高能粒子进动漂移动理学效应对内扭曲模影响的模拟

周利娜; 乔岩; 姜硕; 刘钺强; 苗雨田; 王勇; 段萍; 陈龙; 刘开颖

doi:10.7498/aps.75.20251694

高能粒子进动漂移动理学效应对内扭曲模影响的模拟

1.
大连海事大学理学院, 大连　116026
2.
美国通用原子公司, 圣地亚哥　92186-5608, 美国
3.
核工业西南物理研究院, 成都　610041
4.
大连海事大学航海学院, 大连　116026

通讯作者: E-mail: zhoulina@dlmu.edu.cn.;

Precessional drift-kinetic effects of energetic particles on internal kink mode in tokamaks

1.
School of Science, Dalian Maritime University, Dalian 116026, China
2.
General Atomics, San Diego 92186-5608, USA
3.
Southwestern Institute of Physics, Chengdu 610041, China
4.
Navigation College, Dalian Maritime University, Dalian 116026, China

Corresponding author: E-mail: zhoulina@dlmu.edu.cn.;

摘要: 内扭曲模(internal kink mode, IKM)是托卡马克中一种典型的磁流体不稳定性, 作为锯齿振荡的主要诱因, 其稳定性与高能粒子(energetic particles, EPs)的动理学效应密切相关. 本文利用全环几何位形的MARS-F/K程序, 基于DEMO平衡重点模拟了EPs进动漂移动理学效应对IKM的影响. 结果表明, 该动理学效应对IKM起稳定或解稳作用, 其强弱取决于壁位置$ d/a $(d为壁与磁轴的径向距离, a为小半径)、EPs与背景粒子磁轴处的压强比值$ \hat{\beta }_{0}^{*} $、能量比值$ {\alpha }_{\text{T}} $及EPs压强和密度的径向分布等参数. 在恒定份额的分布下, $ d/a $和$ {\alpha }_{\text{T}} $较小时动理学效应会轻微解稳IKM; 模的动理学增长率随$ \beta _{0}^{*} $单调变化, $ \beta _{0}^{*}=0.5 $时其峰值位于$ d/a $和$ {\alpha }_{\text{T}} $均最大时. 而在非恒定份额的分布下, 动理学效应在整个参数$ \left(d/a, {\alpha }_{\text{T}}\right) $范围内起稳定作用; $ {\alpha }_{\text{T}} $较大时动理学增长率随$ \beta _{0}^{*} $呈非单调变化, $ \beta _{0}^{*}=0.5 $时其峰值位于$ d/a $最大但$ {\alpha }_{\text{T}} $最小值时. 在这两种分布下, 该效应激发的模频率幅值约差1个量级, 且随$ {\alpha }_{\text{T}} $的变化趋势相反. 本研究增进了对EPs动理学效应影响IKM机制的理解, 为未来聚变装置中相关不稳定性的分析提供重要参考.
- 内扭曲模 /
- 高能粒子 /
- 漂移动理学效应 /
- 托卡马克
Abstract: How energetic particles (EPs) affect the internal kink mode (IKM) is a critical question for the stable operation of future burning plasma devices like ITER and the EU DEMO. In this paper, the IKM in tokamak geometry is numerically systematically investigated utilizing the full toroidal magnetohydrodynamic (MHD) code MARS-F and the non-perturbative MHD-kinetic hybrid code MARS-K. Emphasis is placed on the role of kinetic effects arising from the precessional drift motion of trapped EPs. Multiple plasma parameters are scanned including the ideal wall position $ d/a $, the on-axis pressure ratio $ \hat{\beta }_{0}^{*} $ and the energy ratio $ {\alpha }_{\text{T}} $ between the energetic and thermal particles, as well as the radial profiles of EP density and pressure. The kinetic effect is found to be either stabilizing or destabilizing depending on the EP equilibrium profiles. With the latter being constant fractions of the corresponding thermal particle profiles, the EP kinetic effect mildly destabilizes the IKM at low $ d/a $ and $ {\alpha }_{\text{T}} $. The mode growth rate monotonically increases with $ \beta _{0}^{*} $, peaking at maximum $ d/a $ and $ {\alpha }_{\text{T}} $ for $ \beta _{0}^{*}=0.5 $. For radially distributed profiles for the EP density and pressure fractions, however, the effect is stabilizing throughout the parameter spaces in $ \left(d/a,{\alpha }_{\text{T}}\right) $. The enhanced stabilizing effect comes from the more peaked radial profiles for the EP density and pressure, which increases the magnitude of the resonance operator(given in Eq. (8))by increasing the diamagnetic frequency of the EPs. At large $ {\alpha }_{\text{T}} $, the mode growth rate non-monotonically depends on $ \beta _{0}^{*} $, reaching the peak value at maximum $ d/a $ but minimum $ {\alpha }_{\text{T}} $ for $ \beta _{0}^{*}=0.5 $. This non-monotonic behavior stems from the fact that a higher mode frequency is excited by precessional drift resonance of trapped EPs for radially distributed profiles, while the mode frequency appears in both the numerator and denominator of the resonance operator. Significant quantitative differences are also computed, over the entire parameter space, between the aforementioned two profile models for EPs. Comparative analysis shows approximately one order of magnitude difference in the computed mode frequency, and even sign reversal, between the two EP profile models with increasing $ {\alpha }_{\text{T}} $. With the radial profiles of EP density and pressure more representative of the experimentally conditions, the mode frequency excited by EPs reaches an amplitude on the order of 10^–2, corresponding to real frequencies of several to tens of kHz. This result aligns well with experimental and numerical observations from multiple tokamaks, including DIII-D, EAST, and HL-2A. The amplitude increases monotonically with EP energy, further supporting the consistency with these observations. In experiments, the radial profiles of EP pressure and density can serve as key control parameters. By tailoring the deposition profile of auxiliary heating, the radial distribution of EPs can be actively manipulated, thereby enabling effective control of sawtooth oscillations. These results enhance understanding of the EP kinetic effects on IKM and provide physics insight for analyzing associated instabilities in future fusion devices.
- internal kink mode /
- energetic particles /
- drift-kinetic effect /
- tokamak .
图 1 等离子体平衡的径向剖面　(a)安全因子剖面; (b)由$ B_{0}^{2}/{\mu }_{0} $归一化的压强剖面和磁轴处归一化为1的密度剖面

