Turbulence in Elastic Media:A New Look at Classic Themes

XiangFan1,PHDiamond1,LuisChacon21 UniversityofCalifornia,SanDiego2 LosAlamosNationalLaboratory

APS DPP 2017

ThisresearchwassupportedbytheU.S.DepartmentofEnergy,OfficeofScience,OfficeofFusionEnergySciences,underAwardNumberDE-FG02-04ER54738andCMTFO.

”Tour Guide”

• This talk is NOT a traditional study of plasma physics.• It is about a new system that is related to systems you are familiarwith in plasma physics

• Therearemanysimilarities, butsomeimportantdifferences. Watchforthese!

• We studied thefundamentalphysicsofcascades and self-organizationin this system and in MHD

• It provides a new look at classic themes in plasma physics.

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Elastic Media? -- WhatIstheCHNSSystem?

• Elastic media – Fluid with internal DoFsà “springiness”• TheCahn-HilliardNavier-Stokes(CHNS)system describes phase separationfor binary fluid (i.e.Spinodal Decomposition)

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AB

Misciblephaseà Immisciblephase

[Fan et.al. Phys. Rev. Fluids 2016] [Kim et.al. 2012]

Elastic Media? -- WhatIstheCHNSSystem?

• How to describe the system: the concentration field

• 𝜓 𝑟, 𝑡 ≝ [𝜌) 𝑟, 𝑡 − 𝜌+ 𝑟, 𝑡 ]/𝜌 : scalar field

• 𝜓 ∈ [−1,1]

• CHNS equations:

𝜕1𝜓 + �⃑� 4 𝛻𝜓 = 𝐷𝛻8(−𝜓 + 𝜓: − 𝜉8𝛻8𝜓)

𝜕1𝜔 + �⃑� 4 𝛻𝜔 =𝜉8

𝜌 𝐵? 4 𝛻𝛻8𝜓 + 𝜈𝛻8𝜔

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Why Care?

• Usefultoexaminefamiliarthemesinplasmaturbulencefromnewvantagepoint

• Somekey issuesinplasmaturbulence:1. Electromagnetics Turbulence

• CHNS vs 2D MHD: analogous, with interesting differences.• Both CHNS and 2D MHD are elastic systems• Mostsystems = 2D/ReducedMHD+manylineareffects

ØPhysicsofdualcascadesandconstrained relaxationà relativeimportance,selectivedecay…

ØPhysicsofwave-eddyinteractioneffectsonnonlineartransfer(i.e. Alfveneffectßà Kraichnan)

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MHDßà CHNS

Why Care?

2. Zonalflowformationà negativeviscosityphenomena• ZF can be viewed as a “spinodaldecomposition” of momentum.

• What determines scale?

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[Porter 1981]

Spinodal Decomposition

Arrows:𝜓 for CHNS;flow for ZF.

http://astronomy.nju.edu.cn/~lixd/GA/AT4/AT411/HTML/AT41102.htm

Zonal Flow

[J.A.Boedo et.al. 2003]

Why Care?

3. “BlobbyTurbulence”• CHNS is anaturallyblobbysystem ofturbulence.

• What is the role of structure ininteraction?

• Howtounderstandblobcoalescenceandrelationtocascades?

• How to understand multiple cascades ofblobs and energy?

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• CHNSexhibitsalloftheabove,withmany newtwists

Outline

• ABriefDerivationoftheCHNSModel• 2DCHNSand 2DMHD• LinearWave• IdealQuadraticConservedQuantities• Scales,Ranges,Trends• Cascades• Power Laws• Single EddyMixing• Conclusions

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ABriefDerivationoftheCHNSModel

• Secondorderphasetransitionà Landau Theory.• Orderparameter:𝜓 𝑟, 𝑡 ≝ [𝜌) 𝑟, 𝑡 − 𝜌+ 𝑟, 𝑡 ]/𝜌• Free energy:

F 𝜓 = B𝑑𝑟(12𝐶F𝜓

8 +14𝐶8𝜓

H +𝜉8

2 |𝛻𝜓|8)

�

�

• 𝐶F(𝑇),𝐶8(𝑇).• Isothermal𝑇 < 𝑇M .Set𝐶8 = −𝐶F = 1:

F 𝜓 = B𝑑𝑟(−12𝜓

8 +14𝜓

H +𝜉8

2 |𝛻𝜓|8)

�

�10/21/17 APSDPP2017 9

Phase Transition GradientPenalty

ABriefDerivationoftheCHNSModel

• Continuityequation:N?N1+ 𝛻 4 𝐽 = 0. Fick’sLaw:𝐽 = −𝐷𝛻𝜇.

