c

Kenji Imadera1), Yasuaki Kishimoto1,2) Kevin Obrejan1), Takuya Kobiki1), Jiquan Li1)

1) Graduate School of Energy Science, Kyoto University 2) Institute of Advanced Energy, Kyoto University E-mail address: imadera@center.iae.kyoto-u.ac.jp

Non-local Structure and Transport in Toroidal Flux-Driven ITG Turbulence

2014.1.13(Tue)-1.14(Wed) 20th NEXT Workshop Kyoto, Japan Poster Number: 14

1. Background: Self-organized criticality in toroidal ITG turbulence

• Recent flux-driven full-f GK simulations revealed that stiff ion temperature profiles are sustained with a critical gradient, and a significant part of the heat flux is carried by avalanches.

• Such a type of simulation captured a new class of flow pattern called as E×B staircase, which is a localized E×B shear flow and steepens temperature profiles. [Y. Kishimoto, et.al., Phys. Plasmas 3, 1289 (1996).]

• A radially extended structure imposes a strong constraint on the functional form of temperature profile.

• Two-scale relaxation process provides a self-organized transport.

Profile-driven simulation (without ZF/MF) Flux-driven simulation (with ZF/MF)

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

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

Why self-organized critical transport is dominant even in flux-driven turbulence with mean/zonal flows? What is the role of flow (mean/zonal flows and E×B staircase) to such a self-organized critical transport?

2. Purpose of our research

• We newly developed a 5D toroidal global gyrokinetic code and checked the validity through several benchmark tests (neoclassical, R-H, linear, nonlinear ITG tests). Main features are

Vlasov solver (r, θ, φ, v||, μ) : 4th-order Morinishi scheme Field solver (r, θ, φ, μ): 4th-order FDM (r) + FFT (θ, φ) -> Real space filed solver is also developed (see poster 15) Time integration : 4th-order Runge-Kutta-Gill scheme Parallelization: 3D(r-θ-μ) MPI decomposition Collision and heat source/sink are implemented • Based on this code, we investigated the prominent non-local characteristics of flux-driven ITG

turbulence and transport coupled with neoclassical physics. In particular, we focus on

(A) Dominant turbulent transport process in flux-driven ITG turbulence (B) Role of mean/zonal flows in explosive global transport (C) Profile stiffness/resilience in power scan test

3. Simulation condition

*

0

0

0

1150

0.36

10

2.22

ti

T

n

aaR

RLRL

ρρ

ε

= = = = = =

( )

( )20

1 0 0*

( , , , , )

128,128,64,32,16( , , , , )

150 ,2 , ,12 ,18 /

4[ ], 16[ ]

0.25, 0.5

r v

r v

ti ti ti

in

snkti ti

N N N N N

L L L L L

v v B

P MW MWR Rvv v

θ ϕ µ

θ ϕ µ

ρ π π

τ −

=

=

=

= =

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

( ) ( ) ( ){ }( ) { }

10 0

1

2

( ) ( 0)src src src M M

snk snk snk

S A x f T f T

S A x f t f t

τ

τ

−

−

= −

= − − =

( ) ( ) [ ]

( ) ( ) ( ) ( )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 fu v v q R u v n

v v v vv e dx v e dx ev v

π ε ν ξ

π π π− − −

Φ −Ψ ∂ ∂= + − + + ∂ ∂

Φ − Φ Φ = Ψ = = − ∫ ∫

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

3

0

1

2

1

0.8

0.6

0.4

0

0.2

0 50 100 150 r/ρi

4. Non-local heat transport - avalanches and explosive global transport -

dEr /dr (16MW)

R0 /LT (16MW)

0 200 400 600 800 tvti /R0

Qturb (16MW)

Various types of transport (1) Fast-scale avalanches • Meso-scale • Radially propagate to

both in and outwards • V~2vB~Csρi* (2) Explosive global transport • Meso-Macro scales • Simultaneous event • Coupled with radially

extended vortices (3) Slow-scale propagation of E×B staircase • Meso-scale • Radially propagate to

outward • V~0.2vB

5. Profile resilience/stiffness

Ion temperature profile (790<t<800)

• While temperature corrugation propagates in fast-scale(~2ρiR0/vti), global temperature profile is tied to a functional form.

-> Self-organized critical transport

• E×B shear propagates in slow-scale(~0.2ρiR0/vti) to outer region, where a break of resilience is observed.

Formation of staircase by localized E×B shear

Deviation from self-organized profile

propagates in fast scale (~2ρivti /R0)

Globally tied to a functional form

4MW 16MW R0 /LT

(25<r<75) 4.4 5.0

R0 /LT (100<r<125)

