3.1. Comparison with Analytical Solutions
 We first consider a near-inviscid case, in which an analytical solution is possible (see the appendix) and can be used to test the numerical code. We consider a case in which there is no eddy momentum flux divergence (), εu is very small, and the thermal forcing is
In addition, for the sake of analytical tractability, we neglect the vertical advection term in the zonal momentum equation (the first term on the RHS of (2.1)). The forcing parameters are θ00 = 330 K, Δy = 100 K, Δz = 60 K, y1 = 9439 km. The dynamical parameters are H = 16 km, T0 = 300 K, and β = 2 × 10−11 m−1 s−1.
 Figure 2 shows a comparison between the numerical solution and the analytical solution presented in the appendix. The agreement is very good, giving us confidence in our numerical code.
3.2. Sensitivity to Rayleigh Drag
 In this section, we present the sensitivity of our axisymmetric solutions to the Rayleigh drag coefficient εu, with no EMFD (). This is of interest in its own right, but also necessary for a proper understanding of our results with . When the θE maximum is a significant distance from the equator, it will turn out that we need a somewhat larger value of εu than that for which the solution converges to the inviscid analytical one for the hemispherically symmetric forcing (the only case for which we have computed the analytical solution). We need to understand how this larger value of εu influences the solution.
 Figure 3 shows results from calculations using the same forcing as used for Figure 2, but with three different values of εu: the smallest is 10−9 s−1, a factor of 10 greater than that used in Figure 2, while the other two are larger by factors of 10 and 100, respectively. Results for εu = 10−10 s−1, (not shown) are indistinguishable from those for εu = 10−9 s−1, within the thickness of the lines on the plots. The maximum value of v is 50% larger than that for the near-inviscid solution when εu = 10−8 s−1, and much larger when εu = 10−7 s−1. This is expected since in steady state, the Rayleigh drag can only be balanced by advection of angular momentum, requiring meridional flow. In the linear regime, in which only planetary angular momentum is advected, the balance in (2.1) is βyv ∼ , which is known in the stratospheric literature as “downward control” [Haynes et al., 1991]. The solutions of interest here are nonlinear, so that in steady state v (βy − ∂yu) ∼ , but it is still the case that v increases with , at least for hemispherically symmetric thermal forcing (with asymmetric thermal forcing, v may decrease with because the zonal flow may weaken with increasing [Walker and Schneider, 2005]). In fact, the dependence is stronger in the nonlinear regime than in the linear, because the absolute vorticity βy − ∂yu is invariably smaller than the planetary component βy alone; in inviscid theory, the absolute vorticity vanishes within the Hadley cell.
Figure 3. Numerical solutions for meridional velocity (left) and zonal velocity (right) with no imposed EMFD (), no vertical momentum advection, the same forcing as in Figure 2, and Rayleigh drag coefficient εu equal to 10−9 s−1 (blue), 10−8 s−1 (red), and 10−7 s−1 (black). Results for εu = 10−10 s−1 (not shown) are indistinguishable from those for εu = 10−9 s−1, within the thickness of the curves.
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 Next we present calculations evaluating the sensitivity to εu using the model configuration which we will use for the rest of the study. We now include the vertical advection term in (2.1) and specify the thermal forcing, following Schneider and Bordoni , as
Here, θ00 and y1 are as above, but we set Δy = 50 K. Figure 4 shows results for equatorially symmetric forcing, y0 = 0, and the same three values of εu used for Figure 3.
 Comparing Figures 3 and 4, we see that when the circulation is stronger, the percentagewise impact of small Rayleigh drag is smaller. The maximum value of v is now only 16% larger when εu = 10−8 s−1 than that for the near-inviscid solution (εu = 10−9 s−1); it is about twice as large when εu = 10−7 s−1. This is further supported by Figure 5, which shows results from a set of calculations identical to those in Figure 4, except that the thermal forcing is centered well off the equator, y0 = 1000 km. The results feature a very strong winter subtropical jet and relatively weak summer jet, and a single-celled meridional circulation for all but the largest value of εu shown. It is now only for this largest value, 10−7 s−1 (corresponding to a damping timescale of 116 days), that the meridional circulation departs significantly (20%) from that of the less dissipative solutions.
3.3. Calculations with Varying Off-equatorial Forcing
 Figure 6 presents results from a set of calculations in which y0 is varied from 0 to 2000 km in intervals of 200 km. Each calculation is carried out until a steady state is reached. The zonal and meridional velocity fields in the steady state are contoured as functions of y and y0; any horizontal cut gives u or v as a function of y from a single calculation with a given y0. As y0 increases, the winter jet strengthens and moves poleward, the summer jet weakens, and the equatorial easterlies strengthen, while the meridional circulation also strengthens. These qualitative features are familiar from previous work [e.g., Lindzen and Hou, 1988].
