Turbulence:fromhydrodynamicstothesolarwind

plasma-AnIntroduc:on

PinWu(Penny)

Kolmogorov

(googleimage)

The5thSOLARNETsummerschool,Belfast,UnitedKingdom,August5,2016

Whystudyturbulenceinsolarphysics?

•  Coronalhea:ngproblem•  Solarwindhea:ngproblem

Matthaeus et al., 1999

WhatisTurbulence?

?Googleimage

Startwithhydrodynamicsdescrip:ons

•  Notepar:cularlytheNavier-Stokesequa:on*(momentumequa:on)

•  ReynoldsnumbersR=Lu/νRa:oofiner:altoviscousforceDimensionless *Note,usuallyanalyzedwithcon:nuityequa:on

andincompressibleassump:onunderspecifiedini:alcondi:onandboundarycondi:on(B.C.).

∂u∂t

+ u ⋅∇u= −∇Pρ

+ν∇2uNonlinear!

∇⋅u = 0

∂ρ∂t

+∇⋅(ρu) = 0

R=1.54

PhotosfromVonDyke(1982)

R=9.6

R=13.1

R=26

R=41

R=140

VonKarmanVortexStreet

AsReynoldnumberincreases,thesymmetriespermidedbytheNavier-Stocksequa:onandboundarycondi:onaresuccessivelybroken.

R=1800R=240

Frisch(1995)Grid

Twocylinders

FullydevelopedTurbulence:symmetriesrestored.LordKelvin(1887):homogeneousandisotropicturbulence.

WhatisTurbulence?TurbulentorLaminar?Iner:alForcesv.s.Viscousforce?Reynoldnumberistheessen:al.

Chao:cIrregularMixingRota:onal,vor:cityDissipa:veSta:s:calordeterminis:c?

ω = ∇× u

WhathavewelearnedfromHydrodynamics?

Richardsoneddycascadephenomenology(1922)

Outerscale(Integratedscales)AnisotropicIner:alscale(Taylormiscroscale)Innerscale(KolmogorovScales)Isotropicandhomogeneous

L

η<<l<<L

Kolmogorov’sthreehypotheses.AthighR,1.  thesmall-scaleturbulentmo:onsaresta0s0callyisotropic(rota0on

invariant).2.  thesmal-scaleturbulentsta:s:csareuniversallyanduniquely

determinedbyνandenergydissipa:onrateε.Bydimensionalanalysis,Kolmogorovlengthscale,

3.  theiner:alrange(η<<l<<L)ishomogeneous(transla0oninvariant).Sta:s:cshereareuniversallyanduniquelydeterminedbythescalel(1/k)andenergydissipa:onrateε,independentofν.Bydimensionalanalysis,

E(k)~ε2/3k-5/3 Kolmogorov(1941),K41

WhathavewelearnedfromHydrodynamics?

K41dimensionalanalysis

Thus,E(k)hasdimensionL3/T2

Dimensionofε(energydissipa:onrateperunitmass)isL2/T3

K41assumesE(k)onlydependsonεandk,Then,wemusthaveE(k)~ε2/3k-5/3

12< u2 >= E(k)dk

0

∞

∫

OneexampleoftheexperimentalsuccessofK41

Champagne,1978

R=626

E(k)~ε2/3k-5/3

Self-similar

1-Dexample:brownianmo:on

The“generalaspect”(sta:s:calproper:es)withinthemagnifica:onwindowisindependentofwherethewindowisposi:oned!

Frisch,1995

Self-similar:preserva:onofstructurefunc:on

•  Self-Similar(symmetries:isotropicandhomogeneous)intheiner:alrange(equivalentoftheuniversalassump:oninK41).Thereexistsascalingexponenthforthe1storderstructurefunc:onδu(l)suchthat

whereincrementThep-thorderstructurefunc:onthuswhereζp=p/h.K41-3statesthatSponlydependsonεandl.Bydimensionalanalysis,thesecondorderstructurefunc:onS2~ε2/3l2/3.Thereforeh=1/3andζp=p/3.AndinfactSp(l)~εp/3lp/3.

