DPP15-2015-000573
SymmetryBreakingbyParallelFlowShear:
DynamicsofIntrinsicAxialFlow inaLinearDevice,
and
ItsImplicationforIntrinsicRotation inTokamaks
J.Li,P.H.Diamond,UCSD
X.Q.Xu,LLNL
Y.Kosuga,KyushuUniversity,Japan
O.D.Gurcan,Ecole Polytechnique,France
Ackn:R.Hong,A.Ashourvan,S.C.Thakur,L.Cui,G.R.Tynan,P.Vaezi,UCSD
ThismaterialisbaseduponworksupportedbytheU.S.DepartmentofEnergy,Officeof
Science,OfficeofFusionEnergySciences, underAwardNumber DE-FG02-04ER54738
Outline
• Motivation
• LinearDeviceConfiguration andResults:CSDX&PANTA
• Problem: Originofaxialflow?
• DynamicalSymmetryBreakingMechanism
•
•
•
•
Introduction toresidualstress,problem withapplicabilityofconventional wisdom
Dynamicalsymmetrybreaking
Residualstress
Comparetostandardmechanism(negativeviscosityvsintrinsictorque)
• NegativeViscosityPhenomena
• Modulational instabilityforatestflowshear<->𝜒" vs|𝜒 $%& |
• Whatstops 𝑣( ) growth? – ParallelShearFlowInstability(PSFI)
• Flowstructure
• Turbulent pipeflowmodel: Δ𝑃( ,neutralboundary layer
• Flowprofile: Including PSFIeffectà 𝑣( ) structure
• Significancefortokamaks
• Conclusion
2
Summary
• Existenceofintrinsicflowsuggestedbyexperiments
• Dynamicalsymmetrybreaking: Atestflowshearseeds
symmetrybreakingà 𝛿 𝑣( ) feedsbackonitself
•
𝛿Π $%& ∼ 𝜒 $%& 𝛿⟨𝑣( ⟩′ inducesanegativeviscosityincrement
𝜒 $%& ,totalviscosity 𝜒 tot = 𝜒 − |𝜒 $%& |
"
"
• 𝑣( ) staysunderParallel ShearFlowInstabilitythreshold,
totalviscositystayspositive
• Asynergyofstandardresidualstressdrivenby𝛻𝑇, 𝛻P, etc.
and𝛿Π $%& inducednegativeviscosityincrementisimpliedfor
tokamaks.
3
Experiments:configuration
Gasinput
• PANTA[2]
• CSDX[1]
• Gasinputfromthesourceend
• Gasinputfromside
àAxialmomentuminput
àNoaxialmomentuminput
Parameters
CSDXTypical Values
PANTA TypicalValues
Source
<5kW
3kW
Pressure
0.1~1.3Pa
0.1Pa,0.4Pa
B field
Upto2400G
900G
𝑇%
3~6eV
3eV
𝑛%
0.5~2x1019m3
1x1019m3
𝑇E
0.3~0.8eV
---
*[1]S.Thakuretal,PlasmaSourcesSci.Technol.23(2014)044006;
[2]T.Kobayashi (2014,Jun).Parallelflowstructureformationbyturbulentmomentumtransportinlinearmagnetized
plasma. AsiaPacificTransportWorking Group,Kyushu University, Japan.
4
ProfileofAxialFlow
Endplate
cs ~ 3 km/s
• CSDX[1]
• Noaxialmomentuminput
• FlowprofilesteepensasBincreases
Source
• PANTA[2]
• Flowreversal
• Inputflow:insufficientmomentuminput
Intrinsicaxialflow!
Origin,physics?
*[1]L.Cui (2015,Nov). SpontaneousProfileSelf-OrganizationinaSimpleRealizationofDrift-WaveTurbulence.Invitedtalk,
Session BI3,57th APS-DPPmeeting,Savannah,Georgia.
[2]T.Kobayashi (2014,Jun).Parallelflowstructureformationbyturbulentmomentumtransportinlinearmagnetized
5
plasma. AsiaPacificTransportWorking Group,Kyushu University, Japan.
EvidenceofIntrinsicFlow
Device
CauseofDrivenFlow
CSDX
Neutralgas
EvidenceofIntrinsicFlow
Flow profile,densityprofilesteepenasB
increases:
à Ionized&heatedduringhelicon
discharge
𝛻𝑁 ↑ à DriftWave (DW)turbulence ↑
à Ionpressuregradientinaxial
direction Δ𝑃(
à Drivenflow
PANTA
à Turbulenttransportofaxialmomentum↑
à Intrinsicflowinteractswithdrivenflow
à Totalflowstructurechanges
Gasthruput intothesourceend
Flowreversal
à Externalmomentumsource
à Intrinsicflowinteractswithdrivenflow
à Drivesflow fromsourceto
endplate
à Globalnetflowdirection:
Sourceà Endplate
6
Problem
• Axialflows
• CSDX,PANTAbothsuggesttheexistenceofintrinsicaxialflows
• Questions:
• (1)Whatgeneratestheintrinsicflow?
