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  • View in gallery

    Block diagram of AMEDA showing the main computation (gray boxes) and the conditional calculations (light gray diamond shapes). Notice that some input or output variables (white boxes) can be used in more than one sequence. User parameters (white circles) are defined in Table 1.

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    (a) Surface circulation in the western Mediterranean Sea according to Millot and Taupier-Letage (2005), showing permanent currents (black arrow) with the AC, which are known as a place of eddies and meanders formation (dotted line arrow); Study area [illustrated in (b)] is indicated by the black rectangle]. (b) Snapshot of the surface geostrophic velocity field (vectors), gridded at 1/8°, derived from the AVISO ADT (background colors) in the Algerian Basin on 19 Feb 2013. Only half of the vectors are represented on this map. Bathymetry of 500 and 2000 m (black contours).

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    The surface velocity (vectors) and relative vorticity (ζ/f, colors) fields illustrating the unstable evolution at (a) t = 30 days and (b) t = 50 days of an idealized coastal current computed by Cimoli et al. (2017).

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    Surface velocity (vectors) derived from PIV in the laboratory experiments of N. Géhéniau et al. (2017, unpublished manuscript). Velocities and vorticity (ζ/f, colors) exhibit unstable evolution at (a) t = 8 days and (b) t = 26 days of a density front in the cylindrical tank.

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    Map of the OW parameter (gray scale) computed from the AVISO ADT geostrophic velocity field (black vectors) on 19 Feb 2013. Thresholds (left) OW0 = −0.2σω and (right) −2 × 10−12 s−2 are used to assign the closed contours of OW = OW0 to eddies (black solid lines).

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    Illustration of the method used for 19 Feb 2013. (a) First step of the detection with the LNAM parameter masked when LOW is >0 (colors). These values are computed from the velocity field (black vectors) with b = 1 (Lb = 2.4Rd). Eddy centers [black dots in (a) and (b)] correspond to the extremum of LNAM’s inside contour computed with K as |LNAM(LOW < 0)| = K = 0.7. (b) Second step of the detection with the velocity field (black vectors) and the speed magnitude (colors). The method filters eddy centers that are not enclosed by streamlines (white dots) and computes the characteristic eddy contours for the other centers, with cyclones (red lines) and anticyclones (blue lines). Eddy (solid lines) and gyre (dashed lines) types, as explained in section 3c.

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    Sensitivity of the number of detected eddies to the relative size of Lb/Rd and K for the various surface velocity fields described in section 2: (a),(b) 3 months of daily AVISO velocity fields with a grid parameter Γ ≃ 0.8, (c),(d) the full period of ROMS numerical simulation with Γ ≃ 3, and (e),(f) the full period of daily velocity fields for the laboratory experiment with Γ ≃ 10.5. The optimal value for AMEDA for each case [i.e., K = 0.7, and box size Lb ≃ 2.4Rd (= 2dX) for AVISO and Lb ≃ 1.2Rd for ROMS and PIV] is denoted the black dots.

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    Mean averaged velocity along each streamline (white dots) as a function of the corresponding radius 〈R〉 for the (a) anticyclone A, and cyclones (b) C1 and (c) C2 identified in Fig. 6b. Positive (negative) streamline velocities correspond to a cyclonic (anticyclonic) circulation. Corresponding Vmax = 〈V〉(Rmax) = max(|〈V〉|) are identified on the curves (black dot), and location of the largest closed streamline (i.e., Rend, Vend; black square) is denoted. Vortex Rossby numbers of A, C1, and C2 are equal to Ro = 0.11, 0.06, and 0.05, respectively; a typical eddy period is τe = 4, 8 and 11 days. Generic fit of the velocity profiles (solid lines) as described in section 3d with a steepness parameter α = 1.15 for (a) anticyclone A and (b) α = 1.72 for the cyclone C1.

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    (a) Probability distribution function of the number of detected eddies as a function of their relative size Rmax/Rd for the three datasets, (b) number of detected eddies for the numerical simulation (ROMS), and (c) laboratory experiment for various grid parameters. Shown in (b) and (c) are the eddy size distribution of datasets at the native resolution (solid lines) and those obtained with the artificially reduced grid resolution (gray lines and dashed lines). These eddy size distributions are computed in intervals of 0.1Rd, and the plotted curves are smoothed with a moving average of 0.5Rd.

