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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
$q_{{\rm{min}}} \approx 2$ 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 q-profile with$q_{{\rm{min}}} \approx 2$ . 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 m/n = 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$1 \leqslant n \leqslant 5$ , respectively, which are located at$1.98\ {\rm{m}} \leqslant R \leqslant 2.07\ {\rm{m}}$ with normalized minor radius$0.2 \leqslant \rho \leqslant 0.45$ . The excitation conditions are investigated for the condition of$q_{{\rm{min}}} \approx 2$ , 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$\alpha \propto (1 + \tau) $ $ (1 + \eta_{\rm{i}})$ ,$\tau = T_{\rm{e}}/T_{\rm{i}}$ ,$\eta_{\rm{i}} = L_{n_{\rm{i}}}/ L_{T_{\rm{i}}}$ . In the end, the suppression of OAS and establishment of ITBs are achieved. Three important processes appear under the condition of$q_{{\rm{min}}} \approx 2$ 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 q-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:$1 \leqslant n \leqslant 5$ ), the sustainment of ITBs is accompanied by the thermal ion temperature gradient induced instability (median size:$5 \leqslant n \leqslant 10$ ). 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.-
Keywords:
- double tearing mode /
- Alfvén eigenmodes /
- low-frequency instability /
- internal transport barriers
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图 1 EAST托卡马克装置反磁剪切
$q_{{\rm{min}}} \leqslant 2$ 条件下双撕裂模(DTM)和离轴锯齿(OAS)、快离子激发低频阿尔芬波(BAEs, RSAEs)、热压力梯度激发低频模(LFMs)和内部输运垒(ITBs)等物理过程之间的相互关系Figure 1. Sketch of different physics phenomena, namely, Double tearing modes (DTM) and “off-axis” sawteeth (OAS), fast ions induced low-frequency Alfvén waves, thermal pressure gradient induced low-frequency mode (LFM) and internal transport barriers (ITBs) under the condition of
$q_{{\rm{min}}} \leqslant 2$ in the EAST tokamak.
图 2 EAST上离轴锯齿的不同激发条件 (a)—(d) ICRH辅助条件下离轴锯齿和边界局域模共存现象(#62863); (e), (f)提高ECRH功率阈值(#62085)和(g), (h)降低环向磁场
$B_\phi$ 实现离轴ECRH加热(#66465)条件下激发的离轴锯齿现象Figure 2. Several different conditions for the excitation of OAS: (a)–(d) Coexistence of OAS and ELM instabilities during the ICRH (#62863); (e), (f) effect of ECRH with power threshold (#62085); (g), (h) effect of toroidal field
$B_\phi$ on the deposition of ECRH (#66465)
图 3 离轴锯齿可以分为中心(Central)、环形(Annular)和“广义锯齿”崩塌事件三大类 (a) 不同径向
$T_{\rm{e}}$ 信号随时间演化; (b) 中心崩塌事件前后的$T_{\rm{e}}$ 剖面的变化; (c)—(e)三类崩塌事件发生前后的温度变化量$\Delta T_{\rm{e}}/T_{\rm{e}}$ (磁轴$R_0 \approx 1.9\ {\rm{m}}$ )Figure 3. The OAS can be divided into three categories: Central crash, annular crash and “generalized-sawteeth” crash events. (a)
$T_{\rm{e}}$ for different radial positions; (b)$T_{\rm{e}}$ -profiles before and after the central crash event; (c)–(e) the relative alterations of$\Delta T_{\rm{e}}/T_{\rm{e}}$ for the three cases (magnetic axis:$R_0 \approx 1.9\ {\rm{m}}$ )
图 4 离轴锯齿崩塌前激发m/n = 2/1先兆振荡模(图2 #66465阴影区间放大结果) (a) 不同径向ECE信号; (b) SXR阵列相对扰动信号(
${\text{δ}} I_{{\rm{sx}}}/I_{{\rm{sx}}}$ )沿着Z方向随时间演化分布图; (c) 边界磁探针频谱图; (d) ECE阵列相对扰动信号(${\text{δ}} T_{\rm{e}}/T_{\rm{e}}$ )沿着R方向随时间演化分布图. 说明: 离轴锯齿崩塌前可以观察到两次不同的崩塌事件($t\approx 4.043\ {\rm{s}}$ 和$t\approx 4.05\ {\rm{s}}$ )Figure 4. The m/n = 2/1 precursor mode is taken place before the final collapse of OAS: (a) ECE signals of different radial positions; (b) relative fluctuation of
${\text{δ}} I_{{\rm{sx}}}/I_{{\rm{sx}}}$ for SXR array along Z direction; (c) spectrogram of edge magnetic signal; (d) relative fluctuation of${\text{δ}} T_{\rm{e}}/T_{\rm{e}}$ for the ECE array along R direction. Note: two different collapse events are observed successively ($t\approx 4.043\ {\rm{s}}$ and$t\approx 4.05\ {\rm{s}}$ )
图 5 BAEs和RSAEs的径向位置
$1.98\ {\rm{m}} \leqslant R \leqslant 2.07\ {\rm{m}}$ (归一化小半径$ 0.2 \leqslant \rho \leqslant 0.45$ ),$q_{{\rm{min}}}$ 位置为$R \approx 2.025\ {\rm{m}}$ ($\rho \approx 0.3$ )Figure 5. Radial coverage of the pairs of BAEs-RSAEs is located at
$1.98\ {\rm{m}} \leqslant R \leqslant 2.07\ {\rm{m}}$ (the radial position of$q_{{\rm{min}}}$ should be located at$R \approx 2.025\ {\rm{m}}$ ).
