Resonant excitations of kinetic ballooning modes by the energetic ions/alpha particles in tokamaks have been analyzed theoretically. The theoretical analysis 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 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.
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Tsai et al. (1993) studied this question.
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