The weak temperature gradient (WTG) approximation, in which the temperature tendency and advection terms are neglected in the temperature equation so that that equation reduces to a diagnostic balance between heating and vertical motion, is applied to nonlinear shallow water model with the heating parameterized as a Newtonian relaxation on the layer thickness (the relevant temperature-like variable). Layer thickness variations are diagnosed from the horizontal flow using a balance constraint obtained from the divergence of the momentum equation. For this particular heating parameterization, if all terms are retained in that equation, the balance condition transforms into a time-dependent, diffusion-type equation for the layer thickness. If the tendency term is dropped, a more standard elliptic balance equation is obtained. In either case, in the present implementation, the resulting thickness variations are then used in the heating parameterization, although they have been neglected in the thickness equation itself.

The equations are solved numerically. As tests of the solution algorithm, the solutions are used to reproduce earlier results using the same system in particular limits, namely a linear Gill-type problem and a nonlinear axisymmetric Hadley cell model. Then general nonlinear solutions are obtained for forcings with two-dimensional structure. These are compared with the full shallow water solutions. The comparison sheds light on the relative roles of balanced vs. unbalanced motions in the tropical circulation.