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The resonant excitations of high-n magnetohydrodynamic instabilities by the energetic ions/alpha particles in tokamaks are theoretically analyzed. Here, n is the toroidal mode number. The magnetohydrodynamic eigenmodes, typically, consist of two-scale structures; one corresponds to the singular (‘‘inertial’’) region and the other the regular (ideal) region. Due to the finite-size orbits, the energetic particle contributions in the singular region are suppressed. Analytical dispersion relations can be derived via the asymptotic matching analysis. The dispersion relations have the generic form of the ‘‘fishbone’’ dispersion relation Phys. Rev. Lett. 52, 1122 (1984) and demonstrate, in particular, the existence of two types of modes; that is, the discrete gap mode and the energetic-particle continuum mode. Specific expressions are given for both the kinetic ballooning modes and the toroidal Alfvén modes.
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