On Magnetohydrodynamic Gauge Field Theory

Abstract

Clebsch potential gauge field theory for magnetohydrodynamics is developed based in part on the theory of Calkin (1963). It is shown how the polarization vector ${\bf P}$ in Calkin's approach, naturally arises from the Lagrange multiplier constraint equation for Faraday's equation for the magnetic induction ${\bf B}$, or alternatively from the magnetic vector potential form of Faraday's equation. Gauss's equation, (divergence of ${\bf B}$ is zero), is incorporated in the variational principle by means of a Lagrange multiplier constraint. Noether's theorem, coupled with the gauge symmetries is used to derive the conservation laws for (a)\ magnetic helicity (b)\ cross helicity, (c) fluid helicity for non-magnetized fluids, and (d) a class of conservation laws associated with curl and divergence equations, which applies to Faraday's equation and Gauss's equation. The magnetic helicity conservation law is due to a gauge symmetry in MHD and not due to a fluid relabelling symmetry. The analysis is carried out for the general case of a non-barotropic gas, in which the gas pressure and internal energy density depend on both the entropy $S$ and the gas density $ρ$. The cross helicity and fluid helicity conservation laws in the non-barotropic case, are nonlocal conservation laws, that reduce to local conservation laws for the case of a barotropic gas. The connections between gauge symmetries, Clebsch potentials and Casimirs are developed. It is shown that the gauge symmetry functionals in the work of Henyey (1982) satisfy the Casimir determining equations.