Scaling and intermittency in Burgers turbulence

SCALING AND INTERMITTENCY IN BURGERS’ TURBULENCE J.P. Bouchaud arXiv:cond-mat/9503144v1 28 Mar 1995 Service de Physique de l’Etat Condensé, CEA-Saclay, Orme des Merisiers, 91 191 Gif s/ Yvette CEDEX, France M. Mézard Laboratoire de Physique Théorique de l’ Ecole Normale Supérieure rue Lhomond, 75231 Paris Cedex 05, France 1 , 24 G. Parisi Universita di Roma I, Piazza A.Moro 2, 00185 Roma, Italy Abstract: We use the mapping between Burgers’ equation and the problem of a directed polymer in a random medium in order to study the fully developped turbulence in the N dimensional forced Burgers’ equation. The stirring force corresponds to a quenched (spatio temporal) random potential for the polymer. The properties of the inertial regime are deduced from a study of the directed polymer on length scales smaller than the correlation length of the potential, which is not the regime usually considered in the case of polymers. From this study we propose an Ansatz for the velocity field in the large Reynolds number limit of the forced Burgers’ equation in N dimensions, which should become exact in the limit N → ∞. This Ansatz allows us to compute exactly the full probability distribution of the velocity difference u(r) between points separated by a distance r much smaller than the correlation length of the forcing. We find that the moments < uq (r) > scale as r ζ(q) with ζ(q) ≡ 1 for all q ≥ 1 (in particular the q = 3 moment agrees with Kolmogorov’s scaling ζ(3) = 1). This strong ‘intermittency’ is related to the large scale singularities of the velocity field, which is concentrated on 1 Unité propre du CNRS, associée à l’Ecole Normale Supérieure et à l’Université de Paris Sud a N − 1 dimensional froth-like structure, which is turn related to the onestep replica symmetry broken nature of the associated disordered problem. We also discuss the similarities and differences between Burgers turbulence and hydrodynamical turbulence, and we comment on the anomalous tracer fluctuations in a Burgers turbulent field. This replica approach being rather unusual in turbulence problems, we provide all the necessary details of the method. LPTENS preprint 95/12 1 1 Introduction The theory of statistical turbulence is more than fifty years old, but its status is still not fully satisfactory. The simplest and surprisingly robust approach is Kolmogorov’s dimensional analysis, which leads to the celebrated k −5/3 law for the velocity cascade [1, 2]. However, analytical calculations immediately lead to difficulties: the simplest closure scheme to deal with the non linear term in the Navier-Stokes equation (‘DIA’) does not reproduce Kolmogorov’s scaling [3, 2, 4]. A lot of effort has been devoted, in particular by Kraichnan, to understand why this was so. More refined schemes were proposed to recover the k −5/3 law and even to calculate adimensional constants [5, 6, 2], but they are based on uncontrolled assumptions (like the RG approach), or lead to enormous calculations, which are difficult to manipulate and are not fully transparent from a physical point of view [5]. More recently, a tremendous activity has developed on the intermittency (or multifractal) corrections to Kolmogorov’s scaling: higher moments of the velocity field do not seem to scale with the one originally predicted by Kolmogorov. Many interesting suggestions have been put forward to describe and explain this feature, starting by Kolomgorov himself [7, 8, 9, 10, 11, 12, 13, 14]. In recent years, non linear partial differential equations with noise have been the focus of quite a number of studies in the context of growing interfaces, with the ‘KPZ’ equation standing out as a new paradigm [15, 16, 18, 17]. This equation is in fact a variant of Burgers’ equation, and has (together with its many siblings) a wide range of applications in different physical contexts [17]. Interestingly, this field is also related (through an appropriate ‘Cole-Hopf’ mapping) to the physics of disordered systems, in particular elastic strings in random media (‘directed polymers’). The Burgers problem in N dimensions is then equivalent to the problem of a directed polymer in N +1 dimensions, which is the space time of the original problem. The forcing term in the Burgers equation translates into a random potential for the directed polymer, which is quenched in space time: to each realization of the stirring force in the Burgers language corresponds one sample of a directed polymer. This enables one to adapt techniques originally devised for with spin-glasses and obtain original results on the underlying non-linear equation (or vice-versa) [19, 20, 21, 22, 23, 24]. It is the aim of the present paper to exploit in detail this mapping, which allows us to propose an original Ansatz for the velocity field for the randomly stirred Burgers equation, which should become exact solution in high dimensions and in the limit of large Reynolds number. Scaling in the inertial range can then be precisely discussed: we obtain in closed form the full probability distribution for velocity differences. In particular, the third moment of the velocity difference grows linearly with distance, i.e à la Kolmogorov. In fact, Kolmogorov’s dimensional analysis should be directly applicable to Burgers’ turbulence. We find however very strong intermittency effects, which we relate in a quantitative manner to the existence of large scale structures, in the form of singularities 2 concentrated on time dependent hypersurfaces. Although not exact in finite dimension, we expect that our (variational) description remains qualitatively correct even in one dimension. We also obtain the dynamical evolution of the velocity field: we find that the field is convected away by the largest structures, corresponding to a dynamical exponent equal to z = 1 rather than the one obtained from Kolmogorov’s scaling, zK = 2/3. In the bulk of the paper, we primarily focus on the ‘dictionary’ between Burgers’ turbulence and disordered systems and discuss a number of physical points, relegating more technical points to various appendices. We show that Kolmogorov’s scaling has a counterpart in the ‘directed polymer’ language, where it is known as the Larkin-Ovchinnikov scaling. It is however easy to see that this ‘naive’ scaling cannot hold, as is confirmed by the full calculation. We describe in physical terms the nature of the velocity field, and argue that there should be a large distance regime (beyond the ‘injection’ scale) characterized by a non-trivial exponent. We briefly discuss the problem of a passive scalar in such a velocity field. In the conclusion, we compare our results to other approximation schemes, and comment on the possible differences with ‘true’ Navier-Stokes turbulence. 2 From Burgers turbulence to directed polymers: dictionary and dimensional analysis The problem we shall consider is that of a randomly forced potential flow in N dimensions, which follows the Burgers equation 2 : ∂~v ~ ~v = ν∇2~v + f~(~x, t) + (~v · ∇) ∂t (2.1) where ~v is minus the gradient of a velocity potential h(~x, t), and f~ is a randomly fluctuating force, which is also minus the gradient of a potential φ(~x, t). ~v and ~x are N-dimensional vectors, and there is no constraint on ~ v . Here, we wish to describe the problem of a fluid which is randomly ∇.~ stirred only at very large lengthscales. We shall thus take φ to be Gaussian, with fluctuations given by: φ(~x, t)φ(~x′ , t′ ) (~x − ~x′ )2 . = ǫ∆ N δ(t − t ) exp − 2N∆2 ′ 2 # " (2.2) Here ∆ is the ‘injection’ length, over which the forcing is roughly constant. The correlation time of the forcing was set to zero; a finite (small) correlation time would however not affect the following conclusions. The stirring force correlation thus reads: ′ " f µ (~x, τ )f ν (~x′ , τ ′ ) = ǫδ(τ − τ ) δ 2 µν (~x − ~x′ )µ (~x − ~x′ )ν − N∆2 The density of the fluid is taken equal to one. 3 # (~x − ~x′ )2 exp − . 