Competition ofMeanPerpendicularandParallelFlowsinaLinearDevice

J.C.Li,P.H.Diamond,R.Hong,G.TynanUniversityofCaliforniaSanDiego,USA

ThismaterialisbaseduponworksupportedbytheU.S.DepartmentofEnergy,OfficeofScience,OfficeofFusionEnergySciences,underAwardNumbersDE-FG02-04ER54738.

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FestivaldeThéorie 2017,Aix-en-Provence,France

Background• Intrinsic axialandazimuthalflowsobservedinlineardevice(CSDX)• IncreaseBà scanmeanflows-𝑉" and𝑉∥• Dynamical competitionbetweenmeanperpendicularandparallelflows• [SeeGeorgeTynan’stalkearlier]

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– Dynamical: 𝑉" and𝑉∥ exchangeenergywiththebackgroundturbulence,andeachother.à Energybalancebetween𝑉" and𝑉∥à TradeoffbetweenV" andV∥

KeyQuestionsandWhy

• What’sthecouplingbetweenmean perpendicularandparallelflows(𝑉" and𝑉∥)?– Howdotheyinteract?à Howdotheycompeteforenergyfromturbulence?

– Canwehaveareducedmodelofthecouplingbetween𝑉" and𝑉∥?• Whyshouldwecare?– Lineardevice(CSDX)studiessuggestapportionmentofturbulenceenergybetween𝑉" and 𝑉∥

– RelevanttoL-Htransition• Both𝑉"& and𝑉∥ increase,duringtransition.• Thecouplingofthetwoisrelevanttotransitionthresholdanddynamics.

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OutlineoftheRest

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• Currentstatusofmodel• Explorationof𝑉" and𝑉∥ competition– Turbulentenergybranching between𝑉∥ and𝑉"– Reynoldspowerratio𝑃∥( 𝑃"(⁄ decreasesas𝑉" increasesà tradeoffbetween𝑉" and𝑉∥

– 𝑃∥( 𝑃"(⁄ maximumoccurswhen 𝛻𝑉∥ isbelowthePSFI(parallelshearflowinstability)thresholdà saturationofintrinsic𝑉∥

• Wave-flowresonanceeffects– Areshearsuppression“rules”alwayscorrect?– 𝑉"& weakensresonanceà flowshearenhancesinstability

– Implicationforzonalflowdynamics

Currentstatusofmodel

• Conventionalwisdomof𝑉" → 𝑉∥ coupling:– 𝑉"& breaksthesymmetryin𝑘∥,butrequiresfinitemagneticshear– Notapplicable inlineardevice(straightmagneticfield)

• 𝑉∥ → 𝑉" couplingviaparallelcompression:– 3Dcoupleddrift-ionacousticwavesystem [Wangetal,PPCF2012]– CouplingbetweenfluctuatingPVandparallelcompression 𝑞.𝛻∥𝑣.∥breaksPVconservationà Sink/sourceforfluctuatingpotentialenstrophy densityà Zonalflowgeneration

SectionII:Explorationof𝑉"-𝑉∥ Coupling

• Hasegawa-Wakatani driftwaveà nearadiabaticelectron:

𝑛. = 1 − 𝑖𝛿 𝜙, 𝛿 ≪ 1

• Slabgeometry

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• Goal:studyhowextrinsicflowsaffectReynoldspowers

à generationofintrinsicflows

à turbulentenergybranchingbetweenintrinsicV" andV∥

• Analogoustoincrementstudy

BottomLine:𝛻𝑛9 isthePrimaryInstabilityDrive

• KHisnotimportant– 𝑉"&& driveweakerthan𝛻𝑛9drive,i.e. 𝑘:𝜌<=𝑉"&& ≪ 𝜔∗@

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• PSFIstableinCSDX

• Otherpotentialdrives:– 𝑉"&& à Kelvin-Helmholtzinstability– 𝛻𝑉∥ à Parallelshearflowinstability

Definition:ReynoldsPower

• MeanflowevolutionisdrivenbyReynoldspower12𝜕 𝑉∥

=

𝜕𝑡 ∼ −𝜕𝜕𝑥 𝑣.F𝑣.∥ 𝑉∥

– ParallelReynoldspowerofasingleeigenmode

𝑃∥( = G 𝑑𝑥 −𝜕𝜕𝑥 𝑣.F,I∗ 𝑣.∥,I 𝑉∥

JK

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– PerpendicularReynoldspowerofasingleeigenmode

