Nonlinear shallow water model with WTG approximation

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 parameterized as a Newtonian relaxation on the layer thickness towards a prescribed function of latitude and longitude, containing an isolated maximum or minimum, as in the classic linear Gill problem. 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. A boundary separating regions of parameter space in which steady solutions are and are not found is also delineated. Differences between the WTG solutions and those from the full shallow water system will be discussed; over much of the parameter space, the two are quite similar.