^{1,a)}and Shane D. Ross

^{2,b)}

### Abstract

We generalize the concepts of finite-time Lyapunov exponent (FTLE) and Lagrangian coherent structures to arbitrary Riemannian manifolds. The methods are illustrated for convection cells on cylinders and Möbius strips, as well as for the splitting of the Antarctic polar vortex in the spherical stratosphere and a related point vortex model. We modify the FTLE computational method and accommodate unstructured meshes of triangles and tetrahedra to fit manifolds of arbitrary shape, as well as to facilitate dynamic refinement of the FTLE mesh.

Riemannian manifolds are ubiquitous in science and engineering, being the more natural mathematical setting for many dynamical systems. For instance, transport along isopycnal surfaces in the ocean and large-scale mixing in the atmosphere are processes taking place on a curved manifold, not a vector space. In this paper, we generalize the notion of finite-time Lyapunov exponent (FTLE) and Lagrangian coherent structures (LCS) to arbitrary Riemannian differentiable manifolds. We show that both notions are independent of the coordinate system but depend on the chosen metric. However, we find that LCS do not depend much on the metric. The FTLE measures separation and tends to be large and positive along LCS and very small elsewhere. For sufficiently large integration times, the steep variations of the FTLE field cannot be modified much by smooth changes in the metric and the LCS remain essentially unchanged. Approximating, or even ignoring, the manifold metric does not influence the result for large integration times, and therefore, computing the FTLE field on manifolds is robust. Aside from these conclusions, we present a general algorithm for computing the FTLE on manifolds covered with meshes of polyhedra. The algorithm requires knowledge of the mesh nodes, the image of the mesh nodes under the flow, as well as information about node neighbors (but not the full connectivity). We also used the same algorithm in Euclidian spaces where the unstructured mesh permits efficient adaptive refinement for capturing sharp LCS features. We illustrate the results and methods on several systems: convection cells in a plane, on a cylinder, and on a Möbius strip, as well as atmospheric transport resulting from the 2002 splitting of the Antarctic ozone hole in the spherical stratosphere and a related point vortex model on the sphere.

I. INTRODUCTION

II. FINITE-TIME LYAPUNOV EXPONENTS IN EUCLIDEAN SPACES

A. Computation with a Cartesian mesh

B. Computation with an unstructured mesh (in )

C. Computation with an unstructured mesh (in )

D. Example 1: Rayleigh–Bénard convection cells

III. FINITE-TIME LYAPUNOV EXPONENTS IN RIEMANNIAN MANIFOLDS

A. FTLE in manifold coordinates

B. Orthogonal coordinates

C. Orthonormal coordinates

D. Example 2: Convection cells on a torus

E. Example 3: Convection cells on a Möbius strip

IV. FINITE-TIME LYAPUNOV EXPONENTS ON DIFFERENTIABLE MANIFOLDS EMBEDDED IN

A. Finding the local basis

B. Exploiting the embedding

C. Example 4: The Antarctic polar vortex splitting event of 2002

D. Example 5: A multivortex flow on a sphere

V. CONCLUSIONS

### Key Topics

- Manifolds
- 79.0
- Liquid crystals
- 69.0
- Rotating flows
- 35.0
- Singular values
- 18.0
- Riemannian manifolds
- 12.0

## Figures

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.

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.

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.

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.

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.

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.

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).

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).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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