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Intermittency, synthetic turbulence and wavelet structure functions

Intermittency, synthetic turbulence and wavelet structure functions
Piero Olla1 and Paolo Paradisi2

arXiv:chao-dyn/9803039v1 26 Mar 1998


ISIAtA-CNR, 73100 Lecce Italy FISBAT-CNR, 40129 Bologna Italy Abstract


Some techniques for the study of intermittency by means of wavelet transforms, are presented on an example of synthetic turbulent signal. Several features of the turbulent ?eld, that cannot be probed looking at standard structure function scaling, become accessible in this way. The concept of a directly measurable intermittency scale, distinct from the scale of the ?uctuations, is introduced. A method for optimizing the analyzing wavelets, which exploits this concept, and allows to minimize non-local contributions in scale to wavelet correlations, is described. The transition from a wavelet to a Fourier transform based description of an intermittent random ?eld, and the possibility of using Fourier correlations to measure intermittency are discussed. Important limitations in the ability of structure functions to give a local in scale description of intermittent random ?elds, are observed. PACS numbers: 47.27.Jv, 02.50.Ey, 05.45.+b, 47.27.Gs

Submitted to Physica D 3/27/98


I. Introduction Structure functions and scaling arguments, are basic tools in the study of turbulence. One of the key issues in the ?eld, intermittency, is expressed in this language through the statement, that the scaling of the structure functions Sl (q) = |?l v|q , with ?l v(x) = v(x + l) ? v(x), is non-trivial. This means simply, that in high Reynolds number turbulence, there is a large ”inertial” range of scales where Sl (q) obeys a power law: Sq (l) ? lζq , with ζq a nonlinear function of the order q. Many explanations have been proposed for this phenomenon, but to date, nothing de?nitive is still available (see [1] for a review). There are several reasons for this state of a?airs; one is perhaps, that scaling by itself is not able to provide a su?ciently complete description, of what is happening in the turbulent ?eld. Within Kolmogorov 1941 theory [2], energy conservation is able by itself to (0) ?x a value of the scaling exponent: ζq = q/3, which is an acceptable lowest order approximation for ζq . However, no comparable symmetry based argument has been proposed, which was able to (0) ?x the value of the intermittency correction ζq ? ζq . The fact is that, although the mechanism of intermittency generation is likely to be universal, possibly associated with some property of the nonlinear energy transfer in the inertial range, it is also likely that its e?ect is not exhausted in the production of scaling corrections. In fact, very di?erent kinds of intermittency, like e.g. the one associated with the long and thin vortices observed in numerical simulations [3], and the one that would be obtained in a random beta-model picture of turbulence [4], produce equally acceptable spectra of scaling corrections. Wavelet analysis [5] has often been proposed as an alternative tool in the study of turbulent intermittency. The possibility of having an additional degree of freedom associated with the wavelet shape, beyond position in space and scale, makes these objects, particularly appropriate to study coherent structures and the geometrical properties of intermittency [6]. However, the great freedom to describe geometrical features in two- or three-dimensional settings, thus becoming available, has hindered perhaps an extensive use of these techniques in turbulence theory. An alternative use of wavelets has been to employ them as building blocks in the generation of arti?cial turbulent signals, trying to reproduce the kind of velocity time series one gets in experiments. This kind of technique, ?rst introduced in [7] to study connections between dissipation and velocity intermittency, and in [8], has been later developed systematically in [9] (see [10] for recent references). More recently [11], this approach has been used to provide a kinematic explanation for the kind of energy spectra developing in wall turbulence. The picture of a superposition of eddies at di?erent scales obtained in this way, appears particularly natural to study intermittency e?ects. However, even in such one-dimensional settings, there is a great freedom in the choice of the eddy generation mechanism. It is clear, for example, that the same spectrum of anomalous exponents can be obtained, both from a ”random eddy model” with inclusion of intermittency, like the one considered in [11], and from a multiplicative cascade of the kind described in [9]. (To get such an identically scaling signal, out of the ?rst model, it is enough to randomly permute the wavelets at the di?erent scales in the second). Now, di?erent mechanisms of turbulence synthesis imply, to some degree, di?erent assumptions on the real turbulent dynamics. It is therefore of some relevance, to devise methods which allow to identify these mechanisms, from the statistical properties of the signal. In light of the present discussion, synthetic turbulence appears to be the appropriate ”training facility”, in which to test di?erent techniques of wavelet analysis, and their ability to reveal speci?c intermittency features [12]. Recently, spatial [13] and scale [14] correlations between wavelet components, have been used to probe the cascade structure of turbulent signals. Here, the interest is focused on two di?erent issues: the choice of the wavelet in both the signal generation and analysis, and the amount of phase space available to each ”building block” wavelet in the generation algorithm. This phase space is given by the relative position and scale of the generated wavelet, with respect to the parent one, and in typical algorithms of signal generation [9], it consists just of a single point. It turns out that both issues of wavelet shape and phase space availability, have important consequences


