A geometric look at MHD and the Braginsky dynamo

TL;DR

This paper employs differential geometry to analyze ideal and non-ideal MHD, deriving governing equations, exploring wave dynamics, and providing a geometric formulation of the Braginsky dynamo, enhancing understanding of fluid and magnetic field interactions.

Contribution

It introduces a geometric framework for MHD equations, including a novel formulation of the $oldsymbol{ extalpha}$-tensor and detailed analysis of Alfvén waves and dynamo mechanisms.

Findings

Derived MHD equations using an action principle and Lie derivatives.

Provided a geometric formulation of the $oldsymbol{ extalpha}$-tensor.

Connected geometric insights to classical dynamo theory.

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 the mathematical structure of classical mechanics. 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 a fluid parcel to a background location, is detailed. This is then used to parameterise Alfv\'en 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 $\alpha\omega$-dynamo in a general setting. Expressions for the $\alpha$-tensor are obtained, including a novel geometric formulation in terms of connection coefficients, and related to formulae found elsewhere in the literature.