Numerical Study on Ideal MHD Stability and RWM in Tokamaks Speaker: Yue Liu Dalian University of Technology, China Co-Authors: Li Li, Xinyang Xu, Chao

Numerical Study on Ideal MHD Stability and RWM in Tokamaks Speaker: Yue Liu

Dalian University of Technology, China

Co-Authors:Li Li, Xinyang Xu, Chao Liu, Chang Liu, Xinnian Xia, Sihong Nian*, Shaoyan Cui**, Shu Zheng, Lulu Zhao, Yuling He, Guoliang Xia

*School of Mathematical Science, Dalian University of Technology, China**School of Mathematics and Information, Ludong University, China

2011-5-25 International West Lake Sympoium on Fusion Plasma Physics, May 25-27, 2011, Hangzhou, China 2

OUTLINE

Background and Motivation Model Effect of Current Profiles in Equilibrium on

Ideal MHD Modes Effect of Shear Flow on Evolutions of

Perturbations in Linearized MHD System with Resistive Wall

Summary


Background and Motivation From Troyon scaling law, the

maximum achievable value of beta in a tokamak is predicted to be a constant for all tokamaks.

Kessel C et al. Phys. Rev. Lett., 72, 1212(1994).

Advanced tokamak was proposed which promise a high bootstrap current fraction, yet still with high enough beta N, and also to realize stabilization of external kink mode with a perfect wall close to plasma region.


Background and Motivation

External kink mode is the most danderous MHD instability in tokamaks, and can be stabilized by a perfect wall.

In fact, there is finite resistivity in an actual wall, and magnetic perturbation can be diffused by it. Then, another instability, which is named RWM (Resistive Wall Mode), appears.

In 1989, Alper et al. founded a new instability in RFP where placed a thin shell around plasma, named it thin shell instability. In 1993, Greene et al. observed resistive wall mode in Reversatron (RFP), and then Taylor et al. Ⅱalso founded RWM in DIII-D (Tokamak) in 1995.

plasma

vacuum

wall

Kink Mode



Plasma flow can not be ignored in tokamaks as results of heating by neutral beam and other reasons, and it has effect on MHD modes and resistive wall mode.

It was found that resistive wall mode could be stabilized by plasma rotation. This stabilization mechanism was first discovered by Bondeson and Ward in their numerical study.

Then, Betii and Freidberg studied this mechanism in analytical way, and plasma velocity approximate to sound speed.

Recent DIII-D experiments suggested that resistive wall mode remain stable when plasma rotation is lower.

Bondeson A. et al. Phys. Rev Lett., 72, 2709 (1994).


Eriksson found that the location and width of stable window depend on current profile in equilibrium (Eriksson, PoP1995)


Current Profiles Distribution of Stable Windows


Background and Motivation There must be shear velocity in tokamak plasma, due to need

heating by neutral beam to enhance plasma characters. Shear flow has effect on many MHD modes, such as ballooning

mode, tearing mode and so on. It is found it is important that effect of shear flow profiles on stabilities of these modes. (F. L. Waelbroeck 1991, L. Ofman 1992.)

In 1995, Chu et al. studied effect of shear flow on resistive wall mode by MARS code, and found that shear flow could stabilized resistive wall mode.

Then, Gimblett et al. used analytic method to study effect of shear flow. They let shear flow just as a step function of radius.



Nowadays, there are many codes to study the resistive wall mode, such as MARS, AEGIS and so on.

MARS is a linear code by solving eigenvalue problem of MHD equations in tokamaks to study MHD stabilities and resistive wall mode with complex eigenvalues. (Bondeson 1994, Chu 1995.)

AEGIS (Y. Q. Liu et al. 2008) and AEGIS-K (L. Zheng et al. 2008) could study linear stability problem (including RWM).



What is mechanism of stabilization of resistive wall mode with lower plasma rotation?

What are effects of shear velocities both in poloidal and toroidal directions on resistive wall mode?

In this work, we use numerical method to solve an initial-value problem and obtain evolutions of all perturbations and growth rate of the system for analysis of MHD stabilities and resistive wall mode.


Model

Geometric Model

Plasma Region

Internal Vacuum Region

Wall Region

External Vacuum Region


Model Physical Model Plasma Region

Vacuum Region

Wall Region

Connection Condition on Interface between Plasma and Vacuum

11 0 0 1 0 1 1 0 0

t

V V V V

10 1 0 0 0 1 0 0 0 1

21 1 0 0 1 1

1t

p

V V V V V V V

B B B B V

211 0 0 1 1t

B V B V B B

10 1 1 0 0 1 1 0 0p p p p p

t

V V V V

1 10 0 B B，

211

0

1t

B B

0 1 0 11

0 0a a

p

B B B B


Model

Equilibrium

If given, from , can be obtained, , .

Then, from , can be obtained, .

0 0 ( )r

0 0 ( )V r V e

0 0 ( )p p r

0 0 ( )J r J e

0 0 0( ) ( )B r B r B e e

0 ( )J r 0 0 0 B J 0B

0 00

1( ) ( )r

B r J r drr 0 ( ) TB r B

0 0 0 0p J B 0p 0 0 0( ) ( ) ( )a

rp r J r B r dr


Effect of Current Profiles in Equilibrium on Ideal MHD Mode Equilibrium

0 0 ( )r

0 0 ( )V r V e

0 0 ( )J r J e

0 0 0( ) ( )B r B r B e e

0 0 ( )p p r

0 ( ) Cr 0 ( ) 0V r

2

0 2( ) 1CsrJ r Ja

2

02

0 20

1 02 2

( )

12 2

C

C

J r sr r aa

B rJ as r ar

0 ( ) TB r B

2 2 2 2 4

2 200 2 2 4

3( ) 1 1 14 4 6TJ s r s r rp r a r

a a a

02

20

20 0 0

2

01

2( )( )

( )

12

q r asrarB r

q rR B r q r r a

s a

00 0

2 T

C

BqJ R


Effect of Current Profiles in Equilibrium on Ideal MHD Mode Equilibrium


Effect of Current Profiles in Equilibrium on Ideal MHD Mode





1 1P 1VW W W 22

1P P0

12 1 2

p BW V d

r

2

1V V02

BW d

r




Effect of Shear Flow on Evolutions of Perturbations in Linearized MHD System with Resistive Wall Equilibrium

1s


Effect of Shear Flow on Evolutions of Perturbations in Linearized MHD System with Resistive Wall

1.0 AT



1.0 AT



10 AT



After , system enters linear growing stage.

The perturbation in the wall varies with plasma perturbation in the edge.

20 AT

15 A


Summary

Effect of current profiles in equilibrium on ideal MHD modes is studied numerically.

A MHD instability was found. But, we are not sure whether it is RWM.


Thank you for your attention.

Question and comment please.

Documents

Numerical Study on Ideal MHD Stability and RWM in Tokamaks Speaker: Yue Liu Dalian University of Technology, China Co-Authors: Li Li, Xinyang Xu, Chao