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As is well known, large-scale type-I edge localized modes (ELMs) may pose serious risks to machine components in future large fusion devices. The resonant magnetic perturbation (RMP), generated by magnetic coils external to the plasma, can either suppress or mitigate ELMs, as has been shown in recent experiments on several present-day fusion devices. Understanding the ELM control with RMP may involve various physics. This work focuses on the understanding of the roles played by three key physical quantities: the edge safety factor, the RMP coil current, and the particle drift kinetic effects resulting from thermal and fusion-born α-particles. Full toroidal computations are performed by using the MARS-F/K codes. The results show that the plasma response based figures-of-merit i.e. the pitch resonant radial field component near the plasma edge and the plasma displacement near the X-point of the separatrix,consistently yield the same periodic amplification as
q95 varies. The number of peaks, y, is positively correlated with the toroidal number n, i.e.y≈nΔq95 withΔq95=3.5 . The peak window inq95 occurs when a new resonant surface passes through a specific region of the plasma edge. Two-dimensional parameter scans, for the edge safety factor and the coil phasing between the upper and lower rows of coils, yield a linear relationship between the optimal/worst current phase difference andq95 , which can be well fitted by a simple analytic model. The optimal value of coil current amplitude is sensitive to n. Compared with the same current amplitude assumed for the two/three rows of coils, the optimal current amplitude can increase theξX but does not change the prediction of the relative toroidal phase difference. More advanced response model, including kinetic resonances between the RMP perturbation and drift motions of thermal particles and fusion-born alphas, shows that the modification of kinetic effects should be considered in order to better describe the plasma response to RMP fields in high-β plasmas. The fluid response model with a strong parallel sound wave damping (κ∥=1.5 ) can well predict the plasma response for the ‘DEMO-like’ equilibria. For low β plasma, the kinetic response is consistent with the fluid response, whether a strong parallel sound wave damping exists or not.1. 引 言
为了推动社会的可持续发展, 开发清洁新能源是解决能源和环境问题的关键途径. 核聚变能具有燃料充分、产能高、利于环保等优点, 是人类未来最理想的能源之一, 将为中国实现“双碳”目标做出重大贡献. 磁约束核聚变是实现核聚变能的一种重要途径, 已被中国列为探索新型能源的关键科学问题之一. 其中, 托卡马克在磁约束核聚变研究中逐渐突显出其独特的优势, 成为研究磁约束核聚变的主流装置. 托卡马克装置在高约束运行模式下不可避免地产生边缘局域模(edge-localized modes, ELMs), 尤其对于未来大型托卡马克装置, 边界台基区的碰撞频率很低, 所以会产生大振幅、小频率的I型ELMs[1-3]. I型ELMs在一次爆发过程中会释放大量的粒子和能量, 远远超出面向等离子体材料的承受能力, 对材料造成极大的损坏, 大大缩减材料的寿命[4,5], 同时会引起第一壁上高Z杂质的溅射, 从而影响芯部磁流体不稳定性, 降低装置的约束性能. 所以, 控制I型ELMs对未来聚变堆的稳态运行是不可或缺的.
由外加三维磁扰动线圈产生的共振磁扰动(resonant magnetic perturbations, RMP)作为一项控制ELMs的成熟技术, 不仅可以缓解ELMs(使其频率增大、幅度下降)[6,7], 还可以抑制ELMs(使其完全消失)[8,9], 已经在ASDEX Upgrade[10], DIII-D[8], JET[11]和MAST[10], 以及中国的EAST[12,13], HL-2A[14]和J-TEXT[15]等托卡马克装置上得到了成功的应用. 近年来, 为理解RMP控制ELMs的物理机制开展了广泛的研究, 结果发现, 对于固定环向模数
n 的线圈位形, RMP的极向谱在ELMs的控制[16,17]以及相关的等离子体输运[18-20]过程中起关键作用, 而该极向谱又与ELM控制线圈[21]和等离子体参数[22,23]密切相关.本工作基于MARS[24,25]系列程序, 以‘类-DEMO’平衡为例, 针对RMP控制ELMs的物理机制对以下几个重要方面进行研究. 首先, 因为RMP实现ELMs的控制只出现特定的安全因子
q95 (95%归一化极向磁通量处的安全因子)窗口内[7,26], 所以在不同环向模数(n = 1—4)下模拟计算了安全因子q95 对等离子体流体响应的影响. 然后, 针对上下两组和上中下三组RMP线圈, 利用MARS-F程序和解析方法计算了线圈电流幅值和相位差的最优值, 该方法简单、准确、高效, 可用于不同平衡、不同线圈组优化的比较研究. 最后, 采用更先进的模型, 研究了背景粒子和聚变产生α粒子的动理学效应对等离子体响应的影响, 其研究结果验证了考虑强平行声波阻尼(κ∥=1.5 )的流体响应对‘类-DEMO’平衡等离子体响应预测的可靠性.本文第2节介绍了本文模拟等离子体对RMP响应所采用的基本物理模型和多组RMP线圈电流优化的解析方法; 第3节概述了等离子体的平衡和RMP线圈位形, 并展示了等离子体对RMP流体和动理学响应的数值模拟结果; 第4节对本工作进行总结与讨论.
2. 计算模型
2.1 MARS-F/K计算模型
本文将采用MARS-F[27]和MARS-K[24,25]模拟研究等离子体对RMP的流体响应和动理学响应. MARS-F/K是基于环几何位形, 求解电阻单流体模型的线性程序. 该程序包括真实的等离子体位形, 广泛用于等离子体对RMP响应的模拟研究, 其模拟结果的可靠性已在多个托卡马克装置上得到了验证, 如: MAST[28], DIII-D[29], ASDEX Upgrade[30], 以及中国的EAST[31]和HL-2A[32]等. 为了更好理解模拟结果, 将简要介绍MARS-F/K模型. 该模型可以从MARS-F中的单流体模型连续过渡到MARS-K中磁流体力学-动理学混合模型, 具体方程如下:
i(ΩRMP+nΩ)ξ=v+(ξ⋅∇Ω)R2∇ˆϕ, (1) iρ(ΩRMP+nΩ)v=−∇⋅p+j×B+J×b−ρ[2ΩˆZ×v+(v⋅∇Ω)R2∇ˆϕ]−∇Π, (2) i(ΩRMP+nΩ)b=∇×(v×B)+(b⋅∇Ω)R2∇ˆϕ−∇×(ηj), (3) i(ΩRMP+nΩ)p=−v⋅∇P−(1−αk)ΓP∇⋅v, (4) j=∇×b, (5) p=pI+αkp∥ˆbˆb+αkp⊥(I−ˆbˆb), (6) p∥=∑iMiv2∥f1LdV,p⊥=∑i12Miv2⊥f1LdV, (7) 其中,
ΩRMP 是RMP旋转频率, 本文考虑外加静态RMP的作用, 即ΩRMP=0 .Ω 是环向旋转角频率.n 是环向模数, 下文将分别研究n = 1—4的三维响应场.ρ ,B 和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(包括背景粒子和高能粒子)总平衡压强.ξ ,v ,b 和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.η 为等离子体电阻率,Γ=5/3 表示绝热系数.αk 是从单流体方程(αk=0 时)过渡到磁流体力学-动理学混合方程(αk=1 时)的系数.(2)式右侧最后一项为平行声波阻尼效应, 即
∇Π=ρκ|||k||vth,i|v|| , 其中,κ|| 表示平行声波阻尼强度系数,k|| 为平行波数,vth,i 为热离子速度. 在物理上, 这一项描述了平行声波引起的朗道阻尼, 可看作动理学效应对磁流体力学模型的修正[33]. 研究表明, 对于高β (等离子体比压值)等离子体, 考虑强平行声波阻尼的等离子体响应与实验结果更为吻合[34,35]. 本文在3.2和3.3节的流体响应中考虑强平行声波阻尼(κ||=1.5 ); 因为模型中平行声波阻尼系数κ|| 是可以调节的, 所以为了理解平行声波阻尼对等离子体响应的影响, 在3.4节的动理学响应中比较了强平行声波阻尼(κ||=1.5 )和无平行声波阻尼(κ||=0 )两种情况.p 表示等离子体的扰动压强张量, 漂移动理学效应通过p 自洽地耦合到等离子体动量方程(2)中形成磁流体力学-动理学混合模型. 如(6)式所示, 扰动压强张量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⊥ 组成.I 为单位张量,ˆb=B/B|B||B| . 漂移动理学扰动压强p∥ 和p⊥ 由(7)式计算, 其中V 为粒子的速度空间, i为粒子种类, 本研究中包括背景离子、背景电子和聚变产生的α 粒子. Mi是相应的粒子质量.v∥ 和v⊥ 分别表示粒子导向中心漂移运动的平行速度和垂直速度.f1L 是粒子的分布函数[24]. 研究表明, 当等离子体归一化比压βN (βN=β(%)a(m)B0(T)/IP(MA). )接近或超过Troyon无壁极限βNWN 时, 单流体响应与实验结果不一致, 而考虑动理学效应的磁流体力学-动理学混合模型能定量解释实验结果[29,36]. 因此, 下文也采用磁流体力学-动理学混合模型开展了动理学响应的研究.2.2 多组RMP线圈电流优化的解析方法
在MARS程序模拟研究等离子体对RMP场响应中, 有两个优化参数对于模拟和实验结果的联系非常重要. 这两个优化参数分别为(i)等离子体最外层有理面处的径向扰动场
b1res 和(ii)等离子体边界X点附近法向扰动位移的最大值ξX , 本文以这两个优化参数作为下文研究的基础. 其中, 等离子体最外层有理面处径向扰动场b1res 与有理面处形成磁岛的大小有关, 这些磁岛的增大和重叠会增加等离子体的径向输运, 从而实现对ELMs的控制[37]. 在直场线坐标系中, 径向扰动场的振幅为b1res=|(b⋅∇ψB⋅∇ϕ)nm|1R20, (8) 式中,
B 是平衡磁场,b 是扰动磁场,Ψ 是平衡极向磁通量,n 和m 为沿等离子体环向角和极向角的傅里叶谐波. 另一个优化参数—等离子体边界X点附近法向扰动位移的最大值ξX 表示等离子体边界的三维扭曲, 在实验上容易测量. 这个量与实验上实现ELMs的控制[30]和伴随ELMs控制发生的密度泵出[28,38]密切相关. 实际上, 模拟研究表明两个优化判据对电流相位差的优化具有高度的一致性[39].RMP线圈电流优化的目标是使上述两个优化参数最大化. 当RMP线圈多于一组时, 位于同一环向、不同极向的线圈之间的电流会产生一个相位差, 这个环向相位差会直接影响扰动磁场的极向谱. 本文基于上述的两个优化参数, 用解析方法对多组(多于一组)线圈情况下线圈电流的幅值和相位进行优化研究. 下面以X点附近等离子体扰动位移
ξX 为例说明优化过程, 具体的计算方法如下:Max(ξX)=|∑k(CkξXk)|, (9) 式中
k 表示不同的线圈组, 如只有上下两组线圈k=1,2 , 有上中下三组线圈k=1,2, 3 , 以此类推.ξXk 为每组线圈在等离子体分界面X点附近产生扰动位移的最大值,Ck 为对应线圈组的优化电流. 注意ξXk 和Ck 均为复数, 多组线圈总电流的限制条件为A2=∑k|Ck|2 . 每组线圈的优化电流Ck 由Cauchy-Bunyakovsky-Schwarz不等式确定:|ξXk|2=|∑k(CkξXk)|2⩽∑k|Ck|2∑k|ξXk|2. (10) 当
Ck=conj(ξXk)A/ABB ,B2=∑k|ξXk|2 时,Ck 为最优电流. 这一优化线圈电流幅值和相位的解析方法即高效又适合多组RMP线圈. 下文中将此解析方法与单一相位逐个扫描方法进行比较[40], 其结果具有很好的一致性.3. 数值模拟结果
3.1 等离子体平衡和位形
参考EU DEMO1 2015[41]设计的等离子体平衡, 采用CHEASE[42]程序, 在固定真空环向磁场下, 将等离子体电流从
Ip=20.4 MA (q95=3 )扫描到Ip=9.8 MA (q95=6.5 )共产生了71个安全因子剖面(相邻安全因子剖面的q95 相差0.05), 图1(a)展示了其中8个安全因子径向剖面. 实验研究表明RMP实现ELMs的缓解或抑制与边界安全因子q95 [7,26]密切相关, 下文将详细研究安全因子对等离子体三维响应场的影响. 图1(b)是等离子体的压强剖面, 它的归一化常数为B20/B20μ0μ0 , 其中, 磁轴处真空环向场B0=5.56 T. 图1(c)是磁轴处归一化为1的等离子体密度剖面. 等离子体的压强剖面和密度剖面均参考EU DEMO1 2015设计[41]. 旋转剖面参考ITER设计[21,43], 图1(d)是由磁轴处阿尔芬频率归一化的旋转剖面. 等离子体的关键平衡参数如下: 磁轴处大半径R0=9.07 m, 拉长比A=3.1 . 此系列平衡简称为‘类-DEMO’平衡.![图 1 等离子体平衡的径向剖面 (a)在固定的磁场下扫描等离子体电流时安全因子剖面的变化; (b)由${{B_0^2} \mathord{\left/ {\vphantom {{B_0^2} {{\mu _0}}}} \right. } {{\mu _0}}}$归一化的压强剖面; (c)磁轴处归一化为1的密度剖面; (d)由磁轴处阿尔芬频率${\omega _A} = {{{B_0}} \mathord{\left/ {\vphantom {{{B_0}} {\left[ {{R_0}\left( {{\mu _0}{\rho _0}} \right)} \right]}}} \right. } {\left[ {{R_0}\left( {{\mu _0}{\rho _0}} \right)} \right]}}$归一化的旋转频率剖面\r\nFig. 1. Plasma equilibrium radial profiles: (a) Variation of the equilibrium safety factor profile while scanning the plasma current at fixed field; (b) the plasma pressure normalized by ${{B_0^2} \mathord{\left/ {\vphantom {{B_0^2} {{\mu _0}}}} \right. } {{\mu _0}}}$; (c) the plasma density normalized to unity at the magnetic axis; (d) the (assumed) plasma toroidal rotation frequencies normalized by the on-axis Alfven frequency${\omega _A} = {{{B_0}} \mathord{\left/ {\vphantom {{{B_0}} {\left[ {{R_0}\left( {{\mu _0}{\rho _0}} \right)} \right]}}} \right. } {\left[ {{R_0}\left( {{\mu _0}{\rho _0}} \right)} \right]}}$.](/fileWLXB/journal/article/wlxb/2023/7/PIC/7-20222196-1.png)
图 1 等离子体平衡的径向剖面 (a)在固定的磁场下扫描等离子体电流时安全因子剖面的变化; (b)由B20/B20μ0μ0 ωA=B0/B0[R0(μ0ρ0)][R0(μ0ρ0)] Fig. 1. Plasma equilibrium radial profiles: (a) Variation of the equilibrium safety factor profile while scanning the plasma current at fixed field; (b) the plasma pressure normalized byB20/B20μ0μ0 ωA=B0/B0[R0(μ0ρ0)][R0(μ0ρ0)] 图2所示为等离子体的边界、真空双壁以及RMP线圈的几何位形. 本文借鉴ITER设计, 假设有三组RMP线圈. 其中, 上下两组线圈位于第一真空壁外侧
rUc = rLc = 1.46a (a是小半径), 极向位置分别为θUc=+20∘ 和θUc=−20∘ , 极向宽度为ΔθU=ΔθL=15∘ ; 中线圈位于真空双壁外侧rMc = 1.72a , 极向位置为θMc=0∘ , 极向宽度为ΔθM=45∘ . 上中下三组RMP线圈电流分别定义为IUexp(iΦU) ,IMexp(iΦM) 和ILexp(iΦL) . 对于固定的环向模数n , 线圈的几何位形和线圈电流都会直接影响外加场的极向谱, 因此对ELMs的控制起着重要的作用. 先前的工作已经采用MARS-F程序开展了RMP线圈几何位形优化的研究[21,39]. 本文只以一种RMP线圈几何位形为例, 开展等离子体对RMP响应和线圈电流优化的模拟研究.
