A Geometric Heat-Flow Theory of Lagrangian Coherent Structures

TL;DR

This paper introduces a geometric heat-flow approach to identify Lagrangian coherent structures by transforming advection-diffusion equations into a Lagrangian framework, revealing metastable sets as diffusion barriers.

Contribution

It presents a novel geometric heat-flow theory that connects various methods for detecting coherent structures through a Lagrangian diffusion perspective.

Findings

LCSs are characterized as boundaries of metastable sets under Lagrangian diffusion.

The approach links dynamic isoperimetry, variational methods, and graph Laplacians.

Visualization of the geometry of mixing enhances understanding of diffusion barriers.

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.