Exact Equilibrium Solutions of the Magnetohydrodynamic Plasma Model

Abstract

The use of plasma descriptions in areas such as space sciences and thermonuclear fusion devices are of great importance. Of these descriptions, the most widely used are the fluid descriptions which view plasma as a continuum medium and out of these fluid descriptions, the idealized isotropic magnetohydrodynamics (MHD) system of equations is the most used and arguably the most important. Due to the complex nonlinear structure of this system of equations, very few exact solutions are known, and of the know ones, even fewer have physically relevant behaviour. In most cases, solutions are sought for simpler forms of the MHD equations such as the time independent and static equilibrium simplifications. In this work, new exact solutions are derived for the incompressible axially and helically symmetric static and dynamic equilibrium MHD equations. The static equilibrium MHD equations with axial or helical symmetry reduce to a single partial differential equation (PDE). In the case of axial symmetry this is known as the Grad-Shafranov equation and in the case of helical symmetry this is the JFKO equation. New families of separated solutions are found for both of these PDEs and in both cases, the two separate families of solutions arise depending on the type of pressure profile. As most literature focuses on a pressure profile which is lower in the centre of the plasma and goes to a higher ambient pressure at the boundary (that is, the plasma configuration is supported by external pressure), such as those found in [11, 12] emphasis in this work is directed towards the other type of pressure profile where the pressure is higher inside the plasma domain and lower or vanishing outside. Such solutions are relevant to modelling plasma in a vacuum. Using a transformation described in [13, 14], the new static solutions are transformed into dynamic solutions which satisfy the incompressible equilibrium MHD equations. In the last chapter, a modern derivation of Hill’s spherical vortex [31] is presented that employs the Galilean invariance and the axially symmetry reduction to the Grad-Shafranov equation. Along with this, a similar and more general MHD spherical vortex-type solution is derived. Stability analysis of the localized vortex-type solutions is considered.