• Chemical potential: 𝜇 = ST ?S?

= −𝜓 + 𝜓: − 𝜉8𝛻8𝜓.

• Combiningaboveà CahnHilliardequation:N?N1= 𝐷𝛻8𝜇 = 𝐷𝛻8(−𝜓 + 𝜓: − 𝜉8𝛻8𝜓)

• 𝑑1 = 𝜕1 + �⃑� 4 𝛻.Surfacetension:forceinNavier-Stokes equation:

𝜕1�⃑� + �⃑� 4 𝛻�⃑� = −𝛻𝑝𝜌 − 𝜓𝛻𝜇 + 𝜈𝛻8�⃑�

• Forincompressiblefluid,𝛻 4 �⃑� = 0.

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2DCHNSand 2DMHD

• 2DCHNSEquations:

𝜕1𝜓 + �⃑� 4 𝛻𝜓 = 𝐷𝛻8(−𝜓 + 𝜓: − 𝜉8𝛻8𝜓)

𝜕1𝜔 + �⃑� 4 𝛻𝜔 =𝜉8

𝜌 𝐵? 4 𝛻𝛻8𝜓 + 𝜈𝛻8𝜔

With �⃑�=𝑧W×𝛻𝜙, 𝜔 = 𝛻8𝜙, 𝐵? = 𝑧W×𝛻𝜓, 𝑗? = 𝜉8𝛻8𝜓.• 2DMHDEquations:

𝜕1𝐴 + �⃑� 4 𝛻𝐴 = 𝜂𝛻8𝐴

𝜕1𝜔 + �⃑� 4 𝛻𝜔 =1𝜇]𝜌

𝐵 4 𝛻𝛻8𝐴 + 𝜈𝛻8𝜔

With �⃑�=𝑧W×𝛻𝜙, 𝜔 = 𝛻8𝜙,𝐵 = 𝑧W×𝛻𝐴,𝑗 = F^_𝛻8𝐴.

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−𝜓:Negative diffusion term𝜓::Self nonlinear term−𝜉8𝛻8𝜓:Hyper-diffusion term

𝐴:Simplediffusion term

Linear Wave

• CHNS supports linear “elastic” wave:

𝜔 𝑘 = ±𝜉8

𝜌�

𝑘×𝐵?] −12 𝑖 𝐶𝐷 + 𝜈 𝑘

8

Where• Akintocapillarywaveatphaseinterface.Propagatesonly along theinterface of the two fluids, where |𝐵?| = |𝛻𝜓| ≠ 0.

• Analogue of Alfven wave.• Important differences:

Ø𝐵? in CHNS is large only in the interfacial regions.ØElasticwaveactivitydoesnotfillspace.

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Air

Water

CapillaryWave:

IdealQuadraticConservedQuantities

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• 2DCHNS1.Energy

𝐸 = 𝐸f + 𝐸+ = B(𝑣8

2 +𝜉8𝐵?8

2 )𝑑8𝑥�

�

2.MeanSquareConcentration

𝐻? = B𝜓8�

�

𝑑8𝑥

3.CrossHelicity

𝐻M = B �⃑� 4 𝐵?

�

�

𝑑8𝑥

• 2DMHD1.Energy

𝐸 = 𝐸f + 𝐸+ = B(𝑣8

2 +𝐵8

2𝜇])𝑑8𝑥

�

�2.MeanSquareMagneticPotential

𝐻) = B𝐴8�

�

𝑑8𝑥

3.CrossHelicity

𝐻M = B �⃑� 4�

�

𝐵𝑑8𝑥

Dual cascade expected!

