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In order to explore the conductive wall effect of plasma magnetohydrodynamic (MHD) instability and the wall designing idea, the various forms of ideal conductive walls based on divertor equilibrium configurations in the HL-2A Tokamak and their role in suppressing kink modes are studied. The MHD instabilities and the ideal MHD operational β limits under free boundary or ideal wall conditions are compared. In the stability calculation, n = 1 kink mode is considered, which has a decisive influence on the MHD instability of Tokamak plasma. The research focuses on verifying the effectiveness of various shapes of conductive walls in suppressing internal and external kink modes, and observing the operational β limit changes, and discussing and analyzing related physics. It is found that an ideal conducting wall placed at a suitable distance from the plasma can effectively suppress the external kink modes. Under the condition that the average distance between the wall and the plasma surface is the same and small enough, the circular cross-section wall is not necessarily the best option. Setting an optimized polygonal conductive wall can more effectively suppress the MHD instability. It makes the ideal MHD operational β limit of the device, βN, increase to 2.73, which is about 6.5% higher than that for the device with a wall assumed to be set at infinity (
$ \sim $ 2.56). This implies that it is necessary to optimize and make a polygonal conductive wall as close as possible to the average distance from the plasma surface according to the poloidal-section shape of the elongated and shaped plasma, so as to achieve the suppression of external kink mode and increase the operational β limits. The physical mechanism of the stabilizing effect of the ideal wall on external kink modes is analyzed. With the development of the kink mode, when the plasma column is twisted closely to the wall, the plasma column will squeeze the magnetic field in the vacuum area, making the magnetic field line compressed and bent. At this time, the magnetic pressure and the component force of the magnetic tension in the opposite direction of the radial direction push the plasma back, thus stabilizing the kink mode. Finally, a conclusion is given.-
Keywords:
- Tokamak /
- ideal conductive wall /
- magnetohydrodynamic instability /
- kink mode /
- operational $ \beta $ limit
[1] Ferron J R, Casper T A, Doyle E J, et al. 2005 Phys. Plasmas 12 056126
Google Scholar
[2] Holcomb C T, Ferron J R, Luce T C, et al. 2009 Phys. Plasmas 16 056116
Google Scholar
[3] Petty C C, Kinsey J E, Holcomb C T, et al. 2016 Nucl. Fusion 56 016016
Google Scholar
[4] Petty C C, Nazikian R, Park J M, et al. 2017 Nucl. Fusion 57 116057
Google Scholar
[5] Huysmans G T A, Hender T C, Alper B, Baranov Yu F, Borba D, Conway G D, Cottrell G A, Gormezano C, Helander P, Kwon O J, Nave M F F, Sips A C C, Söldner F X, Strait E J, Zwingmann W P, JET Team 1999 Nucl. Fusion 39 1489
Google Scholar
[6] ITER Physics Expert Group on Disruptions, Palsma Control, and MHD and ITER Physics Basis Editors 1999 Nucl. Fusion 39 2251
Google Scholar
[7] Phillips M W, Todd A M M, Hughes M H, Manickam J, Johnson J L, Parker R R 1988 Nucl. Fusion 28 1499
Google Scholar
[8] Kerner W, Gautier P, Lackner K, Schneider W, Gruber R, Troyon F 1981 Nucl. Fusion 21 1383
Google Scholar
