Local coupling of shell models leads to anomalous scaling


 Leon Owen
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1 Z. Phys. B 12, (1997) ZEITSCHRIFT FÜR PHYSIK B c SpringerVerag 1997 Loca couping of she modes eads to anomaous scaing C. Uhig, J. Eggers Fachbereich Physik, Universität Gesamthochschue Essen, D Essen, Germany Received: 14 November 1996 / Revised version: 2 January 1997 Abstract. We consider cascade modes of turbuence which are obtained by restricting the NavierStokes equation to oca interactions. By combining the resuts of the method of extended sefsimiarity and a nove subgrid mode, we investigate the inertia range fuctuations of the cascade. Significant corrections to the cassica scaing exponents are found. The dynamics of our oca NavierStokes modes is described accuratey by a simpe set of Langevin equations proposed earier as a mode of turbuence [2]. This aows for a prediction of the intermittency exponents without adjustabe parameters. Exceent agreement with numerica simuations is found. PACS: Jv; 2.5.Ey; Eq; j I. Introduction Much of our intuitive understanding of turbuence is based on the concept of interactions which are oca in kspace. Physicay, it is based on the notion that most of the distortion of a turbuence eement or eddy can ony come from eddies of comparabe size. Turbuent features which are much arger ony uniformy transate smaer eddies, which does not contribute to the energy transfer. This immediatey eads to the idea of a chain of turbuence eements, through which energy is transported to the energy dissipating scaes. Accepting such a cascade structure of the turbuent veocity fied, it is natura to assume that the statistica average of veocity differences δv(r) = v(x+ r) v(x) over a distance r foows scaing aws D (p) (r) v(x+r) v(x) p r ζp (1) in the imit of high Reynods numbers. By taking veocity differences over a distance r, one probes objects of corresponding size. In addition to this assumption of sefsimiarity, Komogorov [1] aso made the seemingy intuitive assumption that the oca statistics of the veocity fied shoud be independent of argescae fow features, from which it is widey separated in scae. Because the turbuent state is maintained by a mean energy fux ɛ, the ony oca scaes avaiabe are the ength r and ɛ itsef, which eads to the estimate δv (ɛr) 1/3 or ζ p (cass) = p/3. (2) At the same time, one obtains an estimate for the Komogorov ength η =(ν 3 /ɛ) 1/4 (3) where viscosity is important. However, it was ony appreciated ater [2] that in turbuence ongrange correations aways exist in spite of oca couping. Namey, argescae fuctuations of the veocity fied wi resut in a fuctuating energy transfer, which drives smaer scaes. As a resut, the statistics of the smascae veocity fuctuations wi be infuenced by the energy transfer and fuctuations on widey separated scaes are correated, vioating the fundamenta assumption impicit in (2) and (3). Indeed, Komogorov [3] and Obukhov [4] ater proposed the existence of corrections to the scaing exponents (2), ζ p = p/3+δζ p,δζ p /= (4) which were subsequenty confirmed experimentay [5 8]. On one hand, carefu aboratory experiments have been performed at ever higher Reynods numbers [5, 9]. On the other hand, a new method of data anaysis [6, 7] has been successfu in eiminating part of the effects of viscosity. In particuar, for the highest moments up to p = 18 significant corrections to cassica scaing were found, a currenty accepted vaue for the socaed intermittency parameter µ being [5] µ = δζ 6 =.2, (5) which is a 1 % correction. The existence of corrections ike (5) impies that on sma scaes arge fuctuations are much more ikey to occur than predicted by cassica theory. This intermittent behavior is thus most noticeabe in derivatives of the veocity fied such as the oca rate of energy dissipation ɛ(x,t)= ν ( ) 2 ui / x k + u k / x i. 2 Much of the research in turbuence has been devoted to the study of the spatia structure of ɛ(x,t) [3, 1], but which
2 514 wi not be considered here. The statistica average of this quantity is what we simpy caed ɛ before. Owing to energy conservation, it must be equa to the mean energy transfer. The oca couping structure of turbuence has inspired the study of socaed she modes, where each octave in wavenumber is represented by a constant number of modes, which are ony ocay couped. This aows to focus on the impications of oca couping for intermittent fuctuations, disregarding effects of convection and mixing. The mode representation of a singe she serves as a simpe mode for the coherent structures a turbuent veocity fied is composed of, and which to date have ony been poory characterized, both experimentay and theoreticay. We caution that this eaves out two important aspects of turbuence, both of which have recenty been proposed to ie at the heart of intermittent behavior. First, we have assumed that the coherent structures are simpe in the sense that they ony possess a singe scae. But experimenta [11], numerica [12], and theoretica [13] evidence points to the importance of ong and skinny threads of vorticity. Athough their rea significance to turbuence is not without dispute [14], they have ed to quantitative predictions of intermittency exponents [15], in exceent agreement with experiment. Second, nonoca interactions are disregarded in she modes. On the other hand it has recenty been proposed [16] in the framework of perturbation theory that nonoca interactions can in fact be resummed to yied correction exponents. Given the compexity of the turbuence probem, this makes it a the more interesting to carefuy assess the possibiities for intermittent fuctuations in the case where both effects are eiminated. Thus the aim of this paper is to combine previous numerica [17, 18] and anaytica [19, 2] efforts to gain insight into intermittent fuctuations in modes with oca couping aone. We focus on a particuar cass of she modes, introduced in [17], which estabishes a direct connection with the NavierStokes equation. To this end the NavierStokes equation is projected on a sefsimiar seection of Fourier modes, which enforce oca couping. We wi adopt the name REduced Wave vector set Approximation (REWA) here [18]. In the present paper, most of our effort is devoted to separating inertia range fuctuations from other effects reating to the fact that scae invariance is broken either by an externa ength scae L or by the Komogorov ength η. In the neighborhood of either scae deviations from the power aw (1) are to be expected. However, for dimensiona reasons L or η appears even for a pure power aw as soon as δζ p /=. In the case of the stochastic mode considered in [2], the form of the rdependence of the structure function was shown to be D (p) (r) =b (p) (ɛr) p/3 (r/l) δζp, (6) where the b (p) are constants independent of Reynods number. This impies that the existence of an outer ength L has to be taken into account in any cacuation of the structure function, whie dependences of power aw form (6) need to be distinguished from nonuniversa effects. To this end we simuate extremey ong cascades, covering up to 5 decades in scae. To eiminate viscous effects as much as possibe, we use the method of extended sefsimiarity (ESS) [6, 7]. In addition, to make sure our resuts are independent of the method of data anaysis, we deveop a new subgrid mode using fuctuating eddy viscosities [21]. The resuts are in exceent agreement with those found from ESS. Finay, and