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The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds
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Image of FIG. 1.
FIG. 1.

Computation of FTLE on a Cartesian mesh: A mesh of points is initialized (left panel), then integrated for a given integration time (right panel). The derivative of the flow at one grid point is evaluated using finite differences with other grid points. If one were to evaluate the flow derivative using off grid test points (gray dots), a LCS (dashed line) passing between grid points could go unnoticed.

Image of FIG. 2.
FIG. 2.

Deformation of an unstructured mesh under the flow. Left panel: the node has neighbors . Right panel: under the action of the flow, the node moves to the position . The deformed edges are used to approximate the deformation tensor.

Image of FIG. 3.
FIG. 3.

Using elements that are too large can lead to an underestimated FTLE. Starting with a square element at , the stretching is well captured by the motion of the four grid points at a later time . Nevertheless, the deformation is nonlinear and the element can fold at a later time . At this time, the algorithm that uses the position of the four grid points to evaluate FTLE will underestimate the actual stretching.

Image of FIG. 4.
FIG. 4.

Computation of FTLE for the aperiodic convection cells of Solomon and Gollub (Ref. 56) using an unstructured mesh and dynamic mesh refinement. The first row is the initial computation (rough estimate). A mesh with 3000 triangles of equal size (upper left panel) is used to evaluate the FTLE (upper right panel). The second row shows the first refinement step. The domain is covered by 8000 triangles whose maximum diameter is set to 0.02 wherever the FTLE estimate (upper right panel) is above 2.5. The third row uses the FTLE estimate from the second row to further refine the mesh. The new mesh has 20 000 triangles and the maximum triangle diameter is set to 0.01 for all the points where the previous FTLE estimate was above 2.9. In the last row, a final mesh of 55 000 triangles is created by setting the maximum diameter to 0.004 for all points, where the previous FTLE estimate is above 4.0. The resulting FTLE field shows a crisp LCS with roughly constant FTLE values (lower right panel).

Image of FIG. 5.
FIG. 5.

FTLE on a cylinder for a positive integration time (upper panel) and a negative integration time (middle panel). Corresponding attractive and repulsive LCS (lower panel) (enhanced online). [URL: http://dx.doi.org/10.1063/1.3278516.1]10.1063/1.3278516.1

Image of FIG. 6.
FIG. 6.

FTLE (upper panel) LCS (lower panel) on a Möbius strip (enhanced online). [URL: http://dx.doi.org/10.1063/1.3278516.2]10.1063/1.3278516.2

Image of FIG. 7.
FIG. 7.

Approximation of the deformation tensor (map from to ) using an unstructured mesh on a manifold embedded in .

Image of FIG. 8.
FIG. 8.

The splitting of the Antarctic ozone hole, September 2002, as evident in total column ozone concentrations. September 2001 and 2003 are shown for comparison (from Ref. 61). An example of the computed LCS around the time of the splitting and reformation event is shown in Fig. 9 (enhanced online). [URL: http://dx.doi.org/10.1063/1.3278516.3]10.1063/1.3278516.3

Image of FIG. 9.
FIG. 9.

The left column shows the superposition of the attractive and repulsive LCS on the 650 K isentrope on the days surrounding the Antarctic polar vortex splitting event of September 2002 (based on NCEP/NCAR reanalysis data). The attracting (repelling) curves, analogous to unstable (stable) manifolds, are shown in blue (red). Before and after the splitting event in late September, we see an isolated blob of air, bounded by LCS curves, slowly rotating over Antarctica. In the days leading up to the splitting, LCS curves form inside the vortex. The vortex pinches off, sending the northwestern part of the ozone hole off into the midlatitudes while the southwestern portion goes back to its regular position over Antarctica. Note the formation of lobe at the edges where chaotic stirring occurs across the LCS. The right column shows the corresponding daily ozone concentration (based on NASA TOMS satellite data) (enhanced online). [URL: http://dx.doi.org/10.1063/1.3278516.4]10.1063/1.3278516.4

Image of FIG. 10.
FIG. 10.

Prior to the splitting event, repulsive LCS appear in the core of the polar vortex indicating the onset of a bifurcation and a separation line between particles that will end up in different vortices. To verify the dynamics we initiated two parcels of particles on 20 September, one on each side of the nascent LCS. The simulation shows that the green parcel (northwest of the LCS) will remain in the core vortex while the purple parcel (southeast of the LCS) is dragged into the secondary vortex and disintegrate at higher latitude (enhanced online). [URL: http://dx.doi.org/10.1063/1.3278516.5]10.1063/1.3278516.5

Image of FIG. 11.
FIG. 11.

Left: Unstructured mesh on the Earth. Center: FTLE for (forward time). Right: FTLE for (backward time).

Image of FIG. 12.
FIG. 12.

Superposition of the attractive and repulsive LCS for a perturbed four vortex ring near the South Pole. As in the LCS shown in Fig. 9 from atmospheric reanalysis data, the model shows a hyperbolic region and the formation of filamentary structures that extend over large portions of the sphere (enhanced online). [URL: http://dx.doi.org/10.1063/1.3278516.6]10.1063/1.3278516.6



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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds