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

1. 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

2. 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

Our point of departure is the symmetric model used by Charney (1973) in the limit of a thin spherical shell. If a is the radius of the sphere, φ is the latitude, and

yaφ,cyφ1 − y²/a²(2.1)

then the zonal momentum equation can be written in the form

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In (2.2) ψ is the meridional streamfunction defined by

_yψw,_zψcυ,(2.3)

and the Jacobian is J(A, B) = A_yB_z − A_zB_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.

Instead of the zonal momentum u we will use the angular momentum

Mac²uc(2.4)

as a dependent variable. Rewriting (2.2) in terms of M produces the more transparent conservation equation

M_tJψ, Mν_VM_zzν_Hc⁴c⁻²M_yy(2.5)

The meridional momentum equation and the hydrostatic relation are combined into a single equation for the zonal vorticity, which in the thin shell approximation reads

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Once again, the last term in (2.6) is negligible when 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⁻²ψ_zz.

Finally, the potential temperature evolves according to

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where τ is the relaxation time toward the radiative–convective equilibrium distribution θ_E(y, z).

A stress-free, no-flux rigid lid is assumed at the top of the spherical shell; that is,

ψM_zψ_zzθ_zzH.(2.8)

At the rigid bottom the stress is parametrized by a drag-type law, linearized around constant velocity, and a no-flux condition is imposed on the potential temperature:

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The bottom drag relaxes the angular momentum M to the local planetary value Ωac².

The system (2.5) through (2.9) is nondimensionalized using a scaling that follows that of Schneider and Lindzen (1977), except that the zonal velocity u is implicitly scaled with Ωa:

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We have used a standard notation for the thermal Rossby number R, the Ekman number E, and the aspect ratio ϵ, defined as

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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.

Dropping the carets from now on, the nondimensional equations for the angular momentum, zonal vorticity, and potential temperature are

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The nondimensional boundary conditions are

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Thus, in addition to the three parameters defined in (2.12), there are four more parameters:

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Here 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

The distinguished limit (3.1) is suggested by the balance obtained integrating (2.13a) and (2.13c) vertically:

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where the overbar indicates the vertical average.

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(R²). The choice γ = O(R²) 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(R²) and ∂_t = O(R) all the terms in (3.2a) have the same asymptotic order, namely, O(R²).

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(R²). The choice α = O(R²) 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 α₂ and γ₂ are large.

Although we will be mainly concerned with the steady circulation, the asymptotic reduction detailed below can be applied also to time-dependent flows whose timescale of evolution is slow; that is, ∂_t = R∂_t1. With this choice, an expansion is possible in which M, θ, and ψ are expanded in powers of R; that is,

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a. The O(R⁰) balance

The terms of O(R⁰) in the system (2.13) are

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At this order both the top and bottom boundary conditions are M_0z = θ_0z = 0. Thus the solution of (3.4) is

M₀My, t₁θ₀y, t₁(3.5)

To a first approximation, the angular momentum and the potential temperature are independent of height but are undetermined at this order. It will be apparent at next order that, even though the leading order balance is viscous and diffusive, the vertically uniform fields in (3.5) are also exact solutions when the nonlinear terms are included. Our goal is to find the evolution equations for M and Θ.

b. The O(R) balance

At the next order, we have from (2.13):

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Although the last term in (3.6b) is negligible if 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, is

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The meridional circulation, characterized to a first approximation by ψ₀, 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

E²c²ϕ_zzzzyM²_yϕc⁴(3.8)

with boundary conditions 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(M²)_y/(cE)²]^−1/4, instead of (4y²/E²)^−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 M is homogenized laterally. Thus, at least as far as the meridional circulation is concerned, angular momentum homogenization is equivalent to a weakening of the rotation.

In the inviscid limit, E ≪ 1, the streamfunction, ψ₀ in (3.7b), is independent of height, outside the thin top and bottom boundary layers. Thus the zeroth-order angular momentum and temperature, M and Θ, which are also depth independent, are exact nonlinear solutions in the interior region; that is,

Jψ₀MJψ₀(3.9)

except in the top and bottom Ekman layers. In this sense, the solutions (3.5), although springing from a viscously dominated balance, also satisfy an advectively dominated balance in the interior. Remarkably, with the scaling (3.1), the exact nonlinear solutions M(y, t₁) andΘ(y, t₁) 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 M and Θ very simple.

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 N², 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 M from the vertical position. The constraint that the zonal shear is in cyclostrophic balance, outside the Ekman layers, determines the meridional circulation in the interior.

