Plasma elongation effects on energetic particle-induced geodesic acoustic modes in tokamaks

Abstract

Plasma elongation effects on energetic particle-induced geodesic acoustic modes (EGAMs) are theoretically investigated by using gyro-kinetic equations and the Miller local equilibrium model. Including an arbitrary elongation κ and a finite radial derivative sκ=r∂rκ/κ, a general EGAM dispersion relation is obtained for an arbitrary energetic particle (EP) distribution. In particular, we obtain analytical EGAM dispersion relations for both the double-shifted Maxwellian distribution and the standard slowing-down distribution of EPs. In both cases, the frequency of the unstable EGAM branch decreases slowly with increasing elongation, while its growth rate decreases rapidly with κ when the ratio of the EP to the bulk ion density, nh/ni ≥ 0.1. These trends agree well with previous GENE and ORB5 simulations [A. Di Siena et al 2018 Nucl. Fusion 58 106014], but differ significantly from the elongation effects on geodesic acoustic modes (GAMs) [Zhe Gao et al 2009 Nucl. Fusion 49 045014]. The portion of the EGAM dispersion relation accounting for the first-order finite-orbit-width shows greater sensitivity to frequency compared to that of GAM, which explains the smaller variations in the frequency of EGAM as κ changes. When the EP number is small (typically, nh/ni≈ 5%) in the double-shifted Maxwellian case, the growth rate of EGAMs first increases with the increasing elongation and then decreases, while it monotonically increases with κ in the slowing-down case. Furthermore, the effects of sκ on EGAMs are similar to the elongation κ effects but weaker.

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

“…In addition, it should also be pointed out that the above analyses are based on the ideal circular crosssection, and do not include the non-circular shaping effects. Although the elongation and its radial derivative do not introduce additional EP-wave resonance [39], other shaping parameters (such as triangularity, Shafranov gradient, and inverse aspect ratio) can introduce additional particle-wave resonance at ω = nω j t (n = 1, 2, 3, 4) [15], similar to the high-order FOW effects. Therefore, it can be inferred that realistic non-circular shaping effects can also induce the instability of the HFB even in the absence of second-order FOW effects.…”