Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena

337 Pages·1999·2.755 MB·English

by  Majda A.J., Kramer P.R.

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.:[phone];fax:[phone];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