\documentclass[dc]{svjour} \usepackage{latexsym} \usepackage[fleqn]{amstex} \usepackage{graphics} %\usepackage{script} %\usepackage{mytimes} %\usepackage{mymathtimes} \usepackage{graphicx} \usepackage{psfig} \makeatletter \@mathmargin\z@ \makeatother % %\input{totalin.dc} % %\idline{Distrib. Comput (1997) 10: 65--78}{65} \begin{document} \def\bsub#1{\def\theequation{#1\alph{equation}}\setcounter{equation}{0}} \def\esub#1{\def\theequation{\arabic{equation}}\setcounter{equation}{#1}} \title{Local coupling of shell models\thanks{see also paper by Claudia Schiffer, Valeria Mazza, Cindy Crawford, Naomi Campbell, Helena Christensen, Elle McPherson, Eva Herzigova, Kathy Ireland, Kate Moss, Stephanie Seymour} leads to anomalous scaling} \subtitle{This is a sample article for Distributed Computing} \author{C. Uhlig\inst{1} \and J. Eggers\thanks{Professor emeritus}\and I.~Abt\inst{1} \and T.~Ahmed\inst{2} \and V.~Andreev\inst{3}\and B.~Andrieu\inst{4}\and R.-D. Appuhn \inst{5}} \mail{Springer-Verlag} \institute{Fachbereich Physik, Universit\"at -- Gesamthochschule -- Essen, D-45117 Essen, Germany\and I. Physikalisches Institut der RWTH, Aachen, Germany\thanks{Supported by the Bundesministerium f\"ur Forschung und Technologie, FRG under contract numbers 6AC17P, 6AC47P, 6DO57I, 6HH17P, 6HH27I, 6HD17I, 6HD27I, 6KI17P, 6MP17I, and 6WT87P} \and III. Physikalisches Institut der RWTH, Aachen, Germany \and School of Physics and Space Research, University of Birmingham, Birmingham, UK\thanks{Supported by the UK Science and Engineering Research Council} \and Inter-University Institute for High Energies ULB-VUB, Brussels, Belgium\thanks{Supported by IISN-IIKW, NATO CRG-890478} \and Rutherford Appleton Laboratory, Chilton, Didcot, UK \and Institute for Nuclear Physics, Cracow, Poland\thanks{Supported by the Polish State Committee for Scientific Research, grant No. 204209101} \and Physics Department and IIRPA, University of California, Davis, California, USA} %\date{Received: 14 November 1996 / Revised version: 2 January 1997} \maketitle \begin{abstract} This article demonstrates (almost) all possibilities offered by the new document class \verb|SVJour| in connection with the journal specific class option \verb|[dc]|. For detailed instructions please consult the accompanying documentation. \keywords{Routing schemes -- Universal routing schemes -- Implementation}\end{abstract} %%%%%%%%%%%%%%%%% % Introduction % %%%%%%%%%%%%%%%%% \section{Introduction} \label{sec:intro} Much of our intuitive understanding of turbulence is based on the concept of interactions which are local in k-space. 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 ${\bf \delta v(r) = v(x+r) - v(x)}$ over a distance $|{\bf r}|$ follows scaling laws \begin{equation} \label{vscaling} D^{(p)}(r) \equiv \left<|{\bf v(x+r) - v(x)}|^p\right> \sim r^{\zeta_p} \end{equation} in the limit of high Reynolds numbers. By taking velocity {\it differences} over a distance $r$, one probes objects of corresponding size. \begin{claim} This is a claim. Claims are unnumbered and the appearance is exactly the same as is proofs. \end{claim} \begin{proof} This is a proof. Proofs are unnumbered and the appearance is exactly the same as is claims. \end{proof} \begin{case}\label{romadur} This is a case. Cases are unnumbered and the appearance is exactly the same as is claims. \end{case} In Case \ref{romadur} you want a different numbering system for your theorem like environments please see Sect. 2 of the documentation of the general Springer journals document class. \begin{remark} This is a remark. Remarks are unnumbered and the appearance is exactly the same as is claims. \end{remark} \begin{theorem} This is a theorem. This environment is automagically numbered and the layout