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Attracting and repelling Lagrangian coherent structures from a single computation
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View: Figures


Image of FIG. 1.
FIG. 1.

Schematic illustration of approach I (a) and approach II (b) in the extended phase space.

Image of FIG. 2.
FIG. 2.

The errors in the computation of a repelling LCS grow exponentially as the LCS is advected forwards in time. The same statement holds for the backward-time advection of an attracting LCS.

Image of FIG. 3.
FIG. 3.

(a) A forward strain-surface evolves into a backward stretch-surface. (b) A forward stretch-surface evolves into a backward strain-surface.

Image of FIG. 4.
FIG. 4.

Trajectories of system (9) . The homoclinic orbits are shown in red.

Image of FIG. 5.
FIG. 5.

(a) Forward stretchline through the origin for three integration times  = 0.5 ( ),  = 1 ( ) and  = 2 ( ). (b) Forward strainline for the same integration times, as in panel (a). (c) The asymptotic position of the strainline (−°−) and the stretchline (−°−) compared to the classic stable and unstable manifolds (black).

Image of FIG. 6.
FIG. 6.

(a) Classical stable and unstable manifolds (black) are shown together withthe stretchline through the origin (magenta). Three blobs of tracers with radii (blue), (yellow), and (red) are centered at the origin. The tracers and the manifolds are then advected to time  = 0.1 (b),  = 0.2 (c), and  = 0.4 (d). Over the time interval [0,2], the stretchline is the de facto unstable manifold for spreading tracers. For larger advection times, this de facto unstable manifold practically coincides the classic unstable manifold of the origin.

Image of FIG. 7.
FIG. 7.

Stretchline (blue) and the advected image of an attracting LCS (red) at  = 0. The exponential growth of errors in backward-time advection of the LCS results in a jagged curve that deviates from the true attracting LCS.

Image of FIG. 8.
FIG. 8.

(a) The concentric tracers with radii 0.05 (blue), 0.1 (yellow), and 0.2 (red). The stretchline (black) passing through the center is computed from the time interval [0, 50] (i.e.,  = 0 and  = 50). The tracers and the stretchline are then advected forward in time to = 10 (b),  = 15 (c),  = 25 (d).

Image of FIG. 9.
FIG. 9.

(a) Forward stretchlines at  = 0. The attracting LCSs (i.e., locally most-stretching stretchlines) are highlighted in red. The green closed curves show the boundaries of elliptic regions. Tracers (blue circles) are used to visualize the overall mixing patterns. (b) Advected image of the attracting LCSs, tracers, and elliptic barriers at time  = 50.

Image of FIG. 10.
FIG. 10.

(a) A spherical tracer surface (blue) at time  = 0 and the corresponding approximate stretch-surface (red) passing through its origin. (b) The advected positions of these surfaces at the final time  = 4.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Attracting and repelling Lagrangian coherent structures from a single computation