Riemannian geometry of the motion of an ideal incompressible magnetohydrodynamical fluid

Abstract

This paper presents a Riemannian geometrical formalism of the hydrodynamics of an ideal incompressible fluid, originally proposed by Arnold, and then extends it to include the magnetohydrodynamics (MHD) of such a fluid. For the motion of an ideal incompressible fluid, a right-invariant metric, the induced Riemannian (Levi-Civita) connection, and then the geodesics are firstly defined on the Lie group D ( M) of all diffeomorphisms of an N-manifold M, being the ambient space of the Lie group D ν( M) of volume-preserving diffeomorphisms of M. Next, the associated covariant derivatives on the tangent space T θ D ( M) to D ( M) at θ ϵ D ν( M) ⊂ D ( M) are orthogonally projected onto T θ D ν( M), and finally the geodesic equation on D ν( M) is obtained. In the same way, the equation of motion for an ideal incompressible MHD fluid is obtained, too. In this case, the appropriate configuration space is the semidirect product of D ν( M) with the space Λ 1( M) of 1-forms on M, denoted S ν( M) = D ν ( M) × Λ 1( M), and as its ambient space the semidirect product group S ( M) = D ( M) × Λ 1( M) is considered. In addition, as on a finite-dimensional manifold, classical tensor calculus is performed on the considered infinite-dimensional Lie groups when M is an N-/three-dimensional flat torus. The curvature tensors responsible for the instability of these systems are also obtained.