Effects of plasma toroidal rotation on tearing mode instability in

TokamakS. Wang, Z.W. Ma and W. Zhang

IFTS, Zhejiang University

Hefei , 2015.3.23

Outline• Introduction to CLT code

– Model and Equations– Numerical scheme and boundary handling

• Benchmark results– Internal kink mode(m/n=1/1)– Resistive kink mode(m/n=1/1)– Resistive tearing mode(m/n=2/1)

• Resistive tearing mode with toroidal rotation– Effect of rotation (uniform)– Effect of rotation shear

• Summary and discussion

CLT code

• 利用我们早期开发的直柱位的三维MHD程序，开发一个柱坐标系（ R， φ， Z）下的三维环形MHD程序，用于模拟托卡马克中等离子体的演化，研究等离子体的不稳定性等。

• 在柱坐标系（ R， φ， Z）下，方程显得简单直接，并自动包含环形效应；相对于环坐标系（ ψ， θ， φ），它可以避免中心零点奇异性的处理；并且可以设计各种截面形状的位形。

• 难点在于边界条件处理比较麻烦。

R

Z

3 3.1 3.2 3.33.4 3.5 3.63.7 3.8 3.9 4 4.1 4.2 4.34.4 4.5 4.64.7 4.8 4.9 55-1

-0.9-0.8

-0.7-0.6-0.5

-0.4-0.3-0.2

-0.10

0.1

0.20.30.4

0.50.60.7

0.80.9

11

Model and Equations

0

0

0

0[ ( )]( )

( )

[ ( )

[ ( )]/

]

( )

tp

p pt

p p

pt

t

D

v

v v

vv v J B

BE

E v B

J

κ

J J

v v

B

2 1 11

2 1 11

1 1

1( ) ( )

1

(

( ) ( )

)

( )

R R ZR Z

R R ZR Z

R R RR Z

R Z

R Z

D D DR Z

p pp

vv v vv v v

t R R R R Z Z

vv v vp p p pv p p v p v p

t R R R R Z Z

vv v vv

t R R

Z

v

R

11 11 0 1 1 0 1

21 1 0 0 00 0 0 0 0 01

12 1 1 11 2 2

1 1 0 0

1 1

1

2( )

[ ]

( )

R R RR R

RZ Z Z Z

R R R R R RR Z R

R

Z

Z

v vv pJ B J B J B J B

Z R R

v v v v vv v v v v vv v v v

R R Z R R R Z R

v v v vv

vv v vv

R Z

t R R

R R

1 1 11 0 1 1 0 1

0 1 0 0 1 0 0 0 0 0 0 011 1 0 0

1

1 1 12 11 2 2

2

1 1[ ]

(

)

)

(

RZ Z R Z R R Z R Z

R RR

RR Z

Z R Z

Z ZR

v v v pv J B

v v vvv

J B J B J BZ R R

v v v v v v v v v v v vv v v v

R R Z R R R Z R

vv vv

t

R

R

R R Z

1 1 11 0 1 1 0 1

1 00 0 0 0 0 011 1

2

0 0

1 11

1[ ]

( )

( ) Z ZZ R

Z ZZ R R R R

Z Z Z Z Z ZR Z R Z

Z

v v pv J B J B J B J B

R Z Z

v vv v v v v vv v

v

v vR

vv

R

R Z R Z

Z

R

Detail in (R, φ, Z) components

1 0f f f

here use

0 0 0 0 0 0p u u J B

1

1

1

1 1 0 1 1 0 1 1

1 1 0 1 1 0 1 1

1 1 0 1 1

1

1

1

0 1 1

1

11

1 1

1

1 1

1

R

R Z Z Z Z R

R Z

Z R Z R R Z R Z

Z R R R R Z

R

R

Z

Z R

Z

EB E

t R Z

B E E

t Z RREB E

t R R

E v B v B B B

E v B v B B

R

v v J

v B

E B B v B v B

BB

R Z

B

v J

v v J

J

J

1

111

1 1

Z

RZ

B

Z RR

JR

B B

R R

0 0 E

0 Bkeep

Numerical scheme

• 空间上，目前主要采用有限中心差分的方法，在φ方向也可以选用 pseudospectral方法。

, , ,( )here e kini j i j k

k

f n f

, , 1, , 1, ,

, , , 1, , 1,

, , , , 1 , , 1

, ,,

1[ ( )e ]

