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 $ with $n = 1$. 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 kAt

Figure 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 kAt

Figure 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 (${\kappa _\parallel } = 0$).