To obtain the O(R) angular momentum and temperature corrections, M₁ and θ₁, (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 M₁|_z=0 = 0 and ∫¹₀ θ₁ dz = 0. Thus M is normalized to be the surface angular momentum, and Θ is the vertically averaged potential temperature. With this choice of gauge the bottom boundary condition for the angular momentum becomes M_z = R²γ₂(M − c²).

c. The O(R²) balance

The O(R²) angular momentum and potential temperature equations are

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with boundary conditions θ_2z|_z=0,1 = M_2z|_z=1 = 0 and M_2z|_z=0 = γ₂(M − c²). The equations in (3.10) have a“solvability condition,” which is obtained by integrating vertically over the whole depth:

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The parameter μ₂, 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.

In appendix A approximate expressions for the integrals involving ϕ(z, y, M) are given. Using these approximations, (3.11) becomes

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Thus, in this weakly nonlinear expansion, the advection of angular momentum takes the form of downgradient lateral diffusion of M with a “diffusion” coefficient, F²Θ²_y, that is always positive and proportional to the square of the meridional temperature gradient. The role of the meridional momentum transport is indeed to homogenize the angular momentum, while the viscosity tends to homogenize the rotation rate, and the bottom drag brings the angular momentum toward the planetary value, c². 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 N²: 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(M²)_y = π⁴E²: 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 λFα₂N²Θ_y on the right-hand side of (3.12b)] becomes negative, even for positive prescribed stratification N². 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.

Both the nonlinear and the viscous terms in (3.11a) are in flux form and vanish when integrated over the domain. Thus, we have the kinematic condition

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which implies that, to first order, the horizontally averaged surface angular momentum spins down to the planetary value on a timescale inversely proportional to the surface drag. This implies that the net relative rotation rate of the atmosphere, ω_r, is O(R) and is given by

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Using the expressions for M₁ and ψ₀ in (3.7a,b) the net relative rotation rate is given by

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As long as the circulation is thermally and “angular momentum” direct, M₁ is positive everywhere in the domain and the net relative rotation of the atmosphere, ω_r, is eastward.

A kinematic constraint analogous to (3.13) exists for the vertically averaged potential temperature. Thus, integrating (3.11b) over the domain gives

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Equation (3.16) simply states that advective and diffusive processes cannot change the volume-averaged potential temperature from that imposed by the radiative–convective equilibrium (the “equal area” rule).

4. Solutions with imposed stratification: N² ≠ 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, μ₂, E ≪ 1 and α₂, γ₂ ≫ 1, and N² ≲ 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.

As a specific example we use the same form for the radiative–convective equilibrium temperature given by HH, that is, Θ_E = 4/3 − y², and consider a set of parameter values examined in that reference, except that we have a nonzero horizontal diffusivity, ν_H ≠ 0:

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This choice is in the middle of the parameter range examined by HH (where 25 > ν_V > 0.5 m² 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:

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Figure 1 shows M, u, and Θ obtained by solving the steady version of (3.11) numerically.¹ Despite the large nondimensional relaxation constant, α₂, 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 M from the planetary value, c², 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 ψ₁ in terms M and Θ is somewhat more laborious and can be found in appendix C. The agreement with the numerical results presented in HH (cf. the bottom panel in Fig. 4a of HH) is good: the structure of the streamfunction and of the zonal wind is well captured, and the maximum values of the meridional transport, the top and the bottom zonal winds in Fig. 2, are within 10% of the solution presented by HH (cf. the bottom panel in Fig. 4a of HH). However, there is a small region where closed angular momentum contours are found (see Fig. 2b), violating Hide’s theorem.

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 (M₀, T₀, ψ₀) is much greater than R(M₁, T₁, ψ₁) is violated. In the following we will explain the parametric restrictions necessary to avoid this breakdown.

a. Steady solutions of (3.12) for α₂N² ≫ 1, γ₂ ≫ 1, E ≪ 1, μ₂ ≪ 1, and N² ≪ 1

With the hindsight afforded by the numerical solutions, it is not difficult to obtain approximate analytical solutions of (3.12). The first thing to notice is that for large γ₂ the surface angular momentum M is approximately clamped to the planetary value in the whole domain. Thus, the solution of (3.12a) is just

My²Oγ⁻¹₂(4.3)

This result implies that the surface zonal velocity is O(γ⁻¹₂) ≪ 1. The reader might question the validity of (3.11) in this limit of large surface drag. We examine this issue in section 7 by returning to the full equations in (2.13) and making the expansion in R ≪ 1, but with γ = O(1), rather than γ = O(R²) as in (3.1). Based on this alternative expansion we conclude that (3.11) is valid even when γ₂ ≫ 1. Thus, encouraged by the results of section 7, we continue with the solution of (4.3) in the limit γ₂ ≫ 1. All the interesting information about the structure of the surface zonal wind is in the O(γ⁻¹₂) term of the expansion (4.3). In order to obtain this term we must first calculate the vertically averaged potential temperature Θ.

