Modelling saturated tearing modes in tokamaks.

Abstract

In this thesis a model for saturated tearing mode islands is developed. The equations for the mode amplitudes are essentially those of R B White et al,after a pertubation expansion has been made. It is well known that these equations are not then analytic at the mode rational surface. In our model this problem is overcome when a suitable choice of the axisymmetric current density perturbation is added to the unperturbed equilibrium current density profile. The modelled axisymmetric current density perturbation flattens the unperturbed profile locally at the rational surface and is sufficient to induce an island. No modelling in the interior of the island is necessary. The axisymmetric perturbation has a free variable which adjusts the amount of local flattening. However, when the boundary conditions are taken into account, this free parameter is determined, and the problem becomes an eigenvalue problem. The boundary condition thus determines the amount of local flattening at the rational surface. The saturated island widths are determined using D.' (W) criterion. The model allows for non axsymmetric plasma surface in a simple way, requiring careful choice of D (W). The different criteria are compared to establish the validity of the use of such criteria for perturbed boundaries. In the cylindrical approximation, one or two modes may be included in the model. In the case of two modes, non-linear coupling via the current density profile is introduced. Toroidal coupling between modes can also be simply introduced. Two modes that are toroidally coupled are considered, but mode-mode coupling is ignored.

The emphasis falls in large part on the boundary conditions. Various boundary conditions can be considered because distortion of the plasma surface can be fixed by wall effects, plasma rotation, external DC coil currents, plasma rotation with external coil currents, etc. Of particular interest is the case of toroidally coupled modes, coupled in turn to these external conditions as this is the first study of such a nature. Results flowing from the study include among others that: for the special case of circular boundaries the model agrees reasonably with the results of R B White et al. No significant difference was found between the D. I (W) criterion of P H Rutherford, which is valid for circular boundaries, and that of A H Reiman, which is also valid for perturbed boundaries, when the boundary is perturbed significantly. Toroidally coupled islands do not increase in size if the boundary condition of that particular mode is not changed. If a coil current of particular helicity is switched on, it will only affect the mode of that particular helicity. Toroidally induced sideband islands have approximately the same width as natural tearing islands when the size of the natural island is large.