perhaps most importanty, we use an anaytica cacuation [2] to predict the correction exponent without adjustabe parameters. Again, we find the same vaues within error bars. We show that the stochastic mode introduced in [19] gives an exceent description of a REWA cascade, both in an equiibrium and nonequiibrium state. We compare the equiibrium properties to adjust a singe free parameter in the stochastic mode. Thus we reach two objectives: first, we gain anaytica insight into the origin of intermittent fuctuations in a oca cascade. Second, we demonstrate for a simpe exampe how equiibrium information about the interactions of Fourier modes can be used to compute intermittency exponents. Unike she modes with ony one compex mode per she [22, 23, 24] exponent corrections δζ p for REWA cascades are quite sma [17, 18]. Carefu studies of inertia range fuctuations have found them to be significant [18], but their numerica vaue is ony about 1/2 of the numbers found experimentay [5]. This has ead to the idea that perhaps the experimentay observed exponent corrections are not genuine inertia range properties, but resut from extrapoations of (1) into regimes where stirring or viscous effects are important [17, 18, 25]. Possibe dynamica effects of the viscosity have been expored in [26]. We wi not foow up on this idea here. However, we note that the REWA cascade misses an essentia feature of threedimensiona turbuence and thus can hardy be expected to yied quantitative predictions. Namey, in rea turbuence, the number of modes within a she proiferates ike r 2 as one goes to smaer scaes, whie in cascade modes this number is constant. This can be remedied by aowing a particuar she to branch out into eight subshes, which represent eddies of haf the origina size. It has been argued [27, 19] that the competition between eddies of the same size is responsibe for the much arger growth of fuctuations observed in threedimensiona turbuence. The difficuty with this approach is that one aso has to take convection of spatiay ocaized structures into account. Aso, it is not obvious how to disentange interactions which are oca in kspace from those oca in rea space, as to systematicay reduce the couping of the NavierStokes equation to a tree of turbuence eements. Recenty deveoped waveet methods are a promising step [28], but they have so far been used ony for data anaysis [29]. The paper is organized as foows: In the next section we introduce both the modereduced approximations of the NavierStokes equation and the corresponding Langevin modes. The inertia range properties of the atter have ony one adjustabe parameter, as expained in the third section. This parameter is determined for a given seection of Fourier modes by considering the equiibrium fuctuations of both modes. We are thus abe to predict intermittency exponents of the turbuent state without adjustabe parameters. The resut is compared with numerica simuations of the REWA cascades in the fourth section. Exponents are determined by carefuy examining various sources of error, and are in exceent agreement with the theoretica prediction. In the fifth
3 515 section, we investigate tempora correations in REWA modes to inquire further in the origin of intermittency in modes with oca couping. This aso sheds ight on the reasons for the success of the simpe Langevin mode used by us. II. Two cascade modes A. Reduced wave vector set approximation The REWA mode [17, 18] is based on the fu Fouriertransformed NavierStokes equation within a voume of periodicity (2πL) 3. In order to restrict the excited Fouriermodes of the turbuent veocity fied to a numericay tractabe number, the NavierStokes equation is projected onto a sefsimiar set of wave vectors K = K. Each of the wave vector shes K represents an octave of wave numbers. The she K describes the turbuent motion of the arge eddies which are of the order of the outer ength scae L. This she is defined by N wave vectors k () i : K = {k () i : i =1,...,N}. Starting with the generating she K, the other shes K are found by a successive rescaing of K with a scaing factor 2: K = 2 K. Thus each K consists of the N scaed wave vectors 2 k () i, i = 1,...,N. The she K represents eddies at ength scaes r 2 L, i.e. to smaer and smaer eddies as the she index increases. At scaes r η the fuid motion is damped by viscosity ν, thus preventing the generation of infinitey sma scaes. Hence we ony need to simuate shes K,< ν, where ν og 2 (L/η) is chosen such that the ampitudes in K ν are effectivey zero. In this representation the NavierStokes equation for incompressibe fuids reads for a k K = ν = K : t u i(k,t)= ım ijk (k) u j (p,t)u k (q,t) p,q K k=p+q νk 2 u i (k,t)+f i (k,t) (7a) k u(k,t)=. (7b) The couping tensor M ijk (k) = [ k j P ik (k)+k k P ij (k) ] /2 with the projector P ik (k) =δ ik k i k k /k 2 is symmetric in j, k and M ijk (k) = M ijk ( k). The inertia part of (7a) consists of a triadic interactions between modes with k = p + q. They are the same as in the fu NavierStokes equation for this triad. The veocity fied is driven by an externa force f(k,t) which simuates the energy input through a argescae instabiity. Within this approximation scheme the energy of a she is E (t) = 1 u(k,t) 2, (8) 2 k K and in the absence of any viscous or externa driving the tota energy of the fow fied E tot (t) = ν = E (t) is conserved. The choice of generating wave vectors k () i determines the possibe triad interactions. This choice must at east guarantee energy transfer between shes and some mixing within a she. In [17, 18] different choices for wavenumber sets K Fig. 1. A twodimensiona projection of the kvectors in she K for both the REWA modes considered here. The sma set ( ) contains a vectors with 1,, and 1 as components. The arge set ( ), in addition, contains combinations with ±1/2 and ±2 are investigated. The arger the number N of wave numbers, the more effective the energy transfer. Usuay one seects directions in kspace to be distributed eveny over a sphere. However, there are different possibiities which change the reative importance of intrashe versus intershe coupings. In this paper, we are going to investigate two different wave vector sets, with N = 26 and N = 74, which we ca the sma and the arge wave vector set, respectivey. In Fig. 1 a twodimensiona projection of both sets is potted. The arge wave vector set aso contains some nexttonearest neighbor interactions between shes, which we put to zero here, since they contribute itte to the energy transfer. The sma set aows 12 different interacting triads, the arge set 51 triads, 333 of which are between shes. Since in the modes we consider energy transfer is purey oca, the she energies E (t), =,..., ν ony change in response to energy infux T 1 from above and energy outfux T +1 to the ower she. In addition, there is a rate of viscous dissipation T (diss) (t) which is concentrated on sma scaes, and a rate of energy input T in (t), which feeds the top eve ony, cf. Fig. 2. From (7a) we find an energy baance equation which governs the time evoution of the she energies E (t) d dt E (t) =T 1 (t) T +1 (t)+t (diss) (t)+t (in) (t)δ.(9) The different transfer terms are found to be T +1 (t) =2ı M ijk (k)u i (k,t)u j (p,t)u k (q,t) (1a) (+1) () T (in) (t) = ( Re u (k,t) f(k,t) ) (1b) k K T (diss) (t) = ν k 2 u(k,t) 2. (1c) k K In (1a) indicates the summation over a nextneighbor triads k = p + q with k K, p K +1 and (+1) () q K K+1.
