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Cascade

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Cascade

    Cascade

    Commun.Theor.Phys.(Beijing,China)46(2006)pp.10741080

    c?InternationalAcademicPublishersVol.46,No.6,December15,2006

    CascadeofRandomRotationandScalinginaShellModelIntermittentTurbulence? SUNPeng,1,2,?

    CHENShi-Gang,3

    andWANGGuang-Rui3

    1

    GraduateSchoolofChinaAcademyofEngineeringPhysics,P.O.Box2101,Beijing100088,China

    2

    PhysicsDepartment,AnshanNormalCollege,Anshan114005,China

    3

    InstituteofAppliedPhysicsandComputationalMathematics,P.O.Box8009(28),Beijing100088,China

    (ReceivedFebruary15,2006)

    AbstractThetimebehaviorsofintermittentturbulenceinGledzerOhkitani

    Yamadamodelareinvestigated.Two

    kindsoforbitsofeachshellwhichisintheinertialrangearediscussedbyportraitanalysisinphasespace.We?nd

    intermittentorbitpartswanderingrandomlyandthedirectionsofunstablequasi-periodicorbitpartsofdi?erentshells

    formrotational,reversalandlockedcascadeofperiodthreewithshellnumber.Wecalculatethecriticalscalingof

    intermittentturbulenceandtheextendedself-similarityofthetwopartsoforbitandpointoutthatnonlinearscalingin

    inertial-rangeisdecidedbyintermittentorbitparts.

    PACSnumbers:47.27.Gs,47.52.+j,05.45.Jn,05.45.Pq Keywords:intermittentorbit,unstablequasi-periodicorbit,criticalscaling,extendedself-si

    milarity(ESS)

    1Introduction

    Oneoftheintriguingproblemsinthree-dimensional turbulenceisrelatedtotheunderstandingofthedynam- icalmechanismtriggeringandsupportingtheenergycas- cadefromlargetosmallscales.TheGlederOhkitani

    Yamada(GOY)[1,2]

    modelcanbeseenasatruncationof

    NavierStokesequation.Themostoutstandingproperty oftheGOYmodelisthat,forasuitablechoiceoffreepa- rameter,thesetofscalingexponentsζpcoincideswiththat

    measuredintrueturbulence?ows.Itisstillinterestingto investigatetimebehaviorsofintermittentturbulencewith GOYmodel,becauseitmaybeimportanttosimulatethe fastcourseoflaserbeampropagationintheatmosphere. Recently,KatoandYamada[3]

    studiedunstableperi-

    odicsolutionandturbulencesolutionofsystemof12shells inGOYmodel.Theydescribedstatisticssimilaritybe- tweentwosolutionsandfoundthatthescalingexponents ofstructurefunctionoftwosolutionshaveasimilarnonlin- earscalingatthesameparametervalues.Buttheydidn't givedetailedstructureandanalysisofunstableperiodso- lutioninphasespace.Inthispaper,westudythephase portraitofshellvelocityininertialrangeofsystemof22 shellsinGOYmodeland?ndthattherandomwandering ofintermittentorbit(IO)partsandthedirectionofunsta- blequasi-periodicorbit(UQO)partsinphasespaceform

therotational,reversalandlockedcascadeofperiodthree

    withshellnumber.Wealsocalculatethecriticalscaling

    ofintermittentturbulenceandtheextendedself-similarity

    (ESS)ofIOandUQOrespectively.

    GOYmodeldescribesaone-dimensionalcascadeofen-

    ergiesamongasetofcomplexvelocities.Itisanordinary

    di?erentialequationsystem,

    dUn

    dt

    =?νk2

    nUn+fδn,4+ikn

    ?

    U?

    n+1U?

    n+2?

    δ

    λ

    U?

    n?1U?

    n+1

    ?

    1?δ

    λ2

    U?

    n?1

    U?

    n?2

    ?

    ,(1)

    whereνisviscosity,wavenumbersknaregivenbykn= k0λn

    (n=1,...,N),andNisshellnumber.External forcingfisappliedtothefourthshell,?standsforcom- plexconjugate,andδstandsfornonlinearcouplingpa- rameter,whichisanadjustableparameter.Thismodel holdsthemainsymmetryofNavierStocksequation.

    Thetime-averageenergy?uxthroughthenthshellis Πn=Im?Δn+1+(1?δ)Δn?,(2)

    wheretripleproductsΔn=kn?1Un?1UnUn+1.Wethen

    canpictureasteadystateofthedynamicalsystemascas- cadeofenergyfromlarge"eddies"tosmallerones,where theenergyisdissipatedthroughviscousdi?usion.Itis inthissensethatthedynamicsmaysimulaterealtur- bulence.Inthenumericalimplementation,weuseda slavedAdamsBashforth-schems.[4]

    Theresultisqualita-

    tivelyconsistentwithRungeKuttamethod.Throughout

    thispaperwemakechoicesN=22,k0=2?4

    ,λ=2,

    ν=1×10?7

    ,f=5×(1+i)×10?3

    δn,4,andsteph=10?4

    .

