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Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

Part of: PLA Topological Methods in MHD, Fluids, and Plasmas

Published online by Cambridge University Press:  10 October 2019

G. M. Webb Open the ORCID record for G. M. Webb [Opens in a new window] ,
A. Prasad Open the ORCID record for A. Prasad [Opens in a new window] ,
S. C. Anco Open the ORCID record for S. C. Anco [Opens in a new window]  and
Q. Hu Open the ORCID record for Q. Hu [Opens in a new window]
G. M. Webb*
Affiliation:
Center for Space Plasma and Aeronomic Research, The University of Alabama in Huntsville, Huntsville, AL 35805, USA
A. Prasad
Affiliation:
Center for Space Plasma and Aeronomic Research, The University of Alabama in Huntsville, Huntsville, AL 35805, USA
S. C. Anco
Affiliation:
Department of Mathematics, Brock University, St. Catharines, ON L2S 3A1, Canada
Q. Hu
Affiliation:
Center for Space Plasma and Aeronomic Research, The University of Alabama in Huntsville, Huntsville, AL 35805, USA Department of Space Science, The University of Alabama in Huntsville, Huntsville, AL 35899, USA
*
†Email address for correspondence: [email protected]
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Abstract

The Godbillon–Vey invariant occurs in homology theory, and algebraic topology, when conditions for a co-dimension 1, foliation of a three-dimensional 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 $h_{m}=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}=\boldsymbol{A}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{A})=0$, where $\boldsymbol{A}$ is the magnetic vector potential and $\boldsymbol{B}$ is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon–Vey field $\unicode[STIX]{x1D6C8}=\boldsymbol{A}\times \boldsymbol{B}/|\boldsymbol{A}|^{2}$ and the Godbillon–Vey helicity density $h_{\text{gv}}=\unicode[STIX]{x1D6C8}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D6C8})$ in general MHD flows in which either $h_{m}=0$ or $h_{m}\neq 0$. A conservation law for $h_{\text{gv}}$ occurs in flows for which $h_{m}=0$. For $h_{m}\neq 0$ the evolution equation for $h_{\text{gv}}$ contains a source term in which $h_{m}$ is coupled to $h_{\text{gv}}$ via the shear tensor of the background flow. The transport equation for $h_{\text{gv}}$ also depends on the electric field potential $\unicode[STIX]{x1D713}$, which is related to the gauge for $\boldsymbol{A}$, which takes its simplest form for the advected $\boldsymbol{A}$ gauge in which $\unicode[STIX]{x1D713}=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{u}$ where $\boldsymbol{u}$ 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.

Keywords

astrophysical plasmasplasma flowsspace plasma physics
Type
Research Article
Information
Journal of Plasma Physics , Volume 85 , Issue 5 , October 2019 , 775850502
Copyright
© Cambridge University Press 2019 

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