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Multi-symplectic magnetohydrodynamics

Published online by Cambridge University Press:  09 June 2014

G. M. Webb*
Center for Space Plasma and Aeronomic Research, The University of Alabama in Huntsville, Huntsville, AL 35805, USA
J. F. McKenzie
Center for Space Plasma and Aeronomic Research, The University of Alabama in Huntsville, Huntsville, AL 35805, USA Department of Mathematics and Statistics, Durban University of Technology, Steve Biko Campus, Durban, South Africa, and School of Mathematical Sciences, University of Kwa-Zulu Natal, Durban, South Africa
G. P. Zank
Center for Space Plasma and Aeronomic Research, The University of Alabama in Huntsville, Huntsville, AL 35805, USA Department of Space Science, The University of Alabama in Huntsville, Huntsville, AL 35805, USA
Email address for correspondence:


A multi-symplectic formulation of ideal magnetohydrodynamics (MHD) is developed based on the Clebsch variable variational principle in which the Lagrangian consists of the kinetic minus the potential energy of the MHD fluid modified by constraints using Lagrange multipliers that ensure mass conservation, entropy advection with the flow, the Lin constraint, and Faraday's equation (i.e. the magnetic flux is Lie dragged with the flow). The analysis is also carried out using the magnetic vector potential à where α=Ã⋅dx is Lie dragged with the flow, and B=∇×Ã. The multi-symplectic conservation laws give rise to the Eulerian momentum and energy conservation laws. The symplecticity or structural conservation laws for the multi-symplectic system corresponds to the conservation of phase space. It corresponds to taking derivatives of the momentum and energy conservation laws and combining them to produce n(n−1)/2 extra conservation laws, where n is the number of independent variables. Noether's theorem for the multi-symplectic MHD system is derived, including the case of non-Cartesian space coordinates, where the metric plays a role in the equations.

Copyright © Cambridge University Press 2014 

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