A conserved local cross helicity for non-barotropic MHD

Yahalom, Asher

doi:10.1080/03091929.2017.1281410

Cited by 26 publications

(41 citation statements)

References 33 publications

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“…and In this appendix we discuss the work of Yahalom (2013Yahalom ( , 2016Yahalom ( , 2017a for B and u (we use r ≡ −σ in our formulation). It is straightforward to write down the other variational equations by varying ρ, S and the Clebsch variables in the variational principle (see e.g.…”

Section: Appendix Dmentioning

confidence: 99%

“…(E.12) shows that for a closed field line, the jump in [ζ] is non-zero for a non-trivial magnetic helicity. Yahalom (2013Yahalom ( , 2016Yahalom ( , 2017a refers to (E.12) as the MHD 'magnetic Aharonov-Bohm effect', in analogy with the Aharonov-Bohm effect in quantum mechanics. Yahalom (2013Yahalom ( , 2016Yahalom ( , 2017a and Webb et al (2014a,b) developed conservation laws for cross helicity and a generalized cross helicity for both barotropic and nonbarotropic MHD.…”

Section: (E4)mentioning

confidence: 99%

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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: Appendix Dmentioning

confidence: 99%

Section: (E4)mentioning

confidence: 99%

On magnetohydrodynamic gauge field theory

Webb

Anco

2017

J. Phys. A: Math. Theor.

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“…Webb et al. 2014 a , b ; Yahalom 2017 a , b ). Cross-helicity describes the linkage of the vortex tubes and magnetic flux tubes.…”

Section: Introductionmentioning

confidence: 97%

“…2007; Yahalom 2013; Webb et al. 2014 a , b ; Yahalom 2017 a , b ). These Lie dragged invariants in many cases are related to fluid relabelling symmetries and Casimirs for non-canonical Hamiltonian brackets (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In ideal fluid dynamics and magnetohydrodynamics (MHD), there is a class of invariants that are Lie dragged by the flow (e.g. Moiseev et al (1982); Tur and Yanovsky (1993); Kats (2003); Moffatt (1969Moffatt ( , 1978; Salmon (1982Salmon ( , 1988; Moffatt and Ricca (1992); Cotter et al (2007); Holm et al (1998); Padhye and Morrison (1996a,b); Yahalom (2013Yahalom ( , 2017a; Webb et al (2014a,b)). These Lie dragged invariants in many cases are related to fluid relabelling symmetries and Casimirs for non-canonical Hamiltonian brackets (e.g.…”

Section: Introductionmentioning

confidence: 99%

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Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

et al. 2019

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The Godbillon-Vey invariant occurs in homology theory, and algebraic topology, when conditions for a co-dimension 1, foliation of a 3D manifold are satisfied. The magnetic Godbillon-Vey helicity invariant in magnetohydrodynamics (MHD) is a higher order helicity invariant that occurs for flows, in which the magnetic helicity density h m = A·B = A·(∇ × A) = 0, where A is the magnetic vector potential and B is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon-Vey field η = A × B/|A| 2 and the Godbillon-Vey helicity density h gv = η·(∇ × η) in general MHD flows in which either h m = 0 or h m = 0. A conservation law for h gv occurs in flows for which h m = 0. For h m = 0 the evolution equation for h gv contains a source term in which h m is coupled to h gv via the shear tensor of the background flow. The transport equation for h gv also depends on the electric field potential ψ, which is related to the gauge for A, which takes its simplest form for the advected A gauge in which ψ = A · u where u is the fluid velocity. An application of the Godbillon-Vey magnetic helicity to nonlinear force-free magnetic fields used in solar physics is investigated. The possible uses of the Godbillon-Vey helicity in zero helicity flows in ideal fluid mechanics, and in zero helicity Lagrangian kinematics of three-dimensional advection are discussed.

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Introduction

Webb¹

2018

Magnetohydrodynamics and Fluid Dynamics: Action Principles and Conservation Laws

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A conserved local cross helicity for non-barotropic MHD

Cited by 26 publications

References 33 publications

On magnetohydrodynamic gauge field theory

On magnetohydrodynamic gauge field theory

Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

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