图 6 EAST上离轴锯齿振荡期间阿尔芬波不稳定性的激发条件对比 (a) NBI功率; (b) 中子产额
$S_{\rm{n}}$ ; (c) 等离子体储能$W_{{\rm{dia}}}$ ; (d) 芯部电子温度$T_{{\rm{e}}0}$ ; (e) 径向$R \approx 2.02\ {\rm{m}}$ 附近ECE信号功率谱; (f) #60223下ECE信号频谱图Figure 6. Excitation conditions of Alfvén waves during the oscillation of OAS in EAST: (a) Input powers of NBI; (b) neutron yield
$S_{\rm{n}}$ ; (c) plasma stored energy$W_{{\rm{dia}}}$ ; (d) core electron temperature$T_{{\rm{e}}0}$ ; (e) power spectra of ECE signal at$R \approx 2.02\ {\rm{m}}$ ; (f) spectrogram of ECE signal for #60223
图 7 影响阿尔芬本征模激发的3个相关因素 (a) 温度剖面(
$T_{\rm{e}}$ 为点划线;$T_{\rm{i}}$ 为实线); (b) 温度归一化梯度标长($R/L_{T_{\rm{e}}}$ 为点划线,$R/L_{T_{\rm{i}}}$ 为实线); (c) RNC诊断测量的中子计数率空间分布(高能离子密度); (d), (e) NUBEAM/TRANSP代码计算不同NBI束方向下的高能离子的经典分布函数; (f) CXRS诊断测量的环向速度归一化梯度标长($R/L_{v_\phi}$ )Figure 7. Three correlated factors for the excitation of Alfvén eigenmodes: (a) Profiles of
$T_{\rm{e}}$ and$T_{\rm{i}}$ ; (b) normalized temperature gradients of$R/L_{T_{\rm{e}}}$ and$R/L_{T_{\rm{i}}}$ ; (c) counts of neutron flux measured by RNC; (d), (e) classical distribution functions for the two conditions are estimated by the NUBEAM/TRANSP; (f) the normalized gradient of$R/L_{v_\phi}$ measured by CXRS
图 8 两种不同类型离轴锯齿崩塌前RSAEs向上扫频斜率对比结果
Figure 8. Upward sweeping rates of RSAEs different branches before the central/annular collapse events.
图 9 低频模和阿尔芬本征模之间的共存关系 (a), (b) BAEs-RSAEs和LFMs的频谱图; (c) LFMs和BAEs的共存时间正比于离轴锯齿的振荡周期
Figure 9. Coexistence between LFMs and Alfvén eigenmodes: (a), (b) The spectrogram of the pairs of BAEs-RSAEs and LFMs; (c) the coexistence time between LFMs and BAEs versus the OAS period.
