agranglan turbulmce and t rownian moti ” on paradox.pdf

7/27/2019 agranglan turbulmce and t rownian moti on paradox.pdf

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agranglan turbulmce and t++ rownian motion paradoxJ. A. \iiecelliLawrerzce ivermore National Laboratory, Livemore, California 94550(Received 16 April 1991; accepted 25 June 1991)The unique properties of three-dimensional hydrodynamic turbulence depend on the nature ofthe long-range time correlations as well as the spatial correlations. Although KoImogorovssecond simil@y hypothesis predicts a power-law spatial scaling exponent for the Eulerianvelocity fluctuations in agreement with experiments, it also leads, via the Lagrangian velocitytime structure function relationship, to particle dispersion predictions that are inconsistentwith enhanced diffusion. Recently, a new computational technique has been developed whichcan generate random power-law correlated fields in any number of dimensions with unlimitedscale range. This new method is used to explore the consequences of a proposed set ofassumptions about the nature of the time correlations and their relationship to the spatialcorrelations. In particular, the Brownian motion paradox is examined and it is~shown that itcan be resolved if the time domain constraint part of Kolmogorove second hypothesis isrelaxed and replaced with an assumption of space-time isotropy+ The proposed modificationpreserves the observed one-dimensional k - w3 spatial energy spectrum, allows for enhanceddiffusion consistent with Richardsons law, is consistent with Taylors frozen turbulenceassumption under the appropriate conditions, and yields an w- frequency spectrum for thenvelocity fluctuations in a frame at rest with respe& to the turbulence.

1. NTRODUCTIONIt is not generally appreciated that the inclusion of timein Kolmogorovs second similarity hypothesis makes the hy-pothesis somewhat more restrictive than necessary to obtaina k i- 5n turbulent kinetic energy spectrum. Onsagers inde-pendent derivation of the k - 53 spatial scaling law empha-sizes that no time assumption need be made, and that the lawessentially rests on the empirical observation that at suffi-ciently high Reynolds numbers the overall energy loss rate Eis fixed by the outer dimension of the flow and otherwisedepends only on a power of the mean velocity of the outerscaIe of the motion.13 Since the overall dissipation rate isfound not to depend on the viscosity, the smaller scales of theflow must readjust in such a way as to provide the exactamount of overall dissipation fixed by the outer scales of themotion. From this it follows that a spatial spectra1 scalinglaw consistent wi th an energy cascade process from larger tosmaller scales can involve only E. Dimensional consistencythen requires that the one-dimensional spectral energy den-sity folIow E(k) = C, tY3k - S/3where the dissipation E s avolume average time-independent quantity. Time does notenter in the scaling argument since a time-independent aver-age rate of energy transfer from smaller to larger wavenumbers is assumed.A theoretical prediction that has become almost as wellknown as the k s3 aw is the Lagrangian velocity structurefunction relationship attributed to Obukhov and Landau.This result states that the time structure function for thevelocity fluctuations of a Lagrangian particle scales propor-tional to the first power ofthe time. This sealing follows fromthe second part of Kolmogorovs second hypothesis, the partthat postulates that the time dependence for the velocityfluctuation probability distribution function also- depends

only on the dissipation rate. Under the assumption that tfluctuationsin velocity can depend only on the energy dipation rate, dimensional consistency requires that the grangian velocity structure function scale proportional towhich is equivalent to an asymptotic ecti ~scaling for Lagrangian power spectrum in frequency space. Althouthere have been many experimental studies confirming k - 5n spatial scaling relationship, the Lagrangian frequcy scaling relationship has not been subjected to direct perimental tests.A Lagrangian velocity structure function scaling pportional to et is equivalent to the assumption of normstatistics for the particle velocity fluctuations, such that particle follows Brownian motion- in velocity space with mlecular diffusivity replaced by the dissipation rate per umass.B However, Brownian motion in velocity space ayields a normal Brownian random walk for the particle placements in physical space because incremental displaments driven by velocities with a normal distribution satthe conditions of the central limit theorem.7 Here we hwme to what we call the Brownian motion paradox, becathere is much evidence that the turbulent dispersion of grangiarrparticles and passive scalars is not correctly scribed by normal Brownian motion.*- In Brownian tion mean-square displacements grow proportional to tiwhereas dispersion experiments in turbulent fluids show hanced diffusion with mean-square displacements growproportional to time raised to a power in the neighborhoof three.Recognizing that the inclusion of time in Kolmogorovsecond postulate is not necessary to the derivation of k _ t/3 law, and that experimental support for a Lagrangstructure function scaiing proportional to et is lacking, s


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a possible way of resolving the paradox. That intermit- of the four coordinate axes. With this method one can simu-turbulent statistics might require some correction to late enhanced diffusion experiments for whatever Eulerianscaling has been appreciated for a long time, velocity field correlation assumptions one may wish tothe nature of what it is that sets the overall scale of the make. Lagrangian statistical properties can be obtained byfluctuations has not been much explored. It seems vis- integrating the motion of particles over time, using velocitiesplay a role because, ust as the overall scale of the generated from the correlated random field function. Foris set by physical cutoffs, time cutoffs example, one can measure frequency spectra for the velocitythe overall scale in the time domain. Viscos- fluctuations of Lagrangian particles moved about by veloc-is explicitly included in Kolmogorovs first similarity hy- ity field fluctuations correspdnding to a specific choice forto the characteristic space, time, and ve- the Eulerian space-time co&lations while simultaneouslycutoffs q, r, and v, which set the smallest units of the measuring the corresponding enhanced diffusion properties

fluctuations. Onsagers derivation of the k - 53 of the particles. In the following we use this method to ex-that the time fluctuations in modal energy are plore the consequences of the proposed alternative hypothe-sis for the time fluctuations within inertial-range turbulence.d la(k) 1 =dt - g~4k121a&) I2 + C Q(W), (1)k

s the kinematic viscosity and Q is the cascade trans-function. Equation ( 1) indicates that the fluctuations ofcoefficients a(k) in a cascade process depend oncosity, even though the total rate of dissipation, theover all of the modes, does not.it is the part of Kolmogorovs second hypothesis deal-the time domain that leads to normal BrownianObukhovs Lagrangian velocity structure func-

hypothesis to apply only to the spatial domain, con-One must then seek some other means for deali ng with

