9th IAEA TCM on H-mode physics and transportCatamaran Resort Hotel, Sep.24-26,2003

Y. KishimotoNaka Fusion Research Establishment

Japan Atomic Energy Research Institute (JAERI)

Theoretical understandings of transport barrier as a complex system

Transport within transport barriers : theorist’s view of the feature

1. Introduction Background and motivation

2. Fluctuation dynamics in wide frequency and wave number space Key issues of nonlinear fluctuation dynamic essential for transport barrier physics Possibility of “control” of fluctuation and related transport

3. Summary

Contents

Acknowledgement :

T.S. Hahm, K. Itoh, S-I. Itoh, M. Yagi, Z. Lin, P. Diamond, E-J. Kim, C.Holland,A.K. Wong, R. White, D.R. ErnstJ.Q. Li, Y. Idomura, N. Miyato, T. Matsumoto

Discussion

• Prof. P. Diamond : Furture direction in transport barrier Physics• Prof. K. Itoh : Prospect of transport physics in science• Dr. X. Q. Xu : Prospect for Edge physics

High performance is realized by having “structure”

• H-mode : ASTEX (1982)• CNTA-NBI mode• Core H-mode• DH-mode• Reversed (Negative) Shear mode• Enhanced Reversed Shear mode• High Density H-mode• Helical Electron ITB• High p mode• High p-H mode• High li-mode• High Ti mode• I-mode• Improved Ohmic Confinement mode• Lower-Hybrid Heating mode• Pellet mode• Pellet Ehhanced Performance H-mode• Radiation Improved Mode• Super shot• VH-mode• etc ……….

[Itoh S.I., et al., J.Nucl.Materials, ’95, Zohm, PPCF, ’96, Burrell, PoP, ’97, Ida, PPCF, ’98

Steady state

Understandingthe “selection rule” of the distinct states and the control

0 1.00.5

BS currentdominant

L-mode

Pre

ssur

e0 1.00.5Inductive current

dominant

Cur

rent

H-mode

High p-HRS-mode

High p

High confinement

Complex nonlinear loop system of structural plasma

� �

Self-Organizationrelaxation/cascade in wave number

space

external control 　 　 　 　 system

・heating ・current ・momentum

self-generation of current

structure of magnetic field

self-generation of electric filed

structure of flow and rotation

Ideal and non-ideal macro-scale

MHD fluctuation

Electrostatic and electromagnetic

micro-scale fluctuation

new pressure profile

nonlinear looppressure gradient

of plasma

DT burning

pressure profile

self-generation of current

1. Neoclassical dynamics2. Fluctuation and self-organization dynamics3. Global linkage as nonlinear loop system, and “control”4. Identification of the degree of complexity of the state

11 Contribution papers

E8: Yagi E9: S. Itoh E10: Ernst E11: Xu (E1: Diamond)

3. Global linkage as a nonlinear loop, and control

1. Neoclassical dynamics

2. Fluctuation and self-organization dynamics E1: DiamondE2: HahmE3: KimE4: HollandE5: WangE6: K-ItohE7: White

4. Identification of the degree of complexity of the state (E9 : S. Itoh) (E3: Kim)

(E8: Yagi, E10: Ernst)

Fluctuation dynamics in wide wave number space

1. Linear free energy source in complicated magnetic field

2. Nonlinear free energy source • Normal/inverse spectrum cascade

• Secondary and higher order nonlinear instabilities with different time and

spatial scales

cf. Generalized zonal/streamer mode,Zonal mode driven KH mode, etc

•　 Non-local, non-diffusive “new energy/information transfer channel” using not only spectral-space and “spatial dimension”

[Diamond, Hahm, et al. H-WS, ’03]

3. Interaction and interference among activitieswith different time and spatial scales, and through spatial dimension

• Mode structure in reversed/weak magnetic shearcf. Failure of ballooning picture

• Fluctuation due to nonlinear/turbulent dissipation. cf. CDBM

e1

i1

a1

a1 i1 e1 xk

yk

MHD

ion

electron

skin size

xk

ykelectron

skin size

ion

r

Linear free energy source

[Smolyakov, Yagi, et al., PRL, ’02]

• Short wavelength ITG mode (shear-less slab)

reversed shear ETG

normal shear ETG

negative shear ITG

normal shear ITG

20k i (※)

[Idomura, et al., NF, ’02]

• Global linear gyro-kinetic dispersion in reversed shear plasma

[Voitsekhovitch, Garbet, et al., PoP, ’02]

[Idomura, et al., PoP,’02, Kishimoto, et al., PPCF, ’99]

Gap structure Slab mode-like structure

e1

i1

a1

a1 i1 e1 xk

yk

ion

electron

skin size

pc

pc

[Wang, H-WS, ’03]

• Short wavelength ITG mode in current carrying plasma

Nonlinear free energy source

e1

i1

a1

a1 i1 e1 xk

yk

MHD

ion

electron

skin size

Various “Zonal modes” are exited through modulational instabilityFlow : Φ Field : //A Pressure :p

[Holland-Diamond, PoP, ’02, Jenco et al., IAEA, ’02, Miyato, et al., PPCF, ’02]

Csinp~iA~

,A~

2

~,

~

t1 n//

2//

e22

“Reynolds stress”

“Maxwell stress” “Collisional damping”

“Pressure anisotropy (Stringer-Winsor term)”

[Lin, et al., PRL, ’99, Kim, et al., PRL, ’03]

[Hallatshek-Biskamp PRL, ’01]

//A//e

//2//2e ACp~

~,A

~

2A~

,~

t

A

2

pCsinp~4A~

,A~

2p~,

~

t

ppn//

2//

e

• Small scale pressure corrugations are hardly controllable SOC dynamics

• Large scale component may change the q-profile

Nonlinear turbulent-convective cell system with complex “activator” and “suppressor” roles

Nonlinear free energy source

Maternal fluctuation

Transport

00

nm

01

nm

11

nm

Low m/n drive

Flow driven tertiary nonlinear instability

• GAM : • Stringer-Winsor :

φ~

p~• Kelvin-Helmholtz mode• GKH mode

collisonal damping

p-profileq-profile

A,p

00n

l

streamer

Neo-classical mean shear flow

[K-Itoh, et al., White, et al., Holland-Diamond, Yagi, et al., H-WS, ’03]

[Kim-Diamond, PoP, ’03]

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

[Miyato-Kishimoto, JPS, ’03][Matsumoto, Naitoh, PoP, ’03]

Nonlinear fluctuation dynamics• Local inverse/normal cascade

Mixed turbulent/zonal fluctuation system

Internal kink event

MHD-drivenEr-field

Ti

Zonal-Zonal-

[Idomura, PoP, ’00]

[Holland-Diamond, et al. H-WS, ’03]

ETG streamers found near threshold are essentially linear structures whose nonlinear interaction is weak. [Dorland, et al., IAEA, ’02]

e1

i1

a1

a1 i1 e1 xk

yk

MHD

ion

electron

skin size

[Jenko-Kendel,PoP, ’02]

Wendelsteien 7ASsimulation

[Kendel, PoP, ’03]

• Nonlinearly generated “convective cell mode”

Zonal pressure and -increase

Reduced MHD equation[Ichiguchi, et al., IAEA,’02]

•　 Resistive interchange modes induce a staircase structure, leading to a

linearly unstable high- profile

[Carreras, et al., PoP,’01]

Impact of zonal flow on transport (1)Gyro-kinetic PIC simulation using global profile effect and canonical Maxwellan particle distribution [Idomura, et al., NF, 2002]

linear saturation Quasi-steady state

Zonal fluctuationTurbulent fluctuation : m/n=0/0GAM fluctuation : m/n=1/0

• Macro-scale “mean flow”, same level as that of the equilibrium, regulated by equilibrium profile

[Lin, et al., Science, ’98]

Modulational instability and zonal flow

qkqx1 kk

x1k x2k

qx2 kk

xk

turbyxITG2

q2 Wk,kGk

turbturb ,, kq

xx k~q note but

• ITG case (adiabatic electron except k||=0)

turbyxETG4

q2 Wk,kGk

• ETG case (adiabatic ion)

2.0 4.0 6.0 8.0 0.10

2.0

4.0

6.0

8.0

0.1

0

yk

2xk

2z 103.4

(b) 7.0kq

Large grow rate forStreamer-like anisotropic pump wave : yx kk

Parameter to change the ratio of “turbulence” part and “zonal” part

TFZFz EE

x broader xk narrow

[White, et al., H-WS, ’03]

[Smolyakov, et al., PoP, ’00, Malkov, et al., PoP, ’01,

Li-Kishimoto, PoP,’02]

(A) S=0.2

(B) S=0.1

Self-organization to flow dominated fluctuations

disappearance of anomaly in high pressure state

• Weak magnetic shear increases linear instability sources, but nonlinearly transfers energy to zonal components

