Detecting the birth and death of finite-time coherent sets

Abstract

Finite-time coherent sets (FTCSs) are distinguished regions of phase space that resist mixing with the surrounding space for some finite period of time; physical manifestations include eddies and vortices in the ocean and atmosphere, respectively. The boundaries of finite-time coherent sets are examples of Lagrangian coherent structures (LCSs). The selection of the time duration over which FTCS and LCS computations are made in practice is crucial to their success. If this time is longer than the lifetime of coherence of individual objects then existing methods will fail to detect the shorter-lived coherence. It is of clear practical interest to determine the full lifetime of coherent objects, but in complicated practical situations, for example a field of ocean eddies with varying lifetimes, this is impossible with existing approaches. Moreover, determining the timing of emergence and destruction of coherent sets is of significant scientific interest. In this work we introduce new constructions to address these issues. The key components are an inflated dynamic Laplace operator and the concept of semi-material FTCSs. We make strong mathematical connections between the inflated dynamic Laplacian and the standard dynamic Laplacian [Froyland 2015], showing that the latter arises as a limit of the former. The spectrum and eigenfunctions of the inflated dynamic Laplacian directly provide information on the number, lifetimes, and evolution of coherent~sets.

References in corpus (7)

• The structure and dynamics of multilayer networks
• Diffusion dynamics on multiplex networks
• Geometry of the ergodic quotient reveals coherent structures in flows
• Topological chaos, braiding and bifurcation of almost-cyclic sets
• A tale of two vortices: how numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems
• A Lagrangian perspective on nonautonomous advection-diffusion processes in the low-diffusivity limit
• Heat-content and diffusive leakage from material sets in the low-diffusivity limit

Cited by in corpus (4)

• Evolutionary clustering of Lagrangian trajectories in turbulent Rayleigh-Bénard convection flows
• Definition, detection, and tracking of persistent structures in atmospheric flows
• A patch in time saves nine: Methods for the identification of localised dynamical behaviour and lifespans of coherent structures
• Scaling Limits of Stochastic Transport Equations on Manifolds