4 516 Fig. 3. Logogpot of the structure function D (2) = E versus eve number for the sma cascade. At arge scaes, where the infuence of dissipation is negigibe, cassica scaing is observed. At sma scaes the turbuent motion is damped by viscosity. The Reynods number is Re = Fig. 2. The structure of a oca cascade. Eddies of size r 2 L are represented by their tota energy E. Ony modes of neighboring shes interact, eading to a oca energy transfer T +1 (t). The cascade is driven by injecting energy into the argest scae with rate T (in) (t). The turbuent motion is damped by viscous dissipation at a rate T (diss) (t) The driving force f(k,t) is assumed to act ony on the argest scaes, and contros the rate of energy input T (in) (t). As in [17] we choose f(k,t) to ensure constant energy input T (in) = ɛ : ɛu(k,t) f(k,t)= N u(k,t) 2 for a k K (11a) f(k,t) = for a k K. (11b) This eads to a stationary cascade whose statistica properties are governed by the compicated chaotic dynamics of the noninear mode interactions. Owing to energy conservation, viscous dissipation equas the energy input on average. The Reynods number is given by Re = LU ν = ɛl2 E ν, (12) since T = E /ɛ sets a typica turnover time scae of the energy on the highest eve. We beieve T to be of particuar reevance, since the argescae fuctuations of the energy wi turn out to be responsibe for the intermittent behavior we are interested in. In [18], for exampe, time is measured in units of L 2/3 ɛ 1/3, which typicay comes out to be 1/1th of the turnover time of the energy T. As we are going to see beow, this is rather a measure of the turnover times of Fig. 4. Time evoution of the energy transfer T 1 (t) from she K to she K 1. Typica excursions are arge compared with the mean vaue T 1 = ɛ. The time is given in units of T = E /ɛ, where E is the mean energy of the top eve the individua Fourier modes. Figure 3 shows the scaing of the mean energy in a ogog pot at a Reynods number of The inertia range extends over three decades, where a power aw very cose to the prediction of cassica scaing is seen. Beow the 1th eve the energies drop sharpy due to viscous dissipation. In Sect. III we are going to turn our attention to the sma corrections to 2/3scaing, hardy visibe in Fig. 3. Sti, there are considerabe fuctuations in this mode, as evidenced by the pot of the energy transfer in Fig. 4. Typica excursions from the average, which is normaized to one, are quite arge. Utimatey, these fuctuations are responsibe for the intermittency corrections we are going to observe. Thus within the REWAcascade we are abe to numericay anayze the infuence of fuctuations on the stationary statistica properties of a cascade with oca energy transfer on the basis of the NavierStokes equation. In Fig. 5 we pot the time evoution of the energy on the second eve of the cascade. One observes shortscae fuctuations, which resut from the motion of individua Fourier modes within one cascade eve. However, performing a foating average reveas a second time scae, which is of the same order as the turnover time of the top eve. As we are going to see in Sect. IV, this disparity of time scaes is even more
5 517 Fig. 5. Energy of a she ( = 2) as a function of time. The rapid fuctuations come from the motion of individua Fourier modes. A much onger time scae is reveaed by performing a foating average over one turnover time of the second eve (bod ine). The time is given in units of T = E /ɛ pronounced on ower eves. The physics idea is the same as in the microscopic foundation of hydrodynamics, where conserved quantities are assumed to move on much sower time scaes than individua partices. This motivates us to consider the energy as the ony dynamica variabe of each she, and to represent the rapid fuctuations of Fig. 5 by a whitenoise Langevin force. In this approximation we sti hope to capture the rare, argescae events characteristic of intermittent fuctuations, since the conserved quantity is the sow variabe of the system. Simiar ideas have aso been advanced for the conservative dynamics of a nonequiibrium statistica mechanica system [3]. B. The Langevincascade In this mode we take a phenomenoogica view of the process of energy transfer. The chaotic dynamics of the REWAcascade is modeed by a stochastic equation. We make sure to incude the main physica features of energy conservation and oca couping. In particuar, the dynamics is simpe enough to aow for anaytica insight into the effects of fuctuating energy transfer [2]. As in the REWAcascade, the turbuent fow fied is described by a sequence of eddies decaying successivey (Fig. 2). The eddies at ength scaes r 2 L are represented by their energy E (t). As before we restrict ourseves to oca energy transfer, and thus the time evoution of the she energies E (t) is governed by (9). The crucia step is of course to choose an appropriate energy transfer T +1 (t). For simpicity, we restrict ourseves to a Langevin process with a white noise force. Thus the oca transfer T +1 (t) is spit into a deterministic and a stochastic part T +1 (t) =T (det) +1 (t)+t(stoch) +1 (t) where both parts shoud depend ony on the oca ength scae 2 L and the neighboring energies E and E +1. The most genera form dimensionay consistent with this has been given in [2]. For simpicity, here we restrict ourseves to the specific form ( ) T (det) +1 (t) =D2 E 3/2 (t) E 3/2 +1 L (t) (13a) ( ) 2 T (stoch) (+1) 1/2 +1 (t) =R (E (t)e +1(t)) 5/8 ξ +1(t) (13b) L T (in) (t) =ɛδ (13c) T (diss) (t) = ν(2 L) 2 E. (13d) The white noise is represented by ξ, i.e. ξ (t) = and ξ (t)ξ (t ) =2δ δ(t t ). We use Ito s [31] definition in (13b). To understand the dimensions appearing in (13a), note that u E 1/2 is a oca veocity scae and k 2 /L is a wavenumber. Thus (13a) dimensionay represents the energy transfer (1a). In (13b) the powers are different, since ξ carries an additiona dimension of 1/time 1/2. It foows from (13a) that the sign of the deterministic energy transfer depends on which of the neighboring energies E or E +1 are greater. If for exampe E is arger, T (det) +1 (t) is positive, depeting E in favor of E +1. Hence the deterministic part tends to equiibrate the energy among the shes. The stochastic part, on the other hand, is symmetric with respect to the two eves and + 1. This refects our expectation that in equiibrium it is equay probabe for energy to be scattered up or down the cascade. The combined effect of (13a) and (13b) is that without driving, energies fuctuate around a common mean vaue. This equipartition of energy in