[Kishimoto,Li, et al., IAEA ’02]

10-2

10-1

100

101

2 3 4 5 6 7

s=0.2s=0.1s=0.4

<|d(z

) /dx|>2 /2

(a)

s=0.1

s=0.2

e

(B)

(A)

s=0.4

[Kendel, Scott, et al., PoP, ’03]

10-2

10-1

100

101

102

103

0

5

10

15

20

0 100 200 300 400 500 600

t (te

/Ln)

ETG saturation

Zonal flow energy

KH-mode likeinstability

• Drift-Alfven turbulence in edge plasma

1.0

0.8

0.6

0.4

0.2

w/o zonal flow

0.8

0.6

0.4

0.2

1.0with zonal flow

Characteristics of flow dominated fluctuationsTime averaged spectrum

DW

ZFKH

Near marginal andquasi-linear process

Condensation to KH-mode [Kim-Diamond, PoP ’02]

ETG

GKH ?

wavelet analysis

Statistical nature of turbulence-zonal fluctuation system

No flow case

10 2 3 41234110

210

210

410

510QQ

rate

9.1d

1.3d

2.4d

“Fractal dimension” and “PFD” : Probability Distribution Function

0 5.0 1 5.15.015.1110

210

210

410

610QQ ra

te

strong flow case

• Shrinking dimensionality due to coherent structure• Coherency increases with near Gaussian PDF of flux

[Matsumoto, et al., Toki-conf, ’03]

• Size distribution of heat pulse from GK simulation [Nevince,’00, Holland, et al., IAEA,’02] hhf like -L flowshear weak : 5.1

like-H flowshear strong : 2.2

• TEXTOR: Signal from Langmuir probes [Budaeev, et al., PPCF, ’93]d= 12-16 (attached) d=6-7 (detached) d=30 (from 15) (induced H-mode)

• CHS : Electron density fluctuation [Komori, et al., PRL, ’94]d~ 6.1 (RF heating) d~6.2 (NBI heating) d~8.4 (RF+NBI)

1. Fractal dimension

2. Probability Distribution PDF of density fluctuation of PISCES-A linear device and SoL of the Tore Supra

[Antar,et al.,PRL,’01]

Statistical nature of turbulence

“Noise forcing by coherent structure”

[S-Itoh, et al., 　 Kim-Diamond, H-WS, ’03]

• Non-Gaussian PDF for the Reynolds stress and hest flux [Kim, et al., IAEA,’02]

• Probabilistic view of L-H transition

Interference among different scale fluctuations

Interaction through micro-scale structure, eddy viscosity, noise, etc.

[Li-Kishimoto, PRL, ’02, Idomura, et al., NF, ’02]

[Hahm-Burrel, PoP, ’02, Hahm, et al., PoP, ’99]

turbxx kq , turb

xx kq ,

Interaction through quasi-coherent zonal modes

• Time varying Random shearing• Scattering to high-k

Open new nonlinear energy transfer channel

Trigger problem

[Itoh, et al.,PPCF,’01, Yagi, et al., IAEA, ’02]

xk

yk

MHD

electron

skin size

ion

a1 i1 e1 pc

e1

i1

a1

pc

ITG transport modulation due to small scale flow

[Li-Kishimoto, PRL, ’02, PoP, ’03]

GF-ITG simulation with micro-scale ETG driven flows

0

0.05

0.1

0.15

0.2

0.25

0.3

0 500 1000 1500 2000 2500 3000

Intermittency

<|dzf/dx|2>/2

i

(b) Upper state

(a) Lower state

Probabilistic damping trigger

10-13

10-9

10-5

10-1

0.1 1 10

a

b

c

d|(k

x)|2 /2

kx

(a)

(b)

high-k qkφ

kφlow-k

(q)cos(qx)A 2100 • Non-local mode coupling and associated

energy transfer to high kx damped region

No flow

• Micro-scale flow intermittently quenches ITG turbulence

L-state

H-state

[courtesy of Miura and JFT-2M group]

turbxx kq ,0 assuming

Multiple-scale turbulence and bifurcation

Langevan approach for 2-scale plasma turbulence system

• For micro-mode dominated solution, semi-micro mode is quenched, and vise-versa.

