  A reduced dynamical model is derived which describes the interaction of weak
inertia-gravity waves with nonlinear vortical motion in the context of rotating
shallow-water flow. The formal scaling assumptions are (i) that there is a
separation in timescales between the vortical motion and the inertia-gravity
waves, and (ii) that the divergence is weak compared to the vorticity. The
model is Hamiltonian, and possesses conservation laws analogous to those in the
shallow-water equations. Unlike the shallow-water equations, the energy
invariant is quadratic. Nonlinear stability theorems are derived for this
system, and its linear eigenvalue properties are investigated in the context of
some simple basic flows.
