2015.3.4(Wed) - 3.6(Fri)The 7th IAEA Technical Meeting on “Theory of Plasma Instabilities”Frascati, Italy

Kenji Imadera1), Yasuaki Kishimoto1,2)

Kevin Obrejan1), Takuya Kobiki1), Jiquan Li1)1) Graduate School of Energy Science, Kyoto

University2) Institute of Advanced Energy, Kyoto University

Non-local profile relaxationand barrier formation

in toroidal flux-driven turbulence

SOL, diverter,sheath, wallphysics

Above marginal

Under marginal

Profile Stiffness/Resilience in Toroidal Plasmas

ü Ion temperature gradient is tied to a constant around the critical value to drive Ion Temperature Gradient (ITG) instability.

-> Profile stiffness/resilience[P. Mantica, et.al., Phys. Rev. Lett., 102, 175002 (2009).]

Turbulence

Marginal profile

r

Ti

Q

Q

Q

External TransportBarrier(ETB) formation

Internal TransportBarrier(ITB) formation

1/25

[Y. Kishimoto, et.al. Phys. Plasmas 3, 1289 (1996).]

ü Full-kinetic global simulation demonstrated that a radially extended structure imposes a strong constraint on the functional form of temperature profile.

ü Any perturbations on self-organized profile are quickly smoothed out.

Zonal Flow level: low

Self-organized profile

Perturbation on self-organized profile

Fast relaxation to self-organized profile

Stiffness/Resilience In Profile-Driven ITG Turbulence

2/25

( )0 exp / TT r L t= −� �

1T

( )0 exp / TT r L t= −� �

ü Recent flux-driven full-f gyrokinetic (GK) simulations also revealed that resilient ion temperature profile is sustained with a critical gradient, and the heat flux is mainly carried by avalanches.

Why profile stiffness/resilience is dominant even in flux-driven turbulence with MF and ZF?

Stiffness/Resilience in Flux-Driven ITG Turbulence

[Y. Idomura, et al. Nucl. Fusion, 49, 065029 (2009).]

ü In the power scan test, double input power does NOT change the temperature gradient, showing strong profile stiffness even if mean flow (MF) and zonal flow (ZF) are properly taken into account.

3/25

Purpose of This Work

ü Understanding of the non-local characteristics in flux-driven ITG turbulence with MF and ZF

ü Control of profile stiffness/resilience

Purpose of this work

Approaches

Fig. Ballooning structure oflinear toroidal ITG mode

Fig. Typical structure of flux-driven toroidalITG turbulence calculated by GKNET

GK Vlasov equation for ion

GK quasi-neutrality condition

Toroidal Full-f Gyrokinetic Code GKNET - 1

[G. Dif-Pradalier, et.al. Phys. Plasmas 18, 62309 (2011).][S. Satake, et.al. Comput. Phys. Comm. 181, 1069 (2010).]

DK collision operator

[K. Imadera, et.al. 25th Fusion Energy Conference, TH/P5-8, Oct. 16, 2014.]ü Full-f (Global)ü Electrostatic limitü Full-order FLRü Conservative

collision operator

( ) *1 ||

0 0

1 1

( ) ( )fe i

f B dv dT r n r αα

µΦ − Φ + Φ − Φ = � � P

{ } { }

{ } ( )

{ } ( )

*|| ||*

||

*|| ||

|| *||

, ,

,

,

source sink collision

i ii

ii

f f fH v H S S C

t v

d cH v e m v B

dt e B

dvv H e B

dt m B

α

α

φ µ

φ µ

� � �+ � + = + +� � �

� � = + � � + � � + ����� � = − � � + ���

RR

RR b b b b

B

PP

( ) ( ) [ ]

( ) ( ) ( ) ( )2 2 2

3/2

* 20 0

2 20 0

3 1( ) ( ) ( ) ( )

2 2

'2 1 2 2,

2 2

coll Mth th

v vx x v

v vn f uC f f aF x bG x cH x f

u v v q R u v n

v v v vv e dx v e dx e

v v

π ε ν ξ

π π π− − −

� Φ − Ψ � �� �= + − + +� � �� �� �� Φ − Φ � �Φ = Ψ = = −� �� �� �

Vlasov solver : 4th-order Morinishi schemeü Non-dissipativeü Density/Energy Conservative

Time integration : 4th-order RK schemeü Now ASIRK scheme is being

introduced for new version

Toroidal Full-f Gyrokinetic Code GKNET - 2

Flow chart of GKNET

Parallelization: 3D (R-Z-μ) MPI decomposition

Field solver : Real space field solverü Field equation is solved in real space (not k-space)ü Full-order FLR effect (without Tayler/Pade

approximation)6/25

Toroidal Full-f Gyrokinetic Code GKNET - 3

●Single averaging: Average of sample points on circle + 2D(or 3D) interpolation

●Double averaging: Two single averaging + μ integration

③

　　　　　　　　　　　　　　　 -> Fixed point method

Real space field solverSingle/double averaging

[K. Obrejan, et.al., accepted for the publication on Plasma and Fusion Res.]

