Physics of Zonal FlowsP. H. Diamond1 , S.-I. Itoh2, K. Itoh3, T. S. Hahm4

1 UCSD, USA; 2 RIAM, Kyushu Univ., Japan; 3 NIFS, Japan; 4 PPPL, USA

20th IAEA Fusion Energy ConferenceIAEA/CN-116/OV/2-1Vilamoura, Portugal

1 November 2004

This overview is dedicated to the memory of Professor Marshall N. Rosenbluth.

Contents2

What is a zonal flow?Basic Physics, Impact on Transport, ZF vs. mean <Er >, Self-regulation

Why ZF is Important for Fusion ?

Assessment: "What we understand" Universality, Damping and Growth, Unifying Concept: Self-regulating dynami

cs

Current Research: "What we think we understand"Existence, Collisionless Saturation, Marginality, ZF and <Er>, Control Knob

s

Future Tasks: "What we do not understand"

Summary

Acknowledgements

What is a zonal flow?

Drift Waves

Drift waves+

Zonal flows

1. Turbulence driven2. No linear instability3. No direct radial transport

ZFs are "mode", but:Paradigm Change

ITER

Poloidal Crosssection

m=n=0, kr = finite

ExB flowsZonal flow on Jupiter

3

Basic Physics of a zonal flow4

VExB

GenerationBy vortex tilting

Damping by collisions Tertiary instability

Suppression of DWby shearing

Drift waveturbulenceZonal flowsShearingCollisional flow damping

SUPPRESSNonlinear flow damping

energyreturn

DRI VE<vx vy>~~∇T, ∇n...<vx p>~~Transport

t

5

Self-regulation: Co-existence of ZF and DW

∂∂t WZF=γdamp⋅⋅⋅WZF+ α WdWZF

∂∂t Wd=γ∇p0, ⋅⋅⋅Wd– α WdWZF

Wd ∼

γdampα

ii, q, geometry

+ rf, etc.

0 γdamp

DW

<U2> <N>

ZF

damping rate of ZFtu

rbul

ence

and

tran

spor

t

Transport coefficient

Wd : drift wave energy

: zonal flow energy WZF

Confinement enhancementIncludes other reduction effects (i.e., cross phase)

"R – Factor"

χ i =R χgB χ i ∼

γdampωeff

χgB

6

Why ZFs are Important for Fusion ?

1. Self-regulation

3. Identify transport control knob

2. Shift for onset of large transport

"Intrinsic" H-factor (wo barrier)

Operation of ITER near marginality

τE =HτE,L – scaling

PLH- threshold

Intermittent flux, e.g., ELMs Peak heat load

4. Meso-scalesand zonal field Impacts on

RWM and NTM

-limit

Steady state

Realizability ofIgnition

Issues in fusion

H ∝ R – 0.6

$∝ H– 1.3

χ =R χgB

$ ∝ R – 0.8

Flow damping ?

7

Assessment I "What we understand"

ZFs are UNIVERSALScales Transport Role of ZF

ITG

TEM/TIM

ETG

Resistiveballooning/interchange

Drift Alfven waves at edge

ρi

ρi

ρe

Δresis. layer

∼ρi

cs/a

cs/a

vTh,e/a

core

core

core χe, χ J

edge, sol,

helical edge

edge, sol

Very important

Very important

Very important

Important

On-going

This IAEA Conference

Falchetto (TH/1-3Rd), Hahm (TH/1-4), Hallatschek (TH/P6-3), Hamaguchi (TH/8-3Ra), Miyato (TH/8-5Ra), Waltz (TH/8-2), Watanabe (TH/8-3Rb)

Lin (TH/8-4), Sarazin (TH/P6-7), Terry (TH/P6-9)

Holland (TH/P6-5), Horton (TH/P3-5), Idomura (TH/8-1), Li (TH/8-5Ra), Lin (TH/8-4)

Benkadda(TH/1-3Rb), del-Castillo-Negrette (TH/1-2)

Scott(TH/7-1), Shurygin(TH/P6-8)

χi, χφ,D

χi, χe, χφ,D

8

Linear Damping of Zonal FlowsNeoclassical process Rosenbluth-Hinton

undamped flow - survives for

Role of bananasBanana-plateau transitionCL

iiγdampε3/2ωtωt

Screening effect if qrρp∼O 1

φqt

φq0= 1

1+1.6ε– 1/2q2

> transport

Frictional damping

even in "collisionless" regime

γdamp∼ ii/ χ i ∝ ii

9

Growth Mechanism

ZFs by

modulational instability

primarydrift wavesecondarydrift wavezonal flowk ,k' ,k"= qr rω̂" = 0~ω'ω

kd+kd-kd0q

Numerical experiment indicates instability of finite amplitude gas of drift waves to zonal shears