2N∆2 (2.3) " # As will be clear below, this ensures that the injected energy per unit time, defined as the increase of kinetic energy 12 ~v 2 , is equal to N2 ǫ (Note that the dimension of ǫ is [x2 ]/[t3 ]). The dependance of the force correlations on the dimension N has been chosen such as to insure the existence of the large N limit. The typical velocity at the injection scale is, from dimensional consider1 ations, v∆ ≡ (ǫ∆) 3 , which allows us to define the Reynolds number as: ǫ∆4 1 v∆ ∆ = ( 3 )3 . (2.4) ν ν We shall be interested in studying the statistics of the velocity field at large Re. Let us now use the standard techniques to transform this problem into ~ one finds the a directed polymer. Integrating once Eq. (1), with ~v = −∇h so-called Kardar-Parisi-Zhang (KPZ) equation: Re ≡ 1 ∂h(~x, t) = (∇h(~x, t))2 + ν∇2 h(~x, t) + φ(~x, t) (2.5) ∂t 2 which describes, in particular, the growth of a surface under a random ‘rain’ of particles, the flux of which is given by φ(~x, t). At this stage, the crucial difference with previous work on the KPZ equation is in the correlations of the noise φ(~x, t). We shall thus be mostly concerned, in the following, with the velocity field statistics at length scales smaller than ∆; for length scales larger than ∆, the KPZ scaling prevails. Now, as is well known, the KPZ equation (2.5) can be transformed (through a ‘Hopf-Cole’ transformation) into a linear problem describing a directed line (polymer) in a random potential. Setting h(~x, t) = 2ν log Z(~x, t), one finds that Z(~x, t) obeys the equation 1 ∂Z(~x, t) = ν∇2 Z(~x, t) + φ(~x, t)Z(~x, t) (2.6) ∂t 2ν which is the equation for the partition function of an elastic string in a 1 φ(~x, t), subject to the constraint that its end random potential V (~x, t) = 2ν point is fixed at (~x, t). Said differently, the solution of Eq. (2.3) can be written as a path integral (sum over all configurations): Z(~x, t) = Z ρ(~x0 )d~x0 Z ~ x(t)=~ x ~ x(t=0)=~ x0 d[~x(τ )] exp(−H) . (2.7) The hamiltonian H is given by: H= Z t 0  c dτ  2 d~x dτ !2  + V (~x(τ ), τ ) . (2.8) The ‘dictionary’ between the two problems is the following: The temperature scale of the polymer has been chosen equal to one. Then the elastic modulus of the polymer is 1 . (2.9) c≡ 2ν 4 The random potential seen by the polymer V has a gaussian distribution, with a second moment given by: (~x − ~x′ )2 . V (~x, τ )V (~x′ , τ ′ ) = W Nδ(t − t′ ) exp − 2N∆2 # " (2.10) where the strength of potential fluctuations, W , is related to the energy density ǫ through: ǫ∆2 W ≡ , (2.11) 4ν 2 and the length scale of fluctuations of the potential is equal to ∆, the lengthscale at which the Burgers fluid is stirred. The probability distribution ρ(~x0 ) of the initial point of the polymer is related to the initial conditions of the velocity through: ~ x) . ~v (~x, t = 0) = −2ν ∇ρ(~ (2.12) The non-linearity has disappeared from this formulation, and has been replaced by the famous problem of disordered systems, which is to average the logarithm of a partition function (in order to calculate various moments of the velocity field ~v). In the next section we shall deal with this problem using the replica trick and a variational method which becomes exact in the limit of very large dimensions N. Before turning to this calculation, it is useful to give some kind of qualitative (more or less dimensional) analysis of the directed polymer, and to enrich our ‘dictionary’ by stating its counterpart in the turbulence language. The study of an elastic structure like the directed polymer in presence of a random potential has been discussed in several works recently [25, 21]. In the phase where the disorder is strong, one expects a scaling behaviour of the lateral fluctuations of the polymer described by a wandering exponent ζ: < (~x(t) − ~x(t′ ))2 > ≃ A|t − t′ |2ζ . (2.13) The thermal fluctuations are irrelevant at large distance and this scaling also holds in the zero temperature limit, or for the disconnected correlation (< ~x(t) − ~x(t′ ) >)2 . As for the free energy differences for two polymers finishing at point t, ~x and t′ , ~x′ , they scale as: 2 (h(~x, t) − h(~x′ , t′ )) ≃ |~x − ~x′ |2ω/ζ gh |~x − ~x′ | . |t − t′ |ζ ! (2.14) The t → t′ limit then implies that the free energy difference at points ~x and ~x′ scales as |~x − ~x′ |2ω/ζ , and therefore the difference of velocities in Burgers equation should scale as: 2 ′ (~v (~x, t) − ~v (~x′ , t′ )) ≃ |~x(t) − ~x(t )| −2) (2 ω ζ gv |~x(t) − ~x(t′ )| . |t − t′ |ζ ! (2.15) If this last scaling form holds, the galilean invariance of Burgers equation ~ ~v should scale in the same implies that the two terms in ∂~v /∂t and (~v · ∇) 5 way under a rescaling ~x → b~x and t → b1/ζ t, which implies ω = 2ζ − 1 [19, 18]. (Another way to argue about this identity directly on the polymer problem is by observing that the fluctuations of elastic energy in the directed polymer scale as x2 /t = t(2ζ−1) = tω ). So we are left with only one scaling exponent. An important point which was discussed in [26, 27] is the existence of two distinct scaling regimes, in the case where the random potential has a large correlation length, ∆. The regime which is most studied for directed polymer is the large time regime where the transverse fluctuations of the polymer are much larger than ∆. But there also exists a short distance regime where the transverse fluctuations of the polymer are smaller than the correlation length of the potential. This is obviously the regime which interests us most for the turbulence problem (although we shall return to the long distance regime later on). This regime holds for time differences shorter than a typical time scale τ ∗ , defined from: < (~x(t + τ ∗ ) − ~x(t))2 > ≃ ∆2 . (2.16) One expects to be allowed to linearize the random potential in this slowly varying regime, and to study the much simpler (linear) problem of a polymer with a random force, defined by the Hamiltonian: H= Z 0 t  c dτ  2 d~x dτ !2  − f~L (τ ) · ~x(τ ) , (2.17) √ (where the random force is of order |fL | ≃ W /∆). This random force problem was studied by Larkin and Ovchinikov long time ago [28]. In this regime there is no metastable state and the problem is easily solved for one given sample. Assuming for simplicity periodic boundary conditions, one obtains the Fourier transform of the average polymer’s position:: < ~x(ω) >= f~(ω) . c ω2 (2.18) A dimensional analysis of the Fourier transform would lead to: | < ~x(t) − ~x(t′ ) > |2 ≃ W |t − t′ |3 , 2 2 c ∆ (2.19) from which one deduces the scaling exponent in this regime, and the value of τ ∗ : 3 ζK = , 2 ωK = 2ζK − 1 = 2 , ∗ τ ≡ c2 ∆4 W !1/3 ≡ ∆ v∆ (2.20) This result looks very nice when translated in terms of turbulence, since it predicts (using (2.15)) that the difference of velocity between two points at distance r will scale like r ωK /ζK −1 = r 1/3 , and also that time scales and length 6 scales are related through tζK ∼ r, with ζK = 3/2 (Richardson diffusion). Note that τ ∗ is simply the convective time across the injection length ∆. These results thus precisely reproduce the Kolmogorov scaling, which is here derived from a very simple argument on the directed polymer problem! However, it turns out that this result is wrong, because we must go beyond the linear approximation. Technically the reason is in the integration which leads from the expression (2.18) of the average position in Fourier space to the scaling expression (2.19). Clearly the corresponding integral over frequencies ω is divergent at small ω, and it turns out that the result is proportional to (t − t′ )2 T ,where T is the total length of the polymer. One sees that, even if t − t′ < τ ∗ , the scaling depends on time scales which are larger than ∆. Therefore one cannot work out the scaling behaviour at small length and time scales within the linear – random force – approximation. In the polymer problem one must study the case of a random potential, which is a non linear problem with many metastable states [21, 26]. Translated into the turbulence language, these correspond to intermittency effects which are crucial and cannot be neglected. The full solution derived in the next section indeed finds a Kolmogorov scaling, but only for the third moment of the velocity difference (~v (~x, t) − ~v (~0, t))3 ≃ |~x|. The other moments do not agree since all the moments (~v (~x, t) −~v (~0, t))q with q larger or equal to 1 scale like |~x|. Furthermore, length and time are related by a convective scaling, corresponding to the dynamical exponent z = 1ζ = 1. 