equiibrium is precisey what has been predicted on the basis of the NavierStokes equation [32, 33]. The ony free parameters appearing in the transfer are thus the ampitudes D and R. IfRis put to zero, the motion is purey deterministic, and one obtains the simpe soution E () = C2 (2/3) with C = ( ) 2/3 2ɛL. (14) D This corresponds to a cassica Komogorov soution with no fuctuations in the transfer. The ampitude D of the deterministic part is a measure of the effectiveness of energy transfer. On the other hand R measures the size of fuctuations. In [2] it is shown that a finite R necessariy eads to intermittency corrections in the exponents. In the next section we are going to determine the mode parameters for the two REWA cascades we are considering. III. Determination of mode parameters The aim of this section is to expain the significance of the parameters appearing in the Langevin mode. We show that ony the combination R/D 1/2 determines the nonequiibrium fuctuations in the inertia range. But it is the same combination which aso sets the width of the equiibrium distribution of energies, if the chain of shes is not driven. Thus we are abe to fix a the parameters necessary to describe the nonequiibrium state soey by measuring equiibrium properties. The physica parameters of the turbuent cascade are the ength scae L of the highest eve, the rate of energy input ɛ, and the viscosity ν. The properties of the energy transfer are determined by the dimensioness strength of the deterministic part D and of the stochastic part R. In the previous section we have seen that E () =(2ɛL/D) 2/3 sets an energy
6 518 Fig. 6. Equiibrium distribution of the energy in the top eve of the arge REWAcascade (N = 74), consisting of 7 shes. By adjusting the ony parameter R/D 1/2, the distribution of the Langevin cascade can be made to agree within statistica error. The resuts for R/D 1/2 for both wave vector sets are given in (16) scae for the highest eve, and thus T = E () /ɛ is a time scae. Both scaes can be used to nondimensionaize (13a), giving ˆT ( ) (det) +1 (t) =2+1 Ê 3/2 (t) Ê 3/2 +1 (t) (15a) ˆT (stoch) +1 (t) = R D 1/22(+2)/2 (Ê (t)ê +1 (t)) 5/8 ˆξ +1 (t) (15b) ˆT (in) (t) = 1 (15c) ˆT (diss) (t) = 2 2 (Re) 1 Ê. (15d) (15e) Hence R/D 1/2 is the ony parameter characterizing the inertia range dynamics, which can be understood as foows: At any eve, a timescae of deterministic transport is set r/d, where E is a typica energy scae and r =2 L. This deterministic transport competes with diffusion of energy introduced by the stochastic part of the energy transfer. Namey a timescae over which the energy E can diffuse away is given by τ R (E /A) 2, by τ D E /ɛ E 1/2 where A = Rɛr 1/2 E 3/4 is the ampitude of the noise term. Pugging in the expression from (13b) one ends up with τ R re 1/2 /R 2, and hence τ R D τ D R 2. In particuar, the reative importance of deterministic transport and diffusion is independent of the eve, and ony depends on the combination of D and R given above. We are now eft to determine R/D 1/2 from an experiment which is independent of the nonequiibrium state. To that end we consider a ong chain of she eements, for which energy input as we as viscous dissipation has been turned off (ɛ = ν = ). As a resut, the energy wi perform fuctuations around some mean vaue E. Using a simiar argument as above, the probabiity distribution p(e / E ) for the she energies wi ony depend on R/D 1/2 for the Langevin mode. Figure 6 shows the probabiity distribution for one eve of a cascade with 7 shes. Except for some end effects at the owest eve, the distribution turns out to be eveindependent. The arge REWA cascade is compared with a simuation of the Langevin cascade. Once R/D 1/2 is adjusted, there is an amost perfect match between the two modes. This shows that the simpe stochastic dynamics proposed here modes the equiibrium distribution of the chaotic fuctuations of Fourier modes very we. Most importanty, we have determined the ony adjustabe parameter of the energy transfer of the Langevin cascade corresponding to the two mode systems we are considering: R = for the sma cascade (N = 26), (16a) D1/2 R = for the arge cascade (N = 74). (16b) D1/2 As seen from (16a), the REWA cascade with a arger number of modes has smaer fuctuations. This is not surprising, since arge excursions of the energy correspond to a coherent motion of the individua Fourier modes. The arger the number of modes, the harder this is to achieve, since the random motion of individua modes tends to destroy correations. In the next section, the fit (16a),(16b) wi be used to compare nonequiibrium properties in the inertia range. A information about possibe differences between the sma and the arge cascade has been condensed into a singe number. IV. Intermittency corrections We now turn to the inertia range fuctuations of the two cascade modes. To aow for a direct comparison, we focus on the scaing of the energies E of one she. The anaogue of the moments of the veocity fied usuay considered in turbuence are the structure functions based on the energy D (p) = E p/2 2 ζp. (17) We concentrate on the sma corrections δζ p to the scaing exponents as predicted on dimensiona grounds. As seen in Fig. 3, these corrections are extremey sma, so that no significant deviation from cassica scaing is seen on the scae of the figure. In order to know what to expect, we use the resut of [2] where we have computed the scaing behavior of a genera cass of stochastic modes in a perturbation expansion. To owest order, the exponent corrections are given by the quadratic dependence δζ p = µ p (p 3). (18) 18 Pugging the specific form of the energy transfer (15a), (15b) into the formuae given in [2], we find ( ) 2 R µ =.42. (19) D 1/2 For the sma cascade this eads to µ =.13, which is about 1/1 of the experimenta vaue accepted for threedimensiona turbuence [5]. Therefore, it is more than ordinariy difficut to measure the scaing exponents with sufficient accuracy to obtain significant answers for the deviations from cassica scaing. On the other hand, the number of modes in a REWA cascade being greaty reduced as compared with the fu NavierStokes equation, we are abe to
