Turbulence and Transportin Elastic Media:

A New Look at Classic Themes

XiangFan1,PHDiamond1,LuisChacon21 UniversityofCalifornia,SanDiego2 LosAlamosNationalLaboratory

Sherwood 2018

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

”Tour Guide”

• This 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

𝜌𝐵? 4 𝛻𝛻8𝜓 + 𝜈𝛻8𝜔

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Outline

• ABriefDerivationoftheCHNSModel• 2DCHNSand 2DMHD• LinearWave• IdealQuadraticConservedQuantities• Scales,Ranges,Trends• Cascades• Power Laws• Transport• 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(12𝐶F𝜓8 +

14𝐶8𝜓H +

𝜉8

2|𝛻𝜓|8)

�

�

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

F 𝜓 = B𝑑𝑟(−12𝜓8 +

14𝜓H +

𝜉8

2|𝛻𝜓|8)

�

�4/23/18 Sherwood2018 6

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

𝜌𝐵? 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?] −12𝑖 𝐶𝐷 + 𝜈 𝑘8

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/:𝜖n

l8/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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20

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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Transport: Something Old

• 𝑀8 = 𝑣¡8 /𝑣)]8

• Higher 𝑣)]8 / 𝑣¡8 →lower𝐷¢ →longer 𝐸£ persistance

• Ultimately 𝜂 asserts itself

• Blue: 𝐵 sufficient forsuppression

• Yellow: Ohmic decay phase

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[Cattaneo andVainshtein ‘91]

• Initial condition: cos(x) for A• Shorter time (suppression phase)

• Domains, and domain boundariesevident, resembles CHNS

• A transport barriers?!

• Longer time (Ohmic decay phase)• Well mixed• No evidence nontrivial structure

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Spatial Structure (Preliminary)

Something New, Cont’d

• For analysis: pdf of A• Suppression phase:

• quenched diffusion• bi-modal distribution

• quenching prevents fill-in• consequence i.c.

• Ohmic decay phase:• uni-modal distribution returns

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Conclusion, of Sorts

• Elastic fluids ubiquitous, interestingly similar and different.Comparison/contrast is useful approach.

• CHNS is interesting example of elastic turbulence where energycascade is not fundamental or dominant.

• Spatio-temporal dynamics of (bi-stable) active scalar transport is apromising direction. Pattern formation in this system is terra novo.

• Revisiting polymer drag reduction would be interesting.

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