Intermittent Transport and Relaxation Oscillations of Nonlinear Reduced Models

for Fusion Plasmas

S. Hamaguchi,1 K. Takeda,2 A. Bierwage,2 S. Tsurimaki,2 H. Sato,2 T. Unemura,2 S. Benkadda, 3 and M. Wakatani2

1Osaka University, Japan 2Kyoto University, Japan

3CNRS/Université de Provence, France

Dedicated to Prof. Masahiro Wakatani

Outline

• Intermittent Thermal Transport due to ITG turbulence– a low degree-of-freedom model: how low can it

be?– full PDE

• Intermittent Particle Transport due to resistive drift turbulence

Toroidal ITG Equations

where

Toroidal ITG mode is described by the following equations, W. Horton, D-I. Choi and W. Tang, Phys. Fluids 24, 1077 (1981)

2 2 2 2 2

2

, 1 ,

, ,

1 , ln ln

i

i

i i i e i n Ti i i

pg K g

t y y

p p K pt y

K T T L L d T d n

iK

: electrostatic potential (fluctuation)

: ion temperature gradient

: viscosity

p

g

: ion pressure (fluctuation)

: effective gravity

: thermal conductivity

Low Degree-of-Freedom Model with 18 ODEs

Low-order modes in a slab geometry

If are neglected, the above model agrees with the 11 ODE model by G. Hu and W. Horton [Phys. Plasmas 4, 3262 (1997)]

, , , ,,0 03 3c s c sp

c c

s s

c

c

s

x x

x y x y

x y x y

x x

x y

x

x

x

x y

x y t t k x t k x

t k x k y t k x k y

t k x k y t k x k y

p x y t p t k x p t k x

p t k x

t k x

t k x k y

t k x k y

p t k x

0 01 2

1 2

1 2

0 01

0

2

1

3

3

3

03

, , sin sin 2

sin cos sin 2 cos

sin sin s

sin 3

sin 3 cos

sin 3 sinin 2 sin ,

, , sin sin 2 3

sin

sin

c

s ss

cx y

x

y x

y

y

x y x y

p t k x k y

p t k

k y p t k x k y

p t k x k y p t k y x k yx k

3

2 3

2

1

cos sin 2 cos

si

sin 3 cos

sin sin si n 3 sin 2 nsin .

Characteristic quantities

0 11 x Vx

i

pvdpNu pv dV

dx K

,x y y x yR y

y R

S v v dy L v v

v x St x

The mean velocity is driven by the Reynolds stress SR.

Nusselt number Nu :

Periodic Oscillation

(a)Phase space‘K0-Nu’ and

(b)Power spectrum of K1.

Time evolutions of (a)kinetic energy K and(b)Nusselt number Nufor Ki=0.4.A periodic oscillation

appears.

Chaotic Oscillations

(a)Phase space‘K0-Nu’ and

(b)Power spectrum of K1.

Time evolutions of (a)kinetic energy K and(b)Nusselt number Nufor Ki=0.6.Chaotic oscillations

appear.

Intermittent Bursts

Time evolution of kinetic energy for Ki=4.Intermittent bursts occur.

Time evolution of(a)Nusselt number Nu(b)Reynolds stress SR

Nu and SR burst when the ITG

modes grow rapidly.

Intermittent Transport: 　 3 Steps

1. Turbulence generates a mean flow through Reynolds stress.

2. The mean flow suppresses instability.

3. The mean flow becomes weak due to viscous damping.

Scaling Law-18 ODEs

3Nu Ki

Transition of transport scaling due to intermittency.

Dependence of the time averaged Nusselt number on the ion temperature gradient is expressed as

Intermittent oscillationsin full PDE model

Time evolutions of kinetic energy K (left) and Nusselt number Nu (right) for Ki=5. Intermittent oscillations appear.

