Conservation laws in magnetohydrodynamics and fluid dynamics: Lagrangian approach

Webb, G. M.; Anco, Stephen C.

doi:10.1063/1.5125089

Cited by 12 publications

(20 citation statements)

References 27 publications

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“…With the 'pre-Maxwell' Ampére's law j = ∇×B/μ 0 and ∇ • B = 0, the Maxwell-Faraday equation (1.3) plays the important role of preserving Galilean invariance (Hosking & Dewar 2015, § 5.4;Webb & Anco 2019), independent of whether or not the IOL equation is enforced. Thus we shall retain it in the following development of a dynamical relaxation theory.…”

Section: Basicsmentioning

confidence: 99%

“…Our derivation shows the Clebsch representation is not needed to apply this polarization representation for j in an action principle if we apply the Lagrangian variational approach of Newcomb (1962). (See also Webb & Anco (2019) for a discussion of the equivalence of Lagrangian and Eulerian variational approaches. )…”

Section: Physical Interpretation Of Estimated Lagrange Multipliermentioning

confidence: 99%

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Relaxed magnetohydrodynamics with ideal Ohm's law constraint

Dewar

Qu²

2022

J. Plasma Phys.

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The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity ‘ideal’ Ohm's law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium.

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Section: Basicsmentioning

confidence: 99%

Section: Physical Interpretation Of Estimated Lagrange Multipliermentioning

confidence: 99%

Relaxed magnetohydrodynamics with ideal Ohm's law constraint

Dewar

Qu²

2022

J. Plasma Phys.

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“…Noether's theorem has been used to obtain conservation laws for ideal MHD (Webb & Anco 2019); see also Padhye & Morrison (1996a,b) and Padhye (1998). A different approach to deriving conservation laws is Lie dragging of differential forms, vector fields and tensors, as developed by Moiseev et al (1982), Sagdeev, Tur & Yanovskii (1990), Tur & Yanovsky (1993), Webb et al (2014a,b), Gilbert & Vanneste (2020), Besse & Frisch (2017) and Anco & Webb (2020).…”

Section: Introductionmentioning

confidence: 99%

“…Our methods are adapted from MHD and ideal fluid mechanics. It is well known how to use a Lagrangian map to relate Eulerian and Lagrangian fluid quantities (Newcomb 1962), and this underpins the EP action principle which is a Lagrangian counterpart of a Hamiltonian formulation (Holm, Marsden & Ratiu 1998; Webb 2018; Webb & Anco 2019). This action principle is based on the Lagrangian map and utilises an associated Lie algebra.…”

Section: Introductionmentioning

confidence: 99%

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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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Group classification of the two‐dimensional magnetogasdynamics equations in Lagrangian coordinates

Meleshko

Капцов

Moyo

et al. 2023

Math Methods in App Sciences

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The present paper is devoted to the group classification of magnetogasdynamics equations in which dependent variables in Euler coordinates depend on time and two spatial coordinates. It is assumed that the continuum is inviscid and nonthermal polytropic gas with infinite electrical conductivity. The equations are considered in mass Lagrangian coordinates. Use of Lagrangian coordinates allows reducing number of dependent variables. The analysis presented in this article gives complete group classification of the studied equations. This analysis is necessary for constructing invariant solutions and conservation laws on the base of Noether's theorem.

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Conservation laws in magnetohydrodynamics and fluid dynamics: Lagrangian approach

Cited by 12 publications

References 27 publications

Relaxed magnetohydrodynamics with ideal Ohm's law constraint

Relaxed magnetohydrodynamics with ideal Ohm's law constraint

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

Group classification of the two‐dimensional magnetogasdynamics equations in Lagrangian coordinates

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