A simple model for perturbative kinetic particle resonances in tokamaks

Resonances driven by particle distribution gradients are studied in a simple statistical model which does not involve advancing individual particles due to the interaction with the mode. The phase-space structure of resonances and the associated modification of density profiles in tokamaks are due to equilibration between the bounce averaging of particles within the resonance, which tends to locally flatten the distribution, particle collisions, which tend to re-establish the original density gradients, and wave background damping, which extracts energy from the mode. Modes are perturbatively driven while the distribution is irreversibly modified due to diffusion within the resonance, where local properties of the equilibrium and particle distribution are sufficient to determine the time evolution of a mode. As a result, simulations replicate the expected scaling with the collision frequency of mode saturation amplitudes and collisional broadening of the density modification and give reasonable agreement with saturation amplitudes given by full guiding center simulations.