    Thecourseoftransient3000n.uiscuto?,whenwecalcu- latescalinglaw.

    2PropertyofOrbitinPhaseSpaceand

    CriticalScaling

    2.1PropertyofStableQuasi-periodicOrbits inPhaseSpace

Ininertialrange,Πn

    =2νQ,whereQ=

    P

    n

    k2

    n|un|2

    /2,

    inertialrangeisadoptedn=513.Foreachshellin

    theinertialrange,theorbitofsolutioninphasespace isplottedbytakingRe(Un)andIm(Un)ascoordinates. ?

    TheprojectsupportedbyNationalNaturalScienceFoundationofChina,theScienceFoundat

    ionofChinaAcademyofEngineering

    PhysicsunderGrantNo.10576076,theMajorProjectsofNationalNaturalScienceFoundati

    onofChinaunderGrantNo.10335010,and

    theScienceFoundationofChinaAcademyofEngineeringPhysicsunderGrantNo.2004043

    0

    ?

    E-mail:sunpeng169@sina.com

    No.6CascadeofRandomRotationandScalinginaShellModelIntermittentTurbulence1075

    Whenδ<δc=0.37502,theorbitsofsolutioncorre- spondtostablequasi-periodictimeseries.Hereδciscrit-

    icalpointfromsteadyquasi-periodictimeseriestointer- mittenttimeseries.Atδ=0.371,theorbitsofshells

    n=5,8,11andn=6,9,12areenvelopecycleswhichare shapedquasi-periodicallybyunclosedloopcurvesprecess- ingclockwiseandcounterclockwisearound(0,0)respec- tively(seeFig.1(a)).Thoseenvelopecyclesaredelayed slightlyfrombigshellstosmallones(seeFig.1(d)).For example,envelopecycleofshell5includesabout172.002

    wholeprecessionswiththeaverageprecessionanglebe- ingabout2.093?

    ,andtheaverageperiodofeachpreces-

    sionisT=556.44withtherelativeerrorbeingabout 8.0×10?4

    .Theeddyturnovertimeofshell5isabout7.47.

    Similarly,envelopecycleofshell6includesabout169.891 wholeprecessionswiththeaverageprecessionanglebeing about2.119?

    ,andtheaverageperiodofeachprecession

    isaboutT=556.45withtherelativeerrorbeingabout 9.5×10?4

    (seeFig.1(b)).Theeddyturnovertimeofshell

    6isabout2.27.Ontheotherhand,thephaseorbitsof shellsn=7,10,13formsuccessivelyquasi-periodiccircles whicharealllockedatthesamedirection(seeFig.1(c)). Thesephenomenahavecharactersofperiodicthreewith shellnumber,whichcanbeexplainedasthefactthatthe GOYmodel(1)inthecaseofin?nitenumberofshells withinviscidandunforcedlimithasstaticsolutionfor 0<δ<1,

    UK41

    n=k?1/3

    nh1(n).(3)

    Hereh1(n)isanyperiodicfunctionofperiodicthree.[5] Fig.1Phaseportraitsatδ

    =0.371.(a)Theunclosedloopcurvesofshells5and6precessaround(0,0)withaverage

    period556.44and556.45clockwiseandcounterclockwiserespectively.Thearrowheadisth

    einitialpositionofenvelope

    cycle;(b)Shells5and6formthesmallandbigenvelopecyclerespectively;(c)Stablequasi-pe

riodiccyclelockedatthe

    samedirectionofshells7,10,and13.(d)Atδ

    =0.371,thetimeseriesofvelocitymodulusofshells5,6,and7are delayedslightly.

    Atδ=0.374,theenvelopecyclesofshellsn=5,8,11 andn=6,9,12changeintostablequasi-periodiccycle. Thedirectionofquasi-periodiccycleisde?nedintermsof thephaseofintersectionofquasi-periodiccycle.Forthe directionofstablequasi-periodiccycleofshelln=5,6, theresultsareabout121.42?

    and76.71?

    withrelative

    errorbeingsmallerthan1.5×10?3

    (seeFig.2).The

    powerspectrumofeachshellshowsthatithasthesame fundamentalandmultiplefrequencies.

    1076SUNPeng,CHENShi-Gang,andWANGGuang-RuiVol.46 Fig.2Phaseportraitsatδ=0.374.(a)Shellsn=5,8,

    and11;(b)Shellsn=6,9,and12.