[9] Park J M, Ferron J R, Holcomb C T, Buttery R J, Solomon W M, Batchelor D B, Elwasif W, Green D L, Kim K, Meneghini O, Murakami M, Snyder P B 2018 Phys. Plasmas 25 012506
Google Scholar
[10] Howl W, Turnbull A D, Taylor T S, Lao L L, Helton F J, Ferron J R, Strait E J 1992 Phys. Fluids B 4 1724
[11] Wesson J A, Sykes A 1985 Nucl. Fusion 25 85
Google Scholar
[12] Shen Y, Dong J Q, Peng X D Han M K, He H D, Li J Q 2022 Nucl. Fusion 62 106004
Google Scholar
[13] Yavorskij V, Goloborod'ko V, Schoepf K, Sharapov S E, Challis C D, Reznik S, Stork D 2003 Nucl. Fusion 43 1077
Google Scholar
[14] Taylor T S, Strait E J, Lao L, et al. 1989 Phys. Rev. Lett. 62 1278
Google Scholar
[15] Troyon F, Gruber R, Saurenmann H, Semenzato S, Succi S 1984 Plasma Phys. Controlled Fusion 26 209
Google Scholar
[16] Ferron J R, Chu M S, Helton F J, Howl W, Kellman A G, Lao L L, Lazarus E A, Lee J K, Osborne T H, Strait E J, Taylor T S, Turnbull A D 1990 Phys. Fluids B 2 1280
Google Scholar
[17] Taylor T S, St John H, Turnbull A D, Lin-Liu V R, Burrell K H, Chan V, Chu M S, Ferron J R, Lao L L, Haye R J La, Lazarus E A, Miller R L, Politzer P A, Schissel D P, Strait E J 1994 Plasma Phys. Control. Fusion 36 B229
Google Scholar
[18] Shen Y, Dong J Q, He H D, Shi Z B, Li J, Han M K, Li J Q, Sun A P, Pan L 2020 Nucl. Fusion 60 124001
Google Scholar
[19] 沈勇, 董家齐, 何宏达, 丁玄同, 石中兵, 季小全, 李佳, 韩明昆, 吴娜, 蒋敏, 王硕, 李继全, 许敏, 段旭如 2021 物理学报 70 185201
Google Scholar
Shen Y, Dong J Q, He H D, Ding X T, Shi Z B, Ji X Q, Li J, Han M K, Wu N, Jiang M, Wang S, Li J Q, Xu M, Duan X R 2021 Acta Phys. Sin. 70 185201
Google Scholar
[20] Garofalo A M, Doyle E J, Ferron J R, et al. 2006 Phys. Plasmas 13 056110
Google Scholar
[21] Turnbull A D, Lin-Liu Y R, Miller R L, Taylor T S, Todd T N 1999 Phys. Plasmas 6 1113
Google Scholar
[22] Bernard L C, Moore R W 1981 Phys. Rev. Lett. 46 1286
Google Scholar
[23] 胡希伟 2006 等离子体理论基础 (北京: 北京大学出版社) 第119—182页
Hu X W 2006 Fundamentals of Plasma Theory (Beijing: Peking University Press) pp119–182 (in Chinese)
[24] Liu Y Q, Bondeson A, Chu M S, Favez J Y, Gribov Y, Gryaznevich M, Hender T C, Howell D F, La Haye R J, Lister J B, de Vries P, EFDA JET Contributors 2005 Nucl. Fusion 45 1131
Google Scholar
[25] Chu M S, Ichiguchi K 2005 Nucl. Fusion 45 804
Google Scholar
[26] Hender T C, Gimblett C G, Robinson D C 1989 Nucl. Fusion 29 1279
Google Scholar
[27] Hao G Z, Liu Y Q, Wang A K, Qiu X M 2012 Phys. Plasmas 19 032507
Google Scholar
[28] Shen Y, Dong J Q, He H D, Turnbull A D 2009 Plasma Sci. Technol. 11 131
Google Scholar
[29] Lao L L, Ferron J R, Groebner R J, Howl W, John H St, Strait E J, Taylor T S 1990 Nucl. Fusion 30 1035
Google Scholar
[30] Lao L L, John H St, Stambaugh R D, Kellman A G, Pfeiffer W 1985 Nucl. Fusion 25 1611
Google Scholar
[31] Gruber R, Troyon F, Berger D, Bernard L C, Rousset S, Schreiber R, Schneider W, Roberts K V 1981 Comput. Phys. Commun. 21 323
Google Scholar
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图 1 (a) HL-2A托卡马克极向截面图, 包括等离子体分界面(红线), 限制器(黑线), HL-2A真空室壁(深蓝线), 以及拟设新导体壁(淡蓝线); 预设4种导体壁, 概略图包含(b)原壁, (c) 圆截面壁, (d)多边形壁, 以及(e)优化多边形壁
Figure 1. (a) Scheme of the poloidal cross-section of HL-2A Tokamak, including plasma separatrix (red line), limiter (black line), HL-2A vacuum chamber wall (dark blue line), and proposed new conductive wall (light blue line). Four kinds of conductive walls are preset, and the sketch is as follows: (b) Original wall, (c) circular section wall, (d) polygonal wall, and (e) optimized polygonal wall.
