A geometric look at MHD and the Braginsky dynamo

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 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 $\alpha \omega$-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.

References in corpus (18)

• The Navier-Stokes-alpha model of fluid turbulence
• A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations
• Helicity is the only integral invariant of volume-preserving transformations
• Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces
• A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes
• Concomitant Hamiltonian and topological structures of extended magnetohydrodynamics
• Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics I: Lie Dragging Approach
• Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics II: Noether's Theorems and Casimirs
• Variational Principles for Lagrangian Averaged Fluid Dynamics
• Relativistic helicity and link in Minkowski space-time
• Variational Integrators for Reduced Magnetohydrodynamics
• Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics
• The Godbillon-Vey Invariant as a Restricted Casimir of Three-dimensional Ideal Fluids
• An Action Principle for Relativistic MHD
• Relabeling symmetry in relativistic fluids and plasmas
• Godbillon-Vey Helicity and Magnetic Helicity in Magnetohydrodynamics
• Potential Vorticity Formulation of Compressible Magnetohydrodynamics
• Geometric Results for Compressible Magnetohydrodynamics