Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics

P. J. Dellar  (2002) Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics    Phys. Plasmas 9 1130-1136   doi:10.1063/1.1463415  (BibTeX entry)

Reprints available as gzipped PostScript (swmhd.ps.gz 50K) or as PDF with hyperlinks (swmhd.pdf 125K)

Copyright (2002) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.

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 non-canonical 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 appropriate divergence-free constraint div (hB)=0 is relaxed.

See P. A. Gilman (2000) Magnetohydrodynamic "shallow water'' equations for the solar tachocline  Astrophys. J. Lett. 544 79-82

Matters arising
It is not perhaps as clear as it could be from the text that the matrices for the symmetric hyperbolic structure in appendix B are given for the (p,ux,uy,Bx,By) ordering of the entropy variables. In the Hamiltonian formulation it is conventional to put the momentum or velocity first, in order to exhibit the semi-direct product structure of the Poisson tensor J, but when dealing with hyperbolic systems from fluid dynamics it is more common to put the density or height first.

The Hamiltonian structure in (u,h,psi) variables given at the end of section III turns out to be the same as the structure given previously by Ripa (1993) for his shallow water equations with a horizontally varying temperature (with psi denoting temperature rather than magnetic flux function). My derivation of this structure from the Lie-Poisson structure in (m,h,hB) variables provides a much more direct proof of the Jacobi identity for the Poisson bracket than Ripa's long "formal proof". For further details see my 2003 followup paper.

P. Ripa (1993) Conservation-laws for primitive equations models with inhomogeneous layers Geophys. Astrophys. Fluid Dynamics 70 85-111

P. J. Dellar (2003) Common Hamiltonian structure of the shallow water equations with horizontal temperature gradients and magnetic fields  Phys. Fluids 15 292-297  doi:10.1063/1.1530576

BibTeX citation information:
author = "P. J. Dellar",
title = "{Hamiltonian} and symmetric hyperbolic structures of shallow water magnetohydrodynamics",
year = "2002",
journal = "Phys. Plasmas",
volume = "9",
pages = "1130--1136",
URL = "http://link.aip.org/link/?PHP/9/1130",
DOI = "doi:10.1063/1.1463415"}

Many journal articles now list a Digital Object Identifier (DOI). This is intended to provide a uniform citing and linking mechanism across journals and publishers,  see www.doi.org for details. Any paper with a listed DOI may be linked to using a URL of the form http://dx.doi.org/DOI. The DOI resolver will translate this URL into a valid URL for the paper.