• (2)Howdoesintrinsicflowinteractwithdrivenflow?
• Approach:
$%&
• IntrinsicflowacceleratedbyΠH∥
à Needssymmetrybreaking
• Standardapproach
$%&
à Intrinsictorque,−𝜕H ΠH∥
acceleratesflow
à DoesnotapplytostraightB fields
• Here:DynamicalSymmetryBreaking
7
à Negativeviscosityincrement
StandardApproach
• Intrinsicflowisacceleratedbytheresidualpieceofthe
momentumflux:
𝑣KH 𝑣K∥ =
L M∥
−𝜒" LH
$%&
+ 𝑉P 𝑣∥ + ΠH∥
• Ignoremomentumpinch𝑉P
• CorrelatedbyB fieldstructure,𝑘∥ =
shearlength
•
$%&
ΠH∥
∼ 𝑘R 𝑘∥ = ∑Y 𝑘R 𝑘∥|𝜙Y
|Z
• 𝑥:distancefromrationalsurface
=
S
𝑘R T ,𝐿&
U
= magnetic
Z S
𝑘R
TU
• Needssymmetrybreaking!
8
SymmetryBreaking
• Summaryofconventionalsymmetrybreakingmechanisms*:
Conventional mechanisms
Keyphysics
Electricfield shear𝐸H)
Centroidshiftà parallelacousticwave
asymmetryà mean⟨𝑘∥ ⟩
Intensity gradient𝐼 )
Spectraldispersionfromintensitygradient
Stress frompolarization
acceleration⟨𝐸^∥ 𝛻_Z 𝜙^⟩
Guidingcenterstressfromaccelerationdue
topolarizationcharge
Stressfrom𝜕H 𝑣KH 𝑣K_ àBR ⟨𝐽H ⟩
𝑱×𝑩 torque frompolarizationflux
• DonotapplytoCSDXorPANTAß StraightBfields
• à Dynamicalsymmetrybreaking
• Driftwaveturbulenceinpresenceofaxialflowshear
• 𝛿 𝑣( ) seedssymmetrybreaking
*P.H.Diamond etal,Nucl.Fusion 53(2013)104019;
P.H.Diamond etal,Nucl.Fusion 49(2009) 045002.
9
ModelEquations
• Hasegawa-Wakatani +Axialflow:
Acousticcoupling
• Acousticcoupling
• CoupleaxialflowfluctuationtoDW
• Familiar:Convertparallelcompressionintozonalflow*
• 𝑖𝛿:nonadiabatic electronresponseà Driftwaveinstability
• Dispersionrelation:
Driftwave
*Wangetal,PlasmaPhys.Control.Fusion 54(2012)095015
Symmetry
breaking
Ionacousticwave
10
DynamicalSymmetryBreaking
• Spectralimbalance:
• Growthrate~frequencyshift:
Infinitesimaltestaxialflow
shear,e.g.𝛿 𝑣( ) > 0
• Frequency:
• Frequencyshift~Flowshear:
kR
𝜙Y
Modeswith𝑘R 𝑘( > 0 grow
fasterthanothermodes,
𝛾Y |Ym Yn op > 𝛾Y |YmYn qp
Z
Spectral
imbalance
0
: {𝑘+}
Spectralimbalance(Fig.1)
k(
:{𝑘−}
{𝑘±}:Domainswheremodes growfaster/slower
Fig.1:Spectralimbalance.
0
k(
𝑘R 𝑘( ≠ 0
11
Quasilinear ReynoldsStress
• Reynoldsstress=diffusiveflux+residualstress
• Turbulentdiffusivity:
• Residualstress:
• Sumover2domains,accountingforthespectralimbalance
Spectralimbalance∼ 𝛿 𝑣(
)
12
Contrastthe2Stories
StandardSymmetry Breaking
Dynamical SymmetryBreaking
Freeenergy source 𝛻𝑇E,𝛻n,…depending onturbulence type Onlydriftwaveturbulence sofar,𝛻n
Symmetrybreaker
Radial electricfieldshear, 𝐸H) ;
Intensitygradient,𝐼 𝑥 ),etc.
All tiedtomagneticfieldconfiguration.
Testaxial flowshear, 𝛿 𝑣( );
No requirement forshearof𝑩
structure.