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    Error estimation as a result of the degradation of the resolution during the ROMS experiment. The relative errors of (a) the eddy center Δrc/Rmax, (b) the eddy radius ΔRmax/Rmax, and (c) the maximal velocity ΔVmax/Vmax are plotted as a function of the relative eddy size Rmax/dX for various grid parameters Γ = 0.5–3. Errors bars only for Γ=1 (gray circles) are represented. Each side of a bar corresponds to the root-mean-square of the relative error distribution.

  • View in gallery

    Merging of two anticyclonic vortices in the numerical simulation ROMS. Snapshots show the surface velocity field (black vectors) and the relative vorticity ζ/f (blue shading: anticyclonic; red shading: cyclonic) at t of (a) 44, (b) 48, and (c) 52 days. Characteristic contours of anticyclones A1 and A2 (blue with black dots at their center), and characteristic shared contour (green contour).

  • View in gallery

    (a) Streamlines (thin black contours) corresponding to the velocity field where anticyclones A1 and A2 are merging at t = 48 days during the numerical simulation ROMS (Fig. 11b). Their respective characteristic contours (black bold line) and shared contour (gray bold line) are also reproduced. (b) Mean averaged velocity along each streamline (+ symbols) as a function of 〈R〉 for eddies A1 and A2 and their shared streamlines. Values of Vmax at Rmax for A1 and A2 (black circles) and for their shared contour (gray circle).

  • View in gallery

    Four cases of eddy–eddy interactions are depicted in this spatiotemporal schematic. (left to right) Splitting, merging, and two neutral interactions. Characteristic contours of individual eddies (black solid or black dashed contours), characteristic shared contours (bold gray), and individual eddy trajectory (gray dashed lines). Depicted are the first detection points (large black filled circles), and the splitting (white square) and merging points (white triangle).

  • View in gallery

    Formation points (black dots), splitting (white squares), and merging points (white triangles) of (left) anticyclonic and (right) cyclonic eddies for the three datasets used in this study. Area of formation, and splitting or merging points are proportional to the eddy lifetime. Trajectory of the longest eddy for each dataset is plotted (gray, gray diamond represents the last position). (a),(b) Algerian Basin analyzed with the AVISO 1/8° gridded products with the AC (dashed line). (c),(d) Periodic coastal channel studied by Cimoli et al. (2017). Initial location of the maximal jet velocity of the unstable coastal current (horizontal dashed line). (e),(f) Rotating tank experiment performed by N. Géhéniau et al. (2017, unpublished manuscript) and initial location of the density front (dashed circle).

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Angular Momentum Eddy Detection and Tracking Algorithm (AMEDA) and Its Application to Coastal Eddy Formation

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  • 1 Laboratoire de Météorologie Dynamique, École Polytechnique, Palaiseau, France
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Abstract

Automated methods are important for the identification of mesoscale eddies in the large volume of oceanic data provided by altimetric measurements and numerical simulations. This paper presents an optimized algorithm for detecting and tracking eddies from two-dimensional velocity fields. This eddy identification uses a hybrid methodology based on physical parameters and geometrical properties of the velocity field, and it can be applied to various fields having different spatial resolutions without a specific fine-tuning of the parameters. The efficiency and the robustness of the angular momentum eddy detection and tracking algorithm (AMEDA) was tested with three different types of input data: the 1/8° Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO) geostrophic velocity fields available for the Mediterranean Sea; the output of the idealized Regional Ocean Modeling System numerical model; and the surface velocity field obtained from particle imagery on a rotating tank experiment. All these datasets describe the dynamical evolution of mesoscale eddies generated by the instability of a coastal current. The main advantages of AMEDA are as follows: the algorithm is robust to the grid resolution, it uses a minimal number of tunable parameters, the dynamical features of the detected eddies are quantified, and the tracking procedure identifies the merging and splitting events. The proposed method provides a complete dynamical evolution of the detected eddies during their lifetime. This allows for identifying precisely the formation areas of long-lived eddies, the region where eddy splitting or merging occurs frequently, and the interaction between eddies and oceanic currents.

