Memory,CascadesandSpectrainModels
of2DMHDandElasticTurbulence
XiangFan1,PHDiamond1,LuisChacon2,andHuiLi2
1
UniversityofCalifornia,SanDiego
2 LosAlamosNationalLaboratory
ThisresearchwassupportedbytheU.S.DepartmentofEnergy,OfficeofScience,Officeof
FusionEnergySciences,underAwardNumberDE-FG02-04ER54738andCMTFO.
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Why Care?
• This is not a traditional study on transport inconfined
plasma. It is about thefundamental physicsofmemory
andcascades in elastic turbulence systems. Why
should we care?
• Virtually allmodelsofdrift-Alfven, EMITG,etc.
turbulence arebaseduponavorticity equation, Ohm‘s
Lawand(usuallymultiple) scalaradvection equations.
Theappearance oftheAlfvenwaveintroduces acrucial
element ofmemorytothedynamics.Such
Alfvenization-induced-memory cansignificantly impact
structure formation andtransport inturbulence.
• Inthisposter,westudythefundamental physicsof
memoryandcascadesinverysimplemodels suchas
2DMHDandelasticturbulence.
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Why Care?
• The2DCahn-HilliardNavier-Stokes (CHNS)model for spinodal
decomposition is an example of generalized elastic system, and is
analogousto2DMHD.
• 2DMHDissimplebutsimilartosomeimportantmodels for
transportstudies.
2DCHNS
𝜕" 𝜓 + 𝑣⃑ ' 𝛻𝜓 = 𝐷𝛻 + (−𝜓 + 𝜓 . − 𝜉 + 𝛻 + 𝜓)
𝜉+
𝜕" 𝜔 + 𝑣⃑ ' 𝛻𝜔 = 𝐵4 ' 𝛻𝛻 + 𝜓 + 𝜈𝛻 + 𝜔
𝜌
2DMHD
𝜕" 𝐴 + 𝑣⃑ ' 𝛻𝐴 = 𝜂𝛻 + 𝐴
1
𝜕" 𝜔 + 𝑣⃑ ' 𝛻𝜔 =
𝐵 ' 𝛻𝛻 + 𝐴 + 𝜈𝛻 + 𝜔
𝜇: 𝜌
Atypical3-field
modelforELM
[Xi2012]
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Why Care?
• Spinodal decomposition is also related
to zonal flow.
Spinodal Decomposition
• The pattern formation is because of
negative diffusion/negative viscosity.
Zonal Flow
http://astronomy.nju.edu.cn/~li xd/GA/AT 4/AT41 1/HTM L/AT4 1102. htm
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[Porter 1981]
4
Overview
• TheCahn-HilliardNavier-Stokes(CHNS)modelforspinodaldecomposition
in2Dsymmetricbinaryliquidmixtureand2DMagnetohydrodynamics
(MHD)arebothexamplesofgeneralized elasticturbulentsystems.
• Since2DMHDturbulence hasbeenwellstudied,itprovidesuswith
potentialinsightandguidanceforexploringthephysicsof2DCHNS
turbulence.
• We compare and contrast the cascades and spectra of the two systems.
• [New] Bydirectnumericalsimulation,wefindthatintheelasticrange,the
4
meansquareconcentrationspectrum𝐻< ofthe2DCHNSsystemexhibits
thesamepowerlaw(−7/3)asthemeansquaremagneticpotential
spectrum𝐻<= in the inverse cascade regime of2DMHD.
• [New] Thekineticenergyspectrumofthe2DCHNSsystemis𝐸<? ~𝑘 B. if
forcedatlargescale,suggestive ofthedirectenstrophycascadepowerlaw
of2DNavier-Stokes (NS)turbulence.It is different from 2D MHD, and the
difference could be explained by the difference of interface packing fraction.