Semi-micro mode amplitude

Semi-Micro(cf. ITG)MicroMode

(cf. skin/ETG)

• Mechanism of ITB formation with different ion and electron dynamics

[Yagi, et al., IAEA, ’02, Itoh-Itoh, PPCF, ’01]

[Koide, et al., PPCF, ’98] cf. Distinct dynamics between ion and electron

Reversed shear

0

1 0

2 0

Ti

Te

ne0.5a= 3x101 9m -3

PNB=1 3M W

0

1 0

2 0Weakly reversed shear

eTiT

Nonlinear transfer channel of fluctuation

In spectral space

In real position space

Energy transfer among different scale fluctuations through local/non-local cascade or inverse cascade process

• Inverse cascade of “radial” shorter wavelength modes

• Radial “diffusion” and/or “convection” from linearly unstable region to stable zone

yk

xk

electron

skin size

ion

r

Radial energy transfer through propagation and/or spread

• Successive excitation of secondary and tertiary instabilities using spatial dimension

[Diamond-Hahm-Lin, et al. H-WS, ’03]

[White, et al., H-WS, ’03]

Modulational approach with spatial dimension

Nonlinear transfer channel of fluctuation

cf. Transport phenomena hardly explained from linear

analysis

JT-60U E29728 t=6.03

1

2

010

1

0.10 0.5 1

(10

5 sec

-1)

x(m

2 /s)

Full code(w/rotation)

xi

xe

P

r/a

kPi=0.53

[Rewoldt-Shirai, et al., NF, ’02]Garbet, et al., NF, ’94

A turbulent zone spreading radially in such a way that its level is no longer directly related to local plasma parameter

• Toroidal linear coupling “convection”• Nonlinear mode coupling “conduction”

Newmann, Diamond, et al.

ITB dynamics based on turbulent-transport equation system

Mattor-Diamond, Rep. UCRL, ’93• Coupling through equilibrium profile

① t=2s

② t=2s

③ t=6s

④ t=8s

①

②

③

④

rdiusq(r) q-min

surface

+2cm-2cm 0

brr

Spatial convection of instability

• Increase of linear instability source for reversed shear plasma

Origin of “structure” is anomalous transport!!

Strong turbulenceTransport suppression[Idomura, et al., PoP, ’00]

• Turbulent energy is “nonlinearly” converted to flow component through “spatial dimension”.

ETG

ETG-ZFKH-ZF

KHKH

Turbulent spreading and diffusionSome evidence from numerical simulationGarbet, et al., NF, ’94, Sydora, et al., PPCF, ’96, Parker et al., PoP, ’96, Lin, et al., IAEA, ’02, also PRL, ’02

[Sydora, et al., PPCF, ’96]

saturation phase

linear phase

ITG GK simulation ETG GF simulation

no rational surface(no damping)

[courtesy of J-Li]

linear phase

steady state phase

Turbulent spreading and size scaling

Nonlinear model of turbulent propagation

)Ir

I(r

II)r()t,r(It 0

2

I

rr0 r0+

Linear damping region

Front like solution

[Lin, et al., IAEA, ’02][Hahm, et al., H-mode WS, ’03]

Discussion about transport size scaling (B or GB)

• Radial spreading of fluctuation into stable zone

PDF of particle diffusion : close to “Gaussian” with no significant tail, suggesting “diffusive transport”

• No device size dependence of radial eddy length : x~7i : Scale size is “microscopic”

i20

Δ/a)41(

χχ GB

Size scaling of transport

133a s

400a s

• GB scaling well above the instability threshold

[Waltz-Candy, et al., PoP, ’02, also IAEA, ’02]

Stabilization effect due to shearing in the ballooning phase velocities due to global profile variation

local

n

ns

shear dr

d

k

Tc

• Break of the GB scaling to Bohm scaling (worse) near threshold

Tc

• Non-local transport where local diffusivity depend on finite radial length : i20~

Summary : prospect for future direction

• Methodology to control the nonlinear loop system is becoming necessary. cf. integration of key element.

• The physics of key elements dominating the transport barrier, specifically nonlinear process, is extensively studied, and the understandings have been developed.

• Interaction and/or interference among different time and scale fluctuations, not only in wide frequency/wave number space, but also real space dynamics, becoming crucially important.

• Statistical approaches to identify the degree of complexity of the state and transition dynamics are becoming necessary.

• Close interplay and interference among theory, simulation and experiment is desirable.

• Numerical approach to handle wider dynamical range is becoming tough, for example, micro-scale electron dynamics, but continuous efforts are crucial.

[Koshyk-Hamilton, JAS, 01] [courtesy of Earth simulator center]

Role of H-mode and ITB physics in science ?