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PhysicsReports314(1999)237}574 Simpli"ed models for turbulent di!usion: Theory, numerical modelling, and physical phenomena Andrew J. Majda*, Peter R. Kramer NewYorkUniversity,CourantInstitute,251MercerStreet, NewYork,NY10012,USA ReceivedAugust1998;editor:I.Procaccia Contents 1. Introduction 240 4.2. Evolutionofthepassivescalarcorrelation 2. Enhanceddi!usionwithperiodicorshort- functionthroughaninertialrangeofscales 427 rangecorrelatedvelocity"elds 243 4.3. Scalingregimesinspectrumof#uctuations 2.1. Homogenizationtheoryforspatio- ofdriven passivescalar"eld 439 temporalperiodic#ows 245 4.4. Higher-ordersmall-scalestatisticsof 2.2. E!ectivedi!usivityinvariousperiodic#ow passivescalar"eld 450 geometries 262 5. Elementarymodelsforscalarintermittency 460 2.3. Tracertransportinperiodic#owsat"nite 5.1. Empiricalobservations 462 times 285 5.2. Anexactlysolvablemodeldisplaying 2.4. Random#ow "eldswithshort-range scalarintermittency 463 correlations 293 5.3. Anexamplewithqualitative"nite-time 3. Anomalousdi!usionandrenormalizationfor correctionstothe homogenizedlimit 483 simpleshearmodels 304 5.4. Othertheoreticalworkconcerningscalar 3.1. Connectionbetweenanomalousdi!usion intermittency 488 andLagrangiancorrelations 308 6. MonteCarlomethodsforturbulentdi!usion 493 3.2. Tracertransportinsteady,randomshear 6.1. GeneralaccuracyconsiderationsinMonte #owwithtransversesweep 316 Carlosimulations 495 3.3. Tracertransportinshear#owwith 6.2. NonhierarchicalMonteCarlomethods 496 randomspatio-temporal#uctuationsand 6.3. HierarchicalMonteCarlomethodsfor transversesweep 342 fractalrandom"elds 521 3.4. Large-scalee!ectiveequationsformean 6.4. Multidimensionalsimulations 545 statisticsanddeparturesfromstandard 6.5. Simulationofpairdispersionintheinertial eddydi!usivitytheory 366 range 551 3.5. Pair-distancefunctionandfractal 7. Approximateclosuretheoriesandexactly dimensionofscalarinterfaces 389 solvablemodels 559 4. Passivescalarstatisticsforturbulentdi!usion Acknowledgements 561 in rapidlydecorrelatingvelocity"eldmodels 413 References 561 4.1. De"nitionoftherapiddecorrelationintime (RDT)modelandgoverningequations 417 *Correspondingauthor.Tel.:(212)998-3324;fax:(212)995-4121;e-mail:[email protected]. 0370-1573/99/$}seefrontmatter ( 1999ElsevierScienceB.V.Allrights reserved. PII: S0370-1573(98)00083-0 A.J. Majda, P.R. Kramer/PhysicsReports314(1999)237}574 239 Abstract Severalsimplemathematicalmodels forthe turbulentdi!usionof a passivescalar "eldaredevelopedherewith an emphasis on the symbiotic interaction between rigorous mathematical theory (including exact solutions), physical intuition,andnumericalsimulations.Thehomogenizationtheoryforperiodicvelocity"eldsandrandomvelocity"elds withshort-rangecorrelationsispresentedandutilizedtoexaminesubtlewaysinwhichthe#owgeometrycanin#uence the large-scale e!ective scalar di!usivity. Various forms of anomalous di!usion are then illustrated in some exactly solvablerandomvelocity"eldmodelswithlong-rangecorrelationssimilartothosepresentinfullydevelopedturbulence. Herebothrandomshearlayermodelswithspecialgeometrybutgeneralcorrelationstructureaswellasisotropicrapidly decorrelating models are emphasized. Some of the issues studied in detail in these models are superdi!usive and subdi!usivetransport,pairdispersion,fractaldimensionsofscalarinterfaces,spectralscalingregimes,small-scaleand large-scale scalar intermittency, and qualitative behavior over "nite time intervals. Finally, it is demonstrated how exactly solvable models can be applied to test and design numerical simulation strategies and theoretical closure approximationsforturbulentdi!usion. ( 1999ElsevierScienceB.V.Allrightsreserved. PACS: 47.27.Qb;05.40.#j;47.27.!i;05.60.