Linear study of the precessional fishbone instability

Abstract

The precessional fishbone instability is an m = n = 1 internal kink mode destabilized by a population of trapped energetic particles. The linear phase of this instability is studied here, analytically and numerically, with a simplified model. This model uses the reduced magneto-hydrodynamics (MHD) equations for the bulk plasma and the Vlasov equation for a population of energetic particles with a radially decreasing density. A threshold condition for the instability is found, as well as a linear growth rate and frequency. It is shown that the mode frequency is given by the precession frequency of the deeply trapped energetic particles at the position of strongest radial gradient. The growth rate is shown to scale with the energetic particle density and particles energy while it is decreased by continuum damping.

Nonlinear dynamics of the fishbone-induced alpha transport on ITER

“…Since the fishbone mode is an hybrid Kinetic-MHD mode, its dynamics in both physical space and phase space were analyzed in order to understand the nonlinear behavior of the instability. In physical space, it was noted in both simulations that the fishbone mode frequency in laboratory frame chirps up and down, accordingly with earlier theoretical and numerical nonlinear results [2][16] [17][21][23] [25]. The effects of the sheared plasma rotation [33][34] were identified for the first time to have a significant impact on the fishbone mode frequency in plasma frame, inducing an important change in the wave-particle resonance condition.…”

“…When the n = 1 mode kinetic energy reaches a local minimum near t ∼ 9700τ A in Figure 2 (b), a second structure forms inside of q = 1 in Figure 5 (b). It is characteristic of a double step MHD displacement, first observed in [16] and [17] during the nonlinear evolution of the near threshold fishbone instability.…”

“…These results stress further the necessity of studying the nonlinear phases of the fishbone instability in ITER like conditions, in order to assess the amount of alpha particles transported out of the plasma core, hence reducing the alpha heating power P α . Reduced kinetic/hybrid models [2][15] [16] [17] [18] [19] were initially considered to analyse the nonlinear phase of the instability. Besides recovering characteristic features of the instability such as mode up/down chirping coupled to particle transport, these models identified two nonlinear regimes.…”

Contributions to the linear and nonlinear theory of the beam–plasma interaction

“…, and (4.2) provides the following expression (specified using the subscript F) (Idouakass et al 2016):…”

“…We easily get , and (4.2) provides the following expression (specified using the subscript ) (Idouakass et al. 2016): with (for the sake of completeness, in our specific case, we have: , , and ), whose roots can be numerically determined.…”