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

@article{Dellar2003CommonHS,
  title={Common Hamiltonian structure of the shallow water equations with horizontal temperature gradients and magnetic fields},
  author={Paul J. Dellar},
  journal={Physics of Fluids},
  year={2003},
  volume={15},
  pages={292-297}
}
  • P. Dellar
  • Published 1 February 2003
  • Physics
  • Physics of Fluids
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… Expand
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References

SHOWING 1-10 OF 31 REFERENCES
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 twoExpand
Hamiltonian theory of relativistic magnetohydrodynamics with anisotropic pressure
This Brief Communication introduces a special relativistic extension of ideal magnetohydrodynamics having anisotropic pressure, and provides its Hamiltonian formulation in a fixed inertial frame. TheExpand
Extended-geostrophic Hamiltonian models for rotating shallow water motion
Abstract By using a small Rossby number expansion in Hamilton's principle for shallow water dynamics in a rapidly rotating reference frame, we derive new approximate extended-geostrophic equationsExpand
Hyperbolic theory of the “shallow water” magnetohydrodynamics equations
Recently the shallow water magnetohydrodynamic (SMHD) equations have been proposed for describing the dynamics of nearly incompressible conducting fluids for which the evolution is nearlyExpand
A New Integrable Shallow Water Equation
Publisher Summary This chapter discusses about a new integrable shallow water equation. Completely integrable nonlinear partial differential equations arise at various levels of approximation inExpand
Hyperbolic theory of the ‘ ‘ shallow water ’ ’ magnetohydrodynamics equations
Recently the shallow water magnetohydrodynamic ~SMHD! equations have been proposed for describing the dynamics of nearly incompressible conducting fluids for which the evolution is nearlyExpand
The Euler-Poincaré Equations in Geophysical Fluid Dynamics
Recent theoretical work has developed the Hamilton's-principle analog of Lie-Poisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: 1. Euler-PoincareExpand
Euler – Poincaré Equations in Geophysical Fluid Dynamics 3 1 Introduction
Recent theoretical work has developed the Hamilton’s-principle analog of Lie-Poisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: 1. Euler–PoincaréExpand
Conservation laws for primitive equations models with inhomogeneous layers
Abstract A general primitive equation model with non-uniform layers is set up by simply vertically averaging the density, horizontal pressure gradient and velocity fields in ech layer; these averagedExpand
HAMILTONIAN FLUID MECHANICS
This paper reviews the relatively recent application of the methods of Hamiltonian mechanics to problems in fluid dynamics. By Hamiltonian mechanics I mean all of what is often called classicalExpand
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