The strategy is to take advantage of the smallness of E, of α⁻¹₂, and of N² to simplify (3.12b) in nested equatorial regions of decreasing size and use matched asymptotics to construct a uniformly valid solution.

Because α₂ 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 α₂ and N²α₂, the term proportional to Θ³_y can be neglected in (3.12b), everywhere in the domain. Using the leading order term for M in (4.3), (3.12b) becomes (after division by α₂)

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For small N², the departures of Θ from the imposed equilibrium are small and confined to a small equatorial region of size δ ≡ N, making the analysis of (4.4) very simple.

Within the region of size 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 to

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In the context of the approximation δ ≪ 1 and E ≪ δ, the boundary conditions that ensure matching with the outer radiative–convective solution are

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Although (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 form

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where 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 = ∞:

hξξ^3/2K_3/42π²ξ²(4.8)

Multiplying (4.5) by h(ξ) and integrating by parts then leads to

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which gives

bδ²π^−5/22δ²(4.10)

This prediction agrees well with the numerical solution shown in Fig. 1.

To complete the analysis it is necessary to find the solution of (4.4) in the equatorial Ekman layer, that is, when y = O(E). To this end, it is useful to introduce the boundary layer variable ŷ ≡ y/(π²E) and solve the rescaled form of (4.4),

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The first terms in an expansion in powers of E² are

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Notice 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, is

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Figure 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(E²), the approximation (4.13) shows that the local behavior of the vertically averaged potential temperature gradient Θ_y near the equator is proportional to y³ 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.

We are now in a position to calculate the O(γ⁻¹₂) correction to the surface angular momentum and the 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 equator

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In the asymptotic limit γ₂ → ∞, the baroclinic term in (4.14) (i.e., the third term in the square bracket) is much larger than the departures of M from the planetary value (the second term inside the square bracket). Notice that the baroclinic term tends to weaken the latitudinal gradient of the angular momentum in a region that expands with height. Thus, the angular momentum emerges from the upper branch of the Hadley cell with a gradient that is reduced from the planetary rest state. However, this quasi homogenization is achieved through viscous processes in this parameter range, as all the streamlines go through a bottom viscous boundary layer, precluding the applicability of Prandtl–Batchelor arguments.

The expression for 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 that

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is satisfied if

Rbπ²N²(4.16)

Using b = −0.172δ² and δ² = N we find

RNπ²N.(4.17)

For the values used in the numerical calculation shown in Figs. 1 and 2, the constraint (4.17) is not satisfied and the approximation M ≪ RM₁ 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 N² 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 ≪ 80E² 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 ψ₀ 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 N² is small but α₂N² and γ₂ are large, Θ is significantly different from Θ_E in a region of thickness N = (RΔ_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 = (RΔ_V/Δ_H)^1/2. Notice that in this limit in which the relaxation time τ is short compared to the viscous time H²/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 (a²gHΔ_V/Ω²)^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 (λFα₂N²Θ_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, Rθ₁ 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: N² = 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, N² = 0. The solution in the previous section depended crucially on the assumption that N² ≠ 0. In this section we examine the qualitative changes contingent upon taking N² = 0.

Figure 5 shows M, 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 N² = 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 N² ≠ 0 (cf. Fig. 1b).

Inside the “cavity” of homogenized M and Θ there is a tight meridional circulation, as shown in Fig. 6a. Because the stratification produced by the meridional circulation is proportional to the square of the latitudinal temperature gradient [see the term Θ²_yϕ in (3.7c)] the homogenized equatorial cavity is unstratified, as illustrated in Fig. 6d.

a. Steady solutions of (3.12) for α₂ = γ₂ ≫ 1, E ≪ 1, μ₂ ≪ 1, and N² = 0

In the following approximate solutions in this range are presented. As in the analysis of section 4, we first consider the outer region of the meridional circulation, corresponding to its descending branch (cf. Fig. 6a). In this region, the terms proportional to E and μ₂ can be neglected and we have the approximate balance

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In the special case when α₂ = γ₂, this formidable-looking system can be solved, at least when Θ_E = 4/3 − y², by noticing that the solution must satisfy M − 1 = Θ − 4/3. Numerical exploration shows that the qualitative character of the solution for α₂ = γ₂ is unchanged when α₂ ≈ γ₂. Then M (or Θ) satisfies