3 The replica variational approach 3.1 The replica solution We now turn to the problem of computing the average ‘free energy’ log Z(~x, t) of the directed polymer over the random forcing V (~x, τ ). The procedure we use is standard, and we follow closely [21, 26]: we first express log Z(~x, t) n as the ‘zero replica’ limit log Z = limn→0 Z n−1 , average Z n which generates an effective attraction between replicas, which we treat using a Gaussian variational Ansatz. The quality of this approximation, and the regimes of dimension and Reynolds number in which it becomes exact, will be discussed in section (3.3). We proceed as usual by writing the average of Z n as the partition function of the n− replica Hamiltonian Hn , which reads: n 1X Hn = 2 a=1 Z 0 t  d~xa dτ c dτ !2  + µ~x2a  (~xa (τ ) − ~xb (τ ))2 WN XZ t dτ exp − − 2 a,b 0 2N∆2 " 7 # (3.1) where we have added a ‘mass’ term µ for regularizing intermediate computations, which we will set eventually to zero. We furthermore consider periodic polymers, for which ~x(t) = ~x(0). This is not exactly the same problem as the one with free ends which is in direct correspondence with Burgers’s equation. We shall first work out the periodic case, and in section (3.3) we discuss the changes for the free case. In order to handle the problems of the metastable states, the idea is to use a variational method and approximate Hn by an effective Gaussian Hamiltonian Hv , which we write in Fourier space as: Hv ≡ 1 XZ ˜ dω ~xa (−ω)G−1 xb (ω) ab (ω)~ 2 a,b (3.2) ˜ stands for +∞ dω . Note that Hv is isotropic in real space; however where dω −∞ 2π its structure in ‘replica’ space is arbitrary. The trial free-energy obtained with Hv depends on Gab and reads: R R Fv [G] =< Hn >v − N T r log G 2 (3.3) where < ... >v means averaging with the Bolzmann weight associated to the trial Hamiltonian Hv . Remember that the temperature scale has been set to one. The calculation of < Hn >v using Gaussian integrals is straightforward and leads to: < Hn >v =  NT  2 XZ a 2 ˜ dω(cω + µ)Gaa (ω) − W X 1+ ab Bab N∆2 − N2   (3.4) R ˜ aa (ω) + Gbb (ω)− where T is the total length of the polymer and Bab ≡ dω[G 2Gab (ω)]. The variational statement is that Fv [G] is greater than or equal to ∂Fv [G] the ‘true’ free-energy. We thus look at the optimal Gab such that ∂G ≡ 0. ab (ω) This leads to the following self-consistent equations: Bab W [G ]ab (ω) = − 2 1 + ∆ N∆2  −1 − N2 −1 and Gaa (ω) + X b6=a Gab (ω) ≡ Gc (ω) = (a 6= b) 1 µ + cω 2 (3.5) (3.6) The task is now to solve these equations using some Ansatz on the structure of Gab . This has been discussed in full details in [21]. We keep here to dimensions N > 2. There are two regimes of Reynolds number, separated by a critical value Rec = [2(1 − 2/N)(1−N/2) ]1/3 . For Re < Rec , the solution is a ’replica symmetric’ one with Gab (ω) = δab Gc (ω)+G(ω). The two propagators Gc and G are easily computed. Translated in terms of the velocity, this just corresponds, for an infinite size system (but for a finite injection q length ∆) to a vanishing velocity field. For a finite ‘box’ of length L = µν , one finds 8 x0 ) , where ~x0 is a random (Gaussian) time dependent variable that ~v ≃ ν (~x−~ L2 depending on the forcing history. When one increases the forcing beyond Rec the correct solution is the so-called ‘one step’ replica symmetry breaking scheme. This amounts to parametrizing the off-diagonal elements Gab (ω) with two functions G0 (ω), G1 (ω), depending on whether the replica indices belong to the same ‘block’ or to different blocks. The size of these blocks are furthermore parametrized by a number m ranging between 0 and 1, fixed by 0 ,G1 ,m] = 0. We refer the reader to [29, 21] for a more imposing that ∂Fv [G∂m detailed discussion of this construction, but will recall below its physical interpretation, on which we shall heavily rely to discuss our results in the turbulence language. The relevant formulae for inverting such matrices are given, for completeness, in Appendix A, together with the basic calculation steps needed to solve Eqs. (3.5-6). One important final equation is the one fixing m. We find that: (1 − 1 ∆2 W 2m 1−N/2 ) = 2m3 = m3 Re3 N ν 2 (3.7) where we have used the definition of the Reynolds number (Eq. (2.4)). Note that Eq. (3.7) is well behaved in the limit N → ∞, where it becomes 2em = (mRe)3 . We shall see later that when Re > Rec , the breaking of replica symmetry corresponds to a non trivial structuration of the flow into large size structures which are‘cells’ of size ∆, inside which the velocity is of order v∆ ∼ Re ∆ν (independently of the box size L). We shall thus identify Rec as the critical Reynolds number for the onset of turbulence. 3.2 Physical description of the solution Let us see how this is encoded in the one-step solution, by first looking at equal time correlations. Within the Gaussian variational Ansatz, one finds that   X X 1 P[{~xa (t)}] = Z −1 [Q−1 ]π(a),π(b) ~xa (t)~xb (t) (3.8) exp − 2 π a,b ˜ ab (ω), and π denotes the sum over all the n! permutawhere Qab = dωG tions of replica indices. Taking the limit where the mass term µ is zero, we find that (see Appendix A): R Q0 ∼ µN/4−1 → 0, P Q1 − Q0 → ∞, Q̃ − Q1 = m∆2 ) (1 − 2m N (3.9) In the very large Reynolds limit, on which we focus now, the parameter m scales as 1/Re and one finds Q̃ − Q1 ≃ m∆2 . The physical interpretation of the replica probability distribution Eq. (3.8) has been worked out in [21], and is particularly simple in the present case where Q0 is zero and Q1 −Q0 is very large. Remember that one is dealing with a polymer in a random environment. For each ‘sample’ Ω (i.e for a particular 9 realisation of the disorder V (~x, t)) the probability distribution PΩ (~x) for the ‘end point’ of the polymer ~x has a certain shape. The 1 step replica symmetry breaking variational approach assumes that this distribution can be written as a weighted sum of Gaussians: PΩ (~x) = X α 1 Wα exp − [~x − ~rα ]2 2δ   (3.10) where δ ≡ Q̃ − Q1 , Wα are random weights chosen with a probability proportional to W −1−m (1 − W )m−1 and ~rα are uniformly distributed in the ‘box’ of size L (A more formal description of the solution is presented in Appendix B, together with useful technical details). The Wα and the ~rα encode the particular features of the sample Ω, and must thus eventually be averaged over. The above construction was restricted to a certain ‘time’ (or length of the polymer). Similar considerations also enable us to construct two-time correlation functions, and the result is very simple: it amounts to let the ~rα acquire a time dependence. More precisely, ~rα are independent Gaussian, time dependent variables such that (again taking the µ = 0 limit): C(t − t′ ) ≡ < [~rα (t) − ~rα (t′ )]2 > = 2S m Z ′ ˜ (1 − cos ω(t − t )) , dω cω 2 (cω 2 + S) (3.11) 2 where S = 1/(4c(Q̃ − Qq 1 ) ). Eq. (3.11) shows that a characteristic time ∆2 , which is nothing but the ‘Larkinscale appears, given by Sc = 21/3 νRe ∗ Ovchinnikov’ time τ encountered above and corresponds to the convective 4 time across the injection length. For τ ≫ τ ∗ , one finds that C(τ ) = 2 3 ∆2 ττ∗ , whereas for τ ≪ τ ∗ , one finds C(τ ) = 12 ∆2 ( ττ∗ )2 . Eqs. (3.10-11), together with the value of m and δ, are the central results of this paper, from which we shall derive in the next section the very interesting statistical properties of the velocity field, in particular the exact calculation of the full probability distribution of ~v (~x, t) − ~v (~x′ , t). Before turning to this computation, we first discuss the validity of this replica solution. 3.3 Discussion of the replica solution One is not able to solve exactly the directed polymer problem. The variational method to which we have resorted is however known to be a good approximation to the real behaviour of the directed polymer ([21, 30]). An interesting advantage of this approach is that it gives exact answers for the thermodynamics in the limit of a large number of dimensions N → ∞. Technically, as shown in [21], this comes from the fact that this variational Ansatz can be seen as a resummation of the ‘Hartree’ diagrams in the perturbation theory for the correlation function, which are the only ones surviving in the large N limit. However beyond the thermodynamic potentials, there are some quantities for which this approach fails to give the right answer, even 10 for N → ∞. It is unfortunately the case of the velocity correlation function defined in (2.15), for which our calculation may only become exact in the large N and large Reynolds limit. The problem can be understood as follows: the replica calculation is expected to reproduce faithfully the thermodynamical behaviour of the system, by correctly describing the low-lying energy states. The weight of these low-lying states is found to be distributed as ∝ W −1−m (1 − W )m−1 . Hence the moments [PΩ (~x)]η which are primarily determined by the low-lying states (i.e. by those with large weights) are expected to be accurate, while those which are sensitive to the W → 0 part of the distribution are not. This is the case when η < m, in particular for log PΩ (~x) which we need in order