A new model for shallow water in the low-Rossby-number limit

We revisit Salmon's 'Dirac bracket projection' approach to constructing generalized semi-geostrophic equations. One of the obstacles to the method's applicability is that it leads to a sign-indefinite energy functional in the computational domain. In some instances this can cause severe failure of the model. We demonstrate in the simple context of shallow-water semi-geostrophy that the Hamiltonian remains positive definite when the asymptotic expansion at the heart of this method is carried to the next order. The resulting new model can be interpreted in the framework of regularization by Lagrangian averaging, which is currently receiving much attention.