Drive of a Vortex-Flow pattern by coupling to Zonal Flows ...€¦ · Drive of a Vortex-Flow...
Transcript of Drive of a Vortex-Flow pattern by coupling to Zonal Flows ...€¦ · Drive of a Vortex-Flow...
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Drive of a Vortex-Flow pattern by coupling toZonal Flows in presence of Resonant Magnetic
Perturbations
M. Leconte and Juhyung Kim
Adv. Physics Research DivisionNFRI, Korea
Acknowledgements: P.H. Diamond, Y.M. Jeon
TTF 2015
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Outline
1 Introduction
2 MotivationRMPs affect turbulence-driven flows
3 1D Zonal Flow - Vortex Flow modelBasic MechanismModel for Zonal Flow shear in 1DEquationsSchematic derivation of the model
4 Our Results/ContributionNumerical resultsImplications
5 DiscussionBasic Ideas for experiments
6 Summary and conclusions
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General Intro
ITER requires control of Edge Localized ModesPopular approach: Resonant Magnetic Perturbation (RMP)
RMPs generate magnetic island chains, stochastic layerThese modify turbulence, flows and gradients
Working hyp. for ELM suppression: RMP generateadditional transport↪→ limits the pedestal below ideal MHD threshold.Why ELM suppression needs - spatial - resonancecondition ? (q95 window)
requires 1D modelcoherent magnetic perturbation
we don’t address the plasma response
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Outline
1 Introduction
2 MotivationRMPs affect turbulence-driven flows
3 1D Zonal Flow - Vortex Flow modelBasic MechanismModel for Zonal Flow shear in 1DEquationsSchematic derivation of the model
4 Our Results/ContributionNumerical resultsImplications
5 DiscussionBasic Ideas for experiments
6 Summary and conclusions
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RMPs damp Zonal flowsZF damping in ballooning modesimulation Beyer ’02
↘ Long Range Correlation during RMP (TEXTOR)Y. Xu ’11
RMPs causedrop influctuationLRCsuggestsreduced ZFshearing
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RMPs modify spatial structure of E × B flows
Contours of the E × B flowv.s. helical angle inRFX-Mod [in ‘tokamak’config) [Vianello ’15] :
Poloidal cuts of 3Dresistive ballooning simul.showing RMP-inducedquasi-static helicalperturbations of potential:
E × B flow Electric potential[Leconte ’10, Marcus ’13]
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Time-delay between RMP penetration and ELMsuppression in KSTAR
[Y.M. Jeon IAEA ‘14]
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Outline
1 Introduction
2 MotivationRMPs affect turbulence-driven flows
3 1D Zonal Flow - Vortex Flow modelBasic MechanismModel for Zonal Flow shear in 1DEquationsSchematic derivation of the model
4 Our Results/ContributionNumerical resultsImplications
5 DiscussionBasic Ideas for experiments
6 Summary and conclusions
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Basic Mechanism: Nonaxisym. Field-line bending
MPs→ distortion of field lines→ δBr component
generates electron transport ⊥ unperturbed fieldIf MP has random phase δbx = bxeiαrand + c.c.↪→ ZF damping ∼ |bx |2 [Leconte & Diamond ‘11]
if MP has coherent phase δbx = bx sin(Kyy)
↪→ spatial resonance y ∼ θ − ϕ/q0
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Basic Mechanism (cont’d)
single-fluid approx.: shows coupling of Zonal Flows toVortex-Flow via resonant current:
⟨j‖ sin(Kyy)
⟩= −D‖bx
[bx∂φzon
∂x− K‖0(x)
⟨φ cos(Kyy)
⟩]
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Basic Mecanism (cont’d)
δBr linearly couples Zonal φ to Vortex φ
ZF
VF
Turb
X
RMP
X
ZF+RMP
Weak coupling→ small correctionStrong coupling→ electron force balance↪→ NL potential becomes a flux-surface function
single-fluid electron force balance
bxVzon − K‖0(x)φVF ∼ 0
Note: also has implications for polarization current oftearing-modes
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Outline
1 Introduction
2 MotivationRMPs affect turbulence-driven flows
3 1D Zonal Flow - Vortex Flow modelBasic MechanismModel for Zonal Flow shear in 1DEquationsSchematic derivation of the model
4 Our Results/ContributionNumerical resultsImplications
5 DiscussionBasic Ideas for experiments
6 Summary and conclusions
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Cahn-Hilliard (CH) model for Zonal Flow shear in 1D
spatial resonance↪→Wave Kinetic theory not applicable here↪→ Reynolds stress↔ Up-Gradient diffusionIn Geofluid dynamics, the Cahn-Hilliardmodel ‘58 (paradigm for phase separation)describes jets/Zonal Flows [Manfroi & Young‘99]:
Figure: CH model
∂C∂t
= − ∂2
∂x2
[νUGC
]− νhyp
∂4C∂x4 −µC
Phase separation
C : concentrationνUG = 1− C2:Up-Gradient diffusivityνhyp: hyper-viscosity
Geo Fluid / Fusion Devices
C = ∂xVZF : ZF shearνUG ∼ νUG0 − (∂xVZF )2:turb. ‘negative’ viscosityi.e. Up-Gradient diffusionµ : bottom drag / neoclass.
