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Local coupling of shell models^{⋆} leads to anomalous scaling

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C. Uhlig^{1}, J. Eggers^{⋆⋆}, I. Abt^{1}, T. Ahmed^{2}, V. Andreev^{3}, B. Andrieu^{4}, R.-D. Appuhn^{5}

1 Fachbereich Physik, Universität – Gesamthochschule – Essen, D-45117 Essen, Germany 2 I. Physikalisches Institut der RWTH, Aachen, Germany^{⋆⋆⋆} 3 III. Physikalisches Institut der RWTH, Aachen, Germany 4 School of Physics and Space Research, University of Birmingham, Birmingham, UK^{†} 5 Inter-University Institute for High Energies ULB-VUB, Brussels, Belgium^{‡} 6 Rutherford Appleton Laboratory, Chilton, Didcot, UK 7 Institute for Nuclear Physics, Cracow, Poland^{§} 8 Physics Department and IIRPA, University of California, Davis, California, USA

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1 Introduction

Much of our intuitive understanding of turbulence is based on the concept of interactions which are local in kspace. Physically, it is based on the notion that most of the distortion of a turbulence element or eddy can only come from eddies of comparable size. Turbulent features which are much larger only uniformly translate smaller eddies, which does not contribute to the energy transfer. This immediately leads to the idea of a chain of turbulence elements, through which energy is transported to the energy dissipating scales. Accepting such a cascade structure of the turbulent velocity field, it is natural to assume that the statistical average of velocity differences δv(r) = v(x + r) − v(x) over a distance |r| follows scaling laws

D^{(p)}(r) ≡ ⟨|v(x + r) − v(x)|^{p}⟩ ∼ r^{ζ_{p}}  (1)

in the limit of high Reynolds numbers. By taking velocity differences over a distance r, one probes objects of corresponding size.

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⋆ see also paper by Claudia Schiffer, Valeria Mazza, Cindy Crawford, Naomi Campbell, Helena Christensen, Elle Mc-Pherson, Eva Herzigova, Kathy Ireland, Kate Moss, Stephanie Seymour ⋆⋆ Professor emeritus ⋆⋆⋆ Supported by the Bundesministerium für Forschung und Technologie, FRG under contract numbers 6AC17P, 6AC47P, 6DO57I, 6HH17P, 6HH27I, 6HD17I, 6HD27I, 6KI17P, 6MP17I, and 6WT87P

† Supported by the UK Science and Engineering Research Council ‡ Supported by IISN-IIKW, NATO CRG-890478 § Supported by the Polish State Committee for Scientific Research, grant No. 204209101 Correspondence to: Springer-Verlag

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In addition to this assumption of self-similarity, Kolmogorov [1] also made the seemingly intuitive assumption that the local statistics of the velocity field should be independent of large-scale flow features, from which it is widely separated in scale. Because the turbulent state is maintained by a mean energy flux ϵ, the only local scales available are the length r and ϵ itself, which leads to the estimate δv ∼ (ϵr)^{1/3} or

ζ_{p}^{(class)} = p/3.  (2)

At the same time, one obtains an estimate for the Kolmogorov length

η = (ν^{3}/ϵ)^{1/4}  (3)

where viscosity is important. However, it was only appreciated later [2] that in turbulence long-range correlations always exist in spite of local coupling. Namely, large-scale fluctuations of the velocity field will result in a fluctuating energy transfer, which drives smaller scales. As a result, the statistics of the small-scale velocity fluctuations will be influenced by the energy transfer and fluctuations on widely separated scales are correlated, violating the fundamental assumption implicit in (2) and (3).

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Indeed, Kolmogorov [3] and Obukhov [4] later proposed the existence of corrections to the scaling exponents (2),

ζ_{p} = p/3 + δζ_{p} , δζ_{p} ≠ 0  (4)

which were subsequently confirmed experimentally [5–8]. On one hand, careful laboratory experiments have been performed at ever higher Reynolds numbers [5,9]. On the other hand, a new method of data analysis [6,7] has been successful in eliminating part of the effects of viscosity.

