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Circular cross-section plasma is the most basic form of tokamak plasma and the fundamental configuration for magnetic confinement fusion experiments. Based on the HL-2A limiter discharge experiments, the magnetohydrodynamic (MHD) equilibrium and MHD instability of circular cross-section tokamak plasmas are investigated in this work. The results show that when $ {q}_{0}=0.95 $, the internal kink mode of $ m/n=1/1 $ is always unstable. The increase in plasma $ \beta $ (the ratio of thermal pressure to magnetic pressure) can lead to the appearance of external kink modes. The combination of axial safety factor $ {q}_{0} $ and edge safety factor $ {q}_{{\mathrm{a}}} $ determines the equilibrium configuration of the plasma and also affects the MHD stability of the equilibrium, but its growth rate is also related to the size of $ \beta $. Under the condition of $ {q}_{{\mathrm{a}}} > 2 $ and $ {q}_{0} $ slightly greater than $ 1 $, the internal kink mode and surface kink mode can be easily stabilized. However the plasma becomes unstable again and the instability intensity increases as $ {q}_{0} $ continues to increase when $ {q}_{0} $ exceeds $ 1 $. As the poloidal specific pressure ($ {\beta }_{{\mathrm{p}}} $) increases, the MHD instability develops, the equilibrium configuration of MHD elongates laterally, and the Shafranov displacement increases, which in turn has the effect on suppressing instability. Calculations have shown that the maximum $ \beta $ value imposed by the ideal MHD mode in a plasma with free boundary in tokamak experiments is proportional to the normalized current $ {I}_{{\mathrm{N}}} $ ($ {I}_{{\mathrm{N}}}={I}_{{\mathrm{p}}}\left({\mathrm{M}}{\mathrm{A}}\right)/a\left({\mathrm{m}}\right){B}_{0}\left({\mathrm{T}}\right) $), and the maximum specific pressure $ \beta \left({\mathrm{m}}{\mathrm{a}}{\mathrm{x}}\right) $ is calibrated to be $ ~2.01{I}_{{\mathrm{N}}},{\mathrm{ }}{\mathrm{i}}. {\mathrm{e}}. $ $ \beta \left({\mathrm{m}}{\mathrm{a}}{\mathrm{x}}\right)~2.01{I}_{{\mathrm{N}}} $. The operational $ \beta $ limit of HL-2A circular cross-section plasma is approximately $ {\beta }_{{\mathrm{N}}}^{{\mathrm{c}}}\approx 2.0 $. Too high a value of $ {q}_{0} $ is not conducive to MHD stability and leads the $ \beta $ limit value to decrease. When $ {q}_{0}=1.3 $, we obtain a maximum value of $ {\beta }_{{\mathrm{N}}} $ of approximately $ 1.8 $. Finally, based on the existing circular cross-section plasma, some key factors affecting the operational $ \beta $ and the relationship between the achievable high $ \beta $ limit and the calculated ideal $ \beta $ limit are discussed.
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Keywords:
- circular cross-section plasma /
- magnetohydrodynamic equilibrium /
- magnetohydrodynamic instability /
- kink mode /
- $ \beta $ limit
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图 1 在4206次放电中, $ {q}_{0}=0.95 $, $ {\beta }_{{\mathrm{p}}}=0.8 $时, 装置与等离子体平衡位形(a)和等离子体平衡磁面结构(b), 以及不同$ {q}_{0} $与$ {\beta }_{{\mathrm{p}}} $下的(c)压强剖面、(d) $ q $剖面和(e), (f)电流密度($ {J}_{{\mathrm{m}}{\mathrm{i}}{\mathrm{d}}} $)剖面
Figure 1. (a) Equilibrium configuration constructed and (b) mapped flux surfaces for $ {q}_{0}=0.95 $ and $ {\beta }_{{\mathrm{p}}}=0.8 $ in HL-2A discharge #4206, and (c) pressure ($ p $) profile, (d) $ q $ profile and (e), (f) current density ($ {J}_{{\mathrm{m}}{\mathrm{i}}{\mathrm{d}}} $) profiles for different $ {q}_{0} $ and $ {\beta }_{{\mathrm{p}}} $.
图 2 在4206次放电中, $ {\beta }_{{\mathrm{p}}}=1.2 $时, 不同$ {q}_{0} $对应的$ n=1 $扭曲模扰动位移在极向截面的投影(a), (c)及模的傅里叶分解图(b), (d) (a), (b) $ {q}_{0}=0.95 $; (c), (d) $ {q}_{0}=1.05 $. 横轴$ \left\langle{\psi }\right\rangle $代表归一化磁通
Figure 2. At $ {\beta }_{{\mathrm{p}}}=1.2 $ for discharge #4206, the mode displacement vectors projected onto the poloidal plane (a), (c) and the Fourier decomposition of the normal displacement (b), (d): (a), (b) $ {q}_{0}=0.95 $; (c), (d) $ {q}_{0}=1.05 $. The horizontal axis $ \left\langle{\psi }\right\rangle $ represents the normalized magnetic flux.
