A geometric heatflow theory of Lagrangian coherent structures
Abstract
We define Lagrangian coherent structures (LCSs) as maximal material subsets whose advective evolution is maximally persistent to weak diffusion. For their detection, we first transform the Eulerian FokkerPlanck equation (FPE) into a diffusiononly equation in Lagrangian coordinates. In this framework, LCSs express themselves as almostinvariant sets under this diffusion process. The EuleriantoLagrangian coordinate transformation separates the reversible effects of advection from the irreversible joint effects of advection and diffusion. We approximate the Lagrangian FPE in two steps: first, we take the timeaverage of the diffusion tensors and identify Froyland's dynamic Laplacian as the associated generator; second, we introduce a deformed Riemannian geometry that is consistent with the averaged anisotropic diffusion. The latter turns the diffusion equation into a geometric heat equation, where the distribution of heat is governed by the dynamically induced intrinsic geometry on the material manifold, to which we refer as the geometry of mixing. We study and visualize this geometry in detail, and discuss the connections to diffusion barriers, i.e., boundaries of LCSs. We demonstrate the consistency with the geometric analysis in several numerical examples. Our approach facilitates the discovery of strong mathematical connections between several prominent methods for coherent structure detection, including the probabilistic transfer operator, the dynamic Laplacian, the variational geometric approaches to elliptic and parabolic LCSs and the effective diffusivity approach.
 Publication:

arXiv eprints
 Pub Date:
 August 2016
 arXiv:
 arXiv:1608.05598
 Bibcode:
 2016arXiv160805598K
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 Nonlinear Sciences  Chaotic Dynamics;
 Nonlinear Sciences  Pattern Formation and Solitons
 EPrint:
 50 pages, 19 figures, submitted. v4: major revision incorporating comments by the referees. v3: major revision with new material. v2: minor revision