Nonlinear shallow-water solutions using the weak temperature gradient approximation

A form of the weak temperature gradient (WTG) approximation, in which the temperature tendency and advection terms are neglected in the temperature equation so that the equation reduces to a diagnostic balance between heating and vertical motion, is applied to a two-dimensional nonlinear shallow-water model with the heating (mass source) parameterized as a Newtonian relaxation on the temperature (layer thickness) towards a prescribed function of latitude and longitude, containing an isolated maximum or minimum, as in the classic linear Gill problem. In this model, temperature variations are retained in the Newtonian heating term, so that it is not a pure WTG system. It contains no free unbalanced modes, but reduces to the Gill model in the steady linear limit, so that steady solutions may be thought of as containing components corresponding to unbalanced modes in the same sense as the latter. The equations are solved numerically and are compared with full shallow-water solutions in which the WTG approximation is not made. Several external parameters are varied, including the strength, location, sign, and horizontal scale of the mass source, the Rayleigh friction coefficient, and the time scale for the relaxation on the mass field. Indices of the Walker and Hadley circulations are examined as functions of these external parameters. Differences between the WTG solutions and those from the full shallow-water system are small over most of the parameter regime studied, which includes time-dependent as well as steady solutions.