2

i j k i j k i j k

i j k i j k i j k

i j k i j k i j k

i j k ini j

n

f f f

R dRf f f

Z dZf f f

d

finf n

或

2, , 1, , , , 1, ,

2 2

2, , , 1, , , , 1,

2 2

2, , , , 1 , , , , 1

2 2

2, , 2

,2

2

( )

2

( )

2

( )

1[ ( ) e ]

2

i j k i j k i j k i j k

i j k i j k i j k i j k

i j k i j k i j k i j k

i j k ini j

n

f f f f

R dR

f f f f

Z dZ

f f f f

d

fn f n

• 时间推动上，采用四阶 Runge-Kutta方法。

boundary handling• 沿 R、 Z划分方网格，实际的物理边界往往不能落在网格点上；此外，在 R、Z坐标下，边界条件也不容易直接给出。这些都对程序在边界附近格点上的值的计算带来了困难。

• 目前这里采用一种插值外推的方法： 1.先将内部网格上的值插值到里面的几圈磁面上，并沿磁面作平滑

2.从里面磁面上的值外推到边界磁面上

3.用这几个磁面上的值插值到最外圈有效格点（ Rb， Zb）上

( , , ) ( , , )f R Z fs

0 or 0 ( , , )a a

afs fs fs

( , , ) ( , , )b bfs f R Z

Rho,p,v,Bboundary Derivative

ofRho,p,v,B

J,E

Right hands of Equations

Step on

Smoothing?

initial

diagn.&output

Process diagram

Simulation results

• Benchmark• Internal kink mode(m/n=1/1)• Resistive kink mode(m/n=1/1)• Resistive tearing mode(m/n=2/1)

• Resistive tearing mode with toroidal rotation• Effect of rotation (uniform)• Effect of rotation shear

1. Internal kink mode(m/n=1/1)

3 3.5 4 4.5 5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

R/a

v

0(z

=0) (

a.u.

)

NOVACLTC

t=419.53ta

a=1R0=4 ρ0=1η0 =0

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

r/a

q

P

0 0.05 0.1 0.150

0.005

0.01

0.015

0.02

0

Lin

ear

gro

wth

rate

NOVACLTC

v0

R/a

Z/a

3 3.5 4 4.5 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1

-0.5

0

0.5

1

1.5

x 10-7

2. Resistive kink mode(m/n=1/1)

η0=1x10-5

3. Resistive tearing mode(m/n=2/1)

η0=1x10-5

Effects of plasma toroidal rotation on tearing mode instability

• Reduced plasma rotation is experimentally found to destabilize NTMs at lower beta in DIII-D, NSTX, and JET…

R. J. La Haye et al, PoP2010 S.P.Gerhardt et al, NF2009

NSTX

• There are a lot of work having been done to the investigatin of the influence of shear flows on tearing modes. But past analytic and numerical studies were mostly carried out in slab or cylindrical (large aspect ratio) geometries with a purely poloidal flow or a helical flow.

– Paris, PoF1983– Chen and Morrison, PoFB1990,1992– Wessen and Persson, JPP1991– Ofman, Chen et al., PoFB1991– Ofman, Morrison et al., PoFB1993– Chandra, Sen et al., NF2007…

• In general, sheared plasma flows can either increase or decrease the instability growth rate, depending on magnetic equilibrium, magnitude of the shear in the flow and plasma viscosity .

• In tokamaks the observed equilibrium flows are primarily toroidal since the poloidal flow experiences a strong neoclassical damping.

• R. Coelho and E. Lazzaro (PoP2007) has found that toroidal shear flow reduces the growth rate for viscous plasmas(Γ=τR/τV>>1), but has a destabilizing effect for low viscosity plasmas (Γ=τR/τV<<1) in a cylindrical geometry by means of numerical MHD simulations.

Γ=0.01Γ=100

• D. Chandra, A. Sen et. al (NF2005) has found that differential toroidal rotation between rational (q=m/n) magnetic surfaces without shear is stabilizing to tearing modes, while toroidal velocity shear at the resonant surface is shown to be destabilizing in toroidal geometry by using NEAR code.

1

2

0

• A. Sen, D. Chandra et. al (NF2013) has derived a flow modified external kink equation for a single helicity mode in a toroidal geometry to find the corrections to Δ' arising from toroidal shear flow contributions. In their results, toroidal shear flow is also seen to make a destabilizing contribution to the tearing mode.