7 519 Fig. 7. Convergence of the structure function D (18) = E 9, on the 11th step of the sma REWA cascade. We show the evoution of D (18) as function of the averaging time T av in comparison with the stationary vaues D (18,stat), which are cacuated by integrating over a initia vaues in the data set. The error bars give the statistica error of the data as estimated by (2). The averaging time T av is given in units of T = E /ɛ simuate cascades with up to 17 eves, corresponding to 5 decades in scae. To obtain reiabe resuts within the accuracy needed, it is essentia to disentange statistica errors from systematic errors, introduced through finitesize effects or viscous damping. To that end we individuay assign statistica errors to every average taken. Figure 7 shows the convergence of the 9th moment of the energy on the 11th step of the sma REWA cascade, which is the owest eve reevant to our fits. We pot the mean vaue, averaged over the time given. Evidenty the fuctuations of this mean vaue get smaer as the averaging time T av is increased. To obtain a quantitative measure of the uncertainty of the T average of some quantity x, we aso consider the ensembe of averages obtained with different initia conditions. The resut for the mean vaue and variance of this ensembe average is potted as circes with error bars in Fig. 7, where E 9 takes the pace of x. The variance is found equivaenty as an integra over the tempora correation function [34]: σx(t 2 av )= 1 Tav Tav [ Tav 2 x(τ 1 )x(τ 2 ) x 2] dτ 1 dτ 2. (2) This variance σ(t av ) is seen to give a reasonabe approximation to the fuctuations of the tempora average. The variance for the argest averaging time avaiabe has been taken as the statistica error of a measured tempora average. Next we consider systematic errors in the computation of the scaing exponents. Deviations from power aw scaing are expected to occur on both ends of the cascade and have been studied extensivey [17, 35, 18]. First, fuctuations on the highest eve are suppressed, since there is no couping to a higher eve, but deterministic energy input instead. Second, in the viscous subrange the energy is increasingy depeted by viscosity, eading to even more drastic effects on the spectra. Both effects are strongest for the highest moments, which are most sensitive to arge fuctuations. The 18th order structure function for a typica run of the sma cascade is potted as diamonds in Fig. 8. The average on the highest eve is consideraby ower than expected from scaing, as fuctuations are suppressed. The cas Fig. 8. Scaing of the structure functions D (p) = E p/2 with p = 3 ( ) and p = 18 ( ) versus eve number (Standard) in comparison with extended sef simiarity (ESS) scaing of D (18) ( ) versus D (3) / D (3). The structure functions are normaized by their power aw fits: D (p,norm) = D (p) /(b(p) 2 ζp ) in the standard pot and D (p,norm) = D (p) /(A(p) ( D (3) / D (3) )ζp/ζ 3) in the ESS pot. Refer to Tabe 1 for the vaues of the exponents. In the ESS pot the scaing range is more than doubed. The numerica cacuation is performed for the sma REWA cascade cade ony graduay recovers from this suppression, which eads to a faster rise in the eve of fuctuations and thus to a decrease in the oca scaing exponent as reported in [18]. Note that we show a scatterpot of the data around the power aw D (p) 2 ζp, which represents our best fit. Thus deviations from this power aw are hugey exaggerated and woud not be visibe on a customary ogogpot. The absoute variation of D (18) over the range of the pot is 21 orders of magnitude. Owing to the extreme sensitivity of our pot, viscous damping is visibe beow the fourth eve, even for a cascade with 17 shes, as seen in Fig. 8. This restricts the inertia range to four eves. We empoy two different strategies to improve on this situation: (i) We pot the higher order structure functions against the third order structure function, as suggested by the ESS method. (ii) We eiminate viscosity atogether by introducing a fuctuating eddy viscosity instead. It is known [6, 7] that extended sefsimiarity eads to a very considerabe improvement of the scaing of experimenta data. Without any viscous corrections, the scaing behavior is unaffected, as seen in Fig. 8 in the fitting range marked standard. Beow the forth eve, however, the third order structure function suffers viscous damping quite simiar to that affecting the other structure functions. Thus by potting D (3) on the abscissa, both effects are hoped to argey cance each other. This is indeed true for the REWA cascade as we, where the scaing range has been extended to amost three decades. In Tabe 1 we have compied various scaing exponents for the sma cascade. Except for the smaest moment, highy significant corrections to the scaing exponents are measured. The errors are based on a east squares fit with weighted averages [36], based on the statistica errors as expained above. To make sure the resuts do not depend on our choice of the inertia range, we aso made fits in ranges other than the one marked ESS in Fig. 8. Namey, we variousy shifted the fitting range by one eve
8 52 Tabe 1. Correction exponents of the structure functions D (p) for p = 2, 6, 12 and 18 for the sma REWA cascade. The exponents in coumn ESS are determined by potting D (p) versus D (3) as suggested by extended sefsimiarity. To obtain the vaues in coumn EV, viscous effects have been removed entirey by introducing a fuctuating eddy viscosity. In the ast coumn the prediction based on the Langevin mode is given p δζ p (ESS) δζ p (EV) δζ p (Theory) ± ± ± ± ± ± ± ± up or down, or took an additiona eve into account on either end. In each case, the vaues of the exponents were within the errors given in Tabe 1. Typica averaging times were 1 turnover times of the argest scae. This is about 1 times as ong as in [18], as we base the turnover time on the time scae of the energy. For comparison, we aso suppy the vaue of the exponent correction as cacuated from (18) and (19) on the basis of the known vaue of R/D 1/2, given in (16a). There is no adjustabe parameter in this comparison with the theoretica prediction, since the noise strength was determined soey from the equiibrium fuctuations of the REWA cascade. The exceent agreement found for a exponents makes us confident that finite size corrections and viscous effects have been successfuy eiminated, and we are measuring genuine properties of the inertia range. However, since there is no theory demonstrating that the ESS method eads to the correct inertia range scaing behavior, we have nevertheess attempted to eiminate viscous effects by a second, and competey independent method. This was done by putting ν =, and instead draining energy from the owest she K max using an eddy viscosity [37, 17]. In order to mimic inertia energy transfer into the subgrid shes K,> max, we preserve the couping structure of the