图 2
$ {q}_{0}=1.05 $ , (a)$ {\beta }_{\rm{p}}=0.45 $ 时与(b)$ {\beta }_{\rm{p}}=2.00 $ 时的平衡位形; (c)$ {\beta }_{\rm{p}}=2 $ 时, GATO计算的安全因子(q)剖面(由图2(a)与图2(b)对比得知,$ {\beta }_{\rm{p}} $ 越大, 等离子体Shafranov位移越大)Figure 2. When
$ {q}_{0}=1.05 $ , the equilibrium configuration of (a)$ {\beta }_{\rm{p}}=0.45 $ and (b)$ {\beta }_{\rm{p}}=2.00 $ , and (c) the safety factor (q) profile calculated by GATO at$ {\beta }_{\rm{p}}=2 $ (According to the comparison between Fig. 2(a) and Fig. 2(b), the larger the$ {\beta }_{\rm{p}} $ , the larger the plasma Shafranov displacement).
图 3
${q}_{0} = 0.95$ 时, 较低$ {\beta }_{\rm{p}} $ 与较高$ {\beta }_{\rm{p}} $ 时模的极向投影与对应模径向扰动X的傅里叶分解结果 (a), (b)${\beta }_{\rm{p}} = 0.45$ ; (c), (d)${\beta }_{\rm{p}} = 2.5$ Figure 3. When
$ {q}_{0}=0.95 $ , the poloidal projection of the mode and the Fourier decomposition of the radial perturbation of the corresponding mode$ \mathit{X} $ at lower$ {\beta }_{\rm{p}} $ and higher$ {\beta }_{\rm{p}} $ : (a), (b)$ {\beta }_{\rm{p}}=0.45 $ ; (c), (d)$ {\beta }_{\rm{p}}=2.5 $ .
图 4
$ {q}_{0}=1.05 $ , 自由边界条件下, 模位移矢量在极向平面的投影及扰动位移径向分量X的傅里叶分解结果图 (a), (b)${\beta }_{\rm{p}}= $ $ 0.45$ , q95 = 4.37,$ \lambda =-0.4492\times {10}^{-7} $ ; (c), (d)$ {\beta }_{\rm{p}}=2 $ , q95 = 5.35,$ \lambda =-0.1247\times {10}^{-6} $ ; (e), (f)$ {\beta }_{\rm{p}}=2.3$ , q95 = 5.67,$\lambda = $ $ -0.1198\times {10}^{-3}$ ; (g), (h)$ {\beta }_{\rm{p}}=2.5$ , q95 = 5.80,$ \lambda =-0.4774\times {10}^{-3} $ Figure 4. When
$ {q}_{0}=1.05 $ , the projection of the mode displacement vector on the poloidal plane and the radial component of Fourier decomposition of the perturbation$ \mathit{X} $ : (a), (b)$ {\beta }_{\rm{p}}=0.45$ , q95 = 4.37,$ \lambda =-0.4492\times {10}^{-7} $ ; (c), (d)$ {\beta }_{\rm{p}}=2$ , q95 = 5.35,$\lambda = $ $ -0.1247\times {10}^{-6}$ ; (e), (f)$ {\beta }_{\rm{p}}=2.3$ , q95 = 5.67,$ \lambda =-0.1198\times {10}^{-3} $ ; (g), (h)$ {\beta }_{\rm{p}}=2.5 $ , q95 = 5.80,$ \lambda =-0.4774\times {10}^{-3} $ .
