The Effect of Helicity on the Effective Diffusivity for Incompressible Random Flows

IncompressibleRandomFlows

D.S.Dean

IRSAMC,LaboratoiredePhysiqueQuantique,

Universit´ePaulSabatier,118routedeNarbonne,31062ToulouseCedex

I.T.DrummondandR.R.HorganDAMTP,CMS

WilberforceRoad,CambridgeCB30WA

February1,2008

Abstract

Theadvectionofapassivescalarbyaquenched(frozen)incompressibleveloc-ityfieldisstudiedbyextensivehighprecisionnumericalsimulationandvariousapproximationschemes.Weshowthatsecondorderselfconsistentperturbationtheory,intheabsenceofhelicity,perfectlypredictstheeffectivediffusivityofatracerparticleinsuchafield.Inthepresenceofhelicityintheflowsimulationsrevealanunexpectedlystrongenhancementoftheeffectivediffusivitywhichishighlynonperturbativeandismostvisiblewhenthebaremoleculardiffusivityoftheparticleissmall.Wedevelopandanalyseaseriesofapproximationschemeswhichindicatethatthisenhancementofthediffusivityisduetoanovelsecondordereffectwherebythehelicalcomponentofthefield,whichdoesnotdirectlyrenormalizetheeffectivediffusivity,enhancesthestrengthofthenonhelicalpartoftheflow,whichinturnrenormalizesthemoleculardiffusivity.Weshowthatthisrenormalizationismostimportantatlowbaremoleculardiffusivity,inagreementwiththenumericalsimulations.

DAMTP-2000-139

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1Introduction

Theadvectionofpassivefieldssubjecttomoleculardiffusionandconvectionbyturbu-lentfluidhasbeenextensivelystudiedbyboththeoreticalandcomputationaltechniques[1,2,3,4,5,6].Bycomparingtheresultsofsimulationwiththetheoreticalpredictionforvariouslong-rangequantities,theefficacyofthetheoreticalmethodscanbetestedalbeit,insomewhatartificialmodels.Theapplicationstothephysicsofcomplexsys-temsandengineeringaremanyfold.Inpracticalproblemsweneedtocalculatethebulkpropertiesofrandommediagivenstatisticalmodelsforthedisorderpresent.Ingeneralthecomplexityoftheserealworldproblemsmeansthatonemustresorttoapproxima-tionschemestocalculatetheselargescalebulkproperties.Itisthereforeessentialtoverifyvariousmethodsofanalysisonmodelproblemsbeforeonecanbeconfidentthattheseorsimilarmethodscanbeappliedtomorerealisticsystems.Thesuccessofanapproachdependsonwhethertheapproximationpreservestheessenceofthephysicalmechanismresponsiblefordeterminingthelong-rangeparametersoftheadvectionintermsoftheparametersdescribingthelocalcharacteristicsoftheflow.InthispaperweconsideradvectioninahelicalGaussianturbulentflowwhichwasoriginallystudiedin[4].Thesurprisingresultobservedonthebasisofsimulationisthatthelong-rangeeffectivediffusivity,κe,isgreatlyenhancedbythepresenceofthehelicitybymorethanafactoroftwo,theeffectbeingstrongestforsmallmoleculardiffusivity,κ0.Intheab-senceofhelicitythecalculationofκetotwoloopsinself-consistentperturbationtheoryagreesaccuratelywiththesimulationforallκ0.However,suchanapproachpredictsthatevenmaximalhelicitywillhaveonlyasmalleffectoftheorderof10%.Thisisinstarkcontrasttotheresultsofsimulation.Thepuzzleistoexplaintheseresultsforwhatisarelativelysimplyposedsystem.Asuccessfultheoreticalapproachwillinvolveinfiniteresummationsofcontributionsanditisinthissensethattheenhancementisnon-perturbative.

Inthispaperwediscussapossibleresolutionoftheconflictbetweentheoryandsimulationbyusingvariousmethodstoidentifythelow-wavenumbereffectivetheorygoverningthediffusivedispersalofparticlesadvectedintheturbulentflowwhenhelicityispresent.Thederivationoftheeffectivetheoryisguidedbytherenormalizationgroup(RG)ideathattheGreenfunctionatlowwave-numberis,insomeapproximation,thesolutiontoaneffectivesecond-orderdifferentialequationwhoseparametersaredeterminedself-consistentlyintermsoftheoriginalor‘bare’definingthemodel.Theeffectofhelicityintheflowcausestheturbulentvelocityfield,u(x,t),tobeadditivelyrenormalizedbyatermproportionaltothevorticity,ω=∇×u.Thecoefficientofproportionalityisaisapseudo-scalarwhichisgeneratedbytheaxial-vectornatureofthehelicalflowandsodependsonthehelicityh,definedintermsofuby

h=󰀒u·∇×u󰀓

(1.1)

where󰀒·󰀓denotestheensembleaverageovertherandomvelocityfield.Inourmodelthemagnitudeofhismeasuredbyaparameterλ,0≤λ≤1,andtheresultsaregivenintermsofλ.Theusualperturbativeresultforthedependenceofκeonλisthatκeisaseriesinλ2forallvaluesofκ0.Thisisself-evidentsincethemagnitudeofκeisindependentofthesignofλ.Thesimulationisseeminglyconsistentwiththisfactforλ<0.2atκ0=0butisnotwellfittedbyanysimpleapproach,andforlargerλthe

2

curveliesfarhigherthanthenaivecalculation.Wediscussanimprovedself-consistentschemewhichexpressestheGreenfunctionandvertexfunctionsassolutionstointegralequationswhicharesolvedinalow-wavenumberapproximation.Thismethodleadstoastrongenhancementofκeforincreasingλand,assuch,isagoodindicationthatweareontherighttrack.However,forsmallλtheeffectisparadoxicallytoostrong,leadingtoanon-analyticdependenceofκeonλwhichispredictedtobeκe∼λ2/3intheone-loopcase.Thisispossiblyduetotheapproximationmadeinobtainingthesolutionbutitisacomplexmattertoascertainwhetherthisisso.AnalternativeapproachistousethefunctionalHartree-FockmethodwhichleadstoanintegralequationfortheGreenfunctionself-energyasafunctionofwave-number.Theresultofthismethodforκe(λ)isbetterbehavedatsmallλbutthepredictedenhancementisnotbigenoughanddoesnotfitthesimulationdata.Ingeneral,theeffectismostpronouncedforsmallκ0andempiricallyfromoursimulationwefindthattheresultsdistinguishtheregionsκ0≪0.2andκ0≫0.2.Thereisapronounceddipinκevsκ0atκ0∼0.2forλ=1.Thisdipisnotpredictedbyeitherofthemethodsmentionedsofar.

