The Berk–Breizman (BB) extension of the bump-on-tail instability includes a finite, fixed wave damping (γd), and a collision operator with drag (νf) and diffusion (νd). The BB model is applied to a one-dimensional plasma, to investigate the kinetic nonlinearities, which arise from the resonance of a single electrostatic wave with an energetic-particle beam. For a fixed value of the linear drive normalized to the linear frequency, γL0/ω0 = 0.1, the long-time nonlinear evolution is systematically categorized as damped, steady-state, periodic, chaotic and chirping. The chirping regime is sub-categorized as periodic, chaotic, bursty and intermittent. Up–down asymmetry and hooked chirping branches are also categorized. For large drag, holes with quasi-constant velocity are observed, in which case the solution is categorized into steady, wavering and oscillating holes. Two complementary parameter spaces are considered: (1) the (γd, νd) space for fixed νd/νf ratios; (2) the (νf, νd) space for fixed γd/γL0 ratios, close to and far from marginal stability. The presence of drag and diffusion (instead of a Krook model) qualitatively modifies the nonlinear bifurcations. The bifurcations between steady-state, periodic and steady-hole solutions agree with analytic theory. Moreover, the boundary between steady and periodic solutions agrees with analytic theory. Nonlinear instabilities are found in both subcritical and barely unstable regimes. Quasi-periodic chirping is shown to be a special case of bursty chirping, limited to a region relatively far from marginal stability.
Nonlinear frequency chirping of toroidal Alfvén eigenmodes in tokamak plasmas