Multi-symplectic magnetohydrodynamics

Abstract

AbstractA 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 potentialÃwhere α=Ã⋅dxis Lie dragged with the flow, andB=∇×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 producen(n−1)/2 extra conservation laws, wherenis 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.