图 2 等离子体的边界(蓝线)、真空双壁(红线)和RMP线圈的几何位形(黑线)Fig. 2. Plasma boundary (bule line), double wall shapes (red lines) and RMP coil geometry (black lines).3.2和3.3节通过扫描安全因子
q95 、环向模数n、多组RMP线圈(包括上下两组线圈和上中下三组线圈)之间的电流相位差和幅值等重要物理参数, 模拟计算了 '类-DEMO' 平衡下包括强平行声波阻尼(κ∥=1.5 )的流体响应. 3.4节在单组线圈位形下以q95=3.27 和q95=6.54 两个平衡为例, 利用MARS-K程序模拟研究背景粒子和聚变产生α粒子的动理学效应对n=1 等离子体响应的影响, 通过比较强平行声波阻尼(κ∥=1.5 )和无平行声波阻尼(κ∥=0 )两种模型, 研究了等离子体响应与平行声波阻尼的依赖关系. 表1所示为扫描参数的相关信息.表 1 在流体响应和动理学响应计算过程中扫描参数的相关信息Table 1. Parameter information used to compute fluid response and kinetic response.流体响应 动理学响应(TP, TP+EP) n = 1—4 n = 1 {\kappa _\parallel } = 1.5 {\kappa _\parallel } = 1.5 {\kappa _\parallel } = 0 中线圈 上下两组线圈 上中下三组线圈 中线圈 {q_{95} } = 3.0—6.5 {q_{95}} = 3.27 \begin{gathered}{q_{95}} = 3.27\left( {{\beta _{\text{N}}} = 2.69} \right)\\{q_{95}} = 6.54\left( {{\beta _{\text{N}}} = 1.35} \right)\end{gathered} — \begin{gathered} \Delta {\varPhi ^{ {\text{UL} } } } = {\varPhi ^{\text{U} } } - {\varPhi ^{\text{L} } } \\ = - {180^{\circ} }—{180^{\circ} } \end{gathered} \begin{gathered}{\varPhi ^{\text{U} } } - {\varPhi ^{\text{M} } } = - {180^{\circ} }—{180^{\circ} }\\{\varPhi ^{\text{L} } } - {\varPhi ^{\text{M} } } = - {180^{\circ} }—{180^{\circ} }\end{gathered} — {I^{\text{M}}} = 90{\text{ kAt}} \begin{gathered}{I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}}\\ {I^{\text{U}}} + {I^{\text{L}}} = 180{\text{ kAt}} \end{gathered} \begin{gathered} {I^{\text{U}}} = {I^{\text{M}}} = {I^{\text{L}}} = 90{\text{ kAt}}\\{I^{\text{U}}} + {I^{\text{M}}} + {I^{\text{L}}} = 270{\text{ kAt}} \end{gathered} {I^{\text{M}}} = 90{\text{ kAt}} 3.2 安全因子对等离子体流体响应的影响
实验研究表明RMP对ELMs的控制与
{q_{95}} 窗口密切相关, 如: DIII-D上RMP在{q_{95}} = 6 的窗口实现了对ELMs的抑制[44], ASDEX Upgrade[30]和EAST[45]等装置上在{q_{95}} = 5—6 的窗口实现了对ELMs的缓解, DIII-D上RMP在{q_{95}} = 5.3—7.1 窗口实现了草丛型(grassy)ELMs区域[46]. 因此, 本节以‘类-DEMO’平衡为例, 模拟计算不同环向模数(n = 1—4 )下安全因子对等离子体流体响应的影响.3.2.1 等离子体对中线圈的响应与安全因子的依赖关系
单组RMP线圈无需考虑多组线圈之间电流的相位差, 能直观观察
{q_{95}} 变化对等离子体响应的影响. 因此, 首先介绍等离子体对单组中线圈响应与{q_{95}} 的依赖关系. 如前文所述, 中线圈的中心位于中平面上, 即极向位置为\theta_{\rm c}^{{\rm{M}}}=0^{\circ} , 极向宽度和径向位置分别为\Delta {\theta ^{\text{M}}} = 45^\circ 和r_{\text{c}}^{\text{M}}{\text{ = }}1.72 a (位于真空双壁外侧), 线圈电流设定为90 kAt. 通过扫描等离子体电流,{q_{95}} 值从3变化到6.5, 即\Delta {q_{95}} = 6.5 - 3 = 3.5 .图3所示为等离子体最外侧有理面处的径向扰动场
b_{{\text{res}}}^{\text{1}} 随{q_{95}} 的变化. 从图3可以看出, 主导环向模数n = {\text{2}}—4 的线圈位形下真空径向扰动场的幅值随{q_{95}} 增大而减小, 这是因为真空场分量一般随极向模数m(m \approx n{q_{95}} )增大而减小. 与真空径向扰动场相比, 由于等离子体的屏蔽作用, 考虑了等离子体流体响应后,n = {\text{2}}—4 的线圈位形下最外侧有理面处的总径向扰动场明显下降. 然而对于n = 1 的线圈位形, 包含等离子体流体响应的总径向扰动场在某些{q_{95}} 处出现明显的放大现象. 更有趣的是, 在n = 1—4 的线圈位形下, 随着{q_{95}} 的增大, 径向扰动场都出现周期性的变化. 真空径向扰动场出现的波动不明显, 主要是由新有理面进入到等离子体区域引起的; 而总径向扰动场的波动现象非常显著, 在特定的{q_{95}} 窗口出现了大的峰值, 这些{q_{95}} 窗口与ELMs的控制密切相关.
图 3 在环向模数 (a)n = 1 n = 2 n = 3 n = 4 \left| {b_{{\text{res}}}^{\text{1}}} \right| {q_{95}} \Delta {\theta ^{\text{M}}} = 4{{\text{5}}^{\circ}} {I^{\text{M}}} = 90{\text{ kAt}} Fig. 3. Comparison of the computed amplitude of the outermost pitch resonant radial field components\left| {b_{{\text{res}}}^{\text{1}}} \right| \Delta {\theta ^{\text{M}}} = 4{{\text{5}}^{\circ}} {I^{\text{M}}} = 90{\text{ kAt}} 同样地, 随着
{q_{95}} 的变化, 等离子体边界X点附近法向扰动位移的最大值{\xi _X} 也出现了明显的周期性波动, 如图4所示. 比较图4(a)—(d)发现, 尽管个别{q_{95}} 处出现的峰值现象不明显, 如: 在n = 4 线圈位形下,{q_{95}} = 5.3 和{q_{95}} = 5.6 处几乎观察不到峰值现象, 这可能有多模耦合或更复杂的物理相关. 但整体而言, 在\Delta {q_{95}} = 3.5 的窗口内, 等离子体边界X点附近扰动位移{\xi _X} 出现峰值的个数y与环向模数n呈正相关, 即y \approx n\Delta {q_{95}} , 这与对ASDEX Upgrade预测的结果一致[30]. 对于低n (n = 1, 2 )线圈位形, 等离子体扰动位移出现的峰值位于\Delta {q_{95}} \approx {m / n} 和\Delta {q_{95}} \approx {{\left( {m{\text{ + }}1} \right)} \mathord{\left/ {\vphantom {{\left( {m{\text{ + }}1} \right)} n}} \right. } n} 之间; 对于高n (n = 3, 4 )线圈位形, 等离子体扰动位移出现峰值的位置略有偏移, 这可能是因为在高n的线圈位形下, 等离子体边缘会出现更多共振极向谐波m, 它们之间的耦合引起了峰值位置的偏移. 另外, 从图4还可以观察到, 相同的线圈电流({I^{\text{M}}} = 90{\text{ kAt}} )在X点附近产生的扰动位移随环向模数的增大而大大减小,n = 1 的扰动线圈在X点附近产生的扰动位移最大.
图 4 在环向模数 (a)n = 1 n = 2 n = 3 n = 4 \left| {{\xi _X}} \right| {q_{95}} \Delta {\theta ^{\text{M}}} = 4{{\text{5}}^{\circ}} {I^{\text{M}}} = 90{\text{ kAt}} Fig. 4. The computed amplitude of the plasma surface displacement near the X-point\left| {{\xi _X}} \right| {q_{95}} \Delta {\theta ^{\text{M}}} = 4{{\text{5}}^{\circ}} {I^{\text{M}}} = 90{\text{ kAt}} n = 1 n = 2 n = 3 n = 4 为了理解安全因子影响等离子体响应的物理机制, 图5以
n = 1 线圈位形为例, 对比了图3(a)和图4(a)中红、橙、蓝和绿点所示的{q_{95}} = 4 , 4.25, 4.4和4.55四种平衡下各有理面处相应的总径向扰动场b_{{\text{res(tot)}}}^{\text{1}} . 其中,{q_{95}} = 4 和{q_{95}} = 4.4 分别对应b_{{\text{res}}\left( {{\text{tot}}} \right)}^{\text{1}} 或{\xi _X} 出现谷值和峰值的平衡,{q_{95}} = 4.25 时新有理面{m \mathord{\left/ {\vphantom {m {n = {6 \mathord{\left/ {\vphantom {6 1}} \right. } 1}}}} \right. } {n = {6 \mathord{\left/ {\vphantom {6 1}} \right. } 1}}} 刚进入等离子体区域. 当有一个新的有理面经过等离子体边缘的某个特定区域(即{q_{95}} = 4.4 )时, 等离子体响应场就会出现一个峰值, 这是因为峰值响应与等离子体边缘剥离模的稳定性有关, 对ASDEX Upgrade[30]和ITER[40]的模拟研究也证明了这一点. 研究表明这个特定区域通常为等离子体边缘台基区, 显然, 新有理面经过等离子体边缘台基区时, 有理面处的大磁岛更容易加速该区域粒子的径向输运, 从而有利于实现ELMs的控制[47].