Figure 1. Equilibrium radial profiles: (a) The safety factor; (b) the plasma pressure normalized by $ B_{0}^{2}/{\mu }_{0} $and the plasma density normalized to unity at the magnetic axis.

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图 2 MARS-F计算$ n=1 $理想内扭曲模　(a)增长率$ \gamma {\tau }_{\text{A}} $随真空理想壁径向位置$ d/a $的变化; (b)理想壁位于等离子体边界$ d/a=1 $和位于$ d/a=3 $处的本征结构

Figure 2. The MARS-F computed (a) growth rate $ \gamma $of for the $ n=1 $ ideal IKM with varying radial location $ d/a $ of the ideal conducting wall, and (b) eigenmode structure, in terms of the poloidal Fourier harmonics of the radial plasma displacement with the wall located at the plasma boundary, $ d/a=1 $, or far away from the plasma surface, $ d/a=3 $.

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图 3 MARS-K计算$ n=1 $理想内扭曲模的增长率$ \gamma {\tau }_{\text{A}} $(左侧)和频率$ \omega {\tau }_{\text{A}} $(右侧)随动理学因子$ {\alpha }_{\text{k}} $和磁轴处EPs与背景粒子压强比值$ \beta _{0}^{*} $的变化, 假设EPs与背景粒子的压强比值和密度比值为恒定值, 分别定义为$ \beta _{0}^{*}={\beta }_{\text{h}}/{\beta }_{\text{th}} $和$ {\alpha }_{\text{N}}={N}_{\text{h}}/{N}_{\text{e}} $, EPs和背景粒子的能量比值$ {\alpha }_{\text{T}}=\beta _{0}^{*}/{\alpha }_{\text{N}} $分别为(a), (b) $ {\alpha }_{\text{T}}=10 $, (c), (d) $ {\alpha }_{\text{T}}=20 $和(e), (f) $ {\alpha }_{\text{T}}=30 $, 真空理想壁位于等离子体边界处$ d/a=1 $