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Outline
• Introduction: What is Spinodal Decomposition
• The CHNSmodelforspinodaldecomposition
• 2D CHNS turbulence compared to 2D MHD turbulence
•
•
•
•
Basicequations
Idealquadraticconserved quantities
Cascades
Linearelasticwave
• Importantlengthscalesandrangesof2DCHNSturbulence
• Simulation Results
•
•
•
•
Simulation Setup
Benchmark
4
𝐻< /𝐻<=spectrum power law
Energyspectrum power law
• Conclusion & discussion
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Introduction:WhatIsSpinodalDecomposition
• What is spinodal decomposition: binaryliquidmixtures
canpassspontaneouslyfromonemisciblephasetotwo
coexistingimmisciblephases followingatemperature drop,
and thissecondorderphasetransitioniscalledthespinodal
decomposition.
• One miscible phase: like water + alcohol.
• Two coexisting immiscible phases: like water + oil.
• A real life example: the cyclohexanemethanol system, with
the critical temperature of spinodal decomposition 𝑇E =
45.14℃. [Huang et al. 1974]
• TheCahn-HilliardNavier-Stokes (CHNS)modelisthe
standardmodelforbinaryliquidmixtureundergoing
spinodaldecomposition.
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Introduction:WhatIsSpinodalDecomposition
A-rich
phase
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B-rich
phase
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TheCHNSModelforSpinodalDecomposition
• Second order phase transition -> Landautheory.
• The order parameter: thelocal relative concentrationfield:
𝜓 𝑟⃑, 𝑡 ≝ [𝜌= 𝑟⃑, 𝑡 − 𝜌O 𝑟⃑, 𝑡 ]/𝜌
• 𝜓 = −1 means A-rich phase, 𝜓 = 1 means B-richphase.
• The free energy functional:
1
1
𝜉+
+
U
F 𝜓 = R 𝑑𝑟⃑ 𝐴𝜓 + 𝐵𝜓 +
𝛻𝜓
2
4
2
+
Phase Transition Curvature Penalty
Where A and B are coefficients of Taylor expansion, 𝐴 = 𝐴: (𝑇 − 𝑇W ); 𝜉 is a
parameter that describes the strength of the interaction.
• Forsimplicity,westudytheisothermalcasewhenthetemperatureisfixed
belowTc,andsetB=−A=1:
1 + 1 U 𝜉+
F 𝜓 = R 𝑑𝑟⃑(− 𝜓 + 𝜓 + |𝛻𝜓|+ )
2
4
2
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TheCHNSModelforSpinodalDecomposition
• Chemical potential: 𝜇 =
]4
YZ 4
Y4
= −𝜓 + 𝜓 . − 𝜉 + 𝛻 + 𝜓. Fick’sLaw:𝐽⃗ = −𝐷𝛻𝜇.
Continuityequation: + 𝛻 ' 𝐽⃗ = 0. Combiningtheabovetogether:
]"
𝑑𝜓
= 𝐷𝛻 + 𝜇 = 𝐷𝛻 + (−𝜓 + 𝜓 . − 𝜉 + 𝛻 + 𝜓) Cahn-HilliardEquation
𝑑𝑡
• Thefluidvelocitycomesin viatheconvectionterm 𝑑" = 𝜕" + 𝑣⃑ ' 𝛻.
• ThesurfacetensionenterstheNavier-Stokes Eq asaforce:
𝛻𝑝
𝜕" 𝑣⃑ + 𝑣⃑ ' 𝛻𝑣⃑ = −
− 𝜓𝛻𝜇 + 𝜈𝛻 + 𝑣⃑
𝜌
• Forincompressiblefluid,itisconvenienttotakethecurlandworkwith
vorticity.Finally,wegettheCahn-HilliardNavier-Stokes (CNHS)Equations:
𝜕" 𝜓 + 𝑣⃑ ' 𝛻𝜓 = 𝐷𝛻 + (−𝜓 + 𝜓 . − 𝜉 + 𝛻 + 𝜓)
𝜉+
𝜕" 𝜔 + 𝑣⃑ ' 𝛻𝜔 = 𝐵4 ' 𝛻𝛻 + 𝜓 + 𝜈𝛻 + 𝜔
𝜌
With 𝑣⃑=𝑧a⃑×𝛻𝜙, 𝜔 = 𝛻 + 𝜙, 𝐵4 = 𝑧a⃑×𝛻𝜓, 𝑗4 = 𝜉 + 𝛻 + 𝜓.
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Comparison & Contrast: Basic Equations
• The2DCHNSEquations:
−𝜓:Negative diffusion term
𝜕" 𝜓 + 𝑣⃑ ' 𝛻𝜓 = 𝐷𝛻 + (−𝜓 + 𝜓 . − 𝜉 + 𝛻 + 𝜓)
𝜉+
𝜕" 𝜔 + 𝑣⃑ ' 𝛻𝜔 = 𝐵4 ' 𝛻𝛻 + 𝜓 + 𝜈𝛻 + 𝜔
𝜌
With 𝑣⃑=𝑧a⃑×𝛻𝜙, 𝜔 = 𝛻 + 𝜙, 𝐵4 = 𝑧a⃑×𝛻𝜓, 𝑗4 = 𝜉 + 𝛻 + 𝜓
𝜓. :Self nonlinear term
−𝜉+ 𝛻+ 𝜓:Hyper-diffusion term
• The2DMHDEquations:
𝜕" 𝐴 + 𝑣⃑ ' 𝛻𝐴 = 𝜂𝛻 + 𝐴
1
𝜕" 𝜔 + 𝑣⃑ ' 𝛻𝜔 =
𝐵 ' 𝛻𝛻 + 𝐴 + 𝜈𝛻 + 𝜔
𝜇: 𝜌
e
With 𝑣⃑=𝑧a⃑×𝛻𝜙, 𝜔 = 𝛻 + 𝜙,𝐵 = 𝑧a⃑×𝛻𝐴,𝑗 = 𝛻 + 𝐴
𝐴:Simplediffusion term
fg
• The force on fluid:
hi
j
𝐵4 ' 𝛻𝛻 + 𝜓
e
fg j
𝐵 ' 𝛻𝛻 + 𝐴
• Note that the magnetic potential 𝐴 is a scalar in 2D.
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Comparison & Contrast:
Idealquadraticconservedquantities
• 2DMHD
• 2DCHNS
1.Energy
1.Energy
2.MeanSquareMagnetic
Potential
2.MeanSquare
Concentration
+ +
+
𝜉
𝐵4 +
𝑣
?
O
𝐸 = 𝐸 + 𝐸 = R( +
)𝑑 𝑥
2
2
+
+
𝑣
𝐵
𝐸 = 𝐸 ? + 𝐸 O = R( +
)𝑑 + 𝑥
2 2𝜇:
𝐻 = = R 𝐴+ 𝑑 + 𝑥
𝐻4 = R 𝜓 + 𝑑+ 𝑥
3.CrossHelicity
3.CrossHelicity
𝐻 E = R 𝑣⃑ ' 𝐵4 𝑑 + 𝑥
𝐻 E = R 𝑣⃑ ' 𝐵𝑑 + 𝑥
• “Ideal”heremeans𝐷, 𝜂 = 0; 𝜈 = 0.
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Comparison&Contrast:Cascades
• Turbulencecascadedirectionsaresuggestedbytheabsolute
equilibrium distributions.
• Thepeakoftheabsoluteequilibrium distributionforeach
quadraticconservedquantityisagoodindicatorofthe
correspondingcascadedirection.
• The spectrum is peaked at high k -> excitation relaxes towards
high k -> direct cascade.
• The spectrum is peaked at small k -> excitation relaxes towards
small k -> inverse cascade.
• Thisapproachonlydependsontheidealquadraticconserved
quantitiesofthesystem, so wecanfollowasimilarapproachas
for2DMHDtoobtainanindicationofthecascadedirectionsin
2DCHNS.
