large-scale lognormality in turbulence modeled by ornstein-uhlenbeck process

7/30/2019 Large-scale lognormality in turbulence modeled by Ornstein-Uhlenbeck process

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arXiv:1212

.5314v1

[nlin.CD]

21Dec2012

Large-scale lognormality in turbulence modeled by Ornstein-Uhlenbeck process

Takeshi Matsumoto1, and Masanori Takaoka2

1Division of Physics and Astronomy, Graduate School of Science, Kyoto University,Kitashirakawa Oiwakecho Sakyoku Kyoto, 606-8502, Japan.

2Department of Mechanical Engineering, Doshisha University,Kyotanabe, Kyoto, 610-0321, Japan.

(Dated: December 24, 2012)

Lognormality was found experimentally for coarse-grained squared turbulence velocity and veloc-ity increment when the coarsening scale is comparable to the correlation scale of the velocity (Mouriet al. Phys. Fluids 21, 065107, 2009). We investigate this large-scale lognormality by using a simplestochastic process with correlation, the Ornstein-Uhlenbeck (OU) process. It is shown that the OUprocess has a similar large-scale lognormality, which is studied numerically and analytically.

PACS numbers: Valid PACS appear here

I. INTRODUCTION

The lognormal distribution appears in a wide range ofnatural and social phenomena (see, e.g., [1]). In fluid tur-bulence, it is well-known that the distribution is an im-

portant consequence of the Kolmogorovs 1962 theory formodeling fluctuations of the energy cascade rate acrossscales [2]. In this 1962 theory, we have a very clear picturewhy the lognormal distribution was the first candidate ofthe cascade fluctuations in his refined phenomenology.Namely, the energy cascade at high Reynolds numbercan be modeled as a multiplicative process consisting of alarge number of independently and identically distributedrandom variables. This large number is important for thecentral limit theorem to be applicable to the logarithmof the multiplicand.

There is a different example of lognormally distributedvariables in turbulence. Laboratory experiments of tur-

bulent boundary layers in 1980s suggest that spanwiseseparations between the low-speed streaks follow the log-normal distribution [3, 4]. In this case, the underlyingmechanism of the lognormally distributed streaks is notas clear as in the Kolmogorov 1962 phenomenology be-cause a multiplicative structure for the streaks is notfound immediately.

In this paper we consider yet another lognormal ex-ample in turbulence, which was recently found experi-mentally [5, 6]. In Ref.[6], it is observed that the coarse-grained squared velocity and squared velocity incrementsover distance r

u2

R(xc) =

1

Rxc+R/2xcR/2 u

2

(x)x. , (1)

u2r,R(xc) =1

R rxc+R/2xcR/2

[u(x + r) u(x)]2x. (2)

are lognormally distributed when the averaging scale Ris set to R Lu. Here Lu is the correlation length

Electronic address: [email protected]

defined as the integral scale of the velocity correlationfunction u(x + r)u(x) and the velocity u can be eitherlongitudinal or transverse velocity component. Since thelognormal property holds when R is comparable to thelarge scale Lu, this property is called large-scale lognor-

mality. For the asymptotic regime R Lu, as a usualresult of the central limit theorem, the distribution of thecoarse-grained quantities become closer and closer to nor-mal (Gaussian) distribution. Notice that the large-scalelognormality, which we consider here, concerns the rangeof the coarse-graining scale R Lu, which is differentfrom the final Gaussian state in R Lu.

The large-scale lognormality was observed in the ex-periments of grid turbulence, turbulent boundary layers,and turbulent jets, suggesting its universality [6]. Butagain as in the case of the streaks in the boundary layerturbulence, why the coarse-grained turbulence data arelognormally distributed is not clear. Our final goal is to

understand its mechanism why they are so.As a first step to this goal, we address the follow-ing question: Is the large-scale lognormality a generalproperty of correlated random variables, not restrictedto large-scale fluctuations of turbulence? The previousresults [6] lead us to recognize that this balance betweenR and L is most important for the lognormality. In orderto answer the question, we use the Ornstein-Uhlenbeckprocess (OU process) as a simplest way to generate cor-related random variables with the correlation length Land check whether or not the coarse-grained quantitieslike Eqs.(1) and (2) follow lognormal distributions in therange R/L 1. Our numerical results suggest that theanswer to the question is yes. Then we study, by analyt-ical calculations of moments, further details on how theOU data become close to the lognormal variables.

