Nonlinear categorization of the energetic-beam-driven instability with drag and diffusion
Résumé
The Berk-Breizman (BB) extension of the bump-on-tail instability includes a finite, fixed wave damping (γ d), and a collision operator with drag (ν f) and diffusion (ν d). 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, γ L0 /ω 0 = 0.1, the long-time nonlinear evolution is systematically categorized as damped, steady-state, periodic, chaotic and chirping. The chirping regime is subcategorized 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 (γ d , ν d) space for fixed ν d /ν f ratios; (2) the (ν f , ν d) space for fixed γ d /γ L0 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.
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