# 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

- Morel, P.; Larceveque, M. Relative Dispersion of Constant-Level Balloons in the 200-mb General Circulation. J. Atmos. Sci.
**1974**, 31, 2189–2196. [Google Scholar] [CrossRef] - Babiano, A.; Basdevant, C.; Le Roy, P.; Sadourny, R. Relative dispersion in two-dimensional turbulence. J. Fluid Mech.
**1990**, 214, 535–557. [Google Scholar] [CrossRef] - LaCasce, J.H. Statistics from Lagrangian observations. Prog. Oceanogr.
**2008**, 77, 1–29. [Google Scholar] [CrossRef] - Aurell, E.; Boffetta, G.; Crisanti, A.; Paladin, G.; Vulpiani, A. Growth of Noninfinitesimal Perturbations in Turbulence. Phys. Rev. Lett.
**1996**, 77, 1262–1265. [Google Scholar] [CrossRef] [PubMed][Green Version] - Artale, V.; Boffetta, G.; Celani, A.; Cencini, M.; Vulpiani, A. Dispersion of passive tracers in closed basins: Beyond the diffusion coefficient. Phys. Fluids
**1997**, 9, 3162–3171. [Google Scholar] [CrossRef][Green Version] - LaCasce, J.H.; Bower, A. Relative dispersion in the subsurface North Atlantic. J. Mar. Res.
**2000**, 58, 863–894. [Google Scholar] [CrossRef] - LaCasce, J.H.; Ohlmann, C. Relative dispersion at the surface of the Gulf of Mexico. J. Mar. Res.
**2003**, 61, 285–312. [Google Scholar] [CrossRef] - Lumpkin, R.; Elipot, S. Surface drifter pair spreading in the North Atlantic. J. Geophys. Res. (Oceans)
**2010**, 115, C12017. [Google Scholar] [CrossRef][Green Version] - Haza, A.C.; Özgökmen, T.M.; Griffa, A.; Poje, A.C.; Lelong, M.P. How Does Drifter Position Uncertainty Affect Ocean Dispersion Estimates? J. Atmos. Ocean. Technol.
**2014**, 31, 2809–2828. [Google Scholar] [CrossRef][Green Version] - Berti, S.; Santos, F.A.D.; Lacorata, G.; Vulpiani, A. Lagrangian Drifter Dispersion in the Southwestern Atlantic Ocean. J. Phys. Oceanogr.
**2011**, 41, 1659–1672. [Google Scholar] [CrossRef][Green Version] - Choi, J.; Bracco, A.; Barkan, R.; Shchepetkin, A.F.; McWilliams, J.C.; Molemaker, J.M. Submesoscale Dynamics in the Northern Gulf of Mexico. Part III: Lagrangian Implications. J. Phys. Oceanogr.
**2017**, 47, 2361–2376. [Google Scholar] [CrossRef] - Corrado, R.; Lacorata, G.; Palatella, L.; Santoleri, R.; Zambianchi, E. General characteristics of relative dispersion in the ocean. Sci. Rep.
**2017**, 7, 46291. [Google Scholar] [CrossRef][Green Version] - Zavala Sansón, L.; Pérez-Brunius, P.; Sheinbaum, J. Surface Relative Dispersion in the Southwestern Gulf of Mexico. J. Phys. Oceanogr.
**2017**, 47, 387–403. [Google Scholar] [CrossRef] - Boffetta, G.; Celani, A.; Cencini, M.; Lacorata, G.; Vulpiani, A. Nonasymptotic properties of transport and mixing. Chaos
**2000**, 10, 50–60. [Google Scholar] [CrossRef][Green Version] - Cencini, M.; Vulpiani, A. Finite size Lyapunov exponent: Review on applications. J. Phys. A Math. Theor.
**2013**, 46, 254019. [Google Scholar] [CrossRef] - Meyerjürgens, J.; Ricker, M.; Schakau, V.; Badewien, T.H.; Stanev, E.V. Relative Dispersion of Surface Drifters in the North Sea: The Effect of Tides on Mesoscale Diffusivity. J. Geophys. Res. (Oceans)
**2020**, 125, e15925. [Google Scholar] [CrossRef] - Haza, A.C.; Poje, A.C.; Özgökmen, T.M.; Martin, P. Relative dispersion from a high-resolution coastal model of the Adriatic Sea. Ocean Model.
**2008**, 22, 48–65. [Google Scholar] [CrossRef] - Koh, T.Y.; Legras, B. Hyperbolic lines and the stratospheric polar vortex. Chaos
**2002**, 12, 382–394. [Google Scholar] [CrossRef] [PubMed] - d’Ovidio, F.; Fernández, V.; Hernández-García, E.; López, C. Mixing structures in the Mediterranean Sea from finite-size Lyapunov exponents. Geophys. Res. Lett.
**2004**, 31, L17203. [Google Scholar] [CrossRef][Green Version] - Bettencourt, J.H.; López, C.; Hernández-García, E. Characterization of coherent structures in three-dimensional turbulent flows using the finite-size Lyapunov exponent. J. Phys. A Math. Theor.
**2013**, 46, 254022. [Google Scholar] [CrossRef] - Letz, T.; Kantz, H. Characterization of sensitivity to finite perturbations. Phys. Rev. E
**2000**, 61, 2533–2538. [Google Scholar] [CrossRef] - LaCasce, J.H. Estimating Eulerian energy spectra from drifters. Fluids
**2016**, 1, 33. [Google Scholar] [CrossRef][Green Version] - LaCasce, J.H. Baroclinic Vortices over a Sloping Bottom. Ph.D. Thesis, M.I.T./W.H.O.I., Cambridge, MA, USA, 1996. [Google Scholar]
- Kraichnan, R.H. Inertial Ranges in Two-Dimensional Turbulence. Phys. Fluids
**1967**, 10, 1417–1423. [Google Scholar] [CrossRef][Green Version] - Lundgren, T.S. Turbulent pair dispersion and scalar diffusion. J. Fluid Mech.
**1981**, 111, 27–57. [Google Scholar] [CrossRef] - Bennett, A.F. Lagrangian Fluid Dynamics; Cambridge University Press: Cambridge, UK, 2006; p. 286. [Google Scholar]
- LaCasce, J.H. Relative displacement PDFs from balloons and drifters. J. Mar. Res.
**2010**, 68, 433–457. [Google Scholar] [CrossRef][Green Version] - Tang, X.Z.; Boozer, A.H. Finite time Lyapunov exponent and advection-diffusion equation. Phys. D Nonlinear Phenom.
**1996**, 95, 285–305. [Google Scholar] [CrossRef] - Lapeyre, G. Characterization of finite-time Lyapunov exponents and vectors in two-dimensional turbulence. Chaos Interdiscip. J. Nonlinear Sci.
**2002**, 12, 688–698. [Google Scholar] [CrossRef][Green Version] - Brunton, S.; Rowley, C. Fast computation of finite-time Lyapunov exponent fields for unsteady flows. Chaos Interdiscip. J. Nonlinear Sci.
**2010**, 20, 017503. [Google Scholar] [CrossRef] - Babiano, A.; Basdevant, C.; Sadourny, R. Structure functions and dispersion laws in two-dimensional turbulence. J. Atmos. Sci.
**1985**, 42, 941–949. [Google Scholar] [CrossRef][Green Version]

**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