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In tokamak plasmas, the resistive wall mode is a very important magnetohydrodynamic instability, and its time scale is on the order of millisecond. For the advanced tokamaks with long-pulse and steady-state operation, the resistive wall mode limits the operating parameter space (the discharge time and the radio of the plasma pressure to the magnetic pressure) of the fusion devices so that it affects the economic benefits. Therefore, it is very important to study the stability of the resistive wall modes in tokamaks. In this work, the influences of the plasma rotations and the feedback controls on the resistive wall modes are studied numerically using MARS code for an ITER 9 MA equilibrium designed for the advanced steady-state scenario. In the equilibrium, the profile of the safety factor has a weak negative magnetic shear in the core region. The safety factor is
${q_0}= 2.44$ on the magnetic axis and${q_a}= 7.13$ on the plasma boundary. And, the minimum safety factor${q_{\min }}$ is 1.60. The structure of this kind of weakly negative magnetic shear can generate higher radio of the plasma pressure to the magnetic pressure and it is the important feature of the advanced steady-state scenario. Using MARS code, for two cases: without wall and with ideal wall, the results of growth rates of the external kink modes for different values of${\beta _{\rm N}}$ are obtained. The limit value of$\beta _{\rm N}^\text{no-wall}$ is 2.49 for the case without wall, and the limit value of$\beta _{\rm N}^\text{ideal-wall}$ is 3.48 for the case with ideal wall. Then, a parameter${C_\beta } = \left( {{\beta _{\rm{N}}} - \beta _{\rm{N}}^{{\text{no-wall}}}} \right)/\left( {\beta _{\rm{N}}^{{\text{ideal-wall }}} - \beta _{\rm{N}}^{{\text{no-wall }}}} \right)$ is defined. The research results in this work show that with the plasma pressure scaling factor${C_\beta } = 0.7$ and plasma rotation frequency${\Omega _{0}} = 1.1\% {\Omega _A}$ , the resistive wall modes can be completely stabilized without feedback control. And, with the plasma pressure scaling factor${C_\beta } = 0.7$ and the feedback gain$\left| G \right| = 0.6$ , only plasma rotation with the frequency${\Omega _{0}} = 0.2\% {\Omega _A}$ can stabilize the resistive wall modes. Therefore, a faster plasma rotation is required to stabilize the resistive wall modes by the plasma flow alone. The synergetic effects of the feedback and the toroidal plasma flow on the stability of the RWM can reduce plasma rotation threshold, which satisfies the requirements for the operation of the advanced tokamaks. The conclusion of this work has a certain reference for the engineering design and the operation of CFETR.-
Keywords:
- resistive wall mode /
- plasma flow /
- magnetic feedback
[1] Haney S W, Freidberg J P 1989 Phys. Plasmas 1 1637
[2] Matsunaga G, Takechi M, Kurita G, Ozeki T, Kamada Y, Team t J T 2007 Plasma Phys. Controlled Fusion 49 95
Google Scholar
[3] Gimblett C G 1986 Nucl. Fusion 26 617
Google Scholar
[4] Pfirsch D, Tasso H 1971 Nucl. Fusion 11 259
Google Scholar
[5] Liu Y, Bondeson A, Gribov Y, Polevoi A 2004 Nucl. Fusion 44 232
Google Scholar
[6] Xia G, Liu Y, Liu Y 2014 Plasma Phys. Controlled Fusion 56 095009
Google Scholar
[7] Xia G, Liu Y, Liu Y, Hao G Z, Li L 2015 Nucl. Fusion 55 093007
Google Scholar
[8] Liu C, Liu Y, Liu Y, Hao G, Li L, Wang Z 2015 Nucl. Fusion 55 063022
Google Scholar
[9] Hao G Z, Wang A K, Liu Y Q, Qiu X M 2011 Phys. Rev. Lett. 107 015001
Google Scholar
[10] Hao G Z, Liu Y Q, Wang A K, Jiang H B, Lu G M, He H D, Qiu X M 2011 Phys. Plasmas 18 032513
Google Scholar
[11] Hao G Z, Liu Y Q, Wang A K, Chen H T, Miao Y T, Wang S, Zhang N, Dong G Q, Xu M 2020 Plasma Phys. Controlled Fusion 62 075007
Google Scholar
[12] Wang S, Liu Y Q, Zheng G Y, Song X M, Hao G Z, Xia G L, Li L, Li B, Zhang N, Dong G Q, Bai X 2019 Nucl. Fusion 59 096021
Google Scholar
[13] Chu M S, Greene J M, Jensen T H, Miller R L, Bondeson A, Johnson R W, Mauel M E 1995 Phys. Plasmas 2 2236
Google Scholar
[14] Ward D J, Bondeson A 1995 Phys. Plasmas 2 1570
Google Scholar
[15] Garofalo A M, Jackson G L, La Haye R J, Okabayashi M, Reimerdes H, Strait E J, Ferron J R, Groebner R J, In Y, Lanctot M J, Matsunaga G, Navratil G A, Solomon W M, Takahashi H, Takechi M, Turnbull A D, Team D D 2007 Nucl. Fusion 47 1121
Google Scholar
[16] Liu Y Q, Bondeson A, Fransson C M, Lennartson B, Breitholtz C 2000 Phys. Plasmas 7 3681
Google Scholar
[17] Liu Y, Chu M S, Chapman I T, Hender T C 2008 Phys. Plasmas 15 112503
Google Scholar
[18] Wang Z R, Guo S C, Liu Y Q, Chu M S 2012 Nucl. Fusion 52 063001
Google Scholar
[19] Chu M S, Greene J M, Jensen T H, Miller R L, Bondeson A, Johnson R W, Mauel M E 1995 Physics Plasmas 2 2236
[20] Li L, Liu Y, Liu Y Q 2012 Phys. Plasmas 19 012502
Google Scholar
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图 1 ITER装置极向截面及反馈控制示意图
Figure 1. Geometry of poloidal cross section and feedback control in ITER.
