Summary of magnetohydrodynamic instabilities and internal transport barriers under condition of <i>q</i><sub>min</sub><inline-formula><tex-math id="Z-20231016185819">\begin{document}$\approx $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_Z-20231016185819.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_Z-20231016185819.png"/></alternatives></inline-formula>2 in EAST tokamak
Authors: Xu Ming , Xu Li-Qing , Zhao Hai-Lin , Li Ying-Ying , Zhong Guo-Qiang , Hao Bao-Long , Ma Rui-Rui , Chen Wei , Liu Hai-Qing , Xu Guo-Sheng , Hu Jian-Sheng , Wan Bao-Nian , the EAST Team ,   Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, China,   Hebei Key Laboratory of Compact Fusion, Langfang 065001, China,   ENN Science and Technology Development Co., Ltd., Langfang 065001, China,   Southwestern Institute of Physics, Chengdu 610041, China
Published: July 2023

Establishment and sustainment of the structure of internal transport barriers (ITBs) is an important guarantee for the magnetic fusion plasma. The related physics processes for the establishing and sustaining of ITBs with <inline-formula><tex-math id="M15">\begin{document}$q_{{\rm{min}}} \approx 2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M15.png"/></alternatives></inline-formula> are simply summarized as follows: the “off-axis sawteeth” (OAS) mode instability and double tearing mode (DTM) instability, fast ions induced Alfvén eigenmode instability, thermal pressure gradient induced low-frequency modes (LFMs) instability, etc. Firstly, the burst of OAS is an important criterion for evaluating reversed <i>q</i>-profile with <inline-formula><tex-math id="M16">\begin{document}$q_{{\rm{min}}} \approx 2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M16.png"/></alternatives></inline-formula>. The excitation conditions, classifications and the structures of precursor modes of OAS are given in detail, and the collapse event is triggered off by the magnetic reconnection of <i>m</i>/<i>n</i> = 2/1 DTM. Secondly, the beta-induced Alfvén eigenmode and reversed shear Alfvén eigenmode are easily excited by the fast ions during the oscillation of OAS. The toroidal mode numbers of the two kinds of Alfvén waves are <inline-formula><tex-math id="M17">\begin{document}$1 \leqslant n \leqslant 5$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M17.png"/></alternatives></inline-formula>, respectively, which are located at <inline-formula><tex-math id="M18">\begin{document}$1.98\ {\rm{m}} \leqslant R \leqslant 2.07\ {\rm{m}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M18.png"/></alternatives></inline-formula> with normalized minor radius <inline-formula><tex-math id="M19">\begin{document}$0.2 \leqslant \rho \leqslant 0.45$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M19.png"/></alternatives></inline-formula>. The excitation conditions are investigated for the condition of <inline-formula><tex-math id="M20">\begin{document}$q_{{\rm{min}}} \approx 2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M20.png"/></alternatives></inline-formula>, and three different physical variables, i.e. thermal pressure gradient, fast ions distribution function, and the toroidal flow or flow shear are considered. Thirdly, the LFMs instabilities are excited by the pressure gradient during the oscillation of OAS. The general fishbone-like dispersion relationship (GFLDR) is adopted for solving the basic features of LFMs: 1) the frequency of LFMs scales with ion diamagnetic frequency; 2) the LFMs has the Alfvén polarization direction; 3) the LFMs are a reactive-type kinetic ballooning mode. The excitation of LFMs does not depend on the fast ions, which is taken place in a higher pressure gradient regime <inline-formula><tex-math id="M21">\begin{document}$\alpha \propto (1 + \tau) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M21.png"/></alternatives></inline-formula><inline-formula><tex-math id="M21-1">\begin{document}$ (1 + \eta_{\rm{i}})$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M21-1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M21-1.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M22">\begin{document}$\tau = T_{\rm{e}}/T_{\rm{i}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M22.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M23">\begin{document}$\eta_{\rm{i}} = L_{n_{\rm{i}}}/ L_{T_{\rm{i}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M23.png"/></alternatives></inline-formula>. In the end, the suppression of OAS and establishment of ITBs are achieved. Three important processes appear under the condition of <inline-formula><tex-math id="M24">\begin{document}$q_{{\rm{min}}} \approx 2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M24.png"/></alternatives></inline-formula> in EAST: 1) the tangential injection (NBI1L) of NBI is easier for the suppression of OAS than the perpendicular injection (NBI1R); 2) the micro-instability can be suppressed during the oscillation of OAS, and the reversed shear <i>q</i>-profile is more favorable in the establishment of the structure of ITBs; 3) the establishment of ITBs is accompanied by the excitation of Alfvén wave instability (bigger toroidal mode number: <inline-formula><tex-math id="M25">\begin{document}$1 \leqslant n \leqslant 5$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M25.png"/></alternatives></inline-formula>), the sustainment of ITBs is accompanied by the thermal ion temperature gradient induced instability (median size: <inline-formula><tex-math id="M26">\begin{document}$5 \leqslant n \leqslant 10$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M26.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20230721_M26.png"/></alternatives></inline-formula>). Therefore, for the establishment of ITBs, it is important to understand the establishment and suppression of OAS, the excitation of Alfvén wave instability and the redistributed fast ions, and the related instability of thermal pressure gradient.