图 5 (a) 优化理想导体壁条件下等离子体与壁的位置关系; (b)
${\beta }_{\rm{p}}=2.3,\; {q}_{95}=5.67$ ,$ \lambda =-0.1418\times {10}^{-5} $ 和(c)${\beta }_{\rm{p}}=2.5, $ $ {q}_{95}=5.8$ ,$ \lambda =-0.4985\times {10}^{-4} $ 时, 扰动位移在极向平面的投影; (d)${\beta }_{\rm{p}}=2.5,\;{ q}_{0}=1.05$ 情形下径向扰动X的傅里叶分解图Figure 5. (a) Under the condition of the ideal optimized conductive wall, the position relationship between plasma and wall; (b) the projection of perturbation displacement on the polar plane when
${\beta }_{\rm{p}}=2.3,\; {q}_{95}=5.67$ ,$ \lambda =-0.1418\times {10}^{-5} $ and (c)${\beta }_{\rm{p}}=2.5, $ $ {q}_{95}=5.8$ ,$ \lambda =-0.4985\times {10}^{-4} $ ; (d) Fourier decomposition diagram of radial perturbation X when$ {\beta }_{\rm{p}}=2.5 $ ,$ {q}_{0}=1.05 $ .
图 6 自由边界与各种壁条件下等离子体的几何位置及位形, 其中(a1) 自由边界位形, (b1) 原壁位形, (c1) 圆截面壁位形, (d1) 多边形壁位形, (e1) 优化壁位形; 在各种位形下,
$ {q}_{0}=0.95 $ (图(a2)—(e2))与1.05 (图(a3)—(e3)) 时模的极向截面投影和对应的模特征值$ \lambda $ , 其中(a2)$ \lambda =-0.2162\times {10}^{-2} $ ; (a3)$ \lambda =-0.4774\times {10}^{-3} $ ; (b2)$ \lambda =-0.2162\times {10}^{-2} $ ; (b3)$\lambda =-0.4315\times {10}^{-3}$ ; (c2)$ \lambda =-0.1846\times {10}^{-2} $ ; (c3)$ \lambda =-0.1991\times {10}^{-3} $ ; (d2)$ \lambda =-0.1761\times {10}^{-2} $ ; (d3)$ \lambda =-0.7246\times {10}^{-4} $ ; (e2)$\lambda = -0.1727\times $ $ {10}^{-2}$ ; (e3)$ \lambda =-0.4985\times {10}^{-4} $ Figure 6. Geometry and configuration of plasma under free boundary and various wall conditions: (a1) Free boundary configuration; (b1) original wall configuration; (c1) circular section wall configuration; (d1) polygonal wall configuration; (e1) optimized wall configuration. For each configuration, the poloidal projection of the mode is given respectively at
$ {q}_{0}=0.95 $ (panel (a2)—(e2)) and 1.05 (panel (a3)—(e3)), and the corresponding mode eigenvalues$ \lambda $ are given under the projection diagrams: (a2)$\lambda = $ $ -0.2162\times {10}^{-2}$ ; (a3)$ \lambda =-0.4774\times {10}^{-3} $ ; (b2)$ \lambda =-0.2162\times {10}^{-2} $ ; (b3)$ \lambda =-0.4315\times {10}^{-3} $ ; (c2)$ \lambda =-0.1846\times {10}^{-2} $ ; (c3)$\lambda = -0.1991\times $ $ {10}^{-3}$ ; (d2)$\lambda = -0.1761\times {10}^{-2}$ ; (d3)$\lambda = -0.7246\times {10}^{-4}$ ; (e2)$\lambda = -0.1727\times {10}^{-2}$ ; (e3)$\lambda = -0.4985\times {10}^{-4}$ 
图 7 模特征值绝对值
$ \left|\lambda \right| $ (归一化增长率)随$ {\beta }_{\rm{p}} $ 的变化关系图 (a) 自由边界位形条件; (b)优化壁条件Figure 7. Absolute value of mode eigenvalue (
$ \left|\lambda \right| $ ) variation with$ {\beta }_{\rm{p}} $ : (a) Free boundary configuration; (b) the condition of optimized wall.表 1