Wealsopresentarenormalizationgroupapproachwhichshowsamechanismfortheenhancingeffectofhelicityonκe.Therenormalizationgroupisnormallymostusefulforcomputinganomalousexponentssincetheyaregenerallyindependentofmuchofthedetailsdefiningthemodel:theideaofuniversality.ItismuchmoredifficulttocontrolastandardRGanalysisifitisusedtocalculatethecoefficientsofscalingbehaviour,i.e.,observableslikeκe.However,inref.[7]wereportedonasuccessfuluseoftheRGinpredictingκeforgradientflowsandbelievethatanRGanalysiscangenerallygiveastrongindicationofthekindofmechanismwhichinfluencesthesizeofparameterscontrollingthelarge-scalecharacteristicsofadvection.Inthispaperweshowthattheflowatlargewavevectorcanstronglyenhanceκewhenκ0issmall.Inparticular,thisapproachdoesprovideamechanismforthedipobservedinκevsκ0atκ0∼0.2forλ=1.

Insection2themodelandtheformalismarereviewed;insection3theperturbationtheoryisbrieflydescribed;insection4theself-consistentintegralequationsfortheGreenfunctionandvertexfunctionsarederivedtoone-loopandthesmallwavevectorapproximationforκeisderived;insection5thefunctionalHartreeFockmethodisexamined;insection6therenormalizationgroupapproachisexplainedandinsection7theconclusionsarepresented.

2TheModelandFormalism

In[4]theproblemofapassivescalaradvectedbyanincompressibleturbulentflowwithamoleculardiffusivitywasstudied.Theturbulentfluidvelocityfield,u(x,t),wasdescribedbyitsstatisticalpropertieswhichwereassumedtobeGaussianandsofullydeterminedbythevelocityauto-correlationfunction.Intheoriginalstudytheflowwastime-dependent,butsincetheenhancementofκebyhelicityintheflowispresentalsofortime-independentflowsweassumehere,forsimplicity,atime-independentflow(i.e.quenchedorfrozenturbulence)forwhichtheauto-correlationfunctioncanbeexpressedinthefollowingform:

󰀒ui(x)uj(x)󰀓=

′

󰀍

d3k

TheensembleofvelocityfieldswastakentobehomogeneousandisotropicandsoforincompressiblefluidsFij(k)canbewrittenas

Fij(k)=Φ(k)(k2δij−kikj)+Ψ(k)iǫimjkm,

(2.2)

whereΨrepresentsthepresenceofhelicityintheflow.In[4]itwasassumedthatΦandΨtookthefactorizedforms:

Φ(k)=

(2π)3

A2kE(k)sin2ψ,

(2.3)

3

whereAischosensothat

󰀍

dkE(k)=1,

󰀒u·u󰀓=u20,

(2.4)

andwhereu0isther.m.s.velocity.Choosingtheangleψtobek-independentmeans

thatthehelicityisofequalstrengthatallwavevectors.Thehelicityparameter,h,hasbeendefinedineqn.(1.1)andwiththedefinitionsineqn.(2.3),wefind

h=

2

dt

=κ0∇2Θ−∇·(uΘ),

󰀍

(2.6)

andtheeffective,orlong-range,diffusivity,κe,isdefinedby

󰀒x·x󰀓(t)=󰀒

whereΘisnormalizedtounity:

󰀍

d3xx·xΘ(x,t)󰀓,

(2.7)

=6κet+O(t0)ast→∞,

d3xΘ(x,t)=1.

(2.8)

Forthepurposesofnumericalsimulationaparticularmemberofthevelocity-fieldensembleisthenrealizedby[1,2,4]

u(x)=A

N󰀉󰀎󰀊

n=1

+

wherethevectorsξnandχnaredistributeduniformlyandindependentlyovertheunit

sphereandthewavevectorknisdistributedaccordingtothedistributionE(k).ForNsufficientlylargethecentrallimittheoremguaranteesthatu(x)isGaussianupto

4

󰀎

ˆnsinψ∧kncos(kn·x)ξncosψ−χn∧k

ˆnsinψ∧knsin(kn·x)χncosψ+ξn∧k

󰀏

󰀏

󰀌

,(2.9)

O(1/N)corrections.WehaveusedN=forwhichtheseeffectsaresufficientlysmallforourpurposes.

TosimulatetheevolutionofthescalarfieldΘ(x,t)weintegratenumericallythestochasticequationfortheevolutionofaparticlewithpathx(t)givenby

˙(t)=u(x(t))+η(t),x

(2.10)

whereη(t)isaGaussianrandomvariablewith󰀒η(t)󰀓=0and󰀒η(t)·η(t′)󰀓=2κ0δ(t−t′).

Theresultingprobabilitydistributionforparticlepositionx(t)isthenΘ(x,t)withtheinitialconditionΘ(x,0)=δ(x).

Thediscreteformofeqn.(2.10)suitablefornumericalintegrationis:

xn+1−xn=u(xn)∆t+(2κ0∆t)

1

M

a=1

=6κet+O(1)

M󰀊

x(a)(t)·x(a)(t),

ast→∞.