δui (l) = ui (x)− ui (x − l)δu(λl) = λ hδu(l)

Sp(l) = 〈(δui (l))p 〉 ∝ lζ p

Intermidency:dissipa:onrange(highk)isnotself-similar!(BatchelorandTownsend,1949)

VelocitysignalfromajetwithR=700

Samesignalsubjecttohigh-passfiltering,showingintermidentbursts

Gagne1980

Example

Hea:ngisbursty,patchy,andnon-uniform

Moreexamplesofintermidentfunc:ons

Devil’sstaircase

Realityisseldomso“blackandwhite”.Howintermident?Needtointroduceintermidencymeasurement.

MeasureofintermidencyQuan:ta:vely:Kurtosis(Flatness)

Visually:PDF(δui)*devia:onfromGaussian

Perfectlyself-similarcase:Gaussiansignals(Normalfunc:on).Caussianfluctua:onshaveaflatnessof3,independentoffilteringfrequency.

Fourthmomentaroundmeandividedbythefourthpowerofstandarddevia:onagain,velocityincrementNotekisscale(l)dependentThelargerthekurtosis,themoreintermident!

δui (l) = ui (x)− ui (x − l)

k(l) = µ4σ 4 =

〈(δui (l)− 〈δui (l)〉)4 〉

〈(δui (l)− 〈δui (l)〉)2 〉2

= 〈(δui (l))4 〉

〈(δui (l))2 〉2

*PDF=probabilitydistribu:onfunc:on Subedietal.,2014

Intermidencyvisualiza:on:vor:cityfilaments

Vor:cityfield(VincentandMeneguzzi,1991)

Vor:cityfilament(highconcentra:onofvor:city)inturbulentwater(Boonetal.,1993)

K41,data,modelsthatmodifiesK41

Frisch,1995Blackcircles,whitesquaresandblacktrianglesaredatafromAnselmet,(1984)

Experimentallyvalidatedinwindtunnelmeasurements(BatchelorandTownsend,1949)

EnergydecayrateεiswridenasdU2/dtIntheplot

Decay(dissipa:on)rateεiscontrolledbyU=<u2>1/2(amplitude)andLintheouterscale,independentofviscosity(ordetaildissipa:onmechanism)!

SimilaritydecaywassuggestedbyTaylor(1935)andmadeprecisedbyvonKarmanandHowarth(1938).Itpostulatesthepreserva:onofshapeof2pointcorrela:onfunc:onsduringDecay(Essen:ally,arephrasingofK41).Deriveε=-aU3/LanddL/dt=bUwhereaandbareconstants.

vonKarmandecayinHydrodymanics(3rdorderlaw)

Fromafluidperspec:ve,howdoesturbulenceinthesolarwindplasmadifferfromhydrodynamicTurbulence?

Navier-stocksequa:onbecomes

Addi:onalvariableBAddi:onalnonlineartermNeedonemoreequa:on

Note,here,BiswrideninAlfvenunit(sameunitasvelocityu)

∂u∂t

+ u ⋅∇u= −∇Pρ

+ν∇2u +B ⋅∇BNonlinear!Nonlinear!

•  Maxwell’sEqua:ons

•  Ohm’slawJ=σ(E+u×B)•  Magne:cReynoldnumberRm=R=Lu/ηη=1/σisthemagne:cdiffusivityEliminateE,àInduc:onequa:on

Again,wecanwriteBinAlfvenunitintheinduc:onequa:onandthereazer.

Fromafluidperspec:ve,howdoesturbulenceinthesolarwindplasmadifferfromhydrodynamicTurbulence?

∂B∂t

+ u ⋅∇B = B ⋅∇u +η∇2BNonlinear! Nonlinear!

Magneto-hydrodynamic(MHD)Turbulence

BcanbesplitintoameanfieldB0andafluctua:ngfieldb,B=B0+b.Definenewvariablestoreplacebandu,theElsässer(1950)variablesz±=u±b,Wecanrewriteourequa:onsintoWhereν±=½(ν±η),Nonlinearinterac:onsoccurbetweenthez±.

