Kinetic-magnetohydrodynamic hybrid simulations were performed to investigate the linear growth and the nonlinear evolution of off-axis fishbone mode (OFM) destabilized by trapped energetic ions in tokamak plasmas. The spatial profile of OFM is mainly composed of m/n = 2/1 mode inside the q = 2 magnetic flux surface while the m/n = 3/1 mode is predominant outside the q = 2 surface, where m and n are the poloidal and toroidal mode numbers, respectively, and q is the safety factor. The spatial profile of the OFM is a strongly shearing shape on the poloidal plane, suggesting the nonperturbative effect of the interaction with energetic ions. The frequency of the OFM in the linear growth phase is in good agreement with the precession drift frequency of trapped energetic ions, and the frequency chirps down in the nonlinear phase. Two types of resonance conditions between trapped energetic ions and OFM are found. For the first type of resonance, the precession drift frequency matches the OFM frequency, while for the second type, the sum of the precession drift frequency and the bounce frequency matches the OFM frequency. The first type of resonance is the primary resonance for the destabilization of OFM. The resonance frequency which is defined based on precession drift frequency and bounce frequency of the nonlinear orbit for each resonant particle is analyzed to understand the frequency chirping. The resonance frequency of the particles that transfer energy to the OFM chirps down, which may result in the chirping down of the OFM frequency. A detailed analysis of the energetic ion distribution function in phase space shows that the gradient of the distribution function along the E′ = const. line drives or stabilizes the instability, where E′ is a combination of energy and toroidal canonical momentum and conserved during the wave-particle interaction. The distribution function is flattened along the E′ = const. line in the nonlinear phase leading to the saturation of the instability.

We have investigated the waveform distortion of energetic particle driven off-axis fishbone mode (OFM) in tokamak plasmas with kinetic magnetohydrodynamic (MHD) hybrid simulations. We extended our previous simulations (Li et al 2022 Nucl. Fusion 62 026013) by considering higher-n harmonics in the MHD fluid, where n is toroidal mode number. The waveform distortion is successfully reproduced in the simulation for both magnetic fluctuations and temperature fluctuations. It is clarified that the waveform distortion arises from the superposition of the n = 2 harmonics on the fundamental n = 1 harmonics of OFM, where the n = 2 harmonics are generated by the MHD nonlinearity from the n = 1 OFM. Two types of waveform distortion can occur depending on the phase relationship between the n = 1 and n = 2 harmonics and the relative amplitude of the n = 2 harmonics to the n = 1 harmonics. Lissajous curve analyses indicate that the wave couplings between the n = 1 and n = 2 harmonics with phase-lock ∼ π and ∼ 0 lead to “rising distortion” and “falling distortion”, respectively. The two types of waveform distortion can be attributed to the strong shearing profile of radial MHD velocity with n = 2 around the q = 2 magnetic flux surface. The dependence of waveform distortion on viscosity is investigated. It is found that the viscosity which is needed to reproduce the waveform distortion is larger than that in the experiment.