In this paper we explore a modified form of the secondleads to enhanced diffusion with the correctscaling, and which preserves the k -53 l aw forspatial spectrum of velocity fluctuations. It also pro-- Eulerian frequency spectrum of velocitys, and an w - 5n velocity frequency spectrum asfrom a probe moving at arbitrary velocity relativesuch as is the case in almost all experimental

to the treatment of time in the second hypothesisstated as a requirement that the exponent dependencethe velocity fluctuation spectrum should be independentvelocity of the measuring probe relative to the turbu-It is equivalent to the physically appealing notion thatvelocity tluctuations may have no preferred axis inthe supposition that the spectrum of velocityis isotropic in four-dimensional space-time.A newly developed computation method allows one toconsiderable resolution; the consequences ofmethod permits one to gener-ensemble Monte Carlo samples of four-dimensional cor-fields that are described by power-law corre-in the program input. The generator isin theof a computational function that yields the value of acorrelated field at any point in a four-dimensionalarray whose size depends only on the integer wordfour-dimensional virtual space-time grids of

( 215)4,with a scale resolution of better than lo4 in each

II. ALTERNATIVE TO THE SECOND HYPOTHESISThe spatial correlations of the vector potential for three-dimensional isotropic turbulence are constrained by the requirement that they satisfy inertial-range scaling, corre-

sponding to a k - 113 ependence fdr the spectral density ofthe velocity components, an r 23 spatial structure functiondependence, and a one-dimensional k -53 kinetic energyspectrum. The scaling relationship for the lateral velocitycomponent structure function,

D,, (r) = C r&3r 23, (2)was obtained by Kolmogorov from two postulates. The firstpostulate assumes that in the high Reynolds number limitthe statistical relationships describing isotropic turbulencecan depend only on the dissipation rate per unit mass 6, andon the kinematic viscosity Y. Given this assumption the char-acteristic length, time, and velocity of the tlow are defined by

37 = $/4c - 1147 (3).r = y1/2E - 2, (4)u = $y4&4* (5)

In practice 77and r define the smallest measurable unit ofspace and time fluctuation within the inertial range becausevariations on smaller space and time scales are damped outby the action of viscosity.Kolmogorovs second hypothesis assumes that, in theinertial range, statistical properties in both space and timecan depend only on the dissipation rate. The 2/3 structurefunction exponent follows from this and the assumption oisotropy. From the first postulate the lateral velocity compo-nent structure function must have the form

where a and b are determined from the requirement that Eq.(6) also satisfy the second postulate. The only way Eq. (6)can be made dimensionally consistent is by choosinga = 2/3, b = 2/3, yielding Eq. (2).With the assumption of isotropy the lateral componentstructure function for a solenoidal velocity field is related tothe kinetic energy spectrum by


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DNN(r)m= 2sin E(k)dk, (kr) > (7)

corresponding to the one-dimensional kinetic energy spec-trumE(k) = C, c?k - 53. (8)

Isotropy with respect to the spatial coordinates implies sym-metry of the spectrum in wave space and invariance of thespectrum with respect to a rotation in wave space coordi-nates (k,,ii,,k, ) . Hence the assumption of isotropy is equiv-alent to the requirement that the spectrum must depend onlyon the spectral radius k where

k=k; +k; +k;. (9)Extending the assumption of isotropy to include time isan appealing notion since there seems no reason to expectthat measurements of inertial-range turbulence over onetime interval should statistically differ from measurementsover any other equal time interval, nor do experiments offer

any evidence that the power-law scaling exponent changesdepending on the velocity of the measuring probe relative tothe turbulence. In fact, as we have pointed out, assumingthat the scaling exponent does change when measured fromLagrangian coordinates so as to be consistent with time fluc-tuations depending only on the dissipation, leads to disper-sion predictions contradicting experimental observation.Equation (9) can be modified to include time by adding aterm proportional to the square of the frequency, generaliz-ing the spectral radius to include four dimensions. The con-stant of proportionality must involve velocity to ensure di-mensional consistency, and the onIy velocity availableconsistent with Kolmogorovs first hypothesis is that givenby Eq. ( 5 ) . Hence we can ensure isotropy in the space-timecase by requiring that all statistical relationships are func-tions of the four-dimensional spectral radius K, where

Kofmogorov eelIs Future, Lagrangian turbulence

and w is the frequency measured along an orthogonal foucoordinate axis. Putting K in dimensionless form by muplying by r from Eq. (3) shows that multiplying the quency by e _^#v - A as in Eq. ( 10) corresponds to scathe wave number by the smallest measurable spatial inment, and frequency by the smallest measurable time inment. This, however, means that we have to abandonpart of Kolmogorovs second hypothesis dealing with time Auctuations, although we can insist that it continueapply to the spatial fluctuations.Breaking space-time into discrete cells by scaling witand rdefines the direction in space-time of a probe movinconstant velocity U relative to the turbulence. Figurshows a schematic of a simplified case where the motiononly in thex-f plane. The angle CJJfthe probe path relativthe x axis on the diagram is