$ {q}_{0}=0.95 $ 时, 自由边界条件下以及各种形状理想壁条件下计算的模特征值Table 1. Eigenvalues calculated under free boundary and ideal wall conditions when
$ {q}_{0}=0.95 $ .βP $ \lambda $/10–3 自由边界 原理想壁 优化多边形壁 多边形壁 圆截面壁 0.10 –0.9944 –0.9944 –0.9944 — — 0.30 –0.8297 –0.8297 –0.8297 — — 0.45 –0.7713 –0.7713 –0.7713 — — 1.05 –0.9110 –0.9101 –0.9001 — — 2.00 –1.2690 –1.2410 –1.1840 –1.190 — 2.20 –1.4730 –1.4620 –1.1301 –1.352 — 2.50 –2.1620 –2.1260 –1.7270 –1.761 –1.846 3.00 –5.9170 –5.6920 –3.6110 — — 4.00 –21.930 –20.700 –9.7990 — — 
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表 2
$ {q}_{0}=1.05 $ 时, 各种条件下计算的模特征值Table 2. Eigenvalues calculated under different conditions when
$ {q}_{0}=1.05 $ .βP $ \lambda $/10–7 自由边界 原理想壁 优化多边形壁 多边形壁 圆截面壁 0.10 –0.4284 –0.4284 –0.4284 — — 0.45 –0.4492 –0.4492 –0.4492 — — 1.05 –0.4647 –0.4647 –0.4647 — — 1.50 –1.2230 –1.2230 –1.2200 — — 2.00 –1.2470 –1.2460 –1.2380 –1.239 1.240 2.20 –393.40 –313.70 –1.2010 –2.246 –1.258 2.30 –1198.0 –1054.0 –14.180 –60.220 –195.20 2.40 –2460.0 –2200.0 –103.00 –211.90 –546.0 2.50 –4774.0 –4315.0 –498.50 –724.60 –1391 3.00 –38850 –35570 –9185.0 –10780 –15430
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[1] Ferron J R, Casper T A, Doyle E J, et al. 2005 Phys. Plasmas 12 056126
Google Scholar
[2] Holcomb C T, Ferron J R, Luce T C, et al. 2009 Phys. Plasmas 16 056116
Google Scholar
[3] Petty C C, Kinsey J E, Holcomb C T, et al. 2016 Nucl. Fusion 56 016016
Google Scholar
[4] Petty C C, Nazikian R, Park J M, et al. 2017 Nucl. Fusion 57 116057
Google Scholar
[5] Huysmans G T A, Hender T C, Alper B, Baranov Yu F, Borba D, Conway G D, Cottrell G A, Gormezano C, Helander P, Kwon O J, Nave M F F, Sips A C C, Söldner F X, Strait E J, Zwingmann W P, JET Team 1999 Nucl. Fusion 39 1489
Google Scholar
[6] ITER Physics Expert Group on Disruptions, Palsma Control, and MHD and ITER Physics Basis Editors 1999 Nucl. Fusion 39 2251
Google Scholar
[7] Phillips M W, Todd A M M, Hughes M H, Manickam J, Johnson J L, Parker R R 1988 Nucl. Fusion 28 1499
Google Scholar
[8] Kerner W, Gautier P, Lackner K, Schneider W, Gruber R, Troyon F 1981 Nucl. Fusion 21 1383
Google Scholar
[9] Park J M, Ferron J R, Holcomb C T, Buttery R J, Solomon W M, Batchelor D B, Elwasif W, Green D L, Kim K, Meneghini O, Murakami M, Snyder P B 2018 Phys. Plasmas 25 012506
Google Scholar
[10] Howl W, Turnbull A D, Taylor T S, Lao L L, Helton F J, Ferron J R, Strait E J 1992 Phys. Fluids B 4 1724
[11] Wesson J A, Sykes A 1985 Nucl. Fusion 25 85
Google Scholar
[12] Shen Y, Dong J Q, Peng X D Han M K, He H D, Li J Q 2022 Nucl. Fusion 62 106004