as regards the ability of a structure function to detect features of the turbulent dynamics that are really local in scale. In the next section, the generation algorithm for the synthetic turbulent signal is introduced, and in section III, the basic wavelet structure function properties are derived. A systematic analysis of the turbulent statistics dependence, on the shape of both the analyzing and the building block wavelets, is carried on in section IV. Given the one-dimensional nature of the signal, geometrical aspects are minimal, and this leaves out only one essential degree of freedom in the choice of the wavelets: their ”number of wiggles”, i.e. the product of their dominant wavevector and spatial extension. Section V contains discussion of the results and conclusions. We leave in the appendix, the analysis of the case in which the cascade is probabilistic in space and discrete in scale. II. Synthesis of an arti?cial turbulent signal Introduce the Gaussian wavepacket: wS (k, y, x) = exp(ik(x ? y) ? (x ? y)2 /λ2 ); S and let: Ψ(x) =
n hn

λS = aS /k

(1) (2)

Ahn wS (khn , yhn , x)

be a real random ?eld, which should mimic the time signal from a ?xed position velocity measurement in a turbulent ?ow. Following standard practice [9], the building block wavelets wS are generated through a cascade process, to model the mechanism of energy transfer in the turbulent ?ow. The vector index hn = (h0 , h1 , ....hn ) identi?es then the position of the wavelet in the cascade through the sequence of its ancestors: the integer hn labels the hn -th daughter wavelet generated at the n-th step in the cascade, by wavelet hn?1 . Notice, however, that an intermittent random ?eld could be generated, without any reference to cascade processes, either by varying appropriately the space density of the wavelets with scale, or by making the distribution of the amplitudes A more intermittent as k grows, but keeping the wavelets randomly distributed in space [11]. From reality of Ψ, for each index hn with positive components, there is a wavelet with index ?hn such that A?hn = A? n yhn = y?hn and khn = ?k?hn . The cascade is assumed to be local in h n in the sense that the probability that a given wavelet hn has a certain value of its parameters ξhn ≡ {ln Ahn , ln khn , yhn }, can be written in terms of transition probabilities, as: P (ξh0 → ξhn ) = dξh1 ...dξhn?1 p(ξh0 → ξh1 )...p(ξhn?1 → ξhn ). (3)

For the sake of simplicity, the transition probabilities are assumed to factorize into their ln A, ln k and y components, with scale invariance forcing the cascade to be governed by a multiplicative random process: p(ξ → ξ ′ ) = pA (A′ /A|k ′ /k)pk (k ′ /k)px (k|y ? y ′ |). (4) The relative phase of A′ and A is assumed random, and we take: d ln y pA (y|x)y p = cp x?ζp , (5)

in order to get power law scaling in the structure functions. Given Eqns. (4-5), the transition probability over n steps is in the form: P (ξhn → ξhp ) = PA (p ? n, Ahp /Ahn |khp /khn )Pk (p ? n, khp /khn )Px (khn |yhp ? yhn |). (6)