7/25

*1 || 2f B dv d A C

ααµΦ − Φ + Φ = � Φ − Φ =� � P

( ) 22 22

2,1

C A AA C

A

ββ

β� �+ + − Φ� −� Φ − Φ = � = Φ =� �� �−� �

( ) ( ){ } 220 0i i

i

x c J x dxα� Γ −� � ��

Non-Local Ballooning Theory - 1

[J. Y. Kim, et.al., Phys. Plasmas, 3, 3689 (1996).][Y. Kishimoto, et.al., Plasmas Phys. Controlled Fusion, 40, A663 (1998).]

Mode width/angle of ballooning structure

*8/25

0

0

m

n0

0

1m

n

+0

0

1m

n

−L L

r

( ) ( ){ }ˆ, ( )expm bm

r r i m m mφ θ φ θ θ α∆∆

= + ∆ − ∆ +� � �

1/3

0

/ /

ˆ ˆ2r f

b

r r

k sθ

ω ωθ

γ� � + � �

= m

( )

1/2

0ˆ2 sin

ˆ / /b

r f

rk s r rθ

γ θω ω

∆ =� � + � �

0ˆ cos bγ γ θ=

0

2~ , ~i r

r D f

T Ek k

eBR eBθ θω ω ω= − −

Ballooningstructure

Non-Local Ballooning Theory - 2

Radial force balance

ü Cancellation by mean flow

ü Impact of toroidal rotation

Eigenfrequency + Doppler shift frequency

Diamagnetic drift Mean flow Toroidal rotation 9/25

1/3

0

/ /

ˆ ˆ2r f

b

r r

k sθ

ω ωθ

γ� � + � �

= m

( )

1/2

0ˆ2 sin

ˆ / /b

r f

rk s r rθ

γ θω ω

∆ =� � + � �

0ˆ cos bγ γ θ=

10i

ri

pE v B v B

n e rθ ϕ ϕ θ�− + − =�

1 1

i

ir

n T

rB T kE U

qR e L L

� �−= − +� �� �� �P

0 0, , exp , expi

ii i i

n T

k T r rv v U n n T T

eB r L Lθ ϕ

� �� ��= = = − = −� �� � � �� � � � �P

0

2 1 1~

i

r f in T

k k erBT U

eB R L L qRθω ω

� � �−+ − − −� � �� �� � �� P

Simulation condition

Linear Global GK ITG Simulation (Case 1)

30 90 150 210 2700

1

2

3

30 90 150 210 2700

0.5

1

2.5

1.5

2

30 90 150 210 270-0.015

-0.010

0.005

-0.005

0

0.010

0.015

10/25

Ballooning structure of n=35 mode

Linear Global GK ITG Simulation (Case 1)

0.07

0.05

0.04

0.02

0

0.06

0.03

0.01

0 0.1 0.2 0.3 0.4 0.5 0.6

300

150

0

-150

-300

300

150

0

-150

-300

300

150

0

-150

-300

11/25

Simulation condition

Linear Global GK ITG Simulation (Case 2)

30 90 150 210 2700

1

2

3

30 90 150 210 2700

0.5

1

2.5

1.5

2

30 90 150 210 270-0.015

-0.010

0.005

-0.005

0

0.010

0.015

12/25

Ballooning structure of n=35 mode

Linear Global GK ITG Simulation (Case 2)

0.07

0.05

0.04

0.02

0

0.06

0.03

0.01

0 0.1 0.2 0.3 0.4 0.5 0.6

300

150

0

-150

-300

300

150

0

-150

-300

300

150

0

-150

-300

13/25

Simulation condition

Nonlinear Flux-Driven GK ITG Simulation

1

0.8

0.6

0.4

0

0.2

Asource(r)

Asink(r)

0 50 100 1500

1

2

3

0

1

4

2

3

0 50 100 150

0 50 100 150

Source operator

Sink operator

ü Constant power input near magnetic axis

ü Krook-type operator to f in boundary region [Y. Idomura, et. al., Nucl. Fusion, 49, 065029

(2009).]

14/25

( ) ( ) ( ){ }10 02src src src M MS A x f T f Tτ −= −

( ) { }1 ( ) ( 0)snk snk snkS A x f t f tτ −= − − =

Time-Spatial Evolutions of Qturb, LT and Er shear

ü Flux-driven turbulent transport is mainly dominated by three process; aa (a) fast-scale avalanches, (b) slow-scale avalanches and (c) global transport.