$=tan -(7j/Ur) = tan-(v/U). (In the following we consider conditions typical of subslaboratory or geophysical fluid turbulence, so that [ U I U [ Qs, and 7 &r&r, where c and s are the fight and sovelocities, respectively. The light cone and the acoustic care shown in Fig. 1, although drawn much out of scale wrespect to the rest of the figure so as to mak e their locavisible. The light cone half-angle is of course extremsmaii; if we take T= 0.3 mm and r = 0.03 set as apprmate representative values for the Kolmogorov cuto& tone obtains light coneand acoustic cone angles in the neborhood of 10 - r1 and 10 - rad. Hence, wi th the spacetime scales set by the Kolmogorov cell discretization, speed of communication between any two spatially sepaed cells is effectively in&rite. Consequently, we can ttime as an independent fourth coordinate orthogonal tothree space coordinates and can define a distance measuwave-number-frequency space as in Eq. ( IO). G,iven t

/ Taylor frozen turbulence

FIG. I. Diagram showing the breakspace-time nto discrete KoImogorovwith the ocation of the Taylor frozen lenceand Lagrangian urbulencelimitsKohnogorov spice and time fluctuatiotoffs are q and 7.

A particle path Past


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a requirement of invariance of the exponent in atical scaling law of turbulent fluctuations, as observedto an

A(K) = C,K -=? (12)_where C-4 and p are to be determined by the constraint thatEq. (12) yield the one-dimensional inertial-range k --53scaling in Eq. (8).Measurements of turbulent velocity tluctuation spectrausually from time samples that are interpreted as spatialby invoking Taylors frozen turbulence hypothe-Sampling is done by an external probe traveling with avelocity Urelative to the turbulence, whi ch i s assumed

have a fluctuating amplitude much smaller than U, suchions essentially represent spatial. The diagram in Fig. 1 divides into two limitingthan the light and acoustic cone angles, such that Tay-assumption is valid, and one where e,in the neighborhood of n/2 corresponding to the turbu-

mean velocity relative to the turbulence. The diagonal1 show the case where the mean velocitythe probe is equal to the Kolmogorov fluctuation velocity.particles move with the flow and have mean ve-the same as the mean velocity of the flow, so theirrelative to the turbulence is on average zero. Corre-the paths of particles on the diagram in Fig. 1 areat angle e, = n-/2 relative to the x axis, but indi-particles wander along random paths across the x-t

A sample of turbulence drawn from an ensemble couldrepresented in Fig. 1 by plotting the velocity fluctuationalong a third coordinate perpendicular to the dia-ith ampli tude in each Kolmogorov cell obtainedcorrelated field-generating function. The diagramit plain that the power-law scaling exponents ob-either space or time samples will differ only if thetical properties of the fluctuations are anisotropic inore, if there were a difference in expo-then one would expect that the fluctuation statisticschange smoothly from one law in the Taylor regime

This would imply a continuous range of scalingmeasured value depending on the ratio of the samplingelocity to the Kohnogorov velocity, since that definesSuch an exponent dependencenot been experimentally observed. In contrast, the pro-

alternate hypothesis of space-time isotropy does notthis difficulty nor, as the computations demonstrate,it conflict with the experimental observations of en-Given space-time isotropy it is easy to show that onlyspecific choice for the vector potential component spec-

spatial fluctuation scaling law. This choice yields areference frame invariant frequency spectrum ex-- 5/3, and in the frame at rest with respect to thea spectrum coefficient with a specific value de-by E and V. Under the modified second hypothesisspectral scaling for the vector potential components can

Since the velocity is the curl of the vector potential, afour-dimensional kinetic energy spectrum is obtained fromthe vector potential spectral density by multiplying by k .The one-dimensional spatial power spectrum can then beobtained by integrating out the frequency and two of thespatial wave-number coordinates; Fourier transformation tophysical space, with the corresponding physical space vari-ables set to zero, yields the one-dimensional spatial powerspectrum

mm(k) = Jk2(4rk2)C, dwo (k2 + E- I/2,,- 1iZw2)8 (13)

A change of variable x = (w/k) 2 yields*(k) = 2n-CAk5-2P s

X - l/20 (1+.-%--2x)~ dx

=,Tr(9r(8-9 CAEl/$1/4k5-2~rw> (14)The corresponding integral for the one-dimensional fre-quency spectrum is

smE(o) = k2(4nk2)C, dk (15)o (k 2 + E- l/2,,- 1/2@2)8 *A change of variable x = ( k /rj) 2 yields

smE(w) = 2rC,w5 - 28 x3/2r(p)

q @3 5,/2V2+3 - 5/2)/2w5 - 28(16)

The only choice for C,, andp that makes Eq. ( 14) consistentwith Eq. (8) yieldsA(K) = cl [r(~)/2~r(~) r(167)1Es'~2Y--~~K -20'3.

