nonlinear instability plasma...

Graduate School of Energy Science Kyoto University, Japan

Joint ICTP‐IAEA College on Advanced Plasma PhysicsInternational Centre for Theoretical PhysicsTrieste, Italy, 18‐29 August 2014

Nonlinear instability and plasma dynamics

• August 19 : Structure/dynamics of fluctuations in fusion plasmas• August 19 : Nonlinear instability and plasma dynamics• August 21 : Interaction between different scales

Y. Kishimotocollaboration with K. Imadera and J.Q. Li

Planetary environment and fusion plasmas

( )

( ) ( ) ( )

ZF

ZF tot ZF LSS

EE E E

( )

( ) ( ) ( )

ZFZF

ZF turb ZF LSSt ZF LSS

EE E E

• Micro‐turbulences• Macro‐flows

(sometime with low mode)• Large scale structure

World dominated by

No external heatingFully self‐sustained

( )ZFE

( )turbE

( )LSSE

PQ Q

What is the role of ZF and LSS on transport ?

Micro‐scale vortices are embedded in macro‐scale flows

ther

mal

diff

usiv

ity

1

0200 400

2

600

zonal flow

ON

OFF

time

Interaction between flows & fluctuations

g

21

2

xV

gk yy

Shearing decorrelationof turbulence

x

xxx k

x

kysfkyk dk

dkkik

tk

222 , 11

Shearing decorrelation is a local interaction in k space

2'222EVktk

Fluctuation with different time and spatial scales

Meso-scaleMacro-scale Micro-scale

810~510~k

e1

i1

a1

a1 i1 xk

yk

Ion scale

Electron scale

eρ1

Ion scale

[courtesy of Kagei (JAEA)] [Y. Kishimoto et al.’96, PoP ] [Y. Idomura et al.’04, IAEA, NF]

• Coexistence of different scale fluctuations :

θ

r rk

θk

n0

t + n

t + n0v+nv =0

nk

t + n0vk + nk vk

k=k +k = 0

n0

t + Re nkvk

* exp 2k t k

= 0

Flux : nD

Quasi-linear transport due to drift wave (1)

),(~)(),( 0 txnxntxn

),(~)(),( 0 txxtx uuu

kkE -ikk

vk = -i ck kb

B = -i cTi

eB ekTi

kbr + krrb

nk(ExB)

n0 = -i

n0 vkn0k

= - *ik

ekTi

= - *ik r

k2

ekTi

1 - i kk r

dibi vk*： dia-magnetic drift frequency 2

00

00

BennTe

d

Bv

vdj = - cTjejB

1Lpj

= - jvjLpj

= - jvjLnj

1+j

rkbk ˆkˆk rb|| Coupling between ion motion and density gradient (drift wave)


DB = 116

cTeB

= nkvk r*

k = - n0 cTi

eB kb

*ik

k2

ekTi

2

exp 2kt k

D = - cTieB

kbLn k*i

nkn0

2exp 2k t k

0

cf. nLDn

2icTcf. eB i i i civ

Step size： gyro-radiuscorrelation time ： gyration period

TB2

B aB

2

B cf.aD


D = - cTieB

kbLn k*i

nkn0

2exp 2k t k

poloidal direction

radial direction

Zero transport 0k

k=tan-1 kk r

nk(ExB)

