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diff --git a/macros/latex/contrib/springer/svjour/dc/layout.tex b/macros/latex/contrib/springer/svjour/dc/layout.tex new file mode 100644 index 0000000000..99efa609a9 --- /dev/null +++ b/macros/latex/contrib/springer/svjour/dc/layout.tex @@ -0,0 +1,860 @@ +\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 % +%%%%%%%%%%%%%%%%% + +\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. Fluid Mech. {\bf + 13}, 77 (1962) + +\bibitem{anselmet84} F.~Anselmet, Y.~Gagne, E.~J.~Hopfinger and R.~Antonia, +J. Fluid Mech. {\bf 140}, 63 (1984) + +\bibitem{benzi93a} R.~Benzi, S.~Ciliberto, R.~Tripiccione, C.~Baudet, + F.~Massaioli and S.~Succi, Phys. Rev. E {\bf 48}, R29 (1993) + +\bibitem{benzi93b} R.~Benzi, S.~Ciliberto, C.~Baudet, G.~R.~Chavarria and + R.~Tripiccione, Europhys. Lett. {\bf 24}, 275 (1993) + +\bibitem{herweijer95} J.~Herweijer and W.~van~de~Water, Phys. + Rev. Lett. {\bf 74}, 4651 (1995) + +\bibitem{castaing90} B.~Castaing, Y.~Gagne and E.~J.~Hopfinger, + Physica D {\bf 46}, 177 (1990) + +\bibitem{nelkin89} M.~Nelkin, J. Stat. Phys. {\bf 54}, 1 (1989) + +\bibitem{douady91} S.~Douady, Y.~Couder, and M.~E.~Brachet, + Phys. Rev. Lett. {\bf 67}, 983 (1991) + +\bibitem{kida92} S.~Kida and K.~Ohkitany, Phys. Fluids A + {\bf 4}, 1018 (1992) + +\bibitem{moffatt94} H.~K.~Moffatt, S.~Kida, and K.~Ohkitany, + {J. Fluid Mech.} {\bf 259}, 241 (1994) + +\bibitem{jimenez93} J.~Jim\'{e}nez, A.~A.~Wray, P.~G.~Saffman, and + R.~S.~Rogallo, J. Fluid Mech. {\bf 255}, 65 (1993) + +\bibitem{she94} Z.-S.~She and E.~Leveque, Phys. Rev. Lett. + {\bf 72}, 336 (1994) + +\bibitem{lvov95} V.~L'vov and I.~Procaccia, + Phys. Rev. E {\bf 52}, 3840 (1995); + Phys. Rev. E {\bf 52}, 3858 (1995) + +\bibitem{eggers91a} J.~Eggers and S.~Grossmann, Phys. Fluids A + {\bf 3}, 1958 (1991) + +\bibitem{grossmann94a} S.~Grossmann and D.~Lohse, Phys. Fluids + {\bf 6}, 611 (1994) + +\bibitem{eggers92} J.~Eggers, Phys. Rev. A {\bf 46}, 1951 (1992) + +\bibitem{eggers94} J.~Eggers, Phys. Rev. E {\bf 50}, 285 (1994) + +\bibitem{domaradzki95} J.~A.~Domaradzki and W.~Liu, Phys. Fluids {\bf 7}, 2025 (1995) + +\bibitem{obukhov71} A.~M.~Obukhov, Atmos. Ocean. Phys. + {\bf 7}, 471 (1971) + +\bibitem{gledzer73} E.~B.~Gledzer, Sov. Phys. Dokl. + {\bf 18}, 216 (1973) + +\bibitem{yamada87} M.~Yamada and K.~Ohkitami, J. Phys. Soc. Jpn + {\bf 56}, 4210 (1987) + +\bibitem{grossmann93} S.~Grossmann and D.~Lohse, Europhys. Lett. {\bf 21}, 201 (1993) + +\bibitem{glr96} S.~Grossmann, D.~Lohse, and A. Reeh, +Phys. Rev. Lett. {\bf 77}, 5369 (1996) + +\bibitem{eggers91b} J.~Eggers and S.~Grossmann, Phys. Lett. A +{\bf 156}, 444 (1991) + +\bibitem{farge92} M.~Farge, Ann. Rev. Fluid Mech. {\bf 24}, + 395 (1992) + +\bibitem{meneveau91} C.~Meneveau, J. Fluid Mech. {\bf 232}, + 469 (1991) + +\bibitem{spohn} P.~L.~Garrido, J.~L.~Lebowitz, C.~Maes, and +H.~Spohn, Physical Rev. A {\bf 42}, 1954 (1990) + +\bibitem{gardiner83} C.~W.~Gardiner, {Handbook of Stochastic Methods} +(Springer, Berlin, 1983) + +\bibitem{kraichnan73} R.~H.~Kraichnan, J. Fluid Mech. {\bf 59}, 745 (1973) + +\bibitem{orszag73} S.~Orszag, ``Lectures on the Statistical + Theory of Turbulence'' in: {Fluid Dynamics 1973, Les + Houches Summer School of Theoretical Physics}, edited by R.~Balian and + J.~L.~Peube (Gordon and Breach, 1977) + +\bibitem{tennekes80} H.~Tennekes and J.~L.~Lumley, {A First Course in + Turbulence} (MIT Press, 6th ed., 1980) + +\bibitem{grossmann92} S.~Grossmann and D.~Lohse, Z. Phys. B {\bf 89}, 11 (1992) + +\bibitem{press92} W.~H.~Press, S.~A.~Teukolsky, W.~T.~Vetterling, + and B.~P.~Flannery, {Numerical Recipes} (Cambridge University + Press, 2nd edition, 1986) + +\bibitem{kerr78} R.~M.~Kerr and E.~D.~Siggia, J. Stat. Phys. {\bf 19}, 543 (1978) + +\bibitem{yeung89} P.~K.~Yeung and S.~B.~Pope, +J. Fluid Mech. {\bf 207}, 531 (1989) + +\bibitem{kraichnan74} R.~H.~Kraichnan, J. Fluid Mech. {\bf 62}, 305 (1974) + +\bibitem{uhlig96} C.~Uhlig and J.~Eggers, Z. Phys. B, to be published + (1997) + +\bibitem{majda90} A.~J.~Majda, SIAM Review {\bf 33}, 349 (1990) + +\bibitem{olla95} P.~Olla, +Phys. Fluids {\bf 7}, 1598 (1995) + +\bibitem{uhlig94} C.~Uhlig and J.~Eggers, {unpublished + manuscript} (1994) + +\bibitem{Siggia77} E.~D.~Siggia, Phys. Rev. A {\bf 15}, 1730 (1977) + + +\bibitem{jensen91} M.~H.~Jensen, G.~Paladin, and A.~Vulpiani, +Phys. Rev. A {\bf 43}, 798 (1991) + +\bibitem{benzi95} R.~Benzi et al, Phys. Fluids {\bf 7}, 617 (1995) + +\bibitem{benzi93c} R.~Benzi, L.~Biferale, and G.~Parisi, Physica D {\bf 65}, 163 (1993) + +\end{thebibliography} +\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} + |