? ? At each step n in the cascade, the wavelets distribution in scale is peaked at kn = k0 exp(n?), with z ?0 distributed around a characteristic large scale L: k0 = L?1 . Each mother the wavevectors k 3

wavelet generates exp(?) daughters; this insures that the mean degree of overlap between wavelets z in k-y space is scale invariant. There are several reasons to consider a mechanism of turbulence synthesis, in which the wavelets are distributed in space and scale in a probabilistic way, rather than on a rigid lattice. The main, rather ”philosophical” motivation, however, is to try considering the building block wavelets, more like eddies (or components of bunches of eddies if aS is large), than like basis functions at ?xed position in space; this also in view of possible extensions of the model to time dependent situations, in which the eddies are mobile. III. Analysis of an arti?cial turbulent signal The choice of analyzing wavelet is in general arbitrary. A Gaussian wavepackets, however, has the minimum spread in k ? y space and allows to retain the maximum simultaneous information possible in space and scale. We thus take from the start the analyzing wavelets to be derivatives of s Gaussian wavepackets: (?k)?s ?x wA (k, y, x), for which in general: aA = λk = aS . The components of Ψ(x) with respect to this set of wavepackets are de?ned as follows:
?1 Ψky = λA k ?s ? s dx wA (k, y, x)?x Ψ(x).


together with the associated structure functions |Ψky |q . In order for these structure functions not to be dominated by the largest scales in Ψ(x), given standard Kolmogorov scaling for Ψ, it is necessary that the parameter s in Eqn. (7) be at least equal to one. To calculate them , we need to evaluate ?rst the component of a building block wavelet on an analyzing wavelet. Because of random phase of A, we will need only its square modulus: C(k, y; k ′ , y ′ ) = ? m ? |k ?m dx wA (k, y, x)?x wS (k ′ , y ′ , x)|2 . If the probabilities Pk vary su?ciently slow, and s is large enough to kill the contribution to the k-integrals from k small, it is possible to write from the start: C(kA , yA ; kS , yS ) = aA aS ?k 2 2k 2 a2 S exp ? 2 A 2 ?y 2 + 2 a2 + a2 aA + aS 4kA A S (8)

with ?k = kA ? kS and ?y = yA ? yS , and the integrals over k, that arise in the averages involved in the correlations can be carried out by steepest descent, disregarding the contribution at k ? L?1 . The simplest correlations are |Ψky |2 and |Ψky |2 |Ψk′ y′ |2 . Given the random phase of A, it is easy to see from Eqns. (2) and (8), that a 2n-order correlation will receive contribution at most by n eddies, and the 4-th order correlation will be in the form: |Ψky |2 |Ψk′ y′ |2 = |Ψky |2 |Ψk′ y′ |2 =
n 1

+ |Ψky |2 |Ψk′ y′ |2


Fn |Ahn |4 +

Gnp |Ahn |2 |Ahp |2 .


From the form of pA , we have power laws for the amplitude correlations; |Ahn |2 = c2 (khn L)?ζ2 and |Ahn |2 |Ah′ |2 = c4 (kh′ L)?ζ4 (khn /kh′ )?ζ2 (kh′ /khp )ζ4 ?2ζ2 m m m m h′ m (11) with khn > kh′ and p the cascade step at which the genealogical tree of hn and branches: m hi = h′ for 0 ≤ i ≤ p and hi = h′ for i > p. Thus, the lower the branching takes place in the tree, i i the closer the correlation gets to its disconnected limit |Ahn |2 |Ah′ |2 = c4 (kh′ khn L2 )?ζ2 . m m From Eqns. (2) and (7) we obtain, for the second order correlation: |Ψky |2 = c2 ? d?d ln k y


? ? ? ? ? kn Pk (n, k/k0 ) (kL)?ζ2 C(k, y; k, y),



? where Pk (n, k/k0 ) is averaged over k0 (from summing over h0 in hn ), and C(k, y; k ′ , y ′ ) the square modulus of the component of wavelet wS (k ′ , y ′ , x) with respect to the analyzing wavelet ? m ? wA (k, y, x): C(k, y; k ′ , y ′ ) = |k ?m dx wA (k, y, x)?x wS (k ′ , y ′ , x)|2 . We have an analogous expression for the one-eddy contribution to |Ψky |2 |Ψk′ y′ |2 : |Ψky |2 |Ψk′ y′ |2 = c4 ? d?d ln k y
n 1

? ? ? ? ? ? ? ? kn Pk (n, k/k0 ) (kL)?ζ4 C(k, y; k, y)C(k ′ , y ′ ; k, y).