#w; 47.27.Eq;02.70.Lq 240 A.J. Majda, P.R. Kramer/PhysicsReports314(1999)237}574 1. Introduction In this review, we consider the problemof describingand understanding the transport of some physical entity, such as heat or particulatematter, which is immersed in a #uid #ow. Most of our attention will be on situations in which the #uid is undergoing some disordered or turbulent motion.Ifthetransportedquantitydoesnotsigni"cantlyin#uencethe#uidmotion,itissaidtobe passive, and its concentration density is termed a passive scalar "eld. Weak heat #uctuations in a #uid, dyes utilized in visualizing turbulent #ow patterns, and chemical pollutants dispersing in the environmentmayallbe reasonablymodelledas passivescalar systemsinwhichthe immersed quantityistransportedintwoways:ordinarymoleculardi!usionandpassiveadvectionbyits#uid environment.The general problem of describing turbulent di!usion of a passive quantity may be stated mathematically as follows: Let (x,t) be the velocity "eld of the #uid prescribed as a function of spatial coordinates x and * time t, which we will always take to be incompressible (+’ (x,t)"0). Also let f(x,t) be a * prescribed pumping (source and sink) "eld, and „ (x) be the passive scalar "eld prescribed at 0 someinitialtimet"0.Eachmayhaveamixtureofdeterministicandrandomcomponents,the latter modelling noisy #uctuations. In addition, molecular di!usion may be relevant, and is represented by a di!usivity coe$cient i. The passive scalar "eld then evolves according to the advection}di+usion equation „(x,t)/ t# (x,t)’+„(x,t)"iD„(x,t)#f(x,t) , R R * „(x,t"0)"„ (x) . (1) 0 The central aim is to describe some desired statistics of the passive scalar "eld „(x,t) at times t’0.Forexample,atypicalgoalistoobtaine!ectiveequationsofmotionforthemeanpassive scalar density, denoted S„(x,t)T. While the PDE in Eq. (1) is linear, the relation between the passive scalar "eld „(x,t) and the velocity "eld (x,t) is nonlinear. The in#uence of the statistics of the random velocity "eld on the * passivescalar"eldissubtleandverydi$culttoanalyzeingeneral.Forexample,aclosedequation for S„(x,t)T typically cannot be obtained by simply averaging the equation in Eq. (1), because S (x,t)’+„(x,t)T cannot be simply related to an explicit functional of S„(x,t)T in general. This is * a manifestation of the turbulence moment closure problem [227]. ‘ a In applications such as the predicting of temperature pro"les in high Reynolds number turbu- lence [196,227,247,248], the tracking of pollutants in the atmosphere [78], and the estimating of thetransportofgroundwaterthroughaheterogeneousporousmedium[79],theproblemisfurther complicated by the presence of a wide range of excited space and time scales in the velocity "eld, extending all the way up to the scale of observational interest. It is precisely for these kinds of problems, however, that a simpli"ed e!ective description of the evolution of statistical quantities suchasthemeanpassivescalardensityS„(x,t)Tisextremelydesirable,becausetherangeofactive scales of velocity "elds which can be resolved is strongly limited even on supercomputers [154]. For some purposes, one may be interested in following the progress of a specially marked particle as it is carried by a #ow. Often this particle is light and small enough so that its presence A.J. Majda, P.R. Kramer/PhysicsReports314(1999)237}574 241 only negligibly disrupts the existing #ow pattern, and we will generally refer to such a particle as a (passive) tracer, re#ecting the terminology of experimental science in which #uid motion is visualizedthroughthe motionofinjected,passivelyadvectedparticles(oftenopticallyactivedyes) [227].Theproblemof describingthestatisticaltransportoftracersmaybeformulatedas follows: Let (x,t) be a prescribed, incompressible velocity "eld of the #uid, with possibly both a mean * component and a random component with prescribed statistics modelling turbulent or other disordered #uctuations. We seek to describe some desired statistics of the trajectory X(t) of atracerparticlereleasedinitiallyfromsomepointx andsubsequentlytransportedjointlybythe 0 #ow (x,t) and molecular di!usivity i. The equation of motion for the trajectory is a (vector- * valued) stochastic di!erential equation [112,257] dX(t)" (X(t),t)dt#J2idW(t) , (2a) * X(t"0)"x . (2b) 0 The second term in Eq. (2a) is a random increment due to Brownian motion [112,257]. Basic statistical functions of interest are the mean