(17)where Cl is the Kolmogorov constant in Eq. (8). The corre-sponding frequency spectrum is

E(o) = ($J) Cl 656~6~ - s3. (18)This is the frequency spectrum for an Eulerian frame at reswith respect to the turbulence. This spectrum is physicallyconsistent with Taylors frozen turbulence hypothesis, sinceone gets the same result as E?q. 18 ) by replacing the probevelocity with the Kolmogorov fluctuation cutoff velocityq/r. As shown by the diagram in Fig. 1, this is physicallyquite a reasonable substitution to make for a frame approaching rest with respect to the turbulence.III. COMPUTED STATISTICAL PROPERTIE S

The method of generating the correlated field samples isdiscussed in detail in the Appendix. The steps in the compu-tation of the field at a given point in space-time are performed in physical space though a resealing process definedby the structure function of the random field. Testing and


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verification of the Eulerian correlation properties of thefields generated by the Monte Carlo method can be done bycomputing estimates of the Eulerian spectral propertiesfrom the field samples. The results of the previous sectionindicated that consistency with the modified hypothesis re-quires the random vecior potential component fields to havesymmetrical K W space-time spectral density functions.The corresponding vaIue of the structure function exponent2Hneeded for generating the vector potential components isdetermined from the relationship between the structurefunction exponent and the spectrum exponent - Zp, whichin four dimensions is W = (2p - 4)/2, yielding X = 4/3 for2p = 20/3. In the computations the scale velocity is chosensuch that if t,, defines the size of the time domain, and r,,defines the size of the spatial domain, then the velocity nor-malizing the time axis is Y= rma/&,aX, such that the meshresolution is the same in nondimeusional units along anyaxis. Hence statistical properties are the same along any ofthe four coordinate axes. In essence, each four-dimensionalcell in the virtual computational grid physically correspondsto a Kolmogorov cell. The computations were done using alinear congruential random number generator on a machinewith 64-bit integer precision, limiting the number of inde-pendent random numbers available for sampling to 262.Thelargest single four-dimensional virtual grid that can be gen-erated under this constraint is (2s)4 = 2m, or 32 768 pointsalong each of the four axes; leaving room for three indepen-dent components of the potential without overrunning therepeat period.The statistical properties of an isotropic vector field canbe characterized in several ways,..but some quantities areeasier to obtain from computations. Structure functions canbe computed directly from samples, but are sensitive to thelargest scales of the turbulent field and begin to deviate sig-nificantly from inertial behavior at distances very much lessthan the outer dimension of the field. Perhaps the best quan-tities for comparison with inertial theory are Fl (k, ) andF2 (k, ), the longitudinal and lateral velocity componentspectra. Good statistical convergence can be obtained bytaking one-dimensional sections spread evenly across thespace-time field and averaging the individual spectra ob-tained from applying a fast Fourier transform to each sec-tion. Another computationally simple variant of this ap-proach is to keep the location of the section fixed but vary therandom number sequence offset for each section by a ran-dom amount. Both methods produce the same result. Figure2 plots estimators of the longitudinal and lateral one-dimen-sional velocity spectra obtained from sections through a( 1024) grid. The average is over 64 individual 1024 pointspectra taken parallel to the x axis at different y, z, and tcoordinates going from the center to the edge of the virtualgrid. A solid line with slope - 5/3 is included in Fig, 2 forreference. The longitudinal spectrum F, is very closely fol-lowing a k - ~3 law over the three decade range, but thelateral spectrum F, is showing some falloff from the k - 53reference line at higher wave numbers.Eulerian rest frame and Lagrangian velocity frequencyspectra were also obtained for the same ( 1024)4 mesh, againusing the power-law correlated vector potential field corre-

FIG. 2. The longitudinal (solid dots) and ateral (open circks) one-dsional velocity spatial power spectraobtained from Monte Carlo samon a 1024 space-timegrid. The reference ine has slope - 53.

spending to Eq. (17). Eulerian spectra were obtainedaveraging the 1024 point frequency spectra of time sesamples taken from 64 points in Ix,y,zj separated by spatial intervals in each direction. Lagrangian frequespectra were derived from time series obtained by integing the motion of a Lagrangian particle over 1024time sthrough (x,v,z) space using the velocity correspondingthe time step coordinate in the four-dimensional velofield sample. The frequency spectra obtained from 64 integrations were averaged to obtain the power spectrEach trial used a different seed in the four-dimensional locity field generator so the average is over an ensemblLagrangian trials. Both the Eulerian and Lagrangian cwere done with all other parameters fixed so that one directly compare the spectral scales. Figure 3 shows that

1rP-m, (II1.,..1,>II*irr5-Timm7--,1IiT.l.TIO 10 1tjl- ii:wFIG. 3. The Eulerian rest frame (octagons with cross) and Lagra(solid dots) velocity frequencypower spectraobtained rom the samespace-timegrid samples.The reference ine has slope -- 5/3.


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rest frame and Lagmngian velocity frequency .spec-identical under the proposed alternate hypothesis, tothe statistical accuracy obtained in the simulations.is in agreement with the hypothesis that the fluctu-isotopic in four-dimensional space-time, such thatmatter what path a particle follows through this space thefluctuation spectra obtained have the same power-

One can also easily measure the enhanced diffusion scal-ties produced by the modified second hypothesis,performing trial integrations of Lagrangian particle mo-components satisfying the scaling in Eq. ( 17). Sta-re gathered from a long series of trial integrations,with a different seed or the four-dimensional field sam-is placed at the center of the spatial part o f thel virtual grid on the first time plane and itsry through space is obtained by integrating the mo-

coordinates. The integration is carried out for al, and the location of the particle saved attimes, using a different field sample for each run, oneobtain a histogram of the probability distribution func-the distance the particle will go for a fixed time inter-show that enhanced diffusion, such that the

