The resonant excitations of highhyphen;nmagnetohydrodynamic instabilities by the energetic ions/alpha particles in tokamaks are theoretically analyzed. Here,nis the toroidal mode number. The magnetohydrodynamic eigenmodes, typically, consist of twohyphen;scale structures; one corresponds to the singular (lsquo;lsquo;inertialrsquo;rsquo;) region and the other the regular (ideal) region. Due to the finitehyphen;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 lsquo;lsquo;fishbonersquo;rsquo; dispersion relation lsqb;Phys. Rev. Lett.52, 1122 (1984)rsqb; and demonstrate, in particular, the existence of two types of modes; that is, the discrete gap mode and the energetichyphen;particle continuum mode. Specific expressions are given for both the kinetic ballooning modes and the toroidal Alfveacute;n modes.
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