Cascade
Commun.Theor.Phys.(Beijing,China)46(2006)pp.1074–1080
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)
AbstractThetimebehaviorsofintermittentturbulenceinGledzer–Ohkitani–
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.TheGleder–Ohkitani–
Yamada(GOY)[1,2]
modelcanbeseenasatruncationof
Navier–Stokesequation.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 holdsthemainsymmetryofNavier–Stocksequation.
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 slavedAdams–Bashforth-schems.[4]
Theresultisqualita-
tivelyconsistentwithRunge–Kuttamethod.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=5，13.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
tothespacetranslationsymmetryofNavier–Stokesequation. 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