Computational investigations of ideal magnetohydrodynamic plasmas with discontinuities.

Abstract

The first part of this thesis addresses the solution of the Grad-Shafranov (G-S) equation from which the equilibrium and stability of a multiregion plasma configuration are obtained. Numerical solutions in spherical geometry are obtained and used to model spheromaks and Magnetically Insulated Inertial Confinement Fusion (MICF) plasmas. The solution of G-S equation produces the magnetic flux function from which the magnetic field, current distributions and pressure profiles can be generated. In this work a contact discontinuity is assumed to exist between the two regions referred to as the hot core and halo; the shape of the boundary can not be prescribed apriori but is obtained during numerical iterations by satisfying the Rankine-Hugoniot conditions at the core-halo interface. A numerical mesh is produced during the these iterations such that the boundary between the hot core and the halo is conformal with the mesh. It is shown that by carefully adjusting a set of free parameters (which are required for solving the G-S Equation) one can obtain equilibria that can model both MICF and Spheromak plasma configurations observed experimentally. The Bernstein's energy integral was used to study the stability of these configurations against the azimuthally symmetric perturbations. The second part of this thesis includes the solutions of Ideal Magnetohydrodynamics (MHD) equations by using a class of first and second order Godunov-type methods. The MHD equations are nonstrictly hyperbolic and some degeneracies exist at points where the normal and tangential magnetic fields vanish. These equations are written in the conservative form that includes the particle, momentum, and magnetic fluxes. The Jacobian matrix of the flux has a complete set of eigenvalues and eigenvectors which describe the wave speeds propagating in the system and their directions. Different. normalizations of the eigensystem (which are required to address the degeneracies) are presented and their effects on the results are discussed. The effectiveness and robustness of the methods are demonstrated by solving several test problems which include cartesian, cylindrical, and spherical geometries in one dimension.