For the two-eddy contribution, we have instead, for k ′ ≥ k: |Ψky |2 |Ψk′ y′ |2 ×
n m=0 p=0 2

= 2c4
n m

? ? y y d ln kd ln k ′ d?d?′

ln k ln L?1

? ? ? ? ? ? d ln k (kL)?ζ4 (k ′ /k)?ζ2 (k/k)ζ4 ?2ζ2

? ? kn km ? ? ? ? ? ? Pk (p, k/k0 ) Pk (n ? p, k/k)Pk (m ? p, k ′ /k) ? kp (14)

?y ? ? ? ? ? ×Px (k|? ? y ′ |)C(k, y; k, y)C(k ′ , y ′ ; k ′ , y ′ ).

? ? ?y ? where (km /kp )Px (k|? ? y ′ |) gives the space density at y ′ of eddies h′ generated from the branching ? m ? ? at hp , given the presence of an eddy hn at y . The factor km /kp = exp((m ? p)?) is the actual ? z ? number of eddies h′ generated from the branching at hp . The factors kn entering Eqns. (12-14), m ? conversely, give the space density of wavelets of typical size aS /kn , at the n-th step in the cascade. The cascade structure is characterized by a discrete component through the sums entering Eqns. (12-14). Since the cascade steps are independent, the width of the cumulative distribution 1 Pk (n, k ′ /k) is n 2 ?z with ?z the width of pk : ?z 2 = d ln k ′ | ln k ′ /k ? z |2 pk (k ′ /k). Thus, if the ? separation of the scales entering the Pk involved in the sums in Eqns. (12-14) is large enough, the e?ect of discreteness will be negligible. The same will occur if ?z/? itself, is large enough. In the z other limit, when ?z/? and n are small, oscillation with period z in ln k ′ /k and ln kL (lacunarity) z ? are to be expected in the scale dependence of the correlations. IV. Structure functions and optimization of the analyzing wavelets The length λS introduced in Eqn. (1) plays an important role in the generation of an intermittent random ?eld, since it identi?es the intermittency scale of ?uctuations of size k ?1 . If aS is large, indeed, it is not the single ?uctuation that is intermittent, but the amplitude of a whole bunch of them, extended over a length λS . As a consequence of this, one expects that, if λS approaches the size of the domain, intermittency is lost and a standard, random phase generated gaussian ?eld is obtained. Conversely, if one carries on the same operation with the length λA , the ?nal result is that, instead of dealing with structure functions, one ends up working with correlations between Fourier components of the random ?eld. Again, intermittency is expected to be lost. If one is interested in studying intermittency by structure function scaling, the optimal choice should be therefore: λA ? λS . For this reason, it becomes necessary to study the dependence of the structure functions |Ψky |p on the parameters aA and aS . If ?z/? is not too small, discreteness e?ects in scale can be neglected and the sums in Eqns. (12z 14) can be approximated by integrals. The transition probability Pk is in the form: Pk (n, k ′ /k) = f (n, ln k ′ /k ? n?) ? f (??1 ln k ′ /k, ln k ′ /k ? n?). Hence: z z z

? ? (kn /k)Pk (n, k/k0 ) ?

dx??1 ex f (??1 ln k/k0 , x) = z z

1 k z k0 ? ?




1 z where ? = limn→∞ n? ln exp(x)|n = O(??3 ?z 2 ) and ...|n indicates average over f (n, x). We z thus see that scale uncertainty in the process of eddy generation contributes to scaling in such a ′ way that ζq → ζq = ζq + ?.


