  We developed kinetic models based on general fishbone-like dispersion
relations. Firstly, a general model for arbitrary magnetic configuration and
ion orbit width is presented. Then, by disregarding ion orbit width and
approximating the magnetic geometry as circular, we introduce a simplified
model that fully incorporates circulating/trapped ion effects. Finally, by
considering the limit of ions being well-circulating or deeply trapped, the
results directly revert to those observed in earlier theoretical studies.