Effect ontheflow
$%&
Intrinsictorque, −𝜕H ΠH∥
Negativeviscosity,
$%&
𝛿ΠH(
= |𝜒 $%& |𝛿 𝑣(
Flowprofile
𝑣∥
Feedbackloop
)
$%&
ΠH∥
=
𝜒"
𝛻𝑇E +geometry
(magneticshear)
Heatflux
Openloop
𝑣∥
)
$%&
ΠH∥
)
$%&
Flowdrive(Π
,Δ𝑃( )
H(
𝑣( ) =
𝜒" − |𝜒 $%& |
Testflow
shear𝛿 𝑣(
)
Intrinsic
flow,
feedbackon
𝛿 𝑣( )
Closed
loop
Breaks the
symmetry,
spectral
imbalance
Residualstress
$%&
ΠH(
13
NegativeViscosityIncrement
$%& ∼ 𝜒 $%& 𝛿 𝑣 ),back-of-envelopestyle
• Calculatenegativediffusion𝛿ΠH(
(
• Quasilinear residualstress:
• 𝑁Y ∼ |𝜙Y |Z/𝜔Y waveactiondensitygovernedbywavekineticequation:
Convectionby
wavepacket
Refraction
Lineargrowth
Self-interaction
• Recall:
14
NegativeViscosityIncrement:cont’d
• Dynamicsofatestflowshear
(From formalcalculation)
• Negativeviscosityincrement:
• Growthrateofflowshearmodulation
15
Limitson 𝑣(
•
tot = 𝜒 − |𝜒 $%& | < 0 à 𝛿 𝑣
𝜒"
"
(
)
)
grows,profilesteepens,until…
• 𝑣( ) hitsParallelShearFlowInstability(PSFI)threshold
• PSFI:recalldispersionrelationofthemodelwithadiabaticelectrons:
• Unstable↔ discriminant
à PSFI
• à
• PSFIturbulenceà 𝜒"PSFI addsontotheambient𝜒"tot
à 𝜒"tot = 𝜒"DW + 𝜒"PSFI Θ
•
𝑣( ) − 𝑣(
)
ƒHE„
− |𝜒 $%& |
𝜒"PSFI switchedon,𝜒"PSFI > |𝜒 $%& | à 𝜒"tot > 0
• 𝑣( ) staysbelowPSFIthreshold;totalviscositystayspositive.
• Similartononlineardampingofzonalflow
16
FlowStructureinLinearDevice
PressuredropΔ𝑃(
Plasma
source,
heating
Neutrallayer
Pipe flow Plasma flow
Drive
Plasmaflow
Momentum outflux∼ 𝜌p ⟨𝑣H 𝑣( ⟩
Pressure
drop Δ𝑃(
Ion pressure
dropΔ𝑃(
Boundary Noslip
condition
Setby
neutrallayer
Viscosity
𝜒" − |𝜒 $%& |
𝜈
• IdeaofModel:Turbulentpipeflow,Prandtl +residualstress
• Prandtl (momentumbalance):
• Reynoldsstress:
• à Flowprofile:
17
FlowStructure:cont’d
• PSFIà Enhanceturbulentdiffusion,
eff = 𝜒 DW + 𝜒 PSFI Θ 𝑣 ) − 𝑣
𝜒"
(
(
"
"
)
ƒHE„
• IncludingPSFIeffect:
•
PSFI
𝜒"
nonlinearin⟨𝑣( ⟩′ à 𝜒"eff > |𝜒 $%&| à Profilerelaxes
• 𝑣( ) staysbelowPSFIthreshold
18
ImplicationforTokamaks
• SynergyofΠ $%& 𝛻𝑇, 𝛻𝑃, 𝛻𝑁 and𝛿Π $%& = 𝜒$%& 𝛿⟨𝑣( ⟩′
• DWturbulence,𝑩 shear
• Symmetrybreaker
(𝐸H) ,𝐼 𝑥 ),…)
$%&
• à residualstressΠH∥
• DWturbulence
• Testflowshear
• à negativeviscosity|𝜒 $%& |
• Flowprofilesetbymomentumfluxbalance:
• Enhancedflowprofile
Applicableto
electronDW’s
à CTEM
à Mechanismto
enhanceintrinsic
rotationpredictions
19
Conclusion
• ResultsfromCSDX,PANTAsuggestintrinsicaxialflow;
• Intrinsicmechanismtogenerateaxialflowsandtobuildupa
meanflowprofileisintroduced:
• Testflowshear𝛿 𝑣( )seedssymmetrybreakingandfeedsbackon
itself;
• Differentfromstandardsymmetrybreakingmechanism:
• Intrinsictorque−𝜕H Π$%& drivenby𝛻𝑇, 𝛻P, … v.s.Negativeviscosity
increment 𝜒 $%& inducedby𝛿Π$%& ;
• Flowstructureinalineardevice:
• Implicationfortokamaks:
• SynergyofΠ $%& 𝛻𝑇, 𝛻P, … and𝛿Π$%& = 𝜒 $%& 𝛿 𝑣( );
• Enhancedintrinsicrotationprofile:
20