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
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