Fast Particle Driven Instabilities in Tokamak Plasmas

The burning plasmas of the next generation tokamak fusion experiment ITER will contain signicant populations of highly energetic ions. Both fusion gener- ated alpha particles and fast ions accelerated by auxiliary heating schemes are capable of exciting instabilities in the Alfven frequency range, which may in turn cause redistribution of energetic particles and lead to deleterious events such as e.g. monster sawtooth crashes. On present day machines, many aspects of fast ion collective eects are observed and well understood, including e.g. ex- citation of toroidal Alfven eigenmodes (TAEs) and frequency sweeping Alfven cascades. However, a currently hot topic is the role of kinetic and nonlinear Alfvenic instabilities, including explanations for rapid frequency sweeping and unambiguous identication of energetic particle modes. This thesis presents theoretical research carried out on the linear and non- linear dynamics of fast particle driven instabilities, with a major focus on such events in tokamaks. In the linear regime, we investigate particle behavior in the presence of a given electromagnetic perturbation. In particular, we formu- late a condition on the prescribed wave amplitude to cause stochastic particle motion and we calculate the rate coecients for the associated diusive ran- dom walks. We also derive analytic expressions for the damping rate of even and odd TAEs, which arises as a result of non-ideal coupling to radially prop- agating waves. The damping is found to be weak for the odd mode, which raises concern for the connement of energetic particles on ITER, where an- tiballooning instabilities are likely to be resonantly driven by passing, fusion born alpha particles. Nonlinearly, we develop a simple one-dimensional model to investigate the impact of signicant frequency shifts on the dynamics of frequency sweeping, weakly driven modes. We include a number of previously neglected long range eects that are all capable of signicantly altering the temporal frequency sweeping patterns. In the case when the fast particle collision operator con- tains both a drag-like slowing-down term, due to binary, small-angle Coulomb collisions, and velocity space diusion, the model predicts transiently hooked and steady state frequency sweeping patterns. The latter scenario turns out to be analytically tractable