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A geometric look at MHD and the Braginsky dynamo

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https://arxiv.org/abs/1911.06592
Original languageEnglish
Number of pages31
JournalGeophysical and astrophysical fluid dynamics
Publication statusAccepted/In press - 4 Oct 2020

Abstract

This paper considers magnetohydrodynamics (MHD) and some of its applications from the perspective of differential geometry, considering the dynamics of an ideal fluid flow and magnetic field on a general three-dimensional manifold, equipped with a metric and an induced volume form. The benefit of this level of abstraction is that it clarifies basic aspects of fluid dynamics such as how certain quantities are transported, how they transform under the action of mappings, (for example the flow map between Lagrangian labels and Eulerian positions), how conservation laws arise, and the origin of certain approximations that preserve much of the structure of the governing equations.
First, the governing equations for ideal MHD are derived in a general setting by means of an action principle, and making use of Lie derivatives. The way in which these equations transform under a pull back, by the map taking the position of fluid parcels to a background position, is detailed. This is then used to parameterise Alfvén waves using concepts of pseudomomentum and pseudofield, in parallel with the development of Generalised Lagrangian Mean theory in hydrodynamics. Finally non-ideal MHD is considered with a sketch of the development of the Braginsky αω-dynamo in a general setting. Formulae for the α-tensor are obtained and related to those elsewhere in the literature.

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