Linear dispersion relation of geodesic acoustic modes driven by trapped and circulating energetic particles

We derive the local dispersion relation of energetic-particle-induced geodesic acoustic modes (EGAMs) for both trapped and circulating ion beams with single pitch angle slowing-down and Maxwellian distributions, as well as a bump-on-tail distribution in tokamak plasmas. For slowing-down and Maxwellian particles, the solutions of the local dispersion relation give the spectrum, growth rate and thresholds of excitation as functions of the pitch angle, beam density and frequency of the energetic particles bounce motion. For circulating ions there is only one unstable branch with frequency below the GAM continuum and a threshold of excitation in the pitch angle, for both the slowing-down and single pitch Maxwellian distributions. Trapped particles cause no excitation of a mode for neither slowing-down nor Maxwellian ion beams, but they can excite a mode with a bump-on-tail distribution when the mean velocity of the beam is larger than the threshold and the energetic particle bounce frequency is high enough.