∂z±

∂t∓ (B0 ⋅∇)z

± + (z∓ ⋅∇)z± = −∇P+ν+∇2z± +ν−∇

2z∓

∂B∂t

+ u ⋅∇B = B ⋅∇u +η∇2B

∂u∂t

+ u ⋅∇u= −∇Pρ

+ν∇2u +B ⋅∇B

OuterscalesL±:e-foldingdefini:onTwo-pointcorrela:onFindL±suchthatR±(L±)=1/e

l

L+L-

R± (l) = 〈z± (x) ⋅z± (x+ l)〉σ z±2

∂z±

∂t∓ (B0 ⋅∇)z

± + (z∓ ⋅∇)z± = −∇P+ν+∇2z± +ν−∇

2z∓

Relevant:mescalesandturbulentmodels•  Alfven:meτA=L±/(kB0)•  Nonlinear:meτNL±=L±/(kzk±)

•  IK:Iroshnikov(1964)andKraichnan(1965)assumedz+andz-interactweaklyandlinearizedtheequa:onswithτAbeingtherelevant:me,theyderivedEu(k)~Eb(k)~(εB0)1/2k-3/2.

•  K41-like:Marsch(1990)assumedfundamentallynonlinear.τNL±istheinterac:on:meforeddies,theyderivedE±(k)~(ε±)4/3(ε)-2/3k-5/3.

•  Cri:calbalance(GoldreichandScridhar,1995):“compound”versionofIKandK41-likedescrip:ons:τA~τNL.Theyderived

andE⊥ (k⊥ )∝ k⊥−5/3 E||(k|| )∝ k||

−2

Spectra:solarwindobserva:ons

-5/3

Magne:cfieldenergyspectrum(MadhaeusandGoldstein,1982)

Velocityspectrum(Podestaetal.,2007)

Complica:on:B0

•  InthestrongB0case(B0>>b),theturbulentspectrumsplitsintotwoparts:anessen:ally2DTurbulencespectrumwithbothuandbperpendiculartoB0,andaweakerandmorenearlyisotropicspectrumofAlfvenwaves(MontgomeryandTurner,1981,MontgomeryandMadhaeus,1995).

•  MHDsimula:ons:MeanMagne:cfieldB0suppressestheenergycascadealongthedirec:onofthemeanmagne:cfieldàanisotropy(Shebalinetal.1983).

2DTurbulencev.s.Slab

BrunoandCarbone(2013)Review:•  Helio(0.3-1AU)dataintheslowwind,Interplanetarysolar

wind,74-95%2Dturbulenceand5-26%slab(Bieberetal.,1996).

•  Inthepolarwind,50%2Dturbulenceand50%ofslab(Smith,2003).

•  Dassoetal.(2005),using5yearsofspacecrazobserva:onsatroughly1AU,showedthatfaststreamsaredominatedbyfluctua:onswithwavevectorsquasi-paralleltothelocalmagne:cfield(slab),whileslowstreamsaredominatedbyquasi-perpendicularfluctua:onwavevectors(2Dturbulence).

∂z±

∂t∓ (B0 ⋅∇)z

± + (z∓ ⋅∇)z± = −∇P+ν+∇2z± +ν−∇

2z∓

MHDperspec:ve:Crosshelicity

Highalignmentofbandu(correspondstomaximumHc)resultsz+andz-alignmentandthusreducesthenonlineartermintheMHDequa:on.Waveandlineartermsmaydominate.Onthecontrary,lowHccorrespondstoamorenonlinearlyturbulentplasma.

Observa:onally,Robertsetal.(1987a,1987b)findthatwhenHcisnearlymaximalinfastwindfrom0.3-1AU,therewaslidleevidenceofturbulentevolu:on.Instead,fluctua:onsarehighlyAlfvenic.Ontheotherhand,MadhaeusandGoldstein(1982)findthatfor(sta:onary)intervalsspanningseveraldays,thespectrumofBisveryclosetok41’s-5/3scaling.

Hc = ∫u ⋅BdV

∂z±

∂t∓ (B0 ⋅∇)z

± + (z∓ ⋅∇)z± = −∇P+ν+∇2z± +ν−∇

2z∓

Alfvenra:orA=Eu/EbSpecialcase:u=bandualignmentwithb(rA=1andmaximumHx))

z-vanishesLezwithz+=2b=2uandasimplerequa:onthatislinearlizableFluctua:onscanbehighlyAlfvenic.

∂z+

∂t− (B0 ⋅∇)z

+ = −∇P +ν+∇2z+

Cau:on:Specialcasedoesnotrepresentsolarwindgeneralcondi:on!

Turbulenceinsolarwindisdynamicallyac:ve,notjustaremnantofturbulenceinthecorona.