to calculate the velocity in the Burgers language. Hence, only when m → 0, i.e. at infinite Reynolds number, can the method be reliably used to determine exactly the statistical properties of the velocity field. To summarize this discussion we expect that this approach may be exact only when N → ∞ and m → 0. This issue can be studied from the solution by a study of the correlation identities, some of which will be checked in section (4.3). In any case the picture which emerges from the discussion in the next section is an appealing one which seems to be a good approximation even at small N. Eventually, we want to discuss an important detail. The previous replica calculation was performed, for simplicity, in the case of a periodic polymer, i.e. one for which x(t) ≡ x(0). From the Burgers equation point of view, however, the end point of the ‘polymer’ must be left free. As argued by Nelson and Vinokur in a different context [31], the difference between points far ‘inside’ the chain and boundary points is the fact that the statistical weight of the former is given, in quantum mechanical language [32], by the product of two propagators, while the end point only needs one. If the ground state wave function of the corresponding quantum mechanical Hamiltonian decays exponentially on a characteristic length scale ℓ, this means that the lateral fluctuations of the polymer’s end point decay on length scale ℓ, while the fluctuations for points in the bulk decay on a length scale ℓ/2. The study of the system with periodic boundary conditions which we performed in Sect.(3.1) only deals with these fluctuations in the bulk. The simple consequence of this analysis for our problem is that formula (3.10) can be used for free end points as well, although the value of δ must be doubled, i.e.: δ = 2m∆2 . (3.12) 4 Structure of the velocity field 4.1 Qualitative Arguments We can now use our ‘dictionary’ between the two problems, in particular the fact that the velocity field is given by the derivative of the free-energy of the polymer problem with respect to ~x. For a given realisation of the forcing, 11 this free-energy is simply the logarithm of the probability distribution PΩ (~x) given in Eq. (3.10); one can thus express ~v as: 2ν ~v = δ P α 2 /2δ Wα (~x − ~rα )e−(~x−~rα ) P −(~ x−~ rα )2 /2δ α Wα e . (4.1) We shall study the statistical properties of this velocity field at large Reynolds numbers. We recall that from the previous analysis, we have δ = 2m∆2 ≃ 24/3 ∆2 /Re. Typical snapshots of such a velocity field in N = 1 or N = 2 dimensions are given in Fig. 1 a,b. Very clearly, a cellular structure appears. Within each cell, the velocity field is radial ~v ∼ 2νδ (~x − ~rα ), with a rapid variation (shocks) across the boundaries of these cells. This structure can be understood qualitatively from Eq. (4.1), using the important property that the weights Wα have a very broad distribution. P More precisely, we know that Wα = exp(−fα )/ γ exp(−fγ ), where the fα are independant random variables with an exponential distribution increasing as exp(mf ). For m small, there is a very strong hierarchy between the smallest f appearing in the sum. For instance the gap between the smallest fα and the next scales as 1/m ∝ Re. So the low lying fα , which are the only terms contributing to the sum, behave at large Re as fα ≃ fˆα Re, and the velocity field in this regime takes the form: 2ν ~v ∼ δ 2 −~ rα ) − ~rα ) exp Re −fˆα − (~2x7/3 2  h i∆ P 2 (~ x −~ r ) α ˆ α exp Re −fα − 7/3 2  x α (~ P h 2 i . (4.2) ∆ For a given ~x and a large Re, the major contribution to the sum comes from the largest term. This dominant term will suddenly switch from –say– α to β when the space dependent Gaussian factor compensates the weight difference, i.e. whenq(~x∗ − ~rα )2 ∼ (~x∗ − ~rβ )2 + mδ . This leads to a typical size of cells of order δ m ∝ ∆, i.e. the injection length itself. The width ℓ ∗ rα |ℓ of the shock separating two cells is obtained by writing that |~x −~ ∼ 1, i.e. δ ∆ ℓ ∝ Re ≪ ∆, which shows that the notion of cells is indeed well defined at large Reynolds. This construction allows one to guess the structure of the probability distribution of the velocity difference between nearby points. Let us present the argument in N = 1 dimension (its generalisation to higher dimensions is straightforward), and call x and x + r these two points. The probability that a cell wall is present within the interval [x, x+ r] is obviously given by p = ∆r . In the limit where p is small, the velocity difference u = v(x+ r) − v(x) is equal to 2νr with probability 1 − p, and of order 2ν∆ ∼ νRe = v∆ δ δ ∆ with probability p. Hence:  Pr (u) ∼ (1 − p)δ u − 1 u 2νr +p f δ v∆ v∆    (4.3) where f (.) is a certain scaling function. This is true in the limit Re → ∞ where the internal structure of the shocks can be neglected. We will see 12 below that precisely such a form is obtained from an exact calculation. Let us now estimate the various moments of u using expression (4.2): uq ∼ (1 − p)( v∆ r q q , ) + p Aq v∆ ∆ (4.4) where Aq = duuq f (u). In the limit where ℓ ≪ r ≪ ∆, one finds that for all q larger than 1, one has R r , ∆ (4.5) r q ) . ∆ (4.6) q uq = Aq v∆ whereas for q < 1, one finds q uq ∝ v∆ ( The velocity field is thus strongly intermittent, with a ‘multifractal’ spectrum given in Fig. 2. Note however that the q = 3 moment scales à la Kolmogorov, i.e as in the real turbulence problem. The presence of large scale structures (shocks) forming an N− dimensional froth-like pattern (see Fig. 1-b) is responsible for such a strong intermittency. Large scale structures in ‘true’ turbulence are similarly thought to be the origin of the experimentally observed intermittency, which is however much milder (see Fig. 2). The deep reason of this difference is probably related to the ‘dimension’ of the large scale singularities, which is N − 1 in the present case and only 1 for vortex lines in hydrodynamical turbulence. This strong intermittency, due to the cell and shock structure, has already been discussed in the one dimensional Burgers turbulence in the decaying (unforced) case ([33, 34, 35]). To the best of our knowledge the forced case has not been studied. Furthermore we are able to derive the exact form of the probability distribution function of velocity differences, to which we now proceed. 4.2 Exact results at infinite Reynolds number Using the direct evaluation of the moments of the velocity difference from Eq. (4.1), one can establish, after rather long manipulations detailed in Appendix B, the following form for the full distribution of longitudinal velocity differences. Let us first start by the equal time case, and write u = v 1 (~x) − v 1 (~y ), 1 1 √ (The other where the component ‘1’ is along the ~x − ~y axis, and l = y∆−x 2 components would be treated in a similar manner). The result then reads:  2ν Pl (u) = δ u − l δ s  δ  −l2 /8 e m  Z ∞ −∞ DhR(h, |l|) 2ν +|l| dh Dt Dsδ u + (t + s) δ −∞ −h−|l|/2 h−|l|/2 Z ∞ Z ∞ Z ∞ 13 s  δ sign(l) R(h, |l|)2 , m (4.7) i−1 h where R(h, l) = e−hl/2 M0 (h − l/2) + ehl/2 M0 (−h − λ/2) (M0 is related −u2 to the standard error function – see Appendix B), and Dx ≡ √dx2π e 2 . Note that Eq. (4.6) has precisely the structure which we guessed using qualitative arguments (Eq. (4.3)). We have tested this formula numerically by generating a one-dimensional velocity field using eq. (4.1), for Re = 100, and directly computed the second moment u2 as a function of ρ, which we compare in Fig 3 with the exact formula for Re → ∞ obtained by inte∆ grating (4.7). For Re ≪ ρ ≪ ∆, corresponding to the ‘inertial range’, one 2 2 |x−y| 16Re ν 2 , or in terms of the ‘traditional’ energy spectrum finds u = √π∆2 ∆ E(k) ≡ k N −1 h~v(~k)~v (−~k)i, E(k) ∝ k −2 for all N. Note the saturation for distances much larger than the cell size ∆, on which we shall comment later. We have also computed the weight of the δ peak and compared it to the one obtained analytically, with good agreement again. We have also extended our analysis to two interesting situations. First of all, one may consider the case of a finite Reynolds number. The calculation (2) ∆ , of the second moment u2 shows that a new length scale appears3 ℓd = Re separating the above linear regime of u2 at larger distances, from the regular, quadratic behaviour at small length scales: u2 = ǫ 2 r ; 2ν (2) r < ℓd (4.8) which is the standard result (up to the numerical prefactor) obtained in turbulence for length scales smaller than the dissipation length (see e.g. [36]). Note that the ‘dissipation length’ ℓd depends on the moment of the velocity one wishes to calculate: this is another consequence of intermittency. Matching the regular behaviour (4.7) with (4.4) or (4.6) suggests that the q th order q (q) dissipation length scales as ℓd = ∆Re 2−2q ; only for q = 3 does one recover (3) the usual Kolmogorov dissipation length ℓd = Re∆3/4 . From our Ansatz for the velocity field (4.1), and the fact that ~rα (t) evolve according to (3.11), one can see that the full velocity correlation function has ∆ ≪ r ≪ ∆): the following scaling form (in the limit Re  r   [~v (~x + ~r, t) − ~v (~x, t′ )]2 = r g  q ′ 2C(t − t) (4.9) where C(τ ) is defined in Eq. (3.11) and g(.) is a certain scaling function. We have obtained the precise form of g in the case where t and t′ are well ‘inside’ the polymer, but did not attempt to compute it for ‘end points’ – where additional numerical factors would appear (see the discussion at the end of section 3.3). q In the limit t−t′ ≪ τ ∗ (see Eq. (3.11)), 2C(t − t′ ) = v∆ |t−t′ |: Eq. (4.9) then means that fluctuations travel in a ballistic way with a ‘velocity’ fixed by 3 (2) Note that ℓd is the width of the shocks ℓ introduced above 14 the injection scale. the dynamics recovers a diffusive q For larger timeqscales, |t−t′ | ′ character, since 2C(t − t ) ∝ ∆ τ ∗ . Note however that the effective 2 diffusion constant νturb. = ∆τ ∗ ∝ Re ν is enormously enhanced compared to its ‘bare’ value ν – as is the case for usual turbulence. From Eq. (4.9) in the limit r → 0, one finds that for coinciding points, the velocity difference grows with time as: [~v (~x, t) − ~v (~x, t′ )]2 ∝ 4.3 q 2C(t − t′ ) (4.10) ‘Sum-Rules’ and the large Reynolds limit There are a number of ‘sum-rules’ that the correct solution of the Burgers equation should satisfy, allowing us to test our prediction for the structure of the velocity field, as given by Eq. (4.1). The first and most interesting one physically is the ‘energy conservation’, i.e. 1 ∂~v 2 ~ v + ν ~v · ∆~v + ~v · f~ = 0 = −~v · (~v · ∇)~ 2 ∂t (4.11) The last term of this expression is the injected energy, and is equal to + N2 ǫ. The two first terms represent the ‘dissipated’ energy – notice that contrarily to the incompressible Navier-Stokes case, this dissipation is non local (i.e. it cannot be expressed in terms of velocity derivatives only). We find (see Appendix C, Eq. C.11), that the total dissipation is given by − N2 ǫ(1 − m). Hence we find that the energy conservation is indeed exactly satisfied in the limit m → 0 (Re → ∞) at fixed ǫ (i.e. for ∆ → ∞ or ν → 0). Interestingly, in this limit, this sum rule is true independently of N. Eq. (4.11) can also be interpreted slightly differently. Suppose that at t = 0 the forcing is switched off. The subsequent evolution of the energy density is then given by the 2/3 2 dissipative terms, i.e. N∆2/3 dǫdt ≃ − N2ǫ , leading to ǫ ∼ ∆t3 , or else for the velocity scale v∆ (t) ∼ ∆t . 2 2 v ) We have also checked that ∂(~∂t = 0 in the limit Re → ∞ for all N, while ∂∇·~ v = 0 identically for all N and Re. The calculation of velocity correlations ∂t ∂ ~v (~x)~v (~y ), are much more intricate, and for non coinciding points, such as ∂t are currently under investigation. Hence these direct checks of certain (local) correlation identities suggest that our Ansatz might be exact for large Re. 5 Discussion There are quite a number of points worth discussing under the light of the previous results, before commenting on the similarities and differences with the ‘real’ turbulence problem. 15 5.1 Burgers turbulence in small dimensions As stated above, our results are a priori only exact for large dimensions and Reynolds number, the method we have used becoming a variational approximation in finite dimensions. By comparison with other problems where this approach was used, however, one expects that our solution describes faithfully the physical situation, even for relatively small values of N. Furthermore, all the sum-rules which we have checked so far (Section 4.3) are satisfied for any N in the limit Re → ∞. Hence, we believe that the forced Burgers equation in say N = 3 dimensions will produce a cellular arrangement of the flow pattern, as described above. It would be extremely interesting to test this prediction numerically, as well as to measure the intermittency corrections – which might be weaker in finite dimensions than the ones obtained within our Ansatz. For N ≤ 2, one however knows from previous studies that the structure of the replica solution changes: instead of a ‘one step’ breaking, a full continuous breaking scheme is needed. This raises the interesting question of knowing whether such a scheme could describe a more complicated behaviour of the moments of the velocity field than the one encountered above. We do not have a complete answer to this question, but it seems that the solution obtained for N ≤ 2 does not lead to a qualitatively different picture, the reason being that all the length scales appearing in recursive construction of the sample dependent measure PΩ [21] (i.e. the width of the Gaussians), as well as the ‘m’ parameters describing the weight distributions, are all of the same order of magnitude. Thus, forced Burgers turbulence in one dimension is expected to look very much like the picture shown above (Fig 1-a), as indeed is well known from studies of ‘decaying’ turbulence from random initial conditions [33, 34, 35]. More work on this aspect would certainly be interesting, in particular to study a variant of the present problem for N = 1, where f – rather that φ – is a random noise (see [37, 38]). In this case also, a full breaking of replica symetry is needed to describe the velocity field. 5.2 Long-distance scaling of the velocity field As discussed in Section 2, the main difference between this work and previous studies on the directed polymer problem lies in the regime of length scales. Usually, in the directed polymer problem, one assumes that the correlation length of the potential ∆ is very small, and one is interested in the longdistance x ≫ ∆ scaling behaviour of – say, the free-energy, where non trivial exponents appear. On the other hand, in line with most studies of the turbulence problem, we focused on the contrary on the small scales x ≪ ∆ regime, where an inertial range appears, characterized by an energy cascade E(k) ∝ k −2 . The existence of these two regimes for the Burgers equation suggests that a similar situation may also occur in ‘real’ turbulence, where a non trivial scaling regime could exist for x ≫ ∆, characterized by a velocity 16 correlation converging towards its asymptotic value as a power law: [~v (~x + ~r) − ~v (~x)]2 ≃ 2~v(~x)2 − B ω r 2(1− ζ ) + ... r≫∆ (5.1) ω or equivalently, by a new exponent for the energy spectrum: E(k) ∝ k 1−2 ζ , where ω and ζ are the generalisation of the exponents defined for the polymer problem in Eqs. (2.13-14), corresponding now to the usual large scale situation r ≫ ∆ studied in directed polymers. Numerical results on directed polymers in 3+1 dimensions [17] give ζ ∼ 0.6, ω = 2ζ − 1 ∼ 0.2. It would be very interesting to analyze experimental data beyond the injection length ∆ along these lines. 5.3 Passive scalar dispersion A subject of recent debate is the behaviour of a passive scalar in turbulent flows, which also shows experimentally ‘anomalous’ density fluctuations – in particular exponential tails [39]. Although a detailed study is beyond the scope of the present paper, it is interesting to discuss qualitatively this problem for Burgers’ turbulence. Let us suppose that the diffusing scalar obeys a Langevin equation of the form d~x = ~v(~x, t) + ~η(t) dt (5.2) where ~v (~x, t) obeys the forced Burgers equation Eq. (2.1), and η(t) describes the molecular diffusion, with diffusion constant equal to D. In the ‘adiabatic’ limit where the evolution of the velocity field can be assumed to be small compared to the equilibration time (i.e D ≫ v∆ ∆), the tracer concentration θ(~x, t) is simply given by the Boltzman equilibrium: θ(~x, t) ∝ exp − h(~x, t) D (5.3) where h(~x, t) is the velocity potential introduced in Eq. (2.5). Using h(~x, t) = 2ν log PΩ (~x, t) + cst., one finds that: 2ν θ(~x, t) ∝ [PΩ (~x, t)]− D (5.4) Now the full distribution function of PΩ can be exactly calculated, and is found to be a totally asymmetric Lévy distribution Lµ with index µ = m. This in turn allows one to obtain the full distribution of θ, as D D P(θ) = θ−1− 2ν Lm (θ− 2ν ) (5.5) Hence, we find that the tail of the tracer density distribution is a power-law for Dm small θ, P(θ) ∼ θ−1+ 2ν , which becomes broader and broader as the h Reynolds i Dm Dm −1 −1 number increases (m ∝ Re ). For large θ, P(θ) ∼ θ 2ν exp −cstθ 2ν . 