    2.2RandomWanderingofIntermittentOrbitand PropertyofRotationofUnstable

    Quasi-periodicOrbit

    Atδ>δc,thetimeseriesofthevelocitymodulus ofeachshellcanbedividedintointermittenttimeseries (chaospart)andunstablequasi-periodictimeseries.Sim- ilarly,fortheorbitsinthephasespace,theorbitscan bedividedintointermittentorbit(IO)partscorrespond- ingtotheintermittenttimeseriesandunstablequasi- periodicorbit(UQO)partscorrespondingtotheunsta- blequasi-periodtimeseries.UQOpartsinoursituation

areunstablequasi-periodcycles.Forthesameshell,one

quasi-periodiccyclewhichhasadirectionaround(0,0)

    graduallyprocessestotheintermittentorbit.Sincethe phasesofintermittentorbitpartwanderrandomlyaround (0,0)intheinterval[0,2π],thedirectionofquasi-periodic

    cyclerotatesrandomlyaround(0,0)aftertheintermit-

    tence(seeFig.3).Thetwopartorbitsevolvewithtime

    alternately.

    Fig.3Atδ

    =0.378,timeseriesofvelocitymodulusevolvingfromunstablequasi-periodtochaosandthei

    rcorresponding

    phaseportrait.(a)Shell5;(b)Shell6.

    No.6CascadeofRandomRotationandScalinginaShellModelIntermittentTurbulence1077

    Thedi?erenceofrotationdirectioncanbede?nedintermsofthedirectiondi?erenceofunstabl

    equasi-periodic

    cycleatthefrontandbackofintermittence.Atδ

    =0.378,thedi?erenceofrotationdirectionofunstablequasi-periodic cycleofshellsn=5,8,11isabout21.88?

    ,21.51?

    ,and20.08?

    withtherelativeerrorbeingsmallerthan1.6×10?3

    ,and

    thedi?erenceofrotationdirectionofquasi-periodiccycleofshellsn=6,9,12isabout?21.71?

    ,?21.25?

    ,and?20.86?

    withtherelativeerrorbeingsmallerthan2.0×10?3

    .Thedi?erenceofrotationdirectionofunstablequasi-periodic cycleofshellsn=7,10,13islockedat45?

    .Thephenomenaalsohavecharactersofperiodicthreewithshellnumber (seeFig.4).

Fig.4Phaseportraitsofunstablequasi-periodiccycleofshellsbeforeintermittence.(a)n=5,8

    ,and11;(b)n=6,9,

    and12;(e)n=7,10,and13.Phaseportraitsofunstablequasi-periodiccycleofshellsafterinter

    mittence:(c)n=5,8,

    and11;(d)n=6,9,and12;(f)n=7,10,and13. BenziandGatetal.[6,7]

    pointedoutthatshellmodelwithoutexternalforcinghasphasesymmetrycorresponding

    tothespacetranslationsymmetryofNavierStokesequation. Let

    Un=k?1/3

    n

    ρnexp(iθn).(4)

    Atf=0,withthischoice,equation(1)becomes ?

    d

    dt

    +νk2

    n

    ?

    ρn=k2/3

    n

    h

    ρn+2ρn+1sin(Δn+2)?

    δ

    2

    ρn+1ρn?1sin(Δn+1)?

    1?δ

    2

    ρn?1ρn?2sin(Δn)

    i

.(5)

    HereΔn=θn?2+θn?1+θnandtheprobabilitydistributionofθ

    nisuniformintheinterval[0,2π].Bysimplytaking

    θn?θn,θn?1?θn?1?α,θn?2?θn?2+α.(6)

    1078SUNPeng,CHENShi-Gang,andWANGGuang-RuiVol.46

    Foranyn,onesolutionoftheequationcanbetransformedintoanothersolution.Inourcase,theexternalforcingin

    Eq.(1)actsonshelln=4.Ifweinterpretθ

    nastherotationdirectionoftheunstablequasi-periodiccycle,therotation directionofshelln=4locksandsodoforshellsn=7,10,and13becauseofcharacterofperiodicthreewithshells

    number.AccordingtoEq.(6),thedi?erenceofrotationdirectionatthefrontandbackofintermittenceofshells7,

    10,and13isΔθ7=Δθ10=Δθ13=0,whileforshells5,8,and11andshells6,9,and12,Δ

    θ5??Δθ6,Δθ8??Δθ9,

    Δθ11??Δθ

    12.Thismeansthatthedirectionofunstablequasi-periodiccycleofdi?erentshellsininertialrangeforms

    therotational,reversalandlockedcascadeofperiodthreewithshellnumber. 2.3CriticalScalingofIntermittentTurbulence

    Withδ

    increasing,thepartsoftheintermittenttimeseriesofthevelocitymodulusincreasegradually,whileunstable

    quasi-periodictimeseriesdecreasecorrespondingly.Wecancalculatethestatisticalaverageofunstablequasi-periodic

    timeserieslength?TUQO

    ?.Theformofcriticalscalingis

    ?TUQO

    ?=C(δ?δC

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