图 10 EAST上LFMs不稳定性激发条件的实验研究 (a) #61960和#61970两炮的功率谱密度; (b), (c)电子和离子温度剖面, 其中低频模的径向位置为黄色阴影区间所示
Figure 10. Experimental investigation of the excitation condition of LFMs instability on EAST: (a) Power spectra densities for the two shots #61960 and #61970; (b), (c) profiles of
$T_{\rm{e}}$ and$T_{\rm{i}}$ , where the radial position of LFMs is demonstrated by the yellow shaded region
图 11 利用GFLDR代码数值求解EAST上LFMs不稳定性的激发条件和基本特征 (a) LFMs增长率
$\gamma$ 和($\eta_{\rm{i}} = $ $ L_{n_{\rm{i}}}/ L_{T_{\rm{i}}}$ ,$\tau = T_{\rm{e}}/T_{\rm{i}}$ )之间的依赖关系; (b) LFMs的频率f和增长率$\gamma$ ; (c) 极化方向$|{\cal{S}}|$ 和$\varOmega_{\ast {\rm{pi}}}\equiv \omega_{\ast {\rm{pi}}}/\omega_{{\rm{ti}}}$ 之间的关系Figure 11. Excitation conditions and basic features of LFMs are numerically calculated by GFLDR in EAST: (a) Growth rate
$\gamma$ of LFMs versus ($\eta_{\rm{i}} = L_{n_{\rm{i}}}/ L_{T_{\rm{i}}}$ ,$\tau = T_{\rm{e}}/T_{\rm{i}}$ ); (b) mode frequency f and growth rate$\gamma$ ; (c) polarization$|{\cal{S}}|$ of LFMs on$\varOmega_{\ast {\rm{pi}}}\equiv \omega_{\ast {\rm{pi}}}/\omega_{{\rm{ti}}}$ 
图 12 利用不同注入方向的中性束(NBI1R为垂直方向, NBI1L为切向注入)实现离轴锯齿的缓解和抑制 (a)储能
$W_{{\rm{dia}}}$ ; (b) #61962 NBI注入源功率; (c)芯部的旋转速度; (d)芯部电子温度$T_{{\rm{e}}0}$ Figure 12. Suppression of OAS by the different injection direction of NBI: (a) Stored energy
$W_{{\rm{dia}}}$ ; (b) source power of NBI in #61962; (c) central rotation velocity$v_{\phi 0}$ ; (d) central electron temperature$T_{{\rm{e}}0}$ 
图 13 离轴锯齿振荡期间微观不稳定性的激发和抑制 (a) POINT诊断不同极向位置测量到的弦平均电子密度
$\langle n_{\rm{e}} \rangle$ ; (b), (c)电子和离子温度的归一化梯度长度$R/L_{T_{\rm{e}}}$ 和$R/L_{T_{\rm{i}}}$ ; (d), (e)不同环向位置POINT和SXR诊断测量到的相对扰动$\Delta n_{\rm{e}}/n_{\rm{e}}$ 和$\Delta I_{{\rm{sx}}} /I_{{\rm{sx}}}$ Figure 13. One kinds of micro-instability is excited and suppressed during the oscillation of OAS: (a) Line-integrated electron densities
$\langle n_{\rm{e}} \rangle$ for different chord position of POINT array; (b), (c) normalized gradient of$R/L_{T_{\rm{e}}}$ and$R/L_{T_{\rm{i}}}$ ; (d), (e) relative fluctuations of$\Delta n_{\rm{e}}/n_{\rm{e}}$ and$\Delta I_{{\rm{sx}}} /I_{{\rm{sx}}}$ respectively for the different toroidal positions of POINT and SXR arrays
图 14 内部输运垒建立过程中伴随快离子或热粒子不稳定性事件 (a1), (a2)归一化
$\beta_{\rm{N}}$ ; (b1), (b2)电子$T_{{\rm{e}}}$ 和离子$T_{{\rm{i}}0}$ 温度; (c1), (c2) ECE诊断测量到的频谱图Figure 14. Energetic ions and thermal pressure gradient instabilities are observed during the establishment of ITBs: (a1), (a2) Normalized
$\beta_{\rm{N}}$ ; (b1), (b2) electron$T_{{\rm{e}}0}$ and$T_{{\rm{i}}0}$ temperatures; (c1), (c2) spectrogram of ECE signal
图 15 内部输运垒建立过程中温度和旋转剖面 (a)
$T_{\rm{i}}$ ; (b)$R/L_{T_{\rm{i}}}$ ; (c)$v_\phi$ ; (d)$\Delta T_{\rm{e}}/T_{\rm{e}}$ Figure 15. Profiles of (a)
$T_{\rm{i}}$ , (b)$R/L_{T_{\rm{i}}}$ , (c)$v_\phi$ , (d)$\Delta T_{\rm{e}}/T_{\rm{e}}$ for the establishment of ITBs.表 1 EAST上离轴锯齿和一般锯齿的对比图
Table 1. Direct comparison between the OAS with conventional sawteeth in EAST.
“锯齿”类型 离轴锯齿 一般锯齿 安全因子q 反磁剪切 单调分布 qmin $q_{\rm{min}} \leqslant 2$ ($q_0 > 1$) $q_0 \leqslant 1$ 先兆模 m/n = 2/1 (D)TM m/n = 1/1 kink 径向位置/m HFS: $1.7 \leqslant R \leqslant 1.8$
LFS: $2 \leqslant R \leqslant 2.1 $$1.8 \leqslant R \leqslant $2 $\Delta T_{{\rm{e0}}}/T_{{\rm{e0}}}$ $\geqslant 30{\text{%}}$ $\leqslant 10{\text{%}} $ “混合半径” 无 有 $D_\alpha$脉冲 有 无 注: EAST上磁轴位置$R_0 \approx 1.9\ {\rm{m}}$, 小半径$a \approx 0.45\ {\rm{m}}$. 
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