Figure 3. The MARS-K computed growth rate $ \gamma {\tau }_{\text{A}} $ (left panel) and mode frequency $ \omega {\tau }_{\text{A}} $ (right panel), of the $ n=1 $ ideal IKM, with 2D parameter scan in the kinetic fraction factor $ {\alpha }_{\text{k}} $ and the on-axis ratio of the EP pressure to the thermal pressure $ \beta _{0}^{*} $, constant equilibrium density and pressure fractions for EPs are specified as $ \beta _{0}^{*}={\beta }_{\text{h}}/{\beta }_{\text{th}} $ and $ {\alpha }_{\text{N}}={N}_{\text{h}}/{N}_{\text{e}} $, three ratios of the EP energy to the thermal-particle energy are assumed: (a), (b) $ {\alpha }_{\text{T}}=10 $, (c), (d) $ {\alpha }_{\text{T}}=20 $, and (e), (f) $ {\alpha }_{\text{T}}=30 $, respectively, an ideal wall is placed at the plasma boundary, $ d/a=1 $.

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图 4 MARS-K计算$ n=1 $理想内扭曲模的(a)增长率$ \gamma {\tau }_{\text{A}} $和(b)频率$ \omega {\tau }_{\text{A}} $随动理学因子$ {\alpha }_{\text{k}} $和磁轴处EPs与背景粒子能量比值$ {\alpha }_{\text{T}}=\beta _{0}^{*}/{C}_{0} $的变化, 假设EPs与背景粒子的压强比值和密度比值为恒定值, 分别定义为$ \beta _{0}^{*}={\beta }_{\text{h}}/{\beta }_{\text{th}} $和$ {\alpha }_{\text{N}}={N}_{\text{h}}/{N}_{\text{e}} $, EPs与背景粒子压强比值$ \beta _{0}^{*}=0.5 $, 真空理想壁位于等离子体边界处$ d/a=1 $

Figure 4. The MARS-K computed (a) growth rate $ \gamma {\tau }_{\text{A}} $ and (b) mode frequency $ \omega {\tau }_{\text{A}} $, of the $ n=1 $ ideal IKM, with 2D parameter scan in the kinetic fraction factor $ {\alpha }_{\text{k}} $ and the on-axis ratio of EPs energy to thermal-particle energy $ {\alpha }_{\text{T}}=\beta _{0}^{*}/{C}_{0} $, constant equilibrium density and pressure fractions for EPs are specified as $ \beta _{0}^{*}={\beta }_{\text{h}}/{\beta }_{\text{th}} $ and $ {\alpha }_{\text{N}}={N}_{\text{h}}/{N}_{\text{e}} $, the EP pressure fraction is assumed to be $ \beta _{0}^{*}=0.5 $, an ideal wall is placed at the plasma boundary, $ d/a=1 $.

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图 5 MARS-K计算$ n=1 $理想IKM的(a)增长率$ \gamma {\tau }_{\text{A}} $和(b)频率$ \omega {\tau }_{\text{A}} $随动理学因子$ {\alpha }_{\text{k}} $和磁轴处EPs与背景粒子能量比值$ {\alpha }_{\text{T}}=\beta _{0}^{*}/{C}_{0} $的变化, 假设EPs与背景粒子的压强比值和密度比值为恒定值, 分别定义为$ \beta _{0}^{*}={\beta }_{\text{h}}/{\beta }_{\text{th}} $和$ {\alpha }_{\text{N}}={N}_{\text{h}}/{N}_{\text{e}} $, EPs与背景粒子压强比值$ \beta _{0}^{*}=0.5 $, 真空理想壁位于等离子体边界处$ d/a=3 $

Figure 5. The MARS-K computed (a) growth rate $ \gamma {\tau }_{\text{A}} $ and (b) mode frequency $ \omega {\tau }_{\text{A}} $, of the $ n=1 $ ideal IKM, with 2D parameter scan in the kinetic fraction factor $ {\alpha }_{\text{k}} $ and the on-axis ratio of EPs energy to thermal-particle energy $ {\alpha }_{\text{T}}=\beta _{0}^{*}/{C}_{0} $, constant equilibrium density and pressure fractions for EPs are specified as $ \beta _{0}^{*}={\beta }_{\text{h}}/{\beta }_{\text{th}} $ and $ {\alpha }_{\text{N}}={N}_{\text{h}}/{N}_{\text{e}} $, the EP pressure fraction is assumed to be $ \beta _{0}^{*}=0.5 $, an ideal wall is placed at the plasma boundary, $ d/a=3 $.