Scales,Ranges,Trends

• Fluidforcingà FluidstrainingvsBlobcoalescence• Scalewhereturbulentstraining~elasticrestoringforce(duesurfacetension): Hinze Scale

𝐿j~(𝜌𝜉)

lF/:𝜖nl8/o

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Howbigisaraindrop?• Turbulentstrainingvscapillarity.

• 𝜌𝑣8 vs𝜎/𝑙.[Hinze1955]

Scales,Ranges,Trends• Elastic range: 𝐿j < 𝑙 < 𝐿N: where elastic effects matter.

• 𝐿j/𝐿N~(rs)lF/:𝜈lF/8𝜖n

lF/Ft à Extentofthe elasticrange

• 𝐿j ≫ 𝐿N required forlargeelasticrangeà caseofinterest

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𝐻? Spectrum𝐻v?

𝑘𝑘wx 𝑘j 𝑘N

ElasticRangeHydro-dynamicRange

(𝐻v? = 𝜓8 v)

Scales,Ranges,Trends

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• Keyelasticrangephysics:Blobcoalescence• Unforced case: 𝐿 𝑡 ~𝑡8/:.(Derivation: �⃑� 4 𝛻�⃑�~ sy

r𝛻8𝜓𝛻𝜓 ⇒ {̇y

{~ }rF{y)

• Forced case: blobcoalescencearrestedatHinzescale 𝐿j.

• 𝐿 𝑡 ~𝑡8/: recovered• Blob growth arrest observed• Blob growth saturation scale

tracks Hinze scale (dashed lines)

Cascades

• Blobcoalescenceinthe elastic range of CHNSis analogoustofluxcoalescenceinMHD.

• Suggestsinversecascade of⟨𝜓8⟩ inCHNS.• Supportedbythe statisticalmechanics studies (absoluteequilibriumdistributions).

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MHD CHNS

Cascades

• So,dualcascade:• Inverse cascadeof 𝜓8 ��• Forward cascadeof𝐸��

• Inversecascadeof 𝜓8 isformalexpressionofblobcoalescenceprocessà generatelargerscalestructurestilllimitedbystraining

• Forward cascade of𝐸 as usual, as elastic force breaksenstrophyconservation

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Cascades• Spectralfluxof 𝐴8 : Spectral flux of 𝜓8 :

• MHD:weaksmallscaleforcingon𝐴 drivesinversecascade• CHNS:𝜓 isunforcedà aggregates naturally• Bothfluxesnegativeà inverse cascades

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MHD

CHNS

PowerLaws• 𝐴8 spectrum: 𝜓8 spectrum:

• Bothsystemsexhibit𝑘l�/: spectra.• Inversecascadeof 𝜓8 exhibitssamepowerlawscaling,solongas 𝐿j ≫ 𝐿N,maintainingelasticrange:Robustprocess.

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CHNSMHD𝑓)

PowerLaws

• Derivation of -7/3 power law:• For MHD, key assumptions:

• Alfvenicequipartition(𝜌⟨𝑣8⟩ ∼ F^_⟨𝐵8⟩)

• Constantmeansquaremagneticpotentialdissipationrate𝜖j), so𝜖j)~

j�

� ~(𝐻v))

�y𝑘

�y.

• Similarly, assume the following for CHNS:• Elastic equipartition (𝜌⟨𝑣8⟩ ∼ 𝜉8⟨𝐵?8⟩)• Constantmeansquaremagneticpotentialdissipationrate𝜖j?, so

𝜖j?~j�

� ~(𝐻v?)

�y𝑘

�y.

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𝑓�

CHNSMore PowerLaws

• Kinetic energy spectrum (Surprise!):

• 2D CHNS: 𝐸vf~𝑘l:;

• 2D MHD: 𝐸vf~𝑘l:/8.• The -3 power law:

• Closertoenstrophycascaderangescaling,in2DHydro turbulence.• Remarkable departurefromexpected-3/2 forMHD.Why?

• WhydoesCHNSßàMHDcorrespondenceholdwellfor𝜓8 v~ 𝐴8 v~𝑘l�/:, yetbreakdowndrasticallyforenergy?

