Numerical Simulation on Flow Generated Resistive Wall Mode

Shaoyan Cui (1,2), Xiaogang Wang (1), Yue Liu (1), Bo Yu (2)

1.State Key Laboratory of Materials Modification by Laser, Ion and Electron Beams, Department of Physics, and College of Advanced Science & Technology, Dalian University of Technology

2. Department of Applied Mathematics, Dalian University of Technology

Outline Introduction Slab geometry The model

Plasma region Inner and outer vacuum region The Wall

Linear expansion Initial and boundary conditions Numerical results Summary

J. A. Wesson, Phys. Plasma 5, 3816(1998) Magnetohydrodynamic flow instability in the presence of resistive w

all

C. N. Lashmore-Davies, J. A. Wesson, and C. G. Gimblett, Phys. Plasma 6, 3990(1999)

The effect of plasma flow ,compressibility, and Landau damping on resistive wall modes

B. M. Veeresha, S. N. Bhattacharyya, and K. Avinash, Phys. Plasma 6, 4479(1999)

Flow driven resistive wall instability

C.N. Lashmore-Davies, Phys. Plasmas 8,151 (2001) The resistive wall instability and critical flow velocity

Introduction

Introduction The uniform flow of an ideal magnetohydrodynamic fluid alo

ng a uniform magnetic field is stable for all velocities, unless there is a boundary.

For incompressible plasma, if v0 > sqr(2)VA the instability is then generated.

For compressible plasma, the resistive wall instability occurred if v0 > cs, for the finite layer width and compressibility of the plasma significantly lowers the flow velocity required for instability to set in.

With various parameters, the linear instabilities are studied in many kinds of instances in our work.

Slab geometry

J. A. Wesson, Phys. Plasma 5,3816(1998)J. A. Wesson, Phys. Plasma 5,3816(1998)

Magnetohydrodynamic flow instability in the presence of resistMagnetohydrodynamic flow instability in the presence of resistive wallive wall

Slab geometry

Physical Models (1) Plasma region The linearized MHD equations:

10 1 1 0

t

0v v

0 0 1 1 1 00

1p

t

v v B B

1 ( ) ( ) ( )t

0 1 0 1 0 1

BB v v B B v

,x z1 yB e

21 v

21 B

Physical Models （ 2 ） In the inner and outer vacuumThe linearized equation for magnetic flux i

s Laplace’s equation:

2 2

2 20

x z

Physical Models （ 3 ） In the resistive wall Magnetic flux satisfies diffusion equation

Assuming that the wall is thin compared to the skin depth, so that the current can be assumed to be approximately constant across the wall, by integrating it across the wall,

where

2

20

1

t x

( | | )w d dct x x

10

wc

Linear Expansion Perturbed variables

1 1 1cos sinc sf f kz f kz

1 1, , v

1x cv 1x sv 1z cv 1z sv

c s

1s1c

Initial conditions Plasma region

Vacuum and wall

( , ) cos sinc sx z kz kz 22 x h

c e 22 x hs e

1 0c 1 0s 1 0x cv 1 0x sv 1 0z cv 1 0z sv

0c 0s

Boundary conditions (1)

The lower boundary

The upper boundary

0x

0f

x

1 1 1 1 1 1 , , , , , , ,c s x c x s z c z s c sf v v v v

x h d a

0f

x

,c sf

Boundary conditions (2)

At the plasma-vacuum interface

the boundary condition satisfies the force balance across the surface:

x h

0 1 0 11 0 0

0 0

| |z zx h x h

B B B BP

1 0 0| 1 |2 x h x hx x

2

22 s

A

c

v

Numerical Results

4 33 10 , 5 10 0.02 0.001wc

Wave number kL/2 = 1, the growth rates versus the initial plasma flow velocities for cases with

,

0 0.07cv v

/ 2 2kL

4 33 10 , 5 10

0.001wc

Wave number kL/2 = 2, the growth rates versus the initial

plasma flow velocities for cases with

0.02 0.001wc

0 0.09cv v

/ 2 2kL

0.02

0.001wc

Wave number kL/2 = 3, the growth rates versus the initial plasma flow velocities for cases with

4 33 10 , 5 10 0.001wc

0 0.36cv v

(1) Unstable critical velocities vs. the wave numbers for cases with , and 4 33 10 , 5 10 0.02 0.001wc

4 33 10 , 5 10 0.02 0.001wc 0 1.0v

(2) Linear growth rates vs. the wave numbers for cases with

.

0.02 0.001wc / 2 2kL

(3) Unstable critical velocities vs. viscosity for cases with

43 10

43 10 , 0.02 0.001wc / 2 2kL 0 0.5v

(4) Linear growth rates vs. viscosity for cases with

4 33 10 , 5 10 0.001wc / 2 2kL 0 1.0v

(5) Linear growth rates vs. plasma beta for the cases with

4 33 10 , 5 10 0.02 / 2 2kL 0 1.0v

Linear growth rates vs. Cw for the cases with

Summary Fixed on some parameters, the critical velocitie

s are calculated for the different wave numbers; With the increase of wave numbers, the mode is

stabilized ; The unstable regions vs. wave number and visco

sity are presented. It is shown that the system tends to stable when the wave number and viscosity increase;

For the parameter beta, it is found that the linear growth rates increase as beta increases in realistic beta regime;

Cw has little effects on the linear growth rate

Thanks !!!