# Optimal Perturbations of an Oceanic Vortex Lens

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## Abstract

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## 1. Introduction

#### 1.1. Meddies

#### 1.2. Optimal Perturbations

## 2. Methods

#### 2.1. Adjoint Methods

#### 2.2. The Numerical Models

#### 2.3. Parameterisation of the Vortex Lens

## 3. Results

#### 3.1. Growth Rates of the Optimal Perturbations

#### 3.2. Spatial Structure of the Optimal Perturbations

#### 3.2.1. Horizontal Structure

#### 3.2.2. Vertical Structure

#### 3.3. Evolution of the Optimal Perturbations in the Non Linear Model

## 4. Discussion and Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Growth rate of the perturbation ($\sigma =\frac{log\left(\lambda \right)}{2\tau}$) against the number of iteration through the tangent linear and adjoint models for different time intervals $\tau $. The additional discontinuous black line represents the growth rate of the most unstable normal mode ($m=2$). (

**b**) Growth rate of the optimal perturbation against time interval after convergence of the solution (80 iterations through the tangent linear and adjoint models). The discontinuous black line represents the growth rate of the most unstable normal mode ($m=2$).

**Figure 2.**Optimal perturbation for the time interval $\tau $ = 4 days. (

**a**) PVA horizontal section through the vortex core. (

**b**) PVA section through the vortex lobes. (

**c**) PVA vertical section. (

**d**) Energy distribution on the azimuthal modes.

**Figure 3.**Optimal perturbation for the time interval $\tau $ = 10 days. (

**a**) PVA horizontal section through the vortex core. (

**b**) PVA section through the vortex lobes. (

**c**) PVA vertical section. (

**d**) Energy distribution on the azimuthal modes.

**Figure 4.**Optimal perturbation for the time interval $\tau $ = 17 days. (

**a**) PVA horizontal section through the vortex core. (

**b**) PVA section through the vortex lobes. (

**c**) PVA vertical section. (

**d**) Energy distribution on the azimuthal modes.

**Figure 5.**Optimal perturbation for the time interval $\tau $ = 174 days. (

**a**) PVA horizontal section through the vortex core. (

**b**) PVA section through the vortex lobes. (

**c**) PVA vertical section. (

**d**) Energy distribution on the azimuthal modes.

**Figure 6.**Projection of the optimal perturbation for the time interval $\tau $ = 17 days on the 3 most energetic azimuthal modes (m = 3, 4, 5). The top panels show PVA horizontal sections through the vortex core; the middle panels show PVA horizontal sections through the vortex lobes and the bottom panels show PVA vertical sections.

**Figure 7.**(

**a**) Distribution of the perturbation energy on the 20 first azimuthal modes for time intervals ranging from 1 to 174 days. For short time intervals, the spectrum is broad and high wave numbers dominate. As the time interval increases, the spectrum becomes narrower and low azimuthal modes dominate. (

**b**) Relative energy of the azimuthal wave numbers against time interval $\tau $. (Energy of each mode normalized by the total energy of the optimal perturbation for each computed time interval). Each azimuthal mode occupies a specific time interval band. High azimuthal modes occupy narrow time interval bands overlapping one another while low azimuthal modes occupy larger time interval bands.

**Figure 8.**Energy distribution on the vertical modes of the optimal perturbations for time intervals $\tau $ = 4, 10, 17 and 174 days (Energy of each vertical mode normalized by the total perturbation energy). As an effect of the vertical symmetry of the vortex lens, even modes have much more energy than odd modes. Whereas the time interval does not have a clear effect on the most energetic vertical modes (centre of the spectral peak), it clearly affects the width of the spectral peak. The longer the time interval, the wider the energy spectrum.

**Figure 9.**Time evolution of the optimal perturbations for time intervals $\tau $ = 4, 10, 17 and 174 days implemented in the non-linear QG model. (

**a**) Amplification factor (non-dimensional) against time (days); (

**b**) Zoom of (a) between 0 and 25 days; (

**c**) Growth rate (s${}^{-1}$) against time(days).

**Figure 10.**Time evolution of the optimal perturbation for a time interval $\tau $ = 17 days in the non-linear QG model. The colour scale shows the PVA perturbation (${q}^{\prime}=q-\overline{q}$) while the black contours show the total PVA q. The top panels show horizontal sections of PVA through the vortex core, the middle panels show horizontal sections of PVA through the vortex lobes and the bottom panels show vertical sections of PVA.

**Figure 11.**Time evolution of the azimuthal decomposition of the optimal perturbation for a time interval $\tau $ = 17 days integrated in the non-linear QG model. The discontinuous black line shows the total energy.

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**MDPI and ACS Style**

Meunier, T.; Ménesguen, C.; Carton, X.; Le Gentil, S.; Schopp, R.
Optimal Perturbations of an Oceanic Vortex Lens. *Fluids* **2018**, *3*, 63.
https://doi.org/10.3390/fluids3030063

**AMA Style**

Meunier T, Ménesguen C, Carton X, Le Gentil S, Schopp R.
Optimal Perturbations of an Oceanic Vortex Lens. *Fluids*. 2018; 3(3):63.
https://doi.org/10.3390/fluids3030063

**Chicago/Turabian Style**

Meunier, Thomas, Claire Ménesguen, Xavier Carton, Sylvie Le Gentil, and Richard Schopp.
2018. "Optimal Perturbations of an Oceanic Vortex Lens" *Fluids* 3, no. 3: 63.
https://doi.org/10.3390/fluids3030063