(2.12)

HereMisthetotalnumberofpathsaveragedoverand(a)labelthememberofthe

ensembleofpaths.InpracticeMisfinitebutlargeenoughtogiveanestimateforκewithsmallerror.Inadditiontmustbelargeenoughsothatthepathevolutionisintheasymptoticregimewheretheevolutioncanbesuitablydescribedintermsoflongrangeeffective,or“renormalized”quantities.Thattislargeenoughistestedbyensuringthattheestimateforκeisindependentoftwithinstatisticalerrors.

3PerturbationTheory

Theperturbativeapproachtosolvingeqn.(2.6)iswellknown[8,9,6]andweonlysummarizeherethenecessaryresults.

SinceweareinterestedintheeffectiveparametersgoverningtheevolutionofthedistributionΘ(x,t),westudytherelatedGreenfunctionG(x)whichsatisfies

κ0∇2G−u·∇G=−δ(x),

(3.1)

˜(k)wheretheincompressibilityofuhasbeenused.Aperturbationseriesinu/k0forG

canbegeneratedbyiteratingtheformalsolutiontoeqn.(3.1)inFourierspace:

˜(k)=G

1

κ0k2

󰀍

dq

κ0k2−Σ(k)5

,(3.3)

wheretheaveragingoverthevelocityensembleisdoneusingWickstheoremtogiveadiagrammaticexpansionandΣ(k)isgivenbyoneparticle-irreduciblediagrams.The

˜󰀓isgivenasymptoticbehaviourineqn.(2.12)impliesthatthesmallkbehaviourof󰀒G

by

κe=κ0−

d

2

u226k20

0

e−k/2k0

,

Ψ(k)=

λkΦ(k),

whereλ=sin2ψ.

Thesimpletwo-loopcalculationforκegivestheresult

κe=κ0󰀄

1+

1

κ20059λ2

0k2+(0.0

−0.00884)

u40

(3.6)

theoryisparametrizedandwhichquantitiesaretreatedself-consistently.Asuccessfulresultwilldependonhowwellthemethodcapturesthedominanteffectsinthisway.

Wefirstdiscussthesimplestapproachwhichtreatsonlyκeself-consistently.Attwo-loopsthisgivesanexcellentfitforκewhenλ=0butfailsforλ=0.Wethengeneralizethemethodandshowthatwecanqualitativelyexplainthelargeenhancementinκeduetohelicityalthoughtheapproachisstillquantitativelydeficient.Furthergeneralizationsarediscussedbuthavenotyetbeencarriedout.

4.1

Self-Consistencyinκe

Togeneratetheself-consistentperturbationseriesinκetheeqn.(3.1)forG(x)isformallyrearrangedtobecome

κe∇2G−∆κ∇2G−u·∇G=−δ(x),

(4.1.1)

where∆κ=κe−κ0.Thesecondtermisacounter-termwhichisincludedaspartoftheperturbation.Itisformallyoffirstorderintheexpansionparameterwhichallowstheexpansionforκetobeconstructedtoaconsistentorder.Theself-consistentperturbationseriesisgeneratedbyiterating˜(k)=G

1

κek2

󰀍

dq

dk2

Σ(k)|k=0=0.

(4.1.3)

22ToN-thorderinu20/κek0itisalwayspossibletowritethisconditionintheform

κe=κ0+κeFN(κe,λ),

FN(κe,λ)=

N󰀊

gn(λ)

n=1

󰀄

u20

9

κ2e

󰀂

=κe

󰀑

1

κ4e

󰀒

0.0059λ2−0.00884

󰀆

󰀃󰀂

.(4.1.5)

Thisresultcanbere-expressedintheformofeqn.(4.1.4)tobecome

κe=κ0+κe

󰀄

1

κ4e

.(4.1.6)

Weshowthetwo-loopself-consistentpredictionforκecomparedwithdatainfigures7–11.Infigures7,8and9κeisplottedagainstκ0forfixedl=0.0,0.4,1.0andinfigures10and11κeisplottedagainstλforfixedκ0=0.0,0.2.Asshouldbeexpected,wesee

7

fromfigures8and9theagreementbetweentheoryandsimulationisacceptableforκ0largeenough.Thisissimplybecausethelargemoleculardiffusivityswampsallothereffects.However,thereisalargedisagreementforsmallκ0whichismostmarkedforκ0=0.ThepredictionforκebehaveslikeO(λ2)andforκ0changesfromκe=0.3697atλ=0toκe=0.4090atλ=1:anincreaseof10%.Incontrast,thesimulationgivesκe=0.3705(1)andκe=0.8018(7)respectivelyatthesetwovaluesofλ:anincreaseofmorethanafactoroftwo.Fromthesimulationforκ0smallenoughwefindthatκeasafunctionofλisstronglyindisagreementwiththeslowpolynomialbehaviourinλpredictedbyself-consistentperturbationtheory.Thiseffectwasfirstobservedin[4]andhasremainedunexplained.

Inaddition,infigure9weobserveamarkeddipinthedataatfixedλ=1forκeversusκ0ataboutκ0=0.2.Themajorfeatureisthatκerisesrapidlywithλatκ0=0whereastheeffectforκ0ofthecurveasλincreasesatκ0≥=00..22butismuchratherlessastrong:rapidrisethewithdipλisatnotκ0a=lowering0.Theself-consistentpredictionofthissectiondoesnotpredictadipofanykind.

4.2Amoregeneralapproach

Inthissectionweproposeanexplanationoftheenhancementofκebyhelicityintheflow.Thetechniqueispresentedindetailattheone-looplevelandtheextensiontwo-loopisthengiven.