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Outline
1 Introduction
2 MotivationRMPs affect turbulence-driven flows
3 1D Zonal Flow - Vortex Flow modelBasic MechanismModel for Zonal Flow shear in 1DEquationsSchematic derivation of the model
4 Our Results/ContributionNumerical resultsImplications
5 DiscussionBasic Ideas for experiments
6 Summary and conclusions
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1D Zonal Flow - Vortex Flow model
ZF - VF model
[∂
∂t+ µ
]VZF = − ∂
∂x
[ε∂VZF
∂x
]− µRMP
[VZF −
xW 2φVF
]+hypervisc. dissipat.
∂
∂t∂2φVF
∂x2 = −µRMPx
W 2
[VZF −
xW 2φVF
]+ visc. dissipat.
VZF : Zonal FlowsφVF : Vortex Flow amplitude
µRMP ∼ b2xν−1ei : RMP-induced friction
W ∼√
bx : vacuum island-widthε = 1− (∂xVZF )2 : turb. intensity
µ ∼ νii : ZF neoclass. friction (ion-ion collisions)
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Outline
1 Introduction
2 MotivationRMPs affect turbulence-driven flows
3 1D Zonal Flow - Vortex Flow modelBasic MechanismModel for Zonal Flow shear in 1DEquationsSchematic derivation of the model
4 Our Results/ContributionNumerical resultsImplications
5 DiscussionBasic Ideas for experiments
6 Summary and conclusions
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Schematic derivation of the model
Charge balance (single fluid)
'Quasilinear' model - (0:0) ZF evolution + Reyn. stress + QL jxb torque
- (M:N) Vorticity evolution + 3rd order jxb torque
QL analysis
Some Algebra- Multiply vorticity eq. by cos(K y)- average over unperturbed flux-surfaces
Zonal Flow – Vortex Flow model
Hyp: 'negative'eddy viscosity
Slave turbulenceto Zonal Flows
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Outline
1 Introduction
2 MotivationRMPs affect turbulence-driven flows
3 1D Zonal Flow - Vortex Flow modelBasic MechanismModel for Zonal Flow shear in 1DEquationsSchematic derivation of the model
4 Our Results/ContributionNumerical resultsImplications
5 DiscussionBasic Ideas for experiments
6 Summary and conclusions
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Numerical results: Spatiotemporal dynamics
(a)
3 2 1 0 1 2 3
x[ρ]0
2
4
6
8
10
t
Vzf
(b)
3 2 1 0 1 2 3
x[ρ]0
2
4
6
8
10
t
ω
RMPs are turned on @ t=5At RMP onset:
enhanced ZF damping drive of Vortex-Flow(M : N vorticity)
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Numerical results: Profiles
−1 −0.5 0 0.5 1−8
−6
−4
−2
0
2
4
6
8
x [ρs]
VZF
@ t=10
0.5 φVF
@ t=10
Zonal Flows setfine-structure ofVortex-Flow pattern
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Timeseries of Zonal Flow energy EZF
(a)
4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
t0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
E(V
zf)
bx 0 5.00 5.02 5.04 5.06 5.08 5.10 5.12 5.14 5.160.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 Drop in the Zonal Flow energy↪→ enhanced ZF damping↪→ increased turbulenceSlow-growth γVF of the VFpattern
µRMPµ ↗ : γVF ↗
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Poloidal section of Vortex-Flow pattern
Snapshot of 3D Vortex flowprofile δφVF (x , y)reconstructed from the 1Dmodel
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Outline
1 Introduction
2 MotivationRMPs affect turbulence-driven flows
3 1D Zonal Flow - Vortex Flow modelBasic MechanismModel for Zonal Flow shear in 1DEquationsSchematic derivation of the model
