# Altimetry-Based Diagnosis of Deep-Reaching Sub-Mesoscale Ocean Fronts

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. LLC4320 Numerical Simulation

#### Finite-Size Lyapunov Exponent

#### 2.2. Altimetry Data

#### 2.3. Southern Elephant Seal Dataset

#### 2.3.1. Buoyancy

#### 2.3.2. Vertical Velocities

## 3. Numerical Results

#### 3.1. Physical Context

**u**= (u,v) the horizontal velocity vector. Rossby and Richardson numbers of order one are indicative of an ageostrophic regime [5].

#### 3.2. Frontal Dynamics

#### 3.3. Recovering Frontal Dynamics at Depth from Finite-Size Lyapunov Exponent at the Surface

## 4. Application to In Situ Data

## 5. Discussion

- A1.
- We recover the orientation of sampled lateral buoyancy gradients, and subsequently their correct magnitude through a normalization scheme. This normalization takes into account the angle between the sampling platform and the front’s orientation deduced from the FSLE’s eigenvector.
- A2.
- We diagnose vertical velocities from the normalized lateral buoyancy gradients using a 2-D QG version of the omega equation.

- B1.
- Identify areas of strong background strain from near-real time satellite observations of geostrophic currents (i.e., when the Okubo-Weiss quantity is positive, Figure 11c).
- B2.
- Use near-real time AVISO-FSLE products to identify the location and orientation of FSLE filaments (Figure 11d).
- B3.
- Pilot the ship/robot perpendicular to this orientation. One should then perpendicularly cross sub-mesoscale fronts embedded in the background flow.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. On the Growth Rate and Orientation of Buoyancy Gradient

**A**:

**A**are purely imaginary. In this case, the gradient vector experiences a simple rotation. On the other hand, in strain-dominated areas where $W>0$, the eigenvalues of

**A**are real and $|\nabla b|$ exponentially grow. Thus, strain-dominated areas are particularly prone to the formation of sub-mesoscale fronts. These mechanisms are further detailed in many studies [32,48,49,50,51] and we summarize some of them below.

**Figure A1.**$\theta $ is the angle between $\nabla b$ and the x-axis and $\alpha $ is the angle between the compressional strain vector (${S}^{-}$) and the x-axis.

## Appendix B. Frequency-Wavenumber Spectrum

**Figure A2.**Frequency-wavenumber spectra computed from October 27 to December 27, 2011 of (

**a**) KE at 39 m, (

**b**) KE at 99 m, (

**c**) KE at 506 m, (

**d**) Strain rate at 39 m, (

**e**) Strain rate at 99 m, (

**f**) Strain rate at 506 m, (

**g**) $|\nabla b|$ at 39 m, (

**h**) $|\nabla b|$ at 99 m, (

**i**) $|\nabla b|$ at 506 m. The dashed line corresponds to the dispersion relation curve (associated with linear internal gravity waves, IGWs) that corresponds to the highest baroclinic mode resolved by the simulation. This curve allows for the partition between balanced motions (below the curve) and IGWs (above the curve). These spectra are presented in a variance preserving form, which allows the direct comparison of the relative contribution of different time and spatial scales to the total variance.

## Appendix C. Filter Cutoff Wavelength

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**Figure 1.**Map of (

**a**) strain rate $\sigma $ normalized by f at 39 m superimposed with contours of $\eta $ ranging from −1.85 to −0.85 m at 0.03 m interval. (

**b**) Buoyancy anomaly b’, defined as the anomaly with respect to the domain-averaged value, at 99 m superimposed with contours of finite-size Lyapunov exponent (FSLE) at 39 m ranging from −2.25 to −0.4 day${}^{-1}$ at 0.5 day${}^{-1}$ interval. (

**c**) Buoyancy gradients $|\nabla b|$ at 99 m superimposed with the same FSLE contours as in (

**b**). The dashed gray line at 87.5${}^{\circ}$ E corresponds to the section in Figure 2. Randomly selected snapshot on 27 October 2011 00:00:00 UTC.

**Figure 2.**Vertical/meridional section taken at a longitude of 87.5${}^{\circ}$ E (dashed gray line in Figure 1) of (

**a**) buoyancy anomaly b${}^{\prime}$. (

**b**) Buoyancy gradient $|\nabla b|$. (

**c**) Rossby number $\zeta /f$. (

**d**) Inverse Richardson number Ri${}^{-1}$. The MLD is shown in gray and corresponds to a density increase of 0.03 kg m${}^{-3}$ from the density at 10 m. Randomly selected snapshot on 27 October 2011 00:00:00 UTC.

**Figure 3.**Wavenumber spectrum of $\nabla b$-amplitude at different depths for a randomly selected snapshot taken on 27 October 2011 00:00:00 UTC.

**Figure 4.**Scatter plots of $|\nabla b|$ and $\zeta /f$ partitioned by Okubo-Weiss/${f}^{2}$ such that ∼20% of the domain points are kept in order to exclude coherent vortices and retain fronts and filaments (see main text, Equation (A3) in Appendix A and Roullet and Klein [27] for more details) at (

**a**) 39 m, (

**b**) 99 m, (

**c**) 209 m, (

**d**) 506 m. Randomly selected snapshot on 27 October 2011 00:00:00 UTC. Dots corresponds to an average over each grid interval on the abscissa (that has a total of 200 grid intervals), and thin vertical lines show std dev around the average. There is a strong asymmetry between positive and negative $\zeta /f$ and $|\nabla b|$, highlighting the ageostrophic regime.

**Figure 5.**Scatter plot between $|\nabla b|$ and strain rate $\sigma $ normalized by f at 39 m between 27 October and 27 December 2011. Gray points represent the average over each grid interval on the abscissa for a given day (that has a total of 200 grid intervals), black points represents the average over the time period and thin vertical lines show std dev around the average.