n0 = -i

n0 vkn0k

= - *ik r

k2

ekTi

1 - i kk r

Phase angle between density and potential

nken0

ekTe

1-ik


Correlation time

BEc

dtd kr ~

2

0

0

BE

Bv

ckk

ckk

r Bcki

BEc

~

Correlation time

generation linear growth rate LkNk

r L Nt t k k kk r steady state

0 NkLk LkNk

ck 1~1~

kk

kc

c

rB

ckD

22 ~~~

annihilation scattering of fluctuation due to various origins

Assuming Markov process

0

1~n r

nn L k

r n r,t +n0 ~ 0

• Mixing length estimate

• Wave breaking estimate

E × B

r

2

2* 0

exp 2 ~b n kr

ncTD k L teB n k

k k k

k

BEc

dtd kr ~

2

0

0

BE

Bv

ckk

ckk

r Bcki

BEc

~Correlation time

kk

kc

c

rB

ckD

22 ~~~

Assuming Markov process 1~ ~b kr

k r

ckB k

0

1 1~ ~ ~k k k

b s r s r n

e nT k k v k L n

*~ ~k kr i 1 1~ ~i Tr

r Lk

1 2

21*2 2~ ~ ~ ~k i i i

i i T i ir r T T

vD k L vk k L L

0 ~ ir

2~ ~k i i

r T

cTDk eB L

0 ~ i Tr L

2~ ~k i

r

cTDk eB

: gyro-Bohm

: Bohm

JT‐60/JET class

ASDEX‐U/JFT‐2Mclass

ITER/ Demo class

ri = 4mma = 2mri /a = 2x10-3

ri = 7mm, a = 1mri /a = 7x10-3

ri = 7mma = 0.35mri /a = 2x10-2

1m 2m 3m 4m 5m 6m

1 2

~ i i

T

cTeB L

2 2 12 1~ ~ s

r n n

eWT k L L

2

~ s

n

WL

~ s

n

WL

Fig.1. Ion heat conductivity vs tokamakminor radius. [Copied from Fig.3 in Ref. [1].

[1] Z. Lin et al., PRL 88, 195004 (2002).

An interesting observation in global gyrokineticsimulations of toroidal ITG turbulence is that forsmall device size plasma, the fluctuations aremicroscopic and local, that transport is diffusive,while the resulting turbulent transport is not gyro-Bohm, shown in Fig.1.[1] The local transportcoefficient exhibits a gradual transition from aBohm-like scaling for small device sizes to agyro-Bohm scaling for future larger devices inthe presence of self-consistent zonal flows.

Fig. 4. Effective heat diffusivity of three global gyrokinetic codes as a function of ρ∗. (copied from Fig.5 in Ref. 3)

[3] Sarazin, et. al, Nucl. Fusion 51 (2011)

tit kk

k rkkArA exp,,ˆa

TB

pB

Weak turbulence ： 1nTturb

Strong turbulence： 1~nTturb

Fluctuating filed in confinement device

tit kk

k rkkr exp,,

22

81 BEturb

φ

rB

θ

02

0E B

E Bv

E~* *

x x k kk

nv n v

* *3 32 2x x k k

k

Q pv p v

*~ ~ sin( )x nnu n v

exp( ), exp( )nn n i v v i

Phase difference causes flux :

Particle motion :

eee pendtdv

mn bEb ˆˆ||

Electron motion (parallel to the magnetic field

jv

j||v

πr2

kk ,kk

k

πR2

→ Adiabatic response (simplest case)

e

nmnm T

exnn

,

0,~

Ion motion (perpendicular to the magnetic field

iiii

ii pendtdmn )( 0BvEv

20

0

BE

Bv

200

00

BennTe

d

Bv

cD B

B 2/22

//2

0

00 B

1th order drift

2th order, polarization drift

)(1

0cEp tBvv

tit kk

k rkkr exp,,

nmitrnm

nm exp,,

,

rm

Lmk

2

Rn

Lmk

2

a

TB

pB

Low frequency ion motion

Drift wave caused by drift

0

iii n

tn v

0ˆ~

0

Bcn

tni b

0

yv

t di *

1 1e e sy d y y y s

n n

cT cT vnk v k k keB n x eB L L

tvy dKey point: Density perturbation is in phase with potential, just drift wave

ie

e nTenn ~~

0

ed kv *

Physical picture of slab ITG mode

y

z

p-

p+

x

Ti,e(x)

drift dir.

B

Ey

Ey

Slab ITG: coupling between and ion sound wave

∆

Ti

yBv ˆ12

00

0

xn

neBcT

Benn i

dp

(1 )ee

en iT

0

0

1 )ey d y *

T nω k υ k ω ( iδeB n x

ei nn ~~

Drift wave becomes can be unstable, leading instability

Drift wave : a plasma oscillation propagating along diamagnetic drift direction

If some mechanism may pressure perturbation is out of phase with the potential perturbation, the drift wave becomes “unstable”

//0 0 // / / 0 / /

ˆ ( ) 0e e eie

bm n en p m nt en

Electron drift wave

iδ model :

Collisional electron drift wave: 2 2*

*/ /

ye ei

ce

ki

k

)(1

0cE

ep tB

vv

20

0

BE

Bv

1

cj

jv

j||v

0

pEii ntn

vv i en n

0)1( 22

byt d

xyyx

222 ],[

Hasegawa-Mima equation

2 2(1 ) [ , ] 0dt y

0ee

en n x

T

for zonal flows : =1for others : =0

Turbulence in quasi-two dimensional bounded system

2 21 0dv bt y

Drift wave : Hasegawa-Mima eq.

