OnthePhysicsofIntrinsic
FlowinPlasmasWithout
MagneticShear
J.C.Li,R.Hong,P.H.Diamond,
G.R.Tynan,S.C.Thakur,X.Q.Xu
Acknowledgment:This material is based upon work supported by the U.S. Department of Energy,
1
Office of Science, Office of Fusion Energy Sciences, under Award Number DE-FG02-04ER54738
Outline
• Meritsofweakmagneticshear+rotation forconfinement
• Question:intrinsicrotationwithweakshear?
• CSDX:atestbedforintrinsicflowsinunsheared magneticfields
• Intrinsicaxialflow
• Newmechanismforintrinsicflows
• Norequirementformagneticsymmetrybreaking
• Buildsonelectrondriftwaveturbulence
• Broaderlessonfortokamaks
• Notlimitedtoweakshearregimes
• Outcome:enhanced𝛻⟨𝑣$⟩
• Futurework:flowboundarylayer↔ ion-neutralcoupling
2
Why“weakshear”profile?
• Longknown:
ERS,TFTR
• weakorreversedshear
beneficialforconfinement(ERS,
NCS,WNS,…)
• B.W.Rice(DIII-D),PPCF, 1996;
• E.J.Synakowski (TFTR),PoP,1997;
• M. Yoshida (JT-60U), NF, 2015...
JET,weakshear
• De-stiffened,ITB-likestate
observed for“weakshear”(JET)
• P. Mantica, PRL, 2011;
• C. Gormazano, PRL, 1998, etc.
• For more, see previous talks by Dr.
Turco, Dr. Pan.
WNS,JT-60U
3
IntrinsicRotationinWeakShear Profiles
• JET:WeakshearAND Rotationà Enhancedconfinement
• ButexternaltorquelimitedinITER
• Soneedunderstand:Intrinsicrotationinweakshearregimes
• Importantfor:
• Totaleffectivetorque
𝜏 = 𝜏)*+ + 𝜏-.+/
4
• Contributionto𝑉1×3
[P. Mantica, PRL, 2011;
Rice, PRL, 2013]
4
IntrinsicRotationinITB
• Intrinsic rotation: self-accelerated
toroidal rotation
• Discovered in JFT-2M, C-Mod (~
95’)
• During ITB formation:
• 𝜏-.+/ in the core flip sign
• Build up from edge
• 𝜏-.+/ ∼ 𝜏)*+
• Strong coupling between heat
transport and momentum transport
• Consistent with
Δ 𝑣$ ∼ 𝛻𝑇, 𝛻𝑃
[Ida(LHD), PRL,2013]
5
IntrinsicRotationinWeakShear
• Weakshear(𝑞 4 → 0)
IntrinsicRotation?
• Externaltorque≅ 0
Status:
• Beneficialforconfinement
andstability.Howmuch?
[G.M.Staebler andH.E.StJohn, NFLett.,2006]
• Results
• GKSimulation:strongerintrinsicrotationat
weakermagneticshear
• Problem
•
•
•
•
Intrinsicrotationrequiressymmetrybreaking
Conventionalsymmetrybreakingmodelsfail
Mostinvolvemagneticshear
Butweakshear
à non-resonantmodestructure!
[Kwon,NF(2012);
Z.X.Lu,NF&PoP,2015]
6
AconceptualModelofIntrinsicRotation:
HeatEngine
• Carmotionvsplasmarotation[Kosuga,PoP,2010]:
Car
IntrinsicRotation
Fuel
Gas
Heatingà 𝛻𝑇, 𝛻𝑛 ?
Conversion
Burn
𝛻𝑇,𝛻𝑛 ? driventurbulence
Work
Cylinder
Symmetrybreaking
Result
Wheel rotation
Flow
• Plasmarotation:
Heatflux
Turbulence
Symmetrybreaking
Usuallyrequiresmagneticshear
Residualstress
B)C
Π/∥
𝛻𝑇, 𝛻𝑛 ?
Intrinsicrotation
𝑣$
4
B)C
Π/∥
𝛻𝑇, 𝛻𝑛 ?