trajectory, SX(t)T, and the mean-square displace- ment of a tracer from its initial location, SDX(t)!x D2T. 0 Itisoftenofinteresttotrackmultipleparticlessimultaneously;thesewilleachindividuallyobey the trajectory equations in Eqs. (2a) and (2b) with the same realization of the velocity "eld but * independentBrownianmotions.Theadvection}di!usionPDEin Eq. (1)and thetracer trajectory equations in Eqs. (2a) and (2b) are related to each other by the theory of Ito di!usion processes [107,257], which is just a generalization of the method of characteristics[150] to handle second- orderderivativesviaarandomnoiseterminthe characteristicequations.Wewillworkwithboth of these equations in this review. In principle, the turbulent velocity "eld (x,t) which advects the passive scalar "eld should be * a solution to the Navier}Stokes equations (x,t)/ t# (x,t)’+ (x,t)"!+p(x,t)#lD (x,t)#F(x,t) , R* R * * * +’ (x,t)"0 , (3) * wherepisthepressure"eld,lisviscosity,andF(x,t)is someexternalstirringwhichmaintainsthe #uid in a turbulent state. But the analytical representation of such solutions corresponding to complex, especially turbulent #ows, are typically unwieldy or unknown. We shall therefore instead utilize simpli"ed velocity "eld models which exhibit some empirical features of turbulent or other #ows, though these models may not be actual solutions to the Navier}Stokesequations.Incompressibility+’ (x,t)"0ishowever,enforcedinallofourvelocity * "eld models. Our primary aim in working with simpli"ed models is to obtain mathematically explicitandunambiguousresultswhichcanbeusedasasoundbasisforthescienti"cinvestigation of more complex turbulent di!usion problems arising in applications for which no analytical solution is available. We therefore emphasize the aspects of the model results which illustrate general physical mechanisms and themes which can be expected to be manifest in wide classes of turbulent#ows.Wewillalsoshowhowsimpli"edmodelscanbeusedtostrengthenandre"nethe 242 A.J. Majda, P.R. Kramer/PhysicsReports314(1999)237}574 arsenal of numerical methods designed for quantitative physical exploration in natural and practical applications. First of all, simpli"ed models o!er themselves as a pool of test problems to assess the variety of numerical simulations schemes proposed for turbulent di!usion [109,180,190,219,291,335]. Moreover, we shall explicitly describe in Section 6 how mathematical (harmonic)analysisofsimpli"edmodelscanbeusedasabasistodesignnewnumericalsimulation algorithmswith superiorperformance[82,84}86].Accurateand reliablenumericalsimulationsin turn enrich various mathematical asymptotic theories by furnishing explicit data concerning the quality of the asymptoticapproximation and the signi"cance of corrections at "nite values of the smallorlargeparameter,andcanrevealnewphysicalphenomenainstronglynonlinearsituations unamenable to a purely theoretical treatment. Physical intuition, for its part, suggests fruitful mathematical model problems for investigation, guides their analyses, and informs the develop- mentofnumericalstrategies.Wewillrepeatedlyappealtothissymbioticinteractionbetweensimpli- "ed mathematical models, asymptotic theory, physical understanding, and numerical simulation. Thoughwedonotdwellonthisaspectinthisreview,wewishtomentionthemoredistantgoal of using simpli"ed velocity "eld models in turbulent di!usion to gain some understanding in the theoreticalanalysisandpracticaltreatmentoftheNavier}StokesequationsinEq. (3)insituations where strong driving gives rise to complicated turbulent motion [196,227]. The advection}di!u- sionequationinEq. (1)hassomeessentialfeaturesincommonwiththeNavier}Stokesequations: they are both transport equations in which the advection term gives rise to a nonlinearity of the statistics of the solution. At the same time, the advection}di!usion equation is more managable sinceitisascalar,linearPDEwithoutanauxiliaryconstraintanalogoustoincompressibility.The advection}di!usionequation,in conjunctionwithavelocity"eldmodelwithturbulentcharacter- istics, therefore serves as a simpli"ed prototype problem for developing