should be exactly the same as that of the corollary, the definition, the lemma, and the proposition. \end{theorem} \begin{proposition} This is a proposition. This environment is automagically numbered and the layout should be exactly the same as that of the corollary, the definition, the lemma, and the theorem. You should try it yourself. \end{proposition} \begin{lemma} This is a lemma. This environment is automagically numbered and the layout should be exactly the same as that of the corollary, the definition, the theorem, and the proposition. \end{lemma} \begin{definition} This is a definition. This environment is automagically numbered and the layout should be exactly the same as that of the corollary, the theorem, the lemma, and the proposition. \end{definition} \begin{corollary} This is a corollary. This environment is automagically numbered and the layout should be exactly the same as that of the theorem, the definition, the lemma, and the proposition. \end{corollary} If this is not enough, simply define your own environment according to Sect. 5.2 of the documentation of the general Springer journals document class. \begin{exercise} This is an exercise. This environment is automagically numbered and the layout should be exactly the same as that of the problem and solution. \end{exercise} \begin{problem} This is an problem. This environment is automagically numbered and the layout should be exactly the same as that of the exercise and solution. \end{problem} \begin{solution} This is an solution. This environment is automagically numbered and the layout should be exactly the same as that of the problem and exercise. \end{solution} \begin{conjecture} This is an conjecture. This environment is automagically numbered and the layout should be exactly the same as that of the example, note, property, and question. \end{conjecture} \begin{example} This is an example. This environment is automagically numbered and the layout should be exactly the same as that of the conjecture, note, property, and question. \end{example} \begin{property} This is an property. This environment is automagically numbered and the layout should be exactly the same as that of the conjecture, note, example, and question. \end{property} \begin{note} This is an note. This environment is automagically numbered and the layout should be exactly the same as that of the example, conjecture, property, and question. \end{note} \begin{question} This is an question. This environment is automagically numbered and the layout should be exactly the same as that of the example, conjecture, property, and example. \end{question} In addition to this assumption of self-similarity, Kolmogorov \cite{kolmogorov41} also made the seemingly intuitive assumption that the local statistics of the velocity field should be {\it 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 $\epsilon$, the only local scales available are the length $r$ and $\epsilon$ itself, which leads to the estimate $\delta v \sim (\epsilon r)^{1/3}$ or \begin{equation} \label{2/3} \zeta^{(class)}_p = p/3 . \end{equation} At the same time, one obtains an estimate for the Kolmogorov length \begin{equation} \label{eta} \eta = (\nu^3/\epsilon)^{1/4} \end{equation} where viscosity is important. However, it was only appreciated later \cite{landau59} 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 (\ref{2/3}) and (\ref{eta}). \begin{itemize} \item This is the first entry in this list. \item This is the second entry in this list. \item This is the third entry in this list. \item This is the fourth entry in the top level of this list. \begin{itemize} \item This is the first entry of the second level in this list. \item This is the second entry of the second level in this list. \begin{itemize} \item