Excitation of 2/1 fishbone-like modes by trapped energetic ions

A new branch of the m/n = 2/1 fishbone-like mode is found to be excitated by trapped energetic ions when the tearing mode is unstable. An energy-principle-based dispersion relation is derived and analyzed, which incorporates the wave-particle resonance and the resistive layer physics. Once the beta amplitude of trapped energetic ions exceeds a critical value, 2/1 fishbone-like instabilities would appear. The mode transition from tearing mode to 2/1 fishbone-like mode will be triggered with the increasing beta of trapped energetic ions. The total effects of the ideal magnetohydrodynamic potential energy and the adiabatic contribution of trapped energetic ions on the real frequency and growth rate of 2/1 fishbone-like modes are investigated; the effects of magnetic Reynolds number and magnetic shear are also studied. Both resistivity and magnetic shear play a stable role on 2/1 fishbone-like modes. An implication of the theory is conducive to understanding the 2/1 fishbone-like activities observed recently in HL-2A.