Exact Equilibrium Solutions of the Magnetohydrodynamic Plasma Model
Date
2022-04-13
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
ORCID
0000-0003-4522-5409
Type
Thesis
Degree Level
Masters
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.
Description
Keywords
Plasma physics, Symmetry methods, Nonlinear problems, Partial differential equations
Citation
Degree
Master of Science (M.Sc.)
Department
Mathematics and Statistics
Program
Mathematics