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+\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
+ $\tilde{D}_{\ell}^{(2)}=\left<E_{\ell}\right>$ 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 $T = \left<E_0\right> / \epsilon$ 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 $T = \left<E_{0}\right>/\epsilon$}
+ \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 %
+%%%%%%%%%%%%%%%%%
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+\vspace*{4mm}
+\begin{biography}
+{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.
+\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}
+