5.1 8.3 Self-organized critical transport

6. Role of zonal/mean flow shear to explosive global transport

7. Power scan test 8. Implementation of kinetic electron (on going)

|| ||1 1 1 1r

n T

rU rUp p T kE v B v B k Tn r n r r qR L L qRθ ϕ ϕ θ

∂ ∂ ∂ −= − + = − + = − + + ∂ ∂ ∂

0 ||1 2 1 1cos

n T

d kf dv d nTdt B R L Lθθ µ θ

−= = − + +

∫∫

r rω∂ ∂ f rω∂ ∂

Neoclassical poloidal flow

( )0 0 0~ cosγ θ γ θ

9. Conclusion and future plans

r/ρ i

15 10

5

0

0

500 600 700 800

Qtu

rb 16MW

4MW

50

100

150

400

150

0

-150 -150 0 150

t=574 t=610

Qturb

tvti /R0

0

150

r/ρ i

100

50

0.03

0.02

0.01

0

-0.01 0 50 100 150

r/ρi

E r

0 0 0

2 1 1 1 1 cos 1 1cosn T n T n T

k RnT nTr R L L L L R R L L

θθ θ ∂ −

= − + + + + + ∂

Radial Er profile just before and after explosive event Effect of Er on ballooning structure

ni(r) Ti(r)

q(r)

Asource(r) Asink(r)

Source/Sink operator

Collision operator

0

150

r/ρ i

100

50

0

150

r/ρ i

100

50

V~0.2vB

V~0.2vB

V~2vB

V~2vB

lc~70ρi

3

1

0.6

0.4

0 50 100 150 r/ρi

T i

• After explosive global transport, meso-scale zonal flow with kr~0.5ρi

-1 grows, which quickly disintegrates radially extended vortices.

21

r∆ ≅2

10 02

k fr

sin

sr rθ

γ θωω ∂ ∂

+ ∂ ∂

0 02k

T i

i d

sin Lsθ

θ γ ρρ ω

≅

( )0 maxθ∆ ≅ 02k

frr r

sθ

ωω

γ

∂ ∂+ ∂ ∂

31

0 12d Tsk Lθ

γω

≅

31

• Flux-driven turbulent transport is dominated by three process; (1) fast-scale avalanches, (2) explosive global transport and (3) slow-scale propagation of E×B staircase.

• Explosive global transport is triggered by the instantaneous formation of radially extended potential vortices, which is enhanced by the neoclassical mean flow shear through the cancelation of global profile variation effect, i.e. diamagnetic shear.

• A self-organized resilient profile keeping the exponential function form is established even in the presence of zonal/mean flow. Such a resilience is also confirmed from step-up/down switching test for heat input power (not shown in this poster).

• A break of resilience is observed in outer region, where radial convection of temperature corrugation coupled with E×B staircase occurs, exhibiting a weak transport barrier formation in high power regime.

Future plans • Implementation of general magnetic

configuration -> Edge boundary effect • Implementation of kinetic electron -> ITG-TEM mode • Study of transport barrier formation in

flux-driven ITG turbulence

16MW

4MW Qtu

rb ω

E×B

16

12

8

20

560 570 580 590 0

0.1

0.2

0.3

0.4

tvti /R0

Qturb

ωE×B

Ballooning mode width/angle

[Y. Kishimoto, et.al., Plasma Phys. Controlled Fusion 41, A663 (1999).]

Er=Eneo (ν*=0.5) 150

0

-150 -150 0 150

0 50 100 150 r/ρi

0.08

0

0.04

-0.04

-0.08

Er=0

θ

(a)Eneo×B drift

(b)diamagnetic drift

(a)+(b)

• The neoclassical mean flow shear can cancel the global profile variation effect, i.e. diamagnetic shear.

Averaged mean flow

1.2100

i

e

ηη

Θ ===

Shear-less slab ITG instability test z

xy

,n T∇ ∇(0, ,1)= ΘΒ

ky ρi

γ R 0

/vti

ZF damping test

0

100/ 0.36

aa R=

=0

2* 0.2

i e

qη η

υ

= ===

Φk

tvti /R0

0 1.2 0.2 0.4 0.6 0.8 1.0 0

0.4

0.2

0.1

0.3

1

0.5

0

-0.5

-1 0 30 20 10

Implementation of implicit scheme with GCR method (2D system)

0

20

40

60

2nd RK 2nd CN+ GCR

2nd CN+ LU

CFL number

k=3

k=4

k=6 k=4 k=6

T i

3

1 0.8 0.6 0.4

0 50 100 150 r/ρi

Qto

tal

R0 /LT

25

20

15

10

5

0 5 6 7 8 9 10 11

50<r<90

110<r<130

Gradient-Flux relation in power scan test Time-averaged temperature profile

10-6

10-8

10-10

10-12

10-14

|Er’|

10-6

10-8

10-10

10-12

10-14

|Er’|

r/ρi=75

r/ρi=125

4MW

16MW

16MW

4MW

ωGAM~1.6

CPU

tim

e

0 6 4 2 8 0

1

2

CPU

tim

e

• Growth rate decreases as mass ratio becomes smaller.

• Implicit scheme with GCR method can reduce CPU time.

• Toroidal collision-less simulation always numerically diverges…

Qto

tal

0 5

10 15

25 20

30

Q

0 5

10 15

25 20

30

0 50 100 150 r/ρi

Qtotal

Qturb

Qneo

16MW

Time-averaged heat flux profile

ωR0 /vti

10-2 10-1 100 101

Spectrum of Er shear

• Temperature profile is tied to an exponential functional form, while temperature scale length largely changes in outer region.

• In outer region, (1) neoclassical heat flux increases due to the effect od sink opearator and (2) Er shear spectrum is piled up especially in ωvti /R0 ~ 1.