Simplified variational principles for barotropic magnetohydrodynamics

Abstract

Variational principles for magnetohydrodynamics have been introduced by previous authors both in Lagrangian and Eulerian form. In this paper we introduce simpler Eulerian variational principles from which all the relevant equations of barotropic magnetohydrodynamics can be derived. The variational principle is given in terms of six independent functions for non-stationary barotropic flows with trivial topologies and three independent functions for stationary barotropic flows. This is less than the seven variables which appear in the standard equations of barotropic magnetohydrodynamics, which are the magnetic field B the velocity field v and the density ρ.For non-trivial topologies it is necessary to assume that some of the variables introduced in the non-stationary formalism are non-single-valued. That is, it is necessary to introduce a number of branch cuts in order to define single-valued branches of the field variables. In turn, these cuts along with the six field variables constitute an extended number of dynamic variables. The number of cuts necessary depends on the flow. The relations between barotropic magnetohydrodynamics topological constants and the functions of the present formalism will be elucidated.The equations obtained for non-stationary barotropic magnetohydrodynamics resemble the equations of Frenkel et al. (Phys. Lett. A, vol. 88, 1982, p. 461). The connection between the Hamiltonian formalism introduced in Frenkel et al. (1982) and the present Lagrangian formalism (with Eulerian variables) will be discussed.