Resonances between high energy particles and ideal magnetohydrodynamic modes in tokamaks

Particle trajectory surfaces in an ideal magnetohydrodynamic high energy particle resonance are studied using kinetic Poincaré plots and through a calculation by perturbing near the resonance and finding canonical variables in the resonance, allowing the study of the distortion of the structure from that of a simple pendulum and to assist in the construction of models for the modification of particle distributions due to the modes. It is found that the narrow structure of an ideal mode eigenfunction can lead to a significant decrease in the resonance width compared to a case in which the eigenfunction does not vary within the resonant island.