Intermittent Dissipation of a Passive Scalar in Turbulence - CNLS, LANL

VOLUME 80, NUMBER 10

PHYSICAL REVIEW LETTERS

9 MARCH 1998

Intermittent Dissipation of a Passive Scalar in Turbulence M. Chertkov,1 G. Falkovich,2 and I. Kolokolov3 1

2

Physics Department, Princeton University, Princeton, New Jersey 08544 Institute for Advanced Study, Princeton, New Jersey and Weizmann Institute, Rehovot, Israel 3 Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia (Received 3 September 1997)

The probability density function (PDF) of passive scalar dissipation P sed is found analytically in the limit of large Peclet and Prandtl numbers (Batchelor-Kraichnan regime) in two dimensions. p The tail of PDF at e ¿ kel is shown to be stretched exponent ln P sed ~ e 1y3 ; at e ø kel, P ~ 1y e. [S0031-9007(98)05479-9] PACS numbers: 47.27. – i, 05.40. + j, 47.10. + g

Probability distribution of the gradients of turbulent fields is probably the most remarkable manifestation of the intermittency of developed turbulence and related strong non-Gaussianity. A typical plot of the logarithm of gradient’s probability density function (PDF) (which would be parabolic for Gaussian statistics) is concave rather than convex, with a strong central peak and slowly decaying tails. This is natural for an intermittent field since rare strong fluctuations are responsible for the tails, while large quiet regions are related to the central peak. In particular, such PDFs were observed for the dissipation field (square gradient) of passive scalar advected by incompressible turbulence which is the subject of the present paper. We consider scalar advection within the framework of the Kraichnan model assuming velocity field to be delta correlated in time [1]. Most of the rigorous results on turbulent mixing have been obtained so far with the help of that model which is likely to play in turbulence the role the Ising model played in critical phenomena. High-order moments of the scalar were treated hitherto by the perturbation theory around Gaussian limits. Clearly, the kind of strongly nonGaussian PDF observed for gradients cannot be treated by any perturbation theory that starts from a Gaussian statistics as zero approximation. Since we consider developed turbulence with large Peclet number Pe (measuring relative strength of advection with respect to diffusion at the pumping scale), it is tempting to use Pe21 as a small parameter. Yet any attempt to treat diffusion term perturbatively is doomed to fail because the PDF of the dissipation is a nonperturbative object with respect to the inverse Peclet number: it is zero at e fi 0 and zero diffusivity yet has nonzero limits as diffusivity goes to 10. Indeed, the main contribution into dissipation is given by the scales around the diffusion scale where advection and diffusion are comparable. Still, the presence of a small parameter calls for finding a proper way to simplify the description. Following [1– 4], we show here that the dynamical formalism of Lagrangian trajectories is a proper language to describe the probability distribution of the dissipation field. Our goal 0031-9007y98y80(10)y2121(4)$15.00

is to express the unknown (statistics of dissipation) via the known (statistics of pumping). Since the Peclet number is the ratio between pumping and diffusion scales, then any piece of passive scalar has a long way to go between birth and death, and our goal is to describe how statistics is modified along the way. Dynamical formalism explicitly reveals the presence of two different time scales, a short one related to stretching and a long one related to diffusion (which eventually restricts the process of stretching); this time separation has been exploited first in solving a similar problem for one-dimensional compressible velocity [4]. The time scale of stretching fluctuations is of order of inverse Lyapunov exponent while the whole time of stretching is ln Pe times larger. Clear time separation in the dynamical history is guaranteed for (but probably not restricted by) trajectories that correspond to the value of larger than that produced by the pumping, kelyPe2 , yet smaller than kel ln Pe (whether it is possible to extend the time-separation procedure for a wider interval of will be the subject of future analysis). At Pe ! `, we are able to calculate PDF P sed rigorously, exploiting time separation and executing explicitly separate averaging over slow and fast degrees of freedom. There are three main steps in deriving P sed: (1) Presenting the PDF as an average of a functional of the time-ordered exponent of the strain matrix; (2) Reparametrization of that average into a path integral over the measure nonlocal in time; (3) Implementing time separation which makes reduction to two subsequent path integrations both local in time: one describing the long evolution and another describing the fast fluctuations. Let us define the problem and make the first step of the derivation. Advection of a passive scalar ust, rd by an incompressible flow vst, rd is governed by the equation s≠t 1 ya =a 2 kDdu f,

