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\title{Entropy-Based Numerical Investigation of Transport and Mixing in Flows}
\tocauthor{K. Padberg-Gehle} \author{} \institute{}
\maketitle
\begin{center}
{\large Kathrin Padberg-Gehle}\\
Technische Universit\"at Dresden\\
{\tt kathrin.padberg@tu-dresden.de}
\end{center}
\section*{Abstract}
Transport and mixing processes in flows can typically only be studied computationally. Finite-time Lyapunov exponents (FTLEs) or related stretching indicators are frequently used as a means to estimate transport barriers \cite{haller_11,shadden_lekien_marsden_05}. Alternatively, dominant eigenvectors \cite{dellnitz_junge_99} or singular vectors \cite{froyland_santi_monahan_10} of numerical transfer operators detect regions in phase space that are minimally dispersed under the dynamics.
Here we attempt to combine these two approaches. We introduce the concept of finite-time entropy (FTE) as a simple and flexible way to capture nonlinear stretching directly from the entropy growth experienced by a small localized density evolved by the transfer operator \cite{froyland_padberg_2012}.
We develop a simple and numerically efficient method of constructing an estimate of the FTE field within a set-oriented framework. Our novel approach is illustrated by several examples and compared to other approaches.
This is joint work with Gary Froyland (University of New South Wales, Australia).
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{haller_11}
{\sc G. Haller}. {A variational theory of hyperbolic Lagrangian coherent structures}. Physica D 240 (2011) 574--598.
\bibitem{shadden_lekien_marsden_05}
{\sc S.C. Shadden and F. Lekien and J.E. Marsden}. {Definition and properties of {L}agrangian coherent structures from finite-time {L}yapunov exponents in two-dimensional aperiodic flows}. Physica D 212 (2005) 271--304.
\bibitem{dellnitz_junge_99}
{\sc M. Dellnitz and O. Junge}. {On the approximation of complicated dynamical behaviour}. SIAM J. Numer. Anal. 36~(2) (1999) 491--515.
\bibitem{froyland_santi_monahan_10}
{\sc G. Froyland and N. Santitissadeekorn and A. Monahan}. {Transport in time-dependent dynamical systems: Finite-time coherent sets}. Chaos 20 (2010) 043116.
\bibitem{froyland_padberg_2012}
{\sc G. Froyland and K. Padberg-Gehle}. {Finite-time entropy: a probabilistic approach for measuring nonlinear stretching}. submitted (2011).
\end{thebibliography}
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