This is the first entry of the third level in this list. \item This is the third entry of the second level in this list. \item This is the third entry of the third level in this list. \end{itemize} \item This is the third entry of the second level in this list. \end{itemize} \item This is the fifth entry in this list. \item This is the last entry in this list. \end{itemize} Indeed, Kolmogorov \cite{kolmogorov62} and Obukhov \cite{obukhov62} later proposed the existence of corrections to the scaling exponents (\ref{2/3}), \begin{equation} \label{correct} \zeta_p = p/3 + \delta\zeta_p \;, \delta\zeta_p \ne 0 \end{equation} which were subsequently confirmed experimentally %[5--8]. \cite{anselmet84,benzi93a,benzi93b,herweijer95}. On one hand, careful laboratory experiments have been performed at ever higher Reynolds numbers \cite{anselmet84,castaing90}. On the other hand, a new method of data analysis \cite{benzi93a,benzi93b} has been successful in eliminating part of the effects of viscosity. \begin{enumerate} \item This is the first entry in this numbered list. \item This is the second entry in this numbered list. \item This is the third entry in this numbered list. \item This is the fourth entry in the top level of this numbered list. \begin{enumerate} \item This is the first entry of the second level in this numbered list. \item This is the second entry of the second level in this numbered list. \begin{enumerate} \item This is the first entry of the third level in this numbered list. \item This is the third entry of the second level in this numbered list. \item This is the third entry of the third level in this numbered list. \end{enumerate} \item This is the third entry of the second level in this numbered list. \end{enumerate} \item This is the fifth entry in this numbered list. \item This is the last entry in this numbered list. \end{enumerate} 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 $\mu$ being \cite{anselmet84} \begin{equation} \label{mu} \mu = -\delta\zeta_6 = 0.2 , \end{equation} which is a 10 \% correction. The existence of corrections like (\ref{mu}) 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 \[ \epsilon({\bf x},t) = \frac{\nu}{2}\left(\partial u_i/\partial x_k + \partial u_k/\partial x_i\right)^2 . \] Much of the research in turbulence has been devoted to the study of the spatial structure of $\epsilon({\bf x},t)$ \cite{kolmogorov62,nelkin89}, but which will not be considered here. The statistical average of this quantity is what we simply called $\epsilon$ 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 {\it 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. %%%%%%%%%%%%%%%%% % section 2 % %%%%%%%%%%%%%%%%% \section{Two cascade models} \label{sec:model} \subsection{Reduced wave vector set approximation} \subsubsection{Test for heading of third order} \paragraph{The REWA model.} The REWA model \cite{eggers91a,grossmann94a} is based on the full Fourier-transformed Navier-Stokes equation within a volume of periodicity $(2\pi L)^{3}$. 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 ${\cal K}=\bigcup_{\ell}{\cal K}_{\ell}$. Each of the wave vector shells ${\cal K}_{\ell}$ represents an octave of wave numbers. The shell ${\cal K}_{0}$ 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 ${\bf k}^{(0)}_{i}$: ${\cal K}_{0}=\{{\bf k}^{(0)}_{i}:i=1,\dots,N\}$. Starting with the generating shell ${\cal K}_{0}$, the other shells ${\cal K}_{\ell}$ are found by a successive rescaling of ${\cal K}_{0}$ with a scaling factor 2: ${\cal K}_{\ell}=2^{\ell} {\cal K}_{0}$. Thus each ${\cal K}_{\ell}$ consists of the $N$ scaled wave vectors $ 2^{\ell}{\bf k}_{i}^{(0)},\ i=1,\dots,N$. The shell ${\cal K}_{\ell}$ represents eddies at length scales $r\sim 2^{-\ell}L$, i.e. to