图 3 在4044次放电中, $ {q}_{0}=0.95 $时, 不同$ {q}_{0} $对应的模扰动位移在极向截面的投影(a), (c)及模的傅里叶分解图(b), (d) (a), (b) $ {\beta }_{{\mathrm{p}}}=0.8 $; (c), (d) $ {\beta }_{{\mathrm{p}}}=1.8 $
Figure 3. Unstable kink mode for limiter discharge with $ {q}_{0}=0.95 $ for discharge #4044, the mode displacement vectors projected onto the poloidal plane (a), (c) and Fourier decomposition of the normal displacement (b), (d) $ : $ (a), (b) $ {\beta }_{{\mathrm{p}}}=0.8 $; (c), (d) $ {\beta }_{{\mathrm{p}}}=1.8 $
图 4 在4044次放电中, $ {q}_{0}=1.05 $时, 不同$ {q}_{0} $对应的模扰动位移在极向截面的投影及模的傅里叶分解图 (a), (b) $ {\beta }_{{\mathrm{p}}}=0.8 $; (c), (d) $ {\beta }_{{\mathrm{p}}}=1.8 $
Figure 4. For discharge #4044, mode displacement vectors projected onto the poloidal plane (a), (c) with $ {q}_{0}=1.05 $and Fourier decomposition of the normal displacement (b), (d) $ :\left({\mathrm{a}}\right), {\mathrm{ }}\left({\mathrm{b}}\right){\beta }_{{\mathrm{p}}}=0.8;{\mathrm{ }}\left({\mathrm{c}}\right), {\mathrm{ }}\left({\mathrm{d}}\right){\beta }_{{\mathrm{p}}}=1.8 $.
图 5 在4206次放电中, $ {\beta }_{{\mathrm{p}}}=2 $时, (a), (b) $ {q}_{0}=0.95 $, (c), (d) $ {q}_{0}=1.05 $和(e), (f) $ {q}_{0}=1.3 $对应的$ n=1 $扭曲模扰动位移在极向截面的投影及模的傅里叶分解图
Figure 5. At $ {\beta }_{{\mathrm{p}}}=2 $ for discharge #4206, the mode displacement vectors projected onto the poloidal plane for $ {q}_{0}=0.95 $ (a), $ 1.05 $ (c) and $ 1.3 $ (e), and Fourier decomposition of the normal displacement for $ {q}_{0}=0.95 $ (b), $ 1.05 $ (d) and $ 1.3 $ (f).
图 6 对典型的限制器类型放电 (a) 固定$ {q}_{0}=1.05 $时, $ 1/{q}_{{\mathrm{a}}} $-$ {\beta }^{*} $平面内的扭曲稳定性; (b)不固定$ {q}_{0} $时, 使$ n=1 $模稳定的最大$ {\beta }^{*} $和$ 1/{q}_{{\mathrm{a}}} $; (c)不固定$ {q}_{0} $时, 使$ n=2 $模稳定的最大$ {\beta }^{*} $和$ 1/{q}_{{\mathrm{a}}} $
Figure 6. For the typical limiter discharges: (a) Kink stabilities in $ 1/{q}_{{\mathrm{a}}} $-$ {\beta }^{*} $plane at fixed $ {q}_{0}=1.05 $; (b) $ {\beta }^{*} $ vs. $ {q}_{{\mathrm{a}}} $ at unfixed $ {q}_{0} $, here $ {\beta }^{*} $ is the maximum achievable one limited by $ n=1 $ kink; (c) $ {\beta }^{*} $ vs. $ {q}_{{\mathrm{a}}} $ at unfixed $ {q}_{0} $, here $ {\beta }^{*} $ is the maximum achievable one limited by $ n=2 $ kink. The solid lines with arrows indicate the change direction of $ {\beta }^{*} $ as $ {q}_{{\mathrm{a}}} $ increases.
图 7 基于4044次放电的计算结果 (a), (b)归一化增长率的平方值$ {\widehat{\gamma }}_{{\mathrm{N}}}^{2} $对$ {\beta }_{{\mathrm{p}}} $和$ {\beta }_{{\mathrm{N}}} $的依赖; (c)边缘最大扰动位移随归一化比压的变化. 注意图中, $ {\beta }_{{\mathrm{p}}} $是计算的平衡位形的实际极向比压. 菱形符号表示临界点
Figure 7. Calculations were based on the data of discharge #4044: (a), (b) Square value of normalized mode growth rate $ {\widehat{\gamma }}_{{\mathrm{N}}}^{2} $ as functions of $ {\beta }_{{\mathrm{p}}} $ and $ {\beta }_{{\mathrm{N}}} $; (c) maximum edge normal displacement as functions of normalized. Note that βp is the actual polar specific pressure of the calculated equilibrium configuration. The rhombus symbol represents the critical point
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