Finite-time Lyapunov exponents in the instantaneous limit and material transport

TL;DR

This paper introduces the instantaneous Lyapunov exponent (iLE) as a faster, Eulerian alternative to traditional Lagrangian FTLE methods for analyzing short-term material transport in fluid flows, validated through numerical and analytical examples.

Contribution

It defines the iLE as the infinitesimal limit of FTLE, connecting Eulerian and Lagrangian frameworks, enabling quick assessment of fluid transport with a single velocity snapshot.

Findings

iLE effectively predicts short-term material transport.

iLE structures align with Lagrangian FTLE in flow analysis.

The method is computationally efficient and suitable for real-time applications.

Abstract

Lagrangian techniques, such as the finite-time Lyapunov exponent (FTLE) and hyperbolic Lagrangian coherent structures (LCS), have become popular tools for analyzing unsteady fluid flows. These techniques identify regions where particles transported by a flow will converge to and diverge from over a finite-time interval, even in a divergence-free flow. Lagrangian analyses, however, are time consuming and computationally expensive, hence unsuitable for quickly assessing short-term material transport. A recently developed method called OECSs [Serra, M. and Haller, G., `Objective Eulerian Coherent Structures', Chaos 26(5), 2016] rigorously connected Eulerian quantities to short-term Lagrangian transport. This Eulerian method is faster and less expensive to compute than its Lagrangian counterparts, and needs only a single snapshot of a velocity field. Along the same line, here we define the instantaneous Lyapunov Exponent (iLE), the instantaneous counterpart of the FTLE, and connect the Taylor series expansion of the right Cauchy-Green deformation tensor to the infinitesimal integration time limit of the FTLE. We illustrate our results on geophysical fluid flows from numerical models as well as analytical flows, and demonstrate the efficacy of attracting and repelling instantaneous Lyapunov exponent structures in predicting short-term material transport.