smaller and smaller eddies as the shell index $\ell$ increases. At scales $r \approx \eta$ the fluid motion is damped by viscosity $\nu$, thus preventing the generation of infinitely small scales. Hence we only need to simulate shells ${\cal K}_{\ell}, \ell < \ell_{\nu}$, where $\ell_{\nu} \approx \log_2(L/\eta)$ is chosen such that the amplitudes in ${\cal K}_{\ell_{\nu}}$ are effectively zero. In this representation the Navier-Stokes equation for incompressible fluids reads for all ${\bf k}\in {\cal K}=\bigcup_{\ell=0}^{\ell_{\nu}}{\cal K}_{\ell}$: \bsub{7} \begin{eqnarray} \label{incompNSeq} \frac{\partial}{\partial t}u_{i}({\bf k},t)&=& -\imath M_{ijk}({\bf k})\sum_{ {\bf p},{\bf q}\in{\cal K}\atop {\bf k}={\bf p}+{\bf q}} u_{j}({\bf p},t)u_{k}({\bf q},t)\nonumber\\ && -\nu k^{2}u_{i}({\bf k},t)+f_{i}({\bf k},t)\label{NSeq}\\ {\bf k}\cdot {\bf u}({\bf k},t)&=&0\label{compress} . \end{eqnarray} \esub{7}% The coupling tensor $M_{ijk}({\bf k})=\left[k_{j}P_{ik}({\bf k})+k_{k}P_{ij}({\bf k})\right]/2$ with the projector $P_{ik}({\bf k})=\delta_{ik}-k_{i}k_{k}/k^{2}$ is symmetric in $j,k$ and $M_{ijk}({\bf k})=-M_{ijk}(-{\bf k})$. The inertial part of (\ref{NSeq}) consists of all triadic interactions between modes with ${\bf k}={\bf p}+{\bf q}$. They are the same as in the full Navier-Stokes equation for this triad. The velocity field is driven by an external force ${\bf f}({\bf k},t)$ which simulates the energy input through a large-scale instability. %\begin{figure*}\sidecaption %\psfig{figure=fig1.eps,width=6.7cm} %\resizebox{0.3\hsize}{!}{\includegraphics*{fig1.eps}} % \caption{A two-dimensional projection of the $k$-vectors % in shell ${\cal K}_0$ for both the REWA models considered here. % The small set ($\times$) contains all vectors with -1, 0, and 1 as % components. The large set ($\bigcirc$), in addition, contains % combinations with $\pm1/2$ and $\pm 2$ % } % \label{fig:sets} %\end{figure*} Within this approximation scheme the energy of a shell is \begin{equation} \label{energy} E_{\ell}(t)=\frac{1}{2}\sum_{{\bf k}\in{\cal K}_{\ell}} |{\bf u}({\bf k},t)|^{2}, \end{equation} and in the absence of any viscous or external driving the total energy of the flow field $E_{tot}(t)=\sum_{\ell=0}^{\ell_{\nu}} E_{\ell}(t)$ is conserved. The choice of generating wave vectors ${\bf k}^{(0)}_i$ determines the possible triad interactions. This choice must at least guarantee energy transfer between shells and some mixing within a shell. In \cite{eggers91a,grossmann94a} different choices for wavenumber sets ${\cal K}_0$ are investigated. The larger the number $N$ of wave numbers, the more effective the energy transfer. Usually one selects directions in ${\bf k}$-space 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. \begin{figure}%f2 %\psfig{figure=fig2.eps,width=3.25cm} \sidecaption \includegraphics[width=2cm,bb=0 0 192 635]{fig2} \caption{The structure of a local cascade. Eddies of size $r\sim 2^{-\ell}L$ are represented by their total energy $E_{\ell}$. Only modes of neighboring shells interact, leading to a local energy transfer $T_{\ell\rightarrow\ell+1}(t)$. 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 $T_{\ell}^{(diss)}(t)$ } \label{fig:modelstructure} \end{figure} Since in the models we consider energy transfer is purely local, the shell energies $E_{\ell}(t),\ \ell=0,\dots,\ell_{\nu}$ only change in response to energy influx $T_{\ell-1\rightarrow \ell}$ from above and energy outflux $T_{\ell\rightarrow \ell+1}$ to the lower shell. In addition, there is a rate of viscous dissipation $T_{\ell}^{(diss)}(t)$ which is concentrated on small scales, and a rate of energy input $T_{0}^{in}(t)$, which feeds the top level only, cf. Fig.