∼
𝜒$
7
Details:ConventionalWisdomof
IntrinsicRotation
• Self-accelerationbyintrinsictorqueduetoresidualstress
(𝜏-.+/ = −𝛻 ⋅ Π B)C)
𝑣G/ 𝑣G∥ = −𝜒$
𝑑 𝑣∥
B)C
+ 𝑉J 𝑣∥ + Π/∥
𝑑𝑟
B)C
• ResidualstressΠ/∥
B)C
• Drivenbyturbulence,i.e.Π/∥
∼ 𝛻𝑃, 𝛻𝑇, 𝛻𝑛?
B)C
• Π/∥
∼ 𝑘Z 𝑘∥ requiressymmetrybreakingink space
• Symmetrybreakingusuallyreliesonmagneticshear
B)C
• Rotationbuildsupfromedge,drivenbyΠ/∥
atedge
[W.X.Wang, PRL,2009]
8
ASimpleExample
• 𝑘∥ =
*
𝑘Z \
]
à 𝑘Z 𝑘∥ ∼
^ ⟨*⟩
𝑘Z \
]
• ⟨𝑥⟩:averageddistancefrommodecenterto
rationalsurface
• 𝑥 set,insimplemodels,by:
•
𝐸/4 :centroidshift
𝑥
• 𝐼 4 𝑥 :spatialdispersionofenvelope
• Whatofweakshear(𝑞 4 → 0)?
[Gurcan etal,PoP,2007&2010]
𝑟
9
IntrinsicParallelFlowinLinearDevice
• ControlledShearDecorrelation Experiment(CSDX)
• Straight,uniformmagneticfieldinaxialdirection
• Idealtestbedforstudyingintrinsicflowsinunsheared magneticfields
DeviceofCSDX.
10
MoreGenerally:
Whystudylineardevice?
• CorrespondencebetweenCSDXandtokamaks:
Tokamaks
CSDX
Toroidal field structureusually
sheared
Intrinsictoroidal rotation
Rotationboundarycondition
setbySOL
L-Htransition
Inwardpinch;densitypeaking
Uniform axialmagneticfield(shearfree)
Intrinsicaxialflow
Axialflowboundaryconditionsetby
boundaryneutral layer
[Cui,PoP,2015, 2016]
Transport bifurcationdrivenby𝛻𝑛?
𝑛? profilesteepening;localizednet
inwardparticleflux
11
Observation
• Intrinsicaxialflowevident(withoutmomentuminput)
• Barrierformedwhen𝐵 > 𝐵c/-+
• 𝛻⟨𝑣d ⟩ steepeningoccurswith𝛻𝑛?,𝛻𝑃- steepeningà barrierformation
<1200G
⟨𝑣d ⟩ profileevident,steepensduringtransition
[Cui,PoP,2015&2016; TTFtalk,2015]
12
Intrinsic𝛻 𝑣d tracks𝐿fg
.
• AxialflowinCSDX:
g
• 𝑣d 4 ∼ . 𝛻𝑛?
h
• 𝛻𝑛? isfreeenergysource
• Recall
• IntrinsicrotationinC-Mod:
• Δ 𝑣$ ∼ 𝛻𝑇
[Rice,PRL,2011]
13
AxialReynoldsPowerTracks 𝐿fg
.
• TotalReynoldspower:
+i+ = −∫ 𝑣
𝑃B)C
G/ 𝑣Gd 4⟨𝑣d⟩𝑟𝑑𝑟
• Totalpowercoupledto
𝛻⟨𝑣d ⟩ fromfluctuations
• AxialReynoldspowerrises
with1/𝐿. (for𝐵 < 𝐵c/-+ )
• Evidencefordirect
connectionoffluctuations
withintrinsicflow
14
Theory
• DynamicalSymmetryBreaking
• Norequirementonspecificmagneticfieldstructure
à aspectsrelevanttobothweakshear,andstandard
configurations
• Electrondriftwaves(CTEMà ITERrelevant)
15
ElectronDriftWaveSystem
• Systemequations:
𝐷
1 1 𝜕𝜙 𝜕𝑣),d
𝑛 +
+
=0
𝐷𝑡 ) 𝐿. 𝑟 𝜕𝜃
𝜕𝑧
𝐷 ^
𝜕
𝛻t 𝜙 =
𝑣d − 𝑣),d
𝐷𝑡
𝜕𝑧
𝐷
1 𝜕𝜙
𝜕𝑛)
4
𝑣d − 𝑣d
=−
𝐷𝑡
𝑟 𝜕𝜃
𝜕𝑧
•
(
•+
=
€
€+
+ 𝐯1 ⋅ 𝛻)
• Non-adiabaticelectrons: 𝑛) ≅ 1 − 𝑖𝛿 𝜙
^
𝜈)- 𝜔∗ − 𝜔
𝑘d^ 𝑣 z{)
𝛿≅
, with1 <
<∞
^
𝑘d^ 𝑣 z{)
𝜈)- 𝜔
• Dispersionrelation
16
DynamicalSymmetryBreaking
• Growthrate~frequencyshift:
• Spectralimbalance:
Infinitesimaltestaxial
flowshear,e.g.
