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

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