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图 6 MARS-K计算$ n=1 $理想IKM的增长率$ \gamma {\tau }_{\text{A}} $(左侧)和频率$ \omega {\tau }_{\text{A}} $(右侧)随动理学因子$ {\alpha }_{\text{k}} $和EPs与背景粒子磁轴处压强比值$ \beta _{0}^{*} $的变化, 假设EPs与背景粒子的压强比值和密度比值$ \beta _{0}^{*}={\beta }_{\text{h}}/{\beta }_{\text{th}} $均为非恒定值, 分别定义为$ {\beta }_{\text{h}}/{\beta }_{\text{th}}=\hat{\beta }_{0}^{*}\left(1-\psi _{\text{p}}^{2}\right) $和$ {N}_{\text{h}}/{N}_{\text{e}}= $$ {C}_{0}\left(1-{\psi }_{\text{p}}\right) $, EPs和背景粒子能量的比值$ {\alpha }_{\text{T}}=\beta _{0}^{*}/{\alpha }_{\text{N}} $分别为(a), (b) $ {\alpha }_{\text{T}}=10 $, (c), (d) $ {\alpha }_{\text{T}}=20 $和(e), (f) $ {\alpha }_{\text{T}}=30 $, 真空理想壁位于等离子体边界处$ d/a=1 $

Figure 6. The MARS-K computed growth rate $ \gamma {\tau }_{\text{A}} $ (left panel) and mode frequency $ \omega {\tau }_{\text{A}} $ (right panel), of the $ n=1 $ ideal IKM, with 2D parameter scan in the kinetic fraction factor $ {\alpha }_{\text{k}} $ and the on-axis ratio of the EP pressure to the thermal pressure $ \beta _{0}^{*}={\beta }_{\text{h}}/{\beta }_{\text{th}} $, analytic specification of radial profiles for equilibrium density and pressure fractions for EPs are specified as $ {\beta }_{\text{h}}/{\beta }_{\text{th}}=\hat{\beta }_{0}^{*}\left(1-\psi _{\text{p}}^{2}\right) $ and $ {N}_{\text{h}}/{N}_{\text{e}}={C}_{0}\left(1-{\psi }_{\text{p}}\right) $, three ratios of the EP energy to the thermal-particle energy are assumed: (a), (b) $ {\alpha }_{\text{T}}=10 $, (c), (d) $ {\alpha }_{\text{T}}=20 $, and (e), (f) $ {\alpha }_{\text{T}}=30 $, respectively, an ideal wall is placed at the plasma boundary, $ d/a=1 $.

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图 7 MARS-K计算$ n=1 $理想内扭曲模的(a)增长率$ \gamma {\tau }_{\text{A}} $和(b)频率$ \omega {\tau }_{\text{A}} $随动理学因子$ {\alpha }_{\text{k}} $和磁轴处EPs与背景粒子能量比值$ {\alpha }_{\text{T}}=\beta _{0}^{*}/{C}_{0} $的变化. 假设EPs与背景粒子的压强比值和密度比值均为非恒定值, 分别定义为$ {\beta }_{\text{h}}/{\beta }_{\text{th}}=\hat{\beta }_{0}^{*}\left(1-\psi _{\text{p}}^{2}\right) $和$ {N}_{\text{h}}/{N}_{\text{e}}= $$ {C}_{0}\left(1-{\psi }_{\text{p}}\right) $, 3种EPs与背景粒子压强比值分别为$ \beta _{0}^{*}=0.1 $, $ \beta _{0}^{*}=0.5 $和$ \beta _{0}^{*}=0.9 $, 真空理想壁位于等离子体边界处$ d/a=1 $