We believe that this simple approach using the stochas-tic process has some value since reproducing the large-scale lognormality by direct numerical simulations of theNavier-Stokes equations can be computationally quite ex-pensive. It requires a very large domain size as comparedwith the correlation length.

In statistics, taking logarithm of a positive randomvariable is known as a common way of symmetrizing
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transformation, which makes the skewness of the trans-formed variable closer to zero [7]. In this language, thelarge-scale lognormality suggests that, even with correla-tion, the log transformation can produce near Gaussianbehaviour successfully already when the averaging scaleR is of the order of the correlation scale L. The presentstudy can be interpreted as a model study how the logtransformation begins to work for the correlated random

variable by changing the coarse-graining scale.This papers organization is the following. In Sec. II,

we present numerical data of the OU process reproducingthe large-scale lognormality. We then in Sec. III provideanalytical calculations of moments of the OU process tostudy how they are close to those of the lognormal dis-tribution. We provide summary and discussion in Sec.IV.

II. NUMERICAL EXPERIMENT OF THEORNSTEIN-UHLENBECK PROCESS

We begin here by listing basic properties of the OU pro-cess (see, e.g., [8] and also [9] in the context of turbulencemodeling). The process is described by the Langevinequation

X. (t)

t.= 1

TLX(t) +

2 (t), (3)

where the Langevin noise (t) is a Gaussian white noisehaving the ensemble-averaged mean and variance

(t) = 0, (t) (t) = (t t). (4)

The linear addition of the Gaussian uncorrelated noise

(t) makes the solution X(t) to Eq.(3) to be a Gaus-sian random variable. However the first term of the righthand side brings temporal correlation. Namely, by writ-ing the initial position as x0, X(t) is characterized asa correlated-in-time Gaussian random variable with themean and variance

X(t) = x0et/TL, (5)[X(t) X(t)]2 = TL(1 e2t/TL). (6)

The correlation function can be calculated analyticallyas

[X(t) X(t)][X(t + s) X(t + s)] =

[X(t) X(t)]2e|s|/TL . (7)

Therefore the integral scale of the OU process is given as0

[X(t) X(t)][X(t + s) X(t + s)][X(t) X(t)]2 s. = TL.(8)

If the large-scale lognormality observed in turbulence[6] is a property of correlated random variables, it is then

107

106

105

104

103

102

101

100

-6 -4 -2 0 2 4 6

PDF

(z)/

(a)

107

106

105

104

103

102

101

100

-6 -4 -2 0 2 4 6

PDF

(z)/

(b)

FIG. 1: (Color online) Probability density functions (PDF)of (a) X2R (Eq.(9)) and (b) its logarithm ln X

2

R. Therandom variables are normalized to have zero mean andunit standard deviation. The averaging scales here areR = 1000TL, 100TL, 10TL, TL, TL/10, which correspond to thecurves from top to bottom. The PDFs are shifted verticallyby being multiplied by factor 0.25 for clarity. The solid curvedenotes the Gaussian distribution.

