Common Hamiltonian structure of the shallow water equations with horizontal temperature gradients and magnetic fields

The Hamiltonian structure of the inhomogeneous layer models for geophysical fluid dynamics devised by Ripa [Geophys. Astrophys. Fluid Dyn. 70, 85 (1993)] involves the same Poisson bracket as a Hamiltonian formulation of shallow water magnetohydrodynamics in velocity, height, and magnetic flux function variables. This Poisson bracket becomes the Lie–Poisson bracket for a semidirect product Lie algebra under a change of variables, giving a simple and direct proof of the Jacobi identity in place of Ripa’s long outline proof. The same bracket has appeared before in compressible and relativistic magnetohydrodynamics. The Hamiltonian is the integral of the three dimensional energy density for both the inhomogeneous layer and magnetohydrodynamic systems, which provides a compact derivation of Ripa’s models.

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