Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics I: Lie Dragging Approach

TL;DR

This paper explores the conservation laws of invariants and helicities in ideal magnetohydrodynamics and gas dynamics, emphasizing Lie dragging, gauge conditions, and both local and nonlocal invariants.

Contribution

It provides a unified Lie dragging framework for understanding advected invariants and helicities, including new nonlocal conservation laws for non-barotropic fluids.

Findings

Derivation of fluid and cross helicity conservation laws for barotropic gases

Identification of gauge conditions for magnetic helicity advection

Development of nonlocal conservation laws using Clebsch variables

Abstract

In this paper we discuss conservation laws in ideal magnetohydrodynamics (MHD) and gas dynamics associated with advected invariants. The invariants in some cases, can be related to fluid relabelling symmetries associated with the Lagrangian map. There are different classes of invariants that are advected or Lie dragged with the flow. Simple examples are the advection of the entropy S (a 0-form), and the conservation of magnetic flux (an invariant 2-form advected with the flow). The magnetic flux conservation law is equivalent to Faraday's equation. We discuss the gauge condition required for the magnetic helicity to be advected with the flow. The conditions for the cross helicity to be an invariant are discussed. We discuss the different variants of helicity in fluid dynamics and in MHD, including: fluid helicity, cross helicity and magnetic helicity. The fluid helicity conservation law and the cross helcity conservation law in MHD are derived for the case of a barotropic gas. If the magnetic field lies in the constant entropy surface, then the gas pressure can depend on both the entropy and the density. In these cases the conservation laws are local conservation laws. We obtain nonlocal conservation laws for fluid helicity and cross helicity for non-barotropic fluids using the Clebsch variable formulation of gas dynamics and MHD. Ertel's theorem and potential vorticity, the Hollman invariant, and the Godbillon Vey invariant for special flows for which the magnetic helicity is zero are also discussed.