At this point, all the terms in Pk and associated sums drop o? Eqns. (12-14) and it is possible to evaluate the integrals for k = k ′ and y = y ′ . From Eqns. (12-13), we get immediately: |Ψky |
′ c2 πaS (kL)?ζ2 ? z aA ?


|Ψky |



c4 πa3 (kL)?ζ4 S ? z aA (a2 + a2 ) ? S A


? Performing some power counting on Eqn. (14), however, we discover that the dk integral is ′ ′ dominated more and more by large scales the closer we are to trivial scaling: ζ4 = 2ζ2 . In this regime, if px is su?ciently well behaved, it will be possible to approximate Px , from central limit theorem arguments, with a Gaussian: Px (k|x ? x′ |) ? k k 2 |x ? x′ |2 exp ? 1 ?2 a2 π2? S ba b S (17)

with the parameter ? characterizing the spatial non-locality of the cascade. Substituting into Eqn. b (14), we obtain: |Ψky |4 = |Ψky |4 1 + |Ψky |4 2 ?
′ aS aA 2π 2 c4 πa2 ?ζ4 S 2 (kL) 2 + a2 + 2 ? z aA ? aS z baS ? A 1

ln kL

dx 1 +

a2 + a2 ?2x A S e ?2 a2 b

1 ?2

e(ζ4 ?2ζ2 )x


This equation is our main result, and tells us how the various eddies in the synthetic turbulent ?eld, contribute to structure function scaling. The second term in square brakets in Eqn. (18) is the two-eddy part of |Ψky |4 , giving ? for each x = ln k/k, the logarithmic distance of the common ancestor of the two eddies from ′ ′ the scale k. For ζ4 ? 2ζ2 small, which is true in the case of turbulence, this integral receives a2 +a2 1 A contributions from max(0, 2 ln ?2 a2 S ) < x < ln kL, where the integrand can be approximated by b

′ ′ ′ ′ exp((ζ4 ? 2ζ2 )x). We have then the important result, that for kL < exp((2ζ2 ? ζ4 )?1 ), which is a very large range of scales, the structure function contains a logarithmic two-eddy contribution, which comes right from the largest scales in the random ?eld. We obtain then, for the kurtosis K4 (k, aS , aA ) = |Ψky |2 ?2 |Ψky |4 :
′ ′ c4 z ? aS aA 2π 2 R2ζ2 ?ζ4 2ζ2 ?ζ4 ln RkL 2 (kL) 2 + a2 + πc2 aS z 2? S ? ba A 1 ′ ′

K4 (k, aS , aA ) ?
?2 a2 b
A 1


′ ′ S where: R = max(1, ( a2 +a2 ) 2 ). Only for kL ? exp((2ζ2 ? ζ4 )?1 ) we reach pure power law scaling:
S ′ ′ aS aA 2π 2 R2ζ2 ?ζ4 c4 z ? + K4 (k, aS , aA ) ? 2 (kL)2ζ2 ?ζ4 ′ ′ πc2 a2 + a2 z 2? S (2ζ2 ? ζ4 ) ? ba S A 1 ′ ′


If, instead of looking at the scaling of K4 (k, aS , aA ), we study the dependence of this quantity on aA for k ?xed, we notice the presence of a maximum at aA = aS . This corresponds to the maximum possible overlap between building block and analyzing wavelets; for aA > aS , an analyizing wavelet will feel the e?ect of many eddies at di?erent position in space, while, for aA < aS , these will be distributed at di?erent scales. It is important to notice, as it is clear from Eqn. (19), that this e?ect will be felt also in the measured scaling exponents, that, because of the logarithm, will be dependent on aA . This e?ect will be minimum only at aA = aS , when the local in scale, one-eddy contribution to the structure function is maximum. Conversely, the importance of the two-eddy contribution goes to zero when aS is large. ′ ′ It is important to stress the importance of the smallness of 2ζ2 ? ζ4 and of the cascade structure of the random ?eld. This causes the slow decay of the two-eddy contribution as aA gets large. In 6

a random eddy model of the kind considered in [11], the two-eddy contribution would scale like ′ (kL)?2ζ2 whatever the value of aA , the reason being the lack of correlations among eddies. When aA /aS becomes large enough, the one-eddy contribution can be disregarded, and the kurtosis K4 begins to scale in R, i.e. in the ratio λS /λA , so that we have, to leading order in k: K4 (k, aS , aA ) ? (L/λA )2ζ2 ?ζ4
′ ′


