

*J. Atmos. Sci.,***51,**1947–1968.Schneider, E. K., 1977: Axially symmetric steady-state models of the basic state for instability and climate studies. Part II. Nonlinear calculations.

*J. Atmos. Sci.,***34,**280–296.——, and R. S. Lindzen, 1976: The influence of stable stratification on the thermally driven tropical boundary layer.

*J. Atmos. Sci.,***33,**1301–1307.

## APPENDIX A

### The Vertical Structure of the Meridional Streamfunction

*ψ*

_{0}in section 3, are given. This amounts to solving the ordinary differential equation (3.8), repeated here for convenience,

*E*

^{2}

*c*

^{2}

*ϕ*

_{zzzz}

*y*

^{2}

_{y}

*ϕ*

*c*

^{4}

*ϕ*=

*ϕ*

_{zz}= 0 at

*z*= 0, 1. The lowest order approximation to the streamfunction

*ψ*

_{0}is then given by

*ψ*

_{0}

_{y}

*ϕ.*

*z*except in the top and bottom Ekman layers of thickness

*η*

^{−1}. Here, however,

*η*depends on the surface angular momentum

*c*

^{2}. Thus the thickness of the Ekman layer becomes infinite not just at the equator, but in the whole region of angular momentum homogenization.

*c*

^{−2}

*y*

^{2}

_{y}

*E*

^{2}

*n*

^{4}

*π*

^{4}

*n*= 0 in (A.4) coincides with the onset of symmetric instability in the original Boussinesq system (2.13). Each successive pole in (A.4) corresponds to the excitation of another unstable mode.

## APPENDIX B

### Symmetric Instability in the Presence of Viscosity

*M*

*y*

*ψ*

*θ*

*μ*= 0 in (2.13) and

*γ*= 0 in (2.14). Infinitesimal perturbations,

*M*′ and

*ψ*′, of the basic state (B.1) satisfy the systemwith boundary conditions

*ψ*′ =

*ψ*

^{′}

_{zz}

*M*

^{′}

_{z}

*z*= 0 and

*z*= 1.

*B*

_{z}=

*B*

_{zzz}= 0 at

*z*= 0 and

*z*= 1. Notice that

*B*is only differentiated with respect to

*z,*and its dependence on

*y*is parametric. In this simple case, where there is no basic stratification and no vertical shear, the growth rate

*s*can be a function of latitude

*y.*Nontrivial solutionsare obtained when the growth rate satisfies the dispersion relationThe roots of (B.6) are

*REs*

*En*

^{2}

*π*

^{2}

*yc*

^{−2}(

^{2})

_{y}

*yc*

^{−2}

^{2}

_{y}

*E*

^{2}

*n*

^{4}

*π*

^{4}

*n*= 1 in (B.8).

## APPENDIX C

### The *O*(*R*) Correction to the Meridional Streamfunction *ψ*_{1}

In this section the first correction to the leading order field for the streamfunction is calculated. The same procedure can be used to calculate the *O*(*R*^{2}) corrections to the angular momentum and temperature fields, *M*_{2} and *θ*_{2} respectively.

*O*(

*R*

^{2}) isThe fields

*θ*

_{1}and

*M*

_{1}are given in terms of the “master” fields

*M*

_{2}is obtained from (3.10) after using (3.11) to eliminate the term

_{t1}

*M*

_{1}and

*ψ*

_{0}. Integrating (C.2) once in

*z*and using the boundary condition

*M*

_{2z}|

_{z=0}=

*γ*

_{2}(

*c*

^{2}) we find

*ψ*

_{1}at the top and the bottom are no normal flow and no stress so that (C.4) is the same type of equation as (3.8) except that the right-hand side depends on

*z*as well as

*y.*In the figures shown in this work, (C.4) is solved numerically using a rapidly converging series expansion of the type given in (A.4).

The essential point is that, in the limit of small Ekman number *E, ψ*_{1} is independent of *E* outside of the thin top and bottom boundary layers, as is the case for *ψ*_{0} and *ϕ.*

^{1}

To solve (3.11) numerically, *ϕ* in (3.11) were performed with the aid of the symbolic manipulation software MAPLE.

^{2}

Indeed, when we forced the steady version of (3.11) with asymmetric temperature distributions a la Lindzen and Hou (1988), the surface angular momentum at the equator satisfied the condition *y*(^{2})_{y} < *π*^{4}*E*^{2}; that is, it was symmetrically unstable.