theories for turbulence itself. Our study of passive scalar advection}di!usion begins in Section 2 with velocity "elds which have either a periodic cell structure or random #uctuations with only mild short-range spatial correlations. We explain the general homogenization theory [12,32,148] which describes the behavior of the passive scalar "eld at large scales and long times in these #ows via an enhanced homogenized di!usivity matrix. Through mathematical theory, exact results from simpli"ed ‘ a models,andnumericalsimulations,weexaminehowthehomogenizeddi!usioncoe$cientdepends onthe#owstructure,andinvestigatehowwelltheobservationofthepassivescalarsystematlarge but "nite space}time scales agrees with the homogenized description.In Section 3, we use simple random shear #ow models [10,14] with a #exible statistical spatio-temporal structure to demon- strateexplicitlyanumberofanomaliesofturbulentdi!usionwhenthevelocity"eldhassu$ciently stronglong-rangecorrelations.Thesesimpleshear#owmodelsare alsousedtoexploreturbulent di!usion in situations where the velocity "eld has a wide inertial range of spatio-temporal scales excitedinastatisticallyself-similarmanner,asinahighReynoldsnumberturbulent#ow.Wealso describesomeuniversalsmall-scalefeaturesof the passivescalar"eldwhichmay bederivedin an exactandrigorousfashioninsuch#ows.Otheraspectsofsmall-scalepassivescalar#uctuationsare similarly addressed in Section 4 using a complementary velocity "eld model [152,179] with a statistically isotropic geometry but very rapid decorrelations in time. In Section 5, we present aspecialfamilyofexactly solvableshear#owmodels[207,233]whichexplicitlydemonstratesthe phenomenon of large-scale intermittency in the statistics of the passive scalar "eld, by which we mean the occurrence of a broader-than-Gaussian distribution for the value of the passive scalar A.J. Majda, P.R. Kramer/PhysicsReports314(1999)237}574 243 "eld „(x,t) recorded at a single location in a turbulent #ow [155,127,146,147,191]. Next, in Section 6,wefocusonthechallengeofdevelopinge$cientandaccuratenumerical MonteCarlo ‘ a methods for simulating the motion of tracers in turbulent #ows. Using the simple shear models from Section 3 and other mathematical analysis [83,87,140], we illustrate explicitly some subtle and signi"cant pitfalls of some conventional numerical approaches. We then discuss the theoretical basis and demonstrate the exceptional practical performance of a recent wavelet- based Monte Carlo algorithm [82,84}86] which is designed to handle an extremely wide inertial range of self-similar scales in the velocity "eld. We conclude in Section 7 with a brief discussion of the application of exactly solvable models to assess approximate closure theories [177,182,196,200,227,285,286,344] which have been formulated to describe the evolution of the mean passive scalar density in a high Reynolds number turbulent #ow [13,17]. Detailed introductions to all these topics are presented at the beginning of the respective sections. 2. Enhanced di4usion with periodic or short-range correlated velocity 5elds In the introduction, we mentioned the moment closure problem for obtaining statistics of the passive scalar "eld immersed in a turbulent #uid. To make this issue concrete, consider the challenge of deriving an equation for the mean passive scalar density S„(x,t)T advected by a velocity "eld which is a superposition of a mean #ow pattern V(x,t) and random, turbulent #uctuations (x,t)withmeanzero.Anglebracketswilldenoteanensembleaverageoftheincluded * quantity over the statistics of the random velocity "eld. Since the advection}di!usionequation is linear, one might naturally seek an equation for the mean passive scalar density by simply averaging it: S„(x,t)T/ t#V(x,t)’+S„(x,t)T#S (x,t)’+„(x,t)T"iDS„(x,t)T#Sf(x,t)T , R R * S„(x,t"0)T"S„ (x)T . (4) 0 Eq. (4) is not a closed equation for S„(x,t)T because the average of the advective term, S ’+„T, * cannot generally be simply related to a functional of S„(x,t)T. Anearlyideaforcircumventingthisobstaclewastorepresentthee!ectoftherandomadvection by a di!usion term: S*(x,t)’+„(x,t)T"!