Free energy associated with Energetic particle (EP) pressure gradient, can drive collective instabilities, e.g., shear Alfvén waves, via wave-particle resonance, and induce EP anomalous transport due to the breaking of toroidal symmetry. One of such Alfvén wave instabilities, which has received considerable interest, is the beta-induced Alfvén eigenmode (BAE) [1]. BAEs have been recently observed in HL-2A during the Ohmic and ECRF heating [2]. Moreover, gyrokinetic simulations using HL-2A parameters [3] show that the precessional resonance of trapped energetic electrons (EEs) can drive BAE (e-BAE) instabilities and induce the typically observed croissant-like up-down asymmetric mode structures. To our knowledge, detailed theoretical understanding of the e-BAE physics, including the excitation mechanism, the global stability and corresponding radial mode structures, are still lacking. On the other hand, the EE finite orbit width normalized to the minor radius of the present-day tokamaks, could be comparable to that of the alpha particles characterized by small dimensionless orbits in reactors, e.g., ITER. Thus, the in-depth understanding of the e-BAE physics based on first-principle-based theory [4] is needed, and this constitutes the main motivation of the present work. In this work, employing the WKB-ballooning mode representation along with the generalized fishbone like dispersion relation [5], the two-dimensional (2D) global dispersion relation of the high-n e-BAEs excited by precessional resonance of magnetically trapped EE is derived in large aspect-ratio, low-β and low magnetic shear tokamaks with shifted circular flux surfaces. It has been proved that [6] the contribution of the trapped EEs to the global e-BAE dispersion relation is limited to the ideal MHD structure of the BAE due to the EE bounce averaging dynamic being governed by normal curvature. Moreover, our numerical results show that, (i) for the local properties of e-BAE: the mode can be destabilized by EEs using the typical equilibrium parameters in HL-2A, and the mode frequency is consistent with the experimental observation. Varying the background plasma parameters can lead to transitions between e-BAEs and energetic particle modes. Moreover, the dependence of the e-BAE frequencies and growth rates on energetic electron parameters shows that the growth rates monotonically increase(decrease) with the energetic electron density(the normalized energetic electron density gradient scale length), and the frequencies are not much affected. The frequency and growth rate are sensitive to the energetic electron temperature, and there exists a maximum growth rate. (ii) For the global properties of e-BAE [6], the mode is radially localized in the potential well generated by the pressure gradient of EEs, and for the parameter regime we interested, (i) at the ground radial eigenstate, the mode growth rate has a maximum with increasing energetic electron density; (ii) the ground and excited radial eigenstates can be unstable simultaneously, and the most unstable mode is related not only to the pressure gradient of energetic electrons, but also to the width of the mode itself; (iii) the corresponding two-dimensional mode structures are twisted due to the anti-Hermitian contribution from wave-energetic electron interaction and show an opposite deformation directions compared with that in the presence of energetic ions, shown in Fig. 1, which agrees with that of the existing numerical simulations. Finally, we have also shown that, the radial symmetry breaking of mode structure with respect to parallel wave-number has a potential impact on toroidal momentum transport, which is with, however, relatively weak importance on transport due to the relatively localized e-BAE mode structure.

中文翻译：

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被俘获高能电子激发的 β 诱导阿尔文本征模的全局理论

与高能粒子 (EP) 压力梯度相关的自由能可以通过波粒共振驱动集体不稳定性，例如剪切阿尔文波，并由于环形对称性的破坏而导致 EP 异常传输。引起相当大兴趣的此类阿尔文波不稳定性之一是β诱导的阿尔文本征模式 (BAE) [1]。最近在欧姆和 ECRF 加热期间在 HL-2A 中观察到 BAE [2]。此外，使用 HL-2A 参数的陀螺动力学模拟 [3] 表明，被俘获高能电子 (EE) 的进动共振可以驱动 BAE (e-BAE) 不稳定性并诱发通常观察到的类似羊角面包的上下不对称模式结构。据我们所知，对 e-BAE 物理学的详细理论理解，包括激发机制，仍然缺乏全局稳定性和相应的径向模式结构。另一方面，EE 有限轨道宽度归一化为当今托卡马克的小半径，可以与反应堆中以小无量纲轨道为特征的阿尔法粒子（例如 ITER）的轨道宽度相媲美。因此，需要基于第一性原理的理论 [4] 深入了解 e-BAE 物理，这构成了当前工作的主要动机。在这项工作中，采用 WKB 气球模式表示以及广义鱼骨状色散关系 [5]，由磁捕获 EE 的进动共振激发的高 n e-BAE 的二维（2D）全局色散关系为衍生于大纵横比、低β和低磁剪切托卡马克，具有移动的圆形通量表面。已经证明 [6] 被困 EE 对全局 e-BAE 色散关系的贡献仅限于 BAE 的理想 MHD 结构，因为 EE 反弹平均动态受法向曲率控制。此外，我们的数值结果表明，（i）对于 e-BAE 的局部特性：使用 HL-2A 中的典型平衡参数，模式可以被 EE 破坏，并且模式频率与实验观察一致。改变背景等离子体参数会导致 e-BAE 和高能粒子模式之间的转换。此外，e-BAE频率和生长速率对高能电子参数的依赖性表明，生长速率随着高能电子密度（归一化高能电子密度梯度尺度长度）单调增加（减少），并且频率没有太大影响。频率和生长速率对高能电子温度敏感，存在一个最大的生长速率。(ii) 对于 e-BAE [6] 的全局特性，模式径向定位在由 EE 的压力梯度产生的势阱中，对于我们感兴趣的参数范围，(i) 在地面径向本征态，模式增长率随着高能电子密度的增加而达到最大值；(ii) 基态和受激径向本征态可以同时不稳定，最不稳定的模式不仅与高能电子的压力梯度有关，还与模式本身的宽度有关；(iii) 由于波-能电子相互作用的反厄米特贡献，相应的二维模式结构被扭曲，与存在高能离子的情况相比，显示出相反的变形方向，如图 1 所示，这与现有的数值模拟。最后，我们还表明，模式结构相对于平行波数的径向对称破坏对环形动量传输有潜在影响，但是由于相对局部化的 e-BAE，其对传输的重要性相对较弱模式结构。