equations as much as possibe. We add a term { d(k,t)2 max u(k,t) u(k,t) : k K D(k,t)= (out) (21) : k K (out) to the inviscid equation of motion (7a). The cutoff she K (out) K max contains a the wave vectors of K max which interact directy with modes of the she which is not resoved. The probem of this procedure is that (i) the fuctuations in the subgrid scaes are not accounted for, and (ii) the ampitude to choose for d(k,t) is not known. Both probems are addressed by using a method inspired by the work in [21]. The idea is to adjust d(k,t) such that u(k) 3 = 1 u(k ) 3 for k K (out), k = 1 k, (22) 2 2 which is vaid exacty in a perfecty scaeinvariant cascade [17], anaogous to the Komogorov structure equation [1]. The wave vector k K max 1 in (22) is a scaed copy of k K (out). In the ong time imit, each d(k,t) wi converge to some average vaue, which is determined sefconsistenty from the dynamics of the cascade. But in that case the fuctuations of the subgrid scaes woud be missing. Therefore, we instead took the averages in (22) over just 1 turnover times of the ast resoved she K max. When u(k) 3 / u(k ) 3 was arger than 1/2, we increased d(k,t) by 5 %, otherwise d(k,t) was decreased by the same percentage. Thus a the Fig. 9. Time evoution of the ampitude d(k,t) of the eddy damping (21) for k =2 max (1, 1, 1). The fuctuations mirror the highy intermittent behavior of the veocity modes on the 14th eve. The numerica cacuation is performed with the sma wave vector set. Time is given in units of T = E /ɛ Fig. 1. Anomaous scaing of the structure function D (18), p = 18 for the sma REWA cascade, with the eddy viscosity (21) on the owest eve. We show the scatter of the data around the fitted power aw by potting D (18,norm) = D (18) /(b (18) 2 ζ18 ). Cassica Komogorov scaing is shown for comparison d(k,t) refect the fuctuations occurring at the end of the cascade, as seen in Fig. 9 for one of the ampitudes of the sma cascade. The average of d(k,t) becomes stationary in the ong time imit, but fuctuations are considerabe, as expected in an intermittent cascade. Figure 1 shows the 9th moments of the energy as in Fig. 8, but for the eddydamped cascade. End effects at the sma scaes are very sma, ess substantia than on the argest scaes. Aso shown are the error bars resuting from the statistica estimate described before. The resut of the fit over the scaing range indicated is given in Tabe 1 for various moments. Again, the range of our fit was aso varied, and the resuts are consistent with the errors given. The vaues of the exponent corrections are in exceent agreement with the vaues obtained from ESS. Furthermore, the error is even ess than before, giving very significant deviations from cassica scaing. This is aso seen from the dashed ine in Fig. 1, which represents cassica Komogorov scaing. Finay it shoud be noted that even for the ong averaging times we use, some imbaances in the cascade remain, which make D (3) deviate from its exact power aw behavior of 2. These deviations decrease in time and turn out
9 521 Tabe 2. Comparison of our best numerica estimate of of the intermittency correction µ = δζ 6 for the REWA cascades with the theoretica prediction. In both cases, with the predicted exponents differing by a factor of three, agreement is within error bars µ = δζ 6 REWA: N= ±1 1 3 Langevin: R/D 1/2 = REWA: N= ±1 1 3 Langevin: R/D 1/2 = Fig. 12. The rea part of the autocorreation function C (u) (k,t) = u (k,) u(k,t) normaized to its equa time vaue C (u) (k, ) for the mode k =2 (1, 1, 1) of the sma cascade. Shown are the three top eves. The correation function decays very quicky, refecting the chaotic behavior of the veocity fied. The time, given in units of T = E /ɛ, is rescaed according to the cassica prediction τ (u) 2 2/3 for the decay time Fig. 11. Intermittency corrections for the two REWA cascades in comparison with the prediction of theory. The error bars of the numerica vaues are based on a weighted east square fit, and are given exempary for the highest moment not to be significant for many of our runs. However, we decided to normaize our resuts so as to substract remaining imbaances. So stricty speaking the vaues given for the correction exponents in Tabe 1 are δζ p = ζ p p ζ 3 3. The same procedure has been adopted in previous work on the subject [18]. Since the best error estimates are obtained by using an eddydamped cascade, we are using this method for our fina comparison between numerica simuations and the theoretica prediction of the Langevin mode. The resut of this comparison for µ = δζ 6 is found in Tabe 2 for the sma and the arge cascade. In both cases, numerics of the REWA cascade and theory agrees within error bars, the size of the exponent corrections differing by more than a factor of three between the sma and the arge cascade. This underscores the consistency of our resuts and demonstrates that the Langevin mode captures a the essentia physics responsibe for the buidup of fuctuations in a oca cascade. The same message is contained in Fig. 11, which summarizes the exponent corrections for both modes. A better understanding of why a mode, which has the energy as its ony mode works so we, is suppied by a study of the tempora correations. At the same time it gives insight into the origin of intermittency itsef, at east in modes with oca couping. V. Tempora correations In the previous sections we have ooked ony at equa time correations. Athough they contain information about the fuctuations of the cascade, their information about the dynamics responsibe for these fuctuations is very indirect. So utimatey one has to ook at tempora correations as we to understand the dynamica origins of intermittency. In the case of the REWA cascade, one can ook at the fuctuations of the individua Fourier modes, C (u) (k,t)= u (k,) u(k,t), (23) just as one does in studies of the fu NavierStokes equation. We aso expect the tempora correations of the tota energy within a cascade step to be of particuar significance, C (E) (t) = E ()E (t) E 2, (24) since the conservation properties of the energy are responsibe for maintaining a turbuent state. The standard guess for the scae dependence of correation times is based on the Komogorov picture. Namey, assuming that the typica correation time is a oca quantity, the ony combination of the ength scae 2 L and the mean energy transfer ɛ having dimensions of time eads to τ [ (2 L) 2 /ɛ ] 1/3. (25) We test this idea by computing the tempora correation (23) of a particuar mode of the REWA cascade, for three different shes, as potted in Fig. 12. The abscissa is rescaed according to (25), which makes a three correation functions fa atop of each other. So it seems that if corrections to (25) exist, they are so sma that they need more sophisticated methods to be reveaed. Next, we ook at the tempora correations of the energy on the highest eve, potted in Fig. 13. Apart from differences in shape, the remarkabe feature is that the typica decay time is more than 1 times as ong as that of individua Fourier modes. The tempora correation is apparenty dominated by the ongtime fuctuations of the energy aready noted in Fig. 5. The rapid fuctuations representative of the individua Fourier modes appear to be competey uncorreated. The physica reason is that many different modes within a she contribute to these rapid fuctuations, which are decorreated through many random coisions. Just