107

106

105

104

103

102

101

100

-6 -4 -2 0 2 4 6

PDF

(z)/

(a)

107

106

105

104

103

102

101

100

-6 -4 -2 0 2 4 6

PDF

(z)/

(b)

FIG. 2: (Color online) Same as Fig. 1 but for thesquared increment (a) X2r,R (Eq.(10)) and (b) its loga-

rithm ln X2r,R. Here r = TL/100. The averaging scales areR = 1000TL, 100TL, 10TL, TL, TL/10, which correspond to thecurves from top to bottom.

likely that the following coarse-grained data of the OUprocess

X2R(t) =1

R t+R/2

tR/2

X(t)2t., (9)

X2r,R(t) =1

R rt+R/2tR/2

[X(t + r) X(t)]2t., (10)

are lognormally distributed when the averaging scale Ris in the similar range, such as 1 R/TL 100. Equa-tions (9) and (10) correspond to the turbulence quantitiesEqs.(1) and (2), respectively.

We now check whether or not the random variables(9) and (10) follow lognormal distributions by doing nu-merical simulation of the OU process (3). We use the


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numerical method called exact updating formula of theOU process proposed in Ref.[10] with the parametersx0 = 0.0, TL = 1.0, = 0.50, t = 1.0 103. Theintegrals in Eqs. (9) and (10) are numerically calculatedas

X2R(tn) =1

NR

NR/2

j=NR/2

X(tn+j)2, (11)

X2r,R(tn) =1

NR

NR/2j=NR/2

[X(tn+j + tNr) X(tn+j)]2,

(12)

where NR = R/t, Nr = r/t and tk = kt. Fig-ures 1 and 2 show probability density functions (PDF)of X2R and X

2r,R with or without taking logarithm for

various averaging scales R. The number of samples forthe R = 1000TL case is 1.34 106 (for smaller R cases,the number is much larger). For the largest values ofR = 1000TL shown here, the PDFs are close to Gaussiandistributions as a consequence of the central limit theo-rem. As we decrease the averaging scale R to the correla-tion scale TL, the distributions of X2R and X

2r,R deviate

from the Gaussian distribution. In contrast, the log vari-ables ln X2R and ln X

2r,R remain to be nearly Gaussian

as shown in Figs.1(b) and 2(b). Hence X2R and X2r,R are

lognormally distributed in this range ofR. This behavioris similar to the turbulence data analyzed in Ref.[6]. Inaddition, for the squared increments X2r,R with variousr < TL, qualitatively same results are obtained.

The approach to the lognormal distribution can be ob-served more quantitatively by looking at how the skew-nesses and flatnesses of the log variables ln X2R, ln X

2r,R

change as functions of R (the skewness S(z) and flatnessF(z) of a random variable z are defined as S(z) = (z z)3/[V(z)]3/2, F(z) = (zz)4/[V(z)]2, where V(z)is the variance (z z)2). In Fig. 3, it is seen that themoments of the variables with logarithm ln X2R, ln X

2r,R

approach already around R = TL to the values of theGaussian distributions (S, F) = (0, 3) whereas the mo-ments of the variables without logarithm are still differentfrom them. This is consistent with the behavior observedin Figs.1 and 2.

For the increments X2r,R, the skewness and the flat-ness can depend on the difference r. Indeed a clear r-dependence is seen in Fig.4. However, the fast conver-

gence to the Gaussian of the ln X2r,R around R L isnot affected by this dependence. In fact, these graphs ofthe different rs can be collapsed to one curve by normal-izing R with a different correlation scale from TL as wewill show at the end of the next section.

In summary, we observe that the large-scale lognor-mality holds also for the OU process as in the turbulencecase [6]. Further details on the behavior of the skewnessand flatness of the OU process are studied analyticallyin the next section.

-1

0

1

2

3

102 101 100 101 102 103

S

R /TL

(a)

X2R

lnX2R

0

1

102 101 100 101 102 103

S

R /TL

(b)

X2r,R

lnX2r,R

0

5

10

15

102

101

100

101

102

103

F

R /TL

(c)

X2R

lnX2R

2

3

4

5

6

7

102

101

100

101

102

103

F

R /TL

(d)

X2r,R

lnX2r,R

FIG. 3: (Color online) The skewness of X2R (a) and of X2

r,R

(r = TL/100) (b); the flatness ofX2R (c) and ofX2

r,R (d) withor without logarithm. They are plotted versus the averagingscale R. The horizontal lines correspond to the skewness andthe flatness of the Gaussian distributions. The thick curvesare asymptotic expressions of the moments of X2R and X

2

r,R,Eqs.(2526) and (2829). The dashed curves correspond theexact moment expressions (not involving any asymptotic ar-gument).

