7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

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

Zhe Chen1, Haijun Ren1,2, Yixiang Li1, and Colin M Roach2
1
CAS Key Laboratory of Geospace Environment and Department of Engineering and Applied
Physics, School of Physical Sciences, University of Science and Technology of China,
2
EURATOM/UKAEA Fusion Association, Culham Science Centre
e-mail (speaker): cz112233@mail.ustc.edu.cn
Geodesic acoustic modes (GAMs) are well-establishedwaves in tokamaks. [1] They can be significantly
influenced by energetic particles (EPs), giving rise toenergetic particle-induced geodesic acoustic modes(EGAMs). EGAMs exhibit an unstable branch with apositive growth rate and a lower frequency than that ofconventional GAMs. [2]This study presents an analytical investigation ofEGAMs within a gyro-kinetic model, incorporating
finite-orbit-width (FOW) effects up to the second order.
We begin by deriving a general dispersion relation
applicable to arbitrary EP distributions. Subsequently, bymodeling EPs with the standard slowing-down beam
distribution, we derive exact expressions and find thesecond-order FOW effects introduce two kinds ofadditional resonances: one at 𝑞𝑅𝜔 = 𝑣|| and the otherat 𝑞𝑅𝜔/2 = 𝑣|| . Here, 𝑞 is the safety factor, 𝑅 is themajor radius of tokamaks, 𝜔 is the EGAM frequency,
and 𝑣|| is the EP parallel velocity with respect to themagnetic field.
Utilizing these exact expressions, we investigate theinfluence of second-order FOW effects on unstable
EGAMs by numerically calculating the EGAM
dispersion relation, as illustrated in figures 1(a) and 1(b).
Our findings reveal that second-order FOW effects haveminimal impact on the original unstable EGAM branch,denoted as low frequency branch (LFB). However, theseeffects introduce a new unstable EGAM branch, denoted

0.160

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LFB

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ti

ti
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References[1] G. Conway et al, Nucl. Fusion 62, 013001 (2022)[2] G. Y. Fu, Phys. Rev. Lett. 101, 185002 (2008)[3] T. Ido et al, Nucl. Fusion 55, 083024 (2015)[4] H. Wang et al, Phys. Plasmas 22, 092507 (2015)

Reference line of GAM frequency
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1.20as the high frequency branch (HFB). The frequency of
the HFB is significantly higher than that of GAMs andexceeds the EP transit frequency, which provides apotential explanation for LHD observations [3]. The
instability of the HFB is driven by inverse Landau
damping, originating from the EP-wave resonance at𝑞𝑅𝜔/2 = 𝑣|| , which is an additional resonanceintroduced by second-order FOW effects. Furthermore,
we observe that increasing the EP concentration, asdepicted in figures 1(c) and 1(d), leads to a higherfrequency and growth rate for the HFB. In contrast, as
the EP concentration increases, the LFB frequency
declines, and its growth rate first increases andsubsequently decreases.
Overall, our investigation underscores the significanceof second-order FOW effects in the analysis of EGAMinstability. These effects give rise to a novel unstable
EGAM branch through the EP-wave resonance at𝑞𝑅𝜔/2 = 𝑣|| , characterized by a considerably higher
frequency than conventional GAMs.

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Figure 1. The dependence of frequency 𝜔 and growth rate 𝛾 of two branches on 𝑘𝑟 𝜌ℎ and 𝑛ℎ /𝑛𝑖 . Here, 𝑘𝑟 is theradial wavenumber, 𝑛ℎ (𝑛𝑖 ) is the EP (bulk ion) density, 𝜌ℎ = √2𝐸0 (1 − Λ)/𝑚ℎ /𝜔𝑐ℎ represents the FOW effects, 𝐸0is the injection energy of EPs, Λ is the normalized injection pitch angle, 𝑚ℎ is the EP mass, and 𝜔𝑐ℎ is the EP
gyro-frequency. Parameters are adopted according to Ref. [3-4] as follows: 𝑇𝑒 = 𝑇𝑖 = 4keV, 𝐸0 = 170keV, Λ =0.3, 𝑞 = 2; 𝑛ℎ /𝑛𝑖 = 1/3 in (a, b); 𝑘𝑟 𝜌ℎ = 0.2 in (c, d).

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