Simulation using CLT

• Equilibrium equation with toroidal flow:

• First, we consider a uniform toroidal rotation, i.e. Ω=const.• If ρ=const. too , , equilibrium with uniform toroidal rotation can keep

the same as static, besides pressure modified as:

• Here, we choose P(ψ)~0,

Ω=0~0.0074(VA/a),

R0=4, a=1,

η0=1x10-5 , ν0=1x10-6

(Γ=τR/τV=0.1: low viscosity)

RP d P

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

r/a

q

• q-profile

* 2 202

2

1

| , ( )

R

R

dg PR g R

R d

P R

2 21( ) ( ) ( )

2P P R

( )

2 21( ) ( )

2P P P P R

Tearing mode with uniform rotation

η0=1x10-5 , ν0=1x10-6 , Ω=0.00617 (Ω’=0)

Effect of rotation (uniform)

• Linear growth rate decreases with increasing rotation!• Usually, in cylindrical geometry, a uniform equilibrium toroidal plasma

flow merely provides a propagating frequency to the tearing mode without affecting its growth rate.

0 1 2 3 4 5 6 7 8

x 10-3

0

1

2

3

4

5

6x 10

-3

toroidal rotation frequency at q=2 surface 2

a

linea

r gr

owth

rat

e

a

CLT

Effect of rotation (uniform)

Effect of rotation (uniform)

Effect of rotation shear• Equilibrium equation with toroidal flow:

η0=1x10-5 , ν0=1x10-6 ( ) 0P

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

r=sqrt(norm

)

/

2

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

qr2=r

q=2=0.4742

profile1:2'='(r

2)=0

profile2:2'='(r

2)=-1.2

2

profile3:2'='(r

2)=-3.6

2

q=2r2=r

q=2=0.4742

• q-profile and rotation profiles used in simulation

• Simulation result:linear growth rate vs. rotation frequency

0 1 2 3 4 5 6 7 8

x 10-3

0

1

2

3

4

5

6x 10

-3

toroidal rotation frequency at q=2 surface 2

a

linea

r gr

owth

rat

e

a

sqrt(02-

22)

profile1:2'/

2=0

profile2:2'/

2=-1.2

profile3:2'/

2=-3.6

2/ ( ) 2.5q q r

(Γ=τR/τV=0.1: low viscosity) • Growth rate increases with increasing flow shear, when rotation become strong.

* 2 202

2

1

| , ( )

R

R

dg PR g R

R d

P R

2 21( ) ( ) ( )

2P P R

( )

