Energetic-particle-driven geodesic acoustic mode (EGAM) is analytically investigated by adopting the bump-on-tail distribution for energetic particles (EPs), which is created by the fact that the charge exchange time () is sufficiently shorter than the slowing down time (). The dispersion relation is derived in the use of gyro-kinetic equations. Due to the finite ratio of the critical energy and the initial energy of EPs, defined as , the dispersion relation is numerically evaluated and the effect of finite is examined. Following relative simulation and experimental work, we specifically considered two cases: and . The pitch angle is shown to significantly enhance the growth rate and meanwhile, the real frequency is dramatically decreased with increasing pitch angle. The excitation of high-frequency EGAM is found, and this is consistent with both the experiment and the simulation. The number density effect of energetic particles, represented by , is checked by fixing the pitch angle to 0.1. It is found that for , the growth rate of scan is not monotonic but the real frequency is monotonically decreased. For , the scan shows the growth rate keeps almost unchange under the condition of low value. As increases, the growth rate is rapidly enlarged.
Non-monotonic growth rates of sawtooth precursors evidenced with a new method on ASDEX Upgrade