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

Abstract

We generalize the concepts of finite-time Lyapunov exponent ͑FTLE͒ 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. © 2010 American Institute of Physics. ͓doi:10.1063/1.3278516͔Riemannian 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.

Trajectory encounter volume as a diagnostic of mixing potential in fluid flows

“…Similar to finite-time Lyapunov exponents (FTLEs) that are commonly used to delineate regions with qualitatively different motion (Haller, 2002;Shadden et al, 2005;Lekien and Ross, 2010), V depends on the trajectory starting time, t 0 , which allows us to track the evolution of oceanic features by repeating the calculation at different t 0 , and on the trajectory integration time, T , revealing different structures that impact the mixing potential of the flow from time t 0 to time t 0 + T . Specifically, longer segments of stable or unstable manifolds emanating from the hyperbolic regions are revealed for longer T in forward or backward time.…”

“…This exclusive link between the forward-and backward-time calculations of the trajectories and the stable and unstable manifolds, respectively, is not specific to the encounter volume diagnostic. It is rather typical for many finitetime methods from the dynamical systems theory, including finite-time Lyapunov exponents (FTLEs), which in forward time approximate segments of the stable manifold as maximizing ridges (Haller, 2002;Shadden et al, 2005;Lekien and Ross, 2010).…”

Investigating the connection between complexity of isolated trajectories and Lagrangian coherent structures

“…Over the last few decades chaotic advection has been shown to provide an insightful paradigm for interpreting stirring processes in fluid flows (Aref, 1984;Kuznetsov et al, 2002;Deese et al, 2002;Shadden et al, 2005;Olascoaga et al, 2006;Mancho et al, 2006;Lekien and Ross, 2010;Rypina et al, 2009Rypina et al, , 2010Rypina et al, , 2011. Hyperbolic trajectories and their stable/unstable manifolds are key to understanding this paradigm.…”

“…Hyperbolic trajectories and their stable/unstable manifolds are key to understanding this paradigm. Numerical estimates of these manifolds are often calculated using finite-time Lyapunov exponents (FTLEs), which measure the maximum rate of separation between a fluid trajectory and its nearby neighbors (Haller, 2002;Shadden et al, 2005;Lekien and Ross, 2010). The objects so obtained are known as Lagrangian Coherent…”

Caging the Blob: Using a Slime Mold to Teach Concepts about Barriers that Constrain the Movement of Organisms

“…For example, a mountain range might create a barrier, effectively isolating a population of microorganisms, or other factors, such as severe weather (e.g., hurricanes) might diminish barriers and accelerate the spread of a potentially devastating disease to crop plants across the country (Isard et al, 2005). Subtler atmospheric phenomena, perhaps due to the long-term effects of global climate change, can also lead to a dynamic landscape of "invisible" atmospheric barriers (Lekien & Ross, 2010), leading to changes in cyclic patterns of large-scale microorganism movement.…”