# Eddy Detection in HF Radar-Derived Surface Currents in the Gulf of Naples

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## Abstract

**:**

## 1. Introduction

## 2. Materials

#### 2.1. Dataset

#### 2.2. Dynamical Parameters Characterizing Recirculations

#### 2.2.1. Okubo–Weiss and Local Okubo–Weiss Parameters

#### 2.2.2. Local Normalized Angular Momentum and Momentum Flux Fields

## 3. Methods

#### 3.1. Eddy Detection Algorithms

#### 3.2. Ameda

- 1.
- Identifies grid points which are local extrema of $\mathrm{LNAM}$ satisfying $\mathrm{LNAM}>\mathrm{K}$ and $\mathrm{LOW}<0$, for a chosen threshold $\mathrm{K}\in (0,1)$;
- 2.
- Verifies the existence of at least one closed streamline around each extremum.

- 2’.
- Confirms that the velocity field constantly rotates along the perimeter of the square domain of edge $2b$ and centered at the extremum, for a chosen distance b.

#### 3.3. Neal

- 1.
- Identifies couples of adjacent grid points $({\underline{x}}_{1},{\underline{x}}_{2})$ such that the meridional component of the velocity field changes sign going westward along the zonal segment of length $2a$, centered at ${\underline{x}}_{i}$, and increases its magnitude away from this point. This computation also provides the expected sign of rotation;
- 2.
- Verifies that, at any such grid point ${\underline{x}}_{i}$, the zonal component of the velocity field changes sign going northward along the meridional segment of length $2a$, centered at ${\underline{x}}_{i}$, and increases its magnitude away from this point. This change must be compatible with the expected rotation;
- 3.
- Identifies the $\mathrm{KE}$ (kinetic energy) local minima inside a square domain of edge $2b$, centered at ${\underline{x}}_{i}$, which are global minima in a square neighborhood ${Q}_{b}$ of the same size;
- 4.
- Confirms that the velocity field constantly rotates along the perimeter $\partial {Q}_{b}$.

#### 3.4. Yada

- 1.
- Identifies the local extrema of a dynamical field like $\mathrm{LNAM}$, $\mathrm{KE}$ or $\mathrm{OW}$;
- 2.
- Analyzes the streamline geometry within some neighborhood ${Q}_{b}$ of each extremum, ensuring the existence of either bounded hyperbolic orbits (characterizing eddies with sink-like cores) or elliptic orbits (in presence of eddies having stable orbits).

#### 3.5. Tuning Strategy

## 4. Results

#### 4.1. Ameda Tuning and Results

- 3.
- Discards those extrema satisfying $\mathrm{LNMF}>0.2$.

#### 4.2. Neal Tuning and Results

#### 4.3. Yada Tuning and Results

- (1)
- The end points belong to the square domain ${Q}_{b-2l}$ (that is: they stay away from the boundary of the reference domain);
- (2)
- Each streamline completes at least one revolution.

#### 4.4. Eddy Boundaries

#### 4.4.1. Sink-Like Cores

#### 4.4.2. Eddies Having Elliptic Orbits

## 5. Discussion

#### 5.1. Detected Eddies

#### 5.2. Equivalent Radii

#### 5.3. Spatial Distribution

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. the Hausdorff Distance

## References

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**Figure 1.**Surface currents data provided by the HF radar system in the Gulf of Naples (

**on the left**) and the interpolated data (

**on the right**). Black arrows denote the velocity field whereas the blue line represents the coastline.

**Figure 2.**Number of eddies detected in the observation period $\mathrm{Ne}$, obtained with the angular momentum eddy detection and tracking algorithm (AMEDA) for different values of the parameters ${a}_{0}$, ${b}_{0}$ and $\mathrm{K}$. In each figure, corresponding to a value of ${a}_{0}$, the colored curves denote the graphs of $\mathrm{Ne}$ as a function of $\mathrm{K}$ for different values of ${b}_{0}$ (labeled as in the legend).

**Figure 3.**Source-like eddy core detected by the algorithm AMEDA (black star), velocity field (black arrows), local normalized angular momentum field $\left(\mathrm{LNAM}\right)$ contour lines (colored), local normalized momentum flux $\left(\mathrm{LNMF}\right)$ = $0.2$ contour (black lines) and coastline (blue line).

**Figure 4.**Number of eddies detected in the observation period, $\mathrm{Ne}$, obtained by the ’Nencioli et al. algorithm’ (NEAL) for different choices of the parameters ${a}_{0}$ and ${b}_{0}$. Colored lines denote the graphs of $\mathrm{Ne}$ as a function of ${b}_{0}$ for different values of ${a}_{0}$ (labeled as in the legend).