图 5 在环向模数n = 1 {q_{95}} = 4 b_{{\text{res(tot)}}}^1 {q_{95}} = 4 {q_{95}} = 4.4 Fig. 5. Comparison of the resonant field amplitude at the corresponding rational surfaces among{q_{95}} =4, {\text{ }}4.25, {\text{ }}4.4, {\text{ }}4.55 n = 1 {q_{95}} = 4 {q_{95}} = 4.4 3.2.2 安全因子和上下两组线圈电流相位差对等离子体响应的影响
为进一步理解安全因子对等离子体响应的影响, 这部分主要介绍安全因子对上下两组线圈产生的扰动磁场和扰动位移的影响. 注意当扰动线圈多于一组时, 位于同一环向、不同极向的线圈之间的电流会产生一个相位差, 这个环向相位差的改变直接影响扰动磁场的极向谱[16], 因此本部分也计算了上下两组线圈电流相位差的影响. 上下两组线圈之间的电流相位差是通过上下线圈的电流定义的, 如前文所述, 上下线圈电流分别定义为
{I^{\text{U}}}\exp \left( {{\text{i}}{\varPhi ^{\text{U}}}} \right) 和{I^{\text{L}}}\exp \left( {{\text{i}}{\varPhi ^{\text{L}}}} \right) , 则上下线圈之间的电流相位差为\Delta {\varPhi ^{{\text{UL}}}} = {\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}} . 在MARS程序中, 电流相位差可以连续变化, 本文电流相位差的扫描范围为–180°—180°.图6比较了最外侧有理面处的真空径向扰动磁场和包括了等离子体响应的总径向扰动磁场随上下两组线圈电流相位差
\Delta {\varPhi ^{{\text{UL}}}} 和安全因子{q_{95}} 的变化. 为了更清楚展示最优的电流相位差(使最外侧有理面处的径向扰动场最大), 对图6中每个{q_{95}} 处的径向扰动场进行线性化处理, 使其在[0, 1]范围内变化, 即{\text{normalized }}\left| {b_{{\text{res}}}^{\text{1}}} \right| = \left| {\left[ {b_{{\text{res}}}^{\text{1}} - \min \left( {b_{{\text{res}}}^{\text{1}}} \right)} \right]}/ {\left[ {\max \left( {b_{{\text{res}}}^{\text{1}}} \right) - \min \left( {b_{{\text{res}}}^{\text{1}}} \right)} \right]} \right| . 与ASDEX Upgrade[30]和ITER[40]相似, 电流相位差的最优值与{q_{95}} 之间满足线性依赖关系, 这一关系受环向模数n 的影响, 随着n 的增大, 最优电流相位差受{q_{95}} 的影响程度增大. 最差的电流相位差(使最外侧有理面处的径向扰动场最小)与最优的电流相位差之间相差180o. 为了更明确表示{q_{95}} 和n 对最优/差的电流相位差的影响, 引进两个简单的线性拟合函数. 在真空径向扰动场和包括了等离子体响应的总径向扰动场两种情况下, 上下线圈电流相位差的最优/差值与q_{95} 和n 的关系分别为![图 6 最外侧有理面处归一化的(a)(c)(e)(g)真空径向场和(b)(d)(f)(h)总径向场(包括等离子体响应)随上下两组线圈电流相位差$ \Delta {\varPhi ^{{\text{UL}}}} = {\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}} $和安全因子${q_{95}}$的变化. 环向模数分别为(a)(b)n = 1, (c)(d)n = 2, (e)(f)n = 3和(g)(h)n = 4. 上下两组线圈的极向位置分别为$\theta _{\text{c}}^{\text{U}} = + {20^{\circ}}$和$\theta _{\text{c}}^{\text{L}} = - {20^{\circ}}$, 极向宽度和电流分别为$\Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}}$和$ {I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}} $\r\nFig. 6. Comparison of the amplitude of the outermost pitch resonant radial field component in 2 D parameter space ($\Delta \varPhi , {q_{95}}$), between the vacuum radial field (left panel) and the total radial field including the plasma response (right panel), for the toroidal number (a)(b) n = 1, (c)(d) n = 2, (e)(f) n = 3, and (g)(h) n = 4, respectively, using upper and lower rows of coils located at $\theta _{\text{c}}^{\text{U}} = + {20^{\circ}}$and $\theta _{\text{c}}^{\text{L}} = - {20^{\circ}}$ with poloidal width $\Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}}$ and current $ {I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}} $. The field amplitude is linearly scaled to the range of [0, 1] for each q95 value.](/fileWLXB/journal/article/wlxb/2023/7/PIC/7-20222196-6.png)
图 6 最外侧有理面处归一化的(a)(c)(e)(g)真空径向场和(b)(d)(f)(h)总径向场(包括等离子体响应)随上下两组线圈电流相位差\Delta {\varPhi ^{{\text{UL}}}} = {\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}} {q_{95}} \theta _{\text{c}}^{\text{U}} = + {20^{\circ}} \theta _{\text{c}}^{\text{L}} = - {20^{\circ}} \Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} {I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}} Fig. 6. Comparison of the amplitude of the outermost pitch resonant radial field component in 2 D parameter space (\Delta \varPhi , {q_{95}} \theta _{\text{c}}^{\text{U}} = + {20^{\circ}} \theta _{\text{c}}^{\text{L}} = - {20^{\circ}} \Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} {I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}} {q_{95}} = \frac{7}{{60n}}\left( {\Delta {\varPhi ^{{\text{UL}}}} + 180{k_{\text{c}}}} \right) - \frac{5}{{2n}} \text{, } (11) {q_{95}} = \frac{7}{{60n}}\left( {\Delta {\varPhi ^{{\text{UL}}}} + 180{k_{\text{c}}}} \right) - \frac{6}{n}. (12) 图6中白色实线和虚线分别为两个解析函数预测的电流相位差的最优值(
{k_{\text{c}}} 为偶数)和最差值({k_{\text{c}}} 为奇数), 除了真空近似时环向模数n{\text{ = }}2 的误差较大, 其他电流相位差的最优值和拟合函数之间的误差较小. 此外, 从图6还可以发现, 真空近似时电流相位差的最优值和包括等离子体响应后电流相位差的最优值之间的差值为15°—45°, 这一差值几乎不受{q_{95}} 的影响, 受环向模数的影响也很小. 需要说明的是, 与单组中线圈的响应结果一致, 在n = 1—4 的线圈位形下, 上下两组线圈产生的总径向扰动场也出现了显著的周期性波动, 峰值的个数和位置也与中线圈一致. 由于总径向扰动场的波动, 在出现峰值的{q_{95}} 窗口附近更容易实现ELMs的控制, 这一结果在图7中更容易被观察到.
图 7 X点附近扰动位移随上下两组线圈电流相位差\Delta {\varPhi ^{{\text{UL}}}} = {\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}} {q_{95}} \theta _{\text{c}}^{\text{U}} = + {20^{\circ}} \theta _{\text{c}}^{\text{L}} = - {20^{\circ}} \Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} {I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}} Fig. 7. Amplitude of the plasma surface displacement near the X-point, with 2D parameter scan in toroidal phase difference\Delta {\varPhi ^{{\text{UL}}}} = {\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}} \theta _{\text{c}}^{\text{U}} = + {20^{\circ}} \theta _{\text{c}}^{\text{L}} = - {20^{\circ}} \Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} {I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}} 图7与图6做了相似的计算, 参数的选择与图6一致, 不同的是, 图7主要研究了上下两组线圈电流相位差
\Delta {\varPhi ^{{\text{UL}}}} 和安全因子{q_{95}} 对分界面X点附近等离子体扰动位移幅值的影响, 4个环向模数如图(a)n = 1, (b)n = 2, (c)n = 3和(d)n = 4所示. 比较图7(a)—(d)发现, 随环向模数的增大, 相同的线圈电流在X点附近产生的扰动位移的峰值不断减小. X点附近扰动位移作为共振扰动磁场控制ELMs实验一个有效的参数, 与等离子体的三维扭曲相关, 但在实验中需要避免在等离子体边界处产生太大的三维扭曲, 因为这可能会使局域的等离子体接触到装置的真空壁, 对壁造成极大的危害. 此外, X点附近的扰动位移与总径向扰动场一样, 对ELMs控制实验设计中{q_{95}} 值的选择也具有重要的指导意义.3.3 RMP线圈电流幅值和相位差的优化
上文只在固定的线圈电流(
{I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}} )下, 研究了上下两组线圈电流相位差的作用, 但实际上不同线圈组之间的电流幅值也可能不同, 因此本节中对RMP线圈电流相位和电流幅值同时进行优化研究. 线圈电流相位和幅值的优化过程如前文2.2节所述, 这种优化方法可用于不同装置、不同平衡下ELMs控制线圈的电流优化设计. 下面以‘类-DEMO’平衡({q_{95}} = 3.27 )为例, 利用MARS-F程序和解析方法, 开展RMP线圈电流幅值和相位差的优化研究.3.3.1 上下两组线圈电流幅值和相位差的优化
本部分主要对上下两组线圈电流进行优化分析, 如前文所述, 上下线圈电流分别定义为
{I^{\text{U}}}\exp \left( {{\text{i}}{\varPhi ^{\text{U}}}} \right) 和{I^{\text{L}}}\exp \left( {{\text{i}}{\varPhi ^{\text{L}}}} \right) , 因此有4个不确定参数({I^{\text{U}}},{I^{\text{L}}}, {\varPhi ^{\text{U}}}, {\varPhi ^{\text{L}}} ). 首先设定{I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}} 和{\varPhi ^{\text{U}}} = {\varPhi ^{\text{L}}} = 0 , 使用MARS-F程序计算等离子体对单组上(下)线圈的响应, 环向模数n = 1—4 的线圈位形下X点附近扰动位移的幅值和相位如表2所示.表 2 不同环向模数下, 电流为90 kAt时单组上/下线圈在X点附近产生扰动位移的幅值和相位Table 2. Amplitude and phase of the plasma surface displacement near the X-point for different toroidal number, using only upper/lower row of coils with 90 kAt current.单组上线圈 单组下线圈 n \left| {{\xi _X}} \right|/mm \varPhi /\left( ^\circ \right) \left| {{\xi _X}} \right|/mm \varPhi /\left( ^\circ \right) 1 39.98 - 155.8 39.41 - 67.3 2 16.19 - 156.4 11.16 - 48.1 3 3.46 - 107.0 11.09 22.4 4 6.90 149.4 1.20 - 98.8 接下来根据Cauchy-Bunyakovsky-Schwarz不等式计算上下两组线圈的优化电流
C_{k} , 也就是说当{C_k} = {\text{con}}j\left( {{\xi _{{\text{X}}k}}} \right){A \mathord{\left/ {\vphantom {A B}} \right. } B} 时,{C_k} 为最优电流, 其中{A}^{2}= \displaystyle\sum\nolimits _{k}{\left|{C}_{k}\right|}^{2} ,{B}^{2}=\displaystyle\sum\nolimits _{k}{\left|{\xi }_{\text{X}k}\right|}^{2} . 在线性计算中, 等离子体扰动位移与扰动线圈的电流成线性关系, 为了与前文中两组电流相等的情况对比, 本部分将上下两组线圈的总电流固定为180 kAt. 表3总结了n = 1—4 的线圈位形下, 上下两组线圈电流幅值和相位差的最优值. 结果表明, 线圈电流幅值的最优值随环向模数n的改变而改变.n = 1 时上下两组线圈电流幅值几乎相等, 随着n 的增大, 两组线圈电流最优幅值的差值也增大. 电流幅值的优化不改变上下两组线圈相位差优化的结果.表 3 上下两组线圈电流幅值和相位差的优化结果Table 3. Optimization results of the current amplitude and phase difference between the upper and lower rows of coils.n {I^{\text{U}}}/{\text{kAt}} {I^{\text{L}}}/{\text{kAt}} ({\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}})/(^\circ ) 1 90.64 89.36 88.5 2 106.56 73.44 108.3 3 42.80 137.20 129.4 4 153.34 26.66 111.8 图8利用单一相位逐个扫描方法[40], 进一步对比了线圈电流幅值相等和优化两种情况下X点附近扰动位移随上下两组线圈电流相位差
\Delta {\varPhi ^{{\text{UL}}}} 的变化, 该方法预测最优电流相位差与表3中解析方法的结果具有很好的一致性. 此外, 与相等的电流幅值相比, 在n = 1—4 的线圈位形下, 电流幅值的优化不影响X点附近扰动位移的最大值(最优相位差处X点附近扰动位移). 但有趣的是, 线圈电流幅值的优化增大了n = {\text{2}}—4 时最差电流相位差附近的扰动位移. 随着n 的增大, 电流幅值的优化作用更显著, 尤其n = 4 时电流幅值的优化几乎可以使所有\Delta {\varPhi ^{{\text{UL}}}} (除了最优相位差)处的扰动位移增大. 这说明对于较高n的线圈位形, 上下两组线圈电流幅值的优化可以增大实现ELMs控制的\Delta {\varPhi ^{{\text{UL}}}} 窗口.
