## 1. Introduction

[2] Solutions of axially symmetric inviscid models of the upper branches of the Hadley circulation capture many of the main features of the zonally averaged circulation, despite the vast simplifications employed in their formulations. Outside the Tropics, heat and momentum fluxes are dominated by macro-turbulence, making axially symmetric models less relevant in these latitudes. Most notable of these models is *Held and Hou*'s [1980, hereafter HH] model, which approximates the circulation in the upper branches of the Hadley cell in the inviscid limit. The solutions of HH follow a 2-region paradigm made up of a tropical, uniform-*M* region, and an extratropical, radiative-equilibrium region. The Angular Momentum Conserving (AMC) solutions of HH were extended to off-equatorial heating by *Lindzen and Hou* [1988, hereafter LH]. When compared with the zonally averaged atmosphere and with axially symmetric 2D models, the AMC solutions capture the observed Hadley circulation width, the relative strength of the subtropical jets, and the relative strength of the winter and summer cells. However, the AMC solutions fail in predicting the non-uniform distribution of angular momentum at the Tropics [cf. *Schneider*, 2006], and in the case of off-equatorial heating, they predict stronger-than-observed tropical temperature gradients and easterlies that are also fixed at the equator, and a winter jet that is farther poleward than the summer jet (opposite to their observed relative latitudes). In addition, the AMC solutions for off-equatorial heating predict a nonlinear amplification of the winter circulation even for small displacements of the heating off the equator, in contrast to the observed atmosphere [*Dima and Wallace*, 2003].

[3] As was shown by *Adam and Paldor* [2009, hereafter AP09] and *Adam and Paldor* [2010, hereafter AP10] a Shallow Water Model (SWM) captures the essential features of the idealized axisymmetric models used by HH and LH. In addition, when Vertical Advection of Momentum (VAM) is assumed into the rising branch of the Hadley circulation from an underlying passive surface layer a tropical mass/heat acquiring, uniform-temperature and non-uniform-*M* region forms in addition to the subtropical uniform-*M* and extratropical radiative-equilibrium regions that exist in the AMC solutions. When VAM is “turned off” the VAM solutions of AP09 and AP10 become exactly the AMC solutions of HH and LH, respectively, by enforcing Hide's theorem (choosing those states for which the zonal velocity vanishes at the latitudes separating the winter and summer cells). The VAM solutions predict realistic temperature gradients and easterlies at the Tropics but (as in the AMC solutions) predict stronger than observed circulation strength amplification with increasing latitude of maximal heating and incorrect relative latitudes of the subtropical jets.

[4] The latitudes of the subtropical jets in the AMC and VAM solutions depend on the latitude of maximal heating and on the thermal Rossby number (equation (1f)). The jet latitudes observed in the numerical results of 2D axially symmetric models [e.g., LH; *Fang and Tung*, 1999, hereafter FT] differ from those predicted by AMC and VAM solutions and exhibit a strong dependence on the pole-to-equator temperature difference (PETD). This suggests that the simple parametric dependence of the AMC and VAM solutions does not fully describe “nearly-inviscid” axially symmetric steady states.

[5] In the 2D axially symmetric models such as the ones used by HH, LH, FT and *Walker and Schneider* [2005, hereafter WS], the “nearly-inviscid” limit was assumed to be reached by using the lowest viscosity allowed by the numerical scheme for which stable solutions were obtained. The underlying assumption in all these studies is that in the limit of vanishing viscosity the mean properties of the circulation become insensitive to the value of the viscosity (this is the “nearly-inviscid” assumption) [*Schneider*, 1977]. This assumption (which also implies that inviscid solutions can be used to predict the mean features of the “nearly-inviscid” circulation), however, has never been validated by a direct comparison of viscous 2D axially symmetric models with inviscid models (i.e., with the viscosity coefficient set equal to zero).

[6] A direct comparison of inviscid and nearly-inviscid results is possible using a shallow water framework, which mimics the upper branch of the Hadley circulation. *Polvani and Sobel* [2002, hereafter PS] studied the inviscid limit in a SWM with Rayleigh friction (but without vertical advection of momentum) on the equatorial *f*- and *β*-planes. PS derived expressions for the circulation strength and width by assuming a Weak Temperature Gradient (WTG) throughout the circulation cell. The effect of viscosity in the WTG solutions of PS is to widen and strengthen the circulation while damping the subtropical jet amplitude, in accordance with results of viscous 2D axially symmetric models. Unlike the VAM and AMC solutions that are purely inviscid the WTG solutions are the only solutions that permit non-vanishing viscosity. These solutions, however, do not include vertical advection of momentum and may not be suitable for studying Hadley circulations outside the Tropics.

[7] This work examines the effect of viscosity on the Hadley circulation using a SWM that, like PS, includes Rayleigh friction. The present SWM adds the following physical elements to PS: 1) Vertical advection of momentum; 2) Spherical geometry; 3) Off-equatorial and annually varying heating. The dependence of the solutions on the PETD is also examined. The numerical solutions in the viscous case are compared with the inviscid solutions of AP09 and AP10.