图 1  n = 3, m = 4—8的低频SAW和声波分支的连续谱, 此处采用了DIII-D第#178631次放电在1200 ms的平衡分布　(a1)—(c1)考虑了逆磁效应和热通行离子可压缩效应, 以及通过波-热离子相互作用和逆磁效应而产生的漂移阿尔芬波和漂移波边带模的耦合^[7]的低频SAW连续谱; (a2)—(c2)包含了逆磁效应与热通行和热捕获离子的可压缩效应^[5,8,28]

Fig. 1.  Continuous spectra of low-frequency shear Alfvén and acoustic branches for n = 3, m = 4–8: (a1)–(c1) Considering the diamagnetic effects and thermal ion compressibility as well as drift Alfvén wave and drift wave sideband coupling via the wave-thermal-passing-ion interaction and diamagnetic effect^[7]; (a2)–(c2) considering the diamagnetic effects and thermal ion compressibility (well passing and deeply trapped particle dynamics)^[8,28]. The equilibrium profiles of DIII-D #178631 at 1200 ms are adopted

图 2  Heidbrink等^[34]对DIII-D参考炮数#178631分析的实验结果　(a)布局在大半径R = 192—201 cm之间的ECE通道的互功率谱图; (b) EFIT重建得到的$ q_{\min} $与时间的关系. 其中频谱图上所示的不同符号分别对应不同$ m/n $值的不同模式: RSAE($ \ast $), BAE($ \diamond $)及LFAM($ \square $)

Fig. 2.  The DIII-D experimental results from Ref. [34] by Heidbrink et al.: (a) Cross-power spectrogram in the reference shot for ECE channels between 192–201 cm; (b) measured $ q_{\min} $ from EFIT reconstructions vs. time. The RSAE ($ \ast $), BAE($ \diamond $), and LFAM($ \square $) symbols represent the values of $ m/n $ shown on the spectrogram

图 6  在不包含EP效应的情况下, 低频SAW的(a)频率、(b)增长率和(c)极化性质对$ \varOmega_{\ast {\rm{pi}}}\equiv \omega_{\ast {\rm{pi}}}/\omega_{{\rm{ti}}} $的依赖关系

Fig. 6.  Dependence of (a) mode frequencies, (b) growth rates and (c) polarization of modes on $ \varOmega_{\ast {\rm{pi}}}\equiv \omega_{\ast {\rm{pi}}}/\omega_{{\rm{ti}}} $ without EP effect.

图 3  热粒子和高能量粒子压强特征尺度($ L_{P_{\rm{th}}} $和$ L_{P_{\rm{E}}} $), 以及在弱和/或零磁剪切($ s=rq'/q $)情况下模宽度($ {\varDelta}_m $)的径向依赖关系

Fig. 3.  Radial dependences of the typical scale lengths of thermal and energetic particle pressure ($ L_{P_{\rm{th}}} $ and $ L_{P_{\rm{E}}} $), magnetic shear (s) as well as the estimated radial mode width ($ {\varDelta}_m $).

图 4  “经典”EP分布下EP归一化压强梯度的径向关系. 其中红线是解析拟合曲线, $ q_{\min} $的归一化径向位置为$ \rho_0 $$ \equiv r_0/a=0.28 $

Fig. 4.  Radial dependence of the normalized pressure gradient of EPs with the classical profile. Here, the normalized radial position of $ q_{\min} $ is $ \rho_0\equiv r_0/a=0.28 $.

图 5  DIII-D第#178631次放电的(a)温度和q的径向分布, (b)密度的径向分布, 以及(c)环向磁场$ B_{\rm{t}} $和环向旋转频率$ f_{{\rm{rot}}} $的径向分布

Fig. 5.  Radial profiles of (a) temperature and q, (b) density and (c) $ B_{\rm{t}} $ as well as toroidal rotation frequency $ f_{{\rm{rot}}} $ of DIII-D shot #178631 used for numerical studies

