  In the presence of energetic particles (EPs), the Long-Lived Mode (LLM)
frequency multiplication with n = 1,2,3 or higher is often observed on HL-2A,
where n is the toroidal mode number. Hybrid kinetic-MHD model simulations of
the energetic particle (EP) driven kink/fishbone modes on a static HL-2A-like
tokamak using NIMROD code find that, when the background plasma pressure is
relatively high, and the EP pressure and the beam energy are relatively low,
the mode frequency increases almost linearly with EP pressure, and the
frequency is proportional to n ("frequency multiplication"), even in absence of
any equilibrium plasma rotation. In addition, the frequency multiplication
persists as the safety factor at magnetic axis q0 varies. In absence of EPs,
the growth rate of the 1/1 mode is the largest; however, as the EP pressure
increases, the growth rate of 2/2 modes or 3/3 modes becomes dominant,
suggesting that the higher-n modes are more vulnerable to EPs. These results
may shed light on the understanding about the toroidal mode number dependence
of kink/fishbone modes in the advanced scenarios of tokamaks with weak or
reversed central magnetic shear.
