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 [Fro15], 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.

A patch in time saves nine: Methods for the identification of localised dynamical behaviour and lifespans of coherent structures

“…In a recent work, Froyland and Koltai [15] introduce an inflated dynamic Laplace operator and semi-material finite-time coherent sets (FTCSs) to investigate related questions regarding the number, lifetimes and evolution of coherent sets, and test these methods in settings different to ours. Our methods identify lifespans associated with dynamically meaningful coherent structures in a variety of systems.…”

Persistence and material coherence of a mesoscale ocean eddy

“…The effect of telescoping windows on the geometry of finite-time coherent sets was investigated in [34] and [3] have recently used a telescoping sequence of windows to determine the life expectancy of an ocean eddy by finding the largest window length that successfully identified a coherent vortex. An alternative approach to detecting the birth and death of finite-time coherent sets was developed in [32], where an inflated dynamic Laplace operator on a time-expanded domain removes the need for multiple telescoping and sliding computations.…”

Definition, detection, and tracking of persistent structures in atmospheric flows

“…The second part of the definition involves the persistence or coherence of the structures. In this context we will discuss the use of feature tracking approaches to directly estimate the lifetime of a structure (Günther et al, 2017;Sakurai et al, 2017;Heinze et al, 2017), Lagrangian coherent set analysis which provides estimates of the diffusive mixing time scale of a given set of Lagrangian trajectories (Froyland, 2013(Froyland, , 2015Banisch and Koltai, 2017;Koltai and Renger, 2018;Froyland and Koltai, 2021), and a new approach called "dynamic spectral clustering", which applies key ideas from the Lagrangian coherent set approach, but applies them to surrogate dynamics that are learned from the data and that may or may not approximate any Lagrangian paths (Koltai et al, 2021).…”