𝛿 𝑣d 4 < 0
kZ
𝜙ƒ
^
Spectral
imbalance
0
:{𝑘+}
kd
:{𝑘−}
0
kd
Modeswith𝑘Z 𝑘d < 0
grow fasterthanother
modes,
𝛾ƒ |ƒ… ƒ† ‡? > 𝛾ƒ | ƒ… ƒ†ˆ?
Spectralimbalancein𝑘Z𝑘d
space
B)C ≠ 0
𝑘Z 𝑘d < 0 à Π/d
{𝑘±}:Domainswheremodesgrowfaster/slower
Spectralimbalance
17
NegativeViscosityIncrement
• Reynoldsstress:
• Turbulentmomentumdiffusivity:
• Residualstressà Negativeviscosityincrement
Ž.c
• 𝛿Π B)C = 𝜒$
𝛿⟨𝑣d ⟩′ [Lietal,submittedtoPoP,2016]
18
Modulational Enhancementof𝛿⟨𝑣d ⟩′
+i+
Ž.c
• 𝛿⟨𝑣d⟩′ à Π B)C à 𝜒$
= 𝜒$ − 𝜒$
• Dynamicsof𝛿⟨𝑣d ⟩′ :
^
𝜕
𝜕
B)C − 𝜒 𝛿 𝑣
𝛿 𝑣d 4 + ^ 𝛿Π/d
$
d
𝜕𝑡
𝜕𝑟
4
=0
• Growthrateofflowshearmodulation
Ž.c
𝛾• = −𝑞/^ 𝜒$ − 𝜒$
+i+
• 𝜒$
< 0 à Modulational growthof𝛿⟨𝑣d ⟩′
• Feedbackloop:𝛿⟨𝑣d ⟩′ à Π B)C à − 𝜒$Ž.c
19
4
UpperLimitof 𝑣d Set byPSFI
• Parallelshearflowinstability(PSFI)
• Drivenby𝛻 𝑣d ,negativecompressibility(similartoITG)
𝜒$+i+ = 𝜒$•‘ − 𝜒$Ž.c < 0
+i+
•‘
J’“Ž
𝜒$
= 𝜒$
+ 𝜒$
Θ 𝑣d 4 − 𝑣d
𝜒$+i+ = 𝜒$•‘ + 𝜒$J’“Ž − 𝜒$Ž.c > 0
4
c/-+
Ž.c
− 𝜒$
20
Parallelshearflowinstability
• Growthrateandresultingturbulentmomentumdiffusivity:
𝛾ƒJ’“Ž ≅
𝜒$J’“Ž
𝑘Z 𝑘d 𝜌C 𝑐C 𝑣d 4 − 𝑣d
1 + 𝑘t^ 𝜌C^
4
c/-+
^ ^
4
1
+
𝑘
𝜌C
t
^ ^ ^
—
≅
𝜙ƒ 𝑘Z 𝜌C
𝜔∗^
ƒ
^
𝑘Z 𝑘d 𝜌C 𝑐C 𝑣d 4 − 𝑣d
1 + 𝑘t^ 𝜌C^
4
c/-+
J’“Ž
+i+
• HitPSFIthresholdà 𝜒$
nonlinearin𝛻 𝑣d à 𝜒$
>0
• 𝛿⟨𝑣d⟩′ à Π B)C à 𝛿⟨𝑣d⟩′ growthß SaturatedbyPSFI
+i+
•‘
Ž.c
𝜒$
= 𝜒$
− 𝜒$
<0
+i+
•‘
J’“Ž
Ž.c
𝜒$
= 𝜒$
+ 𝜒$
− 𝜒$
>0
21
ComparingSymmetryBreakingMechanisms
FreeEnergy
Symmetry
Breaker
Effect on
theFlow
FlowProfile
StandardSymmetry Breaking
Dynamical Symmetry Breaking
𝛻𝑇,𝛻𝑛?