Figure 7. The MARS-K computed (a) growth rate $ \gamma {\tau }_{\text{A}} $ and (b) mode frequency $ \omega {\tau }_{\text{A}} $, of the $ n=1 $ ideal IKM, with 2D parameter scan in the kinetic fraction factor $ {\alpha }_{\text{k}} $ and the on-axis ratio of EPs energy to thermal-particle energy $ {\alpha }_{\text{T}}=\beta _{0}^{*}/{C}_{0} $, analytic specification of radial profiles for equilibrium density and pressure fractions for EPs are specified as $ {\beta }_{\text{h}}/{\beta }_{\text{th}}=\hat{\beta }_{0}^{*}\left(1-\psi _{\text{p}}^{2}\right) $ and $ {N}_{\text{h}}/{N}_{\text{e}}={C}_{0}\left(1-{\psi }_{\text{p}}\right) $, the EP pressure fraction is assumed to be $ \beta _{0}^{*}=0.1 $, $ \beta _{0}^{*}=0.5 $ and $ \beta _{0}^{*}=0.9 $, respectively. An ideal wall is placed at the plasma boundary, $ d/a=1 $.

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图 8 MARS-K计算$ n=1 $理想内扭曲模的(a)流体极限($ {\alpha }_{\text{k}}=0 $)和包括全部动理学($ {\alpha }_{\text{k}}=1 $)时增长率$ \gamma {\tau }_{\text{A}} $和(b) $ {\alpha }_{\text{k}}=1 $时频率$ \omega {\tau }_{\text{A}} $随理想壁位置$ d/a $和磁轴处EPs与背景粒子能量比值$ {\alpha }_{\text{T}}=\beta _{0}^{*}/{C}_{0} $的变化, 假设EPs与背景粒子的压强比值和密度比值均为非恒定值, 分别定义为$ {\beta }_{h}/{\beta }_{th}=\hat{\beta }_{0}^{*}\left(1-\psi _{p}^{2}\right) $和$ {N}_{h}/{N}_{e}={C}_{0}\left(1-{\psi }_{p}\right) $. EPs与背景粒子压强比值分别为$ \beta _{0}^{*}=0.5 $

Figure 8. The MARS-K computed (a) growth rate $ \gamma {\tau }_{\text{A}} $ at the nearly fluid limit ($ {\alpha }_{\text{k}}=0 $) and the full fraction of the kinetic contribution ($ {\alpha }_{\text{k}}=1 $), as well as (b) mode frequency $ \omega {\tau }_{\text{A}} $at $ {\alpha }_{\text{k}}=1 $, of the $ n=1 $ ideal IKM, with 2D parameter scan in the ideal wall radial location $ d/a $ and the on-axis ratio of EPs energy to thermal-particle energy $ {\alpha }_{\text{T}}=\beta _{0}^{*}/{C}_{0} $, analytic specification of radial profiles for equilibrium density and pressure fractions for EPs are specified as $ {\beta }_{h}/{\beta }_{th}=\hat{\beta }_{0}^{*}\left(1-\psi _{p}^{2}\right) $ and $ {N}_{h}/{N}_{e}={C}_{0}\left(1-{\psi }_{p}\right) $, the EP pressure fraction is assumed to be $ \beta _{0}^{*}=0.5 $.

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1. 引　言

内扭曲模(internal kink mode, IKM)不稳定性是托卡马克中一种常见的磁流体不稳定性, 其极向模数m = 1, 环向模数n = 1. IKM主要发生在安全因子q = 1有理面之内, 相当于磁面整体向一个方向平移, 破坏托卡马克的轴对称性, 触发锯齿振荡. 小型的锯齿振荡可以排出等离子体芯部高电荷数的杂质、防止芯部氦灰的积累, 因此它的存在经常有利于托卡马克的运行^[1]. 但是, 巨型锯齿的崩塌通常会通过局部磁流体扰动产生初始种子磁岛, 该种子磁岛在新经典效应与自举电流的共同驱动下进一步发展、增长, 最终演化为新经典撕裂模. 大的新经典撕裂模会严重降低等离子体的约束, 甚至会引起大破裂^[2,3]. 例如, Buttery等^[4]的研究表明随着锯齿周期的延长, m/n = 3/2新经典撕裂模在较低的归一化比压β_N 处触发, 限制了等离子体的性能.