Thephilosophyistosuggestaneffective,lowwavevectordiffusionequationobeyedbythesmootheddistributionfunction.Becausethewavevectorissmallitisassumedthattheequationcanbelimitedtoatmosttwospatialderivatives.Theshortcomingsofthisassertionarediscussedlater.Weproposetheequation

dΘ

Theself-consistentequationsaregivenbysettingthenextrenormalizationsofκeandβRtozeroinperturbationtheory.Thisgivestwoequationswhichsimultaneouslydetermineκe,αRandβRintermsofthebareparametersκ0,α0andβ0.ItisconvenienttodefinethegeneralvertexUi(k′,k)oftheform

Ui(k′,k)=iV(k′,k)k′i+W(k′,k)(k′∧k)i,

(4.2.3)

wheretheform-factorsVandWarescalarfunctionsofkandk′.Thebarevertex

00U0iisdefinedbyV=α0,W=β0.Thereisnoindependentform-factorcoefficient

proportionaltokiinthisexpansionsincethevelocityfieldisincompressible.ThediagrammaticrepresentationofU0iisshowninfigures1cand1d,wherethebarevertexisrepresentedbyanopencirclewhiletherenormalizedvertexcarriesadditionallyan

˜(k)󰀓canbedefinedasinsetletter‘R’.Likewise,thegeneralexpressionfor󰀒G

˜(k)󰀓=󰀒G

1

graph.Weconsiderthecontributionshowninfigure4tothevertexrenormalizationandwillconcentrateonthepartproportionaltok′.Thevalueofthisgraphis

Tβαα=

α2RβR

󰀍

dq

κe(k−q)2κe(k′−q)2

.(4.2.6)

Theapproximationsofeqn.(4.2.5)havebeenimplemented.Onlythehelicalpartof

Fij(q)contributesandwefindtheresult

Tβαα=

α2RβRλ

ik′i

(2π)3

󰀍

dq

ǫlmnkmqn(k′−q)pqǫlpqqqΦ(q)

(k−q)2(k′−q)2

.

(4.2.7)

κ2e

ClearlythecontributiontoVisO(k·k′)andsoαisnotrenormalized.Allcontributions

toVaresimilarlyofhigherorderandtheresultisthatαR=α0.

Thecouplingβisrenormalizedwhenλ=0.Thecalculationfollowsasimilarpathtothatusedintheanalysisoftherenormalizationofα.AgainweshowonecalculationexplicitlyandconsiderthecontributiontoW(k,k′)bycalculatingthecoefficientof(k′∧k)inTααα:

Tααα=−i

α3R

(2π)3

(k−q)l(k′−q)nFln(q)

κ2e

󰀍

dq

(k−q)2(k′−q)2

α3λ

.(4.2.9)

HencewefindthecontributionδβRtotherenormalizationofβfromTαααtobe

δβR=−

6π3

󰀍

dqqnΦ(q)

n=1,2,3.(4.2.11)

Afterevaluatingalltherelevantgraphstheself-consistentequationsare

αR−α0=0,

22

β0−βR+(α3RλI1+2αRβRI2+αRβRλI3)=0.

(4.2.12)

˜(k)isgivenbytheequationforΣ(k)intermsTheapproximateequationforG

oftheone-particleirreduciblegraphsinfigure5atone-looporder.Becauseweareusingthelow-wavenumberapproximationthisreducestosubstitutingtheexpression

10

fortherenormalizedvertexUi(k′,k)givenineqns.(4.2.3)and(4.2.12)intotheone-loopdiagramforΣ(k)infigure5.Weanalyzetheone-loopself-energygraphandkeeponlythetermproportionaltok2.Inobviousnotationthisgivestheresults

Tαα∼−

2α2R

κe

I3.

(4.2.13)

Usingthespectraineqn.(3.6)eqns.(4.2.12)and(4.2.13)fortheone-loopself-consistentconditionsbecome:

∆α≡αR−α0=0,where

B1=

1

󰀄

−∆β+

B1

κe

=0,(4.2.14)

2

α2R+4

󰀐

2

αRβRλ+3βR

2

󰀆

9π

.(4.2.15)

¿Fromtheseequationsitisclearthatnorenormalizationoccursifthereisnopseudo-scalaroraxial-vectorquantityintheproblem:ifβ0=λ=0thentheproblemreducestotheone-loopself-consistentanalysispresentedinsection4.1.However,ifeitherβ0orλarenon-zerothenβisrenormalizedandtheeffectonκeisencodedineqn.(4.2.14).InourcasewesetαR=α0=1,β0=0andλ=0.Theequations(4.2.14),(4.2.15)thengive

3

󰀐

2

2

β3+3βRλ+9

9κe

󰀄

󰀐

2

α2R+4

󰀐

κeκ0βR−π

λ

󰀆

2

αRβRλ+3βR

.(4.2.16)

Forsmallλandκ0=0wededucethat

β∼

󰀑

1

3

+

1

18π

󰀂1/3

λ2/3.

(4.2.17)

Thedataforκeversusλforκ0=0isshowninfigure10andweseethatforsmallλthesimulationresultsarenotcompatiblewithλ2/3behaviour.Weshallseebelowthatthisisnotrectifiedinthetwo-loopself-consistentcalculation.However,inthisone-loopcalculationthereisaconsiderableenhancementinthedependenceofκeonλ,whereasintheself-consistentcalculationofsection4.1,inwhichthegenerationofthenewvertexcoupledtothevorticityωwasnotincluded,thereisnoeffectatallatone-looporderandonlyamildeffectattwo-looporder.Theequations(4.2.16)canbesolvednumerically.Forexample,forκ0=0,λ=1wefindβR=0.3456andtheeffectivevelocityfieldispredictedtobe

uR=u+βRω,

11

(4.2.18)

whichclearlyleadstoanenhancedeffectivediffusivity,κe=0.5207,comparedwithκe=0.4090fromthetwo-loopcalculationoftheprevioussection.Webelievethatwehavequalitativelycapturedthemechanismresponsiblefortheenhancementoftheeffectivediffusivitybyhelicity.

Theone-loopcalculationislimitedbecauseitisnotaccurateatλ=0unlikethetwo-loopcalculation.Wehaveinvestigatedthetwo-loopextensionoftheself-consistentapproachwhenthenewvertexwithcouplingβisincluded.Thisismoreinvolvedandtheintegralsweredonenumerically.Wepresentthefinalresultsbelow.

