Nonlinear evolution of two fast-particle-driven modes near the linear stability threshold

Abstract

A system of two coupled integro-differential equations is derived and solved for the non-linear evolution of two waves excited by the resonant interaction with fast ions just above the linear instability threshold. The effects of a resonant particle source and classical relaxation processes represented by the Krook, diffusion, and dynamical friction collision operators are included in the model, which exhibits different nonlinear evolution regimes, mainly depending on the type of relaxation process that restores the unstable distribution function of fast ions. When the Krook collisions or diffusion dominate, the wave amplitude evolution is characterized by modulation and saturation. However, when the dynamical friction dominates, the wave amplitude is in the explosive regime. In addition, it is found that the finite separation in the phase velocities of the two modes weakens the interaction strength between the modes. V

Energetic particle physics in fusion research in preparation for burning plasma experiments

“…The exact consequences of this interaction still need to be fully explored. However, work has started to address the nonlinear evolution of two waves excited by the resonant interaction with fast ions just above the linear instability threshold [519]. The paper categorizes the evolution into various regimes and concludes that the nature of the nonlinear evolution obtained depends upon the type of relaxation processes at play, on the magnitudes of these various processes, and on the initial conditions.…”

Generalized model for the nonlinear dynamics of plasma waves driven unstable by energetic ions near the stability threshold

“…It turned out, that a numerical solution of Eqs. (11) revealed very similar amplitudes behaviour, as well as new types, in comparison to the two-mode model [12]. When phase shifts between neighbouring modes were sufficiently small, and depending on the value of the collision parameters, we observed the competition for survival between modes, oscillation and chaotic regimes, and the "blow-up" solution.…”

“…In accordance with [3][4][5][6]12], we use the perturbative analysis to solve the kinetic equation describing the evolution of the fast ion velocity distribution function F ( υ) in the presence of an electric field E( ),…”

“…The theory proposed by Berk and Breizman describes only the case of a single mode amplitude evolution, and in fact, it does not cover a more realistic case, when the resonance excitation of many plasma waves takes place. As a first step to this case, we considered in paper [12] the nonlinear dynamics of two interacting plasma modes driven by energetic ions at the linear stability threshold. According to the assumptions of the Berk-Breizman model, we derived the system of two coupling nonlinear integrodifferential equations.…”

The two modes extension to the Berk-Breizman equation: Delayed differential equations and asymptotic solutions

“…9,10 These equations 10 are nonlinear, inhomogeneous, integrodifferential equations. Their solutions depend on 13 parameters: the linear growth rate c 0 L , the linear dumping rate c 0 d , the Krook collision frequency 0 , the drag collision frequency a, the diffusion collision frequency b 0 , four parameters describing the two modes (wavenumbers k 0 1 , k 0 2 , angular frequencies x 0 1 , x 0 2 ), two initial moduli of amplitudes, and two initial phases.…”