Dispersive shallow water magnetohydrodynamics

P. J. Dellar  Dispersive shallow water magnetohydrodynamics    Phys. Plasmas 10 581-590

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Shallow water magnetohydrodynamics (SWMHD) 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, leading to a reduced two dimensional model. The SWMHD equations have been found previously to admit unphysical cusp-like singularities in finite amplitude magnetogravity waves. This paper extends the Hamiltonian formulation of SWMHD to construct a dispersively regularized system, analogous to the Green--Naghdi equations of hydrodynamics, that supports smooth solitary waves and cnoidal wavetrains, and shares the potential vorticity conservation properties of SWMHD.

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

       P. J. Dellar (2002) Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics Phys. Plasmas 9 pp 1130-1136

       A. E. Green and P. M. Naghdi (1976) A derivation of equations for wave propagation in water of variable depth   J. Fluid Mech. 78 pp 237-246

       J. Miles and R. Salmon (1985) Weakly dispersive nonlinear gravity-waves   J. Fluid Mech. 157 pp 519-531

BibTeX citation information:

author = "P. J. Dellar",
title = "Dispersive shallow water magnetohydrodynamics",
year = "2003",
journal = "Phys. Plasmas",
volume = "10",
pages = "581--590",
URL = "http://link.aip.org/link/?PHPAEN/10/581",
DOI = "doi:10.1063/1.1537690"

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