A simplified set of equations is derived systematically below for the interaction
of large scale flow fields and precipitation in the tropical atmosphere.
These equations, the Tropical Climate Model,
have the form of a shallow water equation and an equation
for moisture coupled through a strongly nonlinear source term.
This source term, the precipitation, is of relaxation type in one region of state space f
or the temperature and moisture, and vanishes identically elsewhere in the state space of these variables.
In addition, the equations are coupled nonlinearly to the equations for barotropic incompressible flow.
Several mathematical features of this system are developed below including energy principles for solutions and
their first derivatives independent of relaxation time.
With these estimates, the formal infinitely fast relaxation limit converges
to a novel hyperbolic free boundary problem for the motion of precipitation fronts
from a large scale dynamical perspective. Elementary exact solutions of the limiting dynamics
involving precipitation fronts are developed below and include three families of waves:
fast drying fronts as well as slow and fast moistening fronts.
The last two families of waves violate Lax's Shock Inequalities;
nevertheless, numerical experiments presented below confirm their robust realizability
with realistic finite relaxation times.
From the viewpoint of tropical atmospheric dynamics,
the theory developed here provides a new perspective
on the fashion in which the prominent large scale regions of moisture in the tropics associated
with deep convection can move and interact with large scale dynamics in the quasi-equilibrium approximation.