On the energetic particle-induced geodesic acoustic modes with finite-orbit-width effects

This study presents an analytical investigation of energetic particle-induced geodesic acoustic modes (EGAMs) within a gyro-kinetic model, incorporating finite-orbit-width (FOW) effects up to the second order. The inclusion of second-order FOW effects introduces two distinct types of energetic particle-wave resonances, occurring at omega = omega(h)(t) and omega = 2 omega(h)(t) , respectively, where omega th denotes the transit frequency of energetic particles (EPs). It is found that two unstable EGAM branches coexist: a low frequency branch (LFB) characterized by 0 < omega(LFB )< omega(h)(t,max) , and a high frequency branch (HFB) marked by omega(h )(t,max)<omega(HFB )< 2 omega(h)(t,max). The instability of LFB primarily arises from the resonance omega = omega(h)(t) , mainly introduced by first-order FOW effects. As a result, the instability of LFB always exists regardless of the presence or absence of second-order FOW effects, and is barely modified by these effects. In contrast, the instability of HFB is exclusively attributed to the resonance omega = 2 omega(h)(t) induced by second-order FOW effects. Consequently, the HFB exhibits instability in the presence of these effects.