~\ref{fig:modelstructure}. From (\ref{NSeq}) we find an energy balance equation which governs the time evolution of the shell energies $E_{\ell}(t)$ \begin{multline} \label{energyconservationlaw} \frac{d}{dt} E_{\ell}(t)=T_{\ell-1\rightarrow \ell}(t)-T_{\ell\rightarrow \ell+1}(t)+T_{\ell}^{(diss)}(t)+\\ T_{0}^{(in)}(t)\delta_{\ell0} . \end{multline} The different transfer terms are found to be \bsub{10} \begin{align} \label{FWtransfer} T_{\ell\rightarrow \ell+1}(t) &= 2\imath \sum_{\triangle^{(\ell+1)}_{(\ell)}} M_{ijk}({\bf k}) u_{i}^{*}({\bf k},t)u_{j}({\bf p},t)u_{k}({\bf q},t) \label{FWta}\\ T_{0}^{(in)}(t) &= \sum_{{\bf k}\in{\cal K}_{0}} \mbox{\rm Re}\left({\bf u}^{*}({\bf k},t)\cdot{\bf f}({\bf k},t)\right) \label{FWtb}\\ T_{\ell}^{(diss)}(t) &= -\nu \sum_{{\bf k}\in{\cal K}_{\ell}} k^{2}|{\bf u}({\bf k},t)|^{2} \label{FWtc}. \end{align} \esub{10}% In (\ref{FWta}) $\sum_{\triangle^{(\ell+1)}_{(\ell)}}$ indicates the summation over all next-neighbor triads ${\bf k}={\bf p}+{\bf q}$ with ${\bf k}\in {\cal K}_{\ell},\ {\bf p}\in{\cal K}_{\ell+1}$ and ${\bf q}\in{\cal K}_{\ell}\bigcup{\cal K}_{\ell+1}$. The driving force ${\bf f}({\bf k},t)$ is assumed to act only on the largest scales, and controls the rate of energy input $T_{0}^{(in)}(t)$. As in \cite{eggers91a} we choose ${\bf f}({\bf k},t)$ to ensure constant energy input $T^{(in)}_0 = \epsilon$ : \bsub{11} \begin{eqnarray} \label{force} {\bf f}({\bf k},t)&=&\frac{\epsilon {\bf u}({\bf k},t)}{N|{\bf u}({\bf k},t)|^{2}} \enspace \mbox{for all } {\bf k}\in{\cal K}_0\\ {\bf f}({\bf k},t)&=&0 \enspace \mbox{for all } {\bf k}\not\in{\cal K}_0 . \end{eqnarray} \esub{11}% \begin{equation} \label{re} Re = \frac{L U}{\nu} = \frac{\epsilon L^2}{\left< E_0 \right>\nu} , \end{equation} \begin{figure*}%f3 \sidecaption \resizebox{10cm}{!}{\includegraphics{fig3.eps}} \caption{Log-log-plot of the structure function $\tilde{D}_{\ell}^{(2)}=\left$ 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 $Re = 4.2\cdot10^5$ } \label{fig:scaling} \end{figure*} 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 $T = \left / \epsilon$ sets a typical turnover time scale of the 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 \cite{grossmann94a}, for example, time is measured in units of $L^{2/3} \epsilon^{-1/3}$, 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 \ref{fig:scaling} shows the scaling of the mean energy in a log-log plot at a Reynolds number of $4.2 \cdot 10^5$. 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.~\ref{fig:scaling}. Still, there are considerable fluctuations in this model, as evidenced by the plot of the energy transfer in Fig.~\ref{fig:scaling}. 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. \begin{figure}%f5 \psfig{figure=fig5.eps,width=6.45cm} \caption{Energy of a shell ($\ell=2$) 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 $T = \left/\epsilon$} \label{fig:energy} \end{figure} 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.~\ref{fig:energy} 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 {\it 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.~\ref{fig:energy} by a white-noise Langevin force. In this approximation we still hope to 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 non-equilibrium statistical mechanical system \cite{spohn}. \subsection{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 \cite{eggers94}. As in the REWA-cascade, the turbulent flow field is described by a sequence of eddies decaying successively (Fig.