Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics

Shallow water magnetohydrodynamics is a recently proposed model for a thin layer of incompressible, electrically conducting fluid. The velocity and magnetic field are taken to be nearly two dimensional, with approximate magnetohydrostatic balance in the perpendicular direction. In this paper a Hamiltonian description, with the ubiquitous noncanonical Lie-Poisson bracket for barotropic magnetohydrodynamics, is derived by integrating the three-dimensional energy density in the perpendicular direction. Specialization to two dimensions yields an elegant form of the bracket, from which further conserved quantities (Casimirs) of shallow water magnetohydrodynamics are derived. These Casimirs closely resemble the Casimirs of incompressible reduced magnetohydrodynamics, so the stability properties of the two systems may be expected to be similar. The shallow water magnetohydrodynamics system is also cast into symmetric hyperbolic form. The symmetric and Hamiltonian properties become incompatible when the relevant divergence-free constraint del.(hB)=0 is relaxed. (C) 2002 American Institute of Physics.