B

E

n

Conserving quantities 22W :Energy 222U :Enstrophy

ˆEcv zB

pci

c dvB dt

2 2ˆ1 0Rh hv z h ht y

Rossby wave : Charney equation

zρfv Hρgp

g

ˆcg z Hf

v 02p

gH d hf dt

v

With adiabatic election, simplified 3-field equations for ITG mode

2 2

2/ / / /

1 (1 )

1 2 ( )2

in n

D i D

a at L L

p

/ // / / / ip

t

2 cos sinD rf f with

Toroidal curvature for toroidal ITG mode

Parallel compression for slab ITG mode

2

2/ / / / / /

1(1 ) (1 )3

1 8 4 ( )2

ii i

n n

D i D i

p a at L L

p k p

Discussion: Ignoring toroidal curvature and parallel compression term,

reduced to just drift wave; Ignoring toroidal curvature effects, reduced to slab ITG Ignoring parallel compression, reduced to toroidal ITG;

Fluid equation of ITG in tokamak

+

-(A) S=0.2

(B) S=0.1 : weak magnetic shear

[Kishimoto,Li, et al., IAEA ’02]

0

2

4

6

8

10

2 3 4 5 6 7

e/[(

e/Ln)(

cTe/e

B)]

e

s= 0.1

s=0.2(A )

(B )

s=0.4

1 0 -2

1 0 -1

1 0 0

1 0 1

2 3 4 5 6 7

<|d

(z) /d

x|>2 /2

( a )

s= 0 .1

s= 0 .2

e

(B )

(A )

s= 0 .4

Conditional flow generationin high pressure region

( )

( ) ( )

ZF

Z turb ZF

EE E

Energy partition low

fluctuation energy = turbulent part + laminar part

Turbulence dominated by large scale structure

6η 0.1s e

e1

i1

a1

a1 i1 xk

yk

Ion scale

Electron scale

eρ1

Ion scale

turbulence + zonal flow

turbulence

► Emergence of large scale vortices micro scale ETG

ZFMacro

Scale KH

a1

a1

► Mixed turbulence with• micro-scale ETG• ETG driven ZF• ZF driven Large scale structure

Coherency and phase between Ey and n

s = 0.1, e = 3

f (Vte/Ln) f (Vte/Ln) f (Vte/Ln)

f (Vte/Ln) f (Vte/Ln) f (Vte/Ln)

Turbulent plasma Zonal flow dominated plasma

s = 0.1, e = 6

Micro-scale region Macro-scale region

High coherency, but keeping phase relation that produces no transport

turbulence

Zonal flow

total

Change of Magnetic shear (tilting)

Magnetic shear increase

Magnetic shear decreaseIncrease instability free energy

heating

Anomalous nature

Anomalous nature

total

total

turbulence

turbulenceZonal flow

Zonal flowand large scale

structure

Zonal flow

Zonal flow

Zonal flow

turbulence

turbulence

total

total

total

Self‐organization in high pressure state

Linearly more unstable Nonlinearly more stable

[Koshyk-Hamilton, JAS, 01]

Hasegawa-Mima Equation

0φφbyφv

tφ1 2

d2

Charney Equation

0hhzyhv

th1 2

R2

ˆ

wave number10-3

10-2

10-1

100

101

102

0.1 1

<(k

x,y)2 >/

2

kx or k

y

t=400

kx

ky

kx

zonal flow

Comparison of atmospheric zonal flows

Total fluctuation = turbulent fluctuation + zonal fluctuation

( induce transport)

sun

Earth environment

sun

Earth environment ???

(ZF)(turb)

(ZF)

(tot)

(ZF)

zF EEE

EEη

Self‐organization in high pressure state

nonlinear instability plasma...

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