17 Note however that our adiabatic assumption breaks down precisely when Dm < 1. It would be interesting to understand the nature of the fluctuations ν of θ in the large Reynolds regime. Hence we have argued that the fluctuations of tracer concentration are strongly anomalous in Burgers’s turbulence. In more physical terms, Eq. (5.2) shows that the tracer is convected towards the shock regions, where the concentration piles up. High density ‘sheets’ of particles spontaneously form, revealing the ‘froth-like’ structure of the flow. This is similar to what happens for the Burgers equation with random initial conditions (but no forcing) [40], and might be relevant for astrophysical applications [41]. Finally, it must be noted that the temporal dispersion of tracers is convective at short times, since it essentially follows the velocity field which evolves according to Eq.(3.11). This is, as noted above, in contrast with Richard3 son’s diffusion, which states that x ≃ t 2 . Convective dispersion is however rather commonly observed (see [42]). 5.4 Turbulence versus ‘Burgulence’ The differences between Burgers’ turbulence (coined ‘Burgulence’ by Fournier and Frisch in [33]) and hydrodynamical turbulence have been discussed many times. The most important one is the absence of vortex line singularities in a potential flow, which are thought to play an important role in turbulence [43]. In our case, singularities are concentrated on N − 1 structures, giving rise to much larger ‘intermittency’ corrections. Second, the energy is only dissipated by viscosity at small length scales in turbulence, while there is an additional dissipation term in Burgulence (which is, as discussed above, non-local). This might be another important difference although, as shown above, it does not prevent the existence of a well-defined inertial range where the energy spectrum follows a k −2 decay. A third difference invoked by Kraichnan is that in fluids, the incompressibility condition is maintained by the underlying pressure field, which would play an important role in the dynamics of the fluctuations and be at the origin of the Richardson-Kolmogorov scaling x ∼ t3/2 , instead of the convection law x ∼ v∆ t found here. On the other hand, the Burgers and incompressible Navier-Stokes equations look very similar, in particular from a dimensional analysis point of view, which is at the heart of Kolmogorov’s argument; furthermore, the third moment of the velocity difference indeed scale in the same way (at least for large N). On a technical level, the simplest closure scheme for turbulence is Kraichnan’s DIA, which leads both for the Navier-Stokes equation and for the Burgers’ equation, to a convective dynamics x ∼ v∆ t and to a k −3/2 energy spectrum. However, DIA’s extension to Lagrangian coordinates (LDIA) was argued by Kraichnan [5] to reproduce exactly Kolmogorov’s scaling for the Navier-Stokes equation, but not for the Burgers case, where results similar 18 to some of ours are obtained 4 . It would be interesting to investigate the precise relation between the two seemingly very different approaches, and in particular to compare LDIA to our results. 5.5 Conclusions We have thus used a method inspired from spin-glasses to investigate a toy model of turbulence, and to propose an Ansatz for the structure of the velocity field, which should become exact in high dimensions and when the Reynolds number is also large. We find that beyond a critical Reynolds number, there exists a well defined inertial range where the energy spectrum decreases as k −2 . The scaling variable is v∆x t , where v∆ is the velocity at the injection scale ∆. The third moment of the velocity difference scales linearly with distance, à la Kolmogorov, but strong intermittent corrections come into play, due to the presence of shocks localized on a froth-like, cellular pattern. The full distribution for the velocity difference is obtained exactly. Interesting scaling results are also argued to hold at scales larger than the injection scale, and we suggest that experimental data on grid turbulence could be analyzed accordingly. Our results are presumably qualitatively correct in low dimension; numerical simulations would be welcome. We have discussed qualitatively the passive scalar problem and we have found power-law tails in the concentration distribution, reflecting the localisation of the tracers near the shocks. From a technical point of view, it would be interesting to understand the precise relation between the present approach and Kraichnan’s Lagrangian DIA, which gives the same scaling as the ones obtained here. From a different point of view, one could also generalize this work to the case of a spacecorrelated forcing term f (~x, t) with a power-law correlation function. This is the starting point of the renormalisation group (RG) analysis of turbulence: the exponent describing the decay of the power-law function is chosen as to reproduce Kolmogorov’s scaling and an RG procedure [6] is applied to obtain adimensional prefactors (the Kolmogorov constant). The model studied here could provide an interesting benchmark to discuss the validity of such a procedure. In fact it can be used to test all the various approximate methods which have been introduced in the study of fully developped turbulence. Finally, our Ansatz for the velocity field [Eq(4.1)], which is inspired from our ‘replica’ approach to the problem, has some interesting mathematical properties (see in particular Appendices B and C). It could be fruitful to generalize Eq. (4.1) to describe rotational flows. Acnowledgements MM thanks the SPhT at the CEA Saclay for its kind hospitality. This work has been supported by the EC grant CHRX CT92 0063. 4 Although the situation investigated by Kraichnan was that of ‘decaying’ turbulence; the intermittent corrections were furthermore not discussed 19 Appendix A: Solution of the saddle-point equations. The starting point of the one-step replica symmetry breaking calculation is the expression of the free-energy in terms of G̃(ω), G0 (ω), G1 (ω) and m. Using the expression (AII.11) given in [21] for the trace of the logarithm of a one-step Parisi matrix, we find that: F= W B1 −N/2 B0 −N/2 1Z ˜ dω(cω 2 + µ)G̃(ω) + {(1 − m)(1 + ) + m(1 + ) } 2 2 N∆2 N∆2 1 log(G̃(ω) − mG0 (ω) − (1 − m)G1 (ω)) m # 1−m G0 (ω) − + log(G̃(ω) − G1 (ω)) m G̃(ω) − mG0 (ω) − (1 − m)G1 (ω) − 1 2 Z ˜ dω  (A.1) R ˜ G̃(ω) − Gi (ω)]. (We have set the temperature to T = 1). where Bi ≡ 2 dω[ Differentating F with respect to G̃(ω), G0 (ω), G1 (ω) yields equations (3.5-6) specialized to the one-step solution, with Gc (ω) ≡ G̃(ω) − mG0 (ω) − (1 − m)G1 (ω) = 1 [G−1 ]c = 1 (µ + cω 2 ) [G−1 ]0 G0 (ω) = − (µ + cω 2 )2 and G1 (ω) − G0 (ω) = (A.2), (A.3) S 1 (µ + cω 2) m(µ + cω 2 + S) (A.4) where we have introduced S ≡ m([G−1 ]0 − [G−1 ]1 ). Using the definitions of B0 , B1 , we obtain the equations: B1 − B0 = − 2 m Z ˜ dω 1 S 2 µ + cω µ + cω 2 + S (A.5) and, using (A.2) 1 µ + cω 2 (A.6) from which we can deduce B0 and B1 as a function of S. Inserting these values into Eq. (3.5) then leads, in the limit µ → 0, to: mB0 + (1 − m)B1 = 2 [G−1 ]0 ∼ 0; Z Z ˜ G̃(ω) − mG0 (ω) − (1 − m)G1 (ω)] = 2 dω ˜ dω[ S = −m[G−1 ]1 = mW 1 √ ]−1−N/2 [1 + 2 2 ∆ N∆ Sc (A.7) Differentiating now (A.1) with respect to m, after a few manipulations, leads to s S 1 √ ]−N/2 (A.8) = 2m2 W [1 + 2 c N∆ Sc 20 Eqs (A.7-8) allows us to obtain both S and m as: 4m3 W ∆2 c = (1 − 2m 1−N/2 ) N (A.9) and )2 (1 − 2m N S= 4m2 ∆4 c Using Eqs (A2-7), we thus obtain in the limit µ → 0: 1 Qc = √ , 2 µc Q1 − Q0 = 1 √ , 2m µc 1 Q̃ − Q1 = √ 2 Sc (A.10) (A.11) N and Q0 ∝ µ 4 −1 . From this, we also obtain the inverse R of the matrix Q as: 1 √ = 2 µc, Qc N Q0 ∝ µ4, 2 Qc √ 2 Sc Q0 − Q1 = R1 − R0 = Qc [Qc + m(Q0 − Q1 )] m Rc = R0 = − 21 (A.12) Appendix B: Probability distribution of the velocity difference In this Appendix we compute the probability distribution function (pdf) of the difference of velocities between two points ~x and ~y , projected onto ~x − ~y , at large Reynolds numbers. We shall proceed in three steps: first we show how to compute the first few moments for general Reynolds numbers. Then we show how the expression simplifies for large Re. This simplification is such that we can extract in this large Re limit all moments and deduce from it the pdf. Our starting point is the result from