Scaling law—full PDE simulation

Mode structures in the radial direction that are essential for the formation of intermittency

1. Low degree ODE models (18 ODE)

2. ITG (PDE): Transport scaling:

3. Resistive drift (PDE) : Transport scaling:

Summary

1/ 3Nu Ki

Velocity-Space Structures Velocity-Space Structures of Distribution Function in of Distribution Function in Toroidal Ion Temperature Toroidal Ion Temperature

Gradient TurbulenceGradient Turbulence

T.-H. Watanabe and H. SugamaT.-H. Watanabe and H. SugamaNational Institute for Fusion Science/National Institute for Fusion Science/

The Graduate University for Advanced StudiesThe Graduate University for Advanced Studies

MotivationMotivation How are the velocity-space structures How are the velocity-space structures

of of ff associated with turbulent associated with turbulent transport and zonal flows ?transport and zonal flows ? Kinetics of the zonal flow and GAMKinetics of the zonal flow and GAM Structures of Structures of ff providing the steady ITG providing the steady ITG

turbulent transportturbulent transportfocusing onfocusing on entropy balanceentropy balance and and generation of fine-scale structuresgeneration of fine-scale structures of of f f through the phase mixingthrough the phase mixing..=> Gyrokinetic-Vlasov simulation of => Gyrokinetic-Vlasov simulation of ff in in multi-dimensional phase-space with high multi-dimensional phase-space with high resolution (Flux tube configuration is resolution (Flux tube configuration is employed).employed).

Entropy Balance in the Entropy Balance in the Toroidal ITG SystemToroidal ITG System

dG

dt

d

dtS W iQi Di

S 1

2d3v ˜ f k

2FM

k

W 1

21 0

Ti

Te

k

2 Ti

Te

k

2ky ,0

k

Qi 1

2 ikyk d3vv 2J0

˜ f kk

Di d3v k ˜ f k

FM

C ˜ f k

k

(Entropy Variable)

(Potential Energy)

(Heat Transport Flux)

(Collisional Dissipation)

Steady State v.s. Steady State v.s. Quasi-steady StateQuasi-steady State

In the quasi-steady In the quasi-steady state for the state for the collisionlesscollisionless case, case,

S

time

d

dtS iQi

S

time

iQi Di

The two limiting states of turbulence have The two limiting states of turbulence have been extensively investigated by the slab ITG been extensively investigated by the slab ITG

simulations. simulations. (Watanabe & Sugama, PoP (Watanabe & Sugama, PoP 99, 2002 & PoP , 2002 & PoP 1111, ,

2004)2004) In the steady state for In the steady state for

the the weakly collisionalweakly collisional case,case,

Collisionless Damping of Collisionless Damping of Zonal Flow and GAMZonal Flow and GAM

Initial value problem for Initial value problem for n=n=0 mode with 0 mode with ff||tt=0=0=F=FMM

The residual zonal flow The residual zonal flow level agrees with level agrees with Rosenbluth & Hinton Rosenbluth & Hinton theory.theory.

What is the velocity-What is the velocity-space structure of space structure of ff ? ?

How is the entropy How is the entropy balance satisfied ?balance satisfied ?

kxt

kxt 0

1

11.6q2 1/ 2

Cyclone DIII-D base case

Conservation Law for the Conservation Law for the Zonal Flow (Zonal Flow (n=n=0) 0)

ComponentsComponents From the gyrokinetic From the gyrokinetic

equation for equation for kkyy=0,=0,

… … Subset of the Subset of the entropy balance entropy balance equationequation

dG

dt

d

dtSkx

Wkx 0

S d3v ˜ f kx

22FM

W 1

21 0(k 2) Ti

Te

kx

2 Ti

2Te

kx

2Entropy variable Entropy variable S increases S increases during the zonal during the zonal flow damping.flow damping.

