Nonlinear trapping in wave–particle interactions in tokamaks

Abstract

Abstract The physics and transport consequences of the nonlinear trapping of the wave–particle interactions on the transit and bounce time scale are investigated by solving drift kinetic equation including collisions. The nonlinear particle dynamics is governed by a new set of coupled evolution equations for the phase of the wave, the toroidal component of the canonical momentum (radial motion), and the particle energy. The resonant particles are trapped nonlinearly in the phase space in a pendulum-like motion to resolve the linear resonance. The trapping mechanism is similar to that of the superbananas in the neoclassical theory. The nonlinear particle dynamics is employed to facilitate the solution of the drift kinetic equation. It is shown analytically that the particle distribution in the nonlinearly trapped region is flattened when a term of the order of δ A is neglected. Here, δA is the typical magnitude of the normalized perturbed wave field. The trapped and barely circulating particles diffuse collisionally in the phase space. Transport fluxes, wave–particle energy transfer rate and nonlinear damping rate are calculated. The transport coefficients have a new analytic scaling ν r 2 δ A U in the context of the wave–particle interaction, and can be significant even for a small amplitude wave in tokamaks. Here, ν is the collision frequency, r is the minor radius, and U is a function of the wave frequency, aspect ratio, magnetic shear, and mode numbers. It is estimated that when δA ∼ 10−4 for the magnetic perturbation, alpha particle energy loss rate is comparable to that of the neoclassical theory. The energy transfer rate is proportional to part of the particle flux, a new relation, and scales as ν δ A ; the nonlinear damping or growth rate ν δ A − 3 / 2 .