图 8 在环向模数 (a)n = 1 n = 2 n = 3 n = 4 \Delta {\varPhi ^{{\text{UL}}}} = {\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}} \theta _{\text{c}}^{\text{U}} = + {20^{\circ}} \theta _{\text{c}}^{\text{L}} = - {20^{\circ}} \Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} Fig. 8. Comparison of the amplitude of the plasma surface displacement near the X-point between the equal coil current and optimized coil current, versus the toroidal phase difference\Delta {\varPhi ^{{\text{UL}}}} = {\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}} \theta _{\text{c}}^{\text{U}} = + {20^{\circ}} \theta _{\text{c}}^{\text{L}} = - {20^{\circ}} \Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} 3.3.2 上中下三组线圈电流幅值和相位差的优化
为了进一步验证解析方法的可靠性, 这一部分将对上中下三组RMP线圈电流幅值和相位差进行优化分析. 上下两组线圈的几何位形和电流定义与前文一致, 中线圈的电流定义为
{I^{\text{M}}}\exp \left( {{\text{i}}{\varPhi ^{\text{M}}}} \right) , 首先设定{I^{\text{M}}} = 90{\text{ kAt}} 和{\varPhi ^{\text{M}}} = 0 , 再用MARS-F程序计算等离子体对单组中线圈的响应, 表4为n = 1—4线圈位形下X点附近扰动位移的幅值和相位.表 4 不同环向模数下, 电流为90 kAt时单组中线圈在X点附近产生扰动位移的幅值和相位Table 4. Amplitude and phase of the plasma surface displacement near the X-point for different toroidal number, using only middle row of coils with 90 kAt current.n \left| {{\xi _X}} \right|/mm \varPhi {\text{/(}}^\circ ) 1 70.57 - 110.8 2 5.52 - 102.5 3 12.30 - 39.0 4 6.33 - 178.8 三组线圈电流共有6个不确定参数(
{I^{\text{U}}}, {I^{\text{M}}}, {I^{\text{L}}}, {\varPhi ^{\text{U}}}, {\varPhi ^{\text{M}}}, {\varPhi ^{\text{L}}} ), 根据Cauchy-Bunyakovsky-Schwarz不等式, 利用单组上、下、中线圈的响应结果(如表2和表4所示), 计算三组线圈电流幅值和相位差的最优值,n = 1—4 线圈位形的电流优化结果如表5所示. 上中下三组线圈的总电流固定为270 kAt, 线圈电流幅值和相位差的最优值都与环向模数n密切相关. 为了进一步理解线圈电流优化的作用, 图9利用单一相位逐个扫描方法, 比较了线圈电流幅值相等和优化两种条件下X点附近扰动位移幅值随上、下两组线圈与中线圈电流相位差({\varPhi ^{\text{U}}} - {\varPhi ^{\text{M}}} ,{\varPhi ^{\text{L}}} - {\varPhi ^{\text{M}}} )的变化. 结果表明, 在线性计算中, 线圈电流幅值的优化只改变X点附近扰动位移幅值的大小, 不改变相位差的优化结果. 对于三组线圈位形, 电流幅值的优化可以使整个二维参数空间({\varPhi ^{\text{U}}} - {\varPhi ^{\text{M}}} ,{\varPhi ^{\text{L}}} - {\varPhi ^{\text{M}}} )中X点附近扰动位移增大.表 5 上中下三组线圈电流幅值和相位差的优化结果Table 5. Optimization results of the current amplitude and phasing with three rows of coils.n {I^{\text{L}}}/{\text{kAt}} {I^{\text{M}}}/{\text{kAt}} {I^{\text{U}}}/{\text{kAt}} ({\varPhi ^{\text{L} } } - {\varPhi ^{\text{M} } }) /(^\circ ) ({\varPhi ^{\text{U} } } - {\varPhi ^{\text{M} } }) /(^\circ ) 1 70.95 127.05 72.00 - 43.5 45.0 2 91.68 45.33 132.99 - 54.4 53.9 3 111.52 123.69 34.79 - 61.4 68.0 4 22.44 118.44 129.12 - 80.0 31.8 
图 9 在环向模数n = 1 {\varPhi ^{\text{U}}} - {\varPhi ^{\text{M}}} {\varPhi ^{\text{L}}} - {\varPhi ^{\text{M}}} \theta _{\text{c}}^{\text{U}} = + {20^{\circ}} \theta _{\text{c}}^{\text{M}} = {0^{\circ}} \theta _{\text{c}}^{\text{L}} = - {20^{\circ}} \Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} \Delta {\theta ^{\text{M}}} = 4{{\text{5}}^{\circ}} Fig. 9. Comparison of the amplitude of the plasma surface displacement near the X-point between the (a) equal coil current and (b) optimized coil current for the toroidal number n=1. The relative toroidal phasing, ({\varPhi ^{\text{U}}} - {\varPhi ^{\text{M}}} {\varPhi ^{\text{L}}} - {\varPhi ^{\text{M}}} \theta _{\text{c}}^{\text{U}} = + {20^{\circ}} \theta _{\text{c}}^{\text{M}} = {0^{\circ}} \theta _{\text{c}}^{\text{L}} = - {20^{\circ}} \Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} \Delta {\theta ^{\text{M}}} = 4{{\text{5}}^{\circ}} 3.4 漂移动理学效应对等离子体响应的影响
当等离子体压强接近或超过无壁Troyon极限时, 动理学效应在等离子体响应计算中变得非常重要, 例如, 包含背景粒子的动理学响应解决了DIII-D上高
\beta 等离子体中流体理论预测与实验观测之间的分歧[29]. 最近, 利用包含背景粒子和高能粒子动理学效应的等离子体响应模型, 针对EU DEMO 2019平衡[48]的研究发现, 聚变产生α粒子的动理学效应在降低Troyon无壁极限附近的响应振幅方面起着主要作用[49]. 本节以{q_{95}} = 3.27 ({\beta _N} = 2.69 )和{q_{95}} = 6.54 ({\beta _{\text{N}}} = 1.35 )两个平衡为例, 假设采用单组中线圈的RMP位形, 利用MARS-K程序模拟研究背景粒子和聚变产生α粒子的动理学效应对n = 1 等离子体响应的影响. 背景粒子的平衡遵从麦克斯韦分布, α粒子的平衡遵从慢化分布[50]. α粒子的平衡密度和平衡压强剖面分别如文献 [51]中图2(e)和(f)所示.图10在强平行声波阻尼(
{\kappa _\parallel } = 1.5 )下, 比较了流体响应(fluid)、包含背景粒子的动理学响应(TP)以及包含背景粒子和聚变产生α粒子的动理学响应(TP+EP)3种情况对应的n=1总径向扰动场沿小半径的最大幅值. 结果表明, 对于{q_{95}} = 3.27 和{q_{95}} = 6.54 两个平衡, 背景粒子和聚变产生的α粒子动理学效应对等离子体响应结果的影响非常微弱, 这说明考虑强平行声波阻尼({\kappa _\parallel } = 1.5 )的流体响应模型能够很好地预测‘类-DEMO’平衡的等离子体响应, 这也进一步验证了上文中用MARS-F研究安全因子对等离子体流体响应的影响, 以及线圈电流幅值和相位差的优化结果的可靠性.
图 10 在强平行声波阻尼({\kappa _\parallel } = 1.5 {q_{95}} = 3.27 {q_{95}} = 6.54 Fig. 10. The maximal amplitude (along the minor radius) of all the poloidal Fourier harmonics of the n=1 total radial field. The resonant harmonics are m = 1—5 for (a){q_{95}} = 3.27 {q_{95}} = 6.54 {\kappa _\parallel } = 1.5 在物理上, 平行声波阻尼可看作动理学效应对磁流体力学模型的修正. 因为在模型中平行声波阻尼系数
{\kappa _{||}} 是可以调节的, 所以为了理解平行声波阻尼对等离子体响应的影响, 又研究了无平行声波阻尼({\kappa _{||}} = 0 )时, 流体响应、包含背景粒子的动理学响应(TP)以及包含背景粒子和聚变产生α粒子的动理学响应(TP+EP)对应的n=1总径向扰动场沿小半径的最大振幅, 如图11所示. 对于高β({\beta _{\text{N}}} = 2.69 )等离子体, 在无平行声波阻尼时背景粒子和α粒子的动理学效应都会放大等离子体流体响应模型计算的径向扰动场. 更准确地说, 背景粒子的动理学效应只微弱地放大了流体响应结果; 而包括背景粒子和α粒子的动理学效应使流体响应计算的径向扰动场放大了4倍. 对于低β({\beta _{\text{N}}} = 1.35 )等离子体, 动理学响应始终与流体响应保持一致, 与有无强平行声波阻尼无关.
图 11 在无平行声波阻尼({\kappa _\parallel } = 0 {q_{95}} = 3.27 {q_{95}} = 6.54 Fig. 11. The maximal amplitude (along the minor radius) of all the poloidal Fourier harmonics of the n=1 total radial field. The resonant harmonics are m = 1—5 for (a){q_{95}} = 3.27 {q_{95}} = 6.54 {\kappa _\parallel } = 0 图12基于‘类-DEMO’平衡(
q_{95}=3.27 ), 比较了无平行声波阻尼({\kappa _\parallel } = 0 )时流体响应模型和动理学响应模型计算的n=1等离子体径向扰动位移. 与图12(a)所示的流体响应相比, 背景粒子的动理学效应对等离子体扰动位移具有一定的放大作用, 而背景粒子和α粒子的动理学效应对流体响应结果的放大作用更显著(注意, 图12(c)的颜色比例是12(a)和(b)的2倍). 另外, 动理学效应的放大作用主要集中在等离子体芯部, 这可能与背景粒子和/或高能粒子的动理学效应在{q_0} < 1 时更容易激发{m \mathord{\left/ {\vphantom {m n}} \right. } n} = {1 \mathord{\left/ {\vphantom {1 1}} \right. } 1} 不稳定性有关. 总之, 对于高β等离子体, 在流体响应模型中需要考虑动理学效应的修正作用, 即可以考虑强的平行声波阻尼, 也可以利用磁流体力学-动理学的混合模型考虑粒子的动理学效应.
