A Geometric Heat-Flow Theory of Lagrangian Coherent Structures

Abstract

We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection-diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or heat equation. By this coordinate transformation, the reversible effects of advection are separated from the irreversible joint effects of advection and diffusion. In this framework, LCSs express themselves as (boundaries of) metastable sets under the Lagrangian diffusion process. In the case of spatially homogeneous isotropic diffusion, averaging the time-dependent family of Lagrangian diffusion operators yields Froyland's dynamic Laplacian. In the associated geometric heat equation, 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 connections between geometric features and LCSs viewed as diffusion barriers in two numerical examples. Our approach facilitates the discovery of connections between some prominent methods for coherent structure detection: the dynamic isoperimetry methodology, the variational geometric approaches to elliptic LCSs, a class of graph Laplacian-based methods and the effective diffusivity framework used in physical oceanography.

A Lagrangian perspective on nonautonomous advection-diffusion processes in the low-diffusivity limit

“…We show that leading order asymptotics that hold for functions [Krol, 1991] extend to the dominant nontrivial singular value. This answers questions raised in [Karrasch & Keller, 2020]. The generator of the time-averaged diffusion/heat semigroup is a Laplace operator associated to a weighted manifold structure on the material manifold.…”

Detecting the birth and death of finite-time coherent sets

“…In other words, the dynamic Laplace operator is (twice) the generator of the dominant spatial process y t in (21) as a → ∞. The expression for Σ appears as a harmonic mean of the metrics g t in [KK20]; here it arises naturally in the limit of speeding up time in the temporal diffusion.…”

Rigid Sets and Coherent Sets in Realistic Ocean Flows

“…We refer to the techniques falling into this second category as operator based methods, which have been introduced by Froyland et al (2007Froyland et al ( , 2008. Different types of coherent structures, called "coherent sets", can be extracted from spectral decompositions of the operator L (Froyland (2013); Froyland and Kwok (2016); Karrasch and Keller (2020)), which correspond to sets of simple geometry mixing slowly with their complement (Froyland et al (2010); Froyland (2015)). This approach yields by nature coherent structures that are subdomains and not codimension-one sets (Froyland and Kwok (2016)); in the autonomous case, codimension-one "coherent" surfaces can also be extracted from the invariant sets of (1) by resorting to ergodic theory (Mezić and Banaszuk (2004); Levnajić and Mezić (2010); Budisić and Mezić (2012)).…”