Thetwo-loopself-consistentequationsare

∆κ+C1

∂C1

κeκ3(Ce2−C12

)−∆β+

B1

κ3B+

B2

e

1−∆β1∂β

κ2+

1

)e

∂β

−∆β+

B1

−2B1C1−B1

∂B1

κ4(Be

2=0,

=0,

woulduseafunctionalself-consistentmethodforUi(k′,k)(eqn.(4.2.3))andΩ(k2)(eqn.(4.2.4)).Althoughacomputationallyformidabletask,thisislikelytoencodethecorrectbehaviourmuchmoreaccuratelythandoesourlow-wavenumberapproximation.

Theoriginofthedipinfigure9inthecurveκeversusκ0forλ=1isunexplainedbythetheorypresentedsofar.

5TheFunctionalHartree-FockMethod

Thisapproachgoessomewaytowardsincludingeffectsomittedinthelow-wavenumberapproximation.Theversionpresentedhereisdeficientinthatthepredictionforκewhenκ0=λ=0isnotasaccurateasthetwo-loopself-consistentapproachbuttheadvantageisthatΩ(k2),eqn.(4.2.4),istreatedasfunctiontobedeterminedself-consistentlybytheHartree-Fockequations.Theverticesarestilltreatedinthelow-wavenumberapproximationand,asintheprevioussection,theyareparameterizedbyαandβ.

TheintegralequationtobesatisfiedbyΩ(k2)andtheone-loopequationsatisfiedbythevertexfunction,whichisthesameastheone-loopself-consistentequation,areshowninfigure6.Notethat,unliketheself-consistentcalculationoftheprevioussection,onlyoneoftheverticesintheone-loopself-energyisreplacedwiththefullvertexsincethisgivesthecorrectcountingofdiagramswhentheequationsareiterated.Theself-consistentcaseisdifferentbecausetheaugmentedvertexisalreadypresentintheperturbationtheoryandcorrectionsareimplementedbycounter-terms.Theapproximationfortheverticesineqn.(4.2.5)isusedandβisdeterminedusingeqn.(4.2.14):

β=β0+

B1

2k2

󰀈

e−k2

/22π󰀍

dp

(pkcosh(pk)−sinh(pk))e−p

2/2

(2π)3

|p+k|φ(|p+k|)(k2p2−(k·p)2)

somesimplificationofthefunctionformwereimplemented.Also,whiletheequationforΩ(k2)isalreadyexactatone-loop,thatforthevertexisnotandwecannotprecludethathigherloopcorrectionsmightbeimportant.Wehavenotpursuedthisapproach.

Wenotethatinthisapproach,aswiththoseoftheprevioussections,themarkeddipinκeasafunctionofκ0forthelargervaluesofλisnotreproduced.

6TheRenormalizationGroup

Intheprevioussectionwepresentedananalysisbasedontheassumptionthatthelarge-scaleadvectioniscontrolledbyaneffectivetransportequationdominatedbythetermscontainingonlyoneandtwoderivatives.Thismethodisrelatedtotherenormalizationgroup(RG)methodswhichhaveprovedverysuccessfulinpredictingexponentsincriticalphenomenon.IntheRGapproachalargewavenumbercutoff,Λ,isintroducedandtheadvectiononscaleslargerthanL≡2π/Λisassumedtobedescribedbyaneffectivetransportequation,inprinciplecontainingtermswithanarbitrarilyhighnumberofderivatives.TheparametersinthisequationarefunctionsofΛinordertoaccountfortheeffectofadvectionatthescalessmallerthanLwhichhavebeenexcised.InthelimitΛ→0theeffectiveequation,bydimensionalanalysis,takesasimpleformdominatedbytermswithfewderivativesandwithassociatedeffectiveor“renormalized”parameters.Inthiswaytheeffectiveequationtakesaformsimilartothatusedintheprevioussection.Thereisadifference,though,becauseanypracticalapplicationoftheseschemesrequiresadrastictruncationoftheoperatorspace:especiallyintheRGmethodwhereitisimpossibletocomputetheflowwithchangingΛforverymanyparametersintheeffectivetransportequation.Unlikethesituationincriticalphenomenatherearenoinfra-reddivergencesinthetheoryandthenotionofarelevantoperatorisnotapplicable.Itisthenamatteroftrialanderrortodeterminewhethertheapproachusedcapturesthevitalfeaturescontrollingtheflow.Thesimplestrenormalizationschemeistocalculatetherenormalizationtothediffusivityκ(Λ)andtothevertexassociatedwiththecouplingoftherandomfieldorexternallyapplieddrift.lnthecaseofgradientflowswedemonstratedinreference[7]thatthisschemeyieldsexactresultsinoneandtwodimensionsandanextremelyaccurate,althoughnotexact,resultinthreedimensions.Itis,ingeneral,muchhardertocalculatedtherenormalizedparameterssuchasκethantheassociatedexponents,andsosuccessin[7]suggeststhatsomeinsightmaybegainedusingRGmethodsinothersimilarproblems.

InthissectionwepresentaRGcalculationofκe.ThevertexrenormalizationisdonebutmultiplevertexrenormalizationisneglectedwhichmeansthattherenormalizedvelocityfieldremainsGaussian.Consequently,afterintegratingouttherandomfielddowntowavenumberΛwepostulatethattheequationfortheeffectiveGreenfunctioncanbeapproximated,forallΛ,byanequationofthesameformastheoriginalone(eqn.3.1):

κ(Λ)∇2G(x,Λ)−u(x,Λ)·∇G(x,Λ)=−δ(x),(6.1)whereκ(Λ)istherunningrenormalizeddiffusionconstantanduΛistherenormalized

velocityfield.Sincewerenormalizethevertexfunctionallythefieldcorrelationfunction

14

willflowundertheRGas

󰀒u˜i(k,Λ)u˜j(k,Λ)󰀓=

where

′

󰀁

(2π)3δ(k+k′)Fij(k,Λ)|k|<Λ

0|k|>Λ

(6.2)

Onefindsthattherenormalizedfieldisstillincompressible.Weshallcomputetheflow

equationsforκ(Λ),Φ(k,Λ)andΨ(k,Λ)asΛvaries.