图 2 ITER装置9 MA先进运行平衡 (a)安全因子剖面; (b)等离子体压强剖面, 由
${B_0}^2/{\mu _0}$ 归一化; (c)质量密度剖面, 磁轴处归一化为1; (d)环向电流密度剖面, 由${B_{0}}/\left( {{\mu _0}{R_0}} \right)$ 归一化,$s = \sqrt \psi $ ,$\psi $ 是归一化的极向通量Figure 2. Radial profiles of (a)safety factor; (b)equilibrium pressure normalized by factor
${B_0}^2/{\mu _0}$ ; (c)plasma density normalized to unity at the magnetic axis; (d)toroidal current density normalized by factor${B_{0}}/\left( {{\mu _0}{R_0}} \right)$ and$s = \sqrt \psi $ ,$\psi $ is the normalized poloidal flux for ITER target plasma designed for 9 MA steady state scenario.
图 3 在无壁和有理想壁时, 不同比压下的外扭曲模增长率
Figure 3. Growth rate of external kink versus
${\beta _{\rm{N}}}$ with and without an ideal wall.
图 4 等离子体比压参量为
${C_\beta } = 0.7$ 、平行黏滞$\kappa _{/\!/}$ 分别取0.5, 1和1.5时, 不同等离子体旋转频率下的电阻壁模增长率Figure 4. Growth rate of resistive wall mode versus plasma flow with equilibrium pressure scaling factor
${C_\beta } = 0.7$ , parallel viscous coefficient${\kappa _{/\!/} }$ = 0.5, 1, 1.5.
图 5 平行黏滞
${\kappa _{/\!/} }$ 取1.5时, 不同等离子体比压参数和不同等离子体旋转频率下的电阻壁模增长率Figure 5. With parallel viscous coefficient
${\kappa _{/\!/} }=1.5$ , growth rate of resistive wall mode versus plasma flow for different equilibrium pressure scaling factor${C_\beta }$ .
图 6 没有等离子体旋转频率,
${C_\beta }$ =0.7时, 计算得到的主动线圈产生的磁场分布Figure 6. Without plasma flow and equilibrium pressure scaling factor
${C_\beta } = 0.7$ , the calculated magnetic field distribution of active coil.
图 7 线圈增益幅值为
$|G|$ = 0.5时, 上下两组线圈不同相位下电阻壁模的增长率Figure 7. Growth rate of resistive wall mode with varying phase of feedback gains for upper and lower sets of active coils, feedback gain amplitude
$|G|$ = 0.5.
图 8 在没有等离子体旋转频率、平行黏滞
${\kappa _{/\!/} } =1.5$ 时, 不同的等离子体比压参量, 不同中间线圈的增益下电阻壁模的增长率变化Figure 8. Without plasma flow and with parallel viscous coefficient
${\kappa _{/\!/} }=1.5$ , growth rate of resistive wall mode with varying equilibrium pressure scaling factor versus feedback gains for middle sets of active coils.
图 9 在没有等离子体旋转频率、平行黏滞
${\kappa _{/\!/}} =1.5$ 时, 不同的等离子体比压参量, 不同上下两组线圈的增益下电阻壁模的增长率变化Figure 9. Without plasma flow and with parallel viscous coefficient
${\kappa _{/\!/}}=1.5$ , growth rate of resistive wall mode with varying equilibrium pressure scaling factor versus feedback gains for upper and lower sets of active coils.
图 10 在没有等离子体旋转频率、平行黏滞
${\kappa _{/\!/} } =1.5$ 时, 不同的等离子体比压参量, 不同上中下三组线圈的增益下电阻壁模的增长率变化Figure 10. Without plasma flow and with parallel viscous coefficient
${\kappa _{/\!/} }=1.5$ , growth rate of resistive wall mode with varying equilibrium pressure scaling factor versus feedback gains for all three sets of active coils.