=a ya 0 ,

(1)

where k is the diffusivity and fst, rd is the external source assumed to be Gaussian and d correlated in time: kfst1 , r1 dfst2 , r2 dl dst1 2 t2 dxsr12 d. Here xsr12 d as a function of r12 ; jr1 2 r2 j decays on the scale L, and xs0d is the production rate of u 2 . © 1998 The American Physical Society

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We consider the simplest possible (yet physically realizable) velocity: smooth in space and delta correlated in time. The velocity field is thus a random laminar flow, while the scalar field will be considered fully turbulent and multiscale (Batchelor-Kraichnan problem) [1–3]. That requires the Prandtl number Pr nyk (viscosity-todiffusivity ratio) to be large; that is, our consideration is intended for liquids rather than gases. The scalar pumping scale L lies in the viscous interval where we thus presume the velocity statistics to be described by the pair correlation function kya st1 , r1 dyb st2 , r2 dl given by dst1 2 t2 d fV0 dab 2 Ds2d ab r 2 y2 2 r a r b dg

(2)

for the scales less than the velocity infrared cutoff Lu , which is supposed to be the largest scale of the problem. We presume also the inequality Pe2 dDL2 y2k ¿ 1 which guarantees that the mean diffusion scale rd p 2 kyD is much less than the pumping scale L. It follows from (2) that the correlation functions of the strain field sab ; =b ya are r independent. That property means that sab can be treated as a random function of time t only. To exploit that, it is convenient to pass into the comoving reference frame that is moving to the frame with the velocity of a Lagrangian particle of the fluid [3,5], ≠t u 1 rj sjl std≠l u 2 kDu fst, rd .

(3)

Making spatial Fourier transform and introducing Z t d ˆ ˆ , ˆ W sˆ W (4) Wstd ; T exp sstd ˆ dt, dt 0 one can write the solution of (3) as follows: Z t ˆ 21,T st 0 dkd dt 0 fst 2 t 0 , W uk std 0 ( ) Z t0 21 21,T ˆ stdW ˆ 3 exp 2kkm fW stdgmn dt kn . 0

(5) Averaging over f a simultaneous product of the 2nth replicas of the inverse Fourier transform of kuk , we get the nth moment of the dissipation field e ks=ud2 . The PDF restored from all the moments is R` Z 01 1i` ds ese Z 2m 2 2s 0 dt Q dm e ke ls , (6) P sed 2 01 2i` 2p i #2 " " ∏ Z ˆ ˆ qWstdm qLstdq exp 2 , (7) Q dq xq 2pPe Pe2 Z t ˆ ˆ 21,T st 0 dW ˆ T std , (8) ˆ 21 st 0 dW ˆ std dt 0 W Lstd ; DW 0

where q kL and an extra integration over the auxiliary vector field m takes care of combinatorics and summation over vector indices. Notice that the direction of time in (7) and (8) is opposite to that in (3). The averaging a b1c d is according to over the traceless s ˆ s b2c 2a 2122

9 MARCH 1998

R` D sstd ˆ expf2 0 dt 0 sa2 1 b 2 1 c2 y2dy2Dg. The whole expression (7) is invariant with respect to the global (time ˆ independent) rotation m ! Rm, sˆ ! Rˆ s ˆ Rˆ T , provided the pumping is statistically isotropic. That allows one to get rid of the angular integration in dm, counting all the dynamical angles from the direction of m. Let us do step 2 now. The general tool to average ˆ particularly (7), is Kolokolov transa function of W, formation replacing T exp by a regular function of new ˆ ˆ Rˆ q Vˆ w, fields [6]. We represent W Rt ˆ where Rq is the matrix of rotation by the angle 0 dt 0 qRst 0 dy2, Vˆ is the t diagonal matrix with the elements expf6 0 dt 0 hst 0 dg and 1 wstd ˆ Our ˆ s0d 1. d, ws0d 0, to provide for W wˆ s 0 1 transform ha, b, cj ! hh, q , wj is a hybrid version of those used in [7] and in Appendix B of [3]. The relation between the old and new variables is obtained by differentiating Rˆ q Vˆ wˆ and comparing with (4), # " Z t 0 dt q a 1 ib h exp i 0