4/3 spatial scaling law, yields an r 53 dependence forintegrated particle count I(r), the distance probabilit yion function.i4 This distribution occurs because theof the total number of particles going a distance ra localized release point, over a fixed time interval,as ( (r ) ) 12 (r 43) 12- r 2/3, such that the number

in space s proportional to r 23/r 2. Sincenumber of particles ending their turbulent dispersion tra-within a sphere of radius r centered at the startings proportional to the volume integral up to radius r,to r 53. Many thousands of trials have been runropy hypothesis and theyistently yielded an r 53 scaling for the integratedmonstrating consistency with enhancedand Richardsons 4/3 law. In contrast, Brownianof the particles is not consistent with enhanced diffu-not yield a power-law scaling result. Figure 4

set of three histograms, each obtained from 500 dis-a (512)4 virtual space-time grid. Histo-dispersion over 128, 256, and 5 12 time steps, aregons, respectively. In order to better show both thedependence and the comparison with the 5/3e, the first and second histograms have beenthe right by scaling their r coordinate by 4~ andx , respectively. This helps to quantitatively show that iftime i nterval for the dispersion trials is increased, thenhistogram moves to the right proportional to the increasemaintaining the r 53 spatial dependence, such that the

at distances from the origin following anprobability distribution with distance to the outer cutoff

FIG. 4. Integrated distributions l? of Lagrangeparticles obtainedafter inte-grating the motion ofthe particles over 128 octagonswith cross), 256 (tri-angles), and 5 12 i me steps octagons), using Monte Carlo samplesof solenoidal velocity fields drawn from the ensembleon a 5124space-timegrid.The radial coordinatehasbeen esealed y factors of 4X and 2 X for the 128and 256 cases, espectively.The reference ine ha s slope 5/3.

expanding proportional to the time interval.The linear time scaling of the horizontal axis cutoff onthe r 53 probability distribution function results because thepart of the velocity field derived from the largest space andtime scale has the biggest variance. To the eye the velocityfield in a typical sample appears to roughly consist of a meanvelocity with which the rest of the turbulence is swept acrossthe virtual grid. This appearance is a consequence of thepower-law scaling cutoff imposed by the outer dimension ofthe virtual grid. On the finite size computing grid the largestscale produces the biggest displacements, such that the hori-zontal axis scaling of the probabilit y distribution functionfor the particle dispersion expands proportional to the timeinterval with average velocity corresponding to the rms ve-locity of the largest scale. The mean convection resultingfrom the outer scale cutoff on the turbulent velocity scalingis not an independent external variable, rather its amplitudeis related through the velocity structure function to the Kol-mogorov small-scale cutoff velocity v in Eq. (5), which inturn depends on the dissipation and viscosity. One can fix theparameters of a given case either by specifying the outer scalevelocity, as might be the case n a physical experiment, or byspecifying the dissipation, such as s often done in theoreticalanalysis, but not both.IV. CQNCLUSION

The notion of a discretization of space-time correspond-ing to the smallest possible scales of turbulent fluctuationshas been fundamental to the arguments discussed in this pa-per. Within the context of subsonic laboratory and geophysi-cal fluid turbulence it appears reasonable that this discretiza-tion should be determined by the Kolmogorov space andtime fluctuation cutoffs, smaller scale turbulent fluctuations


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having been erased by the action of molecular diffusion. Theexistence of a small-scale spatial cutoff has long been appre-ciated, but the existence of a corresponding time cutoff hasseemed less easy to accept, although it necessarily followsfrom the same argument leading to the spatial cutoff.Including a time cutoff makes it possible to draw aspace-time diagram showing the relationship between themotion of a probe along a path corresponding to Taylorsfrozen turbulence hypothesis and the trajectories followedby particles moving with the flow. Such a diagram showsthat the only way the exponent of the Eulerian frequencyspectrum, measured from a frame at rest with respect to theturbulence, can differ from the exponent in the Taylor frozenturbulence spectrum is for the statistical law governing thefluctuations to be anisotropic in space-time. Apart fromthere not being any experimental evidence for velocity fluc-tuation spectrum exponents differing from - 5/3 depend-ing on the motion of the probe frame relative to the restframe of the turbulence, a statistical law with an anisotropicexponent dependence leads to Lagrangian particle disper-sion scalings inconsistent with experimental observation. Inparticular, supposing a transition to an w - dependence inthe neighborhood of the time axis yields Brownian motionwith mean-square particle displacements growing propor-tional to c instead of the approximate t dependence ob-served.The Brownian motion paradox results from the Lagran-gian time structure function argument which is based on theassumption that the only relevant physical parameter is thedissipation rate. However, discretization of space-timeshows that one cannot consider both space and time togetherin a consistent way, unless the time cutoff is included, whichnecessarily introduces a viscosity dependence..As Onsagersargument has shown, one can obtain a k 5/3spatial scalingfor the velocity fluctuations without resorting to the secondpart of Kolmogorovs second similarity hypothesis dealingwith time, Given that, consistency with the observed en-hanced diffusion can be obtained if one replaces the secondpart of the second hypothesis with the assumption that iner-tial-range velocity fl uctuations are isotropic in four-dimen-sional space-time. Adopting this change resolves the para-dox, and yields Lagrangian~ velocity fluctuations thatproduce enhanced diffusion in agreement with experi mentalobservations. The change preserves E(k) = C, fi3k - spatial scaling for the one-dimensional velocity componentspectrum, but permits space-time symmetry in the spectraldensity of the vector potential components. Consequently,velocity frequency spectra follow u ; j3 scaling in both Eu-lerian and Lagrangian coordinate frames.Velocity frequency spectra following an w - s/3 scalingare consistent with direct experimental observation, sinceturbulent spectra have almost always been measured in thetime domain, although they are often interpreted as spatialspectra under the assumption that the turbulence is embed-ded in a mean flow which sweeps it past the measuringprobe. In contrast to the spectrum exponent, the measuredfrequency spectrum ampli tude coefficient depends on thevelocity of the measuring probe relative to the rest frame ofthe turbulence, as expressed in Taylors frequency spectrum