+’(KM T’+„(x,t)) , (5) where KM T is some constant ‘eddy di!usivitya matrix (usually a scalar multiple of the identity matrix I) which is to be estimated in some manner, such as mixing-length theory ([320], Section 2.4). From assumption (5) follows a simple e!ective advection}di!usion equation for the mean passive scalar density RS„(x,t)T/Rt#V(x,t)’+S„(x,t)T"+’((iI#KM T)’+S„(x,t)T)#Sf(x,t)T , S„(x,t"0)T"S„ (x)T , 0 244 A.J. Majda, P.R. Kramer/PhysicsReports314(1999)237}574 wherethedi!usivitymatrix(iI#KM T)is(presumably)enhancedoveritsbaremolecularvalueby the turbulent eddy di!usivity KM T coming from the #uctuations of the velocity. The closure hypothesis(5)isthe Reynoldsanalogy ofasuggestion"rstmadebyPrandtlinthecontextofthe ‘ a Navier}Stokes equations (see [227], Section 13.1). It may be viewed as an extension of kinetic theory, where microscopic particle motion produces ordinary di!usive e!ects on the macroscale. There are, however, some serious de"ciencies of the Prandtl eddy di!usivity hypothesis, both in termsoftheoreticaljusti"cationandofpracticalapplicationtogeneralturbulent#ows(see[227], Section 13.1; [320], Ch. 2). First of all, kinetic theory requires a strong separation between the microscaleandmacroscale,buttheturbulent#uctuationstypicallyextenduptothescaleatwhich themeanpassivescalardensityisvarying.Moreover,therecipesforcomputingtheeddydi!usivity KM Tarerather vague,and aregenerallyonlyde"nedup to someunknownnumericalconstant‘of orderunity .Moresophisticatedschemesforcomputingeddyviscositiesbasedonrenormalization a groupideashavebeenproposedinmorerecentyears[243,300,344],buttheseinvolveotheradhoc assumptions of questionable validity. In Section 2, we will discuss some contexts in which rigorous sense can be made of the eddy di!usivity hypothesis (5), and an exact formula provided for the enhanced di!usivity. All involve the fundamental assumption that, in some sense, the #uctuations of the velocity "eld occur on amuchsmallerscalesthanthoseofthemeanpassivescalar"eld.Theserigoroustheoriestherefore are not applicable to strongly turbulent#ows, but they provide a solid, instructive, and relatively simpleframeworkforexamininganumberofsubtleaspectsofpassivescalaradvection}di!usionin unambiguous detail. Moreover, they can be useful in practice for certain types of laboratory or natural #ows at moderate or low Reynolds numbers [301,302]. Overview of Section 2: We begin in Section 2.1 with a study of advection}di!usion by velocity "elds that are deterministic and periodic in space and time. Generally, we will be considering passivescalar"eldswhicharevaryingonscalesmuchlargerthanthoseoftheperiodicvelocity"eld in which theyare immersed. Thoughthe velocity"eldis deterministic,one mayformally viewthe periodic #uctuations as an extremely simpli"ed model for small-scale turbulent #uctuations. Averaging over the #uctuations may be represented by spatial averaging over a period cell. After a convenient nondimensionalizationin Section 2.1.1, we formulate in Sections 2.1.2 and 2.1.3 the homogenization theory [32,149] which provides an asymptotically exact representation of the e!ects of the small-scale periodic velocity "eld on the large-scale passive scalar "eld in terms of ahomogenized,e!ectivedi!usivitymatrixKH whichisenhancedabovebaremoleculardi!usion. Various alternative ways of computing this e!ective di!usivity matrix are presented in Sec- tion 2.1.4.Weremarkthat,incontrasttousualeddydi!usivitymodels,theenhanceddi!usivityin the rigorous homogenization theory has a highly nontrivial dependence on molecular di!usivity. WewillexpressthisdependenceintermsofthePeHcletnumber,whichisameasureofthestrengthof advectionby the velocity"eld relative to di!usion by molecular processes(see Section 2.1.1). The physicallyimportant limitof high PeHclet number will be of centralinterest throughoutSection 2. InSection 2.2,weapplythehomogenizationtheorytoevaluatethetracertransportinavariety of periodic #ows. We demonstrate the symbiotic interplay between the rigorous asymptotic theories and numerical computations in these