" # Z t wÙ dt 0 s2h 1 iq d , 1 i exp 2 0 # " Z t 0 dt h . 2c 2q 1 wÙ exp 2

(9)

0

The Jacobian of theQtransformation Rt J Jul Jl has a standard product Jul ,R Tt0 expf2 0 dt 0 hg and nontrivial JaT cobian Jl , expf 0 dt hstdg associated with the positive Lyapunov exponent, describing the exponential stretching of trajectories. The rotation matrix Rˆ q , which is the same for all the dynamical matrix processes, can be removed by collective transformation of all the external vectors in the problem; i.e., one may explicitly integrate over D q since we average q -independent objects in (6). Averagˆ sTd is then reduced to averaging ing some function of W the same function of Vˆ sTd to the measure Rt Q Twˆ with respect 0 expf2S gD hstdD wstd t0 expf2 0 dt hg, where ( " # ) Z t 1 Z T wÙ 2 . S ; dt h 2 2 2Dh 1 exp 4 dt h 2D 0 4 0 (10) Finally, we apply the time-separation procedure to capture the term in (6) dominant at large Pe. Two different time scales areRclearly seen in (6)–(8) at least for e ¿ kelyPe2 when Q dt has to be large, which is achieved on such realizations of hˆ where the integrand grows exponentially due to W std until the last exponential factor restricts the growth at the time of order D 21 lnsPed. Well before, on a time scale of the inverse stretching rate D 21 , the exponentially growing Wst 0 d makes the integral over dt 0 in (8) saturated. A dynamical picture in r space would be as follows: For the Lagrangian trajectories giving the main contribution into ke n l, fluid particles start very close. Diffusion



9 MARCH 1998

The temporal separation makes it possible to substitute remains dominant until the particles separate by a dist by t0 as an upper limit in integration over t 0 in (8). tance comparable to the diffusion scale rd . This phase of the dynamics takes place on times of order D 21 ; noAlso, an important manifestation of time separation is a distinctively different behavior of the w field (responsible tice that this time is diffusion independent since rd is for the “rotation” in the pseudospace) at time intervals a scale where diffusion and stretching are comparable. smaller and larger than the separation time. The action Once the distance between particles outgrows rd , ran(10) shows that for t . t0 ¿ 1yD the w field is frozen— dom multiplicative stretching due to velocity becomes dominant. Because of a multiplicative nature of the dyno dynamics at all, wstd wst0 d, which has a clear physical meaning since stretching proceeds along one namics, the time to go from rd to L is proportional to direction. At the smallest times the w-field D 21 lnsPed. Let us now introduce a separation time t0 satRt dynamics is isfying 1 ø Dt0 ø lnsPed; that time will disappear from essential. Indeed, let us denote rstd 0 hst 0 d dt 0 and ˆ the final answer. explicitly transform Lstd according to (9), √ √ !√ ! ! 0 Z t e24rst d 1 fwstd 2 wst 0 dg2 wstd 2 wst 0 d e2r 0 e2r 0 0 0 ˆ dt expf2rst d 2 2rstdg Lstd D . (11) 1 0 1 wstd 2 wst 0 d 0 1 0 For trajectories with predominantly positive h we replace wstd by wst0 d and neglect the integral from 0 to t0 in (6) at t ¿ t0 . Since fwstd 2 wst 0 dg2 expf2rst 0 dg and expf22rst 0 dg decrease exponentially as t 0 increases, then the integral over dt 0 saturates, which allows for time separation too. Finally, nondiagonal and lower diagonal ˆ are exponentially small. For time sepaelements in L ration to be complete, the integral rstd at t ¿ t0 has to be counted from t0 ; i.e., we replace rstd ! rstd 1 rst0 d, then in the dominant order in the Peclet number we get Z ` 2 2r m2 b Z ` Z 22 dt Q ø dt dq xq q12 e2r2q1 e mbPe , 2 s2pPed t0 0 (12) Z t0 0 0 dt 0 e2rst d he24rst d 1 fwstd 2 wst 0 dg2 j , (13) m;D 0

