Excitation of high-n toroidicity-induced shear Alfvén eigenmodes by energetic particles and fusion alpha particles in tokamaks

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Abstract
The stability of high‐n toroidicity‐induced shear Alfvén eigenmodes (TAE) in the presence of fusion alpha particles or energetic ions in tokamaks is investigated. The TAE modes are discrete in nature, and thus can easily tap the free energy associated with energetic particle pressure gradient through wave particle resonant interaction. A quadratic form is derived for the high‐n TAE modes using gyrokinetic equation. The kinetic effects of energetic particles are calculated perturbatively using the ideal magnetohydrodynamic (MHD) solution as the lowest‐order eigenfunction. The finite Larmor radius (FLR) effects and the finite drift orbit width (FDW) effects are included for both circulating and trapped energetic particles. It is shown that, for circulating particles, FLR and FDW effects have two opposite influences on the stability of the high‐n TAE modes. First, they have the usual stabilizing effects by reducing the wave particle interaction strength. Second, they also have destabilizing effects by allowing more particles to resonate with the TAE modes. It is found that the growth rate induced by the circulating alpha particles increases linearly with the toroidal mode number n for small kθρα, and decreases as 1/n for kθρα≫1. The maximum growth rate is obtained at kθρα on the order of unity, and is nearly constant for the range of 0.7≤vα/vA≤2.5. On the other hand, the trapped particle response is dominated by the precessional drift resonance. The bounce resonant contribution is negligible. The growth rate peaks sharply at the value of kθρα, such that the precessional drift resonance occurs for the most energetic trapped particles. The maximum growth rate due to the energetic trapped particles is comparable to that of circulating particles. Finally, the effect of the two‐dimensional wave structure of TAE modes is considered by using the Wentzel–Kramers–Brillouin (WKB) method.