χi

Zonal flow (log)

R-factor

10

Regime

1 kr Diffusion Zakharov, PHD, Itoh, Kim, Krommes2 Turbulent trapping Balescu3 Single wave modulation Sagdeev, Hasegawa, Chen, Zonca4 Reductive perturbation Taniuti, Weiland, Champeaux5 DW trapping in ZF Kaw

Keywords References

V ExBDrift wave packet

Zonal flowCh

iriko

v pa

ram

eter

DW lifetimePacket-circul. time

110SKStochasticTurbulent trappingCoherentBGK solutionParametric modulational instability

∞∞13452

Kubo number

Close Relationship: DW + ZF and Vlasov Plasma

(i) DW + ZF :

(ii) 1D Vlasov Plasma :

∂∂t VZ +γdampVZ =– ∂

∂x VxVy

∂∂t N+vgxN– ∂

∂x kyVZ∂N∂kx

=γk N+Ck N

∂E∂x =4πn0e dv f

∂ f∂t +v d f

dx+ e

mE ∂ f∂v

=C f

‘Ray’ Trapping

Particle Trapping

Ω =qxvg xV ExBDrift wave packet

ω =kv

dxdt =vgxk dy

dt=vgyk

dkxdt =– ∂

∂x kyVZ

dkydt =0

dvdt = e

mE dx

dt =v

11

φ∼

particle

12

Note: Conservation energy between ZF and DW

RPA equations

Coherent equations

∂∂t Wd

byZF

=– ∂∂t WZF

byDW

DW

ZF

2B2 d2k

qx2kθ

2 k xVZF,q2

1+k⊥2ρs

22 R k, qx

∂ N∂k x

Σqx

∂∂t VDW

2+ γL, k+Ck N VDW,k

2Σk

=

∂∂t +γdamp VZF

2 =

– 2B2 d2k

qx2kθ

2 k xVZF,q2

1+k⊥2ρs

22 R k, qx

∂ N∂k x

Σqx

DW

ZF

(S: beat wave)

dZ2

dτ=–

γdampγL

Z2+2P ZScos Ψ

dP2

dτ=P2 – 2P ZScos Ψ

13

Self-regulating System Dynamics

Simplified Predator-Prey modelStable fixed point

Cyclic bursts

Single burst(Dimits shift)

Wave number

Zonal flow

Drift waves

Wave number

Zonal flow

Drift waves

Wave number

time

Zonal flow

Drift waves

+

0

<N>

<U2>

DW

ZF

DW

<N>

<U2>

γL/γ2

γL/α ZF

∂∂t N =γL N – γ2 N 2 – α U 2 N

∂∂t U2 =– γdampU

2 + α U2 N

DW

ZF

γ2 → 0

γdamp→ 0

(No ZF friction)

(No self damping)

II Current research: "What we think we understand"

Zonal flows really do exist

GAM?

Z. F.

-0.01 -0.005 0 0.005 0.01

ω/ Ωi

-1

-0.5

0

0.5

1

10 11 12 13 14r

2 (cm)

r1=12cm

Radial distance

HIBP#1observation points

HIBP#2observation points

ExB flowErφctrφinφoutPoloidal Crosssection 2ErPoloidal Crosssection 1φctrφinφoutExB flow

CHS Dual HIBP System

90 degree apart

Er(r,t)

High correlation on magnetic surface,Slowly evolving in time,Rapidly changing in radius.

Er(r,t)

A. Fujisawa et al., PRL 93 165002(2004)

14

A. Fujisawa et al., EX/8-5Rb

15

Candidates for Collisionless Saturation

Trapping

Tertiary Instability

Higher Order Kinetics

VZF ∝ ωbounce∼Δω, γL

VZF ∼ φ+

Te2Ti

T qr

VZF ∼Δωkθ– 1

Plateau ink-space N(k)

Returns energyto drift waves

Analogous to interaction at beat wave resonance

Partial recovery of the dependence of on drift wave growth rate

χ i

VZF

φ

collisional

Collision-less

χ =

γnlcollisionless

ωeffχgB

drift wavedrift wavezonal flowk ,k' ,k"= qr rω̂" = 0~ω'ω

16

"Near" Marginality

Shift exists

ITER plasma near marginality Importance of ZF

Where the energy goes

What limits the "nearly marginal" region: (Dimits shift)

Mechanisms: Trapping, Tertiary, Higher order nonlinearity,….

Route to the shift is understood, but "the number" is not yet obtained.