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Comparison&Contrast:Linear Elastic Wave
• Alfven wave in 2D MHD:
1
1
𝜔 𝑘 =±
𝑘×𝐵: − 𝑖 𝜂 + 𝜈 𝑘+
𝜇: 𝜌
2
• Linear elastic wave in 2D CHNS:
CapillaryWave:
Air
Water
𝜉+
1
𝜔 𝑘 =±
𝑘×𝐵4: − 𝑖 𝐶𝐷 + 𝜈 𝑘+
𝜌
2
Where
• Thelinear elastic wave in 2D CHNS is like a capillary wave:it only
propagates along the boundary of the two fluids, where the gradient
of concentration 𝐵4: ≠ 0. Surfacetensiongenerates restoringforce.
• The wave is similar to Alfven wave: they have similar dispersion
relation; they both propagates along 𝐵: field lines; both magnetic
field and surface tension act like an elastic restoring force.
• Important difference: B fills the whole space; 𝐵4: is large only in the
interface regions.
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Importantlengthscalesandranges
• Hinze scale: the balance between turbulentkineticenergyand
j⟨qi ⟩
surfacetensionenergy: s/t ~1 where 𝜎 is surface tension and
𝜎=
u
8/9𝜉 [Kendon et al. 2001].
• Inthe2DNSdirectenstrophycascaderegime,thevelocity
qi
distributionis < ~𝜖z+/.𝑘{B.. So
u
𝜌 Be/. B+/}
𝐿{ ~( )
𝜖z
𝜉
• We define thescalesbetween𝐿{ anddissipationscale 𝐿] to be
theelasticrange,wheretheblobcoalescenceprocessdominates.
• Define a dimensionless number:
•
tu
t~
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tu
t~
j
h
Be/e•
= 𝐻𝑑~( )Be/. 𝜈 Be/+ 𝜖z
= 𝐻𝑑 ≫ 1 isrequiredtoformalong enoughelasticrange.
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Importantlengthscalesandranges
𝐻 4 Spectrum
4
𝐻<
Hydrodynamic
Range
𝑘•‚
ElasticRange
𝑘{
𝑘]
𝑘
• Intheelastic rangeofthe2DCHNSsystem,theblobcoalescenceprocessis
analogoustothemagneticfluxblobcoalescence processin2DMHD.
• Theformerleadstotheinversecascadeof𝐻 4,andthelatterleadstothe
inversecascadeof𝐻 = .
• Intheelastic rangeofthe2DCHNSsystem,surfacetensioninduceselasticity
andplaysamajorroleindefiningarestoringforce.Similarly,in2DMHD,the
magneticfieldinduceselasticityandmakeMHDdifferentfromapurefluid.
• The2DCHNSsystemismoreMHD-likeintheelasticrange.
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Simulation Setup
• The PIXIE2D code [Chacon 2002,2003] is used tosimulatethe
system.PIXIE2D originally solves the 2D MHD equation, and now
is modified to be able to solve the Cahn-Hilliard Navier-Stokes
(CHNS) equations, too.Itis a Direct Numerical Simulation (DNS)
code that solves thefollowingequationsinrealspace:
𝜕" 𝜓 + 𝑣⃑ ' 𝛻𝜓 = 𝐷𝛻 + −𝜓 + 𝜓 . − 𝜉 + 𝛻 + 𝜓 + 𝐹4
𝜉+
𝜕" 𝜔 + 𝑣⃑ ' 𝛻𝜔 = 𝐵4 ' 𝛻𝛻 + 𝜓 + 𝜈𝛻 + 𝜔 + 𝐹„
𝜌
With 𝑣⃑=𝑧⃑a ×𝛻𝜙, 𝜔 = 𝛻 + 𝜙, 𝐵4 = 𝑧⃑a×𝛻𝜓, 𝑗4 = 𝜉 + 𝛻 + 𝜓
• Initialcondition:𝜓 ineachcellis assignedto1 or−1 randomly;
𝜙 = 0 everywhere.