图 7  在不包含(w/o)和包含(w/)EP效应情况下, 低频SAW的(a)频率、(b)增长率和(c)极化性质对$ \varOmega_{\ast {\rm{pi}}}\equiv $$ \omega_{\ast {\rm{pi}}}/\omega_{{\rm{ti}}} $的依赖关系. 图中所示的垂直虚线表示$ \varOmega_{\ast{\rm{ pi}};{\rm{exp}}} $的实验值, 约为0.35

Fig. 7.  Dependence of the (a) real frequencies, (b) growth rates and (c) polarization of the low-frequency SAWs on $ \varOmega_{\ast {\rm{p}}{\rm{i}}}\equiv \omega_{\ast {\rm{p}}{\rm{i}}}/\omega_{{\rm{ti}}} $ for the cases without (w/o) and with (w/) EP effects. Here, a dashed vertical line represents the experimental value of $ \varOmega_{\ast {\rm{pi}};{\rm{exp}}} $ of about 0.35

图 8  (a)无EP效应和(b)有EP效应的情况下, 色散关系方程(1)在复平面上的本征值分布

Fig. 8.  Eigenvalues of the dispersion relation Eq. (1) in the complex-ω plane for the cases (a) without and (b) with EPs

图 9  KBM (红色曲线)和BAE (蓝色、绿色、紫色和橙色曲线)的模频率(带标记的实线)和增长率(带标记的虚线)在不同(m, n)下对q_min的依赖关系. 图中还给出了实验观测的频率. 对于BAE, 由于模跨越了一个频率范围, 因此这些线表示不稳定区域的上下限; 对于LFAM, 实验频率变化小于0.5 kHz. 横坐标为$ q_{\min} $的变化, 是依据参考文献[34]的图8所示的实验测量的$ q_{\min}(t) $, 将时间t转换为$ q_{\min} $的变化, 与此相关的不确定度为$ \Delta q_{\min}\approx 0.01 $. 纵坐标为理论实验室框架下的频率, 已将多普勒频移合并到计算的等离子体框架下的频率$ nf_{\rm rot} $, 相关的不确定度为$ \sim0.5\times n $ kHz

Fig. 9.  Dependence of mode frequencies (solid curves with markers) and growth rates (dashed curves with markers) on $ q_{\min} $ of the KBMs (red curves) and the BAEs (blue, green, purple and orange curves) for different (m, n). The experimentally observed frequencies are also shown. For the BAE, since the modes span a range of frequencies, the lines indicate the upper and lower limits of the unstable bands; for the LFAM, the experimental frequency variation is $ <0.5 $ kHz. In the abscissa, the experimentally measured $ q_{\min}(t) $ fit shown in Fig. 8 of Ref. [34] is used to convert time to $ q_{\min} $, with an associated uncertainty of $ \Delta q_{\min} \approx 0.01 $. In the ordinate, the theoretical lab-frame frequency incorporates a Doppler shift to the calculated plasma-frame frequency of $ nf_{\rm rot} $, with an associated uncertainty of $ \sim0.5\times n $ kHz.

图 10  模频率($ {\rm{Re}}(\omega/\omega_{{\rm{ti}}}) $)和增长率($ {\rm{Im}}(\omega/\omega_{{\rm{ti}}}) $)对$ T_{\rm{e}} $的依赖关系, 其中$ T_{\rm{i}}=2.45 $ keV

Fig. 10.  Dependence of mode frequency and growth rate on $ T_{\rm{e}} $. Here, $ T_{\rm{i}}=2.45 $ keV

图 11  KBM(三角形标记)和BAE(带标记的曲线)的频率(蓝色标记)和增长率(红色标记)与径向模数L的依赖关系

Fig. 11.  Dependence of the real frequencies (blue markers) and growth rates (red markers) of the KBM (triangle markers) and BAE (line with markers) on the radial mode number L

图 12  $ L=0 $的BAE的径向模结构$ {\text{δ}} \phi_{m} $(r). 图中还给出了BAE模结构的近似实验测量结果, 如带‘$ \times $’的红线所示

Fig. 12.  Radial mode structure $ {\text{δ}} \phi_{m} $(r) for the $ L=0 $ BAE. The approximate experimental measurement of the mode structure of BAE is also shown.