图 12 在无平行声波阻尼({\kappa _\parallel } = 0 {q_{95}} = 3.27 Fig. 12. Comparison of the amplitude of the normal component of the computed plasma radial displacement, due to the plasma response to the applied n = 1 RMP field for{q_{95}} = 3.27 {\kappa _\parallel } = 0 4. 总结与讨论
本文采用MARS-F/K程序, 以‘类-DEMO’平衡为例, 模拟研究了等离子体对RMP的流体响应和动理学响应. 本工作主要开展了以下几方面的研究: 1)利用MARS-F程序, 详细计算了
n = 1—4 线圈位形下安全因子对等离子体流体响应的影响; 2)利用MARS-F程序和解析方法, 开展了n = 1—4 RMP线圈电流幅值和相位差的优化研究; 3)利用MARS-K程序, 模拟研究了背景粒子和聚变产生α粒子的动理学效应对n = 1 等离子体响应的影响.在n=1—4的线圈位形下, 随着
q_{95} 的增大, 最外层有理面处总径向扰动场b_{{\text{res}}\left( {{\text{tot}}} \right)}^{\text{1}} 和分界面上X点附近等离子体扰动位移{\xi _X} 两个优化参数都出现周期性的变化, 在特定的q_{95} 窗口出现了大的峰值. 结果表明, 优化参数出现峰值的q_{95} 窗口与新有理面出现并移动到等离子体边缘的台基区有关. 在本文研究的q_{95} 范围内(\Delta {q_{95}} = 3.5 ), 优化参数出现峰值的个数y与环向模数n呈正相关, 即y \approx n\Delta {q_{95}} , 这与对ASDEX Upgrade预测的结果一致[30]. 对于低n(n = 1, 2)线圈位形, 优化参数出现的峰值位于\Delta {q_{95}} \approx {m \mathord{\left/ {\vphantom {m n}} \right. } n} 和\Delta {q_{95}} \approx {{\left( {m{\text{ + }}1} \right)} \mathord{\left/ {\vphantom {{\left( {m{\text{ + }}1} \right)} n}} \right. } n} 之间, 对于高n(n = 3, 4)线圈位形, 峰值的位置略有偏移. 另外, 根据两个优化参数对上下两组线圈在二维参数空间(\Delta {\varPhi ^{{\text{UL}}}} ,{q_{95}} )的优化结果表明, 电流相位差的最优/差值与q_{95} 之间满足线性依赖关系, 其拟合函数如(12)式所示.本工作还针对上下两组和上中下三组RMP线圈, 以‘类-DEMO’平衡(
q_{95}=3.27 )为例, 利用MARS-F程序和Cauchy-Bunyakovsky-Schwarz不等式计算了线圈的优化电流. 结果表明, 线圈电流幅值的最优值随环向模数n的改变而改变. 在线性计算中, 与相等的电流幅值相比, 线圈电流幅值的优化只改变X点附近扰动位移幅值的大小, 不改变不同线圈组之间电流相位差的最优值. 对于上下两组线圈位形, 线圈电流幅值的优化增大了n = 2—4 时最差电流相位差附近的扰动位移, 随着n 的增大, 电流幅值的优化作用更显著; 对于三组线圈位形, 电流幅值的优化可以使整个二维参数空间({\varPhi ^{\text{U}}} - {\varPhi ^{\text{M}}} ,{\varPhi ^{\text{L}}} - {\varPhi ^{\text{M}}} )中X点附近扰动位移增大.包含背景粒子或/和高能粒子动理学效应的等离子体响应结果表明, 考虑强平行声波阻尼(
{\kappa _\parallel } = 1.5 )的流体响应模型能够很好地预测‘类-DEMO’平衡的等离子体响应, 这也进一步验证了文中流体响应结果的可靠性. 对于高β({\beta _{\text{N}}} = 2.69 )等离子体, 在无平行声波阻尼({\kappa _\parallel } = 0 )时, 背景粒子和α粒子的动理学效应都会放大等离子体流体响应的结果, 动理学效应的放大作用主要集中在等离子体芯部, 这可能与背景粒子和/或高能粒子的动理学效应在{q_0} < 1 时更容易激发{m \mathord{\left/ {\vphantom {m n}} \right. } n} = {1 \mathord{\left/ {\vphantom {1 1}} \right. } 1} 不稳定性有关. 总之, 对于低β等离子体, 动理学响应与流体响应保持一致, 与有无强平行声波阻尼无关; 而对于高β等离子体, 在流体响应模型中需要考虑动理学效应的修正作用.最后, 本工作中利用MARS-F程序和Cauchy-Bunyakovsky-Schwarz不等式计算了线圈的优化电流, 该方法简单、准确、高效, 可用于不同平衡、不同线圈组优化的比较研究. 但需要指出, 等离子体旋转在等离子体对RMP响应中起着重要作用, 本工作只参考ITER设计的等离子体旋转剖面开展了等离子体对RMP线圈的流体响应和动理学响应, 未来的工作中将结合输运模型开展等离子体旋转影响等离子体响应的研究. 另外, 本文只考虑了背景粒子和聚变产生α粒子的动理学效应, 而其他类型的高能粒子(例如中性束注入产生的高能粒子)也会影响等离子体响应, 未来将采用更精确的动理学响应模型考虑辅助加热产生高能粒子的作用.
感谢东华大学的李莉副教授, 重庆工商大学的杨旭副教授, 核工业西南物理研究院的郝广周研究员、王硕副研究员、张能副研究员、陈海涛在MARS-F程序调试的过程中给予的帮助和支持.
[1] Suttrop W 2000 Plasma Phys. Contr. F. 42 A1
Google Scholar
[2] Loarte A, Lipschultz B, Kukushkin A S, Matthews G F, Stangeby P C, Asakura N, Counsell G F, Federici G, Kallenbach A, Krieger K, Mahdavi A, Philipps V, Reiter D, Roth J, Strachan J, Whyte D, Doerner R, Eich T, Fundamenski W, Herrmann A, Fenstermacher M, Ghendrih P, Groth M, Kirschner A, Konoshima S, LaBombard B, Lang P, Leonard A W, Monier-Garbet P, Neu R, Pacher H, Pegourie B, Pitts R A, Takamura S, Terry J, Tsitrone E, Group D P T 2007 Nucl. Fusion 47 S203
Google Scholar
[3] Ryter F, Fuchs J, Schneider W, Sips A, Stäbler A, Stober J, Team A U 2008 J. Phys. Conf. Ser. 123 012035
Google Scholar
[4] Loarte A, Saibene G, Sartori R, Becoulet M, Horton L, Eich T, Herrmann A, Laux M, Matthews G, Jachmich S, Asakura N, Chankin A, Leonard A, Porter G, Federici G, Shimada M, Sugihara M, Janeschitz G 2003 J. Nucl. Mater 313 962
Google Scholar
[5] Wenninger R P, Bernert M, Eich T, Fable E, Federici G, Kallenbach A, Loarte A, Lowry C, McDonald D, Neu R, Pütterich T, Schneider P, Sieglin B, Strohmayer G, Reimold F, Wischmeier M 2014 Nucl. Fusion 54 114003
Google Scholar
[6] Liang Y, Koslowski H R, Thomas P R, Nardon E, Alper B, Andrew P, Andrew Y, Arnoux G, Baranov Y, Becoulet M, Beurskens M, Biewer T, Bigi M, Crombe K, De La Luna E, de Vries P, Fundamenski W, Gerasimov S, Giroud C, Gryaznevich M P, Hawkes N, Hotchin S, Howell D, Jachmich S, Kiptily V, Moreira L, Parail V, Pinches S D, Rachlew E, Zimmermann O 2007 Phys. Rev. Lett. 98 265004
Google Scholar
[7] Suttrop W, Eich T, Fuchs J C, Gunter S, Janzer A, Herrmann A, Kallenbach A, Lang P T, Lunt T, Maraschek M, McDermott R M, Mlynek A, Putterich T, Rott M, Vierle T, Wolfrum E, Yu Q, Zammuto I, Zohm H, Team A U 2011 Phys. Rev. Lett. 106 225004
Google Scholar
[8] Evans T E, Moyer R A, Thomas P R, Watkins J G, Osborne T H, Boedo J A, Doyle E J, Fenstermacher M E, Finken K H, Groebner R J, Groth M, Harris J H, La Haye R J, Lasnier C J, Masuzaki S, Ohyabu N, Pretty D G, Rhodes T L, Reimerdes H, Rudakov D L, Schaffer M J, Wang G, Zeng L 2004 Phys. Rev. Lett. 92 235003
Google Scholar
[9] Jeon Y M, Park J K, Yoon S W, Ko W H, Lee S G, Lee K D, Yun G S, Nam Y U, Kim W C, Kwak J G, Lee K S, Kim H K, Yang H L, Team K 2012 Phys. Rev. Lett. 109 035004
Google Scholar
[10] Kirk A, Suttrop W, Chapman I T, Liu Y, Scannell R, Thornton A J, Orte L B, Cahyna P, Eich T, Fischer R, Fuchs C, Ham C, Harrison J R, Jakubowski M W, Kurzan B, Pamela S, Peterka M, Ryan D, Saarelma S, Sieglin B, Valovic M, Willensdorfer M 2015 Nucl. Fusion 55 043011
Google Scholar
[11] Liang Y, Koslowski H R, Thomas P R, Nardon E, Alper B, Andrew P, Andrew Y, Arnoux G, Baranov Y, Bécoulet M, Beurskens M, Biewer T, Bigi M, Crombe K, De La Luna E, de Vries P, Fundamenski W, Gerasimov S, Giroud C, Gryaznevich M P, Hawkes N, Hotchin S, Howell D, Jachmich S, Kiptily V, Moreira L, Parail V, Pinches S D, Rachlew E, Zimmermann O 2007 Physical Review Letters 98 265004
[12] Sun Y, Liang Y, Liu Y Q, Gu S, Yang X, Guo W, Shi T, Jia M, Wang L, Lyu B, Zhou C, Liu A, Zang Q, Liu H, Chu N, Wang H H, Zhang T, Qian J, Xu L, He K, Chen D, Shen B, Gong X, Ji X, Wang S, Qi M, Song Y, Yuan Q, Sheng Z, Gao G, Fu P, Wan B 2016 Phys. Rev. Lett. 117 115001
Google Scholar
[13] Wan B N, Liang Y, Gong X Z, Xiang N, Xu G S, Sun Y, Wang L, Qian J P, Liu H Q, Zeng L, Zhang L, Zhang X J, Ding B J, Zang Q, Lyu B, Garofalo A M, Ekedahl A, Li M H, Ding F, Ding S Y, Du H F, Kong D F, Yu Y, Yang Y, Luo Z P, Huang J, Zhang T, Zhang Y, Li G Q, Xia T Y 2019 Nucl. Fusion 59 112003
Google Scholar
[14] Duan X R, Liu Y, Xu M, Yan L W, Xu Y, Song X M, Dong J Q, Ding X T, Chen L Y, Lu B, Liu D Q, Rao J, Xuan W M, Yang Q W, Zheng G Y, Zou X L, Liu Y Q, Zhong W L, Zhao K J, Ji X Q, Mao W C, Wang Q M, Li Q, Cao J Y, Cao Z, Lei G J, Zhang J H, Li X D, Bai X Y, Cheng J, Chen W, Cui Z Y, Delpech L, Diamond P H, Dong Y B, Ekedahl A, Hoang T, Huang Y, Ida K, Itoh K, Itoh S I, Isobe M, Inagaki S, Mazon D, Morita S, Peysson Y, Shi Z B, Wang X G, Xiao G L, Yu D L, Yu L M, Zhang Y P, Zhou Y, Cui C H, Feng B B, Huang M, Li Y G, Li B, Li G S, Li H J, Li Q, Peng J F, Wang Y Q, Yuan B S, Liu Y 2017 Nucl. Fusion 57 102013
Google Scholar
[15] Ding Y, Chen Z, Chen Z, Yang Z, Wang N, Hu Q, Rao B, Chen J, Cheng Z, Gao L, Jiang Z, Wang L, Wang Z, Zhang X, Zheng W, Zhang M, Zhuang G, Yu Q, Liang Y, Yu K, Hu X, Pan Y, Gentle K W, Team J T 2018 Plasma Sci. Technol. 20 125101
Google Scholar
[16] Haskey S R, Lanctot M J, Liu Y Q, Hanson J M, Blackwell B D, Nazikian R 2014 Plasma Phys. Contr. F. 56 035005
Google Scholar
[17] Schaffer M J, Menard J E, Aldan M P, Bialek J M, Evans T E, Moyer R A 2008 Nucl. Fusion 48 024004
Google Scholar
[18] Liu Y Q, Kirk A, Sun Y, Cahyna P, Chapman I T, Denner P, Fishpool G, Garofalo A M, Harrison J R, Nardon E 2012 Plasma Phys. Contr. F. 54 124013
Google Scholar
[19] Liu Y, Li L, Loarte A, Pinches S D, Polevoi A 2021 Nucl. Fusion 61 106029
Google Scholar
[20] 苏春燕, 牟茂淋, 陈少永, 郭文平, 唐昌建 2021 物理学报 70 095207
Google Scholar
Su C Y, Mou M L, Chen S Y, Guo W P, Tang C J 2021 Acta Phys. Sin. 70 095207
Google Scholar
[21] Zhou L, Liu Y, Liu Y, Yang X 2016 Plasma Phys. Contr. F. 58 115003
Google Scholar
[22] 陈撷宇, 牟茂淋, 苏春燕, 陈少永, 唐昌建 2020 物理学报 69 195201
Google Scholar
Chen X Y, Mou M L, Su C Y, Chen S Y, Tang C J 2020 Acta Phys. Sin. 69 195201
Google Scholar
[23] 李春雨, 郝广周, 刘钺强, 王炼, 刘艺慧子 2022 物理学报 71 075202
Google Scholar
Li C Y, Hao G Z, Liu Y Q, Wang L, Liu Y H Z 2022 Acta Phys. Sin. 71 075202
Google Scholar
[24] Liu Y, Chu M S, Chapman I T, Hender T C 2008 Phys. Plasmas 15 112503
Google Scholar
[25] Liu Y, Kirk A, Nardon E 2010 Phys. Plasmas 17 122502
Google Scholar
[26] Sun Y, Liang Y, Liu Y Q, Gu S, Yang X, Guo W, Shi T, Jia M, Wang L, Lyu B, Zhou C, Liu A, Zang Q, Liu H, Chu N, Wang H H, Zhang T, Qian J, Xu L, He K, Chen D, Shen B, Gong X, Ji X, Wang S, Qi M, Song Y, Yuan Q, Sheng Z, Gao G, Fu P, Wan B 2016 Physical Review Letters 117 115001
[27] Liu Y Q, Bondeson A, Fransson C M, Lennartson B, Breitholtz C 2000 Phys. Plasmas 7 3681
Google Scholar
[28] Liu Y, Kirk A, Gribov Y, Gryaznevich M P, Hender T C, Nardon E 2011 Nucl. Fusion 51 083002
Google Scholar
[29] Wang Z R, Lanctot M J, Liu Y Q, Park J K, Menard J E 2015 Phys. Rev. Lett. 114 145005
Google Scholar
[30] Li L, Liu Y Q, Kirk A, Wang N, Liang Y, Ryan D, Suttrop W, Dunne M, Fischer R, Fuchs J C, Kurzan B, Piovesan P, Willensdorfer M, Zhong F C 2016 Nucl. Fusion 56 126007
