Plane one-dimensional MHD flows: symmetries and conservation laws

TL;DR

This paper analyzes symmetries and conservation laws in one-dimensional magnetohydrodynamic flows, providing classifications for conductivity cases and deriving conservation laws using symmetry and variational methods.

Contribution

It offers a comprehensive Lie group classification for MHD equations with variable conductivity and derives conservation laws via Noether's theorem in Lagrangian coordinates.

Findings

Classification of conductivity functions with symmetry extensions

Derivation of conservation laws for finite and infinite conductivity cases

Variational formulation enabling Noether's theorem application

Abstract

The paper considers the plane one-dimensional flows for magnetohydrodynamics in the mass Lagrangian coordinates. The inviscid, thermally non-conducting medium is modeled by a polytropic gas. The equations are examined for symmetries and conservation laws. For the case of the finite electric conductivity we establish Lie group classification, i.e. we describe all cases of the conductivity $ \sigma ( \rho , p)$ for which there are symmetry extensions. The conservation laws are derived by the direct computation. For the case of the infinite electrical conductivity the equations can be brought into a variational form in the Lagrangian coordinates. Lie group classification is performed for the entropy function as an arbitrary element. Using the variational structure, we employ the Noether theorem for obtaining conservation laws. The conservation laws are also given in the physical variables.