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 ofProfessor Marshall N. Rosenbluth.

PlasmaTurbulence

- Itoh Project -

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 understood"Universality, Damping and Growth, Unifying Concept: Self-regulating dynamics

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

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:ParadigmChange

ITER

PoloidalCrosssection

m=n=0, kr = finite

ExB flowsZonal flow on Jupiter

3

Basic Physics of a zonal flow4

GenerationBy vortex tilting

Damping by collisions Tertiary instability

Suppression of DWby shearing

Drift waveturbulence

Zonal flows

Shearing

Collisional flow damping

SUPPRESSNonlinear flow damping

energyreturn

DRIVE

<vx vy>~~

∇T, ∇n...

<vx p>~~Transport

5

Self-regulation: Co-existence of ZF and DW

∂∂t WZF = γdamp ⋅ ⋅ ⋅ WZF + α WdWZF

∂∂t Wd = γ ∇ p0, ⋅ ⋅ ⋅ Wd – α WdWZF

Wd ∼

γ dampα

νii, q, εgeometry + rf, etc.

0 γdamp

DW

<U2> <N>ZF

damping rate of ZF

turb

ulen

cean

d tra

nspo

rt

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

6Why 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 ITERnear 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

β-limit

Steady state

Realizability ofIgnition

Issues infusion

H ∝ R – 0.6

$ ∝ H – 1.3

χ = R χgB $ ∝ R – 0.8

Flow damping ?

7

Assessment I "What we understood"ZFs are UNIVERSAL

Scales Transport Role of ZF

ITG

TEM/TIM

ETG

Resistiveballooning/interchangeDrift Alfvenwaves atedge

ρ i

ρ i

ρe

Δresis. layer

∼ ρ i

c s/a

c s/a

vTh,e/a

core

core

core χe, χ J

edge, sol,

helical edge

edge, sol

Veryimportant

Veryimportant

Veryimportant

Important

On-going

This IAEA ConferenceFalchetto (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)

χ i, χφ, D

χ i, χe, χφ, D

8

Linear Damping of Zonal FlowsNeoclassical process Rosenbluth-Hinton

undamped flow - survives for

Role of bananas Banana-plateau transitionCL

Screening effect if qrρp ∼ O 1

φq tφq 0

= 11 + 1.6ε– 1/2q2

τ > τ transport

Frictional damping

even in "collisionless" regime γdamp ∼ ν ii/ε χ i ∝ ν ii

9

Growth Mechanism

ZFs by

modulational instability χ i

Zonal flow (log)

R-factor

Numerical experiment indicatesinstability of finite amplitude gas ofdrift waves to zonal shears

10

Regime1 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

Zonal flow

Chiri

kov

para

met

er

DW lifetimePacket-circul. time

1

10

S

K

Stochastic

Turbulent trapping

Coherent

BGK solution

Parametric modulational instability

∞

∞

1

3 45

2

Kubo number

Close Relationship: DW + ZF and Vlasov Plasma

(i) DW + ZF :

(ii) 1D Vlasov Plasma :

∂∂t VZ + γdampVZ = – ∂

∂x V xV y

∂∂t N + vgx N – ∂

∂x k yVZ∂N∂k x

= γ k N + Ck N

∂E∂x = 4πn0e dv f

∂ f∂t + v d f

d x + em E ∂ f

∂v= C f

‘Ray’ Trapping

Particle Trapping

Ω = q x vg x

φ∼

particle

ω = k v

d xdt = vgx k d y

dt = vgy k

dk xdt = – ∂

∂x k yVZ dk ydt = 0

dvdt = e

m E d xdt = v

11

12

Note: Conservation energy between ZF and DW

RPA equations

Coherent equations

∂∂t Wd

by ZF

= – ∂∂t WZF

by DW

DW

ZF

2B2 d 2k

q x2k θ2 k x VZF, q2

1 + k⊥2ρs

2 2 R k, q x∂ N∂k x

Σq x

∂∂t VDW

2+ γ L, k + Ck N VDW, k2Σ

k=

∂∂t + γdamp VZF

2 =

– 2B2 d 2k

q x2k θ2 k x VZF, q2

1 + k⊥2ρs

2 2 R k, q x∂ N∂k x

Σq x

DW

ZF

(S: beat wave)

dZ2dτ = –

γ dampγL

Z 2 + 2 P Z S cos Ψ

dP2dτ = P2 – 2 P Z S cos 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 – γ 2 N 2 – α U 2 N

∂∂t U 2 = – γ damp U 2 + α U 2 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

P E (a.

u.)