大量的研究工作已经表明, 减小等离子体电阻或在等离子体边界附近安装理想真空壁都会稳定IKM^[5]. 另外, 对于背景粒子^[6,7], 辅助加热产生的高能离子^[8,9]以及氘氚聚变反应产生的阿尔法粒子^[10], 它们产生的动理学效应也会稳定IKM^[11]. 在JET中通过ICRF加热He⁴离子束证实了高能阿尔法粒子会引起巨型锯齿振荡^[12]. 据预测, ITER中氘氚聚变反应产生3.52 MeV的阿尔法粒子会对IKM有很强的稳定作用, 从而会引起巨型锯齿振荡. 因此, 燃烧等离子体中的阿尔法粒子被认为是引起巨型锯齿振荡的主要原因^[13,14]. 对于未来的聚变堆, 高能粒子(energetic particles, EPs)很有可能也会强烈地稳定IKM, 从而导致巨型锯齿振荡^[15,16]. 因此, 深入理解EPs动理学效应对IKM的影响, 是未来ITER和DEMO等聚变装置研究中亟待解决的关键问题之一.

本文利用全环几何位形的MARS-F/K程序, 基于DEMO平衡重点研究了EPs进动漂移动理学效应对IKM的影响, 系统分析了IKM不稳定性与真空壁位置、EPs与背景粒子磁轴处压强比值、能量比值, 以及EPs压强和密度的径向分布等关键参数之间的依赖关系. 全文结构安排如下: 第2节介绍计算模型; 第3节概述等离子体平衡, 并展示EPs动理学效应对IKM影响的数值模拟结果; 第4节对本工作进行总结与讨论.

2. 计算模型

本文利用MARS-F/K程序^[17,18], 模拟计算了IKM的稳定性. MARS-F通过在全环几何位形下求解扰动的单磁流体力学(magnetophydrodynamic, MHD)方程组, 得到IKM本征值和本征结构. MARS-K是基于MARS-F 程序将动理学效应通过扰动动理学压强张量耦合到等离子体动量方程中形成MHD-动理学模型, 运用有限元差分方法自洽地求解出MHD方程组本征值. MARS-K程序已成功与其他程序进行对比测试^[19,20], 并被广泛用于漂移动理学效应对IKM^[5,21,22]、外扭曲模/电阻模^[23–26]影响的研究. 为了更好地理解下文的模拟结果, 本节将简要介绍MARS-F/K模型, 详细的MARS-F/K模型可参考[17,27]. 在等离子体区域, 主要方程如下:

式中, $ \rho $, $ \boldsymbol{B} $和$ \boldsymbol{J} $分别表示等离子体的平衡密度、平衡磁场和平衡电流密度. P为(包括背景粒子和EPs)总平衡压强. $ \boldsymbol{\xi } $, $ \boldsymbol{v} $, $ \boldsymbol{b} $和$ \boldsymbol{j} $分别表示等离子体的扰动位移、扰动速度、扰动磁场以及扰动电流密度. 本征值$ \tilde{\gamma } $的实部和虚部分别表示IKM的增长率和频率. $ {\alpha }_{\mathrm{k}} $是从单流体方程($ {\alpha }_{\mathrm{k}}=0 $时)过渡到MHD-动理学混合方程($ {\alpha }_{\mathrm{k}}=1 $时)的系数.

如(6)式所示, 扰动压强张量$ \boldsymbol{p} $由描述绝热贡献的扰动压强标量$ p $(表示流体扰动压强)和描述非绝热贡献的各向异性张量$ {p}_{/ / } $和$ {p}_{\bot } $组成. $ \boldsymbol{I} $为单位张量, $ \hat{\boldsymbol{b}} = \boldsymbol{B}/|B| $. 漂移动理学扰动压强$ {p}_{/ / } $和$ {p}_{\bot } $由(7)式计算, 其中Г为粒子的速度空间, j为粒子种类, 本研究中主要包括EPs. $ {M}_{j} $是相应的粒子质量, $ {v}_{/ / } $和$ {v}_{\bot } $分别表示粒子导向中心漂移运动的平行速度和垂直速度. $ f_{\text{L}}^{1} $是粒子的分布函数^[10], 包含一个关键的因素, 即模与粒子的共振算符^[17]:

