Potential vorticity in magnetohydrodynamics

Webb, G. M.; Mace, R. L.

doi:10.1017/s0022377814000658

Cited by 27 publications

(27 citation statements)

References 39 publications

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“…The multi-symplectic approach to fluid dynamics has been explored by Hydon (2005), Bridges et al (2005Bridges et al ( , 2010, Cotter et al (2007), Webb (2015), and Webb and Anco (2016). The exact connection of these approaches to the present approach remains to be explored.…”

Section: Discussionmentioning

confidence: 99%

On magnetohydrodynamic gauge field theory

Webb

Anco

2017

J. Phys. A: Math. Theor.

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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:

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Section: Discussionmentioning

confidence: 99%

On magnetohydrodynamic gauge field theory

Webb

Anco

2017

J. Phys. A: Math. Theor.

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“…The Euler vorticity equations for ω are as follows: ∆ = ω t + ∇ × ( ω × u) = 0, ∆ 4 = ∇ • ω = 0, (6.23) which is the system (6.1) for M = ω × u, where, as earlier, u is the velocity vector. Ertel's theorem [31] states that the following relationship holds on solutions of (6.23):…”

Section: Ertel's Theoremmentioning

confidence: 99%

Second Noether theorem for quasi-Noether systems

Shankar¹

2016

J. Phys. A: Math. Theor.

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Quasi-Noether differential systems are more general than variational systems and are quite common in mathematical physics. They include practically all differential systems of interest, at least those that have conservation laws. In this paper, we discuss quasi-Noether systems that possess infinite-dimensional (infinite) symmetries involving arbitrary functions of independent variables. For quasi-Noether systems admitting infinite symmetries with arbitrary functions of all independent variables, we state and prove an extension of the Second Noether Theorem. In addition, we prove that infinite sets of conservation laws involving arbitrary functions of all independent variables are trivial and that the associated differential system is under-determined. We discuss infinite symmetries and infinite conservation laws of two important examples of non-variational quasi-Noether systems: the incompressible Euler equations and the Navier-Stokes equations in vorticity formulation, and we show that the infinite sets of conservation laws involving arbitrary functions of all independent variables are trivial. We also analyze infinite symmetries involving arbitrary functions of not all independent variables, prove that the fluxes of conservation laws in these cases are total divergences on solutions, and demonstrate examples of this situation.

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“…(2014 a , b ) emphasize that the generalized cross-helicity conservation law in MHD, and the generalized helicity conservation law in non-barotropic fluids, are non-local in the sense that they depend on the auxiliary non-local variable , which depends on the Lagrangian time integral of the temperature . Note that a potential vorticity conservation equation for non-barotropic MHD was derived by Webb & Mace (2015) by using Noether's second theorem.…”

Section: Introductionmentioning

confidence: 99%