Turbulentspectrallowk(1/l)breakpointevolu:on:SolarwindturbulenceisAc:ve

Varia:onofspectralbreakpointv.s.solardistance(Horburyetal.,1996)

-5/3

Magne:cfieldenergyspectrum(MadhaeusandGoldstein,1982)

Solarwindspectralbreakathighk(s):Dissipa:on

Goldsteinetal.,2015

Dissipa:on

1.   IntermiSentdissipa0onbynon-linearcoherentstructures(MadhaeusandMontgomery,1980):primarilycurrentsheets(andrelatedreconnec:on).Hea:ngisbursty,patchy,andnon-uniform.

2.  ResonantdampingofIncoherentWaves-LandaudampingofKine:cAlfvenWave(e.g.,Chandranetal.,2010,Howesetal.,2011)-Whistler(e.g.,Changetal.,2011)Orboth1and2?AndEachdominatesatdifferentcondi:ons?

Dissipa:on:requireskine:cdescrip:ons

•  MHDisnotadequatetoaddressdissipa:on•  Needinves:ga:onsthatcanresolvetheionandelectronscales.1.Simula:ons:-Gyrokine:cCaptureAlfvenicfluctua:ons,howeveritoperatesatlowfrequencylimitandmisshighkphysics(dissipa:onscaleintermidency,whistler,magnetosonicwaves).Italsoaveragesoutcyclotronmo:ons.-HybridCaptureionkine:cs.However,itmisseselectronkine:cs.-Fullyelectromagne0cpar0cle-in-cell(PIC)simula0onsSelf-consistent,solvebothionandelectronkine:cs,computa:onallyexpensive2.Observa:ons:highcadentspacecrazdata

PICsimula:on:Spectrumresolvedtoelectronscales

Wuetal.,2013,APJL

Fully electromagnetic kinetic Simulation

az Jz

vi n

The eddies interact nonlinearly, merge, stretch, attract, and repel each other, similar to a previous MHD simulation by Matthaeus and Montgomery (1980) and Servidio et al. (2009, PRL)

Moviemadefromasimula:oninWuetal.,2013,APJL

Reconnection X-points

az

Hint:intermidency

Simula:onfromWuetal.,2013,APJL

Intermidency

Brunoetal.,2001

Intermidency

Localvariability:underthesamesolarwindcondi:ons,thereisabroadrangeoflocalcascaderatesthatdeviatesfromGaussian.(Coburnetal.2014)

Intermidency:PICsimula:onsandobserva:ons

Kurtosis>3andincreasedwithdecreasedscaleinthedissipa:onrange

PICsimula:ons.PDF(δb(l))deviatesmorefromGaussianasscalel(inthefiguredonatedbyδr)isreduced.

Wuetal.,2013,APJL

Thereisasuddenincreaseatscale~5di.Waves?

CoherentstructuresandwaveExcita:on(VPICsimula:on)

Karimabadietal.,2013,PoP

Hea:ngatcoherentstructures(currentsheet)isordersofmagnitudemoreefficientthanwavedamping!

Enhanceddissipa:on@enhanced“filaments”(strongercurrentsheet)

Intermidentdissipa:on!

PVIl =|δb(l) |σ 2

δb(l )

Wuetal.,2013,APJL

vonKarmanenergydecayinMHDPolitanoandPouquet,1998,PREandWanetal.,2012,JFM

WriteElsasserenergiesZ±2=<|z±|2>=<|u±b|2>,hereZ±istheturbulentamplitude.DeriveandandEnergycontainingeddies(Z+,Z-,L+,L-)controlsdecay(dissipa:on)ε=d(Z+2+Z-2)/dt,independentofviscosityandresis:vity(detailsofdissipa:onmechanism)!

dZ+2

dt= −α+

Z+2Z−

L+

dZ−2

dt= −α−

Z−2Z+

L−

dL+

dt= β+Z−

dL−

dt= β−Z+

MHDequivalentof3rdorderlaw

vonKarmansimilaritydecayinfullyeletromagne:cpar:cle-in-cellsimula:ons

PlasmaenergydecayappearstobeconsistentwithMHDextensionofvonKarmansimilaritydecay,independentofmicrophysics!

Wuetal.,2013,PRL

Remarks

Kinetic scale intermittency not only shares basic properties with its MHD and hydrodynamic counterparts, but also admits interesting differences associated with plasma effects. The coexistence of dissipative coherent structure and incoherent plasma waves makes the study of turbulence in a plasma more challenging than in ordinary fluid.