Figure 6. Plots of (left) zonal velocity, u (m s−1), and (right) meridional velocity, v (m s−1), as a function of latitudinal position, y, on the horizontal axis and of the θE maximum, y0, on the vertical axis. In the plot of v, heavy black contours enclose regions of Ro = −ζ/(βy) > 0.6.
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 The region in which the local Rossby number Ro = −ζ/(βy) > 0.6, where ζ = −∂u/∂y is the relative vorticity, is indicated by the heavy black contours. We see that the maximum absolute value of v lies within the region where Ro > 0.6 for all y0. In a region of constant angular momentum, as occurs within the Hadley cell in near-inviscid theory, Ro = 1, while in the linear downward control regime, Ro ≪ 1. Thus the region in which Ro > 0.6 roughly delineates the region in which the flow is reasonably close to conserving angular momentum. In these calculations with = 0 and small , this region encloses most of the region in which the meridional flow is significant, as expected.
3.4. Parameterized Eddy Momentum Flux Divergence
 We present a set of calculations in which , the eddy momentum flux divergence due to baroclinic eddies, parameterized according to (2.5), is nonzero.
 Figure 7 presents, in the top two panels, a set of calculations identical to that in Figure 6, except that is nonzero and parameterized according to (2.5), with vd = 2.5 m s−1 as in Figure 1 for the idealized GCM. The lower two panels show results from the idealized GCM [Schneider and Bordoni, 2008], averaged over the vertical sigma levels 0.15 < σ < 0.35 as in Figure 1. In both cases, results are plotted as functions of meridional distance y and distance of the θE maximum from the equator y0; recall, however, that the axisymmetric model is formulated on the β-plane while the GCM is on the sphere.
 A broad qualitative resemblance between the results from the two models is apparent. At a finer level of detail a number of differences are evident, but some key qualitative features are nonetheless captured by the axisymmetric model.
 Both the tropical easterlies and extratropical winter westerlies are stronger and have broader maxima in the axisymmetric model than in the GCM, particularly at large y0. In the case of the westerlies, this may be in part due to the lack of eddy heat fluxes in the axisymmetric model, as these tend to reduce the meridional temperature gradient and thus also the winds in the subtropics. The summer westerly jet is stronger in the GCM than in the axisymmetric model at large y0.
 In the axisymmetric model, the maximum v (in absolute value) lies in a region of small local Rossby number when y0 is small; in those cases, Ro > 0.6 only in a very small region near the equator. At larger y0, the region in which Ro > 0.6 expands into the winter hemisphere, while simultaneously the location of the tropical peak in v moves equatorward, so that eventually the two overlap at large y0. In the GCM, qualitatively similar results hold. The region in which Ro > 0.6 is smaller and the peak values of meridional velocity are smaller in the GCM; the location of peak meridional velocity moves equatorward more strongly in the GCM as y0 increases.
 The GCM has an extratropical equatorward-flowing Ferrel cell in the winter hemisphere, which the axisymmetric model largely lacks. This is explained by stronger eddy momentum flux divergence in the GCM, as can be seen by comparing Figures 1 and 8, and also by the larger Coriolis parameter at high latitudes in the axisymmetric model, since it is on an equatorial β-plane as opposed to the spherical GCM, and at high latitudes v is determined by the linear balance fv ≈ . This difference does not concern us, as we are interested in the tropical circulation here.
Figure 8. Eddy momentum flux divergence (m s−2) from the single-layer model. The white color refers to negative values less than 6 × 10−5 m s−2.
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 Figure 8 shows the eddy momentum flux divergence from the axisymmetric model, plotted as a function of y and y0. There is broad qualitative agreement with the GCM results in Figure 1, though the regions of strong positive EMFD in the axisymmetric model's winter hemisphere are broader, occur further poleward, and increase in magnitude more strongly as y0 increases compared to the GCM. The negative EMFD at high latitudes is stronger in the GCM, particularly at small y0.
 Given the better agreement between the two panels of Figure 1 than between the single-layer results shown here and either one of the former, the proximate cause of the disagreement is apparently the different zonal mean zonal winds in the two models, rather than the functional form of the parameterization (2.5) itself. However, the winds themselves are determined by strong feedbacks between the eddies (whether explicit or parameterized) and the mean flow, so it is quite possible that discrepancies which appear relatively small in comparisons such as that in Figure 1 may lead to larger ones in interactive calculations such as that shown in Figure 8. Despite these differences, the similarities are apparently strong enough to reproduce at least qualitatively, and to some degree quantitatively, key features of the GCM solutions, such as the transitions in the dynamics of the circulation as y0 increases.
Figure 9. Log-log plot of maximum meridional velocity as a function of y0, for the calculations with eddy momentum flux divergence (black) and without (red). Reference power laws of and are shown by the blue and black dot-dash lines, respectively.
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