Nonlocality of relative diffusion O. V. Tel’kovskaya and K. V. Chukbar Kurchatov Institute, 123182 Moscow, Russia

~Submitted 31 October 1996! Zh. E´ksp. Teor. Fiz. 112, 163–166 ~July 1997! On the basis of numerical modeling, the hypothesis of the nonlocal character of Richardson relative diffusion is tested and confirmed. © 1997 American Institute of Physics. @S1063-7761~97!01307-3#

Interest in the process of relative diffusion, i.e., the divergence in turbulent flow of two initially close-lying particles of a passive impurity, is chiefly associated with the possibility of avoiding here the fundamental difficulties blocking the path of a complete analytic description of turbulent mixing. Actually, since even turbulent flow, as was already pointed out in the classical papers1,2 ~see also Ref. 3!, is a regular flow on a scale of the order of the spatial period of the given fluctuation, the dynamics of an impurity in an actual turbulent field with a wide spectrum ~the inertial interval! possess the property of memory, which makes it extremely hard to use analytic approaches to the problem. On the other hand, the kinetics of T(l,t)—the probability density that, in a given experiment, two initially close-lying particles are at the ends of vector l at time t—as a consequence of being used to define averaging over different implementations of turbulent motion ~over different experiments!, loses memory and, conversely, must possess the property of information loss ~the mixing process occurs, and entropy increases!. Nevertheless, even this kinetic equation cannot be derived from first principles and has to be introduced into the theory by simply postulating it ~which, as B. Russell once noted, has many advantages, coinciding with those inherent to stealing by comparison with honest labor!. Usually, according to a tradition going back to Richardson ~who himself introduced the very concept of relative diffusion!, everyone reduces it to the usual diffusion equation and only argues about how its coefficient depends on the parameters of the problem.1,2,4 It was proposed in Ref. 5 that the analytic possibilities be broadened by going to the region of integral equations of the convolution type. ~The most obvious physical reason for nonlocality is that the main contribution to the divergence rate of two particles lying in the inertial interval comes from

a turbulent harmonic of the same scale as the current distance between them. Reference 6 was probably the first to call attention to the possibility in principle of a nonlocal variant ~see also Ref. 7!.! Specifically, the following model equation ~in dimensionless units! was written for T: ) ] T ~ l,t ! 5 G ~ 2/3! D ]t 4p2

E

T ~ l8 ,t ! 3 d l8 . u l2l8 u 5/3

~1!

The exponent in the kernel of the integral, equal to the Kolmogorov–Obukhov exponent, arises from the necessity of satisfying Richardson’s law ^ l & } t 3/2, which describes the variation of the characteristic diameter of a cloud of impurity in a turbulent medium,1–4 while the complicated numerical coefficient ~G is Euler’s gamma function! arises from the unity in the Fourier representation. This paper is devoted to testing this hypothesis, since only experiment ~including numerical modeling! can confirm or disprove a theoretical postulate. Let us discuss this feature. Equation ~1!, like the classical diffusion equation, is easy to solve because of its locality in Fourier space, and describes a similar information loss—the evolution of any initial impurity-distribution profile to a finite-parameter selfsimilar solution. Its fundamental difference, however, consists in the presence of an exponential tail in this solution as l→`, i.e., a divergence of certain exponential moments in the self-similar function T, which is a special case of the Levi function.5,7,8 In practice, this property denotes the exponential smallness of the probability of detecting in a given experiment a cloud with a diameter that substantially exceeds the mean Richardson value, but not at all the exponential character of the falloff of the impurity concentration at the periphery of the given cloud—as already mentioned, Eq. ~1! does not strictly describe the process of turbulent mixing, just as one plane projection does not give a complete repre-

FIG. 1. Realization of Richardson’s law: ~a! original data, ~b! analysis.

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FIG. 2. Number of pairs with an interparticle distance greater than a fixed l 0 ~1— l 0 580, 2—100, 3—150! vs. time: a! original data, b! analysis.

sentation of a three-dimensional body ~and a more appropriate analogy in the given case is one with an infinitedimensional body!; see also Refs. 2–4. It is especially simple to see the presence of such a tail by rewriting Eq. ~1! in the desired limit, taking u l2l8 u 5/3.l 5/3 out from under the integral sign on the left-hand side and using the normalization condition * Td 3 l[1:

SD

) ]T 2 1 1 5 D 5/3 } 11/3 , 2 G ]t 4p 3 l l

l@t 3/2.