In fact, for aA large, wA probes eddies coming from common ancestors, which can be very distant in scale from k, and which become uncorrelated when λA > L. From Eqn. (21), |Ψky |4 obeys ′ trivial scaling: |Ψky |4 ? (kL)?2ζ2 and the intermittent nature of the random ?eld is lost. In this regime, the ratio of the one- to two-eddy contribution to |Ψky |4 scales, again to leading order in k: ′ ′ |Ψky |4 1 (22) ? (kL)?1?ζ4 +2ζ2 |Ψky |4 2 The factor (kL)?1 in this equation has a very important interpretation when we consider that, for ?xed λA ? L, and kL large, λA Ψky → Ψk , which is the Fourier transform of Ψ(x) in a box of size ′ L. We have then, that Eqns. (21-22) take the form: |Ψk |4 ? aLk ?3 (Lk)?ζ4 + bL2 k ?2 (Lk)?2ζ2 which is nothing else than the expression Ψk1 Ψk2 Ψk3 Ψk4 = δ(k1 + k2 + k3 + k4 )C4 (k1 , k2 , k3 , k4 )+ δ(k1 + k2 )δ(k3 + k4 )C2 (k1 )C2 (k3 ) for k1 = ?k2 = k3 = ?k4 with δ(0) ? L. Thus, |Ψky |4 2 kill |Ψky |4 1 by a factor kL that is the term kδ(0) that gives the ratio of the disconnected to the connected Fourier 4-point correlation of a uniform random ?eld. All this suggests that, perhaps, structure functions are not the most appropriate objects to probe features of the ?eld which are local in scale; rather, the multiscale Fourier correlation: Ψk1 Ψk2 Ψk3 Ψk4 with ki = ?kj ? i, j and k1 + ... + k4 = 0 should be taken into consideration: Ψk1 ...Ψk4 ? 2πδ(k1 + ... + k4 ) π 2 a4 S z ?
′ a2 dk (kL)?ζ4 exp ? S2 k4 4k


(kn ? k)2 .


Only in this way, it would be possible to avoid logarithmic corrections in the scaling of correlation functions, coming from two-eddy e?ects. V. Conclusions The motivation for the interest in synthetic turbulence has often been, more in the ”output”, i.e. in the turbulent ?eld or turbulent signal being produced, than in the dynamical meaning of the adopted algorithm. A typical application has been, for instance, the possibility of controlling velocity spectra, in the study of turbulent di?usion [15]. When it comes to an issue like intermittency, however, the interest is more in the algorithm itself and in the e?ect that di?erent choices for it, would have on the turbulent statistics. In the present research, the main result is the ease, with which some innocent looking algorithms for the generation of synthetic turbulence, lead to strongly non-local e?ects in structure function scaling. This, despite the local nature of the cascade mechanism on which the algorithms are based. The only way this phenomenon can be explained is through the interaction of the cascade nature of the algorithm, with the probabilistic distribution in space and scale of the eddies. In particular, had the eddies been distributed at random, without a cascade structure, the two eddy contribution ? ? y y ? ? ? ? to a 4-th order structure function, would have been trivially: dkdk ′ d?d?′ C(k, y; k, y )C(k, y; k ′ , y ′ ) ? ? ?? ×(k k ′ )?ζ4 /2 ? k ?ζ4 , with C(k, y; k, y) the square wavelet component of an eddy [see Eqn. (9)]. On the other hand, if the cascade had been rigid, organized on a lattice structure as in [9], this ? ? ? contribution would have been simply N C(k, y; k, y )2 k ?ζ4 ? k ?ζ4 , with N the number of wavelets ?y wS in the lattice, overlapping with the wavelet wA , and k? the typical scale and coordinate of the wavelets wS .


