Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate

Version 1 : Received: 9 August 2021 / Approved: 9 August 2021 / Online: 9 August 2021 (15:07:07 CEST)

How to cite: Meunier, T.; LaCasce, J. The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate. Preprints 2021, 2021080203 (doi: 10.20944/preprints202108.0203.v1). Meunier, T.; LaCasce, J. The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate. Preprints 2021, 2021080203 (doi: 10.20944/preprints202108.0203.v1).

## Abstract

The Finite size Lyapunov exponent (FSLE) has been used extensively since the late 1990’s to diagnose turbulent regimes from Lagrangian experiments and to detect Lagrangian coherent structures in geophysical flows and two-dimensional turbulence. Historically, the FSLE was defined in terms of its computational method rather than via a mathematical formulation, and the behavior of the FSLE in the turbulent inertial ranges is based primarily on scaling arguments. Here we propose an exact definition of the FSLE based on conditional averaging of the finite amplitude growth rate (FAGR) of the particle pair separation. With this new definition, we show that the FSLE is a close proxy for the inverse structural time, a concept introduced a decade before the FSLE. The (in)dependence of the FSLE on initial conditions is also discussed, as well as the links between the FAGR and other relevant Lagrangian metrics, such as the finite time Lyapunov exponent and the second order velocity structure function.

## Keywords

Finite size Lyapunov Exponent; Finite amplitude growth rate; Two-dimensional turbulence; Lagrangian fluid dynamics

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