𝑃"( = G 𝑑𝑥 −𝜕𝜕𝑥 𝑣.F,I∗ 𝑣.:,I 𝑉"

JK

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• Effectsofextrinsic𝑉∥ and𝑉" ontheratio𝑃∥( 𝑃"(⁄ arestudied

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Couplingof𝑉" and𝑉∥ ↔ RatioofReynoldsPowers

• Ratio𝑃∥( 𝑃"(⁄ decreaseswith𝑉"à Energybranchingof𝑉∥ reducedà 𝑉" reducesgenerationof𝑉∥à Competition between𝑉" and𝑉∥

• Increase𝑉∥ à 𝑃∥( 𝑃"(⁄ turnoverbefore 𝛻𝑉∥ hitsPSFIthresholdàMaxenergybranchingof𝑉∥ belowPSFIthresholdà 𝑉∥ saturatesbelow PSFIthreshold

SectionIII:RevisitingWave-FlowResonance

• Areconventionalshearsuppression“rules”alwayscorrect?– 𝐸×𝐵 flowshearsuppressesinstabilityß Isitcorrectwithresonance?– Wave-flowresonanceeffectisoftenoverlooked,thoughwasmentionedin

pastworks.• Findings:

– Wave-flowresonancestabilizesdriftwaveinstability– Perpendicularflowshearweakenstheresonance,andthusdestabilizes the

instability• Implicationsforzonalflowsaturation:

– Collisionless zonalflowsaturation(withoutinvolvingtertiaryinstabilities,suchasKH)setbyresonance,𝐷X ∼ 𝜔I − 𝑘:𝑉"

Y=

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[Li&Diamond,manuscriptinpreparation]

Wave-flowresonance

• Hasegawa-Wakatani driftwavemodel,withextrinsic𝑉"

• KHdrivenegligibleà Driftwaveinstabilitydominant• Nearadiabaticelectron:𝑛. = 1 − 𝑖𝛿 𝜙,𝛿 ≪ 1• 𝛿 = 𝜔∗@ − 𝜔I + 𝑘:𝑉" 𝑘∥=𝐷∥=[ = 𝜈@] 𝜔∗@ − 𝜔I + 𝑘:𝑉" 𝑘∥=𝑣^_@=[

• Resonancereducestheeigenmode scaleà Suppressesinstability

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• Resonance:𝜔I − 𝑘:𝑉" − 𝑘∥𝑉∥𝑘∥ 𝑘:[ ≪ 1à Resonancedominatedby𝜔I − 𝑘:𝑉"

(Widthofeigenmode)

Perpendicularflowsheardestabilizes turbulence

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• Meanperpendicularflowshearincreasesmodescale𝐿a 𝜌<⁄ àWeakensresonanceà Enhancesinstability

• KHdrivenegligible comparedto𝛻𝑛9

ImplicationsforZonalFlowSaturation• Connectiontocollisionless saturationofZF• ZonalflowevolutionàMeanenstrophy equation:

• Vorticity(𝜌 ≡ 𝛻"=𝜙)flux: 𝑣.c𝜌. = −𝐷Xd Xdc

+ ΓX(@<

𝜕𝜕𝑡 G𝑑𝑟

𝜌 =

2

�

�= G𝑑𝑟 𝑣.c𝜌.

𝑑 𝜌𝑑𝑟

�

�− 𝜈] G𝑑𝑟 𝜌 =

�

�+ ⋯

Conservesenstrophy betweenmeanflowandfluctuations

𝜕𝜕𝑡G𝑑𝑟

𝜌 =

2

�

�= −G𝑑𝑟𝐷X

𝑑 𝜌𝑑𝑟

=�

�+ G𝑑𝑟ΓX(@<

𝑑 𝜌𝑑𝑟

�

�− 𝜈] G𝑑𝑟 𝜌 =

�

�+ ⋯

• 𝜈] → 0à Dimitsshiftregimeà ResonancesaturatesZF,w/oKH• Collisionless dampingbyturbulentviscosity:𝑑 𝜌 𝑑𝑟⁄ ∼ ΓX(@< 𝐷X[• Resonancesets𝐷X à ZFsaturation

ΓX(@< =j𝑘:𝑐<= 𝜙I = 𝛾I𝜔∗@ + 𝛼n 𝜔∗@ − 𝜔I + 𝑘:𝑉"𝜔I − 𝑘:𝑉" + 𝑖𝛼n

= −𝛾I 𝜔∗@

𝜔I − 𝑘:𝑉"=

�

I

, 𝐷X =j𝑘:=𝑐<= 𝜙I = 𝛾I𝜔I − 𝑘:𝑉"