where we denoted R b expf2rst0 dg. Next, one needs to average expf2s dt Qg over the short-time D h, D w and a long-time D h. separately. The corresponding weights of averaging with respect to h. and h, are completely decomposed: S S, 1 S. . The great advantage of (12) is that, in the long-time averaging, both b and m are just external parameters, depending neither on time t nor on h. . Once the average over h. is performed, we are left with a function of m. The final result for P sed will then be obtained by averaging over h, and w. R We first average expf2s dt Qg over h. using (10) Ps. sm, md lim e2sT2t0 dy2 FsT 2 t0 ; r 0d , (14) T!` RT Z 2 fs2Dd21 rÙ 2 1sQg dt . FsT , rd ersTd D rstddfrs0dge 0 Here e2sT2t0 dy2 is a normalization factor R (the inverse value of the path integral without expf2s dt Qg). Exactly in a way that Schrödinger equation appears from the pathintegral representation of quantum mechanics [8], from ˆ . dFst, rd 0 with (14) one gets s≠t 2 H Z 2 2 2r 22 ˆ . ; 2 1 ≠2r 1 sm b dq xq q12 e2r2q1 e mbPe . H 2 2 s2pPed (15)

Here sQ, with Q from (12), plays the role of potential, the h 2 term of (10) gives ≠2r , while the linear in the h part of (10) gives the initial condition of t 0: Fs0; rd er . Since sQ vanishes at r ! ` and Fs0; rd does not, one obtains the asymptotic behavior of F at t ! ` Fst; rd ! ety2 yPeFp s yd,

Ps. Fp s sPembd21 d , (16)

f2y 23 ≠y y 3 ≠y 1 2sm2 m21 Us ydgFp s yd 0 , Z Usxd ; s2pd22 dk xk k12 exps2k12 x 2 d ,

(17) (18)

where the new variable y er sPembd21 has been introduced. Fp ! 1 at y ! ` and Fp y should vanish at y ! 0. Ps. is a function of a single argument sm2 ym. The potential Usxd is everywhere positive; it has to turn into a constant at x ! 0 and behave as x 23 at large x. Under such a general assumption on the pumping function xq , one can show that the only singularities of Ps. szd are poles on the negative semiaxis at a finite distance from zero. The asymptotic condition on Fp s yd gives the norp malization Ps. s0d 1 and at z ! ` lnf1yPs. szdg , z. For example, in the particular case Ω 1, x , 1, (19) Uc sxd 1yx 3 , x . 1 , p p p onep finds 2Ps. szd zfI20 s2 z dI1 s z d 1 I2 s2 zd 3 p I10 s z dg21 and Ps. szd ! sz 3y2 y4pd expf23 z g at z ! 1`. Now we have to average Ps. s2sm2 ymd over h, and w, which is equivalent to averaging over the random variable m (responsible for the fluctuations of the diffusion scale due to strain variations): ke2sQ lh kPs. s2sm2 ymdlm . A convenient auxiliary object to calculate is the generating function Pl kexpf2lmgl, which can be expressed via the solution of another differential equation: Z ` ˆ , 1 1dC 0 , dw Csr 0, wd, s2H Pl 2`

(20) 2123

VOLUME 80, NUMBER 10 " Z


#

` n

dw w Ce

2`

2r

! dn0 , r!1` 1 2 2 ≠r

ˆ , ; 22e24r ≠w2 2 H

Cjr!2` ! 0 .

1 le22r s1 1 w 2 e4r d ,

where the static equation (20) follows at t0 ¿ 1 from the respective dynamical one similar to the way (16) pfollows from (14). The exact p solution of (20), p Cp 1 1 w 2 e4r 2l e2r K1 s e2r 2ls1 1 w 2 e4r d d , gives P smd s2pm3 d21y2 exps21yf2mgd .

(21)

21y2

(which can be interpreted as inverse local Note that m diffusion scale) has exactly Gaussian PDF that seems to be a consequence of strain Gaussianity. Integration of Ps. s2sm2 ymd with P smd gives the final answer, Z ` 1 p P sed ds Ps. sisd 2p e 2` µ p ∂ Z ` e 3 dx exp isx 2 2 . (22) x 0 That formula is our main result. It expresses dissipation PDF in terms of the function Ps. determined by (16) and (17) with the potential (18) given by the pumping. For any pumping, all that is necessary to get P sed is to solve an ODE of the second order. In particular, one can find Ps. szd as a series in z iterating Fp in (17) over U. That expresses dissipation moments directly via the pumping ke n l 2n fs2n 2 1d!!g2 n! Z ` dy Z y1 1 3 dy2 y23 Us y2 d y13 0 0 Z ` dy Z y2n 2n21 3 ··· dy2n y2n Us y2n d . 3 0 y2n22 y2n21