0

150

r/ρ

i

100

50

0

150

r/ρ

i

100

50

0 200 400 600 800

tvti /R0

dEr /dr (16MW)

R0 /LT (16MW)

Qturb (16MW)

V~2vB

V~2vBlc~70ρi

V~0.2vB

V~0.2vB

0

150

r/ρ

i

100

50

15

10

5

0500 600 700 800

Qtu

rb

400

tvti /R0

r/ρ

i

0

50

150

100

Explosive Global Transport

Various type of transport(a) Fast-scale avalanches• Meso-scale• Radially propagate to both

inward and outward• V~2vB~Cs ρi*

(b) Slow-scale avalanches of E×B shear (E×B staircase)• Meso-scale• Radially propagate to

outward• V~0.2vB

(c) Explosive global transport • Meso-Macro scales• Simultaneous event• Coupled with radially

extended vortices

16MW

4MW

150

0

-150

t=574 t=586

Qturb

[G. Dif-Pradalier, et.al. Phys. Rev. E, 82, 025401 (2010).]

16/25

Role of Mean and Zonal Flow

ü Mean flow component satisfies the radial force balance roughly, which does not work to stabilize the turbulence, as is demonstrated by the non-local ballooning theory and linear simulations.

ü After the explosive global transport, meso-scale zonal flow with kr~0.5ρi-1 grows, which quickly disintegrates radially extended vortices.

1450 1480

6

4

2

0

0.08

0.04

0

0.12

0 50 100 150

0

-0.015

-0.030

-0.045

Mean flow determined by radial force balance

*[T. S. Hahm and K.H. Burrell, Phys. Plasmas 2, 1648 (1995).]

tvti /R01460 1470

17/25

8

0 200 400 600 800

tvti /R0

0

150

r/ρ

i

100

50

0

150

r/ρ

i

100

50

0

150

r/ρ

i

100

50

0

150

r/ρ

i

100

50

dEr /dr (16MW)

R0 /LT (16MW)

dEr /dr (4MW)

R0 /LT (4MW)

V~0.2vB

V~0.2vB

V~2vB

LT and Er shear in Power Scan Test

V~2vB

ü E×B shear propagates to outer region in high-input power regime.

18/25

Ti

3

10.8

0.6

0.4

0 50 100 150

r/ρiQ

tota

lR0 /LT

25

20

15

10

5

05 6 7 8 9 10 11

Profile Stiffness/Resilience in Power Scan Test

50<r<90

110<r<130

Nonlinear criticalthreshold

Gradient-Flux relation in power scan testTime-averaged temperature profile

ü Temperature profile is tied to an exponential functional form due to explosive global transport triggered by the instantaneous formation of radially extended vortices.

ü A break of profile stiffness is observed in outer region, where slow-scale avalanches of E×B shear are active.

Globally tied to a functional form

Profile resilience

19/25

[Y. Kishimoto, et.al. Phys. Plasmas 3, 1289 (1996).]

Simulation condition

0 50 100 1500

1

2

3

0

1

4

2

3

0 50 100 150

Flux-Driven ITG Simulation with Toroidal Rotation

ü Strong rotation shear is set initially in outer region, which is relaxed by momentum transport.

t=0t=1600

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 30 60 15090 12020/25

　 2

3 4

　 2 3 4

Impact of Toroidal Rotation on Radial Force Balance

Radial force balance:

Strongcorrelation

0 30 60 15090 120-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 30 60 15090 120-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 30 60 15090 120-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 30 60 15090 120-0.06

-0.04

-0.02

0

0.02

0.04

0.06

21/25

Weak Barrier Formation by Toroidal Rotation

-0.06

-0.04

-0.02

0

0 30 60 15090 120

4

3

2

1

0

0 30 60 15090 1200

2

3

4

5

6

0 30 60 15090 120

1

tvti /R0

Time-spatial evolution of turbulent heat flux

ü Transport property is changed by the introduction of toroidal rotation.ü This indicates the disintegration of radially extended vortices.

0

150

r/ρ

i

100

50

400 1000 1600600 1400

Impact of Toroidal Rotation on Non-local Transport

800 12000

150

r/ρ

i

100

50

400 1000 1600600 1400800 1200

23/25

Summary & Future Plans

ü We found that flux-driven turbulent transport is dominated by not only avalanches but also explosive global transport triggered by the instantaneous formation of radially extended ballooning structure, leading to profile stiffness/resilience even in the presence of ZF/MF.

ü Mean flow shear, which is determined by radial force balance, recovers the symmetry or just reverses the ballooning angle so that its stabilization effect is small.

-> One origin of radially extended vortices

Summary - 1ü We have newly developed

toroidal full-f gyrokinetic code GKNET.

24/25

Summary & Future Plans

Future plansü Enhancement of toroidal rotation impact on profile stiffness -> Safety factor profile, momentum source, large a0 , etc…ü Implementation of realistic magnetic configuration, kinetic

electron

Summary - 2ü We also found that a toroidal rotation can modulate such a

radial mean flow shear through the radial force balance, leading to a weak barrier formation.

ü In this case, the transport property is changed from explosive global transport to avalanches, indicating the disintegration of radially extended ballooning structure by the modulated radial mean flow shear.

Diamagnetic drift Mean flow Toroidal rotation

0

2 1 1~

i

r f in T

k k erBT U

eB R L L qRθω ω

� � �−+ − − −� � �� �� � �� P