10 522 Fig. 13. The autocorreation function C (E) (t) of the energy on the top eve, normaized to its equa time vaue C (E) () for the sma cascade. The energy decorreates by more than a factor of 1 more sowy than a singe veocity mode. The time is given in units of T = E /ɛ Fig. 15. Scaing of the correation times τ (E) of the energy correation function C (E) (t) for the sma cascade. The fit corresponds to τ (E) 2.94, which is a very sma decrease compared to the cassica prediction of τ (E) 2 2/3 Fig. 14. Correations times τ (u,e) for the sma cascade in equiibrium, as a function of the eve number. The dashed ine corresponds to the theoretica prediction of (26). The time scae of the energy is onger than that of an individua mode by a factor of 4 on a eves ike in a gas of partices, the compicated interaction between many modes tends to randomize the individua motions. If the smascae fuctuations were a true random wak, the energy woud sowy diffuse to take arbitrary vaues. But eventuay the tendency of the dynamics to restore equiibrium wi drive the energy back to its mean vaue. This requires a coherent motion of many individua Fourier modes, whose correations need some time to buid up. Thus the ong time scae in the motion of the energy. So far the same argument woud appy for a cascade in equiibrium, in the absence of driving. This is shown in Fig. 14 for a cascade of 8 shes, which a perform fuctuations whose static distribution was shown in Fig. 6. Since the energies of a shes fuctuate around some common vaue E av, and the oca ength scae is 2 L, for dimensiona reasons the time scae must be τ (equ) Eav 1/2 2 L. (26) The correation times shown in Fig. 14 for both the energy and one of the veocity modes is cacuated according to τ (E,u) = C (E,u) C (E,u) () (t)dt, (27) and are found to corroborate the scaing aw (26). As in the case of the top eve of a Komogorov cascade, the correation time of the energy is sower compared to an individua Fourier mode. This changes fundamentay when ooking at ower shes of a nonequiibrium cascade. For a turbuent cascade, one expects to recover the scaing (25), since the correation of the energies are defined in terms of oca quantities ony. But remarkaby, the scaing aw (25) does not even approximatey describe the scaing of τ (E), which rather foows the power aw τ (E) 2 α,α=.9 ±.2 (28) as seen in Fig. 15. This means the time scae of the motion of the energy hardy gets shorter at sma scaes. The reason ies within the nonequiibrium properties of the cascade: Once a arge fuctuation of the energy has buit up on the highest eve, it can ony reax by being transported to a ower eve. Thus the same ong time dependence is imprinted on the ower eve, which again can ony be transferred to the next eve. Thus on a given eve, a time scaes of the eves ying above appear, and apart form the singe rescaing (28) the shape of the correation function changes as we. This corresponds directy to the origin of intermittent fuctuations itsef: fuctuations ampify because ower shes are driven by sowy varying energy input of the higher shes. Consequenty fuctuations ride atop of the ongscae fuctuations and ampify. By ooking at correations between individua Fourier modes ike (23) none of these ongrange correations are reveaed. Correations between the coherent structures of a turbuent veocity fied wi aways be masked by the incoherent fuctuations of the individua modes. We checked the behavior of tempora correations for other mode seections as we, in particuar for the arge cascade with N = 74 modes instead of 26. A the above observations on correation times appy equay we. In fact the disparity between the time scae of the Fourier modes and of the energy is even greater on the highest eve, since there are more interactions to decorreate the individua modes. Our best fit for the correation times of the energy eads to
11 523 α =.11 ±.2 for the arge cascade, which agrees with the sma cascade within the error. Further evidence for the correation exponent of the energies being very sma compared with the Komogorov estimate is provided for by a simuation of the Langevin cascade. Here we find α =.18 ±.4, with no significant dependence on the vaue of R/D 1/2. The smaness of α is thus a fundamenta property of such a turbuent cascade, at east in the case of a inear chain of turbuence eements. Ony a cascade which aso aows for a spatia resoution coud potentiay decorreate through the interaction of turbuence eements with its neighbors in space. It is interesting to note that in a numerica simuation of isotropic turbuence Yeung and Pope [38] have found very strong deviation from cassica scaing as we. They ooked at the Reynods number dependence of the acceeration variance, which is a Lagrangian quantity. Like in our case, incuding fuctuations of the energy transfer according to Koomogorov s refined simiarity hypothesis [3] cannot account for the corrections found. However, a meaningfu definition of a Lagrangian quantity within our oca approximation is difficut, since convection cannot be described propery. Therefore we do not know how to reate our findings with the resuts of [38] directy. The second consequence of the extremey sow decorreation of the energy is that the ratio of time scaes of the energy and that of individua modes rapidy becomes arger on smaer scaes. In the Langevin description the white noise term represents the irreguar motion of individua Fourier modes, whie our main interest ies with the argescae fuctuations of the energy. In the scaing imit we are interested in the disparity between these scaes becomes infinitey arge. Thus a whitenoise description shoud become better and better in the