Extended-geostrophic Hamiltonian models for rotating shallow water motion

By using a small Rossby number expansion in Hamilton's principle for shallow water dynamics in a rapidly rotating reference frame, we derive new approximate extended-geostrophic equations with pressure, or surface height, as the dynamical variable. By introducing modeling parameters into the relation between the ageostrophic velocity and surface height, we obtain a family of models for which the functional form of the potential vorticity, conserved on fluid particles, and of the kinetic energy contribution to the globally conserved energy can be prescribed. The particular approximate equations which result when the ageostrophic velocity is chosen to be in full agreement with an asymptotic small Rossby number expansion of the rotating shallow water equations give a new model with apparently higher-order dynamical consistency than either quasigeostrophy or geostrophic momentum. The geostrophic momentum model is recovered as a special case among these new models.