All these e?ects manifest themselves in the behavior of structure functions, through logarithmic scaling corrections carrying information on the largest scales of the signal. One result of the present study could therefore be that, if one wanted simply to generate a random ?eld with prescribed multifractal statistics, it would not be a good idea to include probabilistic e?ects in the space and scale distribution of eddies. If, on the other hand, one thinks that all this may have some relevance for real turbulence, the conclusion is that there exists a kinematic e?ect, which could limit the ability of objects like structure functions, to detect scale-local features of the turbulent ?eld. This con?rms the current thinking on the subject, which leads to expect non-locality, among the other things, from the convolution nature, in Fourier space, of correlations of order greater than two. It is worth stressing the kinematic nature of our result, which is due to the way in which contributions from di?erent eddies sum up in structure functions: the dynamics of the eddy generation mechanism remains strictly local in scale. The question at this point is how to verify that the non-locality e?ects we have discussed are of any relevance in high Reynolds numbers turbulence. An experimental test could be the dependence of the wavelet structure function scaling exponents, on the parameter aA : the number of wiggles of the wavelet. In this case, the suggestion from the present research, is that there should be a value of aA , for which non-local e?ects become minimal, and which is identi?ed by the maximum at ?xed scale of the generalized kurtosis [see Eqns. (19-20)]. This maximum has a nice physical interpretation in terms of resonance between the analyizing wavelet and intermittency, with the wavelet dominant wavevector identifying the scale of the turbulent ?uctuations, and the wavelet extension giving the lengthscale over which these ?uctuations, act coherently to generate intermittency. If the picture of a probabilistic cascade is right, however, only a description based on Fourier correlation scaling, could allow elimination of this kind of non-local e?ects, once all disconnected contributions to the correlation, are eliminated by appropriate choice of the wavevectors. The results of this study are of rather general validity, showing that there is a rather broad class of intermittent random ?elds, for which techniques based on the analysis of structure function scaling, are of limited use. If the interest is more in the modelling of turbulence intermittency, however, our non-locality e?ects may be considered more as an artifact of the algorithm of turbulence generation. On the other hand, it is di?cult to a priori exclude a probabilistic cascade, as opposed to a ”rigid” one, and several arguments in favor and against both are easy to ?nd. A drawback of a probabilistic cascade is that the eddies can overlap in k ?y space, while being treated as independent objects in the random multiplicative process. One may argue back, however, that we are dealing with a one dimensional section of a three-dimensional turbulent ?eld, and that these overlaps are therefore irrelevant. Conversely, it is more aesthetically pleasing to distribute the wavelets in the random ?eld, freely in k ? y space, but then one looses the property of the wavelets, of being base functions for the random ?eld. In any case, there are practical reasons for being interested in probabilistic cascades. One is the possibility of studying the e?ect of space and scale non-locality in the eddy generation [16], which have a direct interpretation in terms of properties of the energy transfer in real turbulence. The second is, that the lack of constrains over the eddy position allows, in a time dependent situation, to put these ”eddies” in motion, accounting for sweep in time correlations, in a much easier way than using ?xed wavelets. Aknowledgements: I would like to thank Jean-Fran?ois Pinton, Sergio Ciliberto and Jens Eggers c for interesting and stimulating discussion. Appendix: The e?ect of discreteness in the cascade For the sake of completeness, we examine the discrete limit of Eqn. (14), analyzing how, restriction of the phase space available to the eddies at their birth, modi?es the scale non-local character of the struture functions.


An argument analogous to the one used to arrive at Eqn. (17), leads to the expression for Pk : Pk (n; k ′ /k) = 1 (nπ) ?z
1 2

exp ?