Thechangeinκ(Λ)onintegratingoutwavevectorsintheshell(Λ,Λ−δΛ)is

δκ(Λ)=−

1

Fij(k,Λ)=Φ(k,Λ)(k2δij−kikj)+Ψ(k,Λ)iǫimjkm.

(6.3)

(2π)3

󰀍

Λ

′uFij(q′,Λ)kiqjqk˜k(q,Λ)

Λ−δΛ

(2π)3κ2(Λ)

qju˜k(q,Λ)

󰀍

ΛΛ−δΛ

′Fij(q′Λ)qk

6π2κ2(Λ)

ǫijkqju˜i(q,Λ)Ψ(Λ,Λ)δΛ

(6.7)

Inrealspacetherefore,therenormalizationisoftheform

u→u+δΛβ(Λ)∇×u.

(6.8)

Usingtherenormalizationofu˜onemaycomputetheflowofFijandthusΦandΨto

obtaintheoneloopfunctionalRGequations:

∂κ∂Φ(q,Λ)∂Ψ(q,Λ)

3π2κ(Λ)

Λ2Φ(Λ,Λ)Ψ(q,Λ)Ψ(Λ,Λ)q2Φ(q,Λ)Ψ(Λ,Λ)

(6.9)

3π2κ2(Λ)3π2κ2(Λ)

Theintegrationoftheeqns.(6.9)isfromΛ=∞to0withtheinitialconditions

κ(∞)=κ0

15

Φ(q,∞)=Φ(q)Ψ(q,∞)=Ψ(q).

(6.10)

Whenthereisnohelicitythereisnovertexrenormalizationattheorderweareconsideringandthereforewemayintegratetheequationsdirectlytoobtain

2

κe=(κ20+2u0/9)

1

2/3=0.47140whichisquantitativelynotveryclosetothe

numericallymeasuredresult,κe=0.3697.However,thediscrepancyissensitivetotheformassumedfortheeffectivediffusionequation.Inourcasethisisgivenbyeqn.(6.1)whichisclearlyinadequatesinceuisnotrenormalizedwhenλ=0.Animprovementcanonlybemadebyincludingtermswithhigherderivativesofu.Thisissimilartoparameterizingthenon-helicalformfactorVRofeqn.(4.2.3)withafunctionofexternalmomentaratherthanapproximatingitbyaconstant,αR,whichisnotrenormalized.Thisisapossibleavenueofresearchbutwehavenotyetfollowedit.

Incontrast,forλ=0,uisrenormalizedandtheeffectonκeissignificantbecausethehelicalformfactorWR,eqn.(4.2.3)isrenormalizedatlowwavenumberasparametrizedbyβ(Λ)above.TheRGequationsmaybeintegratednumericallyandiscomparedwithsimulationinfigures7–11.Althoughtheresultsarenotquantitativelyaccurate,theycapturethequalitativebehaviourseeninthesimulations.Inparticular,theRGpredictsthelargeenhancementasafunctionofλseeninthedataandalsopredictsthedipobservedinthegraphofκeversusκ0forsufficientlylargeλ.

Indeed,thequalitativesuccessofthemethodsuggeststhatthedifficultyinobtainingpredictionsthataremoreaccuratemightliewiththeinadequacyofthesimpleansatzwhenappliedtothecasewhenλ=0.Theeffectofhelicityisneverthelesswellcapturedinthisapproachbecausethiseffectisdominatedbytherenormalizationofβ(Λ).

Atechnicalpointinthenumericalintegrationisthatκ0=0⇒κ(∞)=0,andtheevolutionequationsareill-definedinthelimitΛ→∞.Thisproblemiseasilyrectifiedbymakingκ0verysmallbutnon-zero.Theintegrationprocedureisthenwell-definedandtheresultsareinsensitivetotheexactvalueofκ0inthiscase.

Wethereforebelievethatalthoughtherenormalizationprocedureisnotquanti-tativelyaccurate(asshouldbeexpectedasitdoesnotgiveveryaccurateresultsintheabsenceofhelicity),itsuccessfullyincorporatestheunderlyingmechanismfortheenhancementofthediffusivitybyhelicityatsmallbaremoleculardiffusivity.

7DiscussionandConclusions

Inthispaperwehavestudiedtheproblemofturbulentadvectionofascalarfieldbyanincompressibleflowwithhelicityλ,0≤λ≤1.0,andbackgroundmoleculardiffusivity,κ0.Wehaveperformedcomputersimulationsoftheadvectionforflowswithpropertiesdescribedineqns.(2.1)to(2.4),andcomparedthelong-rangeeffectiveparametersdescribingthetimeevolutionofthescalarfieldwithvariousschemesofcalculation.Inparticular,wehaveconcentratedonhowtheeffectivediffusivity,κe,dependsonκ0andλ.Inearlierworkwefoundanstronganomalousenhancementofκeasafunctionofλforκ0=0.0[4]whichwasunexplainedtheoretically,andthisisthemotivationfor

16

thepresentstudy.Inthatearlierworktheturbulentvelocityfieldwastimedependentwhereashereitisnot.Thisallowsforeasiercalculationwhilststillreproducingtheeffect.