~2!

One more characteristic property of the relative diffusion process described by Eq. ~1! immediately follows from Eq. ~2!—the probability of finding a pair of particles separated by a distance significantly greater than the Richardson distance is a linear function of time. It is easy to see that this property, unlike the exponent on l, depends neither on the turbulence spectrum nor the dimensionality of the problem, but is associated only with the hypothesis of the nonlocality of the process. It is precisely this, as the most stable property, that is convenient to choose as a verifiable parameter. Starting from the possibilities of the authors, such a test was made on the basis of numerical modeling. The turbulence was modeled by a given two-dimensional incompressible flow with a quasi-Kolmogorov spectrum. Specifically, the flux function C(v5 $ v x , v y % 5ez 3¹C) was chosen in the form of a sum of 140 harmonics combined into groups of seven each in twenty scale classes; i.e., 20

C5

(

i51

7

Ci ,

C i 5A i

(

j51

sin~ v i j t2ki j •r1 a i j ! ,

where the moduli of all the wave vectors in each group are identical, u ki j u 5k i , and their ratio in adjacent classes is k i /k i11 51.4.& (k 1 51). Such a choice made it possible to cover the scale range of the turbulent fluctuations in three orders of magnitude ~i.e., 1.420!. Amplitudes A i , according to the Kolmogorov–Obukhov law, equalled k 24/3 . The i angles of rotation of ki j relative to the x and y axes, as well as phases a i j , were chosen randomly in the interval ~0,2p!, and the number of harmonics in each class—seven—was considered sufficient to ensure that the turbulence is isotropic. The frequencies v i j were of the order of k i v i ; i.e., they were determined from v i j 5 b i j k 2/3 i , where b i j are random quantities in the interval ~1/2, 3/2!.

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The motion of 1004 pairs of points was studied in this given field of velocities, with the initial distance in each pair being equal to 3 ~i.e., of the order of a half-wave of the smallest-scale harmonic C!. To average over the implementations of turbulent flow, the pairs were placed at a distance of 3000 from each other ~i.e., of the order of the half-wave of the largest-scale harmonic C!—here, by the way, the difference between the process being modeled and the blurring of any fixed cloud is again seen. The results of numerical modeling are shown in Figs. 1 and 2. The former demonstrates the accuracy of the agreement of Richardson’s law for the given model. It can be seen that the self-similar regime is not reached too rapidly—in a time greater than 30, when the particles in pairs diverge on the average by a distance of about 40. The second, key figure reflects the degree of correspondence of the hypothesis concerning the nonlocality of the numerical calculation process ~the time shift by 10 in the processing does not contradict the linearity and is evoked by the initial data!. The theory of Ref. 5 corresponds to constancy of the functions in Fig. 2b ~and their proportionality to l 22/3 ! at least until l 0 @ ^ l & ~see Fig. 0 1!. The agreement obviously looks rather persuasive. In other words, Eq. ~1! gives an extremely good description of the process of relative ‘‘diffusion.’’ The authors are grateful to V. V. Yan’kov, discussions with whom stimulated the performance of this work. It was also supported by the Russian Fund for Fundamental Research ~Project No. 96-02-17249a! and by the ‘‘Nonlinear Dynamics’’ program of the Ministry of Science.

L. F. Richardson, Proc. R. Soc. London, Ser. A 110, 709 ~1926!. G. K. Batchelor, Proc. Cambridge Philos. Soc. 48, 345 ~1952!. A. S. Monin and A. M. Yaglom, Statistical Hydromechanics, Nauka, Moscow ~1967!, part 2, sect. 24. 4 H. G. E. Hentshel and I. Procaccia, Phys. Rev. A 29, 1461 ~1984!. 5 K. V. Chukbar, JETP Lett. 58, 90 ~1993!. 6 M. F. Shlesinger, B. J. West, and J. Klafter, Phys. Rev. Lett. 58, 1100 ~1987!. 7 J. Klafter, M. F. Schlesinger, and G. Zumofen, Phys. Today 49„2…, 33 ~1996!. 8 E. W. Montroll and M. F. Shlesinger, in Studies in Statistical Mechanics, vol. 11, J. Leibowitz and E. W. Montroll ~eds.! ~North-Holland, Amsterdam 1984!, p. 1. 1 2 3

Translated by W. J. Manthey

O. V. Tel’kovskaya and K. V. Chukbar

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