• Whatphysics underpinsthissurprise?

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!

Interface Packing Matters!• Needto understanddifferences,aswellassimilarities,betweenCHNSandMHDproblems.

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23

MHD CHNS

2D CHNS:ØElasticback-reactionislimitedtoregionsofdensitycontrasti.e. |𝐵?| = |𝛻𝜓| ≠ 0.

ØAsblobscoalesce,interfacialregiondiminished.‘Activeregion’ofelasticitydecays.

2D MHD:Ø Fields pervadesystem.

Interface Packing Matters!

• Definetheinterfacepackingfraction 𝑃:

𝑃 =#�����������������|+�|�+�

���

#�����������������

Ø𝑃 for CHNS decays;Ø𝑃 for MHD stationary!

• 𝜕1𝜔 + �⃑� 4 𝛻𝜔 = sy

r𝐵? 4 𝛻𝛻8𝜓 + 𝜈𝛻8𝜔: small 𝑃à local back reaction

is weak.

• Weak back reactionà reduce to 2D hydro

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WhatAretheLessons?

• Avoidpowerlawtunnelvision!• Realspace realizationoftheflowisnecessarytounderstandkeydynamics.Trackinterfacesandpackingfraction𝑃.

• Oneplayerindualcascade(i.e. 𝜓8 )canmodifyorconstrainthedynamicsoftheother(i.e.𝐸).

• Againstconventionalwisdom, 𝜓8 inversecascadeduetoblobcoalescenceistherobustnonlineartransferprocessinCHNSturbulence.

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BroaderImplications&Speculations

• What,really,istheessentialtransferprocessinMHD?i.e.theoreticalfocusisoverwhelminglyonEnergy

• Followsfluids,examineenergywithforcingin�⃗� equation• but

• AlfventheoremiskeyconstraintinMHD.So,isinversecascade𝐴8 (or 𝐴 ⋅ 𝐵 )actually fundamental?

• Candualcascadeprocesses interact?• Can2DMHDturbulencebethoughtofasfluxaggregationvs.fragmentationcompetition? Is blob dynamics the key?

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Single Eddy Mixing

• Structures are the keyà need understand how a single eddyinteracts with 𝜓 field

• 3 stages, topological change observed. Nontrivial evolution.• More in poster PP11.00113 Wednesday pm!

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(a) t=10

(b) t=70

(c) t=75

(d) t=80

(e) t=85

(f) t=400

(g) t=1500

(h) t=4000

A B C

[Fan et.al. Phys. Rev. ERap. Comm. 2017]

Single Eddy Mixing

• Structures are the keyà need understand how a single eddyinteracts with 𝜓 field

• Mixing of 𝛻𝜓 by a single eddyà characteristic time scales?• Evolution of structure?• AnalogoustofluxexpulsioninMHD(Weiss,‘66)

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?

Single Eddy Mixing

• 3 stages:(A) the ”jelly roll” stage, (B)the topologicalevolution stage,and (C)the target pattern stage.

• 𝜓 ultimately homogenized in slow time scale,butmetastabletargetpatternsformedandmerge.

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(a) t=10

(b) t=70

(c) t=75

(d) t=80

(e) t=85

(f) t=400

(g) t=1500

(h) t=4000

A B C

[Fan et.al. Phys. Rev. ERap. Comm. 2017]

[Ashourvan et.al. 2016]

Single Eddy Mixing

• The bands merge on a time scale long relative to eddy turnover time.• The 3 stages are reflected in the elastic energy plot.• The target bands mergers are related to the dips in the target pattern stage.• Thebandmergerprocessissimilartothestepmergerindrift-ZFstaircases.

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Conclusions

• TurbulentspinodaldecompositiondynamicsilluminatesfamiliarthemesinphysicsofMHDcascades,relaxation,andselectivedecay,fromanovelperspective

• Theories for MHD can attract interest in other fields outside plasmaphysics

• Blobcoalescence and inverse cascade are dominantprocesses inCHNS

• Realspaceconfigurationandpackingofinterfacesareessentialtophysicsofdualcascade

• Single eddy mixing can exhibit unexpected nontrivial dynamics

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