0

0.5

1

1.5

102

101

100

101

102

103

S

R /TL

(a)

X2r,R(r=TL/100)lnX2r,R(r=TL/100)

X2r,R(r=TL/50)lnX2r,R(r=TL/50)X2r,R(r=TL/10)

lnX2r,R(r=TL/10)

2

3

4

5

6

7

102

101

100

101

102

103

F

R /TL

(b)



lnX2r,R(r=TL/10)

FIG. 4: (Color online) The skewness (a) and flatness (b) ofthe increments X2r,R and ln X

2

r,R for three different valuesof r: r = TL/10, TL/50, TL/100.

III. ANALYTICAL EXPRESSIONS OF THE

MOMENTS OF THE ORNSTEIN-UHLENBECK

PROCESS

The OU process is amenable to analytical calculation

in many ways. However it is difficult to determine an-alytically the PDFs of the quantities of our interest,Eqs.(9) and (10), or all their moments, since they in-volve integrations. Instead we calculate the low-ordermoments, namely the skewness and the flatness of X(t)2Rand X(t)2r,R. For them, full analytic results are obtainedbut we present only the asymptotic expressions of themfor R/L 1 since they are sufficient for our purpose.

Then we compare them with the expressions of theskewness and flatness of the lognormal distribution with


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the mean e and the variance v:

SLN =

(3 + ), (13)

FLN = 3 + (16 + 15 + 62 + 3), (14)

where = v/e2. As we have seen, the PDFs of thevariables X(t)2R, X(t)

2r,R become close to the lognormal

distributions for R/TL 1. We study this behavior bycomparing the skewness and flatness of X(t)2R, X(t)2r,Rwith SLN and FLN. However, as we shall see below, itturns out that small corrections exist, which may be hardto be detected in numerical calculations in the previoussection.

In our analytical calculation of the moments, we firstput the formal solution of the Langevin equation (3)

X(t) = et/TLx0 +

t0

e(ts)/TL

2(s) s.. (15)

into the coarse-grained variables Eqs.(9) and (10). Forthe correlation or the variance, we take a suitable powersof them and take the ensemble average with respect to theLangevin noise by using Eq.(4). So that we can reducetheir calculations into multiple integrals of certain expo-nential functions. For the skeweness and the flatness, weuse more compact integral representations proposed inRef.[11], which is explained in Appendix. To calculatethe resultant integrals we use a symbolic computationsoftware, Maple. The integrated result still contains de-pendence on time t. For example, the mean of X(t)2Rreads

X(t)2

R = TL 1 +TL

2R (e

R/TL

eR/TL

) e

2t/TL.(16)

Remember that this time t is much larger than the cor-relation time TL to be consistent to the simulations inSec.II. We hence regard et/TL = 0. From now on, wedrop the dependency on t of all the moments of X(t)2Rand X(t)2r,R calculated here. This does not affect ourstudy since the OU process becomes steady for t TL.

We now list the expressions of the moments. Here wewrite = R/TL for brevity. For X

2R, the mean, variance,

skewness, and flatness are respectively,

E(X2

R) = TL, (17)

V(X2R) = (TL)2

21 + 2(e2 1) , (18)S(X2R) = 1 2W

3/20 [e

2 1 + (e2 + 1)], (19)F(X2R) = 3 + 6W

20 [e

4 + 28e2 29+162e2 + 40e2 + 20], (20)

where W0 = 21+2(e21). For the coarse grained

increment X2r,R, by writing r/TL = , we give the results

in the leading order of 1 = (R/TL)1 to avoid lengthy

expressions:

E(X2r,R) = 2TLe(e 1), (21)

V(X2r,R) = ( 2TL)21e2

3e2 4( + 1)e

+2 + 1

, (22)

S(X2r,R) = 3 2e3W

3/21

10e3 5(2 + 3 + 3)e2

+2(42 + 6 + 3)e

(32 + 3 + 1), (23)

F(X2r,R) = 3 + 3e4W21 [525e

4

(563 + 3362 + 840 + 840)e3+(2243 + 6722 + 840 + 420)e2

(2163 + 4322 + 360 + 120)e+(643 + 962 + 60 + 15)], (24)

where W1 = 1e2[3e2 4( +1)e + 2 + 1]. Notice

that we obtain analytic expressions of S and F, whichare shown as the dashed curves in Fig.3.

With these asymptotic expressions, we next re-writethe skewness Sand flatness F as functions of the variance

over the squared mean = V/E2

to compare them withthe lognormal ones SLN and FLN.For X2R, in the limits of = R/TL , we have

S(X2R) 0 +

3 3

162

, (25)

F(X2R) 3 +

15 +3

8 39

162

, (26)

where

=V(X2R)

E(X2R)2

2TLR

TLR

2. (27)

The asymptotic expression Eqs.(25) and (26) agree withdata shown in Fig.3 for R 10TL.

For the velocity increments X2r,R, we have =

V/E2 = 1

3e2 4( + 1)e + 2 + 1/(e 1)2. Bytaking the limit of = r/TL 0, we obtain expressions

S(X2r,R) 0 +

9940

, (28)

F(X2r,R) 3 + 1359

140, (29)

where

=

V(X2r,R)

E(X2r,R)2

4

3 =

4r

3R . (30)

The equations (28) and (29) agree with the data shown inFig.3 for R 103/2TL (covering all data points), thanksto the small value r = 102TL.

We now compare the analytic expressions of the OUquantities Eqs.(2526) and (2829) with the log-normalones Eqs.(1314). We can observe the following: (i) In-deed as R ( 0), S and F of both X2R and X2r,Rtend to the values of Gaussian distribution, S = 0 and


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F = 3. Before reaching this state, we see an approachto the lognormal distribution. (ii) The sub-leading termsof S and F have the same powers of as SLN and FLN,which is in favor of the large-scale lognormality of the OUprocess. However, the constant in the subleading termis slightly different from those of the lognormal distri-bution. In this sense, the coarse-grained quantities X2Rand X2r,R of the OU process becomes nearly lognormally

distributed when R/TL is large but not exactly so.So far we focus on how the moments vary as a func-

tion of R/TL, where R is normalized by the correlationscale TL of the OU process. We close this section witha digression by pointing out another normalization scaleof the problem. This is indicated by the above analyticresults. For X2R, let us go back to the definition Eq.(9).The integral scale of the integrand of Eq.(9) is

(X2) =

0

[X2(t) X2][X2(t + s) X2]s.[X2(t) X2]2

=TL2

. (31)

For X2r,R, the integral scale of its integrand defined sim-ilarly is calculated as

(X2r ) = TL3e2 4( + 1)e + 2 + 1

4(e 1)2 r

3(32)

for small = r/TL. Notice that the variables definedin Eqs.(27) and (30) contains the corresponding integralscales of the integrands (31) and (32), namely, = 4/Rfor both X2R and X

2r,R. Hence it implies that a suitable

normalization scale of R for collapsing the moment dataplotted in Figs.3 and 4 are respectively the integral scales(X2) and (X2r ) of the integrands. Indeed this normal-

ization yields a better collapse as shown in Fig.5 for bothX2R and X

2r,R with different r. This suggests that, when

we look into a coarse-grained quantity over scale R of a

function fR(y) = (1/R)y+Ry f(y

)y. of a correlated fluc-

tuation y, the correlation scale (integral scale) of f(y)in certain cases can be a better characteristic scale thanthe correlation scale of the fluctuation y itself. We be-lieve that this is the case not only for the OU process butalso for a general fluctuation with a correlation. Howeverthis normalization blurs the fact that the large-scale log-normality of the OU process occurs at R/TL 1. Thiskind of normalization will be reported elsewhere.

