Drive of a Vortex-Flow pattern by coupling to Zonal Flows ...€¦ · Drive of a Vortex-Flow pattern by coupling to Zonal Flows in presence of Resonant Magnetic Perturbations M. Leconte

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

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

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

Outline

1 Introduction






RMPs damp Zonal flowsZF damping in ballooning modesimulation Beyer ’02

↘ Long Range Correlation during RMP (TEXTOR)Y. Xu ’11

RMPs causedrop influctuationLRCsuggestsreduced ZFshearing

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]

Time-delay between RMP penetration and ELMsuppression in KSTAR

[Y.M. Jeon IAEA ‘14]

Outline

1 Introduction






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

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)

⟩]

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

Outline

1 Introduction






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.

Outline

1 Introduction






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)

Outline

1 Introduction






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

Outline

1 Introduction






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)

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

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 ↗

Poloidal section of Vortex-Flow pattern

Snapshot of 3D Vortex flowprofile δφVF (x , y)reconstructed from the 1Dmodel

Outline

1 Introduction






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)

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

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

Outline

1 Introduction






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?

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?

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).

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

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

Documents

Drive of a Vortex-Flow pattern by coupling to Zonal Flows ...€¦ · Drive of a Vortex-Flow pattern by coupling to Zonal Flows in presence of Resonant Magnetic Perturbations M. Leconte