Numerical studies of two-dimensional magnetohydrodynamic initial value problems, with particular application to the plasma focus

The increasing sophistication of the magneto-fluid equations permits their application to a wide variety of physical problems in thermonuclear fusion research, in astrophysics and in geophysics. The equations are in general non-linear, and of mixed hyperbolic, parabolic and ordinary differential type, and are frequently supplemented by elliptic equations, and for their full elucidation, they must clearly be studied numerically. While one-dimensional calculations have successfully been applied for a number of years, severe difficulties have limited the extension to multi-dimensional problems. In this thesis the fully ionised magnetohydrodynamic equations have been studied numerically in the R-Z plane, and the difficulties of such an approach are analysed. For many problems explicit methods may be adapted, and it is shown that a sophisticated fluid model can readily be solved on existing machines. The problems posed particularly, by the advection of fluid variables on a general Eulerian mesh without corrupting physical diffusion, by large variations in the Alfven velocity, and by the necessity of solving elliptic equations at each timestep are overcome. Two physical and contrasting problems are considered. The first problem and main application of the work has been to the study of the dense plasma focus - a non-cylindrical z-pinch phenomenon where the magnetic field is restricted to the azimuthal direction, so that the space remains iso-tropic. The plasma is described by six conservation equations for the density (p), momentum (pv), magnetic flux (Be), electron and ion pressure. Transport processes include electron and ion heat conduction, the resistive and Hall electric fields and ion viscosity. Cyclotron orbiting effects for both electrons and ions are included so that a mixed collision-less, collision-dominated regime is permitted. Good agreement with the limited experimental knowledge of the plasma focus has been obtained. The dynamics of the shock formation of the plasma focus and particularly the axisymmetric nature of the shock are well described. The extremely high kinetic energy densities obtained in the numerical fluid experiment as the result of adiabatic compression and viscous heating agree well with experiment. Three features in the plasma focus are isolated: an anode cold source; a hot pinch region; and an axial shock. The 'anomalously' long lifetime of the plasma focus is shown to be the result of axial flow, with stabilisation of magnetohydro-dynamic modes through the ion stress tensor in the intermediate collision-less, collision-dominated regime. Estimates of the neutron yield based on the numerical fluid experiment concur with experimental yields, and are the result of thermally reacting deuterons in the hot pinch region. The plasma parameters of interest determined from the hot pinch region suggest that the ion distribution function will not have a simple Maxwellian form and this in particular may account for the discrepancy with experiment on the anisotropy in space of the neutron yield.