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2004 年 9 月，成都

托卡马克等离子体的弛豫态分析. 2004 年 9 月，成都. 2 0 0 4 年 9 月. 摘 要 应用最小耗散原理，以磁螺旋平衡和能量平衡作为限制条件，研究任意纵横比托卡马克等离子体的弛豫态特性。 首先应用变分原理得到体系的欧拉 - 拉格朗日方程，然后解析求解以对等离子体弛豫态性质进行分析，并进一步应用数值方法，对给定的参数和边界条件，得到欧拉 - 拉格朗日方程组及磁螺旋平衡和能量平衡的自洽解以及一些重要的等离子体参数。 应用我们的理论结果，研究了 NSTX 实验中电流分布的突变现象。. 主要研究结果.

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2004 年 9 月，成都

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  1. 托卡马克等离子体的弛豫态分析 2004年9月，成都 2 0 0 4 年 9 月

  2. 摘 要 应用最小耗散原理，以磁螺旋平衡和能量平衡作为限制条件，研究任意纵横比托卡马克等离子体的弛豫态特性。 首先应用变分原理得到体系的欧拉-拉格朗日方程，然后解析求解以对等离子体弛豫态性质进行分析，并进一步应用数值方法，对给定的参数和边界条件，得到欧拉-拉格朗日方程组及磁螺旋平衡和能量平衡的自洽解以及一些重要的等离子体参数。 应用我们的理论结果，研究了NSTX实验中电流分布的突变现象。

  3. 主要研究结果 • (1)理论计算结果表明，球环与常规托卡马克具有不同形态的典型最小耗散态，其特征与各自的典型实验电流分布相符。 • 对所选定的装置几何，存在不同的参数区域，对应着不同类型的弛豫态，并有突变现象存在。着重研究了球环托卡马克的三类电流分布，第一类峰值在强场侧边界区，与典型实验分布符合；第二类峰值在中心区；第三类为中空型或反场型。后两类形态可由第一类形态突变得到；反之亦然。 • 各类电流分布模式可以通过调节等离子体电阻、纵场强度、边界电场等可控参数来实现。特别重要的研究结果是发现存在一个决定弛豫态模式的关键参数E0/B0，当此参数增大到高于其临界值时，等离子体将由典型实验电流分布突变到其他形态。相反的过程也会在此关键参数由高于到低于其临界值时出现。

  4. 分析NSTX电流分布及其突变现象 应用以上理论结果，研究了普林斯顿实验室的球环托卡马克NSTX电流分布及其突变现象，发现与理论预言一致的实验事实，包括： (a)上述第一类电流分布正是NSTX实验的典型电流分布。 (b)在一定的实验条件下观察到第二类形态。 (c)实验观察到第一类到第二类的突变现象，以及反过程。 (d)发生突变现象的实验条件与理论预言的关键参数一致。 (e)除电流剖面外，突变前后的磁场性质与安全因子剖面 与实验结果有可比性。 研究表明我们的理论结果得到NSTX实验的支持。

  5. O U T L I N E • 引言 • 应用变分原理得到体系的Euler_Lagrange 方程 • Euler_Lagrange方程 等离子体电流与环向磁场的解析解 及其分析 • Euler-Lagrange 方程的自恰解 • 主要理论结果与对NSTX弛豫态的分析 • 结论与讨论

  6. 引言 __等离子体弛豫行为 • 托卡马克等离子体是个复杂的非线性体系。实验表明在很多情况下，它将趋向于发展到一个 ‘self- consistent’ natural profile。而且在某些条件下会突变到另外的状态[1-3]. 这意味着托卡马克等离子体可能存在着某种弛豫机制。 • 弛豫理论研究的成功之例：泰勒应用最小能量原理研究理想情况下等离子体的完全的弛豫，J/B比值空间均匀，并成功地预言了Z－Pinch等离子体的关键性质。 • 某些物理研究必须考虑弛豫性质，比如 DC-HICD (helicity injection current drive)，其理论基础即是建立在等离子体弛豫理论上。

  7. 引言 __HICD Helicity作为反映磁场拓扑性质的一个量度，表示了磁通交连 的程度。托卡马克磁螺旋正比于tp, 因此所有的电流驱动机制 都必须形成并持续补充helicity 以补偿欧姆损失。 DC-HICD以直流电压来维持一个持续的螺旋注入，偏压线圈电流形 成的角向磁通贯穿于两个电极之间，电压加于电极，持续注入与 角向磁通交连的环向磁通，实现持续的螺旋注入。 基于湍动等离子体的弛豫性质，由最小能量原理，等离子体弛豫 过程将使J/B比值趋向于空间均匀化，因而在等离子体内部区域得 以产生并维持一个环向的驱动电流。即螺旋注入电流驱动的理论 分析是建立在等离子体弛豫理论基础上。