(ln(k ′ /k) ? n?)2 z . n?z 2


The continuous approximation, used to arrive from Eqns. (12-14) at Eqns. (16) and (18), can still be applied to the sums in n of Eqns. (12-13) and in p of Eqn. (14), also for ?z/? small. z This, provided ?z 2 ln kL/? > 1. The e?ect of discreteness remains therefore in the sums over z ? ? n and m in Eqn. (14), which, for ?z/? small, are dominated by n = p + Int(??1 ln k/k) and z z ?1 ?′ /k). From Eqn. (14), we thus obtain the following limit expression for ? m = p + Int(? ln k z |Ψky |4 2 : ′ ′ ? k ? ? ′ ? ? ′ ? ? 2c4 dk (kL)?ζ4 (k ′ /k)?ζ2 (k/k)ζ4 ?2ζ2 ? ? y y |Ψky |4 ? dkdk ′ d?d?′ ? ? ? ? 1 ? π?z 2 (ln(k/k) ln(k ′ /k)) 2 L?1 k 2 ?2 k ?2 k′ z3 ? ? ? k? k? ?y ? ? ? ? ? ×Px (k|? ? y ′ |)C(k, y; k, y)C(k, y; k ′ , y ′ ) exp ? + 2 ? k ln k ′ /k ? ? ? ?z ln k/ (A2)

where ?kk′ = z ?1 ln k ′ /k ? int(??1 ln k ′ /k) is the decimal part of z ?1 ln k ′ /k. This equation di?ers ? z ? from the corresponding formula for the continuous limit, Eqn. (18), because of the exponential term ? in ?kk and ?kk′ . This term has a fast dependence on k, which can be treated by decomposing the ?? ?? ? dk ? ? n? ? d?kk where the integration limits in d?kk are approximated dk integration as: ?? ?? ? ? z k kk by ±∞ thanks to the smallness of ?z. After the Gaussian integral in d?kk is carried out, the ?? ? remaining sum can be approximated back to an integral: z n? ? ? dk . This means that in Eqn. ? ? k kk (A2) we can substitute: exp ? ?2 k ?2 k′ z3 ? ? ? k? k? + ? ? ? ? ?z 2 ln k/k ln k ′ /k → π?z 2 z ln kk ′ /k 2 ? ?? ? ? ? ? ? ln(k/k) ln(k ′ /k)
1 2

exp ?

z 3 ?2 k′ ? k? ? . 2 ln k k ′ /k 2 ?? ? ?z


Substituting into Eqn. (A2) and using Eqn. (17) we obtain: |Ψky |4 ? 22 π 3 z 2 ?z? S ? ba
1 ′ aS aA c4 πa2 ?ζ4 S 2 (kL) 2 + a2 z aA ? aS A 1 ?2

ln kL


dx 1 +

a2 + a2 ?2x A S e ?2 a2 b


z (a2 + a2 ) ? A S 2a2 a2 ?z 2 A S

1 ?2

e(ζ4 ?2ζ2 )x


This expression di?ers from the continuous limit described by Eqn. (18), because of the factor z (a2 +a2 ) ? S (x + 2a2 A 2 ?z2 ). If the discrete approximation must work over the whole domain of integration a
A S S in x, it is necessary however that 2a2 A 2 ?z2 > ln kL. (Incidently, this tells us that this limit is of A aS scarce practical interest in a probabilistic cascade, since it is not particularly interesting to have an error k?z in the cascade step, which is much smaller than the wavelet spectral width k/aS ). In ′ ′ the ?z → 0 limit we get then the following expression for the kurtosis, for ln RkL < (2ζ2 ? ζ4 )?1 : ′ ′ aS aA 2πaS aA R2ζ2 ?ζ4 c4 z ? . K4 (k, aS , aA ) ? 2 (kL)2ζ2 ?ζ4 1 ln RkL 2 + a2 + πc2 aS z 2 ? 2 + aS ) 2 ? b(aA A ′ ′

z (a2 +a2 ) ?


Logarithmic corrections to scaling remain therefore, also in the discrete limit.


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