coefficient formula. Although Taylors formula is normused with a probe velocity much greater than the Kolmgorov velocity fluctuation cutoff, if one neverthelesstrapolates to the rest frame frequency spectrum by slowthe mean probe velocity relative to the turbulence to neighborhood of the Kolmogorov cutoff velocity, onetains, except for a small difference in the numerical muplier, the same frequency spectrum coefficient dependeas the space-time isotropy assumption, again demonstratthat it is necessary to introduce a viscosity dependethrough the Kolmogorov cuto$ to lix the time scale ofrest frame fluctuations.One can perhaps draw no additional conclusion fr6he resulting 2 9~/6 dissipation and kinematic viscosity pendence of the frequency spectrum coefficient, but thispendence is suggestive of other properties of the flow aciated wirh enhanced diffusion. The exponent of viscosity turns out to be equal to the intermittency exponinferred for strictly inviscid three-dimensional turbuleunder the assumption that it evolves following Richardso4/3 law. When the flow is inviscid one can show that becaof Kelvins circulation theorem vorticity cannot be cpletely mixed throughout Euclidean space, but insteadmains confined on a nearly volume-filling fractal surfwith dimension 3 - 8 = E or in sampling along a onemensional section, a fractal set of dimension 1 - &= 2real fluids the vorticity can fill all of Euclidean space becaviscous diffusion of the vorticity fills in the gaps, so_8%1/6 dependence may be a consistent manifestaticnthis.

This work was performed under the auspices of the UDepartment of Energy by the Lawrence Livermore NatioLaboratory under Contract No. W-7405-Eng-48.APPENDIX: GENERATING MONTE CARLO SAMPLESOF CtX?RELATED FIELDS

The problem of how to generate fluctuating randtields with specified long-range space-time correlationclosely tied to that of generating random fractal curves surfaces. The method of successive random additions been successfully used to generate fractals that are quanttively described by a structure function. The process sof successive grid retlnement, with field values at new inmediate grid points obtained by interpolation from the vious grid. Successive mesh refinements are performedrepeated cycles, and at each level of resolution random inments are added to the existing geld points, with variaresealed to the given level. A drawback of the method isit requires a storage array big enough to contain the entireof field points at the highest level of resolution, Howethis limitation-can be overcome by combining the methosuccessive random additions with an algebraic random nber sequence function and random number recovery tniques.16 This makes it possible to write a computatiofunction that generates statistically consistent sample


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8 A

5. The vertex numbering scheme or the four-dimensional hypercube.along northogonalourth coordina te xis, ut t is schematica l-y nested ells.

elated random fields at arbitrary pointsany number of dimensions without using mesh stor-such that the scale range available is limited only by theshows how the method can be used to generateCarlo samples of four-dimensional space-time fluctu-

between arbitrary sampling points in space andThe method of successive random additions generates a

a scalar or component of a vector field, C i is there constant, and His the Hurst exponent. A set ofgrids with each grid a fabtor of 2 smaller in size isfrom one grid to the next finer grid, with the addition atstep of a random perturbation of appropriate ampli-the four-dimensional case the mesh lines define theof a hypercube as shown in Fig. 5; a single mesh cellof a three-space-dimensional cube with time index nanother spatial cube at time 12 1. The numbers la-

array containing the values of A at the ver-At the start of a computati on, the values of the field at16 corners of a hypercube covering the largest distance Rerest are initialized by drawing values from a probabili-with .variance u g = CgR H. This single hy-the boundaries and must be large enough todesired outer dimensions of the computing region,the physical outer l ength scale of the turbulence.The comptitation proceeds by dividing the single hyper-into 16 additional hypercubes obtained by bisectinggrid line, and locating which one of the smaller hyper-contains the sampling point. Values ofA at the verticesthis smaller cube are obtained by linear interpolation from

larger hypercube. In addition, a random perturbation SAeach of the vertices. This process can be repeated

ad nfinitum each time the sizeof the hypercube ontainingthe samplingpoint is cut in half, with values ofA interpolat-ed from the larger hypercube, to which additional randomperturbations are added. At the qth level of this resealing thevariance of the random perturbation for the qth grid is givenby

a; = (&>fi$. (A21The accuracy with which the location of the sampling pointis fixed and A specified can reduce to the level of round-offerror. In practice, for dimension two or greater the numberof calls to the random number generator will usually exceedthe repeat interval of the generator before round-off preci-sion is obtained. For linear congruential generators the re-peat period is fixed by the precision of the integer wo rds usedin the generator. For example, with 64-bit single precisionintegers the repeat. period for a linear congruential generatorwithout an additive term is 262.With this interval the scalerange for a single scalar field variable is limited to 262 n onedimension, 231 n two dimensions, and approximately 2*and 2 in three and four dimensions, respectively.

