Exact Solutions of Stationary Equations of Ideal Magnetohydrodynamics in the Natural Coordinate System

Abstract

Equations of ideal magnetohydrodynamics that describe stationary flows of an inviscid ideally electrically conducting fluid are considered. Classes of exact solutions of these equations are described. With the use of the natural curvilinear coordinate system, where the streamlines and magnetic force lines play the role of the coordinate curves, the model equations are partially integrated and converted to the form that is more convenient for the description of the magnetic lines and streamlines of particles. As the coordinate system used is related to the initial coordinate system by a nonlocal transformation, the group admitted by the system can change. An infinite-dimensional (containing three arbitrary functions of time) group of symmetries is calculated for the system in the natural coordinates. An optimal system of subgroups of dimensions 1 and 2 is constructed for this group. For one of the optimal system subgroups, an invariant exact solution is found, which describes the electrically conducting fluid flow of the vortex source type with swirling magnetic lines and streamlines.