Theoretical studies of low-frequency shear Alfvén waves in reversed shear tokamak plasmas

Abstract

The low-frequency Alfvénic fluctuations in the kinetic thermal-ion gap frequency range have aroused the interest of researchers since they can interact with background thermal particles and/or energetic particles. In the theoretical framework of the general fishbone-like dispersion relation (GFLDR), we theoretically investigate and delineate the linear wave properties of the low-frequency shear Alfvén wave excited by energetic and/or thermal particles observed in tokamak experiments with reversed magnetic shear. These low-frequency shear Alfvén waves are closely related to the dedicated experiment on energetic ion-driven low-frequency instabilities conducted on DIII-D in 2019. Therefore, adopting the representative experimental equilibrium parameters of DIII-D, in this work we demonstrate that the experimentally observed low-frequency modes and beta-induced Alfvén eigenmodes (BAEs) are, respectively, the reactive-type unstable mode and dissipative-type unstable mode, each with dominant Alfvénic polarization, thus the former being more precisely called low-frequency Alfvén modes (LFAMs). More specifically, due to diamagnetic and trapped particle effects, the LFAM can be coupled with the beta-induced Alfvén-acoustic mode (BAAE) in the low-frequency range (frequency much less than the thermal-ion transit frequency and/or bounce frequency), or with the BAE in the high frequency range (frequency higher than or comparable to the thermal-ion transit frequency), resulting in reactive-type instabilities. Moreover, due to different instability mechanisms, the maximal drive of BAEs occurs in comparison with LFAMs, when the minimum of the safety factor (<inline-formula><tex-math id="M1">\begin{document}$ q_{\rm min} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230255_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230255_M1.png"/></alternatives></inline-formula>) deviates from a rational number. Meanwhile, the BAE eigenfunction peaks at the radial position of the maximum energetic particle pressure gradient, resulting in a large deviation from the <inline-formula><tex-math id="M2">\begin{document}$ q_{\rm min} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230255_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230255_M2.png"/></alternatives></inline-formula> surface. The ascending frequency spectrum patterns of the experimentally observed BAEs and LFAMs can be theoretically reproduced by varying <inline-formula><tex-math id="M3">\begin{document}$ q_{\rm min} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230255_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230255_M3.png"/></alternatives></inline-formula>, and they can also be well explained based on the GFLDR. In particular, it is confirmed that the stability of the BAAE is not affected by energetic ions, which is consistent with the first-principle-based theory predictions and simulation results. The present analysis illustrates the solid predictive capability of the GFLDR and its practical applications in enhancing the ability to explain experimental and numerical simulation results.