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Effects of circulating energetic particles on the geodesic acoustic modes(GAMs) with finite wave numbers are analyzed by using the hybrid kinetic-fluidmodel. The dispersion relation is derived by adopting the slowing beam ionsdistribution function in the presence of finite parallel wave number (FPWN),which is shown to affect the GAM as the toroidal rotation to the classical GAMswith zero wave numbers. The effects of CEIs can be divided into two parts. Oneis contributed by the poloidal angle and kinetic energy dependencies of theunperturbed distribution function, and is independent of the poloidal andtoroidal wave numbers. The other related to the radial gradient of the CEIspressure is proportional to the poloidal wave number, and is shown to playdestabilizing effect on the GAMs. The critical condition responsible for aninstability is analytically presented.

Effects of Circulating Energetic Ions on Geodesic Acoustic Modes with Finite Wave Numbers