global transport simulation with radial electric ﬁeld sourceglobal transport simulation with...

Global transport simulation with radial electric field source

Shinsuke TOKUNAGA, IGSES Kyushu University, JapanMasatoshi YAGI, RIAM Kyushu University (JAEA), JapanSanae-I ITOH, RIAM Kyushu University, JapanKimitaka ITOH, NIFS, Japan

2009年12月15日火曜日

Chapter 1: 　 Introduction

• The origin of nonlocal transport property has not been clarified.

• The whole systemic mechanism of ITB formation and sustainment is still not revealed.

• Understanding of transport properties related to nonlocality or barrier dynamics is strongly demanded.

• In this study,

‣ Introducing heat source, parallel (to magnetic field) momentum source and radial electric field source, simulations of ITG turbulence are performed.

‣ Interaction among turbulence (micro), transport (macro) and intermediate scale structure (meso) associated with collapse of barrier structure is investigated.


Landau damping as collisionless dissipation(closure)

Neo-classicalflow damping

Model equations

Energy balance

Parallel flow

Vorticity (Continuity)

32

!dT

dt+ !T

1r

"!"#

"!

!dn

dt+ !n

1r

"!"#

"= ! 9

5"

!

!1"A|#|||T +

25A#||V + #!#2

!T

,

Grad-B drift frequency (Curvature operator)

Gradient scale length of equilibrium temperature T0 , equilibrium density n0

Yagi M et.al. , Plasma Phys. Control. Fusion 48 A409 (2006).

n = !! !̄Density fluctuation (No phase shift) Generalized potential

Poloidal flow

-> No particle flux

d(n!"2!F )

dt+ !n

1r

"!"#

+ A""V = $%̂dF + &#q

$µNC 1

r

"(r Up)"r

! &2#µ"4

!F

F = ! +p̃

!

Up = V + !!q!

"F"r

Chang Z, Callen J D, Phys. Fluids B 4 1167 (1992).

!T0 = !d lnT0

dr

dV

dt= !A"!F + 4µ"2

"V ! µNCUp !A

!1!

25#

"|"!|V +25A

1!"!T

!̂d ! 2!

1r cos " !

!" + sin " !!r

"

Chapter 2: Ottaviani M, Manfredi G, Phys. Plasmas 6 3267 (1999).

!n0 = !d lnn0

dr

00


!! ! 0.01 µ = 1! 10!3 !! = 1! 10"3Other constants

STi(r) = !8.0" 10!3 2r2 ! 1

(1! r2s)2

Heat source term

Ion density profile[Fixed] takes peak value around r~0.45 in t~60.!̂→

Heat source

ITG driving term :

(rs = 0.6)

T̄Flux surface average of temperature fluctuation (equivalent to (0,0) component in Fourier space)

, ,

Source, relaxation↓

:

Profile modification effect appears through the coupling : (m,n)+(0,0) → (m,n).

!T̄ (r,t) ! "d ln T̄dr

-0.02

-0.01

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1

Heat

sou

rce

radius

with (!T0 ! 0 )in this simulation.

Chapter 2:

!̂(r,t) ! 32 (!

T0(r) + !T̄ (r,t))" !n0(r)

n0(r) =(1! r2)(1! r2

s)


Magnetic configurationGarbet X et al., Phys. Plasmas 8 (2001) 2798.

Reversed shear profile

qmin = 1.35rmin = 0.6

q(r) = qmin + C2(r2 ! r2min)2+C3(r2 ! r2

min)3

C2 = 4.66 C3 = !0.987,q(0) = 2.0 , q(1) = 3.0 !,

Chapter 2:


Growth of meso-scale modes and barrier collapse

• After destabilization of ITG mode (30,20) at t~60, until t~78, a barrier-like quiescent domain around q-minimum is observed.

• Meso-scale (4,3) and (7,4) modes dominate the system from t~78.

• The barrier region begins to decay gradually from t~79 from inside and abruptly collapsed at t~84 with energy burst.

• Saturation of the (4,3) mode coincides with the burst. After the collapse, the system changes to low confinement state being accompanied with intermittent small burst.

Chapter 4:


Heat flux provided by meso-scale modes

• Almost of heat flux around r~0.25 and r~0.55 are provided by (7,4) and (4,3) modes, respectively.

• Apparently, the (4,3) mode plays important role in the collapse mechanism.

• Such meso-scale modes develop in outside of ITG excited region.

• They have radially wide correlation length. Their profiles are overlapped each other and indicate existence of non-local interaction via three-wave coupling.

Chapter 4:


ITG excitation around barrier foot position

• Transport provided by the (4,3) mode gives rise to the steepening of temperature gradient around r~0.7.

• This steepening destabilizes micro ITG modes at barrier foot position.

• Consequently, these modes modify zonal flow which has sustained barrier foot, and trigger barrier collapse.

Chapter 4:

dV!

dr


Energy cascade and propagation in r-m space

ITBregion

(4,3)

ITGsInverse Cascade

Normal Cascade

(log)

ITGs• After collapse, thick barrier structure as

before collapse is never recovered again. And the system shows intermittent behavior dominated by meso-scale eddies.

(7,4)Meso scale modes

before collapse (4,3) growing phase

collapse

Chapter 4:mode

mode

mode

|!|


Analysis of energy transfer via three-wave coupling

• At first, the (4,3) mode gains energy mainly from three-wave coupling between micro-scale turbulence.

• About t~68, the main contribution for growth of the mode switches to profile effect, i.e. calculated as coupling of (4,3)=(4,3)+(0,0).

!NLk0 T

= !T !k0

[!k1 , Tk2 ]"

k0 = k1 ± k2

• Instantaneous internal energy transfer via three-wave coupling

for

Quasi-linear

Chapter 4:


Profile modification effect

• Excited ITG modes flatten temperature profile around their resonant surface. Beat interaction between them excites meso-scale modes in this phase.

• Local flattening due to ITG causes rise of temperature gradient in neighboring region.

• It results acceleration of quasi-linear drive of meso-scale modes. Meso-scale modes affect on temperature profile via transport.

• At last, steepening of temperature gradient destabilizes ITG modes at r~0.7 and global collapse is triggered.

steepen

flatten

steepen

Chapter 4:

Domino effect


Zonal flow and meso-scale eddies

• Meso-scale modes grow in the area where zonal flow is absent.

• In this case, linearly unstable modes driven by imposed toroidal flow shear exist at r~0.15.

• Zonal flow is formed around micro ITG mode excited region.

Chapter 4:


Multi-scale interaction in global relaxation scenario

• Profile modification given by transport causes next step of changes and radial propagation of fluctuating energies inward and outward.

• As a consequence of interactions among ITG mode (pump), meso-scale structures and global profile, the system reaches to marginal state.

• A meso-scale mode located in vicinity of q-minimum plays important role in this global relaxation scenario.

• To consider such non-local dynamics introduced by meso-scale mode or profile modification, global simulation study is quite important.

Chapter 4:


-1-0.8-0.6-0.4-0.2

0 0.2

0 0.2 0.4 0.6 0.8 1

Phi

radius

Radial electric field source

• In this study, imposing vorticity source to the model, the effect of strong ExB flow shear on such meso-scale mode is investigated.

vort

icity

sou

rce

Phi

t =!

!t=!0,0 (r) = AE "E

!tanh

"r ! rc

"E

#! !

$Ert=!

0,0 (r) = !AE sech2

!r ! rc

!E

"

SVol(r) = !µ"2!!t="(r)

Er

rc = 0.55 AE = 4.5

ExB flow shear takes peak value at r = rc ± 10.0! !! " rc ± 0.1

-5

-4

-3

-2

-1

0

0 0.2 0.4 0.6 0.8 1radius

-0.03-0.02-0.01

0 0.01 0.02 0.03 0.04

0 0.2 0.4 0.6 0.8 1radius

t =!