E =0 t=637.59

R

Z

3.5 4 4.5

-0.5

0

0.5

-4

-3

-2

-1

0

1

2

3

x 10-6E

=0 t=637.6

RZ

3.5 4 4.5

-0.5

0

0.5

-4

-3

-2

-1

0

1

2

3

x 10-6E

=0 t=637.57

R

Z

3.5 4 4.5

-0.5

0

0.5

-4

-3

-2

-1

0

1

2

3

4x 10

-6

200 400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

-5

time/a

Em

ax

2'/

2=0

2'/

2=-1.2

2'/

2=-3.6

2=0.00247

2 0.00247

Ω2’/Ω2=0 Ω2’/Ω2=-1.2 Ω2’/Ω2=-3.6

Mode structure

E =0 t=1275.3

RZ

3.5 4 4.5

-0.5

0

0.5

-1

-0.5

0

0.5

1

x 10-6E

=0 t=1275.4

R

Z

3.5 4 4.5

-0.5

0

0.5

-8

-6

-4

-2

0

2

4

6

x 10-7E

=0 t=1275.1

R

Z

3.5 4 4.5

-0.5

0

0.5

-8

-6

-4

-2

0

2

4

6

8x 10

-7

500 1000 1500 2000 25000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

-5

time/a

Em

ax

2'/

2=0

2'/

2=-1.2

2'/

2=-3.6

2=0.00617

2 0.00617

Ω2’/Ω2=0 Ω2’/Ω2=-1.2 Ω2’/Ω2=-3.6

E =0 t=531.32

R

Z

3.5 4 4.5

-0.5

0

0.5

-3

-2

-1

0

1

2

3

x 10-6

Ω=0

J =0 t=0

R

Z

3.5 4 4.5

-0.5

0

0.5

0.05

0.1

0.15

0.2

0.25

0.3

J =0 t=1062.6

R

Z

3.5 4 4.5

-0.5

0

0.5

0.05

0.1

0.15

0.2

0.25

0.3

J =0 t=743.85

R

Z

3.5 4 4.5

-0.5

0

0.5

0.05

0.1

0.15

0.2

0.25

0.3

J =0 t=0

R

Z

3.5 4 4.5

-0.5

0

0.5

0.05

0.1

0.15

0.2

0.25

0.3

J =0 t=0

R

Z

3.5 4 4.5

-0.5

0

0.5

0.05

0.1

0.15

0.2

0.25

0.3

J =0 t=850.13

R

Z

3.5 4 4.5

-0.5

0

0.5

0.05

0.1

0.15

0.2

0.25

0.3

J =0 t=1168.9

R

Z

3.5 4 4.5

-0.5

0

0.5

0.05

0.1

0.15

0.2

0.25

0.3

J =0 t=2125.5

R

Z

3.5 4 4.5

-0.5

0

0.5

0.05

0.1

0.15

0.2

0.25

0.3

J =0 t=2656.8

R

Z

3.5 4 4.5

-0.5

0

0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

2 2

2

/ 3.6

0.00617

0

2 2

2

/ 3.6

0.00247

Tearing mode with nonuniform rotation

η0=1x10-5 , ν0=1x10-6 , Ω2=0.00617, Ω2’/Ω2=-3.6

Rotation shear (Ω2’/Ω2) Linear growth rate γτa

0 0.000367-1.2 0.000350-3.6 0.000336

Cases for high viscosity

• Linear growth rate decreases when rotation shear increases for high viscosity, agreeing with (Coelho and Lazzaro, PoP2007)

• Ω2=0.00617, η0=1x10-6 , ν0=1x10-4 (Γ=τR/τV=100)

E =0 t=3400.8

R

Z

3.5 4 4.5

-0.5

0

0.5

-5

0

5x 10

-9E =0 t=3401.1

R

Z

3.5 4 4.5

-0.5

0

0.5

-4

-3

-2

-1

0

1

2

3

4

x 10-9E

=0 t=3400.2

R

Z

3.5 4 4.5

-0.5

0

0.5

-4

-3

-2

-1

0

1

2

3

4

x 10-9

-3.6-1.201.4

1.6

1.8

2

2.2x 10

-3

2'/

2

a

2=0.00617,

0=1x10-5,

0=1x10-6

(=0.1)

Low viscosity

E =0 t=1275.3

R

Z

3.5 4 4.5

-0.5

0

0.5

-1

-0.5

0

0.5

1

x 10-6

Γ=0.1

Γ=100

Ω2’/Ω2=0 Ω2’/Ω2=-1.2 Ω2’/Ω2=-3.6

105

106

10-3

10-2

S

linea

r gr

owth

rat

e

=0.0

2=0.0012

2=0.0025

2=0.0037

2=0.0049

~S-0.6

~S-0.4

Scaling of γ vs. S for different rotations

• Profile2: Ω=Ω0(1-ψ) (Ω2’/Ω2=-1.2) , ν0=1x10-6

• The dependence of γ on S is weakened, when Ω2 increases

Summary and discussion• We are developing a new MHD code in toroidal geometry under cylindrical

coordinate system for studying the MHD stabilities and plasma evolution. Through a series of benchmark tests, the code is proved feasible and reliable.

• We then use this code to examine the effect of toroidal plasma rotation on resistive tearing mode in tokamaks. The simulation results show that, for low viscosity plasmas(τR/τV<<1), toroidal rotation itself can suppress the tearing instability in range, whereas rotation shear exerts little influence when flow is small but diminishes this stabilizing effect when flow becomes strong enough, For high viscosity (τR/τV>>1), rotation shear has a stabilizing effect .

• The stabilizing influences of rotation may primarily arise from the equilibrium modifications of pressure profile, the centrifugal effect and Coriolis effect due to toroidal geometry. Among them, Coriolis effect may have a considerable even dominant influences on the stabilization effect due to rotation.

• The destabilization effect of flow shear, which is consistent with the findings of earlier studies --- destabilizing for low viscosity but stabilizing for high viscosity, possibly results from the distortion of mode structure (magnetic island structure) and mode coupling.

Thank you！