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In particular, for the highest moments up to p = 18 significant corrections to classical scaling were found, a currently accepted value for the so-called intermittency parameter μ being [5]

μ = −δζ_{6} = 0.2,  (5)

which is a 10 % correction. The existence of corrections like (5) implies that on small scales large fluctuations are much more likely to occur than predicted by classical theory.

This “intermittent” behavior is thus most noticeable in derivatives of the velocity field such as the local rate of energy dissipation

ϵ(x, t) = \frac{ν}{2} (∂u_{i}/∂x_{k} + ∂u_{k}/∂x_{i})^{2} .

Much of the research in turbulence has been devoted to the study of the spatial structure of ϵ(x, t) [3,10], but which will not be considered here. The statistical average of this quantity is what we simply called ϵ before. Owing to energy conservation, it must be equal to the mean energy transfer.

The local coupling structure of turbulence has inspired the study of so-called shell models, where each octave in wavenumber is represented by a constant number of modes, which are only locally coupled. This allows to focus on the implications of local coupling for intermittent fluctuations, disregarding effects of convection and mixing. The mode representation of a single shell serves as a simple model for the “coherent structures” a turbulent velocity field is composed of, and which to date have only been poorly characterized, both experimentally and theoretically.

2 Two cascade models

2.1 Reduced wave vector set approximation

2.1.1 Test for heading of third order

The REWA model. The REWA model [17,18] is based on the full Fourier-transformed Navier-Stokes equation within a volume of periodicity (2πL)^{3}. In order to restrict the excited Fourier-modes of the turbulent velocity field to a numerically tractable number, the Navier-Stokes equation is projected onto a self-similar set of wave vectors K = ⋃_{ℓ} K_{ℓ}. Each of the wave vector shells K_{ℓ} represents an octave of wave numbers. The shell K_{0} describes the turbulent motion of the large eddies which are of the order of the outer length scale L. This shell is defined by N wave vectors k_{i}^{(0)}: K_{0} = {k_{i}^{(0)} : i = 1, …, N}. Starting with the generating shell K_{0}, the other shells K_{ℓ} are found by a successive rescaling of K_{0} with a scaling factor 2: K_{ℓ} = 2^{ℓ}K_{0}. Thus each K_{ℓ} consists of the N scaled wave vectors 2^{ℓ}k_{i}^{(0)}, i = 1, …, N. The shell K_{ℓ} represents eddies at length scales r ∼ 2^{−ℓ}L, i.e. to smaller and smaller eddies as the shell index ℓ increases. At scales r ≈ η the fluid motion is damped by viscosity ν, thus preventing the generation of infinitely small scales. Hence we only need to simulate shells K_{ℓ}, ℓ < ℓ_{ν}, where ℓ_{ν} ≈ log_{2}(L/η) is chosen such that the amplitudes in K_{ℓ_{ν}} are effectively zero. In this representation the Navier-Stokes equation for incompressible fluids reads for all k ∈ K = ⋃_{ℓ=0}^{ℓ_{ν}} K_{ℓ}:

\frac{∂}{∂t}u_{i}(k, t) = −ıM_{ijk}(k) ∑_{{p,q∈K \atop k=p+q}} u_{j}(p, t)u_{k}(q, t)

−νk^{2}u_{i}(k, t) + f_{i}(k, t)  (7a)
k · u(k, t) = 0.  (7b)

The coupling 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 inertial part of (7a) consists of all triadic interactions between modes with k = p + q. They are the same as in the full Navier-Stokes equation for this triad. The velocity field is driven by an external force f(k, t) which simulates the energy input through a large-scale instability.

Within this approximation scheme the energy of a shell is

E_{ℓ}(t) = \frac{1}{2} ∑_{k∈K_{ℓ}} |u(k, t)|^{2},  (8)

and in the absence of any viscous or external driving the total energy of the flow field E_{tot}(t) = ∑_{ℓ=0}^{ℓ_{ν}} E_{ℓ}(t) is conserved. The choice of generating wave vectors k_{i}^{(0)} determines the possible triad interactions. This choice must at least guarantee energy transfer between shells and some mixing within a shell. In [17,18] different choices for wavenumber sets K_{0} are investigated. The larger the number N of wave numbers, the more effective the energy transfer. Usually one selects directions in kspace to be distributed evenly over a sphere. However, there are different possibilities which change the relative importance of intra-shell versus inter-shell couplings. In this paper, we are going to investigate two different wave vector sets, with N = 26 and N = 74, which we call the small and the large wave vector set, respectively. In Fig. 1 a two-dimensional projection of both sets is plotted. The large wave vector set also contains some next-to-nearest neighbor interactions between shells, which we put to zero here, since they contribute little to the energy transfer. The small set allows 120 different interacting triads, the large set 501 triads, 333 of which are between shells.

