An idealized model of a Walker circulation based on the Weak Temperature Gradient (WTG) approximation and a single baroclinic vertical mode for all fields is analyzed. The circulation is forced by a sinusoidal variation of sea-surface temperature (SST). A simple feedback of deep convective cloud-radiative forcing on tropospheric radiative cooling is included, and a moist convective adjustment is used to interactively specify the location and intensity of deep convection. A goal is to understand how the fraction of the domain undergoing deep convection depends on the SST difference across the domain.

The WTG approximation greatly simplifies the calculation of the circulation. For small SST differences, convection occurs everywhere and a fully analytic solution is possible; for larger SST differences, a simple nonlinear algorithm is used to determine the edge of the convective region and the tropospheric temperature. The solution is invariant to changes of domain size. The divergent circulation is independent of the Coriolis parameter as long as the domain is sufficiently narrow so WTG remains accurate.

The convective region narrows and intensifies considerably when cloud-radiation feedbacks are considered. As the SST difference increases, the convective region shrinks and the troposphere warms; simple approximate formulas for these trends in terms of the control parameters are derived. For large SST differences, the convection over the warmest water becomes susceptible to a radiative-convective instability and there is no steady-state solution.