式中, $ {\omega }_{*\text{N}} $和$ {\omega }_{*\text{T}} $分别表示由密度梯度和温度梯度引起的抗磁漂移频率, $ {\omega }_{\text{E}} $是由平衡静电势引起的$ \boldsymbol{E}\times \boldsymbol{B} $漂移频率, $ {\omega }_{\text{d}} $为弹跳轨道平均的粒子环向进动漂移频率, $ {\hat{\varepsilon }}_{\text{k}} $是由温度归一化后的粒子动理学能量, $ \tilde{\omega }=\text{i}\tilde{\gamma } $是模频率. 假设背景粒子服从麦克斯韦平衡分布函数, EPs服从各向同性的慢化分布函数^[13].

4. 总结与讨论

本文基于全环几何位形的MARS-F/K程序, 重点研究了EPs进动漂移动理学效应对IKM的影响, 系统分析了IKM不稳定性与真空壁位置$ d/a $, EPs与背景粒子的磁轴处压强比值$ \hat{\beta }_{0}^{*} $、能量比值$ {\alpha }_{\text{T}} $以及EPs压强和密度径向分布等关键参数之间的依赖关系. 研究结果表明, 在EPs与背景粒子的压强和密度比值为恒定和非恒定值两种情况下, EPs的进动漂移动理学效应均会激发模频率, 且模频率幅值随理想壁位置外移(即$ d/a $增大)而逐渐下降. 同时, 在广泛的参数空间中, 该动理学效应对IKM起稳定作用. 尽管在某些参数空间中两种分布在定性上趋势一致, 但具体影响效果在整个参数空间上存在明显差异.

具体而言, 在EPs与背景粒子密度和压强比值为恒定值时, IKM的动理学增长率随$ \beta _{0}^{*} $单调变化; 而在密度和压强比值为非恒定值的径向分布下, $ {\alpha }_{\text{T}} $较大时增长率随$ \beta _{0}^{*} $的增大呈现先减小再增大的非单调变化. 这是因为共振算子的分子和分母均与模频率相关, 而在非恒定分布条件下, EPs的进动漂移动理学效应会激发更大的模频率. 进一步比较IKM流体和包含EPs进动漂移动理学的增长率发现, 在恒定密度和压强份额的径向分布下, $ d/a $和$ {\alpha }_{\text{T}} $均较小时, EPs的进动漂移动理学效应会增大IKM的增长率, 而随着两个参数的逐渐增大, 在更广泛的参数范围内, 该动理学效应对IKM表现出稳定作用; 而在非恒定密度和压强份额的径向分布情况下, EPs的进动漂移动理学效应在整个二维参数$ \left(d/a, {\alpha }_{\text{T}}\right) $范围内都会降低IKM的增长率. 此外, 恒定密度和压强份额的径向分布下, 动理学与流体增长率随$ d/a $和$ {\alpha }_{\text{T}} $的增大一致增大, 并在$ d/a $和$ {\alpha }_{\text{T}} $均取最大时达到峰值; 而非恒定密度和压强份额的径向分布下, 由于能量更高($ {\alpha }_{\text{T}} $更大)的EPs对IKM具有更强的稳定作用, 动理学增长率随$ {\alpha }_{\text{T}} $的增大而减小, 在$ d/a $取最大但$ {\alpha }_{\text{T}} $取最小值时达到峰值. 两种分布下都通过增大$ {\alpha }_{\text{T}} $提升了共振算子中粒子动理学能量, 而非恒定径向分布会进一步提高EPs的抗磁频率, 因此对IKM有更强的稳定作用. 在恒定密度和压强份额的均匀分布下, EPs进动漂移动理学效应激发的模频率量级约为10^–3, 且其幅值随$ {\alpha }_{\text{T}} $的增大逐渐下降; 而在更贴近实验实际工况的非恒定密度和压强份额的径向分布下, EPs激发的模频率幅值更大, 量级达10^–2, 对应实际频率约为几到十几kHz, 且该幅值随$ {\alpha }_{\text{T}} $的增大持续升高, 这与DIII-D、EAST和HL-2A等多个托卡马克装置上的实验及模拟观测高度符合.

