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S.C. Shadden, F. Lekien, J.E. Marsden, 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, 2005.
Cited: 8.
Visited: 5164.
In this paper, Lagrangian Coherent Structures (LCS) are defined as
ridges of finite-time Lyapunov exponent fields, also known
as "Direct" Lyapunov Exponent (DLE) 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 DLE
fields and the associated LCS for revealing the Lagrangian behavior
of systems with general time dependence. However, ridges of the DLE field need not be exactly advected with the flow.
The main result of this paper is an expression for the flux
across an LCS, which shows that the flux is small, and in most cases
negligible, for well defined LCS that rotate at a speed comparable
to the local Eulerian velocity field and are computed from DLE
fields with a sufficiently long integration time. Under these
hypotheses, the structures represent nearly
invariant manifolds even in systems with arbitrary time dependence.
New 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 LCS govern transport
and it is verified numerically that the flux of particles through these
distinguished lines is indeed negligible.
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Received 24 Feb 2005; Revised 10 Sep 2005; Accepted 13 Oct 2005.
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| 9.3 Mb | PDF 1.3 |  | color |
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Lagrangian Coherent Structures in a Simplified Double-Gyre Flow (see source code)
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Here we show a single period of motion of the periodic velocity field of
the double gyre, which is described in the paper. In this animation, the parameters were set to A=0.1,
ε=0.25, ω=2π/10. |
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1.6 Mb
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This animation shows a single period of motion of the DLE field for the double gyre shown above with an integration time T=15. The LCS extends from the bottom of the domain and loops back and forth toward the top. The attachment point of this LCS does not exactly correspond to the instantaneous stagnation point in the Eulerian velocity field, a phenomenon which is expected in time-varying flows.
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In this animation, the color contours of the above DLE field are appropriately adjusted
to highlight the LCS. Additionally, a fluid particle, which is denoted by the black
X and initially located
on the LCS, is advected in the flow (note the integration of the fluid particle and the LCS are
completely independent). Notice the LCS truly behaves as an invariant manifold. |
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This animation is very similar to the previous, except here we have superimposed the advection
of two parcels of fluid particles with the evolution of the LCS. Each parcel is initially located on one side of the LCS
to demonstrate how LCS act as separatrices, which divide the flow into dynamically distinct regions. |
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This animation shows how the integration time, T, affects the DLE field. Notice that as the integration
time increases, the LCS "grows" and further looping is revealed. This field
corresponds to time t = 0.1, and again, the parameters were set to A=0.1,
ε=0.25, ω=2π/10. |
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Lagrangian Coherent Structures along the Coast of Florida (see source code)
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DLE plot based on the surface currents measured by high-frequency radar stations near Fort Lauderdale. The green line correspond to the main ridge (LCS) attached to the coastline. |
Cinepak
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The red lines are LCS extracted from DLE at different time (every 30 mins) and the blue lines are integrated LCS, i.e. the first red line integrated using the velocity. If the LCS was perfectly Lagrangian, the red and green lines would be
identical in the close-up window. The distance between the two can be
used to compute the flux.
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Cinepak
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Longer animations and more information about the coherent structures extracted in the high-frequency radar data collected along the coast of Florida can be found in F. Lekien, C. Coulliette, A.J. Mariano, E.H. Ryan, L.K. Shay, G. Haller, J.E. Marsden, Pollution Release Tied to Invariant Manifolds: A case Study for the Coast of Florida, Physica D, 210(1-2), pp 1--20, 2005
Separation profile over a GLAS-II airfoil (see source code)
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This animation shows how LCS computed from DLE can capture the unsteady separation profile over an airfoil
(velocity field data provided by Jeff Eldredge  )
Fluid particles are colored based on the initial location with regard to the LCS to help
demonstrate the Lagrangian behavior of the
structure. |
Cinepak
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528 Kb
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Figures |
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In this paper, we developed a simplified model of a jet flow. Fluid separates at the Southern boundary before particles are injected in either the left or right gyre. DLE reveals the barrier between the two gyres and the separation point. |
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When the gyres oscillates, there is a still a ridge between the two gyres. The LCS is oscillating as it approaches the Northern boundary. To show that the ridge is an invariant manifold, we followed a particle starting on the ridge. See animation above. |
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Surface current data was collected near Fort Lauderdale by A. Mariano, L. Shay and E. Ryan. In this paper, we use the data collected to compute the DLE field and extract its ridges (or LCS). |
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The paper defines "ridge" in a mathematical sense and show that Lagrangian coherent structures, or ridges in the DLE field, are quasi invariant manifolds. Particles of fluid can hardly go through them. |
| We extracted the LCS near Fort Lauderdale at several times using the high-frequency radar data available. In addition, we took the first LCS (July 22nd 14:30 GMT) and we let it evolve as if it was a line of particles. |
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The small difference between the actual LCS and the line of particles moving in the flow gives an idea of the flux of particle across the LCS. These results are shown in an animation above and flux is plotted along the LCS on this figure. |
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The use of DLE and LCS is not restricted to geophysical flows. On this example, an airfoil is subject to a background flow and a controlled insertion of air through an open hole. The DLE plot on that figure reveals separation of the fluid on a point at the surface. |
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The separation on the airfoil is best viewed by advecting particles of air. The particles have been assigned a color depending on where they started (above or below the DLE ridge). At a later time, no particle has crossed the LCS because it is a quasi-invariant manifold. |
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Citations |
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This paper is cited by:- T. Inanc, S.C. Shadden, J.E. Marsden, Optimal Trajectory Generation in Ocean Flows, American Control Conference, Portland, OR, 2005.
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| - K. Padberg, Numerical Analysis of Transport in Dynamical Systems, Ph.D. Dissertation, Universitat Paderborn, 2005.
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| - S.C. Shadden, J.O. Dabiri, J.E. Marsden, Lagrangian Analysis of Fluid Transport in Empirical Vortex Ring Flows, , submitted, 2006.
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| - F. Lekien, C. Coulliette, A.J. Mariano, E.H. Ryan, L.K. Shay, G. Haller, J.E. Marsden, Pollution Release Tied to Invariant Manifolds: A case Study for the Coast of Florida, Physica D, 210(1-2), pp 1--20, 2005
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| - P. Bhatta, E. Fiorelli, F. Lekien, N.E. Leonard, D.A. Paley, F. Zhang, R. Bachmayer, R.E. Davis, D. Fratantoni, R. Sepulchre, Coordination of an Underwater Glider Fleet for Adaptive Sampling, International Workshop on Underwater Robotics, (in press), pp 61--69, 2005
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