Although these grids are of enormous size, only the 16values of A at the vertices of the hypercube containing thesampling point at the @h level of resealing need be retainedin memory. Hence the value of A at any vertex in the ( 29)4mesh of hypercubes contained in the largest outer hypercubecan be reproduced, with negligible working storage. Thissupposes, however, that we have some way of recovering thespecific random perturbations involved in determining thevalue of A at a specific vertex. Fortunately, the ordering ofthe sequence of random number generator calls can be relat-ed to 4 and the grid indices by an algebraic formula, so thatstandard random number recovery methods can be used toregenerate any numbers in the sequence of calls as needed.If the sampling point is (x~,JQ,z,J, ) where (k,Z,m,n)are the indices of the hypercube containing (x,y,z,t) on thefinest resolution grid, corresponding to q = qmax, then themesh interval S, of the 4th grid, measured in units of thesmallest grid interval, is

4 = 2qm q- (A3)We compute the @h-level indices of the hypercube contain-ing (X,JJ,ZJ) on the 4th grid from

k, = 1+ [(k- U/S,],zq = 1+ [(I- 1)/S,], (A4)m,=I+[(m-1)/S,],n,=l+[(PI-WSq],

where the brackets signify the integer part. For q,,, = 3,corresponding to the division of the outer hypercube into 163smaller hypercubes, the index k = 1,2,3,4,5,6,7, 8 for thefinest resolution mesh lines. At the first level q = 1, the indexkg takes on values 1 and 2. At the second level q = 2 theindex k, takes on values 1, 2, 3, and 4. At the third levelq = 3, the index k, covers the range 1,2,3,4,5,6,7, and 8.We have indexed the hypercubes at each level starting at 1increasing by increments of 1, because this may be morehelpful for computer programming purposes.Each time the grid interval is reduced by 2, it is neces-


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sary to determine which of the 16 new hypercubes containsthe point (k,,Z, ,mq,nq ). Once this is known, field values atthe corners of the q + 1st hypercube can be interpolatedfrom the vertices of the qth hypercube. Storage for only onehypercube is needed, since old values can be overwritten assoon as the interpolation is completed. The relative indices(k,Z,m,n j on the q + l-level grid square containing(k,Z,m,n)~arek 9+1 = l-f [(k- 1) modS,/S,+,],iii1 =lf[(Z-11)modS,/S,,,],~$-bl =l+ [(m-11modS,/S,+,],%,I = 1 + [(n- 1) modS,/S,+,],

where the brackets signify the integer part. With this choiceki+,,~Zi+,, m;,,,andrt;+, takeoneithervalues 1 or2.

These relative indices tell which of the I6 hypercubes withe qth-Ievel hypercube contains the field point (k,Z,mon subdividing the @h-level mesh.In one dimension or even in two and three dimensithe process of linearly interpolating from one mesh tonext is relatively straightforward, but in four dimenswith the 16 possibilities, each with 16 formulas, the probbecomes more complex and the chance for error increaThe task can be made easier by working out the 16 nterption formulas for the case where the sampling point is loed in the first ofthe 16 smaller hypercubes$ and then perming indices to obtain formulas for the cases where sampling point is in one of the other 15 smaller hypercuHence one obtains a 16 x 16 matrix ofindex values whereentries Pzr give the location of the ith node for the samppoint located in the jth smaller hypercube. For the venumbering scheme shown in Fig. 5 the permutation matr

i2345678 9 10 11 12 13 14 15 162 6 1 3 1 5 5 4 10 14 3 II 9 13 13 123442783 6 11 12 12 IO I5 I6 II 144 8 2 1 3 7 I 2 12 16 10 9 11 15 9 105 1 7 8 6 2 8 7 13 9 15 16 14 10 16 156 5 5 7 2 1 6 3 14 13 13 15 10 9 14 117386844 5 f5 11 16 t4 16 l2 12 138 7 6 5 4 3 2 I 16 15 14 13 12 11 10 99 10 11 12 13 14 15 16 1 2 3 4 5 6 3 8

10 14 9 11 9 13 13 12 2 6 1 3 1 5 5 411 12 12 10 15 16 I1 14 3 4 4 2 7 8 3 612 16 10 9 11 15 9 10 4 8 2 1 3 7 1 213 9 15 16 14 10 16 15 5 1 7 8 6 2 8 714 13 13 15 10 9 14 11 6 5 5 7 2 1 6 3I5 11 16 14 16 12 12 13 7 3 8 6 8 4 4 516 15 14 13 12 11 10 4 8 7 6 5 4 3 2 Is

i

Except for values at the vertices of the single hypercube con-taining the sampling point at each level of therescaling, noneof the other mesh values need be generated or stored. Theindex of the smaller hypercube containing the samplingpoint is

j=k+2(E,--1)+4(m-1)+8(n-l), (A71where the relative indices (k ,I ,m,n j areobtained from Eq.(A5). The interpolated values of A p + at the vertices of thejth smaller hypercube containing the sampling point are

A+=AP5.i fw-G~,1==~4(A$,,, t-4&G4 ; = 1/A%,,, -t-A p?,,),A $a& = $(A Sle,+ A $J,

il~~,l,l=(,~*,,+A~,.j),A4 y,, --F i{A SC,, A &, -t A $3>j A ll,.j),A &+,I +(A$~~ +A& +A%s, +A4,,),A j57+fl i(A?:~tA~~:/iA?:,+A4,.,),A $&- g(A ST., I- A &, + A &., + A %o.,)~/j g;++ $(A$., f A $s,l t ~4 *., +- A p,~.,)~rd $;.; = $(A & f A Qp, -t- ~4$9.1 t- A p,d;; = &(A Bptej+A:~~ +%3.j +A%.,

..* +i4q pe, +&,-+A&,,, +-%.,j,


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A 5;; = &(A%,,+ A g, + A I%,, A 4pg.J$ A $,, f A $,,., + A zh., + A h%~)~