Theimportantregionfordiscussioncanbeseenfromthesimulationdata,figures7–11,tobeκ0<0.2;forlargerκ0themoleculardiffusivitybeginstodominateandnotonlyistheeffectofhelicitysuppressedbutalsothemanyschemesofcalculationgivegoodapproximationsforκe.Forλ=0.0wefindthatthetwo-loopself-consistentcalculationofκereproducesthesimulationdataforallκ0verycloselyindeed,asisseeninfigure7anddescribedinsection4.Theotherschemesalsoplottedaremuchlessaccurateintheregionofinterest.Ordinaryperturbationtheoryisnotconvergentinthisregionandwillbediscussednofurther.ThereasonwhytheHartree-Fockand(RG)methodsarelessaccurateisthatthevertexfunctionV(k′,k),eqn.(4.2.3),isnotrenormalizedforlowwavenumberwhichmeansthattheassociatedcouplingαisnotrenormalized.TheHartree-Fockmethodatλ=0sumsrainbowdiagramsbutdoesnotincludeanydiagramscorrespondingtoavertexcorrection,unliketheself-consistenttheory.Intheself-consistenttheorytheone-looppredictionforκ0=0isκe=1/3andthetwo-looptermsmodifythisbyδκe∼0.04,ofwhichthetwo-loopcrossdiagraminfigure3contributesonly10%,orδκe∼0.004.InomittingsimilartermstothislatteronetheHartree-Fockapproximationshouldthereforenotbeexpectedtobetoodiscrepantandthisisseentobethecase.TheRGcalculationgivesaformwhichmustyieldthesimpleone-loopperturbationtheoryexpressionatlargeκ0butallowscontinuationtoκ0=0;thisisgivenineqn.(6.11).InthecaseofgradientflowstheRGapproachisremarkablysuccessful[6]andthisisattributedtothefactthatinthatcasetheprimitivevertexisrenormalizedatlowwavenumber.

Thereasonforexaminingschemesalternativetoself-consistentmethodsisthatforλ>0agreementbetweensimulationdataandtheoryispooranditisnecessarytoinvestigatedifferentapproachesinordertotestdifferenthypothesesforasimpledescriptionoftheobservedanomalouseffect.

Forκ0>0.2allschemesexceptordinaryperturbationtheorybegintoshowrea-sonableagreementwiththedata,andforκ0>0.5allschemesclearlyreproducetheresults.Weconcentrateonresultsforκ0<0.2andtheanomalousenhancementofκebyhelicityinthisregionisseeninfigure10whereκeisplottedagainstλforκ0=0,andischaracterizedbyarapidriseinκeforλ>0.2.Analternativeaspectisseeninfigure9whereκeisplottedagainstκ0forλ=1.0.Thesignificantdipatκ0=0.2isduetothelargeeffectofhelicityonκesmallerκ0comparedwiththemuchreducedeffectatκ0∼0.2.Ithasprovedverydifficulttoconvincinglyexplainthesefeatures.However,wehavebeenabletosuggestmechanismswhichshowthepresenceofhelicityintheflowcanproducealargechangeinκefromthenon-helicalvalue,andeventhoughthesehavenotyetyieldedquantitativepredictionstheydopointtowardsareasonableexplanation.

Thebasicideaistorecognizethattheeffectiveequationgoverningtheadvectionshouldcontaintermsnotpresentintheoriginalequation.Thetermscanbethoughtofasbeinginducedinthelow-wavenumbereffectivetheorybyintegratingouthigherwavenumbers.Thismayalsobeviewedastherenormalizationoftherelatedvertexfunctionsofthetheorycorrespondingtoaselectiveresummationofdiagrams.Inourapproachwehaveassumedthatalow-wavenumberapproximationwillbevalidandso

17

suchtermswillcontaintheminimumnumberofderivatives.Theseideascanbeimple-mentedindifferentwaysandwetriedself-consistent,Hartree-Fockandrenormalizationgroupapproaches.Theself-consistentandHartree-Fockmethodsarebasedontheef-fectiveevolutionequation(4.2.1)whichcorrespondstoalow-wavenumberenhancementoftheflowvelocityuR=u+βRω,whereωisthevorticity.Weperformedatwo-loopcalculationself-consistentinbothκeandβRasdescribedineqn.(4.2.19)andfigure5.Theresultsshowthatastrongenhancementinκeispredictedbutthatthemagnitudeforλ=1.0,κ0=0.0istoosmallandtheformofthedependenceofκeonλdisagreeswiththedata.Thisisparticularlytrueatsmallλforκ0=0.0wherefromfigure10weseethatκeisonlyweaklydependentonλ,λ<0.3whereaswepredictκe∼a+bλpforfractionalp:theone-loopresultisp=2/3.

TheHartree-Fockmethod,showndiagramaticallyinfigure6,computesthecom-pletepropagatorintermsofΩ(k2)(eqn.(4.2.4))asasumoftherainbowdiagramsgeneratedfromtheeffectiveequation(4.2.1)withβRgivenbytheone-loopresultβR=−B1(βR)/κ2ewhereB1(βR)isgivenineqn.(4.2.15)andκe=κ0+Ω(0).Ariseinκewithλispredictedbutnotonebigenoughtoagreewiththedata.However,thebehaviouratsmallλisamuchslowerrisewhichismoreinkeepingwiththedatathantheself-consistentpredictioninthisregion.

Therenormalizationgroupmethodisadifferentapproachinthatitconsidersarunningdiffusivityκ(Λ)andvelocityfieldu(x,Λ)whichsatisfyκ(∞)=κ0,κ(0)=κe,u(x,Λ)=u(x).TherearethreecoupledRGflowequations,(6.9),forκ(Λ)andthetworunningspectralfunctionsΦ(q,Λ),Ψ(q,Λ)whichcorrespondtothedefinitionsineqn.(6.3).TheimportantfeatureoftheRGflowequationsisthatκ(Λ)appearsinthedenominators.ThenumeratorsaresuppressedatlargeΛbythespectralfunctionsandsothemajorcontributionisfromintermediatevaluesofΛ:Λ∼k0.Thiscontributionisstronglyenhancedforsmallκ0andresultsinthepredictionofthedipstructureobservedinthedatabutnotpredictedbytheothermethods.