the replica computation, which provides the following random process to build up the velocity distribution: at a given time, the velocity field is given by: 2ν ~v (~x) = δ P α 2 /2δ Wα (~x − ~rα )e−(~x−~rα ) P −(~ x−~ rα )2 /2δ α Wα e (B.1) The sum over α goes from 1 to M. The points ~rα are uniformly distributed in the volume [−L/2, L/2]d . The weights Wα are random numbers drawn as follows: First one chooses M “energies” fα at random, they are identically distributed independent random variables with a probability distribution: P({) = m⌉m({−{⌋ ) θ({⌋ − {) (B.2) Then the weights are given by: e−fα Wα = P −fβ βe (B.3) We shall let M, L, and fc go to infinity together, keeping the density of states Me−mfc /L fixed. The ~rα and fα are uncorrelated. We shall denote respectively by Er and Ef the expectation values with respect to these two sets of random variables. Let us first evaluate the second moment. We take ~x − ~y along the first axis, and compute the longitudinal correlation: 2ν v1 (x)v1 (y) = δ  Er,f P 2 2 α,β 2 ]/2δ e−fα −fβ (~x − ~rα )1 (~y − ~rβ )1 e−[(~x−~rα ) +(~y−~rβ ) P −fα −fβ e−[(~ x−~rα )2 +(~ y−~rβ )2 ]/2δ α,β e ! . (B.4) We shall compute separately the two contributions to this correlation which come from the terms α = β and α 6= β in the numerator of (B.4). We thus write:  2 2ν (g11 + g12 ), (B.5) v1 (x)v1 (y) ≡ δ where g11 and g12 respectively contain the terms α = β and α 6= β in (B.4). 22 To compute g11 we write the denominator of (B.4) in integral form, which gives: g11 = MEr,f Z ∞ dλ 0 Z ∞ 0 h 2 +(y−~ r dµe−2fα (x − ~rα )1 (y − ~rα )1 e−[(x−~rα ) 2 /2δ  2 /2δ exp e−fα λe−(x−~rα ) Y  h + µe−(y−~rα ) 2 /2δ  exp e−fβ λe−(x−~rβ ) β(6=α) i 2 /2δ + µe−(y−~rβ ) 2 α ) ]/2δ i (B.6) As for the mean over the free energies, we shall use repeatedly in this Appendix the following formula, valid for large fc : Z fc −∞ m(f −fc ) df me −kf −Ae−f e = ( me−mfc Am−k Γ(k − m), if k ≥ 1; 1 − e−mfc Am Γ(1 − m), if k = 0. (B.7) After taking this average over the free energie, (B.6) becomes: g11 = MEr Z ∞ 0 dλ Z ∞ 0 2 +(~ y −~r dµ (~x − ~rα )1 (~y − ~rα )1 e−[(~x−~rα )  2 /2δ me−mfc Γ(2 − m) λe−(~x−~rα ) Y h β(6=α) 2 /2δ + µe−(~y−~rα ) 2 /2δ  m−2 2 /2δ 1 − e−mfc Γ(1 − m) λe−(~x−~rβ ) + µe−(~y−~rβ ) 2 α ) ]/2δ m i (B.8) We now average over the values of ~rα and ~rβ , with a uniform measure in a box of size LN . In the limit of large L, M, fc at fixed density we get: g11 = mΓ(2 − m) Z Me−mfc L 2 +(~ y −~ r )2 ]/2δ d~r(~x − ~r)1 (~y − ~r)1 e−[(~x−~r) Me−mfc Γ(1 − m) exp − L " Z  Z ∞ 0 dλ ∞ Z 0 2 /2δ  λe−(~x−~r) −(~ x−~ r )2 /2δ d~r λe dµ 2 /2δ + µe−(~y−~r)  −(~ y −~ r)2 /2δ m + µe # m−2 (B.9) It is convenient to rewrite µ → λµ′ , and integrate over λ, which gives: g11 = (1 − m) R 2  2 Z 0 ∞ dµ 2 2 /2δ −[(x−r) +(y−r) ]/2δ e−(x−r) /2δ + µe−(y−r)  dr(x − r)(y − r)e R  m dr (e−(x−r)2 /2δ + µe−(y−r)2 /2δ ) m−2   . (B.10) R (In this expression the dr is over a single r variable, namely the component of ~r in the direction of ~x − ~y , the other components have already been integrated out, their contributions cancel between the numerator and the denominator. We denote x = ~x1 , and similarly y = ~y1 ) 23 It is straightforward to perform the same steps for the second contribution g12 in (B.5). One finds: g12 = m Z ∞ 0 Z Z dµ 2 /2δ dr(x − r)e−(x−r) −(y−r)2 /2δ dr(y − r)e Z    −(x−r)2 /2δ e −(x−r)2 /2δ dr e 2 /2δ e−(x−r) 2 /2δ + µe−(y−r)  −(y−r)2 /2δ m−1 + µe  −2 −(y−r)2 /2δ m + µe m−1   . (B.11) It should be clear that these techniques allow to write the velocity correlations of low order in a relatively closed form (this means reduced to some finite dimensional integrals, where all the averages over r and f have been taken care of, as well as the limits M → ∞, L → ∞ and fc → ∞). However the expressions are complicated enough, especially when one goes to high moments, and we have not found a closed form for the pdf in general. Fortunately the situation simplifies in the limit of large Reynolds numbers. Let us first work out the expression of the second moment in this Re → ∞ limit. We know from the replica solution that in this limit the width δ scales as δ ≃ 2m∆2 , where ∆ is the scale at which energy is injected, and m, the breakpoint in Parisi’s RSB solution, behaves as m ≃ 21/3 /Re. Hereafter we shall use m instead of Re, and we want to understand the small m limit of the velocity pdf. We proceed and first work out the small q m limit of theq two point correlation. In g11 we change variable to x = x̃ δ/m, y = ỹ δ/m; We shall take the m → 0 (large Re) limit keeping x̃, and ỹ fixed. The algebraic√distance between the two points is measured by l = (x̃ − ỹ) = (~x − ~y )1 /(∆ 2). We q also change the dummy integration variables in (B.10) from r to z = r δ/m − (x̃ + ỹ)/2, and from µ to h = −m ln(µ)/|l|. This gives: g11 ! δ = (1 − m) |l| m2 Z ∞ −∞ dhe−h|l|/m I1,1 (h) I0,0 (h) , (B.12) where we have introduced the functions (defined for integer k1 , k2 ): Ik1 ,k2 (h) ≡ e−z 2 /2+lz(k 1 −k2 )/2m dz √ (l/2 − z)k1 (−l/2 − z)k2 2π Z  elz/2m + e−h|l|/m−lz/2m m−(k1 +k2 ) . (B.13) Their small m limit is easily worked out by a saddle point integration. The result for k1 ≥ 1, k2 ≥ 1 is: 2 Ik1 ,k2 (h) ∼m→0 e−h /2 m sign(l)k1 +k2 eh|l|(k2/m−1/2) √ |l| 2π !k1 |l| +h 2 !k2 |l| − +h 2 24 Γ(k1 )Γ(k2 ) Γ(k1 + k2 ) (B.14) While the results for k1 k2 = 0 read: l2 /8 Ik1 ,0 (h) ∼m→0 e l2 /8 I0,k2 (h) ∼m→0 e sign(l) sign(l) k2 k1 Mk∞ Mk∞ " ! |l| −h − , ∈ ! |l| h|l|(k∈ /m−∞) ⌉ , h− ∈ # |l| |l| e−h|l|/2 M′ (h − ) + ⌉h|l|/∈ M′ (−h − ) . I0,0 (h) ∼m→0 e ∈ ∈ (B.15) In these equations we have used the definition: l2 /8−h|l|/2 Mk (§) ≡ Z ⌈‡ ∈ √ ‡k ⌉−‡ /∈ . ∈π ∞ § (B.16) Therefore we get for the m → 0 limit of g11 : g11 δ = m s 2 π Z ∞ 0 dhe−h 2 /2 h2 − l2 /4 h el2 /8 e−h|l|/2 M′ (h − |l| ) ∈ + ⌉h|l|/∈ M′ (−h − |l| ) ∈ i . (B.17) The small m limit of the second contribution to this second moment can be worked out with the same technique. One gets: g12 δ −l = m π Z ∞ 0 dhe−h 2 1 h el2 /4 e−h|l|/2M′ (h − |l| ) ∈ + ⌉h|l|/∈ M′ (−h − i |l| 2 ) ∈ . (B.18) We remember that from two point correlations between √ (B.5) the velocity 2 points at distance l∆ 2 equals (2ν/δ) (g11 + g22 ) . It is therefore finite when m → 0. It is interesting to work out its small l limit, which gives the correlation between two points whose distance is small with respect to the injection scale ∆ but large with respect to the dissipation scale (since we have taken the limit Re → ∞ -or m → 0- first): < v1 (x)v1 (y) >≃ − √ 8ν 2 x−y 2 2 πm ∆ ∆ (B.19) We now turn to the computation of higher order moments. We are interested in arbitrary moments of the velocity difference ~v(~x) − ~v (~y ) projected onto the direction of ~x − ~y (chosen as the ‘1’ direction. The starting point is analogous to (B.4): v1 (~x)p v1 (~y )p′ = ( X X 2ν p+p′ −(f +...+fαp +fβ1 +...+fβ ′ ) p e α1 ) Er,f δ α1 ,...,αp β1 ,...,βp 25 (~x − ~rα1 )2 + ... + (~x − ~rαp )2 (~x − ~rα1 )1 ...(~x − ~rαp )1 exp − 2 ! (~y − ~rβ1 )2 + ... + (~y − ~rβp′ )2 (~y − ~rβ1 )1 ...(~y − ~rβp′ )1 exp − 2 ! " X e−fα −(~x−~rα )2 /2δ α −p′ #−p  X 2  e−fβ −(~y−~rβ ) /2δ  (B.20) β We shall proceed as for the second moment. The first step is to exponentiate the denominator in (B.20) using 1 = p A B p′ Z 0 ∞ dλ Z ∞ 0 ′ λp−1 µp −1 −(λA+µB) dµ e . Γ(p) Γ(p′ ) (B.21) In order to perform the averages over the values of the free energies fα and the positions rα in (B.20) we need to distinguish how many indices in α1 , ..., αp , β1 , ..., βp′ are distinct one from another. Let us suppose that there appear in this sequence k different indices, which we call γ1 , ..., γk . We shall call qj the number of indices in α1 , ..., αp which are equal to γj , and qj′ the number of indices in β1 , ..., βp′ which are equal to γj . Such a configuration of indices, characterized by the number k ∈ {1, 2, ..., p + p′ }, and the sequences q1 , ..., qk , q1′ , ...qk′ (such that q1 + ... + qk = p, q1′ + ... + qk′ = p′ and for every j: (k) qj +qj′ ≥ 1) appears a certain number of times, which we call C∐∞ ,...,∐k,∐′∞ ,...∐′ , k in the sum over α1 , ..., αp , β1 , ..., βp of (B.20). For each configuration of indices, we just do the same transformations as for the second moment. We shall not repeat them here, and just give the final result which generalizes (B.12): v1 (x)p v1 (y)p′ X q1 ,...,qk ,q1′ ,...qk′ 2ν 1 ′ δ ′ = ( )p+p ( )(p+p )/2 δ m m (k) C∐∞ ,...,∐k,∐′∞ ,...