Velocity-Space Velocity-Space Structures of Structures of ff in the in the Zonal Flow DampingZonal Flow Damping

Decrease of Decrease of WWkxkx with the invariant with the invariant G G means means increase of increase of SSkx kx as well as generation of as well as generation of fine-scale structures of fine-scale structures of ff due to phase due to phase mixing by passing particles mixing by passing particles <=><=> Entropy Entropy transfer in the transfer in the vv-space-space

Coherent structures for trapped particlesCoherent structures for trapped particles <=> <=> Neoclassical PolarizationNeoclassical Polarization

(2(2

vv////

passing

trapped Bounced Averaged Solution

ITG Turbulence ITG Turbulence SimulationSimulation

kkx,minx,min= 0.1715, = 0.1715, kky,miny,min= 0.175, = 0.175, = 0.001 = 0.001

(Lenard-Berstein collision model)(Lenard-Berstein collision model)

Cyclone DIII-D base case

Entropy Balance in Entropy Balance in Toroidal ITG TurbulenceToroidal ITG Turbulence

Entropy balance is satisfied (Entropy balance is satisfied (/D/Dii <~ 8%). <~ 8%).

Statistically steady state of turbulence:Statistically steady state of turbulence: ii QQii~= ~= DDii => => Entropy production balances Entropy production balances with dissipation.with dissipation.

Velocity-Space Velocity-Space Structures of Structures of ff in in

Toroidal ITG TurbulenceToroidal ITG Turbulence

Fine-scale structures generated by the phase mixing Fine-scale structures generated by the phase mixing appear in the stable mode, and are dissipated by appear in the stable mode, and are dissipated by collisions, while the transport is driven by macro-scale collisions, while the transport is driven by macro-scale vorticesvortices..

=> => Entropy variable transferred in the phase Entropy variable transferred in the phase spacespace

High-velocity space resolution is necessary.High-velocity space resolution is necessary.

Long wavelength mode Linearly stable mode

(2(2

vv////

Production, Transfer and Production, Transfer and Dissipation of Entropy Dissipation of Entropy

VariableVariable

Entropy variable produced on macro-scale is transferred Entropy variable produced on macro-scale is transferred in the phase space, and is dissipated in micro velocity in the phase space, and is dissipated in micro velocity scale.scale.

The result confirms the The result confirms the statistical theory of the entropy statistical theory of the entropy spectrumspectrum for the slab ITG turbulence (Watanabe & for the slab ITG turbulence (Watanabe & Sugama, PoP Sugama, PoP 1111, 2004)., 2004).

Production

Dissipation

Transfer

Phase Mixing

No

nlin

eari

ty

Small Velocity Scale

Small Spatial Scale

Transport

Collisionless

Slab ITG Transport (WS 2004)

SummarySummary Gyrokinetic-Vlasov simulations of ITG Gyrokinetic-Vlasov simulations of ITG

turbulence and ZF with turbulence and ZF with high velocity-space high velocity-space resolutionresolution..

Collisionless damping process of ZF & GAM is Collisionless damping process of ZF & GAM is represented by the represented by the entropy transfer from entropy transfer from macro to micro velocity scalesmacro to micro velocity scales..

Neoclassical polarization effect is identified.Neoclassical polarization effect is identified. Statistically steady stateStatistically steady state of toroidal ITG of toroidal ITG

turbulence is verified in terms of turbulence is verified in terms of the entropy the entropy balancebalance..

Production, transfer, and dissipation processes Production, transfer, and dissipation processes of the entropy variableof the entropy variable are also confirmed for are also confirmed for the toroidal ITG turbulent transport.the toroidal ITG turbulent transport.

Flux Tube CodeFlux Tube Code Toroidal Flux Tube CodeToroidal Flux Tube Code

Linear Benchmark TestA B

CD F E

0

2

B

zy

x r r0

y r0

q0

q(r) z

The gyrokinetic equation is The gyrokinetic equation is directly solved in the 5-D directly solved in the 5-D phase space.phase space.

