Dynamics of Fast Ions in Tokamaks

Fast ions play a prominent role in the heating of tokamak plasmas by, e.g., neutral- beam injection, ion-cyclotron-resonance heating, and alpha-particle heating. In this thesis, a number of physical and mathematical problems concerning the dynamics of fast ions in tokamaks are addressed. First, the motion under adiabatic perturbations is studied. The frequencies of instabilities excited in tokamaks sometimes vary slowly with time. The existence of an adiabatic invariant of particle motion in such circumstances is shown to lead to a rapid convection of particles in the radial direction. Generalized adiabatic invariants are constructed for systems where the slowly varying parameter is subjected to small, but rapidly varying, fluctuations. Second, the onset of stochastic motion under resonant perturbations is considered. It is shown that the finite width of fast-ion drift orbits significantly affects the threshold for stochastic motion caused by magnetic field ripple or ion-cyclotron-resonance heating. Finite-orbit-width effects are also shown to reduce the strength of resonant interaction between alpha particles and internal kink modes. Third, the diffusive motion in the stochastic regime is analysed mathematically. Monte Carlo operators for the motion on long time-scales are constructed, and the validity of the quasilinear diffusion coefficient is examined. Finally, the effects of close ion collisions are investigated. It is demonstrated that close encounters with fast ions produce a high-energy tail in the distribution functions of impurity ions, and that close collisions between fusion-generated alpha particles give rise to a population of such particles with energies extending up to twice the birth energy