Gravity wave dynamics in rotating shallow water

At small Rossby numbers, geophysical fluids display dynamics on two timescales: slow balanced motions and fast gravity waves. For zero Rossby number, quasigeostrophy (QG) is a theory for the leading-order dynamics of just the slow modes, in the absence of gravity wave interactions. Within the context of the shallow water model, an extension of QG to finite Rossby number corrections has been developed which allows for the calculation of balanced corrections to QG, while still retaining interactions with the gravity waves. This same perspective also illuminates an exact theory for the nonlinear dynamics of just the fast gravity waves, in the absence of balanced motions.

In one spatial dimension, the gravity wave evolution is governed by the usual linear dispersion, in addition to the nonlinear tendency to steepen gradients. Not surprisingly, these dynamics admit nonlinear travelling wave solutions, including a solitary wave of depression. In two spatial dimensions, free gravity wave turbulence can be forced without the generation of potential vorticity.