A HAMILTONIAN WEAK-WAVE MODEL FOR SHALLOW-WATER FLOW

A reduced dynamical model is derived which describes the interaction o f weak inertiagravity waves with nonlinear vortical motion in the cont ext of rotating shallow-water flow. The formal scaling assumptions are (i) that there is a separation in timescales between the vortical mot ion and the inertia-gravity waves, and (ii) that the divergence is wea k compared to the vorticity. The model is Hamiltonian, and possesses c onservation laws analogous to those in the shallow-water equations. Un like the shallow-water equations, the energy invariant is quadratic. N onlinear stability theorems are derived for this system, and its linea r eigenvalue properties are investigated in the context of some simple basic flows.