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 P in Calkin's approach naturally arises from the Lagrange multiplier constraint equation for Faraday's equation for the magnetic induction B, or alternatively from the magnetic vector potential form of Faraday's equation. Gauss's equation, (divergence of 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 nonbarotropic 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.PACS numbers: 95.30. Qd,47.35.Tv,52.30.Cv,45.20.Jj,96.60.j,96.60.Vg P·dS = P x dy ∧ dz + P y dz ∧ dx + P z dx ∧ dy,( 1.5) is the polarization 2-form. In the usual MHD, non-relativistic limit ∇·P = 0 (i..e. ρ c = 0), and (1.2) simplifies to:

Hidden symmetries, trivial conservation laws and Casimir invariants in geophysical fluid dynamics

“…As will be shown in subsection 5.1.1, local gauge freedom does not exist for the dynamical equations describing density and the Clebsch potential, at least when the governing equations are unapproximated or only geometrically approximated. The associated conservation laws expressed by the symmetry transformations (3.30)-(3.35) are compatible with what is presented in Webb and Anco (2017).…”

“…The transformations (3.30)-(3.35) are called a gauge symmetry by e.g. Henyey (1982) and Webb and Anco (2017). Notice however that they represent a global, not a local, gauge symmetry as ò is a constant.…”

Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

“…Calkin (1963) and Webb & Anco (2017) derived the conservation law for the magnetic helicity density via gauge field theory. The symmetry responsible for the magnetic helicity conservation law, for an electric field and electric potential , where and is not a fluid relabelling symmetry.…”

“…The symmetry responsible for the magnetic helicity conservation law, for an electric field and electric potential , where and is not a fluid relabelling symmetry. It is due to a gauge symmetry, involving the Lagrange multipliers that enforce Faraday’s equation and Gauss’s equation ( ), in the variational principle (Webb & Anco 2017).…”

Using topology to locate the position where fully three-dimensional reconnection occurs

“…The product of the winding and the magnetic flux B z dA (dA = dxdy) is the magnetic helicity, whose fieldline density may be defined in terms of the winding angle [56,64] as The branch cut discontinuity in the function H also manifests itself in the helicity and can be linked to the Aharonov-Bohm effect [65,66]. By removing the magnetic flux it was shown by Prior and Yeates [56] that the field line distribution of L will change only if the field changes connectivity.…”