Gyrokinetic EquationGyrokinetic Equation GK ordering + Flute ReductionGK ordering + Flute Reduction

Co-centric & Circular Flux Surface with Co-centric & Circular Flux Surface with Constant Shear and GradientsConstant Shear and Gradients

Quasi-Neutrality + Adiabatic Electron Quasi-Neutrality + Adiabatic Electron ResponseResponse

t v||

ˆ b vd ˆ b v||

f

c

B0

,f v vd v||ˆ b e

Ti

FM C f

vd v||

2 R0

cos z ˆ s zsin z y sin z

x

,

v cTi

eLnB0

1i

mv 2

2Ti

3

2

y , v

2

2

J0 kv f d3 v 1 0 k2 e

Ti

e

Te

, k2 kx ˆ s zky 2 ky

2

Selected Parameters for our simulations

・ box size ： Lx=Ly=10.0 Lz=100.0

・ initial perturbations:

23

0)1,2( 2exp100.1

xLxS

24

0)2,3( 2exp100.2

xLxS

・ # of grid points in the x direction : 200

・ mode numbers: 　 -10 m 10 , -2 n 2 　（ total: 105modes ）

-15 m 15 , -3 n 3 　（ total: 341modes)

、

・ density profile:

nLn

n 1

0

0

（ Ln ： density gradient scale length）

Quasi-Linear saturation

Time evolutions of (a)kinetic energy K and(b)Nusselt number Nufor Ki=0.3.

(a)Phase space‘K0-Nu’ and

(b)Power spectrum of K1.

Periodic Oscillations

(a)Phase space‘K0-Nu’ and

(b)Power spectrum of K1.

Time evolutions of (a)kinetic energy K and(b)Nusselt number Nufor Ki=0.5.More modes are

excited.

Linear Analysis

The most unstable wave number is estimated as

In this study, the following wave number is assumed,

From the dispersion relation, the linear growth rate is as follows.

2 2 2 1 x y ik k k g K

2 2 2 4 2

22 2 2 4 2 2

1 1 2 1 ,

1 1 4 1

y i k

k y i i y

k g K k i k k k D k

D k g K k i k k k k gK k

2 1 5 , 2 1 5 x y x i y ik k k g K k g K

Normalization

0 0 0 0 0

0 0 0 0

, , ,

, ,

, .

s s n s

n e s n i i e s

n e s n e s

x x y y t t L c

eL B T p pL T p T

eL B T eL B T

The standard drift wave units are used for normalization.

Linear growth rate

1 5 , 2 1 5x i y ik g K k g K

0.21icK

Increase of Accuracy

1. 18 ODE: Two modes (m=0 and 1) in the y direction and 3 modes (l=1,2, and 3) in the x direction.

2. 1D: Two modes (m=0 and 1) in the y direction and full modes in the x direction.

3. PDE: full modes in the x and y directions.

Intermittent oscillationsin 1D model

Time evolutions of kinetic energy K (left) and Nusselt number Nu (right) for Ki =5.

Scaling LawsODE, 1D, and Full Simulations

1D model Full PDE simulation

18 ODE

Intermittent Particle Transport in Resistive Drift Turbulence

Particle Flux

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 2000 4000 6000 8000 10000

Part

icle

Flu

x 3

Time

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000

Part

icle

Flu

x 3

Time

0 2000 4000 6000 8000 10000

Time

0

Lx/2

Lx

0 10000 20000 30000 40000 50000

Time

0

Lx/2

Lx(a)’ 0 < t < 10000 (b)’ 0 < t < 50000

(a) 0 < t < 10000 (b) 0 < t < 50000

Particle flux vs. Resistivity

4.0E-3

5.0E-3

6.0E-3

7.0E-3

8.0E-3

9.0E-3

1.0E-2

1.1E-2

0.1 0.2 0.3 0.5 0.8 1 2 3 5

Part

icle

Flu

x 3

Resistivity

xnv

part

icle

flu

x Γ

resistivity η