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

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## Abstract

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## 1. Introduction

## 2. Numerical Experiment

## 3. The Finite Amplitude Growth Rate

#### 3.1. Definition

#### 3.2. Properties

## 4. Finite Size Lyapunov Exponents

## 5. Dependence on Initial Conditions

## 6. Concluding Remarks

- Since the method relies on averaging all positive FAGR, it does not require arbitrary choices between the first, fastest, or average crossing times.
- The FAGR suppresses the need for higher frequency interpolation at small separation scales, and short-separation scales are more reliably represented.
- The FAGR can be computed and averaged over any given separation set, and the latter is not required to increase geometrically nor to be regular.
- The negative FSLE can be easily obtained by changing the averaging condition from $\langle \gamma \rangle >0$ to $\langle \gamma \rangle <0$.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The kinetic energy spectra of the three model runs. Run A, which is characterized by a an enstrophy inertial range is shown in panel (

**a**). Run B, which exhibits an energy inertial range, is shown in panel (

**b**). Run C, in which the spectrum exhibits an energy inertial range between approximately k = 10 and 30 (${k}^{-5/3}$ spectrum), and an enstrophy inertial range at wavenumbers exceeding k = 40, is shown in panel (

**c**).

**Figure 2.**Schematic representation of the initial separations used in this study. The reference initial condition consists of a Heaviside distribution between 0.01 and 5 (black/grey dashed line). The Dirac distributions (approximated as narrow Gaussian distributions), centred at some varying reference separations are used in the study of the dependence of FSLE to initial conditions (Section 5), and are represented as colored lines.

**Figure 3.**Comparison between the FSLE computed with Equation (30) (${\lambda}_{\gamma}$, green circles), the FSLE computed from the pair separations using the first crossing time ($\lambda $, black diamonds), and the CVE ($\tilde{\lambda}$, red asterisks). The ${k}^{-2/3}$ slope is indicated by the dotted gray line. The panels correspond to the experiments A (

**a**), B (

**b**) and C (

**c**).

**Figure 4.**Comparison between the instantaneous dispersion coefficient X (green diamonds), and the scale-averaged positive relative diffusivity ${K}_{s}$ (blue squares). The solid black line represents a ${r}^{2}$ slope and the dashed grey line a ${r}^{4/3}$ slope. Panels (

**a**–

**c**) show the results for the model runs A, B, and C, respectively.

**Figure 5.**Comparison between FSLE computed using the assymptotic limit Equation (34) (black circles and line) and Babiano et al. [31]’s inverse structural time (grey squares and line). The left hand side panel (

**a**) is for the model run A, the center panel (

**b**) for model run B, and the right-hand side panel (

**c**) for model run C.

**Figure 6.**(

**a**) CVE ($\tilde{\lambda}$) computed using different initial conditions (Dirac $\delta $ distributions centred on the reference separation thresholds) for model run A. The initial separation is color coded. The dots and diamonds represent the FSLE and CVE, respectively, computed using the reference (Heaviside distribution) initial separation. The colored ticks on the top x-axis correspond to the color-coding of the plain lines. (

**b**) Same as (

**a**) for run B. (

**c**) Same as (

**a**) for run C.

**Figure 7.**(

**a**) FSLE ($\lambda $) computed using different initial conditions (Dirac $\delta $ distributions centred on the reference separation thresholds) for model run A. The initial separation is color coded. The dots and diamonds represent the FSLE and CVE, respectively, computed using the reference (Heaviside distribution) initial separation. The colored ticks on the top x-axis correspond to the color-coding of the plain lines. (

**b**) Same as (

**a**) for run B. (

**c**) Same as (

**a**) for run C.

**Figure 8.**(

**a**) Probability density function of the values of $\gamma $ used in the computation of the CVE $\tilde{\lambda}$ at the reference separation threshold ${r}_{i}=0.35$ for model run A. Each line represents a different initial separation. Green and red lines represent initial separations that are smaller and larger than the reference separation ${r}_{i}$, respectively. (

**b**) Same as (

**a**) for model run B. (

**c**) Same as (

**b**) for model run C.

**Figure 9.**(

**a**) Probability density function of the values of $\gamma $ used in the computation of the FSLE $\lambda $ at the reference separation threshold ${r}_{i}=0.35$ for model run A. Each line represents a different initial separation. Green and red lines represent initial separations that are smaller and larger than the reference separation ${r}_{i}$, respectively. (

**b**) Same as (

**a**) for model run B. (

**c**) Same as (

**b**) for model run C.

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**MDPI and ACS Style**

Meunier, T.; LaCasce, J.H.
The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate. *Fluids* **2021**, *6*, 348.
https://doi.org/10.3390/fluids6100348

**AMA Style**

Meunier T, LaCasce JH.
The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate. *Fluids*. 2021; 6(10):348.
https://doi.org/10.3390/fluids6100348

**Chicago/Turabian Style**

Meunier, Thomas, and J. H. LaCasce.
2021. "The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate" *Fluids* 6, no. 10: 348.
https://doi.org/10.3390/fluids6100348