Google Scholar
[31] Yang X, Sun Y, Liu Y, Gu S, Liu Y, Wang H, Zhou L, Guo W 2016 Plasma Phys. Contr. F. 58 114006
Google Scholar
[32] Liu Y, Kirk A, Li L, In Y, Nazikian R, Sun Y, Suttrop W, Lyons B, Ryan D, Wang S, Yang X, Zhou L 2017 Phys. Plasmas 24 056111
Google Scholar
[33] Bondeson A, Iacono R 1989 Phys. Fluids B Plasma Phys. 1 1431
Google Scholar
[34] Liu Y, Chapman I T, Saarelma S, Gryaznevich M P, Hender T C, Howell D F 2009 Plasma Phys. Contr. F. 51 115005
Google Scholar
[35] Liu Y, Ryan D, Kirk A, Li L, Suttrop W, Dunne M, Fischer R, Fuchs J C, Kurzan B, Piovesan P, Willensdorfer M 2016 Nucl. Fusion 56 056015
Google Scholar
[36] Wang Z, Park J-K, Menard J E, Liu Y, Kaye S M, Gerhardt S 2018 Nucl. Fusion 58 016015
Google Scholar
[37] Hu Q M, Nazikian R, Grierson B A, Logan N C, Paz-Soldan C, Yu Q 2020 Nucl. Fusion 60 076001
Google Scholar
[38] Wang S X, Liu H Q, Jie Y X, Ding W X, Zeng L, Sun Y W, Lian H, Zhu X, Zou Z Y, Lyu B, Li Y Y, Zang Q, Du H F, Zhou C, Liu A D, Zhang T, Gao W, Gao X 2018 Nucl. Fusion 58 112013
Google Scholar
[39] Zhou L, Liu Y, Wenninger R, Liu Y, Wang S, Yang X 2018 Nucl. Fusion 58 076025
Google Scholar
[40] Li L, Liu Y Q, Wang N, Kirk A, Koslowski H R, Liang Y, Loarte A, Ryan D, Zhong F C 2017 Plasma Phys. Contr. F. 59 044005
Google Scholar
[41] Wenninger R, Kembleton R, Bachmann C, Biel W, Bolzonella T, Ciattaglia S, Cismondi F, Coleman M, Donné A J H, Eich T, Fable E, Federici G, Franke T, Lux H, Maviglia F, Meszaros B, Pütterich T, Saarelma S, Snickers A, Villone F, Vincenzi P, Wolff D, Zohm H 2017 Nucl. Fusion 57 016011
Google Scholar
[42] Lütjens H, Bondeson A, Sauter O 1996 Comput. Phys. Commun. 97 219
Google Scholar
[43] Liu Y, Akers R, Chapman I T, Gribov Y, Hao G Z, Huijsmans G T A, Kirk A, Loarte A, Pinches S D, Reinke M, Ryan D, Sun Y, Wang Z R 2015 Nucl. Fusion 55 063027
Google Scholar
[44] Fenstermacher M E, Evans T E, Osborne T H, Schaffer M J, Aldan M P, deGrassie J S, Gohil P, Joseph I, Moyer R A, Snyder P B, Groebner R J, Jakubowski M, Leonard A W, Schmitz O, Team D 2008 Phys. Plasmas 15 056122
Google Scholar
[45] Gu S, Wan B, Sun Y, Chu N, Liu Y Q, Shi T, Wang H H, Jia M, He K 2019 Nucl. Fusion 59 126042
Google Scholar
[46] Nazikian R, Petty C C, Bortolon A, Chen X, Eldon D, Evans T E, Grierson B A, Ferraro N M, Haskey S R, Knolker M, Lasnier C, Logan N C, Moyer R A, Orlov D, Osborne T H, Paz-Soldan C, Turco F, Wang H Q, Weisberg D B 2018 Nucl. Fusion 58 106010
Google Scholar
[47] Hu Q M, Nazikian R, Logan N C, Park J K, Paz-Soldan C, Yang S M, Grierson B A, In Y, Jeon Y M, Kim M, Kim S K, Orlov D M, Park G Y, Yu Q 2021 Phys. Plasmas 28 052505
Google Scholar
[48] Federici G, Bachmann C, Barucca L, Baylard C, Biel W, Boccaccini L V, Bustreo C, Ciattaglia S, Cismondi F, Corato V, Day C, Diegele E, Franke T, Gaio E, Gliss C, Haertl T, Ibarra A, Holden J, Keech G, Kembleton R, Loving A, Maviglia F, Morris J, Meszaros B, Moscato I, Pintsuk G, Siccinio M, Taylor N, Tran M Q, Vorpahl C, Walden H, You J H 2019 Nucl. Fusion 59 066013
Google Scholar
[49] Zhou L, Liu Y, Hu H, Siccinio M, Maviglia F, Zohm H, Pigatto L, Wang Y, Li L, Hao G Z, Yang X, Zhang H, Duan P, Chen L 2023 Plasma Phys. Contr. F. 65 035008
Google Scholar
[50] Hu B, Betti R, Manickam J 2006 Phys. Plasmas 13 112505
Google Scholar
[51] Zhou L, Liu Y, Siccinio M, Fable E, Wu T, Kurki-Suonio T, Varje J, Liu D 2021 Plasma Phys. Contr. F. 63 065007
Google Scholar
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图 1 等离子体平衡的径向剖面 (a)在固定的磁场下扫描等离子体电流时安全因子剖面的变化; (b)由
{{B_0^2} \mathord{\left/ {\vphantom {{B_0^2} {{\mu _0}}}} \right. } {{\mu _0}}} 归一化的压强剖面; (c)磁轴处归一化为1的密度剖面; (d)由磁轴处阿尔芬频率{\omega _A} = {{{B_0}} \mathord{\left/ {\vphantom {{{B_0}} {\left[ {{R_0}\left( {{\mu _0}{\rho _0}} \right)} \right]}}} \right. } {\left[ {{R_0}\left( {{\mu _0}{\rho _0}} \right)} \right]}} 归一化的旋转频率剖面Figure 1. Plasma equilibrium radial profiles: (a) Variation of the equilibrium safety factor profile while scanning the plasma current at fixed field; (b) the plasma pressure normalized by
{{B_0^2} \mathord{\left/ {\vphantom {{B_0^2} {{\mu _0}}}} \right. } {{\mu _0}}} ; (c) the plasma density normalized to unity at the magnetic axis; (d) the (assumed) plasma toroidal rotation frequencies normalized by the on-axis Alfven frequency{\omega _A} = {{{B_0}} \mathord{\left/ {\vphantom {{{B_0}} {\left[ {{R_0}\left( {{\mu _0}{\rho _0}} \right)} \right]}}} \right. } {\left[ {{R_0}\left( {{\mu _0}{\rho _0}} \right)} \right]}} .
图 2 等离子体的边界(蓝线)、真空双壁(红线)和RMP线圈的几何位形(黑线)
Figure 2. Plasma boundary (bule line), double wall shapes (red lines) and RMP coil geometry (black lines).
图 3 在环向模数 (a)
n = 1 , (b)n = 2 , (c)n = 3 和(d)n = 4 线圈位形下, 中线圈在最外层有理面处产生的径向扰动场幅值\left| {b_{{\text{res}}}^{\text{1}}} \right| 随安全因子{q_{95}} 的变化. 中线圈的极向宽度和电流分别为\Delta {\theta ^{\text{M}}} = 4{{\text{5}}^{\circ}} 和{I^{\text{M}}} = 90{\text{ kAt}} Figure 3. Comparison of the computed amplitude of the outermost pitch resonant radial field components
\left| {b_{{\text{res}}}^{\text{1}}} \right| , between the vacuum field (solid) and the total field including the plasma response (dashed), versus the edge safety factor q95, for the (a) n = 1, (b) n = 2, (c) n = 3 and (d) n = 4 coil configurations, respectively, using the middle row of RMP coils with the poloidal width\Delta {\theta ^{\text{M}}} = 4{{\text{5}}^{\circ}} and current{I^{\text{M}}} = 90{\text{ kAt}} .
图 4 在环向模数 (a)
n = 1 , (b)n = 2 , (c)n = 3 和(d)n = 4 线圈位形下, 中线圈在X点附近产生的扰动位移幅值\left| {{\xi _X}} \right| 随安全因子{q_{95}} 的变化, 中线圈的极向宽度和电流分别为\Delta {\theta ^{\text{M}}} = 4{{\text{5}}^{\circ}} 和{I^{\text{M}}} = 90{\text{ kAt}} Figure 4. The computed amplitude of the plasma surface displacement near the X-point
\left| {{\xi _X}} \right| versus the edge safety factor{q_{95}} , using the middle row of RMP coils with the poloidal width\Delta {\theta ^{\text{M}}} = 4{{\text{5}}^{\circ}} and current{I^{\text{M}}} = 90{\text{ kAt}} , for the (a)n = 1 , (b)n = 2 , (c)n = 3 and(d)n = 4 coil configurations, respectively.
图 5 在环向模数
n = 1 线圈位形下,{q_{95}} = 4 , 4.25, 4.4和4.55时有理面处相应总径向扰动场b_{{\text{res(tot)}}}^1 的对比. 红线({q_{95}} = 4 )和蓝线({q_{95}} = 4.4 )分别对应最外层有理面处总径向扰动场或X点附近扰动位移的谷值和峰值, 如图3(a)或图4(a)中红点和蓝点所示Figure 5. Comparison of the resonant field amplitude at the corresponding rational surfaces among
{q_{95}} =4, {\text{ }}4.25, {\text{ }}4.4, {\text{ }}4.55 withn = 1 . The red line ({q_{95}} = 4 ) and blue line ({q_{95}} = 4.4 ) indicate the valley and peak values of the amplitude of the outermost pitch resonant radial field components in Fig. 3(a) or the plasma surface displacement near the X-point in Fig. 4(a), respectively.
图 6 最外侧有理面处归一化的(a)(c)(e)(g)真空径向场和(b)(d)(f)(h)总径向场(包括等离子体响应)随上下两组线圈电流相位差
\Delta {\varPhi ^{{\text{UL}}}} = {\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}} 和安全因子{q_{95}} 的变化. 环向模数分别为(a)(b)n = 1, (c)(d)n = 2, (e)(f)n = 3和(g)(h)n = 4. 上下两组线圈的极向位置分别为\theta _{\text{c}}^{\text{U}} = + {20^{\circ}} 和\theta _{\text{c}}^{\text{L}} = - {20^{\circ}} , 极向宽度和电流分别为\Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} 和{I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}} Figure 6. Comparison of the amplitude of the outermost pitch resonant radial field component in 2 D parameter space (
\Delta \varPhi , {q_{95}} ), between the vacuum radial field (left panel) and the total radial field including the plasma response (right panel), for the toroidal number (a)(b) n = 1, (c)(d) n = 2, (e)(f) n = 3, and (g)(h) n = 4, respectively, using upper and lower rows of coils located at\theta _{\text{c}}^{\text{U}} = + {20^{\circ}} and\theta _{\text{c}}^{\text{L}} = - {20^{\circ}} with poloidal width\Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} and current{I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}} . The field amplitude is linearly scaled to the range of [0, 1] for each q95 value.
图 7 X点附近扰动位移随上下两组线圈电流相位差
\Delta {\varPhi ^{{\text{UL}}}} = {\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}} 和安全因子{q_{95}} 的变化. 环向模数分别为(a) n = 1, (b) n = 2, (c) n = 3和(d) n = 4.上下两组线圈的极向位置分别为\theta _{\text{c}}^{\text{U}} = + {20^{\circ}} 和\theta _{\text{c}}^{\text{L}} = - {20^{\circ}} , 极向宽度和电流分别为\Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} 和{I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}} Figure 7. Amplitude of the plasma surface displacement near the X-point, with 2D parameter scan in toroidal phase difference
\Delta {\varPhi ^{{\text{UL}}}} = {\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}} (vertical axis) between the two offmidplane rows and q95 (horizontal axis), for the toroidal number (a) n = 1, (b) n = 2, (c) n = 3, and (d) n = 4, respectively, using upper and lower rows of coils located at\theta _{\text{c}}^{\text{U}} = + {20^{\circ}} and\theta _{\text{c}}^{\text{L}} = - {20^{\circ}} with poloidal width\Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} and current{I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}} .
