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.

arXiv comment43 pages, 24 figures, postprint

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Cited by in corpus (8)

• A Critical Comparison of Lagrangian Methods for Coherent Structure Detection
• Detecting the birth and death of finite-time coherent sets
• Lagrangian heat transport in turbulent three-dimensional convection
• Evolutionary clustering of Lagrangian trajectories in turbulent Rayleigh-Bénard convection flows
• Transfer Operators from Optimal Transport Plans for Coherent Set Detection
• Deep Lagrangian connectivity in the global ocean inferred from Argo floats
• Lagrangian studies of coherent sets and heat transport in constant heat flux-driven turbulent Rayleigh-Bénard convection
• Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions