2004 年 9 月，成都

2004 年 9 月，成都

托卡马克等离子体的弛豫态分析托卡马克等离子体的弛豫态分析

2 0 0 4 2 0 0 4 年年 9 9 月月

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。

(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。

(2) 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。

(3) 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。

主要研究结果

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。

O U T L I N EO 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

引言引言 ____ 等离子体弛豫行为等离子体弛豫行为• 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。

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 tp, 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。

引言引言 ____ HICD

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 ?

引言引言 ____ 最小能量原理对 HICD 不适用

引言引言 ____ 三种变分原理应用于等离子体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 Tok

amak plasmas. • ----The Principle of Minimum rate of Energy Dissipation (Mont

egomery, 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).

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 flux Fig.1

Variation function

Variation: W = 0Euler-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。

dV

BV

dBdBjdjV

W

2

01 222

2

j

r

j

z

jrj

rrr

0122

2

j

rB

zB

rBr

rr

0112

2

j

zrrrr

由最小耗散原理出发得到 EULER – LAGRANGE 方程

Typical Current Profiles from E-L EquationTypical Current Profiles from E-L Equation

Fig.2 Typical mid-plane current profile for low ( <7.1) agree well with HIT experiment when =2.91

实验计算

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.

Compare of Calculated Parameters Compare of Calculated Parameters with Experiment on HITwith Experiment on HIT

Calculated Exp. R 0.300m 0.32m

a 0.178m 0.19m

A 1.68 1.68 k 1.82 1.85

0.41 0.62(up)0.91(low)

Itclosed 169.21 kA 171 kA

Itotal 238.51 kA 222kA

• 求解 E-L 方程，应用实验参数 Vinj=700V, ed=2.110-

2m, =3.810-6m, 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。

数值解与实验的符合以及多种形式解的发现

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

Three Typical Current Profiles(2)

0.1 0.2 0.3 0.4 0.5 0.60

2

4

6

8

10

Beta=8.1 Beta=7.25

j T(M

A*m

-2)

R(m)

Fig.3. Typical mid-plane j profile profile for =7.1- 9.65

ASIPP

Three Typical Current Profiles(3)

0.1 0.2 0.3 0.4 0.5 0.6

-30

-20

-10

0

10

20

BT(T

esla

)

BT

jT

j T(M

A*m

-2)

R(m)

-3

-2

-1

0

1

2

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

ASIPP

Different State (Different State () Achieved by Adjusting ) Achieved by Adjusting Plasma TemperaturePlasma Temperature

0 2 4 6 8 10 12

2

4

6

8

10

12

14

16

Temperature (0.1KeV)

Reversed field state

Fig.5 Different state () achieved by adjusting plasma temperature. There is the critical temperature value for mode transition to RFS.

ASIPP

Different State Different State (()) Achieved by Adjusting Achieved by Adjusting Bias Voltage VBias Voltage Vinjinj

400 600 800 1000 1200 14004

6

8

10

12

14

16

Vinj (V)

RF-STATE

Fig.6 Different state () achieved by adjusting bias voltage Vinj. There is the critical Vinj value for mode transition to RFS.

Different State (Different State () Achieved by Adjustin) Achieved by Adjustingg

vacuum toroidal magnetic field vacuum toroidal magnetic field BBVV

ASIPP

Fig.6 Different state () achieved by adjusting BV on r= R0. There is the critical Bv value for mode transition to RFS.

0.8 1.2 1.6 2.07

8

9

10

11

BV ( Tesla )

RF-state

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。

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

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:

V

djW 2

VV

djEdj

2

VV

dBEdBj

VVVVV

djEdjdBEdBjdjW

22 ~~

Variational Functional and the Variational Functional and the Euler_Lagrange Equation (2)Euler_Lagrange Equation (2)

• and are Lagrangian multiplies. Taking the first variation, we have:~ ~

S

V

dBEBj

dBEjjW

]~~

)~

1(2[

]~~

2)~

1(2[

• E-L equation and natural boundary condition are obtained if both the volume integral and surface integral are zero. • E-L equation

0~~

2)~

1(2 Ejj

Variational Functional and the Euler_LagrangVariational Functional and the Euler_Lagrange Equation (3)e Equation (3)

• Natural boundary condition

• Redefining and as and ， we obtain the equation and boundary condition as following:

• Equation

• Boundary condition

0]~~

)~

1(2[ nEBj

)~

1/(~ )

~1(2/

~

02

Ejj

0ˆ2

nEBj


• The cylindrical coordinates for Tokamak-axi-symmetric system

• The minor cross section is assumed a rectangle

• The plasma resistivity is assumed a homogeneous scalar

Fig. 1. Coordinate system

• 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 Bb = B1r1 / rb = B0 r0 /rb


• 得到柱坐标下的欧拉 _ 拉格朗日方程

02

1 22

22

2

Ej

r

j

z

j

r

jr

rr （ 1.1）

（ 1.2）0

2

122

2

Ej

r

B

z

B

r

Br

rr

(1.3)011

2

2

jzrrrr

2

1),( 00

00

rBrE

rzrjj

bbb

自然边条件

电流密度与磁场仍为亥姆霍兹方程，但又增加了非齐次项

• 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)

01 2

22

2

Yr

Y

z

Y

r

Yr

rr （ 3）

（ 4）

21 YYY

Analytical Solution of the Euler_Lagrange Equation for Plasma Current Density

r

rEzrYzrj

2),(),( 00

r

rarBrErE

rYb

0000000

),(

22

1

(2)

Analytical Analytical Solution of the Euler_LagrangeSolution of the Euler_LagrangeEquation for Plasma Current Density (2)Equation for Plasma Current Density (2)

• For ，

with um2 = 2-(k1

m)2 for 1m n, um2 = (k1

m)2 - 2 for m>n. In which k1

m = x1m/ r00 , is the mth zero point of Bessel funct

ion of order 1.

• Using the boundary condition of Y1 , we have

1mx

11

1 nn kk

1

11

1

11

1

))(sinh()sinh()sin(

)(

))(sin()sin()sin(

)(

nmmm

m

mm

n

mmm

m

mm

zhuhuhu

rkJa

zhuzuhu

rkJaY

)(/)()0,(2 012

22

00

111

0

rkJrdrkJrrYa m

r

mm

(7)

Analytical Analytical Solution of the Euler_LagrangeSolution of the Euler_LagrangeEquation for Plasma Current Density (3)Equation 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.

zvrkKdrkIc

zvrkNdrkJcY

nmn

nnnn

m

nnnnnn

sin

sin

111

1112

hnn / 1 mm

(8)

•The analytical solution for plasma current density is obtained:The analytical solution for plasma current density is obtained:

jj(r,z) = (r,z) = YY11(r,z, (r,z, , , ((,,)))) + Y + Y22(r,z, (r,z, , , ((,,))+))+ E E0 0 rr00/2/2rr •Two balance conditions are needed to determine Two balance conditions are needed to determine and and self- self-consistently. However, the process can not be accomplished using anaconsistently. However, the process can not be accomplished using analytical method.lytical method.

/),,(),(),,0(]/),([),( 0 zRFzRFBzRB

由 Euler-Lagrange equations 得到 B 的解析解

将方程 (2)代入 (1.2) ， B 方程化为

01

22

2

YR

B

z

B

R

BR

RR

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

• Equations are solved numerically employing Bunemanmethod. 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.

Numerical Self-Consistent Solutions of Whole Euler-Lagrange Equations

• The 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 as well as both helicity and energy balance equations are obtained numerically for a set of given parameters and boundary conditions.numerically for a set of given parameters and boundary conditions.

A global analysis for current profileA 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 / 2r

• 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 / 2r 的相对量值决定

Main Results (1)Main Results (1)

• 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.


Some forms of Some forms of YY on equatorial plane on equatorial plane

FIG.2 Some forms of Y on equatorial plane for NSTX- like.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

= 1.5

= 2.6

= 5.5 = 3.4

F(r)

R ( m )

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.

Main Results (3)Main Results (3)Different typical minimum dissipation state for low and

general aspect ratio tokamaksThe region between c1 and c2 is getting smaller as R0/a decreases,

and becomes a very narrow region for a low aspect tokamak.

Main Results (4)Main Results (4) Meanwhile, the numerical solutions show thatMeanwhile, 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 / 2r , the second part of j, is always dominantalways 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.


is achieved by adjusting controllable parameters such as plasma resistivity, boundary toroidal magnetic field or electric field.

Three forms of current profile are presented under different Three forms of current profile are presented under different experimental conditions for a low aspect ratio tokamakexperimental conditions for a low aspect ratio tokamak

The first similar with the typical experimental form peaks in the edge region of the high field side as shown in Fig.4.

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 otherpossible types

Each current profile mode

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

The typical form with < c2 for NSTX-like.

The current peaks in the edge region on the high field side .


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

The second form with a negative value and c2 < < c3

for NSTX-like. The current peaks in the central region.



The third form with a positive value and c2 < < c3

for NSTX-like. The current may have a hole as shown inFIG 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.


Plasma current profile with a hole or reversed in the central region for general tokamaks.

Fig.7. The current profile reversed in the central region for 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


• 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 obtained violently 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.

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

Experimental results on 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

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

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

B (

R)

(Te

sla

)

R (m)

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

实验表现了对理论预言的定性的支持实验 (shot100857 J.E. Menard)

由 Euler-Lagrange 方程解计算的 q-profiles 与实验结果很好相符

理论计算实验 (shot100857 J.E. Menard)

(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。

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。

结论与讨论

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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谢谢大家

2004 年 9 月，成都

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