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Paper: Advected Invariants in Magnetohydrodynamics and Gas Dynamics
Volume: 484, Outstanding Problems in Heliophysics: From Coronal Heating to the Edge of the Heliosphere
Page: 228
Authors: Webb, G. M.; Hu, Q.; McKenzie, J. F.; Dasgupta, B.; Zank, G. P.
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 kinetic helicity, cross helicity, magnetic helicity, Ertel's theorem and potential vorticity, the Hollman invariant, and the Godbillon Vey invariant for special flows for which the magnetic helicity is zero.
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