Nonlinear Fishbone Dynamics in Spherical Tokamaks with Toroidal Rotation

Fishbone is ubiquitous in tokamak plasmas with fast ions. A numerical study of nonlinear dynamics of fishbone has been carried out in this work. Realistic parameters of NSTX are used to understand instability and nonlinear frequency chirping in tokamak plasmas. First, the effects of shear toroidal rotation are considered for fishbone instability. It's shown that with low $q_{min}$, it has small effects on the mode; while with high $q_{min}$, a new unstable region with a strong ballooning feature in mode structure appears. Second, a detailed study of nonlinear frequency chirping and energetic particles' dynamics is carried out. Linearly, the mode is driven by both trapped and passing particles, with dresonance condition $\omega_{d}\simeq\omega$ for trapped particles and $\omega_{\phi}+\omega_{\theta}\simeq\omega$ for passing particles. As the mode grows, resonance particles oscillate and move outward in $P_{\phi}$ space, which reduces particles' frequency. We believe that this is the main reason for the mode frequency chirping down. Finally, as the mode frequency chirping down, particles with lower orbit frequencies, which are non-resonant linearly, can turn into resonant particles in the nonlinear regime. This effect can sustain a quasi-steady state mode amplitude.