Title:

Resistive evolution of a forcefree plasma to equilibrium

Using the magnetohydrodynamic description, the evolution of a resistive plasma can be represented as a relaxation through a sequence of forcefree equilibrium states. We consider laboratory based plasmas confined in closed vessels and numerically simulate this evolution process using the finite difference method. The work is motivated by the nuclear fusion project. We proceed by showing that the forcefree problem can be reduced, in 2D, to solving a nontrivial ID diffusion equation subject to the forcefree constraint. Next, the diffusion of magnetic field lines is considered in a stationary 'mathematical' solid in which the magnetic field lines evolve such that the ratio of the conductivity perpendicular to the field lines to that parallel is much smaller than unity. The two processes are shown to be equivalent. Solutions to the latter problem are much easier to obtain and will be considered in chapters which follow. We initially consider 2D and 3D cylindrical containment devices possessing rectangular crosssections and develop algorithms to model numerically the evolution of a plasma until it reaches a relaxed state. The relaxed or equilibrium profile is the most suitable state for thermonuclear fusion to proceed. A 2D code for obtaining solutions over arbitrary crosssections is also developed. This can be used to make comparisons with experimental results for devices which have circular, Dshape and the more elaborate multipinch crosssections. The 2D and 3D rectangular crosssection cases are generalised to take account of toroidal effects. In the final chapter we present the results predicted by the suite of codes which total ~20,000 lines of source written in standard FORTRAN 77. We make comparisons with experimental data, Taylor's theory and any relevant simulations. Initially we consider a square crosssection and a variety of universal curves are found which are qualitatively similar to Taylor s theory and experiment. One of these is the well known Ftheta profile. The field reversal value theta is found to be a factor of two greater than that predicted by Taylor's theory but is in good agreement with a recent simulation. The universal curves predict that the final equilibrium state is completely defined once the axial flux and driving field are prescribed. These are just initial conditions and correspond to the axial flux and global helicity which are the only two quantities that need to be prescribed in Taylor's theory. Another universal curve predicts that at a critical axial flux there are states which are inaccessible by the plasma. About this critical value there are two modes at which the plasma can be driven, one at high current and the other at low current. This corresponds to the stable high and low current mode of operation arising in Tokamaks. We have also found the existence of degenerate relaxed states and equilibria which carry the same current but different axial flux. Choosing a particular energy or current profile to obtain an equilibrium state may not therefore give a unique solution. Changing the dimensions of the rectangular crosssection is found to have little effect on the universal Ftheta profiles. The boundary conditions, obtained using Ohm's Law, are found to play a critical role in defining relaxed states. If we allow tangential currents to flow at a boundary, we find that the results correspond to the forcefree paramagnetic model. The model where currents are removed is able to yield equilibrium states in which the field is reversed at the boundary. These are the states which arise in reversed field pinches. Toroidal effects are found to have very little effect on Ftheta profiles. Other profiles do differ and from these we have found a critical aspectratio at which toroidal devices should be built. At this value, the maximum current can be generated for a fixed driving field. The 3D code gives rise to axisymmetric relaxed states and docs not predict current limitation for any theta. The curved boundary code is found to be numerically unstable for curved but noncircular crosssections.