IV. SUMMARY AND DISCUSSION

This study was motivated by the large-scale lognor-mality of turbulence that was recently observed experi-mentally in grid, boundary-layer, and jet turbulences [6].In this lognormality, the correlation scale plays a pivotalrole. Namely when the averaging scale to the correlationscale are of order unity, the averaged squared velocityand velocity increments become lognormally distributed

0

0.5

1

1.5

101

100

101

102

103

104

105

106

S

R /

(a)



lnX2r,R(r=TL/10)X

2R

lnX2R

2

3

4

5

6

7

101

100

101

102

103

104

105

106

F

R /

(b)



lnX2r,R(r=TL/10)X

2R

lnX2R

FIG. 5: (Color online) The collapsed skewness (a) and flatness(b) ofX2R and X

2

r,R as functions ofR normalized with (X2)

and (X2r ) defined in (31) and (32).

fluctuations. We here anticipated that this large-scalelognormality is a property of correlated random variablesin general.

To test this idea, we took the Ornstein-Uhlenbeck pro-cess as a simplest means to generate correlated randomvariables. Our numerical simulation indicates that theOU process also exhibits the large-scale lognormality forabout the same range of R/L as observed in turbulence.Then, to further investigate the approach to the lognor-mal distribution, we calculated analytically the skewnessand flatness of the coarse-grained quantities of the OUprocess. The first subleading term of the asymptotic ex-pressions for large R/L of the moments revealed thatindeed the moments behave nearly as those of the log-normal distribution. However there is small deviationfrom the lognormal distribution.

Now we speculate on behavior of the higher order mo-ments than forth order. The analytical calculation ofthem for the coarse-grained OU variables becomes socomplicated that we did not try even with Maple. Forthe lognormal variable xLN, the moments can be written

with , the ratio of the variance to the squared mean, as

H(q)LN =

[xLN xLN]q[V(xLN)]q/2

= q2

qk=0

q

k

( + 1)

1

2k(k1)(1)qk, (33)

whereqk

is the binomial coefficients. By expanding

Eq.(33) with , the moment can be written as

H(q)LN =

0 + s

(q)1

1

2 + s(q)2

3

2 + . . . + 1

2q(q2) (q: odd),

(q 1)!! + f(q)1 + f(q)2 2 + . . . + 1

2q(q2) (q: eve

Here the terms of the negative powers of are shown tovanish and s

(q)1 , f

(q)1 , . . . are suitable coefficients, which

can be written with the binomial coefficients. The 0

terms in Eq.(34), namely 0 and (q 1)!! = (q 1)(q3) 5 3 1, correspond to the values of the Gaussiandistribution. We speculate that, for the coarse-grainedquantities of the OU variable, their higher-order mo-ments have the similar expression as Eq.(34). As we sawin the previous section, the behavior for small (corre-sponding to R/TL being larger than unity) is relevant for


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the large-scale lognormality. Their first term should co-incide with that of Eq.(34) because of the usual centrallimit theorem. It is tempting to speculate also that thesecond term has the same power of but the value of the

the coefficient, s(q)1 or f

(q)1 are different. We have no idea

how much they are different.

In relation to the real turbulence data, the OU pro-cess is not a good representation as a whole since, for

example, it does not have deviation from being Gaussian(intermittency effect) and equivalence of the energy cas-cade nor the energy dissipation rate. However, as longas we focus on the large-scale fluctuations of turbulencewhere the single point velocity or the velocity incrementsare close to being Gaussian, the OU process serves a use-ful and analytically tractable model. In this study weregarded that the correlation at the integral scale is themost important aspect to be modeled by the OU process.

