physics of zonal flows - pppl
TRANSCRIPT
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
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