investigations, and how they can reveal some important and subtle physical transport mechanisms. We "rst examine periodic shear #ows with various types of cross sweeps (Sections 2.2.1 and 2.2.2), where exact analytical formulas can be derived.Nextweturnto#owswithacellularstructureandtheirperturbations(Section 2.2.3),and A.J. Majda, P.R. Kramer/PhysicsReports314(1999)237}574 245 thesubtlee!ectswhichtheadditionof ameansweepcanproduce(Section 2.2.4).Wediscusshow other types of periodic #ows can be pro"tably examined through the joint use of analytical and numerical means in Section 2.2.5. Animportantpracticalissueistheaccuracywithwhichthee!ectivedi!usivityfromhomogeniz- ation theory describes the evolution of the passive scalar "eld at "nite times. We examine this question in Section 2.3 by computing the mean-square displacement of a tracer over a "nite intervaloftime.Forshear#owswithcrosssweeps,anexactanalyticalexpressioncanbeobtained (Section 2.3.1).The"nitetimebehavioroftracersinmoregeneralperiodic#owsmaybeestimated numericallythroughMonte Carlosimulations(Section 2.3.2).In allexamplesconsidered,the rate of change of the mean-square tracer displacement is well described by (twice) the homogenized di!usivityafteratransienttimeintervalwhichisnotlongerthanthetimeitwouldtakemolecular di!usion to spread over a few spatial period cells [230,231]. InSection 2.4,webeginourdiscussionofadvection}di!usionbyhomogenousrandomvelocity "elds.Weidentifytwo di!erentlarge-scale,long-timeasymptoticlimitsin whicha closede!ective di!usionequationcanbe derived forthe meanpassive scalardensityS„(x,t)T. Firstis the Kubo ‘ theory [160,188,313],wherethetimescaleofthevelocity"eldvariesmuchmorerapidlythanthat a ofthepassivescalar"eld,butthelengthscalesofthetwo"eldsarecomparable(Section 2.4.1).The Kubodi!usivity appearingin thee!ectiveequationis simplyrelatedtothecorrelationfunction ‘ a of the velocity "eld. Next we concentrate on steady random velocity "elds which have only short-range spatial correlations, so that there can be a meaningfully strong separation of scales between the passive scalar "eld and the velocity "eld. A homogenization theorem applies in such cases [12,98,256], and rigorously describes the e!ect of the small-scale random velocity "eld on the large-scale mean passive scalar "eld through a homogenized, e!ective di!usivity matrix (Section 2.4.2). Homogenization for the steady periodic #ow "elds described in the earlier Sec- tions 2.1, 2.2 and 2.3 is a special case of this more general theory for random "elds. We present various formulas for the homogenized di!usivity in Section 2.4.3, and discuss its parametric behavior in some example random vortex #ows in Section 2.4.4. We emphasize again that high Reynolds number turbulent #ows have strong long-range correlations which do not fall under the purview of the homogenization theory discussed in Section 2. The rami"cations of these long-range correlations will be one of the main foci in the remaining sections of this review. 2.1. Homogenization theory for spatio-temporal periodic -ows Herewe presentthe rigoroushomogenizationtheorywhichprovides aformulaforthe e!ective di!usion of a passive scalar "eld at large scales and long times due to the combined e!ects of molecular di!usion and advection by a periodic velocity "eld. We "rst prepare for our discussion with some de"nitions and a useful nondimensionalizationin Section 2.1.1. Next, in Section 2.1.2, westatetheformulaprescribedbyhomogenizationtheoryforthee!ectivedi!usivityofthepassive scalar"eld on large scales and longtimes, and show formally howto derive it througha multiple scaleasymptoticanalysis[32,205].WeindicateinSection 2.1.3howtogeneralizethehomogeniz- ation theory to include large-scale mean #ows superposed upon the periodic #ow structure [38,230]. In Section 2.1.4, we describe some alternative formulas for the e!ective di!usivity, involvingStieltjesmeasures[9,12,20]andvariationalprinciples[12,97].Theserepresentationscan 246 A.J. Majda, P.R. Kramer/PhysicsReports314(1999)237}574 be exploited to bound and estimate the e!ective di!usivity in various examples and classes of periodic #ows [40,97,210], as we shall illustrate in Section 2.2. 