% summary.tex       NSF Applied Math proposal
% Predrag (Sara)    Nov  4 2003
% Marsha            Sep 23 2003
% Predrag           Sep 23 2003
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% summary.tex               PFCNS proposal

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\newcommand{\PFCNS}{Physics Frontier Center in Nonlinear Science}
\newcommand{\CNS}{Center for Nonlinear Science}
\newcommand{\NSP}{Nonlinear Science Program}
\newcommand{\NS}{Nonlinear Science}
\newcommand{\NSC}{Nonlinear Science Course}
\newcommand{\program}{``Dynamics of Complex Systems in Science and Engineering'' program}

\newcommand{\GT}{Georgia Institute of Technology}
\newcommand{\COS}{College of Sciences}
\newcommand{\COE}{College of Engineering}

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\begin{document}

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\noindent
{\large \bf Project Summary}

%
\file{Nov 3 2003 version, printed \today}
%
\PC{There are two general merit review criteria approved by the National Science Board (NSB) and listed in the GPG: (1) the intellectual merit of the proposed activity, and (2) the broader impacts resulting from the proposed activity.
All proposals must separately address both of the merit review criteria
in the Project Summary.
% and should describe the broader impacts as an integral part of the narrative in the Project Description.
   }

A dynamical theory of turbulence
is one of {\em the} grand outstanding challenges of classical physics:
how are we to describe unstable dynamics of very many degrees of freedom?
%
The {\bf recurrent patterns program} proposed here  is
%SAS in its spirit
an undertaking very different from most current
turbulence research.
Here everything follows from deterministic
dynamics, with no probabilistic velocity distribution,
external stochastic forcing, or homogeneity of turbulence assumptions.
The theory emphatically does {\em not} seek
%SAS anything
to be universal; its goal is to accurately predict measurable
effects of a turbulent flow, such as the frictional drag,
for a given system with given physical
parameters and boundary conditions.
Inspired by the unstable coherent structures observed in turbulence,
the proposed theory follows a path
also radically different from the long-time PDE simulation approaches.
It postulates, in a nutshell, that
 $(\mbox{mesh size} \times \mbox{time})^{\mbox{\# fields}}
     \approx 10^{10}$
 is a vastly too large a number, and that a finite set of {\em building
blocks}, of order of $10^2-10^4$ recurrent patterns, is {\em all that is
required for the eventual assembly of a predictive capability for
non-equilibrium turbulent flows}.


The key issues are
(1) {\em how} to identify,
and
 (2) {\em what} to do with these patterns?
In the initial phase,
a new method for determining
recurrent patterns will be applied to a turbulent
1-$d$ model system.
With theoretical refinements,  {\em
turbulent regimes of full 3-$d$ hydrodynamic flows should be
computationally within reach, and will be investigated.}
An implementation of
the Sinai-Ruelle-Bowen
ergodic far-from-equilibrium natural measure, the
recurrence frequency
with which a turbulent flow visits a given state space region,
is a great challenge in the  case of
infinite-dimensional recurrent patterns.
% setting.
We propose to build up this measure
from small, computable recurrent patches, assembled using the
PI's {\em cycle expansions} theory.


\medskip

\noindent
{\bf Intellectual Merit:}
%
Turbulence is {\em the} unsolved problem of classical physics;
a big conceptual gap separates what the dynamical
systems approaches have
achieved so far for few degrees-of-freedom chaotic
systems, and what needs to be done
as systems grow large and turbulent.
A
distinctive aspect of the proposed research is its integration of
diverse areas of expertise: The problem of {\em how} to extract
the recurrent patterns is an intensely numerical undertaking, a
domain of fluid dynamicists. The theory for {\em
what} to do with this infinity of patterns
is the domain of ergodic theorists. We bring
to the project both sets of skills.
{\em This proposal addresses the grand challenge of nonlinear
  science:
Describe theoretically and explore numerically and experimentally
the dynamics of high-dimensional nonlinear systems. Furthermore,
this proposal will foster applications of the methods thus developed to
engineering problems, such as turbulent drag reduction.
}

\medskip

\noindent
{\bf Broader Impact:}
The PI intends to assemble a strong
US, European and Japanese collaborative team
from natural sciences (physics, atmospheric sciences) and engineering
(aerospace, fluid dynamics), with complementary
experimental, numerical and theoretical  skills.
The proposed research will be  carried out
in part globally, within the context of the very successful  electronic
{\tt ChaosBook.org} webbook collaboration, and in part locally,
at the {\GT} {\em Center for Nonlinear Science},
a uniquely interdisciplinary environment in
which training of undergraduates, graduate students,  and postdocs
in mathematical methodology of
complex systems is driven by concrete physical problems. The
record of bringing students into front-line
research is excellent: the ``recurrent patterns program'' itself initially
arose from a student contribution to {\tt ChaosBook.org/projects}.

\end{document}