图 8 在环向模数 (a)
n = 1 , (b)n = 2 , (c)n = 3 和(d)n = 4 线圈位形下, 比较线圈电流幅值相等和优化两种情况下X点附近扰动位移随上下两组线圈电流相位差\Delta {\varPhi ^{{\text{UL}}}} = {\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}} 的变化. 上下两组线圈的极向位置分别为\theta _{\text{c}}^{\text{U}} = + {20^{\circ}} 和\theta _{\text{c}}^{\text{L}} = - {20^{\circ}} , 极向宽度为\Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} , 两组线圈的总电流固定为180 kAtFigure 8. Comparison of the amplitude of the plasma surface displacement near the X-point between the equal coil current and optimized coil current, versus the toroidal phase difference
\Delta {\varPhi ^{{\text{UL}}}} = {\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}} between the upper and lower rows of coils located at\theta _{\text{c}}^{\text{U}} = + {20^{\circ}} and\theta _{\text{c}}^{\text{L}} = - {20^{\circ}} with poloidal width\Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} , for the toroidal number (a) n = 1, (b) n = 2, (c) n = 3, and (d) n = 4, respectively. The total current of the two rows of coils is fixed at 180 kAt.
图 9 在环向模数
n = 1 线圈位形下, 比较线圈电流幅值(a)相等和(b)优化两种条件下, X点附近扰动位移随上、下两组线圈与中线圈电流相位差({\varPhi ^{\text{U}}} - {\varPhi ^{\text{M}}} ,{\varPhi ^{\text{L}}} - {\varPhi ^{\text{M}}} )的变化. 上中下三组线圈的极向位置分别为\theta _{\text{c}}^{\text{U}} = + {20^{\circ}} ,\theta _{\text{c}}^{\text{M}} = {0^{\circ}} 和\theta _{\text{c}}^{\text{L}} = - {20^{\circ}} , 极向宽度为\Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} 和\Delta {\theta ^{\text{M}}} = 4{{\text{5}}^{\circ}} , 三组线圈的总电流固定为270 kAtFigure 9. Comparison of the amplitude of the plasma surface displacement near the X-point between the (a) equal coil current and (b) optimized coil current for the toroidal number n=1. The relative toroidal phasing, (
{\varPhi ^{\text{U}}} - {\varPhi ^{\text{M}}} ,{\varPhi ^{\text{L}}} - {\varPhi ^{\text{M}}} ), of the upper and lower rows of coil currents with respect to the middle row, is scanned in the 2 D parameter space. The coils located at\theta _{\text{c}}^{\text{U}} = + {20^{\circ}} ,\theta _{\text{c}}^{\text{M}} = {0^{\circ}} and\theta _{\text{c}}^{\text{L}} = - {20^{\circ}} with poloidal width\Delta {\theta ^{\text{U}}} = \Delta {\theta ^{\text{L}}} = {\text{1}}{{\text{5}}^{\circ}} and\Delta {\theta ^{\text{M}}} = 4{{\text{5}}^{\circ}} . The total current of the three rows of coils is fixed at 270 kAt
图 10 在强平行声波阻尼(
{\kappa _\parallel } = 1.5 )情况下, 比较流体响应、包含背景粒子的动理学响应(TP)以及包含背景粒子和聚变产生α粒子的动理学响应(TP+EP)对应的n = 1总径向扰动场沿小半径的最大幅值. 对于(a){q_{95}} = 3.27 和(b){q_{95}} = 6.54 , 共振谐波分别为m = 1—5和m = 2—10, 其他m为非共振谐波Figure 10. The maximal amplitude (along the minor radius) of all the poloidal Fourier harmonics of the n=1 total radial field. The resonant harmonics are m = 1—5 for (a)
{q_{95}} = 3.27 and m = 2—10 for (b){q_{95}} = 6.54 , and the remaining harmonics are nonresonant. Compared are the response fields obtained assuming the fluid model (fluid), and the MHD-kinetic hybrid model including the non-adiabatic contributions from thermal particles (TP), or both thermal particles and energetic particles (i.e. fusion-born alphas) (EP+TP). A strong parallel sound wave damping model is assumed for both plasmas ({\kappa _\parallel } = 1.5 ).
图 11 在无平行声波阻尼(
{\kappa _\parallel } = 0 )情况下, 比较流体响应(Fluid)、包含背景粒子的动理学响应(TP)以及包含背景粒子和聚变产生α粒子的动理学响应(TP+EP)对应的n=1总径向扰动场沿小半径的最大振幅. 对于(a){q_{95}} = 3.27 和(b){q_{95}} = 6.54 , 共振谐波分别为m = 1—5和m = 2—10, 其他m为非共振谐波Figure 11. The maximal amplitude (along the minor radius) of all the poloidal Fourier harmonics of the n=1 total radial field. The resonant harmonics are m = 1—5 for (a)
{q_{95}} = 3.27 and m = 2—10 for (b){q_{95}} = 6.54 , and the remaining harmonics are nonresonant. Compared are the response fields obtained assuming the fluid model, and the MHD-kinetic hybrid model including the non-adiabatic contributions from thermal particles (TP), or both thermal particles and energetic particles (i.e. fusion-born alphas) (EP+TP). The parallel sound wave damping model is eliminated for both plasmas ({\kappa _\parallel } = 0 ).
图 12 在无平行声波阻尼(
{\kappa _\parallel } = 0 )情况下, 比较DEMO平衡({q_{95}} = 3.27 )的(a)流体响应、(b)包含背景粒子的动理学响应以及(c)包含背景粒子和聚变产生α粒子的动理学响应对应的n = 1等离子体径向位移沿法向分量的振幅Figure 12. Comparison of the amplitude of the normal component of the computed plasma radial displacement, due to the plasma response to the applied n = 1 RMP field for
{q_{95}} = 3.27 , assuming (a) the fluid model, and the MHD-kinetic hybrid model including non-adiabatic contributions from (b) thermal particles, and (c) both fusion-born alphas and thermal particles. The parallel sound wave damping model is eliminated ({\kappa _\parallel } = 0 ).表 1 在流体响应和动理学响应计算过程中扫描参数的相关信息
Table 1. Parameter information used to compute fluid response and kinetic response.
流体响应 动理学响应(TP, TP+EP) n = 1—4 n = 1 {\kappa _\parallel } = 1.5 {\kappa _\parallel } = 1.5 {\kappa _\parallel } = 0 中线圈 上下两组线圈 上中下三组线圈 中线圈 {q_{95} } = 3.0—6.5 {q_{95}} = 3.27 \begin{gathered}{q_{95}} = 3.27\left( {{\beta _{\text{N}}} = 2.69} \right)\\{q_{95}} = 6.54\left( {{\beta _{\text{N}}} = 1.35} \right)\end{gathered} — \begin{gathered} \Delta {\varPhi ^{ {\text{UL} } } } = {\varPhi ^{\text{U} } } - {\varPhi ^{\text{L} } } \\ = - {180^{\circ} }—{180^{\circ} } \end{gathered} \begin{gathered}{\varPhi ^{\text{U} } } - {\varPhi ^{\text{M} } } = - {180^{\circ} }—{180^{\circ} }\\{\varPhi ^{\text{L} } } - {\varPhi ^{\text{M} } } = - {180^{\circ} }—{180^{\circ} }\end{gathered} — {I^{\text{M}}} = 90{\text{ kAt}} \begin{gathered}{I^{\text{U}}} = {I^{\text{L}}} = 90{\text{ kAt}}\\ {I^{\text{U}}} + {I^{\text{L}}} = 180{\text{ kAt}} \end{gathered} \begin{gathered} {I^{\text{U}}} = {I^{\text{M}}} = {I^{\text{L}}} = 90{\text{ kAt}}\\{I^{\text{U}}} + {I^{\text{M}}} + {I^{\text{L}}} = 270{\text{ kAt}} \end{gathered} {I^{\text{M}}} = 90{\text{ kAt}} 
DownLoad: CSV
表 2 不同环向模数下, 电流为90 kAt时单组上/下线圈在X点附近产生扰动位移的幅值和相位
Table 2. Amplitude and phase of the plasma surface displacement near the X-point for different toroidal number, using only upper/lower row of coils with 90 kAt current.
单组上线圈 单组下线圈 n \left| {{\xi _X}} \right|/mm \varPhi /\left( ^\circ \right) \left| {{\xi _X}} \right|/mm \varPhi /\left( ^\circ \right) 1 39.98 - 155.8 39.41 - 67.3 2 16.19 - 156.4 11.16 - 48.1 3 3.46 - 107.0 11.09 22.4 4 6.90 149.4 1.20 - 98.8
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表 3 上下两组线圈电流幅值和相位差的优化结果
Table 3. Optimization results of the current amplitude and phase difference between the upper and lower rows of coils.
n {I^{\text{U}}}/{\text{kAt}} {I^{\text{L}}}/{\text{kAt}} ({\varPhi ^{\text{U}}} - {\varPhi ^{\text{L}}})/(^\circ ) 1 90.64 89.36 88.5 2 106.56 73.44 108.3 3 42.80 137.20 129.4 4 153.34 26.66 111.8
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表 4 不同环向模数下, 电流为90 kAt时单组中线圈在X点附近产生扰动位移的幅值和相位
Table 4. Amplitude and phase of the plasma surface displacement near the X-point for different toroidal number, using only middle row of coils with 90 kAt current.
n \left| {{\xi _X}} \right|/mm \varPhi {\text{/(}}^\circ ) 1 70.57 - 110.8 2 5.52 - 102.5 3 12.30 - 39.0 4 6.33 - 178.8
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表 5 上中下三组线圈电流幅值和相位差的优化结果
Table 5. Optimization results of the current amplitude and phasing with three rows of coils.