• Boundarycondition:doubly periodic.
• Externalforceforeither𝜓 or𝜙:anisotropichomogeneous force
thathasawavenumber𝑘•‚ :
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Benchmark
• We verified the length scale growth is 𝐿 𝑡 ~𝑡 +/. ; and the
growth can be arrested. The saturation length scale is
consistent with the Hinze scale (dashed line).
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4
=
𝐻< /𝐻< spectrum
power law: -7/3
CHNS
MHD
𝑓=
Inverse cascade of 𝐻 4
Inverse cascade of 𝐻 =
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4
=
𝐻< /𝐻< spectrum
power law: -7/3
• Assumingaconstantmeansquaremagneticpotential
dissipationrate𝜖{= ,according totheAlfvenic
equipartition (𝜌⟨𝑣 + ⟩ ∼ fe ⟨𝐵 +⟩), the time scale for the
g
=
=
decay of 𝐻 (𝜖{= ~𝐻 /𝜏) can be estimated by
𝜏~(𝑣𝑘)Be ~(𝐵𝑘)Be.
• Define the spectrum to be 𝐻 = = ∑< 𝐻<= ~𝑘𝐻<= , so
‰
‰ Š
i •
{‹
= i i
= Š i
= i
𝐵~𝑘𝐴~𝑘(𝐻 ) ~(𝐻< ) 𝑘 . Therefore 𝜖{= ~ ~(𝐻< ) 𝑘 .
Œ
+/. BŽ/.
=
𝐻< ~𝜖{= 𝑘
• Similarly, we can obtain the 𝐻<4spectrum by assuming
the elastic equipartition (𝜌⟨𝑣 + ⟩ ∼ 𝜉 + ⟨𝐵4+ ⟩):
4
+/. BŽ/.
𝐻< ~𝜖{4 𝑘
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4
=
𝐻< /𝐻< spectrum
power law: -7/3
• The−7/3powerisrobust.Itdoesnotchangewiththe
magnitudeofexternalforcing,aslongastheseparation
between theHinzescaleandthedissipationscaleis
maintained,sotheelasticrangeislongenough.
𝑓•
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Energy Spectrum Power Law
𝑓•
𝑓•
• 𝐸<? ~𝑘 B. in 2D CHNS turbulence; on the other hand, it is well known that
𝐸<? ~𝑘 B./+ in MHD turbulence.
• The -3 power law is consistent with the direct enstrophy cascade in 2D
NS turbulence.
• The -3/2 power law comes from the Alfven effect (Iroshnikov-Kraichnan
Theory).
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Energy Spectrum Power Law
• 𝐸<? ~𝑘 B. for 2D CHNS; 𝐸<? ~𝑘 B./+ for 2D MHD. Why? Why not -3/2?
• Thisinitiallysurprisingresultisplausiblebecauseinthe2DCHNSsystem,
𝐵4 vanishesinmostregions. Back reaction is apparently limited.
• On the other hand, themagneticfieldsinMHDarenotlocalizedatspecific
regions,andAlfvenwavescanbeeverywhere.
• Definetheinterfacepackingfraction𝑃 tobetheratioofmeshgridnumber
where |𝐵4 | > 𝐵4’“” (or |𝐵| > 𝐵’“”) tothetotalmeshgridnumber.
MHD
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Conclusion & Discussion
• Comparison:
• Contrast:
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Conclusion&Discussion
• Thestudy ofelasticturbulencegives us useful
lessons.
4
• The𝐻< spectrum(inversecascadeof𝐻 4)isrobust.
It is an analogue to 𝐻<=spectrum (inversecascadeof
𝐻 =) in 2D MHD. On the other hand, the energy
spectrum is NOT so robust. This implies we should
perhaps pay more attention on other spectra than
the energy spectrum.
• Thespatialrepresentations- i.e.trackingcontours
ofbubblesurfaces- may be more revealing way of
representingtheturbulence thetraditionalpower
lawspectrum.
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