Since in the models we consider energy transfer is purely local, the shell energies E_{ℓ}(t), ℓ = 0, …, ℓ_{ν} only change in response to energy influx T_{ℓ−1→ℓ} from above and energy outflux T_{ℓ→ℓ+1} to the lower shell. In addition, there is a rate of viscous dissipation T_{ℓ}^{(diss)}(t) which is concentrated on small scales, and a rate of energy input T_{0}^{in}(t), which feeds the top level only, cf. Fig. 1.

Fig. 1. The structure of a local cascade. Eddies of size r ∼ 2^{−ℓ}L are represented by their total energy E_{ℓ}. Only modes of neighboring shells interact, leading to a local energy transfer T_{ℓ→ℓ+1}(t). The cascade is driven by injecting energy into the largest scale with rate T_{0}^{(in)}(t). The turbulent motion is damped by viscous dissipation at a rate T_{ℓ}^{(diss)}(t)

From (7a) we find an energy balance equation which governs the time evolution of the shell energies E_{ℓ}(t)

\frac{d}{dt}E_{ℓ}(t) = T_{ℓ−1→ℓ}(t) − T_{ℓ→ℓ+1}(t) + T_{ℓ}^{(diss)}(t) +

T_{0}^{(in)}(t)δ_{ℓ0}. (9)

The different transfer terms are found to be

T_{ℓ→ℓ+1}(t) = 2ı ∑_{△_{(ℓ)}^{(ℓ+1)}} M_{ijk}(k)u_{i}^{∗}(k, t)u_{j}(p, t)u_{k}(q, t)(10a)

T_{0}^{(in)}(t) = ∑_{k∈K_{0}} Re (u^{∗}(k, t) · f(k, t))  (10b)

T_{ℓ}^{(diss)}(t) = −ν ∑_{k∈K_{ℓ}} k^{2}|u(k, t)|^{2}.  (10c)

In (10a) ∑_{△_{(ℓ)}^{(ℓ+1)}} indicates the summation over all next-

neighbor triads k = p + q with k ∈ K_{ℓ}, p ∈ K_{ℓ+1} and q ∈ K_{ℓ} ⋃ K_{ℓ+1}.

The driving force f(k, t) is assumed to act only on the largest scales, and controls the rate of energy input T_{0}^{(in)}(t). As in [17] we choose f(k, t) to ensure constant energy input T_{0}^{(in)} = ϵ :

f(k, t) = \frac{ϵu(k, t)}{N|u(k, t)|^{2}} for all k ∈ K_{0}  (11a)
f(k, t) = 0 for all k ∉ K_{0}.  (11b)

Re = \frac{LU}{ν} = \frac{ϵL^{2}}{⟨E_{0}⟩ ν} ,  (12)

This leads to a stationary cascade whose statistical properties are governed by the complicated chaotic dynamics of the nonlinear mode interactions. Owing to energy conservation, viscous dissipation equals the energy input on average. The Reynolds number is given by since T = ⟨E_{0}⟩ /ϵ sets a typical turnover time scale of the

Fig. 3. Energy of a shell (ℓ = 2) as a function of time. The rapid fluctuations come from the motion of individual Fourier modes. A much longer time scale is revealed by performing a floating average over one turnover time of the second level (bold line). The time is given in units of T = ⟨E_{0}⟩ /ϵ

energy on the highest level. We believe T to be of particular relevance, since the large-scale fluctuations of the energy will turn out to be responsible for the intermittent behavior we are interested in. In [18], for example, time is measured in units of L^{2/3}ϵ^{−1/3}, which typically comes out to be 1/10th of the turnover time of the energy T. As we are going to see below, this is rather a measure of the turnover times of the individual Fourier modes. Figure 2 shows the scaling of the mean energy in a loglog plot at a Reynolds number of 4.2 · 10^{5}. The inertial range extends over three decades, where a power law very close to the prediction of classical scaling is seen. Below the 10th level the energies drop sharply due to viscous dissipation. In Sect. 3 we are going to turn our attention to the small corrections to 2/3-scaling, hardly visible in Fig. 2. Still, there are considerable fluctuations in this model, as evidenced by the plot of the energy transfer in Fig. 2. Typical excursions from the average, which is normalized to one, are quite large. Ultimately, these fluctuations are responsible for the intermittency corrections we are going to observe.