ω/Ωi

-1

-0.5

0

0.5

1

10 11 12 13 14

C (r 1,r 2)

r2 (cm)

r1=12cm

Radial distance

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 2004 in press

14

15

Candidates for Collisionless Saturation

Trapping

Tertiary Instability

Higher Order Kinetics

VZF ∝ ωbounce ∼ Δω, γL

VZF ∼ φ +

Te2Ti

T qr

VZF ∼ Δω k θ– 1

Plateau ink-space N(k)

Returns energyto drift waves

Analogous tointeraction at beatwave resonance

Partial recovery of thedependence of on driftwave growth rate

χ i

VZF

φ

collisional

Collision-less

χ =γnlcollisionless

ωeffχ gB

16

"Near" Marginality

Shift exists

ITER plasma near marginalityImportance 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 k θ– 2 α

(Higher-order nonlinearity model)

17

Electromagnetic Effect

Possible Z-field inducedtransition, NTM,…

Transport at high-β,L-H transition

Implications for fusion

Random refraction of Alfvenwave turbulence

Modulational instability of adrift Alfven wave

Mechanism for ZF growth

Zonal magneticfield generation

ZF generation byfinite-β drift waves

Subject

1 + k⊥

2ρ s25/2

2 + k ⊥2 ρ s2

k⊥2 k y

2

k ||∂2

∂k x2

ωkNk

1 + k ⊥2ρ s2

f 0Σk

ηZF = – 4πc s2δ e

2

vth, e 1 + q r2δ e2

∂∂tBθ = – ηZF ∇

2Bθ

Current layer generationCorrugated magnetic shear,Tertiary micro tearing mode ?Can seed Neoclassical Tearing ModeLocalized current at separatrix of tearing mode, ……

vrvθ ⇒ vrvθ – BrBθSelected structure

Zonal flow Zonal fieldβ

18

Distinction between ZF and mean Field <Er>

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

ballistic

smoothly varying

changes on transport timescales

Mean Field <Er>

Turbulence

diffusive

oscillating, complexpattern in radius

can change onturbulence time scales

Zonal Flows

Drive

StretchingBehavior

k of waves

Space

Time

δk2 ∝ t

δk2 = t 2 k 2VE' 2

time

∼ 20 ρ i

∇ 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

Pressuregradient

Prediction ofbifurcation, dither,hysteresis, …

dNdt = – c1EN – c2N + Q

dVdt = V – dN 2 + Fnonlinear

dEdt = EN – a1E 2 – a2V 2E – a3VZF

2 E

Nonlinearity; orbit loss,etc. in previous model

dVZFdt = b 1

VZF2 E

1 + b 2V 2 – b 3VZF

Fluctuations

Pressure gradient

Zonal flow

Mean flow

Newcoupling

20

Zonal Flow and GAM: two kinds of secondary flow

S(ω)

ω

Zonal Flow

GAM Drift Wave Fluctuations~ c

s/R

δn ~ 0ZF GAM

GAMS: important near the edgeZF dynamics and GAM dynamics merge

ZF drive GAC: magnetic field sensitiveDW ZF GAM

Landau and collisional damping

damping

Lower temperature: , and ν ii ↑ c s/R ↓

New feature: geodesic acoustic coupling (GAC)

21

Control Knobs

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

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

Choice of wave, Wavepolarity, launching, (e.g.,studies of IBW), rationalsurfaces…

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

(Simple way does not work.)

R-factor Route to ITERHelps along

Barrier Transitionse. g.,

24

Acknowledgements For many contributions in the course of research on topic related to the material of thisreview, 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 andTechnology of Japan, by the Collaboration Programs of NIFS and of the Research Institute forApplied 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 SelectionRules 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

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

At Kyushu, NIFS, Kyoto, UCSD, IPP

PlasmaTurbulence

- Itoh Project -

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

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

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

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

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

Y. Kawai, S. Shinohara, K. Itoh

Other member collaborators

25