An example:

VZF

φ

low γL

high γL

VZF,crit

γL, crit∼qr2 kθ– 2 α

(Higher-order nonlinearity model)

17

Electromagnetic EffectSubject Mechanism for ZF growth Implications for fusionZF generation by finite- drift waves

Modulational instability of a drift Alfven wave

Transport at high- ,

L-H transition

Zonal magnetic field generation

Random refraction of Alfven wave turbulence

Possible Z-field induced transition, NTM,…

1+k⊥

2ρs2 5/2

2+k⊥2ρs

2

k⊥2 k y

2

k||

∂2

∂k x2

ωkNk

1+k⊥2ρs

2f 0Σ

k

ηZF=–

4πcs2δe

2

vth, e 1+qr2δe

2 ∂

∂tBθ=– ηZF ∇2Bθ

Current layer generationCorrugated magnetic shear, Tertiary micro tearing mode ?

BθJζ

Can seed Neoclassical Tearing ModeLocalized current at separatrix of tearing mode, ……

vrvθ ⇒ vrvθ – BrBθSelected structure

Zonal flow Zonal field

18

Distinction between ZF and mean Field <Er>

Zonal Flows Mean Field <Er>

Time can change on turbulence time scales

changes on transport time scales

Space oscillating, complex pattern in radius

smoothly varying

Stretching

Behavior

k of waves

diffusive ballistic

Drive Turbulence equilibrium , orbit loss, external torque, turbulence, etc.

δk2 ∝ t

δk2 =t2k2VE

' 2

timetime

∼20ρi

∇ p

19

Interplay of Zonal Flow and Mean <Er>

Previous: Coupling between DW, mean <Er> and mean profile

Now: Coupling between DW, ZF, mean <Er> and mean profile

Input power Q

Drift wave energy

Zonal flow

Pressure gradient

Prediction of bifurcation, dither, hysteresis, …

dNdt

=– c1EN – c2N +Q

dVdt

= V – dN 2 + Fnonlinear

dEdt

=EN – a1E2– a2V

2E – a3VZF2 E

Nonlinearity; orbit loss, etc. in previous model

dVZFdt

= b 1VZF

2 E

1 + b 2V 2– b 3VZF

Fluctuations

Pressure gradient

Zonal flow New coupling

Mean flow

20

Zonal Flow and GAM: two kinds of secondary flow

S(ω)

ω

Zonal Flow

GAM Drift Wave

Fluctuations~ cs/R

CLVExBn ~ 0

ZF CLVExB-+dn/dtGAM

GAMS: important near the edge

ZF dynamics and GAM dynamics merge

ZF drive GAC: magnetic field sensitive

DW ZF GAMLandau and collisional damping

damping

Lower temperature: , and ii ↑ cs/R ↓

New feature: geodesic acoustic coupling (GAC)

21

Control Knobs

(i) Zonal Flow Damping Collisionality,, q, geometry, …n.b. especially stellarators

(ii) External Drive (e.g., RF)

Choice of wave, Wave polarity, launching, (e.g., studies of IBW), rational surfaces…

22

III Future Research: "What we do not yet understand"

1. Experimentally convincing link between ZFs and confinement

2. Dominant collisionless saturation mechanism: selection rule ?

3. Quantitative predictability

(a) R-factor ? - trends ??(b) Suppression - γE vs γLin ?(c)How near marginality ?(d) Effects on transition ?(e) Flux PDF ?

4. Pattern formation competition

ZF vs. Streamers, Avalanches

5. Efficiency of control

6. Mini-max principle for self-consistent DW-ZF system ?

23

Summary

1. Paradigm Change: DW and turbulent transport

DW-ZF system and turbulent transport

Linear and quasi-linear theory Nonlinear theory

2. Critical for Fusion Devices

3. Progress and Convergence of Thinking on ZF Physics

4. Speculations: More Importance for Wider Issues

e.g., TAE, RWM, NTM; peak heat load problem, etc.

(Simple way does not work.)

R-factor Route to ITERHelps along

Barrier Transitionse. g.,

26q

Mini-Max Principle for Self-Consistent DW-ZF System ?