Thus within the REWA-cascade we are able to numerically analyze the influence of fluctuations on the stationary statistical properties of a cascade with local energy transfer on the basis of the Navier-Stokes equation. In Fig. 3 we plot the time evolution of the energy on the second level of the cascade. One observes short-scale fluctuations, which result from the motion of individual Fourier modes within one cascade level. However, performing a floating average reveals a second time scale, which is of the same order as the turnover time of the top level. As we are going to see in Sect. 4, this disparity of time scales is even more pronounced on lower levels. The physics idea is the same as in the microscopic foundation of hydrodynamics, where conserved quantities are assumed to move on much slower time scales than individual particles. This motivates us to consider the energy as the only dynamical variable of each shell, and to represent the rapid fluctuations of Fig. 3 by a white-noise Langevin force. In this approximation we still hope to

Fig. 2. Log-log-plot of the structure function D̃_{ℓ}^{(2)} = ⟨E_{ℓ}⟩ versus level number for the small cascade. At large scales, where the influence of dissipation is negligible, classical scaling is observed. At small scales the turbulent motion is damped by viscosity. The Reynolds number is Re = 4.2 · 10^{5}

capture the rare, large-scale events characteristic of intermittent fluctuations, since the conserved quantity is the “slow” variable of the system. Similar ideas have also been advanced for the conservative dynamics of a nonequilibrium statistical mechanical system [30].

2.2 The Langevin-cascade

In this model we take a phenomenological view of the process of energy transfer. The chaotic dynamics of the REWA-cascade is modeled by a stochastic equation. We make sure to include the main physical features of energy conservation and local coupling. In particular, the dynamics is simple enough to allow for analytical insight into the effects of fluctuating energy transfer [20].

As in the REWA-cascade, the turbulent flow field is described by a sequence of eddies decaying successively (Fig. 1). The eddies at length scales r ∼ 2^{−ℓ}L are represented by their energy E_{ℓ}(t). As before we restrict ourselves to local energy transfer, and thus the time evolution of the shell energies E_{ℓ}(t) is governed by (9). The crucial step is of course to choose an appropriate energy transfer T_{ℓ→ℓ+1}(t). For simplicity, we restrict ourselves to a Langevin process with a white noise force. Thus the local transfer T_{ℓ→ℓ+1}(t) is split into a deterministic and a stochastic part T_{ℓ→ℓ+1}(t) = T_{ℓ→ℓ+1}^{(det)}(t) + T_{ℓ→ℓ+1}^{(stoch)}(t) where both parts should depend only on the local length scale 2^{−ℓ}L and the neighboring energies E_{ℓ} and E_{ℓ+1}. The most general form dimensionally consistent with this has been given in [20]. For simplicity, here we restrict ourselves to the specific form

T_{ℓ→ℓ+1}^{(det)}(t) = D \frac{2^{ℓ}}{L} (E_{ℓ}^{3/2}(t) − E_{ℓ+1}^{3/2}(t))  (13a)

T_{ℓ→ℓ+1}^{(stoch)}(t) = R (\frac{2^{(ℓ+1)}}{L} )^{1/2} (E_{ℓ}(t)E_{ℓ+1}(t))^{5/8}ξ_{ℓ+1}(t) (13b)

T_{ℓ}^{(in)}(t) = ϵδ_{ℓ0}  (13c)
T_{ℓ}^{(diss)}(t) = −ν(2^{−ℓ}L)^{−2}E_{ℓ}.  (13d)

