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Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows

Shadden, Shawn C. and Lekien, Francois and Marsden, Jerrold E. (2005) Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D, 212 (3-4). pp. 271-304. ISSN 0167-2789. doi:10.1016/j.physd.2005.10.007.

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This paper develops the theory and computation of Lagrangian Coherent Structures (LCS), which are defined as ridges of Finite-Time Lyapunov Exponent (FTLE) fields. These ridges can be seen as finite-time mixing templates. Such a framework is common in dynamical systems theory for autonomous and time-periodic systems, in which examples of LCS are stable and unstable manifolds of fixed points and periodic orbits. The concepts defined in this paper remain applicable to flows with arbitrary time dependence and, in particular, to flows that are only defined (computed or measured) over a finite interval of time. Previous work has demonstrated the usefulness of FTLE fields and the associated LCSs for revealing the Lagrangian behavior of systems with general time dependence. However, ridges of the FTLE field need not be exactly advected with the flow. The main result of this paper is an estimate for the flux across an LCS, which shows that the flux is small, and in most cases negligible, for well-defined LCSs or those that rotate at a speed comparable to the local Eulerian velocity field, and are computed from FTLE fields with a sufficiently long integration time. Under these hypotheses, the structures represent nearly invariant manifolds even in systems with arbitrary time dependence. The results are illustrated on three examples. The first is a simplified dynamical model of a double-gyre flow. The second is surface current data collected by high-frequency radar stations along the coast of Florida and the third is unsteady separation over an airfoil. In all cases, the existence of LCSs governs the transport and it is verified numerically that the flux of particles through these distinguished lines is indeed negligible.

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Additional Information:© 2005 Elsevier. Received 24 February 2005; received in revised form 10 September 2005; accepted 13 October 2005. Available online 28 November 2005. Communicated by U. Frisch. This work has profited from the motivation and support of the Adaptive Ocean Sampling Network (AOSN) team as well as the Adaptive Sampling and Prediction (ASAP) team. The authors are grateful to Naomi Leonard for her valuable support and greatly thank George Haller for his helpful remarks and enlightening discussions. We also thank Rouslan Krechetnikov for his valuable suggestions. Numerical results in this paper were produced using MANGEN (, a software package developed by Francois Lekien and Chad Coulliette at the California Institute of Technology and supported by Program Manager Manuel Fiadeiro at the Office of Naval Research. The authors are grateful to Chad Coulliette for many enlightening and enjoyable discussions about coherent structures in geophysical flows. The high-frequency radar data used was collected by the Rosentiel School of Marine and Atmospheric Science (RSMAS) at the University of Miami. The authors are grateful to Arthur Mariano, Edward Ryan and Lynn Shay for sharing their results and their experience in using experimentally measured footprints in Lagrangian studies. The authors would also like to sincerely thank Jeff Eldredge for providing the velocity data for the flow around the airfoil in Section 8. This research was partially supported by ONR grants N00014-02-1-0826 and N00014-04-1-0534. Shawn Shadden is supported by a National Science Foundation fellowship.
Funding AgencyGrant Number
Office of Naval Research (ONR)N00014-02-1-0826
Office of Naval Research (ONR)N00014-04-1-0534
NSF Graduate Research FellowshipUNSPECIFIED
Subject Keywords:Coherent structures; Transport barriers; Direct and finite-time Lyapunov exponents; Mixing
Issue or Number:3-4
Record Number:CaltechAUTHORS:20100914-104231365
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19941
Deposited By: Ruth Sustaita
Deposited On:15 Sep 2010 18:41
Last Modified:08 Nov 2021 23:56

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