= �

I

Summary• CSDXexperimentssuggestenergyapportionment betweenmean𝑉" and𝑉∥

• Reynoldspowerratio𝑃∥( 𝑃"(⁄ changesinresponsetoexternalflowincrement– 𝑃∥( 𝑃"(⁄ decreaseswith𝑉" à tradeoffbetween𝑉" and𝑉∥– 𝑃∥( 𝑃"(⁄ maximumoccursbefore 𝛻𝑉∥ hitsPSFIthreshold

• Testingmisconceptionsofshearingeffectsonstability– Wave-flowresonancesuppressesinstability– 𝑉"& weakensresonanceà 𝑉"& enhances instability– Resonanceproducesturbulentviscosity

à collisionless saturationofZF,without involvingtertiaryinstabilities

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Backup

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DetailsonAcousticCoupling

• 𝑉∥ → 𝑉" couplingviaparallelcompression:– 3Dcoupleddrift-ionacousticwavesystem [Wangetal,PPCF2012]– CouplingbetweenfluctuatingPVandparallelcompression 𝑞.𝛻∥𝑣.∥breaksPVconservationà Sink/sourceforfluctuatingpotentialenstrophy densityà Zonalflowgeneration

– Perpendicularflowdynamics:

𝜕𝜕𝑡 𝑉" − 𝐿n

𝑞.=

2 ∼ −𝜈]𝑉" + 𝐿n𝜕𝜕𝑟 𝑣.F

𝑞.=

2 + 𝜇 𝛻𝑞. = − 𝑞.𝛻∥𝑣.∥

𝑞.𝛻∥𝑣.∥ ∼ −j𝛥𝜔I𝜔I=

�

I

𝑘∥= 𝜙I = < 0collisionaldamping

PVdiffusion

StationaryZonalFlowProfile• Turbulentviscositysetbyresonance:

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ΓX(@< =j𝑘:𝑐<= 𝜙I = 𝛾I𝜔∗@ + 𝛼n 𝜔∗@ − 𝜔I + 𝑘:𝑉"𝜔I − 𝑘:𝑉" + 𝑖𝛼n

= −𝛾I 𝜔∗@

𝜔I − 𝑘:𝑉"=

�

I

,

• Residualvorticityflux:

• Reynoldsforce(i.e.netproduction)=0à Stationaryflowprofile:

𝐷X =j𝑘:=𝑐<= 𝜙I = 𝛾I𝜔I − 𝑘:𝑉"

=

�

I

∼

ResonanceandInstabilityRelatedtoModeScale

• Eigenmode equationwithresonanteffect:

Effectively,𝑘"=𝜌<=

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• Modescale:𝐿aY=𝜌<= ≡ 𝜌<= ∫ 𝑑𝑥 𝜕F𝜙 =JK9 ∫ 𝑑𝑥 𝜙 =JK

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• Results:

• Strongresonance𝛾I ≪ 𝜔I − 𝑘:𝑉" ≪ 𝜔∗@

• Eigenmodepeaks(𝐿aY=𝜌<=increases)asresonancebecomesstronger

• Resonancesuppressesdriftwaveinstability

AnalogytoLandauDampingAbsorption

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Langmuir TurbulenceCollapse Collisionless ZFSaturation

Players Plasmon-Langmuirwaveandion-acousticwave(caviton)

Driftwaveandzonalflow

FinalState (Nearly)emptycavity Dimits statezonalflowdominant

Freeenergysource Langmuirturbulencedriver 𝛻𝑛,𝛻𝑇 drives

Resonanceeffect Landaudampingascavitycollapses Absorptionby𝐷X ∼𝜔I − 𝑘:𝑉"

Y=

Othersaturationmechanisms

Ion-acousticradiationfromemptycavity

Kelvin-Helmholtzrelaxation

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𝛾I = linearinstability(𝛾J)+resonanceabsorption𝛾( ∼ 𝛾( 𝜔I − 𝑘:𝑉"

Analogytoion-acousticabsorptionduringcollapseofLangmuirwaves

• Resonanceinducescollisionless saturationthrough𝐷X,apartfromKH:

• Revisitpredator-preymodelwithresonanceeffectàMechanismforcollisionless damping,withoutKH

DriftWaveTurbulence

Wave-FlowResonance

𝑉"ΓX(@<

𝐷X

ZFdrive

ZFdissipation

Resonanceeffects

Revisitpredator-preymodel