Landau Damping of Geodesic Acoustic Mode in Toroidally Rotating Tokamaks

Geodesic acoustic mode (GAM) is analyzed by using modified gyro-kinetic (MGK) equation applicable to low-frequency microinstabilities in a rotating axisymmetric plasma. Dispersion relation of GAM in the presence of arbitrary Mach number is analytically derived. Toroidal rotation plays the same effects on the GAM regardless of the orientation of equilibrium flow. It is shown that the toroidal Mach number $M$ increases the GAM frequency and dramatically decreases the Landau damping rate. The valid of classical gyro-kinetic (CGK) equation is also examined. For zero electron temperature, CGK is identical with MGK. For non-zero electron temperature, CGK gives the same real frequency of GAM as MGK but induces an instability with a growth rate proportional to $M^3/q$, where $q$ is the safety factor.