Here we have seen that the correlation plays the es-sential role in the near lognormal behavior of the coarse-grained positive quantities Eqs.(9) and (10). Roughlyspeaking, this near lognormality around the correlation

scale may be regarded as an intermediate state, or arule of thumb before the central limit theorem holdswith much larger averaging scale R. For the turbu-lence cases studied in [6], similar mechanism is likely towork. Other examples of lognormal behavior involving acoarse-graining average include cosmological density fluc-tuations (see e.g., [12]), which may have a similar struc-ture as studied here.

V. APPENDIX

Here we explain briefly how we calculate the momentsof the coarse-grained quantities. In particular we use theRices method given in Section 3.9 of Ref.[11] to reducethe number of the multiple integrals. For illustration ofthe method, let us take as an example the second-ordermoment of X2R(t)

[X2R(t)]2 =1

R2

t+R2

tR/2

t.1

t+R2

tR/2

t.2 X2(t1)X2(t2).

(35)

The idea of the Rices method is to write the mo-ment X2(t1)X2(t2) in terms of the correlation functionX(t1)X(t2) = (t2 t1) by noticing that X(t1) andX(t2) are multivariate Gaussian variables whose covari-ace matrix is known completely. In principle, any mo-ment of X(t1), X(t2), . . . , X (tn) can be expressed with(ti tj). Specifically, with the correlation function forlarge t

() = X(t)X(t + ) = TLe||/TL, (36)

the variance and other central moments can be writtenas follows:

[X2R(t) X2R]2 =2

R2

t+R2

tR2

t.1

t+R2

tR2

t.22(t1 t2)

=4

R2

R0

(R x)2(x)x.

=

2

T

2

L4R2

2TLR + T2L(e2R/TL 1)

(37)

[X2R(t) X2R]3 =8

R3

t+R2

tR2

t.1

t+R2

tR2

t.2

t+R2

tR2

t.3

(t1 t2)(t2 t3)(t3 t1)

=48

R3

R0

x. (R x)(R x)

x0

y. (x y)(y) (38)

[X2R(t) X2R]4 = 3(X2R(t) X2R)22

+48

R4

t+R2

tR2

t.1

t+R2

tR2

t.2

t+R2

tR2

t.3

t+R2

tR2

t.4

(t2 t1)(t3 t1)(t4 t2)(t4 t3)= 3(X2R(t) X2R)22

+768

R4

R0

x. (R x)R0

y. (y)(x y)

y0

z.(x z)(z). (39)

Here we change variables in the integrals by using the

symmetry of to reduce the double integral to a singleintegral (for details, see [11]). The integrals (38) and (39)are calculated analytically with the software Maple.

Concerning the increments, we write its correlationfunction for large t

r() = rX(t + )rX(t)= TL

e||/TL(2 er/TL) e|r|||/TL

.

(40)

Using the same argument as in the case ofXR(t), we canexpress the central moments with the correlation r asfollows:

[X2r,R(t) X2r,R(t)]2 =4

(R r)2Rr0

(R r x)

2r(x)x.[X2r,R(t) X2r,R(t)]3 =

48

(R r)3Rr0

x. (R r x)

x0

y. r(x y)r(x)r(y)(41)


7/7

7

[X2r,R(t) X2r,R(t)]4= 3[X2r,R(t) X2r,R(t)]22

+768

(R r)4Rr0

x. (R r x)

Rr0

y. r(y)r(x y)y0

z.r(x z)r(z).

(42)

The final forms (41) and (42) are calculated with Maple.

Acknowledgments

We gratefully acknowledge discussion with H. Mouri.We are supported by the Grant-in-aid for Scientific Re-search C (22540402) from the Japan Society for the Pro-motion of Sciences. T.M. is supported by the Grant-in-Aid for the Global COE Program The Next Generationof Physics, Spun from Universality and Emergence fromthe MEXT of Japan.

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large-scale lognormality in turbulence modeled by ornstein-uhlenbeck process

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