The interface between active/inactive convection, referred here to as precipitation front, is characterized mathematically by a discontinuity in the precipitation rate and in the vertical velocity. Assuming the limit of infinitely fast relaxation, the system decouples into an equation for the precipitating region and non-precipitating regions, where the solutions match at the interface.
This framework is applied to study the behavior of a disturbance propagating along a narrow precipitation band, similar to the InterTropical Converegence Zone (ITCZ). First, we obtain stationary solutions in which the convection is active in some limited portion of the domain ( the "ITCZ"). We then analyze the propagation of a small perturbation of this basic state, in effect, analyzing the effects of the width and location of the ITCZ on the propagation of Kelvin, Yanai and Rossby waves.
We begin with a background state consisting of a stationary symmetric Hadley circulation and a precipitating region centered at the Equator. In this problem, the width of the precipitating region is controlled by the saturation mixing ratio at the surface. Next, we obtain the equations for a small perturbation of the equilibrium state and derive the relationship between the width of the ITCZ and the speed of propagation of the wave. These analytic solutions are then compared to numerical simulations. We find that for an ITCZ width of the order of the equatorial Rossby radius, Kelvin waves propagate at the moist gravity wave speed (about 15 m/s), whereas for a narrow ITCZ, the propagation speed is comparable to the dry gravity wave speed (about 50 m/s). Similar results are found in the case of the ITCZ off the equator.