  8. 引言 __最小能量原理对HICD不适用 泰勒最小能量原理长期以来作为HICD的理论基础，由于磁螺旋注入及等离子体耗散都是必须考虑的因素，Taylor原理是显然不适用的。 H I T（Helicity Injected Tokamak， 美国华盛顿大学） 在螺旋注入电流驱动方面取得了世界领先的实验成果： 700V电压，纵场0.5T，实现200kA环向驱动电流.实验分析表明等离子体为偏离泰勒状态的非完全弛豫态。 需要回答的问题：现有的作用原理能否成功地运用于HICD ? 对螺旋注入这样一种富有吸引力的电流驱动手段如何由弛豫理论进行分析?如何确定与解释HICD等离子体弛豫结构?

  9. 引言__三种变分原理应用于等离子体 • ----The Minimum Magnetic Energy Principle (Taylor, 1974), widely employed in laboratory and astrophysics plasmas. It successfully predicted the features of RFP experiments. • ----The Principle of Minimum Entropy Production • ( Hameiri and Bhattacharjee, 1987), employed in Tokamak plasmas. • ----The Principle of Minimum rate of Energy Dissipation (Montegomery, et al, 1988 ), employed in description of RFP (Wang, et al, 1991)， helicity injection current drive (Farengo, et al, 1994 ), helicity injection current drive tokamak (Zhang, et al, 1998) and Ohmically driven tokamak (Farengo, Zhang, et al).

  10. 引言__最小耗散原理对HICD的应用HIT-HICD的物理模型建立引言__最小耗散原理对HICD的应用HIT-HICD的物理模型建立 • Low-aspect-ratio tokamak R=0.3 m, a=0.2 m • Square cross section, vertical height h=0.68 m • Toroidal field Bt=0.5T • A, B, C ,D form the container. A,C electrodes, B,D insulators, C consists of C1, C2 and C3. • Bias voltage Vinj applied between A and C. • A couple of vertical field coils (Iv) • Bias coils (Iex)produce initial poloidal fluxFig.1

  11. 由最小耗散原理出发得到EULER – LAGRANGE方程 Variation function Variation: W = 0 Euler-Lagrange equations in cylindrical coordinate 包括亥姆霍斯方程，拉普拉斯方程并涉及到非一类边条件处理的偏微分方程组。拉格朗日因子以磁螺旋注入率与耗散率之差为其目标函数，采用优化方法得到。

  12. Typical Current Profiles from E-L Equation 计算 实验 Fig.2 Typical mid-plane current profile for low ( <7.1) agree well with HIT experiment when  =2.91

  13. Poloidal Flux Contour of State Poloidal flux contour of state in Fig.2. The magnetic surface construction with R=0.3m, a=0.178m, k=1.82,A=1.68,(=0.41) are in good agreement with HIT experiment.

  14. Compare of Calculated Parameters with Experiment on HIT CalculatedExp. R 0.300m 0.32m a0.178m 0.19m A 1.68 1.68 k 1.82 1.85 0.41 0.62(up)0.91(low) Itclosed169.21 kA 171 kA Itotal238.51 kA 222kA

  15. 数值解与实验的符合以及多种形式解的发现 • 求解E-L方程，应用实验参数Vinj=700V, ed=2.110-2m, =3.810-6m, Iex=42kA and BV=0.5T ，在=2.91下得到与实验极好符合的解. • 实验放电条件体现在变分得到的欧拉方程及边界条件上，对参数空间的扫描发现存在不同的参数空间使欧拉方程的解具有不同的结构。这在理论和实验上都具有重要意义，可以探索实现具有高约束性质的等离子体位形。 • 由于多维参数空间扫描难以得到清昕的结论，需解析分析。电流密度及磁场方程为非齐次边条件的齐次和非齐次 亥姆霍兹方程，对其解析求解，并分析解的特性 。

  16. 解析解的重要发现：解的特性主要由拉格朗日因子决定，存在某些临界值，不同的区间的完全不同的电流分布模式，包括中心区电流反向的模式。解析解的重要发现：解的特性主要由拉格朗日因子决定，存在某些临界值，不同的区间的完全不同的电流分布模式，包括中心区电流反向的模式。 (1)For  <7.1 -- 当前HICD实验的典型模式的解 (2)For 7.1 < < 9.65 -- The typical form on general tokamak. The much larger driven current values than the first case are expected. Total driven current value 998.4kA and 1513.1kA for  =7.3 and =8.0 (3)For > 9.65 -- There exists the reversion of both j and B in the central part of plasmas. Their reversion points are quite close to each other when  is near c.中心区电流反向的模式的解

  17. ASIPP Three Typical Current Profiles(2) Fig.3. Typical mid-plane j profile profile for  =7.1- 9.65

  18. ASIPP Three Typical Current Profiles(3) Fig.4 The profiles of toroidal current density(solid line) and magnetic field (dash line) on mid-plane for  =10.5. There exists the reversion of both j and B in central part of plasmas

  19. ASIPP Different State () Achieved by Adjusting Plasma Temperature Reversed field state Fig.5 Different state () achieved by adjusting plasma temperature. There is the critical temperature value for mode transition to RFS.

  20. Different State () Achieved by Adjusting Bias Voltage Vinj RF-STATE Fig.6 Different state () achieved by adjusting bias voltage Vinj. There is the critical Vinj value for mode transition to RFS.

  21. ASIPP Different State () Achieved by Adjusting vacuum toroidal magnetic field BV RF-state Fig.6 Different state () achieved by adjusting BV on r= R0. There is the critical Bv value for mode transition to RFS.

  22. HICD弛豫态研究的小结 （1）本项研究是针对受控聚变研究领域中一种正处于探索性研究阶段极富有吸引力的电流驱动途径的基本原理及理论基础。这是首次将最小耗散原理应用于球环螺旋注入电流驱动TOKAMAK，与实验的符合表明了这一作用原理对耗散系统的成功运用。 （2）研究论证了螺旋注入驱动下等离子体弛豫态结构的特性主要由拉格朗日因子决定，发现了拉格朗日因子存在某些临界值及不同的区间的完全不同的电流分布模式。 （3）实验放电条件体现在变分得到的欧拉方程及边界条件上，研究发现了影响结构特性的敏感参数及其临界值，预言了由装置设计与放电可控参数的匹配，得到不同结构的电流分布以实现高约束模式的可能性。 （4）研究结果得到TS-3/4等实验中电流分布模式突变的新实验现象的验证13th IAEA Meeting(Y. Ono, 日本东京大学），通过降低纵场，实现了ST到compact RFP的突变。

  23. NSTX一个有趣的实验现象 (copy from Ref. J, Menard PPPL, APS_ DPP, 1999) • There exits two typical current profile modes: One peaks close to edge region of high field side and the other peaks in central region on equatorial plane. • There exits rapid transformation from the typical current profile to a central peak form

  24. 应用最小耗散原理，包括磁螺旋平衡和能量平衡作为限制条件，研究任意纵横比托卡马克等离子体的弛豫态特性应用最小耗散原理，包括磁螺旋平衡和能量平衡作为限制条件，研究任意纵横比托卡马克等离子体的弛豫态特性 • The total energy dissipation : • The magnetic helicity balance condition : • The energy balance condition : • We have the variational functional:

  25. Variational Functional and the Euler_Lagrange Equation (2) • and are Lagrangian multiplies. Taking the first variation, we have: • E-L equation and natural boundary condition are obtained • if both the volume integral and surface integral are zero. • E-L equation

  26. Variational Functional and the Euler_Lagrange Equation (3) • Natural boundary condition • Redefining and as and ，we obtain the equation and boundary condition as following: • Equation • Boundary condition

  27. Variational Functional and the Euler_Lagrange Equation (4) • The cylindrical coordinates for Tokamak-axi-symmetric system • The minor cross section is assumed a rectangle • The plasma resistivity is assumed a homogeneous scalar • For stationary plasma, it is reasonable to • assume that the applied electric field are • inversely proportional to the distance from the • symmetric axis: • E = E1r1 / r = E0 r0 /r • Toroidal magnetic field on the • boundaries is determined by TF • coils, it also can be expressed as • Bb = B1r1 / rb = B0 r0 /rb Fig. 1. Coordinate system

  28. Variational Functional and the Euler_Lagrange Equation (5) • 得到柱坐标下的欧拉_拉格朗日方程 （1.1） （1.2） (1.3) 电流密度与磁场仍为亥姆霍兹方程，但又增加了非齐次项 自然边条件

  29. Analytical SolutionoftheEuler_Lagrange Equation for Plasma Current Density (2) • Equation is homogenized when we write is as • Y satisfies the homogenous equation related to (1) as: • and boundary condition • Now we solve equation (3) under boundary condition (4). (Ref. C. Zhang et al. Nuclear Fusion, 2001) • We take Y as the sum of two parts (5) （3） （4）

  30. Analytical Solution of the Euler_LagrangeEquation for Plasma Current Density (2) • For ， with um2 = 2-(k1m)2 for 1m  n, um2 = (k1m)2 -2 for m>n. In which k1m = x1m/ r00 , is the mth zero point of Bessel function of order 1. • Using the boundary condition of Y1 , we have (7)

  31. Analytical Solution of the Euler_LagrangeEquation for Plasma Current Density (3) • For , ( ) • where J1, N1, I1 and K1 are respectively Bessel and modified Bessel functions. Coefficients cn and dn are obtained by applying the boundary conditions. (8) • The analytical solution for plasma current density is obtained: • j(r,z) = Y1(r,z, , (,)) + Y2(r,z, , (,))+ E0 r0/2r • Two balance conditions are needed to determine  and  self- • consistently. However, the process can not be accomplished using analytical method.

  32. 由Euler-Lagrange equations得到B的解析解 将方程(2)代入(1.2)， B方程化为 对Y的分析表明 Y=  (, ) F(, R, z) 于是有 第一项是相应的齐次方程的通解，可知是R的减函数； 第二项是方程的特解，可知与电流分布相关

  33. Numerical Self-Consistent Solutions of Whole Euler-Lagrange Equations • The self-consistent solutions of whole Euler-Lagrange equations • as well as both helicity and energy balance equations are obtained • numerically for a set of given parameters and boundary conditions. • Equations are solved numerically employing Buneman • method. Poloidal flux on the boundaries is considered to be a • constant, without lose of generality, set to be 0. Powell optimization • method is employed to search for the point satisfying both energy and helicity balance conditions in the (, ) space. • Numerical results are in good agreement with analytical ones.

  34. Main Results (1) A global analysis for current profile 对于给定的装置几何，等离子体电流密度分布为 j (r,z) = Y(r,z, , (,)) + E0 r0 / 2r • Y是E-L方程(亥姆霍兹型)相应的齐次方程的解，其形态主要由决定。(ref [11]，Zhang et.al, Nuclear Fusion, 2001) • (,)只决定Y 的量值 • 等离子体电流分布形态由Lagrange 因子, 以及Y and E0 r0 / 2r 的相对量值决定

  35. Main Results (2) Some forms of Y on equatorial plane • There exists some critical values c • Different forms of Y are obtained in different  ranges • The first form transfers smoothly to the second as  increases up • to >c1 • When  increases up to c2 • the distribution changes • violently, like a phase • transition,Y is reversed • in the central part. FIG.2 Some forms of Y on equatorialplane for NSTX- like.

  36. Main Results (3) Different typical minimum dissipation state for low and general aspect ratio tokamaks The region between c1 and c2 is getting smaller as R0/a decreases, and becomes a very narrow region for a low aspect tokamak. Fig. 3. The dependence of c1 and c2 on aspect ratio a) with fixed a = 0.67m, h = 2.7m. b) with fixed R0=0.85 m, h/a=4.

  37. Main Results (4) • Meanwhile, the numerical solutions show that • For a low aspect ratio tokamak, though Y may have a peak in the central part for c1<  <c2, but E0 r0 / 2r , the second part of j,is always dominant, therefore the total current on equatorial plane for  <c2 is always a decreasing function of r as shown in Fig.4. It is the typical minimum dissipation state on low aspect ratio tokamak and similar with the typical experimental result, where the current peaks in the edge region of the high field side [15]. • For a large aspect ratio tokamak, however, the second part of j is almost uniform, therefore we can obtain a typical current profile with an extremum in the central region for  < c2, which corresponds to the typical experimental form for general tokamak.

  38. Main Results (5) Three forms of current profile are presented under different experimental conditions for a low aspect ratio tokamak similar with the typical experimental form peaks in the edge region of the high field side as shown in Fig.4. The first could be transformed violently from the first when  increases to a value higher than c2 (c2 = 2.86 for NSTX-like). (Fig.5 and Fig.6) Two other possible types Each current profile mode is achieved by adjusting controllable parameters such as plasma resistivity, boundary toroidal magnetic field or electric field.

  39. Main Results (6) The typical form with < c2 for NSTX-like. The current peaks in the edge region on the high field side. FIG. 4. Toroidal current on equatorial plane (a) and in minor cross section (b) with the Parameters B0=0.29 T, E0 / =0.38MA/m2,  = 0.1MA/m2, =2.0m-1

  40. Main Results (7) The second form with a negative  value and c2 << c3 for NSTX-like. The current peaks in the central region. FIG. 5. Toroidal current on equatorial plane (a) and in minor cross section (b) with the Parameters B0=0.266 T, E0 / =1.694,  = - 0.371, =3.8

  41. Main Results (8) The third form with a positive  value and c2 << c3 for NSTX-like. The current may have a hole as shown in FIG 6 or reverse in the central part for other parameters. FIG.6. Toroidal current on equatorial plane (a) and in minor cross section (b). Parameters: B0= 0.266 T, E0 / = 1.8,  = 0.212, =4.85.

  42. Main Results (9) Plasma current profile with a hole or reversed in the central region for general tokamaks. Fig.7. The current profile reversed in the central regionfor JT-60U dimensions (R0=3.4 m, a =1.2m , h = 4.6m), with parameters  = 2.3 m-1, B0=3.51 T, E0/ =11.61,  = 1.71

  43. Main Results (10) We found there exits a key parameter in determining the final relaxed state. It is the boundary parameter (E/B)b, or E0/(B0) for our model • Numerical results show that only when E0/(B0) is larger than a • critical value, E0/(B0)~5.8m-1 for NSTX-like, can we obtain • solutions with  larger than critical value c2. • Both the second and the third types could be obtainedviolently • by increasing E0/(B0) to be above its critical value. • The rapid transformation from the typical current profile to a central • peak form has been observed in the experiment with a high loop • voltage on NSTX [15], which seems to agree with our results.

  44. Experimental results on NSTX (copy from Ref. J, Menard PPPL, APS_ DPP, 1999) • There exits two typical current profile modes: One peaks close to edge region of high field side and the other peaks in central region on equatorial plane. • There exits rapid transformation from the typical current profile to a central peak form

  45. NSTX 实验中等离子体电流分布的突变现象 shot100857 Jonathan E. Menard (PPPL)提供

  46. B解析解的第一项是R的减函数； 第二项与电流分布相关。 对于第一类电流分布，B(R) 应是R的减函数 对于第二类电流分布， B(R) 应将在中心区抬高 实验 (shot100857 J.E. Menard) 实验表现了对理论预言的定性的支持

  47. 由Euler-Lagrange 方程解计算的 q-profiles 与实验结果很好相符 实验(shot100857 J.E. Menard) 理论计算

  48. 结论与讨论 (1)理论计算结果表明，球环与常规托卡马克具有不同形态的典型最小耗散态，其特征与各自的典型实验电流分布相符。而对所选定的装置几何，存在不同的参数区域，对应着不同类型的弛豫态。各类电流分布模式可以通过调节等离子体电阻、纵场强度、边界电场等可控参数来实现。特别重要的研究结果是发现存在一个决定弛豫态模式的关键参数E0/B0，当此参数增大到高于其临界值时，等离子体将由典型实验电流分布突变到其他形态。相反的过程也会在此关键参数由高于到低于其临界值时出现。 (2)对于普林斯顿实验室的球环托卡马克NSTX.进行具体计算，发现其存在三类电流分布，第一类峰值在强场侧边界区，与典型实验分布符合；第二类峰值在中心区；第三类为中空型或反场型。 (3) 后两类形态可由第一类形态突变得到；反之亦然。可以预期，特别是E0/(B0)运行在临界值附近, 等离子体电流分布可能会由MHD 扰动等诱发快速改变，如同NSTX实验所示。

  49. 结论与讨论 (4)应用这一理论结果，研究了NSTX实验中电流分布及其突变 现象，发现与理论预言一致的实验事实，包括： (a)上述第一类电流分布正是NSTX实验的典型电流分布。 (b)第二类形态也在一定的实验条件下观察到。 (c)实验观察到第一类到第二类的突变现象，以及反过程。 (d)发生突变现象的实验条件与理论预言的关键参数一致。 (e)除电流剖面外，突变前后的磁场性质与安全因子剖面 与实验结果有可比性。 研究表明我们的理论结果得到NSTX实验的支持。

  50. 等离子体弛豫性质的研究可以帮助人们认识系统的等离子体弛豫性质的研究可以帮助人们认识系统的 GLOBAL STRUCTURE

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