Impact of energetic particle orbits on long range frequency chirping of BGK modes

Long range frequency chirping of Bernstein-Greene-Kruskal modes, whose existence is determined by the fast particles, is investigated in cases where these particles do not move freely and their motion is bounded to restricted orbits. A nonuniform equilibrium magnetic field is included into the bump-on-tail instability problem of a plasma wave. The parallel field gradients account for the existence of different orbit topologies of energetic particles. With respect to fast particles dynamics, the extended model captures the range of particles motion (trapped/passing) with energy and thus represents a more realistic 1D picture of the long range sweeping events observed for weakly damped modes, e.g. global Alfven eigenmodes, in tokamaks. The Poisson equation is solved numerically along with bounce averaging the Vlasov equation in the adiabatic regime. We demonstrate that the shape and the saturation amplitude of the nonlinear mode structure depends not only on the amount of deviation from the initial eigenfrequency but also on the initial energy of the resonant electrons in the equilibrium magnetic field. Similarly, the results reveal that the resonant electrons following different equilibrium orbits in the nonuniform field lead to different rates of frequency evolution. As compared to the previous model (Breizman 2010 Nucl. Fusion 50 084014), it is shown that the frequency sweeps with lower rates. The additional physics included in the model enables a more complete 1D description of the range of phenomena observed in experiments.