A ;;; = i(A ;,,, + A ;%, + A $., + A %I+A,g,+Ap ,,,, +A: ,,,, +A&.,),

A y; = #A SIP,+ A &, -I- A %,, + A k.,+ A ;$,, + A Sk, + A &., + A p8.1+ A &, + A &,, +A k,., + A $12.1+ A &, + A &, + A $,s, + A sIS.1* (A81solenoidal velocity field sample can be generated from therandom vector potential having three independentA,,, and A, with each component generateding from a different point in the random number se-Since the computation of the potential ends with thevalues at the vertices of the smallest hypercube containingpling point, the average value of the velocity at thee hypercube can be computed from first-orderapproximat ing the curl. The velocity com-

on derivatives of the vector potential,constant and the gradient of a scalar can be added topotential without changing the velocity field.Two remaining considerations are the choice of distri-function for the random additions at each of the qand the formula relating the random number call se-stage of the construction. The choice of distri-function affects the higher moments of the statisticaltions in the generated fields. Drawing random addi-from either a normal distribution or an exponentialwill, for the same scaling of the variance with q,with similar structure function andproduces lognormal statistics in the field amplitudes,the normal distribution produces correlated fieldsstatistics skewed toward more uniform backgroundamplitudes. Kolmogorovs third hypothesis postulatesthe dissipation fluctuations follow lognormal statis-Recent theoretical developments suggest that the dis-in hydrodynamic turbulence cannot be exactlyhowever, experiments in three-dimensional tur-10, close to the limit of measuring resolu tion, thehypothesis is a very good approximation. ig~ For

the random additions used in the computationdrawn from an exponential distribution.The variance of the random perturbations SA added atvertices of the$h hypercube after interpolation from thethe fraction of the initial variance specified.Eq. (A2) with q replaced by q + 1. If the entire field werebe no need to keep track of thegiven stage in generating the field. However, ifthe sampling point is to beit is necessary to know the number of calls to getthose particular vertices in the full construction in order

able to regenerate the random perturbations added atvertices without repeating the entire random number

sequence. The sequence number has to be expressed as formula, since storing the sequence numbers in a pointearray corresponding to the spatial grid and level would require a large storage array. The sequence number formulaalso needs to take into account the number of calls requiredto generate the distribution from which the random additions are drawn; for an exponential distribution only one cato the random number generator is necessary; hence, withindices varying most rapidly from k through iz, the callingsequence number g isc$ = kq + (I, - 1)2q+ (m, - 1)22q

+ (n, - 1)234 + 2 24(J- l),j=2

(-49)where the relative coordinate indices ( kq,lq,mq,nq ) are obtained from Eq. (A4). An arbitrary fixed integer constantcan be added to { without affecting the computation andcorresponds to starting from a different point in the randomnumber sequence. In order to generate independent MonteCarlo samples from an ensemble of different fields it is necessary to increment this offset by the maximum possible valueof {, the number of random numbers used to generate everpoint in the given field, each time a new field is needed.Although there is some overhead involved in computingrelative indices, and interpolating between meshes, nearly alof the computing time is spent in regenerating randomnumbers satisfying an exponential distribution. For large nthe number of random numbers required to generate a singlesample of the necessary components of the vector potentialon a virtual mesh with side 2 in d-dimensional space is

N = 3n2d. C-410In practice the amount of computing time can be estimatedquite well by multiplying N in Eq. (AlO) by the time required to regenerate one random number. The time to regenerate a random number in the standard method* is spenin testing the bits of the sequence number and, for nonzerobits, performing a multiply mod 2N, where N is the numberof bits in the machine word length. If the word length iN = 64 bits, the average number of multiplications is 32whereas only one multiplication is needed to generate thenext number in the sequence. However, the number of multi-plications can be reduced if the function storage space iincreased to include precomputed tables of factored powersof the multiplier.23

A. N. Kolmogorov, C. R. Dokl. Acad. Sci. URSS 30,301 (1941).L. Onsager,Phys. Rev. 68,286 (1945).3L. Onsager,Nuovo Cimento 6, Suppl. No. 2,279 ( 1949).4L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, London1959), p. 122.5A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics,edited by JL. Lumley (MITPress, Cambridge, MA, 1987), Vol. 2, Sec.23.4, p. 4866A. S. Monin and A. M. Yaglom, in Ref. 5, Sec.21.3, p. 360. F. Reif, Fundamental sofStatistical and Thermal Physics McGraw-Hill,New York, 1965), p. 560.L. F. Richardson, Proc. R. Sot. London Ser. A 110,709 (1926).A. S. Monin and A. M. Yaglom, in Ref. 5, Sec.24.3, p. 551.M. F. Shlesinger,B. J. West, and J. Klafter, Phys. Rev. Lett. 58, 1 10(1987).J. A. Viecelli, Phys. Fluids A 2,2036 ( 1990).


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12A.S. Monin a nd A. M. Yaglom, in Ref. 5, Sec. 13.3,p. 110.I3A. S. Monin and A. M. Yaglom, in Ref. 5, Sec.21.4, p. 362.J. k Viecellii Phys. Fluids A 1, 1836 1989). R. F. Voss, n Fundamental Algorithms n ComputerGraphics, dited byR A. Earnshaw Springer-Verlag, Berlin, 1985), p. 805.r6J. A. viacelii and E. H. Canfield, J. Comput. Phys. 95,29 (1991).i J. Feder, Fractak (Plenum, Ne w York, 1988)) p. 180.*A. S. Monin and A. M. Yaglom, in Ref. 5, Sea.25.3, p. 611.

19F. Anselmet, Y. Gagne , E. J. Nopfinger, and R. 4. Antonia, J. FlMach. 140,63 ( 1984).C. Meneveauand K. R. Sreenivasan, ucl. Phys. B, Pron. Suppl. 2f1987L D. EL Knuth, The Art of Computer Programming (Addison-WesReadink MA, 1969).A. E. Koniges and C. E Leith, J. Comput. Phys. 81,230 ( 1989).E. H. Cantleld, Jr. and J. A. Viecelli, J. Comput. Phys. (in press).


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