~\ref{fig:modelstructure}). The eddies at length scales $r\sim 2^{-\ell}L$ are represented by their energy $E_{\ell}(t)$. As before we restrict ourselves to local energy transfer, and thus the time evolution of the shell energies $E_{\ell}(t)$ is governed by (\ref{energyconservationlaw}). The crucial step is of course to choose an appropriate energy transfer $T_{\ell\rightarrow\ell+1}(t)$. For simplicity, we restrict ourselves to a Langevin process with a white noise force. Thus the local transfer $T_{\ell\rightarrow \ell+1}(t)$ is split into a deterministic and a stochastic part $T_{\ell\rightarrow \ell+1}(t) = T_{\ell\rightarrow \ell+1}^{(det)}(t)+ T_{\ell\rightarrow\ell+1}^{(stoch)}(t)$ where both parts should depend only on the local length scale $2^{-\ell}L$ and the neighboring energies $E_{\ell}$ and $E_{\ell+1}$. The most general form dimensionally consistent with this has been given in \cite{eggers94}. For simplicity, here we restrict ourselves to the specific form \bsub{13} \begin{align} \label{Letrans} T_{\ell\rightarrow \ell+1}^{(det)}(t)&= D \frac{2^{\ell}}{L}\left(E_{\ell}^{3/2}(t)-E_{\ell+1}^{3/2}(t)\right) \label{Letransa}\\ T_{\ell\rightarrow \ell+1}^{(stoch)}(t)&= R \left(\frac{2^{(\ell+1)}}{L}\right)^{1/2} (E_{\ell}(t)E_{\ell+1}(t))^{5/8} \xi_{\ell+1}(t)\label{Letransb}\\ T_{\ell}^{(in)}(t)&= \epsilon \delta_{\ell 0} \label{Letransc}\\ T^{(diss)}_{\ell}(t) &= -\nu(2^{-\ell}L)^{-2} E_{\ell} . \label{Letransd} \end{align} \esub{13}% The white noise is represented by $\xi_{\ell}$, i.e. $\left<\xi_{\ell}(t)\right>=0$ and $\left<\xi_{\ell}(t)\xi_{\ell'}(t')\right>=2\delta_{\ell\ell'}\delta(t-t')$. We use Ito's \cite{gardiner83} definition in (\ref{Letransb}). To understand the dimensions appearing in (\ref{Letrans}), note that $u_{\ell} \sim E_{\ell}^{1/2}$ is a local velocity scale and $k \sim 2^{\ell}/L$ is a wavenumber. Thus (\ref{Letransa}) dimensionally represents the energy transfer (\ref{FWta}). In (\ref{Letransb}) the powers are different, since $\xi$ carries an additional dimension of $1/\mbox{time}^{1/2}$. It follows from (\ref{Letransa}) that the sign of the deterministic energy transfer depends on which of the neighboring energies $E_{\ell}$ or $E_{\ell+1}$ are greater. If for example $E_{\ell}$ is larger, $T^{(det)}_{\ell\rightarrow\ell+1}(t)$ is positive, depleting $E_{\ell}$ in favor of $E_{\ell+1}$. 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 $\ell$ and $\ell+1$. 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 (\ref{Letransa}) and (\ref{Letransb}) 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 \cite{kraichnan73,orszag73}. 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 \begin{equation} \label{statenerg} E_{\ell}^{(0)}=C 2^{-(2/3)\ell}\ \mbox{with}\ C=\left(\frac{2\epsilon L}{D}\right)^{2/3}. \end{equation} 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 \cite{eggers94} 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. %%%%%%%%%%%%%%%%%%% % acknowledgments % %%%%%%%%%%%%%%%%%%% \begin{acknowledgement} 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). \end{acknowledgement} %%%%%%%%%%%%%%%%% % bibliography % %%%%%%%%%%%%%%%%% \begin{thebibliography}{47} \bibitem{kolmogorov41} A.~N.~Kolmogorov, C.~R.~Akad.~Nauk~SSSR {\bf 30},~301 (1941); C.~R.~Akad.~Nauk~SSSR {\bf 31}, 538 (1941); C.~R.~Akad.~Nauk~SSSR {\bf 32}, 16 (1941) \bibitem{landau59} L.~D.~Landau and E.~M.~Lifshitz, {Fluid Mechanics} (Pergamon, Oxford, 1959; third edition, 1984) \bibitem{kolmogorov62} A.~N.~Kolmogorov, J. Fluid Mech. {\bf 13}, 83 (1962) \bibitem{obukhov62} A.~M.~Obukhov, J. 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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. \end{biography} \newpage \begin{biography} {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. \end{biography} \begin{biography}{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 {\it USS Titanic} in 1986. He is a member of the Marine Technology Society, Oceanography Society, IEEE Computer Society, ACM SIGGRAPH, 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. \end{biography} \end{document}