n {I^{\text{L}}}/{\text{kAt}} {I^{\text{M}}}/{\text{kAt}} {I^{\text{U}}}/{\text{kAt}} ({\varPhi ^{\text{L} } } - {\varPhi ^{\text{M} } }) /(^\circ ) ({\varPhi ^{\text{U} } } - {\varPhi ^{\text{M} } }) /(^\circ ) 1 70.95 127.05 72.00 - 43.5 45.0 2 91.68 45.33 132.99 - 54.4 53.9 3 111.52 123.69 34.79 - 61.4 68.0 4 22.44 118.44 129.12 - 80.0 31.8
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[1] Suttrop W 2000 Plasma Phys. Contr. F. 42 A1
Google Scholar
[2] Loarte A, Lipschultz B, Kukushkin A S, Matthews G F, Stangeby P C, Asakura N, Counsell G F, Federici G, Kallenbach A, Krieger K, Mahdavi A, Philipps V, Reiter D, Roth J, Strachan J, Whyte D, Doerner R, Eich T, Fundamenski W, Herrmann A, Fenstermacher M, Ghendrih P, Groth M, Kirschner A, Konoshima S, LaBombard B, Lang P, Leonard A W, Monier-Garbet P, Neu R, Pacher H, Pegourie B, Pitts R A, Takamura S, Terry J, Tsitrone E, Group D P T 2007 Nucl. Fusion 47 S203
Google Scholar
[3] Ryter F, Fuchs J, Schneider W, Sips A, Stäbler A, Stober J, Team A U 2008 J. Phys. Conf. Ser. 123 012035
Google Scholar
[4] Loarte A, Saibene G, Sartori R, Becoulet M, Horton L, Eich T, Herrmann A, Laux M, Matthews G, Jachmich S, Asakura N, Chankin A, Leonard A, Porter G, Federici G, Shimada M, Sugihara M, Janeschitz G 2003 J. Nucl. Mater 313 962
Google Scholar
[5] Wenninger R P, Bernert M, Eich T, Fable E, Federici G, Kallenbach A, Loarte A, Lowry C, McDonald D, Neu R, Pütterich T, Schneider P, Sieglin B, Strohmayer G, Reimold F, Wischmeier M 2014 Nucl. Fusion 54 114003
Google Scholar
[6] Liang Y, Koslowski H R, Thomas P R, Nardon E, Alper B, Andrew P, Andrew Y, Arnoux G, Baranov Y, Becoulet M, Beurskens M, Biewer T, Bigi M, Crombe K, De La Luna E, de Vries P, Fundamenski W, Gerasimov S, Giroud C, Gryaznevich M P, Hawkes N, Hotchin S, Howell D, Jachmich S, Kiptily V, Moreira L, Parail V, Pinches S D, Rachlew E, Zimmermann O 2007 Phys. Rev. Lett. 98 265004
Google Scholar
[7] Suttrop W, Eich T, Fuchs J C, Gunter S, Janzer A, Herrmann A, Kallenbach A, Lang P T, Lunt T, Maraschek M, McDermott R M, Mlynek A, Putterich T, Rott M, Vierle T, Wolfrum E, Yu Q, Zammuto I, Zohm H, Team A U 2011 Phys. Rev. Lett. 106 225004
Google Scholar
[8] Evans T E, Moyer R A, Thomas P R, Watkins J G, Osborne T H, Boedo J A, Doyle E J, Fenstermacher M E, Finken K H, Groebner R J, Groth M, Harris J H, La Haye R J, Lasnier C J, Masuzaki S, Ohyabu N, Pretty D G, Rhodes T L, Reimerdes H, Rudakov D L, Schaffer M J, Wang G, Zeng L 2004 Phys. Rev. Lett. 92 235003
Google Scholar
[9] Jeon Y M, Park J K, Yoon S W, Ko W H, Lee S G, Lee K D, Yun G S, Nam Y U, Kim W C, Kwak J G, Lee K S, Kim H K, Yang H L, Team K 2012 Phys. Rev. Lett. 109 035004
Google Scholar
[10] Kirk A, Suttrop W, Chapman I T, Liu Y, Scannell R, Thornton A J, Orte L B, Cahyna P, Eich T, Fischer R, Fuchs C, Ham C, Harrison J R, Jakubowski M W, Kurzan B, Pamela S, Peterka M, Ryan D, Saarelma S, Sieglin B, Valovic M, Willensdorfer M 2015 Nucl. Fusion 55 043011
Google Scholar
[11] Liang Y, Koslowski H R, Thomas P R, Nardon E, Alper B, Andrew P, Andrew Y, Arnoux G, Baranov Y, Bécoulet M, Beurskens M, Biewer T, Bigi M, Crombe K, De La Luna E, de Vries P, Fundamenski W, Gerasimov S, Giroud C, Gryaznevich M P, Hawkes N, Hotchin S, Howell D, Jachmich S, Kiptily V, Moreira L, Parail V, Pinches S D, Rachlew E, Zimmermann O 2007 Physical Review Letters 98 265004
[12] Sun Y, Liang Y, Liu Y Q, Gu S, Yang X, Guo W, Shi T, Jia M, Wang L, Lyu B, Zhou C, Liu A, Zang Q, Liu H, Chu N, Wang H H, Zhang T, Qian J, Xu L, He K, Chen D, Shen B, Gong X, Ji X, Wang S, Qi M, Song Y, Yuan Q, Sheng Z, Gao G, Fu P, Wan B 2016 Phys. Rev. Lett. 117 115001
Google Scholar
[13] Wan B N, Liang Y, Gong X Z, Xiang N, Xu G S, Sun Y, Wang L, Qian J P, Liu H Q, Zeng L, Zhang L, Zhang X J, Ding B J, Zang Q, Lyu B, Garofalo A M, Ekedahl A, Li M H, Ding F, Ding S Y, Du H F, Kong D F, Yu Y, Yang Y, Luo Z P, Huang J, Zhang T, Zhang Y, Li G Q, Xia T Y 2019 Nucl. Fusion 59 112003
Google Scholar
[14] Duan X R, Liu Y, Xu M, Yan L W, Xu Y, Song X M, Dong J Q, Ding X T, Chen L Y, Lu B, Liu D Q, Rao J, Xuan W M, Yang Q W, Zheng G Y, Zou X L, Liu Y Q, Zhong W L, Zhao K J, Ji X Q, Mao W C, Wang Q M, Li Q, Cao J Y, Cao Z, Lei G J, Zhang J H, Li X D, Bai X Y, Cheng J, Chen W, Cui Z Y, Delpech L, Diamond P H, Dong Y B, Ekedahl A, Hoang T, Huang Y, Ida K, Itoh K, Itoh S I, Isobe M, Inagaki S, Mazon D, Morita S, Peysson Y, Shi Z B, Wang X G, Xiao G L, Yu D L, Yu L M, Zhang Y P, Zhou Y, Cui C H, Feng B B, Huang M, Li Y G, Li B, Li G S, Li H J, Li Q, Peng J F, Wang Y Q, Yuan B S, Liu Y 2017 Nucl. Fusion 57 102013
Google Scholar
[15] Ding Y, Chen Z, Chen Z, Yang Z, Wang N, Hu Q, Rao B, Chen J, Cheng Z, Gao L, Jiang Z, Wang L, Wang Z, Zhang X, Zheng W, Zhang M, Zhuang G, Yu Q, Liang Y, Yu K, Hu X, Pan Y, Gentle K W, Team J T 2018 Plasma Sci. Technol. 20 125101
Google Scholar
[16] Haskey S R, Lanctot M J, Liu Y Q, Hanson J M, Blackwell B D, Nazikian R 2014 Plasma Phys. Contr. F. 56 035005
Google Scholar
[17] Schaffer M J, Menard J E, Aldan M P, Bialek J M, Evans T E, Moyer R A 2008 Nucl. Fusion 48 024004
Google Scholar
[18] Liu Y Q, Kirk A, Sun Y, Cahyna P, Chapman I T, Denner P, Fishpool G, Garofalo A M, Harrison J R, Nardon E 2012 Plasma Phys. Contr. F. 54 124013
Google Scholar
[19] Liu Y, Li L, Loarte A, Pinches S D, Polevoi A 2021 Nucl. Fusion 61 106029
Google Scholar
[20] 苏春燕, 牟茂淋, 陈少永, 郭文平, 唐昌建 2021 物理学报 70 095207
Google Scholar
Su C Y, Mou M L, Chen S Y, Guo W P, Tang C J 2021 Acta Phys. Sin. 70 095207
Google Scholar
[21] Zhou L, Liu Y, Liu Y, Yang X 2016 Plasma Phys. Contr. F. 58 115003
Google Scholar
[22] 陈撷宇, 牟茂淋, 苏春燕, 陈少永, 唐昌建 2020 物理学报 69 195201
Google Scholar
Chen X Y, Mou M L, Su C Y, Chen S Y, Tang C J 2020 Acta Phys. Sin. 69 195201
Google Scholar
[23] 李春雨, 郝广周, 刘钺强, 王炼, 刘艺慧子 2022 物理学报 71 075202
Google Scholar
Li C Y, Hao G Z, Liu Y Q, Wang L, Liu Y H Z 2022 Acta Phys. Sin. 71 075202
Google Scholar
[24] Liu Y, Chu M S, Chapman I T, Hender T C 2008 Phys. Plasmas 15 112503
Google Scholar
[25] Liu Y, Kirk A, Nardon E 2010 Phys. Plasmas 17 122502
Google Scholar
[26] Sun Y, Liang Y, Liu Y Q, Gu S, Yang X, Guo W, Shi T, Jia M, Wang L, Lyu B, Zhou C, Liu A, Zang Q, Liu H, Chu N, Wang H H, Zhang T, Qian J, Xu L, He K, Chen D, Shen B, Gong X, Ji X, Wang S, Qi M, Song Y, Yuan Q, Sheng Z, Gao G, Fu P, Wan B 2016 Physical Review Letters 117 115001
[27] Liu Y Q, Bondeson A, Fransson C M, Lennartson B, Breitholtz C 2000 Phys. Plasmas 7 3681
Google Scholar
[28] Liu Y, Kirk A, Gribov Y, Gryaznevich M P, Hender T C, Nardon E 2011 Nucl. Fusion 51 083002
Google Scholar
[29] Wang Z R, Lanctot M J, Liu Y Q, Park J K, Menard J E 2015 Phys. Rev. Lett. 114 145005
Google Scholar
[30] Li L, Liu Y Q, Kirk A, Wang N, Liang Y, Ryan D, Suttrop W, Dunne M, Fischer R, Fuchs J C, Kurzan B, Piovesan P, Willensdorfer M, Zhong F C 2016 Nucl. Fusion 56 126007
Google Scholar
[31] Yang X, Sun Y, Liu Y, Gu S, Liu Y, Wang H, Zhou L, Guo W 2016 Plasma Phys. Contr. F. 58 114006
Google Scholar
[32] Liu Y, Kirk A, Li L, In Y, Nazikian R, Sun Y, Suttrop W, Lyons B, Ryan D, Wang S, Yang X, Zhou L 2017 Phys. Plasmas 24 056111
Google Scholar
[33] Bondeson A, Iacono R 1989 Phys. Fluids B Plasma Phys. 1 1431
Google Scholar
[34] Liu Y, Chapman I T, Saarelma S, Gryaznevich M P, Hender T C, Howell D F 2009 Plasma Phys. Contr. F. 51 115005
Google Scholar
[35] Liu Y, Ryan D, Kirk A, Li L, Suttrop W, Dunne M, Fischer R, Fuchs J C, Kurzan B, Piovesan P, Willensdorfer M 2016 Nucl. Fusion 56 056015
Google Scholar
[36] Wang Z, Park J-K, Menard J E, Liu Y, Kaye S M, Gerhardt S 2018 Nucl. Fusion 58 016015
Google Scholar
[37] Hu Q M, Nazikian R, Grierson B A, Logan N C, Paz-Soldan C, Yu Q 2020 Nucl. Fusion 60 076001
Google Scholar
[38] Wang S X, Liu H Q, Jie Y X, Ding W X, Zeng L, Sun Y W, Lian H, Zhu X, Zou Z Y, Lyu B, Li Y Y, Zang Q, Du H F, Zhou C, Liu A D, Zhang T, Gao W, Gao X 2018 Nucl. Fusion 58 112013
Google Scholar
[39] Zhou L, Liu Y, Wenninger R, Liu Y, Wang S, Yang X 2018 Nucl. Fusion 58 076025
Google Scholar
[40] Li L, Liu Y Q, Wang N, Kirk A, Koslowski H R, Liang Y, Loarte A, Ryan D, Zhong F C 2017 Plasma Phys. Contr. F. 59 044005
Google Scholar
[41] Wenninger R, Kembleton R, Bachmann C, Biel W, Bolzonella T, Ciattaglia S, Cismondi F, Coleman M, Donné A J H, Eich T, Fable E, Federici G, Franke T, Lux H, Maviglia F, Meszaros B, Pütterich T, Saarelma S, Snickers A, Villone F, Vincenzi P, Wolff D, Zohm H 2017 Nucl. Fusion 57 016011
Google Scholar
[42] Lütjens H, Bondeson A, Sauter O 1996 Comput. Phys. Commun. 97 219
Google Scholar
[43] Liu Y, Akers R, Chapman I T, Gribov Y, Hao G Z, Huijsmans G T A, Kirk A, Loarte A, Pinches S D, Reinke M, Ryan D, Sun Y, Wang Z R 2015 Nucl. Fusion 55 063027
Google Scholar
[44] Fenstermacher M E, Evans T E, Osborne T H, Schaffer M J, Aldan M P, deGrassie J S, Gohil P, Joseph I, Moyer R A, Snyder P B, Groebner R J, Jakubowski M, Leonard A W, Schmitz O, Team D 2008 Phys. Plasmas 15 056122
Google Scholar
[45] Gu S, Wan B, Sun Y, Chu N, Liu Y Q, Shi T, Wang H H, Jia M, He K 2019 Nucl. Fusion 59 126042
Google Scholar
[46] Nazikian R, Petty C C, Bortolon A, Chen X, Eldon D, Evans T E, Grierson B A, Ferraro N M, Haskey S R, Knolker M, Lasnier C, Logan N C, Moyer R A, Orlov D, Osborne T H, Paz-Soldan C, Turco F, Wang H Q, Weisberg D B 2018 Nucl. Fusion 58 106010
Google Scholar
[47] Hu Q M, Nazikian R, Logan N C, Park J K, Paz-Soldan C, Yang S M, Grierson B A, In Y, Jeon Y M, Kim M, Kim S K, Orlov D M, Park G Y, Yu Q 2021 Phys. Plasmas 28 052505
Google Scholar
[48] Federici G, Bachmann C, Barucca L, Baylard C, Biel W, Boccaccini L V, Bustreo C, Ciattaglia S, Cismondi F, Corato V, Day C, Diegele E, Franke T, Gaio E, Gliss C, Haertl T, Ibarra A, Holden J, Keech G, Kembleton R, Loving A, Maviglia F, Morris J, Meszaros B, Moscato I, Pintsuk G, Siccinio M, Taylor N, Tran M Q, Vorpahl C, Walden H, You J H 2019 Nucl. Fusion 59 066013
Google Scholar
[49] Zhou L, Liu Y, Hu H, Siccinio M, Maviglia F, Zohm H, Pigatto L, Wang Y, Li L, Hao G Z, Yang X, Zhang H, Duan P, Chen L 2023 Plasma Phys. Contr. F. 65 035008
Google Scholar
[50] Hu B, Betti R, Manickam J 2006 Phys. Plasmas 13 112505
Google Scholar
[51] Zhou L, Liu Y, Siccinio M, Fable E, Wu T, Kurki-Suonio T, Varje J, Liu D 2021 Plasma Phys. Contr. F. 63 065007
Google Scholar
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