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

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 entry)
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####

Abstract

The Hamiltonian structure of the inhomogeneous layer models for
geophysical
fluid dynamics devised by Ripa [*Geophys. Astrophys. Fluid Dyn.* **70**
p. 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.
P. Ripa (1993) *Conservation-laws for primitive equations
models
with inhomogeneous layers** *Geophys. Astrophys. Fluid
Dynamics
**70** 85-111

The Hamiltonian structure for shallow water
magnetohydrodynamics was
given in

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

##### Matters arising

The relation between potential vorticity conservation and the
particle relabeling symmetry referred to in section III has a long
history, going back at least to

W. A. Newcomb (1967) Exchange
invariance in fluid systems in "Magneto-fluid and plasma
dynamics" Proc. Symp. Appl. Math. 18
pages 152-161 (published by the American Mathematical Society).

One of the first oceanographic applications, for incompressible fluids,
is

P. Ripa (1981) Symmetries and
conservation laws for internal gravity waves in "Nonlinear
properties of internal waves" edited by B. J. West, AIP
Conf. Proc. 76 pages 281-306.

For a more detailed history see

N. Padhye & P. J. Morrison (1996) Fluid
element relabeling symmetry Phys.
Lett. A 219 287-292 doi:10.1016/0375-9601(96)00472-0

#### BibTeX citation information:

@article{DellarCommonSW03,

author = "P. J. Dellar",

title = "Common {Hamiltonian} structure of the shallow water equations
with horizontal temperature gradients and magnetic fields",

year = "2003",

journal = "Phys. Fluids",

volume = "15",

pages = "292--297",

URL = "http://link.aip.org/link/?PHFLE6/15/292",

DOI = "doi:10.1063/1.1530576"

}

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