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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)
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##### Abstract

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 (h**B**)=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,u_{x},u_{y},B_{x},B_{y})
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:

@article{DellarSWMHD,

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"}
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