Invariant solutions of the singular vortex in magnetohydrodynamics

Abstract

We investigate properties of plasma flow, governed by an exact reduction of the ideal magnetohydrodynamics equations which was obtained in our recent article. The reduction is called a singular vortex. It gives a description of a three-dimensional nonstationary plasma flow in terms of solution of reduced system of PDEs with two independent variables. In this paper we investigate symmetry properties of the reduced system and construct its invariant solutions. Also we give a description of the physical features of plasma motion, governed by the singular vortex. Namely, we prove that in the observed class of solutions the particles trajectories as well as magnetic field lines are flat curves. The position and orientation of the plane containing the particle's trajectory depends on the initial position of the particle. We describe the shape of trajectories and magnetic field lines for the cases of stationary and self-similar plasma flow.