Theory of kinetic ballooning modes excited by energetic particles in tokamaks

We have analyzed theoretically the resonant excitations of kinetic ballooning modes (KBM) by the energetic ions/alpha particles in tokamaks. Our theory includes finite-size orbit effects of both circulating and trapped particles. With energetic-particle contributions suppressed in the singular inertial layer, an analytic.dispersion relation can then be derived via an asymptotic matching analysis. The dispersion relation, in particular, demonstrates the existence of two types of modes; that is, the magnetohydrodynamic (MHD) gap mode and the energetic-particle continuum mode. Specific expressions for real frequencies, growth rates and threshold conditions are also derived for a model slowing-down beam ion distribution function