4 Our Results/ContributionNumerical resultsImplications
5 DiscussionBasic Ideas for experiments
6 Summary and conclusions
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Implications
Relation between RMP-induced ZF damping andturbulence increase
ZFs act as carrier-wave for the Vortex-Flow patternSpatial resonance→ envelopeSlow-growth of Vortex-Flow pattern↪→ possible explanation for delay between coil activationand ELM suppression (KSTAR shot ] 9286)
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RMP↗ particle transport via Vortex Flow
Our analysis neglects coupling to densityHeuristically, Vortex Flow couples to Vortex density:↪→ Long-lived Convective Cells
φVF cos(Kyy)↔ nVF sin(Kyy)
could generate convective transport Γ ∼ φVF nVF↪→ could provide necessary radial transport for ELMsuppression↪→ ongoing work
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Discussion
Can our theory explain formation of mesoscale E × Bvortices ?(N. Vianello ’15)
partly: Zonal Flows drive Vortex Flow pattern in presence ofδBrbut Vianello et al. pattern looks more complicated
Can our theory explain time-delay before ELM suppressionon KSTAR?
Our model shows long-timescale for VF growthZonal Flows are difficult to observe in KSTAR (c.f. Zoletnik)Ongoing work to compare theory / experiment
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Outline
1 Introduction
2 MotivationRMPs affect turbulence-driven flows
3 1D Zonal Flow - Vortex Flow modelBasic MechanismModel for Zonal Flow shear in 1DEquationsSchematic derivation of the model
4 Our Results/ContributionNumerical resultsImplications
5 DiscussionBasic Ideas for experiments
6 Summary and conclusions
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Basic Ideas for experiments
Can the Vortex-Flow pattern be detected in tokamaks?diagnostics: BES?How does Vortex-Flow amplitude φ2
VF scale with δBr/B ?c.f. [N. Vianello ’15]How does time-delay before ELM suppression scale withδBr/B & collisionality?
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Summary and conclusions
RMPs damp Zonal FlowsRMPs mediate energy coupling:ZF ↔ Vortex-Flow pattern[Leconte & J.H. Kim] (submitted to PoP)
Open Questionscoupling to density→ convective transportRMP effect on density, temperature? → particle v.s. heatpartition: How much energy goes back to the turbulence?How much energy goes to drive the Vortex-Flow?
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THANK YOU
This work was supported by R&D Program through NationalFusion Research Institute (NFRI) funded by the Ministry ofScience, ICT and Future Planning of the Republic of Korea(NFRI-EN1541-1).
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Model: single-fluid charge balance
Model
ρ2s∂∇2⊥φ
∂t= ∇‖j‖
Quasilinear analysis: φ = φZF + δφVF ,j‖ = −D‖
[δbxVZF +∇‖0δφVF
]QL Model
ρ2s∂
∂t∂VZF
∂x+ ρ2
s∂
∂x
⟨v turb
x ∇2⊥φturb
⟩= −D‖
∂
∂x
⟨δb2
xVZF − δbx∇‖0δφVF
⟩ρ2
s∂
∂t∂2
∂x2 δφVF = −D‖∇‖0[δbxVZF −∇‖0δφVF
]with VZF = ∂xφZF
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Coupling to the VF pattern
Key point: consider sidebands of Zonal Flowsextension of standard analysis for Geodesic Modes
for Geodesic Modes: consider (0 : 1) pressure sidebandfor the VF pattern, consider (M:N) potential sideband