reevant imit. VI. Discussion We have investigated modereduced approximations of the NavierStokes equation and found them to have anomaous scaing exponents in the inertia range. Intermittent fuctuations come about through rare excursions of the energy from its mean vaue, which originate from the top eves of the cascade. They are thus we described by a whitenoise Langevin process, whose random forcing represents the motion of individua Fourier modes. The anaytica soution shows [2] that such a Langevin cascade necessariy exhibits corrections to the cassica scaing exponents. It is thus hard to see how any cascade with oca couping coud avoid intermittency corrections in the inertia range, since for a chaotic motion there wi aways be fuctuations, and thus R/D 1/2 /=. In fact, as pointed out by Kraichnan [39], the ony mechanism by which such fuctuations coud be avoided is by sufficienty strong mixing in space, which in kspace woud be represented by nonoca interactions. The fact that our simpe Langevin mode gives an exceent description of a compicated network of Fourier modes is aso highighted by the exceent agreement in an equiibrium state. We have recenty extended the comparison between the Langevin and the REWA mode to the formation of singuarities in the absence of viscosity [4]. In this case, which represents a state even farther from equiibrium than a Komogorov cascade, the two modes sti compare quantitativey. This aows for an anaytica description of the singuarities of the Euer equation [41] in the approximation of oca couping. The second centra point of our paper is to demonstrate that intermittency exponents can be cacuated for a compex cascade with noninear, chaotic dynamics by anaytica means. The ony piece of information one needs about the turbuent fow concerns the oca energy transfer. With this information, the goba nonequiibrium properties determine the intermittency exponents, which is taken care of by the anaytica cacuation. In a very interesting paper, Oa [42] has recenty carried out the same program directy from the NavierStokes equation. He assumes that the broadening of the distribution of the veocity fied from a scae r to r/2 is adequatey described in a Gaussian approximation. Thus he is abe to determine the anaogue of the noise strength R/D 1/2 from perturbation theory. However, a major probem of Oa s work is that he aso does not dea adequatey with the spatia structure of turbuence, but rather maps the probem on the same inear cascade structure we are considering here. A remarkabe feature of the cascades studied here is the very sow decorreation of the energy. This is surprising because intermittency corrections are quite sma, so one might have expected to find a tempora correations to be at east approximatey described by Komogorov theory. Instead, the exponent for the decay time of the energy was typicay five times smaer than the vaue predicted by cassica scaing. In perturbative anayses of the NavierStokes equation [33] decay times of Fourier modes are usuay approximated by (r 2 /ɛ) 1/3 as in (25). Our simuations of the REWA cascade confirm that this is a good guess. But by restricting attention to individua Fourier modes, one is missing other very ong time scaes of the system, ike the decay time of coherent structures, which may be dynamicay important. The most pressing probem remaining with the present mode representations is that the intermittency effects are quite sma. Thus some important physica features are sti missing. The most ikey reason is the fact that a given eddy ony feeds a singe substructure, instead of branching out to severa smaer structures overapping with it in physica space. It has been shown [27, 19] that such a tree structure may strongy enhance intermittent fuctuations, owing to competition between different eddies on the same eve. The procedures of the present paper can be carried over directy to this physica situation, as anaytica soutions of the Langevin mode with tree structure are avaiabe as we [43]. Unfortunatey, there is no technique avaiabe which woud ink the NavierStokes equation to a cascade couped ocay in rea and in kspace in a rationa way. Since the vaue of intermittency exponents depends significanty on the properties of the oca coupings [35], no quantitative prediction for rea threedimensiona turbuence is possibe using the more genera tree structure. An interesting fact reveaed by our anaysis is that the vaue of the correction exponents depends consideraby on the mode seection. There is a tendency, observed earier [18], for the size of fuctuations to decrease with the number of modes per she. In [18], mode systems with up to 86 modes per she have been considered, which aso aow for
12 524 more distant interactions between shes. Athough it is interesting to ook at the significance of more distant interactions, it does not remedy the fundamenta probem of the present approximation, which is that the number of modes does not proiferate on sma scaes. In terms of interactions, there is sti no competition between modes of the same scae. Thus no concusions about the true intermittency exponents can be drawn from incuding more modes, as ong as the spatia structure is not respected. In fact for a tree structure the number of ocaized boxes woud increase ike 2 3, each with a constant number of modes N. Comparing that with the tota number of Fourier modes (4π/3)(2 3 1)2 3 within a she for threedimensiona turbuence, one obtains the estimate N 28, which is cose to the number N =26 of our sma cascade [44]. Thus the sma cascade aready represents a physicay reasonabe choice for the number of ocay interacting Fourier modes. Finay we mention the socaed GOY mode [45, 46], which contains ony one compex mode per she, and which has attracted considerabe attention recenty. Athough it has no spatia structure, the intermittency exponents are comparabe to those of rea turbuence, which seems to contradict our above observations. We beieve that the mechanisms at work are somewhat different here, since there exist puses, which run down the cascade. These goba coherent motions are much more effective in transporting fuctuations to smaer scaes than in REWA cascades, where such structures are destroyed by intrashe mixing. A separation of time scaes between the tota energy and the individua modes of a she evidenty does not exist. Another fact which points to goba structures is the dependence of intermittency exponents on the form of viscous dissipation [46]. 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