(24)

Stretched-exponential tail is natural for steady PDF of the gradients [2] contrary to log normal unsteady distribution which takes place without diffusion [1,2]. The value 1y3 for the stretched exponent may be explained as follows: Since the local value of =u is proportional to

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the scalar fluctuation at the diffusion scale du (which has exponential PDF tail [2,3,9]) times an inverse local diffusion scale [which is Gaussian according to (21)], then ke n l , k n ksdud2 ln n2n rd22n nn which corresponds to (24). Note that the asymptotics (24) is determined by the dynamics of stretching (not of rotations), thus it is likely to take place in any dimensions. The data of numerics done directly for the Kraichnan model [10] are satisfactorily fitted by the 1y3 stretched exponent within the window expanding when Pe grows. The exponent 1y3 agrees also with the values 0.3–0.36 given by numerics [11] and 0.37 by experimental data on air turbulence [12]. Moreover, our exponent derived formally at Pr ¿ 1 agrees with the data of [11,12] obtained at Pr . 1 as well. This is surprising because P sed is independent of Pe at large Pr, while at Pr & 1 the statistics of is markedly different since ke n l ~ Pemn where mn are anomalous exponents of the scalar [13]. It may be, however, that the PDF tail is still given by (24) for any Pr with tp and preexponent factor in (24) generally depending on both Pr and Pe; this will be the subject of future studies. To conclude, the agreement of our result with a variety of data suggests that the Kraichnan model is a proper tool for recovering exponents in the theory of turbulent mixing. We thank E. Balkovsky, V. Lebedev, B. Shraiman, and M. Vergassola for interesting discussions. This work was partially supported by the R. H. Dicke Foundation (M. C.), the Israel Science Foundation and Monell Foundation (G. F.), and the Russian Fund of Fundamental Researches under Grant No. 97-02-18483 (I. K.).

(23)

The conservation law identity kel 22≠z2 Ps. s0d xf0gy2 is a consistency check of our procedure. The form of the PDF at e . kel is pumping dependent while the asymptotics at small and large have universal forms. At small (and infinite Pe) the two first terms have the form P sed ! Ae 21y2 1 B ln e so that the gradient’s PDF P s=ud ! A 2 Bj=ujy ln j=uj tends to a constant; notice the nonanalyticity of the first nonequilibrium correction. Since the only singularities of Ps. sizd are poles on the imaginary axis, then large- asymptotics is determined by the pole nearest to the origin in the upper half-plane: z it p , t p , 1. In particular, tp ø 2.7 for (19). The dx integration in (22) is done by a saddle-point method, P sed , e 21y2 expf23s2t p ed1y3 y2g .

9 MARCH 1998

[1] R. H. Kraichnan, J. Fluid Mech. 64, 737 (1974). [2] B. Shraiman and E. Siggia, Phys. Rev. E 49, 2912 (1994). [3] M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedev, Phys. Rev. E 51, 5609 (1995). [4] M. Chertkov, I. Kolokolov, and M. Vergassola, Phys. Rev. E 56, 5483 (1997). [5] G. Falkovich, I. Kolokolov, V. Lebedev, and A. Migdal, Phys. Rev. E 51, 5609 (1995). [6] I. Kolokolov, Ann. Phys. (N.Y.) 202, 165 (1990). [7] B. Shraiman and E. Siggia, C. R. Acad. Sci. 321, 279 (1995). [8] R. P. Feynman and A. B. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965). [9] D. Bernard, K. Gawedzki, and A. Kupiainen, Report No. cond-mat/9706035. [10] E. Balkovsky, A. Dyachenko, and G. Falkovich (unpublished). [11] M. Holzer and E. Siggia, Phys. Fluids 6, 1820 (1994). [12] M. Ould-Rouis, F. Anselmet, P. Le Gal, and S. Vaienti, Physica (Amsterdam) 85D, 405 (1995). [13] M. Chertkov and G. Falkovich, Phys. Rev. Lett. 76, 2706 (1996).