Er

Φ


ResultsEnhancement of

barrier sustainability

• Barrier structure around q-minimum is observed and its collapse is not observed at least until t=125. It is more than double of lifetime in the case without Er source.

• Strong poloidal flow shear is formed in 0.55 < r < 0.7 and steep temperature gradient is kept in corresponding region.

• Obviously the ExB flow shear affects on sustainability of barrier structure.

Er

V!!̂(! 3

2!T " !n)

q

t=125.0


• The (4,3) mode is suppressed compared to the case without imposed Er. The (4,3) mode is, however, still growing gradually.

• Zonal flow component around barrier is reduced in the case with Er source.

• Imposed ExB flow shear can suppress the meso-scale eddy which cannot be suppressed by Zonal flow.

Suppression of meso-scale eddy

Results

(4,3)


• Two compared cases are completely the same until t~60.

• Stripe pattern indicates series of modes coupled by toroidal coupling.

• The (4,3) mode is suppressed, and the (7,4) and ITGs are stronger in case with Er source.

• Even in the case with Er source, the (4,3) mode is growing across the ExB shear flow.

ResultsEnergy cascade and propagation in r-m space

t=85 t=105

t=125

t=85, W/O Er source

W/O Er source With Er source

Contour plot of phi square root (log scale)

(4,3)

(4,3)

(7,4) (7,5)

ITG ITG


Summary

The dynamics of the multi-scale interaction between the ITG turbulence and transport was investigated using the gyro-Landau fluid model. The numerical simulations with the heat source have been carried out, starting from the initial temperature profile which is stable against ITG modes. Reversed magnetic shear configuration is considered to produce ITB.

•It is found that a meso-scale mode located at q-minimum region, which is (4,3) mode in this simulation, plays an important role for ITB collapse. We confirmed that when the mode is artificially suppressed, ITB collapse does not take place.

• The (4,3) mode is destabilized by the quasi-linear effect. It contributes to the transport in the inner half of ITB, which gives rise to the steepening of the temperature gradient at the ITB foot.

•The ITB foot remains for a while during the ITB collapse due to velocity shear stabilization effect. However, it is destroyed at last due to the ITG modes excited near the ITB foot.


Summary

• ITB collapse is attributed to interactions with different spatial and temporal scale. We have clearly shown one example of multi-scale interaction between transport and ITG turbulence.

• It is found that non-linear, non-local interactions with meso-scale structures and profiles are quite important for ITB dynamics in the global simulation.


Future work

• Parameter survey about heat source shape and intensity

• To solve toroidal and poloidal flows consistently.

• Toroidal and poloidal flow shear effect on toroidal coupling and micro off-resonant modes.

• Electromagnetic effect (consistent evolution of safety factor)


Numerical scheme

in cylindrical coordinate

L :Linear part, N :Non-linear part, B :Toroidal coupling

Linear part

predictor

corrector

!fm,n(ri)!t

= L (fm,n(ri)) +!

m=m!+m!!

n=n!+n!!

N (fm!,n!(ri), fm!!,n!!(ri)) +!

m!=±1

B (fm+m!(ri))

f t+!t/2 =!

I ! !t

2L

2

"!1 !I +

!t

2L

2

"(f t) +

!I ! !t

2L

2

"!1 !t

2#N(f t, f t) + B(f t)

$

f t+!t =!

I !!tL

2

"!1 !I + !t

L

2

"(f t) +

!I !!t

L

2

"!1

!t#N(f t+!t/2, f t+!t/2) + B(f t+!t/2)

$

r direction　 ->　Finite difference methodθ , z direction　->　Fourier spectral method

Nonlinear & Toroidal coupling part Crank-Nicolson implicitPredictor-Corrector


global transport simulation with radial electric ﬁeld sourceglobal transport simulation with...

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