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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.