¿Fromourinvestigationwemustexpectthataproperexplanationoftheobservedeffectswillrequirethecorrecteffectiveequationandtheconsequentgenerationofnewverticesbutthatunliketheλ=0casethelow-wavenumberapproximationwillbeinsufficientsincealthoughanenhancementispredictedforλ=0therapidriseisnotreproducedandnodipisobserved.TheRGmethodsuggeststhatthemaincontributionisfromwavenumbersk∼k0supportingthislatterconclusion.AsuccessfulapproachshouldthereforeincludemoretermsintheeffectiveflowequationincombinationwithanRGapproach.Thechallengeistoobtainaccurateresultsforallλincludingλ=0bysuchatechnique.Workinthisdirectioniscurrentlyunderway.

References

[1]R.H.Kraichnan.Phys.Fluids,13:22,1970.[2]R.H.Kraichnan.J.FluidMech.,77:753,1976.[3]R.H.Kraichnan.J.FluidMech.,81:385,1977.

[4]I.T.Drummond,SDuane,andR.R.Horgan.Nucl.Phys.,B220:119,1983.

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[5]I.T.DrummondandR.R.Horgan.Phys.Lett.,B321:246–253,1994.

[6]D.S.Dean,I.T.Drummond,andR.R.Horgan.J.Phys:A:MathGen,27:5135–5144,

1994.[7]D.S.Dean,I.T.Drummond,andR.R.Horgan.J.Phys:A:MathGen,28:1235–1242,

1995.[8]R.PhythianandW.D.Curtis.J.FluidMech.,:241,1978.

[9]I.T.Drummond,SDuane,andR.R.Horgan.J.FluidMech.,138:75–91,1984.

19

qqkk′kk′V

(a)W(b)q=k′kq+α0k′kqkU0

(c)β0k′qkR≈

k′kqq+αRk′kβRk′UR(d)Figure1:Theverticesoccurringintheperturbationschemes:(a)theprimitivevelocityfield

vertex;(b)theprimitivevorticityvertex;(c)thebarecompletevertexoftheeffectivediffusionequation;(d)therenormalizedcompletevertexapproximatedasasumofrenormalizedverticesassociatedwiththevelocityfieldandthevorticity.

20

q

kΣ(k)=qq′kk-q+qq′kk-qk-q-q′k-q+kk-qk-q-q′k-q′˜(k).Figure2:Thegraphsthatcontributetotwo-loopsimpleperturbationtheoryforG

××Figure3:Theperturbationexpansiontotwo-loopsoftheself-consistentrelationforκe

21

qkβRαRq′

equationsshowninfigure5oncetheapproximationforURgivenineqn.(4.2.5)andshowninfigure1hasbeenused.ThisgraphislabelledTβαα.

αRk′Figure4:Anexampleofthekindofvertexgraphthatmustbeevaluatedinthesolutionof

22

Ω(k) k×R2RR×RRRRRRRRRk′RRRRRRR×RRRRRRRRRRRRlatingtheself-energy,Σ(k),andvertex,UR(k,k′),functions.Thevertexfunctionisrepresented

˜(k),bythefilledboxand∆U=UR−U0.bythecirclewithinset’R’,fullGreenfunction,G

Figure5:Theperturbationexpansiontotwo-loopsofthegeneralself-consistentconditionre-

23

-Ω(k) k2RR+RRR˜(k),isdenotedbythefilledbox.TheequationforU0andUR.ThefullGreenfunction,G

Ω(k)iscorrecttoallloopordersbuttheequationforURiscorrecttoone-looporderonly.

Figure6:TheintegralHartree-FockequationsforΩ(k)andURintermsofthegeneralvertices

24

λ=0.0

1.000.900.80effective diffusivity κe0.700.600.500.400.300.20

0.0

0.2

0.4

bare diffusivity κ0

0.60.8

Figure7:κeversusκ0forfixedhelicityλ=0.0.Thesimulationdataareshown(d)to

becomparedwiththepredictionsoftwo-loopself-consistentperturbationtheory(solid),theHartree-Fockcalculation(long-dashed),therenormalizationgroup(dashed)andordinaryper-turbationtheory(dot-dashed)

25

λ=0.4

0.90

0.80

effective diffusivity κe0.70

0.60

0.50

0.40

0.30

0.0

0.2

0.4

bare diffusivity κ0

0.60.8

Figure8:κeversusκ0forfixedhelicityλ=0.4.Thesimulationdataareshown(d)to

becomparedwiththepredictionsoftwo-loopself-consistentperturbationtheoryinκe(solid)andinκe,β(dotted),theHartree-Fockcalculation(long-dashed),therenormalizationgroup(dashed),andordinaryperturbationtheory(dot-dashed)

26

λ=1.0

0.90

0.80

effective diffusivity κe0.70

0.60

0.50

0.40

0.30

0.0

0.2

0.4

bare diffusivity κ0

0.60.8

Figure9:κeversusκ0forfixedhelicityλ=1.0.Thesimulationdataareshown(d)to

becomparedwiththepredictionsoftwo-loopself-consistentperturbationtheoryinκe(solid)andinκe,β(dotted),theHartree-Fockcalculation(long-dashed),therenormalizationgroup(dashed),andordinaryperturbationtheory(dot-dashed)

27

0.90

κ0=0.0

0.80

effective diffusivity κe0.70

0.60

0.50

0.40

0.30

0.0

0.20.4

helicity λ

0.60.8

Figure10:κeversusλforfixeddiffusivityκ0=0.0.Thesimulationdataareshown(d)to

becomparedwiththepredictionsoftwo-loopself-consistentperturbationtheoryinκe(solid)andinκe,β(dotted),theHartree-Fockcalculation(long-dashed),therenormalizationgroup(dashed)

28

κ0=0.2

0.65

effective diffusivity κe0.60

0.55

0.50

0.45

0.40

0.0

0.5helicity λ

1.0

Figure11:κeversusλforfixeddiffusivityκ0=0.2.Thesimulationdataareshown(d)to

becomparedwiththepredictionsoftwo-loopself-consistentperturbationtheoryinκe(solid)andinκe,β(dotted),theHartree-Fockcalculation(long-dashed),therenormalizationgroup(dashed)

29