The white noise is represented by ξ_{ℓ}, i.e. ⟨ξ_{ℓ}(t)⟩ = 0 and ⟨ξ_{ℓ}(t)ξ_{ℓ′}(t′)⟩ = 2δ_{ℓℓ′}δ(t − t′). We use Ito’s [31] definition in (13b). To understand the dimensions appearing in (13a), note that u_{ℓ} ∼ E_{ℓ}^{1/2} is a local velocity scale and k ∼ 2^{ℓ}/L is a wavenumber. Thus (13a) dimensionally represents the energy transfer (10a). In (13b) the powers are different, since ξ carries an additional dimension of 1/time^{1/2}. It follows from (13a) that the sign of the deterministic energy transfer depends on which of the neighboring energies E_{ℓ} or E_{ℓ+1} are greater. If for example E_{ℓ} is larger, T_{ℓ→ℓ+1}^{(det)}(t) is positive, depleting E_{ℓ} in favor of E_{ℓ+1}. Hence the deterministic part tends to equilibrate the energy among the shells. The stochastic part, on the other hand, is symmetric with respect to the two levels ℓ and ℓ + 1. This reflects our expectation that in equilibrium it is equally probable for energy to be scattered up or down the cascade.

The combined effect of (13a) and (13b) is that without driving, energies fluctuate around a common mean value. This equipartition of energy in equilibrium is precisely what has been predicted on the basis of the Navier-Stokes equation [32,33]. The only free parameters appearing in the transfer are thus the amplitudes D and R. If R is put to zero, the motion is purely deterministic, and one obtains the simple solution

E_{ℓ}^{(0)} = C2^{−(2/3)ℓ} with C = (\frac{2ϵL}{D})^{2/3} .  (14)

This corresponds to a classical Kolmogorov solution with no fluctuations in the transfer. The amplitude D of the deterministic part is a measure of the effectiveness of energy transfer. On the other hand R measures the size of fluctuations. In [20] it is shown that a finite R necessarily leads to intermittency corrections in the exponents. In the next section we are going to determine the model parameters for the two REWA cascades we are considering.

Acknowledgements. We are grateful to R. Graham for useful discussions and to J. Krug for comments on the manuscript. This work is supported by the Sonderforschungsbereich 237 (Unordnung und grosse Fluktuationen).

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Son Pham is currently a professor of Computer Science, College of Engineering and Computer Science at the California State University at Northridge. His research interests include Computer Graphics and Software Engineering. He has published a dozen refereed academic/technical papers in Computer Science. He is a member of ACM. Pham received his BA in Mathematics from the University of Saigon, VietNam in 1973; MA in Mathematics from University of Louisville, Kentucky in 1975; Ph.D. in Statistics from the University of Cincinnati, Ohio in 1978; and PD in Computer Science from the University of California, Berkeley in 1980.

Mel Slater joined the Department of Computer Science, Queen Mary and Westfield College (University of London) in 1981. Since 1982 he has been involved in teaching and research in computer graphics. He produced an early implementation of the GKS graphics standard, and was involved in the ISO graphics group, with responsibility for the PASCAL language binding to GKS. He co-authored an undergraduate textbook on computer graphics, which was published in 1987. He is a principal investigator in several funded research projects, in particular the European Community ESPRIT funded SPIRIT Workstation project, and a project on Virtual Reality with the London Parallel Applications Centre. He was visiting professor in the Computer Science Division of Electrical Engineering and Computer Sciences, University of California at Berkeley in the spring semesters 1991 and 1992.

W. Kenneth Stewart is an assistant scientist at the Deep Submergence Laboratory of the Woods Hole Oceanographic Institution. His research interests include underwater robotics, autonomous vehicles and smart ROVs, multisensor modeling, real-time acoustic and optical imaging, and precision underwater surveying. Stewart has been going to sea on oceanographic research vessels for 19 years, has developed acoustic sensors and remotely-operated vehicles for 6000-m depths, and has made several deep dives in manned submersibles, including a 4000-m excursion to the USS Titanic in 1986. He is a member of the Marine Technology Society, Oceanography Society, IEEE Computer Society, ACM SIG-GRAPH, and NCGA. Stewart received a PhD in oceanographic engineering from the Massachusetts Institute of Technology and Woods Hole Oceanographic Institution Joint Program in 1988, a BS in ocean engineering from Florida Atlantic University in 1982, and an AAS in marine technology from Cape Fear Technical Institute in 1972.