图 11 在等离子体旋转频率
${\varOmega _{0}}/{\varOmega _{\rm{A}}}$ = 0.002、平行黏滞${\kappa _{/\!/} }=1.5$ 时, 不同的等离子体比压参量, 加上旋转后中间线圈的增益和增长率的变化Figure 11. With plasma flow
${\varOmega _{0}}/{\varOmega _{\rm{A}}}$ = 0.002 and parallel viscous coefficient${\kappa _{/\!/} }=1.5$ , growth rate of resistive wall mode with varying equilibrium pressure scaling factor versus feedback gains for middle sets of active coils.
图 12 在等离子体旋转频率
${\varOmega _{0}}/{\varOmega _{\rm{A}}}$ = 0.002、平行黏滞${\kappa _{/\!/} }=1.5$ 时, 不同的等离子体比压参量, 加上旋转后上下两组线圈的增益和增长率的变化Figure 12. With plasma flow
${\varOmega _{0}}/{\varOmega _{\rm{A}}}$ = 0.002 and parallel viscous coefficient${\kappa _{/\!/} }=1.5$ , growth rate of resistive wall mode with varying equilibrium pressure scaling factor versus feedback gains for upper and lower sets of active coils.
图 13 在等离子体旋转频率
${\varOmega _{0}}/{\varOmega _{\rm{A}}}$ = 0.002、平行黏滞${\kappa _{/\!/}}=1.5$ 时, 不同的等离子体比压参量, 加上旋转后上中下三组线圈的增益和增长率的变化Figure 13. With plasma flow
${\varOmega _{0}}/{\varOmega _{\rm{A}}}$ = 0.002 and parallel viscous coefficient${\kappa _{/\!/} }=1.5$ , growth rate of resistive wall mode with varying equilibrium pressure scaling factor versus feedback gains for all three sets of active coils. -
[1] Haney S W, Freidberg J P 1989 Phys. Plasmas 1 1637
[2] Matsunaga G, Takechi M, Kurita G, Ozeki T, Kamada Y, Team t J T 2007 Plasma Phys. Controlled Fusion 49 95
Google Scholar
[3] Gimblett C G 1986 Nucl. Fusion 26 617
Google Scholar
[4] Pfirsch D, Tasso H 1971 Nucl. Fusion 11 259
Google Scholar
[5] Liu Y, Bondeson A, Gribov Y, Polevoi A 2004 Nucl. Fusion 44 232
Google Scholar
[6] Xia G, Liu Y, Liu Y 2014 Plasma Phys. Controlled Fusion 56 095009
Google Scholar
[7] Xia G, Liu Y, Liu Y, Hao G Z, Li L 2015 Nucl. Fusion 55 093007
Google Scholar
[8] Liu C, Liu Y, Liu Y, Hao G, Li L, Wang Z 2015 Nucl. Fusion 55 063022
Google Scholar
[9] Hao G Z, Wang A K, Liu Y Q, Qiu X M 2011 Phys. Rev. Lett. 107 015001
Google Scholar
[10] Hao G Z, Liu Y Q, Wang A K, Jiang H B, Lu G M, He H D, Qiu X M 2011 Phys. Plasmas 18 032513
Google Scholar
[11] Hao G Z, Liu Y Q, Wang A K, Chen H T, Miao Y T, Wang S, Zhang N, Dong G Q, Xu M 2020 Plasma Phys. Controlled Fusion 62 075007
Google Scholar
[12] Wang S, Liu Y Q, Zheng G Y, Song X M, Hao G Z, Xia G L, Li L, Li B, Zhang N, Dong G Q, Bai X 2019 Nucl. Fusion 59 096021
Google Scholar
[13] Chu M S, Greene J M, Jensen T H, Miller R L, Bondeson A, Johnson R W, Mauel M E 1995 Phys. Plasmas 2 2236
Google Scholar
[14] Ward D J, Bondeson A 1995 Phys. Plasmas 2 1570
Google Scholar
[15] Garofalo A M, Jackson G L, La Haye R J, Okabayashi M, Reimerdes H, Strait E J, Ferron J R, Groebner R J, In Y, Lanctot M J, Matsunaga G, Navratil G A, Solomon W M, Takahashi H, Takechi M, Turnbull A D, Team D D 2007 Nucl. Fusion 47 1121
Google Scholar
[16] Liu Y Q, Bondeson A, Fransson C M, Lennartson B, Breitholtz C 2000 Phys. Plasmas 7 3681
Google Scholar
[17] Liu Y, Chu M S, Chapman I T, Hender T C 2008 Phys. Plasmas 15 112503
Google Scholar
[18] Wang Z R, Guo S C, Liu Y Q, Chu M S 2012 Nucl. Fusion 52 063001
Google Scholar
[19] Chu M S, Greene J M, Jensen T H, Miller R L, Bondeson A, Johnson R W, Mauel M E 1995 Physics Plasmas 2 2236
[20] Li L, Liu Y, Liu Y Q 2012 Phys. Plasmas 19 012502
Google Scholar
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