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Physics of Zonal Flows P. H. Diamond 1, S.-I. Itoh 2, K. Itoh 3, T. S. Hahm 4 1 UCSD, USA; 2 RIAM,...
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Transcript of Physics of Zonal Flows P. H. Diamond 1, S.-I. Itoh 2, K. Itoh 3, T. S. Hahm 4 1 UCSD, USA; 2 RIAM,...
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