We derive the local dispersion relation of energetic-particle-induced geodesic acoustic modes (EGAMs) for both trapped and circulating ion beams with single pitch angle slowing-down and Maxwellian distributions, as well as a bump-on-tail distribution in tokamak plasmas. For slowing-down and Maxwellian particles, the solutions of the local dispersion relation give the spectrum, growth rate and thresholds of excitation as functions of the pitch angle, beam density and frequency of the energetic particles bounce motion. For circulating ions there is only one unstable branch with frequency below the GAM continuum and a threshold of excitation in the pitch angle, for both the slowing-down and single pitch Maxwellian distributions. Trapped particles cause no excitation of a mode for neither slowing-down nor Maxwellian ion beams, but they can excite a mode with a bump-on-tail distribution when the mean velocity of the beam is larger than the threshold and the energetic particle bounce frequency is high enough.

中文翻译：

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由俘获和循环高能粒子驱动的测地线声学模式的线性色散关系

我们推导了具有单俯仰角减速和麦克斯韦分布的捕获和循环离子束的高能粒子诱导测地线声模式 (EGAM) 的局部色散关系，以及托卡马克等离子体中的尾部凸点分布。对于减速粒子和麦克斯韦粒子，局部色散关系的解给出了作为高能粒子反弹运动的俯仰角、束密度和频率的函数的光谱、生长速率和激发阈值。对于循环离子，对于减速和单螺距麦克斯韦分布，只有一个频率低于 GAM 连续谱和螺距角激发阈值的不稳定分支。被俘获的粒子不会激发慢速或麦克斯韦离子束的模式，