Hyperbolic theory of the “shallow water” magnetohydrodynamics equations

Abstract

Recently the shallow water magnetohydrodynamic (SMHD) equations have been proposed for describing the dynamics of nearly incompressible conducting fluids for which the evolution is nearly two-dimensional (2D) with magnetohydrostatic equilibrium in the third direction. In the present paper the properties of the SMHD equations as a nonlinear system of hyperbolic conservation laws are described. Characteristics and Riemann invariants are studied for 1D unsteady and 2D steady flow. Simple wave solutions are derived, and the nonlinear character of the wave modes is investigated. The ∇⋅(h B)=0 constraint and its role in obtaining a regularized Galilean invariant conservation law form of the SMHD equations is discussed. Solutions of the Rankine–Hugoniot relations are classified and their properties are investigated. The derived properties of the wave modes are illustrated by 1D numerical simulation results of SMHD Riemann problems. A Roe-type linearization of the SMHD equations is given which can serve as a building block for accurate shock-capturing numerical schemes. The SMHD equations are presently being used in the study of the dynamics of layers in the solar interior, but they may also be applicable to problems involving the free surface flow of conducting fluids in laboratory and industrial environments.