M. Oliver,
A variational derivation of the geostrophic momentum approximation,
J. Fluid Mech. 751 (2014), R2, doi:10.1017/jfm.2014.309.

Abstract:

This paper demonstrates that the shallow water semigeostrophic equations arise from a degenerate second order Hamilton principle of very special structure. The associated Euler--Lagrange operator factors into a fast and a slow first order operator; restricting to the slow part yields the geostrophic momentum approximation as balanced dynamics. While semigeostrophic theory has been considered variationally before, this structure appears new. It leads to a straightforward derivation of the geostrophic momentum approximation and its associated potential vorticity law. Our observations further affirm, from a different point of view, the known difficulty in generalizing the semigeostrophic equations to the case of a spatially varying Coriolis parameter.
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