**Figure 6.**Finite-size Lyapunov exponent (FSLE) computed from the full model velocity field at (

**a**) 39 m, (

**b**) 99m, (

**c**) 209 m, (

**d**) 506 m. Randomly selected snapshot taken on 27 October 2011 00:00:00 UTC. Negative FSLE (computed backward in time [10]) indicate that patches of particles are being stretched and elongated by the background strain field. Large negative FSLE values indicate regions of strong stretching and sub-mesoscale fronts are preferentially located around and in between mesoscale eddies.

**Figure 7.**Ratio ${R}_{k}$ of the strain rate explained by eddies of size k and the strain rate explained by larger eddies. The “effective” strain rate is captured by wavelengths larger than 20 km.

**Figure 8.**FSLE computed from filtered horizontal velocities (referred to as FSLE${}_{filt}$ in the main text) at (

**a**) 39 m, (

**b**) 99 m, (

**c**) 209 m, (

**d**) 506 m and $|\nabla b|$ at (

**e**) 39 m, (

**f**) 99 m, (

**g**) 209 m, (

**h**) 506 m. Randomly selected snapshot taken on 27 October 2011 00:00:00 UTC. The dashed gray line corresponds to the sections in Figure 2.

**Figure 9.**Scatterplot of FSLE${}_{filt}$ at 39 m and FSLE${}_{filt}$ at (

**a**) 99 m, (

**b**) 209 m, (

**c**) 506 m from 27 October to 27 December 2011. The scatterplots are binned on the y-axis over 100 bins. FSLE${}_{filt}$ at 39 m strongly correlate with FSLE${}_{filt}$ at depth, highlighting the persistence of the vertical structure of the mesoscale strain field down to depths of 506 m. Thin vertical lines show std dev around the average.

**Figure 10.**Scatter plots between FSLE${}_{filt}$ ($\lambda $) normalized by f at 39 m and $|\nabla b|$ at (

**a**) 39m, (

**b**) 99 m, (

**c**) 209 m, (

**d**) 506 m between 27 October and 27 December 2011. Gray points represent the average over each grid interval on the abscissa for a given day (that has a total of 200 grid intervals), black points represents the average over the time period and thin vertical lines show std dev around the average.

**Figure 11.**Map of physical quantities derived from AVISO SSH on 11 November 2018 (i.e., at the seal’s mid-date trajectory). (

**a**) Relative vorticity normalized by f. (

**b**) Strain rate normalized by f. (

**c**) Okubo-Weiss quantity normalized by ${f}^{2}$. (

**d**) FSLE normalized by f. The seal’s transect is colored by its along-track distance. Note that the mesoscale eddy field considerably evolved during the seal’s transect.

**Figure 12.**Vertical section of (

**a**) ${b}_{x}$ sampled by the seal along its trajectory (in color in Figure 11), (

**b**) ${b}_{x}$ normalized by the angle between the front and the seal’s trajectory (${b}_{x\phantom{\rule{0.222222em}{0ex}}\mathrm{norm}}$, see main text and methods), (

**c**) ${b}_{x\phantom{\rule{0.222222em}{0ex}}\mathrm{norm}}/{b}_{x}$. The MLD is shown in black and is defined as the level of a 0.03 kg m${}^{-3}$ density increase from 15 m depth.

**Figure 13.**Time series of $\lambda $ (blue curve) and ${u}_{x\phantom{\rule{0.222222em}{0ex}}norm}$ (orange curve) along the seal’s track.

**Figure 14.**Vertical section of (

**a**) vertical velocities w along x the curvilinear abscissa following the seal’s trajectory (in color in Figure 11), (

**b**) vertical velocities ${w}_{norm}$ in the across-front direction, (

**c**) ${w}_{norm}/w$ for $\left|w\right|>5$ m day${}^{-1}$. Vertical velocities are derived by solving a 2-D QG omega equation (Equation (2)). The MLD is shown in black and is defined as the level of a 0.03 kg m${}^{-3}$ density increase from 15 m depth.

**Figure 15.**(

**a**) FSLE${}_{filt}$ at 39 m on 27 October 2011 00:00:00 UTC. (

**b**) FSLE${}_{filt}$ at 39 m on 29 October 2011 00:00:00 UTC. The dashed gray line at 87.5${}^{\circ}$ E corresponds to the section in (

**c**). (

**c**) Meridional profile of FSLE${}_{filt}$ at 39 m and 87.5${}^{\circ}$ E on 27 October 2011 00:00:00 UTC, 2011 (blue line) and 29 October 2011 00:00:00 UTC (orange line).

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## Share and Cite

**MDPI and ACS Style**

Siegelman, L.; Klein, P.; Thompson, A.F.; Torres, H.S.; Menemenlis, D. Altimetry-Based Diagnosis of Deep-Reaching Sub-Mesoscale Ocean Fronts. *Fluids* **2020**, *5*, 145.
https://doi.org/10.3390/fluids5030145

**AMA Style**

Siegelman L, Klein P, Thompson AF, Torres HS, Menemenlis D. Altimetry-Based Diagnosis of Deep-Reaching Sub-Mesoscale Ocean Fronts. *Fluids*. 2020; 5(3):145.
https://doi.org/10.3390/fluids5030145

**Chicago/Turabian Style**

Siegelman, Lia, Patrice Klein, Andrew F. Thompson, Hector S. Torres, and Dimitris Menemenlis. 2020. "Altimetry-Based Diagnosis of Deep-Reaching Sub-Mesoscale Ocean Fronts" *Fluids* 5, no. 3: 145.
https://doi.org/10.3390/fluids5030145