∐′ k k Y |=∞   ′ ′ X e−p h|l|/m p+p dh Γ(k)mk−1 Γ(p)Γ(p′ ) k=1 −∞ Z ∞ −(∐| + ∐′| − m)I∐| ,∐′| (h) I′,′ (h)   . (B.22) Let us now work out the leading behaviour of this expression in the small m P (large Re) limit. From the integrals Iqj ,qj′ (h) we get a factor exp(h|l| j qj′ /m) ′ which exactly compensates the explicit e−p h|l|/m . Using δ = 2m∆2 , we find that the contribution to (B.22) from a given value of k scales as ′ ′ (ν/δ)p+p (δ/m)−(p+p )/2 ma(k) . A close look at the behaviours of the integrals Iq,q′ shows that a(k) is zero for k = 1, 2, and it is strictly positive for k ≥ 3. So we can neglect the terms with k ≥ 3 in (5.5). The term k = 1 is easily worked out. The only allowed configuration of indices has q1 = p and (∞) q1′ = p′ , and its degeneracy is C√,√′ = ∞. For the k = 2 term we keep only the leading terms which correspond to q1 = 0, q2 = p, q1′ = p′ , q2′ = 0 or to q1 = p, q2 = 0, q1′ = 0, q2′ = p′ , which also have a degeneracy 1 (*** a 26 verifier ***). Altogether the leading contribution to this moment at large Re is: Z 2ν p+p′ δ (p+p′ )/2 ′ p+p′ p p v1 (x) v1 (y) = ( ) ( ) sign(l) dh δ m  !p |l|  +h 2 !p ′ |l| − +h 2 2 e−(h /2+l √ 2π 2 /8) R(h, |l|) |l| |l| +(−1)p |l|M√ (−h − )M√′ (h − )R(h, |l|)∈ ∈ ∈ # . (B.23) It is a simple exercise to sum these moments and get the qth moment of the longitudinal velocity difference: 2ν δ (v1 (x) − v1 (y)) = ( )q ( )q/2 sign(l)q δ m q dt 2 √ e−t /2 +(−1) |l| −h−|l|/2 2π q Z ∞ Z Z dh " |l| −(h2 /2+l2 /8) qe √ 2π ds 2 √ e−s /2 (t + s)q R(h, |l|)2 h−|l|/2 2π ∞ R(h, |l|) # , (B.24) where: i−1 h R(h, l) ≡ e−hl/2 M0 (h − l/2) + ehl/2 M0 (−h − l/2) (B.25) Under this form the moments can be inverted and we obtain the explicit form of the pdf P (u) of the velocity difference u ≡ v1 (x) − v1 (y) between two √ points at distance x − y = l∆ 2 at large Re:  2ν P (u) = δ u − l δ +|l| Z ∞ −∞ dh  s  δ  −l2 /8 e m ∞ Z dt 2 √ e−t /2 −h−|l|/2 2π Z 2ν δ u + (t + s) δ s ∞ −∞ Z dh 2 √ e−h /2 R(h, |l|) 2π ∞ ds 2 √ e−s /2 h−|l|/2 2π  δ sign(l) R(h, |l|)2 . m 27 (B.26) Appendix C: Correlation identities In this Appendix we check that the solution we have found does satisfy some necessary identities of correlation functions, in the limit of large Reynolds numbers. We shall perform the explicit check in the case of the ”energy” balance. From Burgers’ equation one finds: 1∂ µ µ (v v ) = A + B + C , 2 ∂t 1 µ µ ρ ρ v ∂ (v v ) B ≡ ν v µ ∂ µ ∂ ρ v ρ C ≡ f ρ v ρ , (C.1) 2 with a convention of summation on the repeated vector indices µ, ρ from 1 to N which will be used in this whole Appendix. The overbar denotes the expectation value with respect to various realizations of the force f~. We shall compute successively the three contributions A, B (from our solution), and C (from a direct computation of the energy injected). In the end we shall check that their sum vanishes at large Reynolds number, as implied by the stationnarity of the forced flow. We use the Ansatz for the velocity given in (4.1) and developped in Appendix B. It will be useful to introduce somewhat more compact notations and define: uµ ≡ xµ − r µ (C.2) A≡− where r µ is a random variable which takes value rαµ with a probability Wα exp(−(~x − ~rα )/2δ). We shall denote as before by brackets the expectation values with respect to this process with fixed Wα and ~rα , while the averaging over Wα and ~rα (corresponding to the overline in (C.1)) will be denoted as in Appendix B by Er,f . (In terms of the directed polymer, < O > denotes a thermal average, and the overline denotes an average over quenched disorder). With these definitions we have: vµ = 2ν < uµ > δ 2ν ρµ 1 δ − (< uρ uµ > − < uρ >< uµ >) δ δ which leads to the following expression of A: ∂ρvµ = 2ν A= δ  3   . (C.3) 1 Er,f −< uµ >< uµ > + < uρ >< uµ >< uρ uµ > δ  1 − < uρ >< uµ >< uρ >< uµ > . δ The same steps give the expression for B:  28 (C.4) B=  2ν δ 3 1 Er,f [−< uρ >< uµ > (< uρ uµ > − < uρ >< uµ >) δ 1 1 + < uρ >< uµ uµ uρ > − < uρ >< uµ uµ >< uρ > . 2 2 The partial cancellation leaves:  A+B =  2ν δ 3  Er,f −< uµ >< uµ > + (C.5) 1 < uρ >< uµ uµ uρ > 2δ 1 (C.6) − < uρ >< uµ uµ >< uρ > 2δ We now evaluate each of the terms in this expression . The first term is given by:   x − ~rα )µ (~x − ~rβ )µ e−  α,β Wα Wβ (~ Er,f (< uµ >< uµ >) = Er,f  P P γ Wγ e−(~x−~rγ )2 /2δ (~ x−~ rα )2 +(~ x−~ r β )2 2 2δ    (C.7) Using the technique of Appendix B, one finds after some work that the only non vanishing contribution comes from the α = β term in (C.7) and gives: 1−m δN . (C.8) m The same technique can be applied to each term in (C.6). One obtains: Er,f (< uµ >< uµ >) = Er,f (< uρ >< uµ uµ uρ >) = 1−m 2 δ (N 2 + 2N) m2 (C.9) and: Er,f (< uρ >< uµ uµ >< uρ >) = 1−m 2 δ (N 2 + (2 − m)N) m2 (C.10) The sum gives in the end: 1 2ν A+B =− 2 δ  3 1−m δN m (C.11) One should notice that the terms of order N 2 vanish automatically due to the structure of the velocity field (4.1), and independently from the value of m. We now proceed to the evaluation of the last term, C, in the correlation identity (C.1). This point needs a little care because one must be more precise about the correct prescription of the forcing term, whether it is of Ito or Stratanovitch type. To settle this issue in a pedestrian but safe way, we 29 have discretized the time in units of τ0 ≪ 1. The Hopf Cole mapping can be carried out in this case, and we deduce from this computation: 1 1 C = f µ f µ = Nǫ 2 2 (C.12) Using the value (3.12) of δ: δ = 2m∆2 , the relation (3.7) between m and )1−N/2 = 12 m3 Re3 , together with the definition the Reynolds number: (1 − 2m N (2.4) of Re: Re3 = ǫ∆4 /ν 3 , we finally get: lim A + B + C = 0 Re→∞ (ǫ fixed) (C.13) which establishes the correlation identity for the energy balance (C.1). Using the same techniques, we have also checked that the correlation ∂ ∂ ~ ∇.~v = 0 and ∂t ((~v )2 )2 = 0 also hold for any N identities corresponding to ∂t in the limit of large Re. 30 Appendix D: Replica computation of the velocity two points correlation The aim of this Appendix is to compute the two point correlation v( x, t)v(y, t) directly from the replica method of sect.3. This will provide a check that the intermediate physical representation (B.1) is safe. In the course of this computation we shall also establish useful replica identities which can be of wider interest. To keep the computations simple we consider only the one dimensional case, N = 1. We start from the result (3.8) for the partition function for n replicas of the polymer arriving at time t at the points ~xa , a = 1, ..., n: Z(x1 , t)...Z(xn , t) = c X π  exp  1 2 X a,b  Rπ(a)π(b) xa (t)xb (t) (D.1) We shall compute Z(x, t)n/2 Z(y, t)n/2 which can be written as: Z(x, t)n/2 Z(y, t)n/2 = c ′ X {σ}  1 + σa 1 − σa 1X x+ y Rab exp  2 a,b 2 2     1 + σb  1 − σb x+ y 2 2 (D.2) where the ′{σ} is over n Ising spins σa = ±1, with the constraint that P a σa = 0. As an intermediate step to this computation, we first compute the action P′ of {σ} onto a polynomial in the spins. Let a1 , ..., ak be k different replica indices. We define: P ′ X Ak ≡ σa1 ..σak . (D.3) {σ} Clearly A1 = 0. As for A2 , one can use 0 = a,b ′{σ} σa σb = n+n(n−1)A2 to deduce A2 = 1/(1 − n). The general result can be deduced from an iteration of the above procedure: P Ak = 0 if k is odd 1 k−1 3 ... 1−n3−n k−1−n and this can be written in the form: Ak = Ak = 2n+1 B(−n/2, −n/2) P Z ∞ −∞ if k is even dy (cosh(y))n /2(tanh(y))k (D.4) (D.5) This expression is well defined for n < 0, and can be continued analytically to positive n. B is Euler beta function [44]. From this expression of Ak one 31 can deduce the generating function: g[h1 ...hn ] ≡ n X ′ X X exp X Y ha σa = ( a {σ} Ak ! cosh(ha )). a tanh(ha1 )...tanh(hak ) a1 <...<ak k=0 Z ∞ n Y 2n+1 cosh(y + ha ) dy = B(−n/2, −n/2) −∞ a=1 (D.6) Formula (D.6) is a replica identity which may turn out useful in other contexts. Here we will use it to compute v(x)v(y). Starting from (D.2), and P using the fact that b Rab = 0 (derived in Appendix A), we have: Z(x, t)n/2 Z(y, t)n/2 where α is defined as: ′ X   αX =c exp  Rab σa σb  2 a,b {σ} (D.7) (x − y)2 (D.8) 4 We call as usual r̃, r1 , r0 the various elements of the hierarchical Rab matrix. Using the fact that r0 = 0, we get from (D.6): α= Z(x, t)n/2 Z(y, t)n/2 Z ∞ −∞ dh0 n 2n+1 exp α (r̃ − r1 ) = B(−n/2, −n/2) 2 Z  √ Dh cosh (h0 + h αr1 ) m n/m 2n+1 n = exp α (r̃ − r1 + mr1 ) B(−n/2, −n/2) 2  Z ∞ nh0 Z    n/m √ −2h0 −2h αr1 )−2mαr1 m  dh0 e (D.9) Dh 1 + e √ where Dh ≡ dh/ 2π exp(−h2 /2). Changing variables to µ = exp(−2h0 ) and √ h = m r1 (z − x)sign(x − y), and using the fact that r̃ − r1 + mr1 = 0, we derive: Z ∞ dµ −n/2 2n n/2 n/2 µ Z(x, t) Z(y, t) = B(−n/2, −n/2) 0 µ # " √ Z  n/m  m r1 −mr1 (z−x)2 /2 −mr1 (z−y)2 /2 m dz e + µe . 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