Introduction

Webb, G. M.

doi:10.1007/978-3-319-72511-6_1

Cited by 4 publications

(12 citation statements)

References 51 publications

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“…In this paper, we studied the Godbillon–Vey invariant which arises in magnetohydrodynamics in the case where where is the magnetic induction and is the magnetic vector potential (Tur & Yanovsky 1993; Webb et al. 2014 a ; Webb 2018). The condition , is a necessary and sufficient condition for the Pfaffian to be integrable (Sneddon 1957), which implies that there exists an integrating factor such that , for some potential function .…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…volume integral analogous to , e.g. Tur & Yanovsky (1993) and Webb (2018))? The answer to this question is that, in general, there is a higher-order topological invariant known as the Godbillon–Vey invariant for flows with .…”

Section: Introductionmentioning

confidence: 99%

“…Tur & Yanovsky (1993), Webb et al. (2014 a ), Webb (2018) and Anco & Webb (2018) give discussions of Lie dragged vector fields , one forms , two forms and three forms and scalars, . An example of a Lie dragged two-form in MHD is the magnetic flux .…”

Section: Introductionmentioning

confidence: 99%

“…In appendix A, we provide a detailed derivation of the conservation law for the Godbillon helicity density for the case using the Lie dragging of differential forms (see also Tur & Yanovsky 1993; Webb et al. 2014 a ; Webb 2018). Appendix B, provides a vector calculus derivation of the Godbillon–Vey helicity evolution equation for general MHD flows, both for the case and for the case (we also discuss the gauge potential used for ).…”

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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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

et al. 2019

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“…The HAP formalism has a long history in fluid dynamics and plasma physics – examples of seminal publications prior to the 20th century include Lagrange (1789), Clebsch (1857), von Helmholtz (1858), Clebsch (1859), Hanke (1861) and Kirchhoff (1876). 2 A summary of modern developments in this area can be found in the reviews by Serrin (1959), Truesdell & Toupin (1960), Seliger & Whitham (1968), Arnold (1978), Morrison (1982), Holm et al (1985), Morrison (1998), Arnold & Khesin (1998), Morrison (2005), Holm (2008), Morrison (2009), Lingam (2015 d ), Sudarshan & Mukunda (2016), Morrison (2017), Tassi (2017) and Webb (2018).…”

Section: Introductionmentioning

confidence: 99%

A class of three-dimensional gyroviscous magnetohydrodynamic models

2020

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A Hamiltonian and action principle formalism for deriving three-dimensional gyroviscous magnetohydrodynamic models is presented. The uniqueness of the approach in constructing the gyroviscous tensor from first principles and its ability to explain the origin of the gyromap and the gyroviscous terms are highlighted. The procedure allows for the specification of free functions, which can be used to generate a wide range of gyroviscous models. Through the process of reduction, the noncanonical Hamiltonian bracket is obtained and briefly analysed.

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Action principles and conservation laws for Chew–Goldberger–Low anisotropic plasmas

et al. 2022

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The ideal Chew–Goldberger–Low (CGL) plasma equations, including the double adiabatic conservation laws for the parallel ( $p_\parallel$ ) and perpendicular pressure ( $p_\perp$ ), are investigated using a Lagrangian variational principle. An Euler–Poincaré variational principle is developed and the non-canonical Poisson bracket is obtained, in which the non-canonical variables consist of the mass flux ${\boldsymbol {M}}$ , the density $\rho$ , the entropy variable $\sigma =\rho S$ and the magnetic induction ${\boldsymbol {B}}$ . Conservation laws of the CGL plasma equations are derived via Noether's theorem. The Galilean group leads to conservation of energy, momentum, centre of mass and angular momentum. Cross-helicity conservation arises from a fluid relabelling symmetry, and is local or non-local depending on whether the gradient of $S$ is perpendicular to ${\boldsymbol {B}}$ or otherwise. The point Lie symmetries of the CGL system are shown to comprise the Galilean transformations and scalings.

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Introduction

Cited by 4 publications

References 51 publications

Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

A class of three-dimensional gyroviscous magnetohydrodynamic models

Action principles and conservation laws for Chew–Goldberger–Low anisotropic plasmas

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