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In the large γ₂ limit, we anticipate that the departures of M from the local planetary value will be of small amplitude and confined to a small equatorial region and seek a solution of (5.2) in the form M = 1 + δ²m(ξ), where δ ≡ (γ₂)^−1/4 is the characteristic size of the Hadley cell and ξ ≡ y/δ. Then in the limit of small δ, (5.2) becomes

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which has the same form as (4.5). Using the method outlined in section 4 we find that the local behavior of m is

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The local solution (5.4) shows that the meridional circulation reduces the gradient of the angular momentum M. The local dependence of M within the ascending branch of the meridional circulation is given by a constant, which is less than unity, implying surface easterlies at the equator. The variation with latitude of the angular momentum is proportional to the fourth power of y = δξ rather than the variation imposed by solid-body rotation y². 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).

Because of viscous effects, the close neighborhood of the equator needs a careful treatment. In the range 1 ≫ E ≫ μ₂, the assumption that E² is much less than y(M²)_y in (3.12), with M given by (5.4), breaks down in an equatorial boundary layer of size Eδ. Here we are assuming that 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 rescaling

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The rescaled form of (3.12) then reads

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so the dominant balance is

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It is now clear that close to the equator m₁ ≈ 3(π¹⁰a₀η⁴)^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

Ma₀γ^−1/2₂π¹⁰a₀^1/3E^4/3γ^1/6₂y^4/3(5.8)

This weak singularity in the angular momentum and in the potential temperature is healed by the explicit lateral diffusion, proportional to μ₂, 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: Rψ₁ becomes larger than ψ₀, R²Θ₂ becomes larger than RΘ₁, 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 M is less than that of the rest state. This leads to surface easterlies at the equator, at least in the limited range of parameters explored, even for temperature distributions that are latitudinally symmetric. This result has a certain appeal because in previous work (Lindzen and Hou 1988) surface equatorial easterlies have been attributed to distributions of equilibrium temperatures that are centered off the equator. However, for latitudinally asymmetric temperature distributions, the model (2.13) is very prone to symmetric instabilities.²

Homogenization of M and Θ leads to homogenization of the whole θ and M fields: recall that the corrections to the vertically homogenous fields, M₁ and θ₁ given in (3.7), are proportional to the lateral gradients of M and Θ. Therefore, in the region where M and Θ are homogeneous, no vertical gradients are generated.

The approximate analytic solutions show that the extent of the Hadley cell, defined as the region where the surface angular momentum and the vertically averaged potential temperature depart substantially from the rest states, is proportional to

γ^−1/4₂R²ν_VCH^1/4

and the maximum surface easterlies are proportional to Rν_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, γ₂ = O(1)

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, −∂_yM: internal boundary layers connect the equatorial region of almost zero vorticity with the polar regions where −∂_yM ≈ 2y, 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 ∂_yΘ ≈ ∂_yΘ_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 γ₂ 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 γ₂. 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 M and Θ develop unless regularized by the explicit lateral diffusion term proportional to μ₂.

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 γ₂. 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 = R²γ₂ and held γ₂ 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(R²) 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).

The O(R) balance is as in (3.6) and (3.7), that is,

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The boundary condition at z = 0, M_1z = γM₁, together with (7.2a) implies that both M₁ and M_1z vanish at the bottom, since ψ₀|_z=0 = 0. The vertical structure of the meridional streamfunction ϕ satisfies

E²ϕ_zzzzyc²_yϕc²(7.3)

subject to the boundary conditions 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(R²) balance, we compare in Fig. 10 the vertical average of ϕ² 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.

In parallel with the procedure detailed in section 3c, a solvability condition emerges when the O(R²) angular momentum and potential temperature equations are vertically integrated, namely,

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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 M₂. Indeed, (7.3) shows that ϕ is independent of M₂ 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 M₂.

On the other hand, the solvability condition (7.4a) determines the surface value of M₂ 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.

Despite this important difference, (7.4a) is actually equivalent to taking the limit γ₂ ≫ 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,

Mc²γ⁻¹₂m₁Oγ⁻²₂(7.5)

Sustituting the expansion (7.5) in (3.11a) and collecting the terms of O(1) we find that m₁ satisfies

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Using (7.5) and recalling that γ₂ ≡ R⁻²γ and that M is defined to be the surface angular momentum, M(y, 0, t₁), (7.6) can be rewritten as

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Now it is clear that (7.4a) is equivalent to (7.7) up to O(R²), because

R⁻²My,t₁c²M₂y,t₁

In summary, we find that the choice γ = O(R²) 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 γ₂ ≫ 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 ψ₀, 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.