Predator-Pray model

Lyapunov Function

∂∂tN=γL N– γ2N2– α U 2N

∂∂t U2 =– γdampU

2+α U 2N

N* ≡

γdampα

U*2≡

γLα –

γ2γdamp

α2

Fixed point

This Lyapunov function is an extended form of Helmholtz free energy

F = N – N* ln N + U2– U*

2lnU2 : ∂F∂t =– γ2 N – N*

2 < 0

F: minimumMini-max principle:

SZF = k B ln U 2 + 1 SDW = k B ln N + 1

Teff, D = k B– 1N*

Teff, Z = k B– 1U *

2

N – N* ln N ⇒ EDW – Teff, D SDW

U 2 – U*2 ln U2 ⇒ EZF – Teff, Z SZF

24

Acknowledgements For many contributions in the course of research on topic related to the material of this review, we thank collaborators (listed alphabetically): M. Beer, K. H. Burrell, B. A. Carreras, S. Champeaux, L. Chen, A. Das, A. Fukuyama, A. Fujisawa, O. Gurcan, K. Hallatschek, C. Hidalgo, F. L. Hinton, C. H. Holland, D. W. Hughes, A. V. Gruzinov, I. Gruzinov, O. Gurcan, E. Kim, Y.-B. Kim, V. B. Lebedev, P. K. Kaw, Z. Lin, M, Malkov, N. Mattor, R. Moyer, R. Nazikian, M. N. Rosenbluth, H. Sanuki, V. D. Shapiro, R. Singh, A. Smolyakov, F. Spineanu, U. Stroth, E. J. Synakowski, S. Toda, G. Tynan, M. Vlad, M. Yagi and A. Yoshizawa. We also are grateful for usuful and informative discussions with (listed alphabetically): R. Balescu, S. Benkadda, P. Beyer, N. Brummell, F. Busse, G. Carnevale, J. W. Connor, A. Dimits, J. F. Drake, X. Garbet, A. Hasegawa, C. W. Horton, K. Ida, Y. Idomura, F. Jenko, C. Jones, Y. Kishimoto, Y. Kiwamoto, J. A. Krommes, E. Mazzucato, G. McKee, Y, Miura, K. Molvig, V. Naulin, W. Nevins, D. E. Newman, H. Park, F. W. Perkins, T. Rhodes, R. Z. Sagdeev, Y. Sarazin, B. Scott, K. C. Shaing, M. Shats, K.-H. Spatscheck, H. Sugama, R. D. Sydora, W. Tang, S. Tobias, L. Villard, E. Vishniac, F. Wagner, M. Wakatani, W. Wang, T-H Watanabe, J. Weiland, S. Yoshikawa, W. R. Young, M. Zarnstorff, F. Zonca, S. Zweben. This work was partly supported by the U.S. DOE under Grant Nos. FG03-88ER53275 and FG02-04ER54738, by the Grant-in-Aid for Specially-Promoted Research (16002005) and by the Grant-in-Aid for Scientific Research (15360495) of Ministry of Education, Culture, Sports, Science and Technology of Japan, by the Collaboration Programs of NIFS and of the Research Institute for Applied Mechanics of Kyushu University, by Asada Eiichi Research Foundation, and by the U.S. DOE Contract No DE-AC02-76-CHO-3073.

Research on Structural Formation and Selection Rules in Turbulent Plasmas

Members and this IAEA Conference

Grant-in-Aid for Scientific Research “Specially-Promoted Research” (MEXT Japan, FY 2004 - 2008)

Principal Investigator: Sanae-I. ITOH

Focus

Selection rule among realizable states through possible transitionsSearch for mechanisms of structure formation in turbulent plasmas

At Kyushu, NIFS, Kyoto, UCSD, IPP

M. Yagi: TH/P5-17 Nonlinear simulation of tearing mode and m=1 kink mode based on kinetic RMHD model

A. Fujisawa: EX/8-5Rb Experimental studies of zonal flows in CHS and JIPPT-IIU

A. Fukuyama: TH/P2-3 Advanced transport modelling of toroidal plasmas with transport barriers

P. H. Diamond: OV/2-1, This talk

K. Hallatschek: TH/P6-3 Forces on zonal flows in tokamak core turbulence

Y. Kawai, S. Shinohara, K. Itoh

Other member collaborators

25

s1

Zonal Flow and GAM: two kinds of secondary flow

CLVExB

CLVExB-+dn/dt

S(ω)

ω

Zonal Flow

GAM Drift Wave

Fluctuations~ cs/R

Er

time

(m=0, n=0)

n ~ 0ZF

GAM

s2

Impact on Transport

Shearing

Cross phase

Suppress fluctuation amplitude

Suppress flux, as well

Qr = pVr

∝ φ 2sinα

L-mode and Improved-confinement modes

L-modeImproved-

Confinement

DW + ZFDW + ZF

+Mean Er

Macro

Micro

s3

Old picturebare drift waves

New pictureDW-ZF system

Theoreticalform

User'sformula

Further issues

DgB

γdampωeff

DgB

Dmix

f

γdampωeff

Dmix

α-particle feedback zonal flow effects on TAE zonal field feeding neoclassical tearing mode

R =

γdampω*

s4