𝛻𝑛?
𝐸/4 ,𝐼 𝑥 4 ,etc.
Linkedtomagneticshear.
B)C
Intrinsictorque,−𝜕/ Π/∥
𝑣$
4
B)C
Π/∥
=
𝜒$
Testaxial flowshear,𝛿 𝑣d 4 ;
No requirementof𝑩 structure.
Negativeviscosity increment,
− 𝜒 B)C drivenby𝛻𝑛?
𝑣$
4
𝑅 ∗ 𝜏)*+ + ΠB)C
=
Ž.c|
𝜒$(𝛻𝑛?, 𝛻⟨𝑣∥ ⟩) − |𝜒$
22
Comparison(cont’d)
• FeedbackLoop:
PSFI
DynamicalSymmetryBreaking
(similartozonalflow)
ConventionalModels
23
AlsorelevanttoTokamaks
• 𝛻 𝑣$ steepeningandΠ B)C canactinsynergy
• Rotationprofilegradientenhancedbynegative
viscosityeffect
• 𝑣$
4
∼
•žŸ ¡
¤¥ ¦¢§¨©ª «( ¬ - f ¬ ª²®
¢£
)f
¢
∥
∥
£
®¯°±
£
• Drivecanbeexternal(𝜏³´µ )orintrinsic(Π B)C )
24
FutureWork
• Boundaryconditionmatter
• Flowboundarycontrolledbyneutralparticles
• Forsimpleanalysis:No-slipboundaryconditionduetofriction
• Intokamaks:InteractionwithSOL
25
BoundaryDynamicsImpactsFlowProfile
• Evolutionofnetaxialionflow:
B
B
𝜕 B
Δ𝑃¶ 𝑑𝑟⟨𝑣-,d ⟩ = ¶ 𝑑𝑟
− 𝑣G/ 𝑣Gd · − ¶ 𝑑𝑟𝜈.- 𝑣-,d − ⟨𝑣.,d ⟩
𝜕𝑡 ?
𝜌
𝐿
B
?
?
/¸
(a) Noexternalsource/sink.
Netflow=0.
(b) No-slipatwallà
𝑣d ≅ 0,⟨𝑣G/ 𝑣Gd ⟩ ≅ 0.
Netflow>0.
(c)Δ𝑃- driveatcenter,
outflux atwall.
26
BoundaryLayer
• Partiallyionizedà NeutralflowwithinBL
• Neutralflowdynamics(canbesolvednumericallybyBOUT++):
• Withinneutrallayer: 𝑣Ÿ,d ≅ ⟨𝑣¹,d ⟩
• NeutralflowwithinBLsetsboundaryconditionforplasmaflows
[Z.H.Wangetal,NF,2014]
27
Summary
• Weknow:
• Weakshear+rotationà beneficialforconfinement
• Intrinsictoroidal rotationexistsinweakshearregions
• Question:
• Intrinsicrotationgenerationforweakshearà 𝑞 4 → 0?
• Newmechanismforintrinsicaxialflowgeneration
• CSDX:atestbedforintrinsicflowswithunsheared
magneticfields
• Norequirementonmagneticshearà Broaderlesson
28
Summary(cont’d)
• Results:
• Dynamicalsymmetrybreaking
• NegativeviscosityincrementinducedbyΠ B)C
Ž.c
• 𝛿Π B)C = 𝜒$
𝛿 𝑣d
4
+i+
Ž.c
• Totalviscosity:𝜒$
= 𝜒$ − 𝜒$
+i+
• 𝜒$
< 0 à Modulationalgrowthof𝛿 𝑣d
4
• Flowprofile:pipeflowanalogy
• Balancebetweendriveandviscosity
+i+
• InCSDX: 𝑣d 4 ∼ Δ𝑃- /𝜒$
29
Summary(cont’d)
• Broaderlessonfortokamaks
• Synergyof 𝑣$ 4 self-amplificationandΠ B)C
• 𝑣$ 4 drivenby𝜏Ä3Ž ,Π B)C 𝛻𝑛? , 𝛻𝑇
Ž.c
• 𝑣$ 4 enhancedby− 𝜒$
• Futurework:flowboundarycondition
• Ion-neutralcouplingwithinboundarylayer
30
ListofRelatedTalks/Posters
• April1st:
• A.Ashourvon,“OntheStructureoftheZonalShearLayerFieldandits
ImplicationforMulti-scaleInteractions”
• March31st:
• R.Hajjar,“ModellingTransportBifurcationsintheCSDXLinearDevice”
• P.Vaezi,“NonlinearSimulationofCSDXIncludingSheathPhysics”
• March30th:
• S.C.Thakur,“Spontaneousself-organizationfromdriftwaveplasmas
toamixedITG-driftwave-shearflowsystemviaatransportbifurcation
inalinearmagnetizedplasmadevice”
• R.Hong,“EffectsofDensityGradientonAxialFlowStructuresina
HeliconLinearPlasmaDevice”
31
Back-up
32
HeatEngine
• Efficiencyofintrinsicrotationgenerationbyturbulence
• Entropyevolution:
•
ŹƞÇÈÉÊ¡ËÆžÌÍƟǹÊÌ¡ÆÇÎÏÇÐÑ¡¹¡žÒƟǹ
Efficiency~ ³¡ÆÈžÇÊÌÍƟǹÊÌ¡ÆÇÆÓ¡žÔÒÏž¡ÏÒÕÒƟǹ
33
Non-resonantmodestructure
• Non-resonant
modestructure
[S.Yi,PoP,2012]
34
Intrinsicrotation
• DiscoveredinJFT-2M,Alcator C-Modplasmas
Steadystatetoroidal momentum
incounter-currentinjection of
NBIistwotothreetimeslarger
thanthatinco-currentinjection.
[Ida,JFT-2M,1995]
Timehistories of(impurity) toroidal
rotationfora2.0MWICRF heatedEDA
H-modeplasma[Rice,C-Mod,2004]
35
CSDX Parameters
• TheseexperimentswerecarriedoutintheControlledShear
Decorrelation Experiment,whichisa2.8mlonglinear
heliconplasmadevicewithasourceradius 7.5cm,1.6kW
RFpowerinput(reflectedpowerlessthan30W),andagas
fillpressureof3.2mTorr.
36
EnergyTransferRatio
• Axialtransferratio:
• 𝑅z
• 𝑅z
Ö
Ö
=
J×Ø]
⟨¬G†Ù ⟩
𝜏Úc
decreases
when𝐵 > 𝐵c/-+
• 𝐵c/-+ ≅ 900𝐺
37
ResidualStress
• Reynoldsstress:
• Turbulentdiffusivity:
Drivenbyambient
background turbulence
• Residualstress
Spectralimbalance∼ 𝛿 𝑣d
4
ParallelShearFlowInstability
• PSFI:recalldispersionrelationofthemodelwith
adiabaticelectrons:
• Unstable↔ discriminant=
•à
à PSFI
• Withnon-adiabaticelectrons
39
ParallelShearFlowInstability
• Negativecompressibility
𝑘Z 𝑘d 𝑣d 4 > 0
𝜔^
𝑘Z 𝜌C
∼ 1−
𝑣
𝑘d 𝑐C d
4
𝑘d^ 𝑐^C
Negativecompressibility
40
FlowStructure
• Pipeflowanalogy
Prandtl
Pipeflow:
𝑣G/ 𝑣Gd ≅ 0
atboundary
CSDX:
41
FlowStructure
• WithexternaldriveΔ𝑃• Don’tneedmodulational growthtogenerateintrinsicflow
Ž.c
• − 𝜒$
enhancesflowgradient
• Momentumbalance:
• 𝑣d
4
∼
ÝJ°
¤¥ ¦¢§¨©ª «( ¬ -f ¬ ª²®
¢£
)f
¢
†
†
£
®¯°±
£
• 𝑣d 4 iskeptatorbelow 𝑣d
4
c/-+
dueto𝜒$J’“Ž
42
ApplicationstoTokamaks
• Weakshear
• Initialflowshear(seed)
à Π B)C bydynamicalsymmetrybreaking
à Acceleraterotation
• Needmodulational growthof𝛿 𝑣$
4
• ORweaknetNBItorque(external)+negative
viscosityincrementà 𝑣$ 4 enhancedby− 𝜒$Ž.c
43