Google Scholar
[13] Yavorskij V, Goloborod'ko V, Schoepf K, Sharapov S E, Challis C D, Reznik S, Stork D 2003 Nucl. Fusion 43 1077
Google Scholar
[14] Taylor T S, Strait E J, Lao L, et al. 1989 Phys. Rev. Lett. 62 1278
Google Scholar
[15] Troyon F, Gruber R, Saurenmann H, Semenzato S, Succi S 1984 Plasma Phys. Controlled Fusion 26 209
Google Scholar
[16] Ferron J R, Chu M S, Helton F J, Howl W, Kellman A G, Lao L L, Lazarus E A, Lee J K, Osborne T H, Strait E J, Taylor T S, Turnbull A D 1990 Phys. Fluids B 2 1280
Google Scholar
[17] Taylor T S, St John H, Turnbull A D, Lin-Liu V R, Burrell K H, Chan V, Chu M S, Ferron J R, Lao L L, Haye R J La, Lazarus E A, Miller R L, Politzer P A, Schissel D P, Strait E J 1994 Plasma Phys. Control. Fusion 36 B229
Google Scholar
[18] Shen Y, Dong J Q, He H D, Shi Z B, Li J, Han M K, Li J Q, Sun A P, Pan L 2020 Nucl. Fusion 60 124001
Google Scholar
[19] 沈勇, 董家齐, 何宏达, 丁玄同, 石中兵, 季小全, 李佳, 韩明昆, 吴娜, 蒋敏, 王硕, 李继全, 许敏, 段旭如 2021 物理学报 70 185201
Google Scholar
Shen Y, Dong J Q, He H D, Ding X T, Shi Z B, Ji X Q, Li J, Han M K, Wu N, Jiang M, Wang S, Li J Q, Xu M, Duan X R 2021 Acta Phys. Sin. 70 185201
Google Scholar
[20] Garofalo A M, Doyle E J, Ferron J R, et al. 2006 Phys. Plasmas 13 056110
Google Scholar
[21] Turnbull A D, Lin-Liu Y R, Miller R L, Taylor T S, Todd T N 1999 Phys. Plasmas 6 1113
Google Scholar
[22] Bernard L C, Moore R W 1981 Phys. Rev. Lett. 46 1286
Google Scholar
[23] 胡希伟 2006 等离子体理论基础 (北京: 北京大学出版社) 第119—182页
Hu X W 2006 Fundamentals of Plasma Theory (Beijing: Peking University Press) pp119–182 (in Chinese)
[24] Liu Y Q, Bondeson A, Chu M S, Favez J Y, Gribov Y, Gryaznevich M, Hender T C, Howell D F, La Haye R J, Lister J B, de Vries P, EFDA JET Contributors 2005 Nucl. Fusion 45 1131
Google Scholar
[25] Chu M S, Ichiguchi K 2005 Nucl. Fusion 45 804
Google Scholar
[26] Hender T C, Gimblett C G, Robinson D C 1989 Nucl. Fusion 29 1279
Google Scholar
[27] Hao G Z, Liu Y Q, Wang A K, Qiu X M 2012 Phys. Plasmas 19 032507
Google Scholar
[28] Shen Y, Dong J Q, He H D, Turnbull A D 2009 Plasma Sci. Technol. 11 131
Google Scholar
[29] Lao L L, Ferron J R, Groebner R J, Howl W, John H St, Strait E J, Taylor T S 1990 Nucl. Fusion 30 1035
Google Scholar
[30] Lao L L, John H St, Stambaugh R D, Kellman A G, Pfeiffer W 1985 Nucl. Fusion 25 1611
Google Scholar
[31] Gruber R, Troyon F, Berger D, Bernard L C, Rousset S, Schreiber R, Schneider W, Roberts K V 1981 Comput. Phys. Commun. 21 323
Google Scholar
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