**Figure 5.**Maps showing the functioning of ’yet another eddy detection algorithm’ (YADA) for two eddies with sink-like cores. Once the eddy extreme point (EEP) (black crosses) is detected, YADA identifies a circle (black stars), centered at the extremum, which emanates streamlines (blue lines) with the following property: The streamline has to complete up to a revolution without reaching the domain boundary. Then it evaluates the mean points (yellow stars) and end points (red stars) of such streamlines, choosing the mean point of the second distribution as eddy symmetry center (ESC) (green stars). Black arrows denote the velocity field. In both panels the mean point and the ESC coincide.

**Figure 6.**Maps showing the functioning of YADA for two eddies having elliptic orbits. Once the eddy extreme point EEP (black crosses) is detected, YADA identifies a circle (black stars) emanating streamlines (blue lines) with the following property: Each streamline has to complete up to a revolution without reaching the domain boundary. Then it evaluates the mean points (yellow stars) and end points (red stars) of such streamlines, choosing the mean point of the first distribution as eddy symmetry center ESC (green stars). Black arrows denote the velocity field. In both panels the mean point and the ESC coincide.

**Figure 7.**Map showing the YADA boundary computation of an eddy having a sink-like core. Once the eddy extreme point EEP (black cross) and the eddy symmetry center ESC (green cross) are detected the algorithm draws the ellipses centered at the ESC with increasing radii. The cycle breaks when the black ellipse is drawn due to the existence of inadmissible streamlines (blue lines) leaving the domain. The last computed ellipse (green line) will be considered as boundary. Black arrows denote the velocity field.

**Figure 8.**Maps showing the YADA boundary computation of an eddy having stable orbits. From panel (

**a**–

**d**) ellipses of increasing semi-major axes are drawn; the temporal frame is unchanged. Once the EEP (black crosses) and the ESC (green crosses) are detected the algorithm draws the ellipses centered at the ESC with increasing semi-major axis d (black stars); from panels (

**a**–

**d**) the semi-major axis increases from 2 to 5 pixel lengths. It then evaluates the end points (red stars in panels (

**a**,

**b**,

**d**), and green stars in (

**c**)) of the streamlines emanated by these ellipses. The algorithm selects the semi-major axis ${d}^{*}$ for which the relative end points (green stars in (

**c**)) form the closest deformation of the associated ellipse. Black arrows denote the velocity field.

**Figure 9.**Eddy detected by AMEDA and missed by YADA. The local normalized angular momentum field $\mathrm{LNAM}$ extremum x (black cross) corresponds to an eddy core, but any circle centered at x (black stars) emanates streamlines (blue lines) which complete up to 3 revolutions before reaching the domain boundary (contact points in red). Black arrows denote the velocity field.

**Figure 10.**Sequence of time frames (from panel 1 to 6) showing the evolution of a detected eddy. Red stars denote eddy centers identified by AMEDA, whereas blue stars indicate ESCs computed by YADA. As the eddy changes shape and becomes less centrosymmetric AMEDA misses it (panel 3 to 4). Black arrows denote the velocity field (not in scale).

**Figure 11.**Left panel: Spatial distribution of the detected eddies by means of YADA (colored circles); different colors denote different sizes. Right panel: Detected long-lived eddies (circles) with lifetime $T\ge 2$ h. Initial, mid and final EEPs (black circles, blue circles and red stars respectively) with their relative eddy trajectories (blue dashed lines) and eddy lifetimes $T\ge 3$ h (blue numbers). Shoreline (blue contour) and bathymetric contour lines (black lines) between 100 m and 800 m of depth.

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## Share and Cite

**MDPI and ACS Style**

Bagaglini, L.; Falco, P.; Zambianchi, E.
Eddy Detection in HF Radar-Derived Surface Currents in the Gulf of Naples. *Remote Sens.* **2020**, *12*, 97.
https://doi.org/10.3390/rs12010097

**AMA Style**

Bagaglini L, Falco P, Zambianchi E.
Eddy Detection in HF Radar-Derived Surface Currents in the Gulf of Naples. *Remote Sensing*. 2020; 12(1):97.
https://doi.org/10.3390/rs12010097

**Chicago/Turabian Style**

Bagaglini, Leonardo, Pierpaolo Falco, and Enrico Zambianchi.
2020. "Eddy Detection in HF Radar-Derived Surface Currents in the Gulf of Naples" *Remote Sensing* 12, no. 1: 97.
https://doi.org/10.3390/rs12010097