2.1.1. Nondimensionalization Webeginourdiscussionofconvection-enhanceddi!usivitywithsmoothperiodicvelocity"elds (x,t)de"nedonRdwhichhavetemporalperiodt ,andacommonspatialperiod‚ alongeachof * v v the coordinate axes: (x,t#t )" (x,t) , * v * (x#‚ e ,t)" (x,t) , * vLj * whereMeNd denotesaunitvectorin thejthcoordinatedirection.Moregeneralperiodicvelocity Lj j/1 "eldscanbetreatedsimilarly;theresultingformulaswouldsimplyhavesomeadditionalnotational complexity. We also demand for the moment that the velocity "eld have mean zero , in that its ‘ a average over space and time vanishes: PtP v ‚~dt~1 (x,t)dxdt"0 . v v * 0 *0,L+d v InSection 2.1.3,wewillextendourdiscussiontoincludethepossibilityofalarge-scalemean#ow superposed upon the periodic velocity "eld just described. It will be useful to nondimensionalize space and time so that the dependence of the e!ective di!usivityonthevariousphysicalparametersoftheproblemcanbemostconciselydescribed.The spatial period ‚ provides a natural reference length unit. To illuminate the extent to which the v periodicvelocity "eld enhances the di!usivityof the passive scalar "eld above the bare molecular value i, we choose as a basic time unit the cell-di!usion time t "‚2/i, which describes the time i v scale over which a "nely concentrated spot of the passive scalar "eld will spread over a spatial period cell. This will render the molecular di!usivity to be exactly 1 in nondimensional units. The velocity "eld is naturally nondimensionalized as follows: (x,t)"v 3(x/‚ ,t/t ) , * 0* v v where 3 is a nondimensional function with period 1 in time and in each spatial coordinate * direction,andv issomeconstantwithdimensionofvelocitywhichmeasuresthemagnitudeofthe 0 velocity "eld. The precise de"nition of v is not important; it may be chosen as the maximum of 0 D (x,t)D over a space}time period for example. * Theinitialpassivescalardensity„ (x)willbeassumedtobecharacterizedbysometotal mass 0 ‘ a P M " „ (x)dx 0 0 Rd and length scale ‚ : T M „ (x)" 0„3(x/‚ ) . 0 ‚d 0 T T A.J. Majda, P.R. Kramer/PhysicsReports314(1999)237}574 247 We choose M as a reference unit for the dimension characterizing the passive scalar quantity 0 (which may, for example, be heat or mass of some contaminant), and we nondimensionalize accordingly the passive scalar density at all times: M „(x,t)" 0„3(x/‚ ,t/t ) . ‚d v v v Passingnowtonondimensionalunitsx3"x/‚ ,t3"t/t ,intheadvection}di!usionequation,and v v subsequentlydroppingthesuperscripts3onallnondimensionalfunctions,weobtainthefollowing advection}di!usion equation: „(x,t) v ‚ R # 0 v (x,t(‚2/it ))’+„(x,t)"D„(x,t) , t i * v v R „(x,t"0)"(‚ /‚ )d„ (x(‚ /‚ )) . (6) v T 0 v T Wenowidentifyseveralkeynondimensionalparameterswhichappearinthisequation.The"rstis the Peclet number & Pe,v ‚ /i , (7) 0 v which formally describes the ratio between the magnitudes of the advection and di!usion terms [325]. It plays a role for the passive scalar advection}di!usion equation similar to the Reynolds number for the Navier}Stokes equations. Next, we have the parameter q "it /‚2 , v v v whichis theratioofthe temporalperiodofthe velocity"eldto thecell-di!usiontime.Thirdly,we havetheratioofthelengthscaleofthevelocity"eldtothelengthscaleoftheinitialdata,whichwe simply denote d,‚ /‚ . (8) v T RewritingEq. (6) in terms of these newly de"ned nondimensionalparameters, we obtain the "nal nondimensionalized form of the advection}di!usion equation which we will use throughout Section 2: „(x,t)/ t#Pe (x,t/q )’+„(x,t)"D„(x,t) , R R * v „(x,t"0)"dd„ (dx) . (9) 0 Notice especially how the PeHclet number describes, formally, the extent to which the advec- tion}di!usion equation di!ers from a pure di!usion equation. Wenote thatthe nondimensionalvelocity"eld (x,t/q ) has period 1 in each spatial coordinate * v directionandtemporalperiodq .Itwillbeconvenientinwhatfollowstode"neaconcisenotation v for averaging a function g over a spatio-temporal period: PqP v SgT ,q~1 g(x,t)dxdt . p v 0 *0,1+d

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