基于上述结果, 从实验控制角度, 针对DEMO芯部为避免大锯齿崩塌、保障内部输运垒稳态维持的需求, 本文提出两个层次的控制策略分析. 首先, EPs的压强与密度径向分布是首要的控制变量. 研究表明, 非恒定的径向分布可通过增强抗磁漂移频率激发更大范围的模频率, 对IKM有更强的稳定作用. 因此, 在实验上可通过调节辅助加热的沉积剖面以主动调控EPs的径向分布, 实现对锯齿行为的有效抑制. 其次, 在径向分布确定的基础上, 磁轴处的EPs份额($ \beta _{0}^{*} $)与能量比($ {\alpha }_{\text{T}} $)可作为次级精细调控参数. 在更接近工程实际的非恒定分布下, 增长率峰值出现在“高$ \beta _{0}^{*} $、低$ {\alpha }_{\text{T}} $”区, 意味着能量更高的EPs具有更强稳定效应. 因此, 在非恒定分布运行下, 可通过调节芯部燃料注入率或辅助加热功率(即改变$ \beta _{0}^{*} $和$ {\alpha }_{\text{T}} $)来实现对IKM不稳定性的主动控制. 上述结论为未来托卡马克装置中实现对IKM的主动控制提供了重要的物理依据和调控思路.

最后需要指出, 本工作所采用的安全因子剖面具有芯部q₀接近1且磁剪切较小的特征, 从理论上看具备触发准交换模的潜在条件. 然而, 线性本征分析表明, IKM是当前位形下的主导模式. 鉴于准交换模与内扭曲模在非线性演化阶段可能存在显著差异, 未来有必要结合非线性模拟, 系统开展二者在不同拉长比、三角形变等关键几何参数下的对比研究, 以深化对DEMO芯部锯齿行为的认识. 此外, 本研究建立的动理学分析框架可拓展应用于BEST与CFEDR混合运行模式下的锯齿行为研究, 并为EPs对双撕裂模及反剪切阿尔芬本征模的线性分析奠定基础, 相关成果可为未来反应堆级装置的运行控制提供重要参考.

参考文献 (42)

高能粒子进动漂移动理学效应对内扭曲模影响的模拟

通讯作者: E-mail: zhoulina@dlmu.edu.cn.;

Precessional drift-kinetic effects of energetic particles on internal kink mode in tokamaks

Corresponding author: E-mail: zhoulina@dlmu.edu.cn.;

计量

高能粒子进动漂移动理学效应对内扭曲模影响的模拟

通讯作者: E-mail: zhoulina@dlmu.edu.cn.;

English Abstract

Precessional drift-kinetic effects of energetic particles on internal kink mode in tokamaks

Corresponding author: E-mail: zhoulina@dlmu.edu.cn.;

全文HTML

3.1. 等离子体平衡

3.2. 流体模型下真空壁位置的影响

3.3. 恒定压强和密度份额下EPs的进动漂移动理学效应

3.4. 非恒定压强和密度份额下EPs的进动漂移动理学效应

目录

高能粒子进动漂移动理学效应对内扭曲模影响的模拟

通讯作者: E-mail: zhoulina@dlmu.edu.cn.;

Precessional drift-kinetic effects of energetic particles on internal kink mode in tokamaks

Corresponding author: E-mail: zhoulina@dlmu.edu.cn.;

计量

出版历程

高能粒子进动漂移动理学效应对内扭曲模影响的模拟

通讯作者: E-mail: zhoulina@dlmu.edu.cn.;

English Abstract

Precessional drift-kinetic effects of energetic particles on internal kink mode in tokamaks

Corresponding author: E-mail: zhoulina@dlmu.edu.cn.;

全文HTML

3.1. 等离子体平衡

3.2. 流体模型下真空壁位置的影响

3.3. 恒定压强和密度份额下EPs的进动漂移动理学效应

3.4. 非恒定压强和密度份额下EPs的进动漂移动理学效应

目录