Additional affiliation: UME, ENSTA-ParisTech, Paris, France.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Briac Le Vu, briac.le-vu@lmd.polytechnique.fr

Abstract

Automated methods are important for the identification of mesoscale eddies in the large volume of oceanic data provided by altimetric measurements and numerical simulations. This paper presents an optimized algorithm for detecting and tracking eddies from two-dimensional velocity fields. This eddy identification uses a hybrid methodology based on physical parameters and geometrical properties of the velocity field, and it can be applied to various fields having different spatial resolutions without a specific fine-tuning of the parameters. The efficiency and the robustness of the angular momentum eddy detection and tracking algorithm (AMEDA) was tested with three different types of input data: the 1/8° Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO) geostrophic velocity fields available for the Mediterranean Sea; the output of the idealized Regional Ocean Modeling System numerical model; and the surface velocity field obtained from particle imagery on a rotating tank experiment. All these datasets describe the dynamical evolution of mesoscale eddies generated by the instability of a coastal current. The main advantages of AMEDA are as follows: the algorithm is robust to the grid resolution, it uses a minimal number of tunable parameters, the dynamical features of the detected eddies are quantified, and the tracking procedure identifies the merging and splitting events. The proposed method provides a complete dynamical evolution of the detected eddies during their lifetime. This allows for identifying precisely the formation areas of long-lived eddies, the region where eddy splitting or merging occurs frequently, and the interaction between eddies and oceanic currents.

Additional affiliation: UME, ENSTA-ParisTech, Paris, France.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Briac Le Vu, briac.le-vu@lmd.polytechnique.fr

1. Introduction

The increase of the spatial resolution in both numerical models and remote sensing observations revealed the prevalence of mesoscale eddies throughout the oceans. These coherent structures are able to trap and transport heat, mass, and momentum from their regions of formation to remote areas. Their crucial role in the transport of heat fluxes has been shown in many studies (Jayne and Marotzke 2002; Colas et al. 2012). For instance, the mean trajectories of the long-lived Agulhas Rings control the global transport in the Southern Ocean (Dencausse et al. 2010; Laxenaire et al. 2017, manuscript submitted to Geophys. Res. Lett.). In the Mediterranean Sea, the mean cyclonic pathways of the Algerian eddies (Escudier et al. 2016) have an impact on the regional transport of Atlantic water and Levantine Intermediate Water in the Algerian Basin. Additionally, mesoscale eddies can have a profound influence on biological productivity and on the upper-ocean ecology and biogeochemical cycles (McGillicuddy et al. 1998; D’Ovidio et al. 2010; Lévy et al. 2014; Cotroneo et al. 2016) especially in an oligotrophic area. Even large pelagic species, such as whales, exhibited a preference for the periphery of eddies during the seasonal phytoplankton biomass minimum in summer (Cotté et al. 2011). Finally, eddies can also influence near-surface winds and clouds or rainfall within their vicinity (Chelton et al. 2004; Frenger et al. 2016). Therefore, the dynamics of mesoscale eddies have a significant impact on the surface circulation and oceanic biogeochemistry at both local and regional scales. To investigate a large number of coherent structures for long periods (several years), the development of automatic eddy-tracking algorithms has recently become an important research topic in oceanography.

Different methods have been developed that use either the velocity fields or the altimetric sea surface maps to identify and track eddies. One of the earliest works in automatic eddy detection was developed to quantify the number of coherent vortices that emerge in the numerical simulations of two-dimensional turbulence (McWilliams 1990). To identify the rotating core of vortices, the relative vorticity was used for eddy detection. Doglioli et al. (2007) improves this method by using a wavelet analysis of the vorticity field. However, parallel velocity shears or filaments could have a strong vorticity signature, and geometrical constraints are generally added to exclude strongly asymmetric structures from such eddy detection. The Okubo–Weiss (OW) parameter, which quantifies the relative importance of the rotation with respect to deformation (Okubo 1970; Weiss 1991), was also widely used to identify and track oceanic eddies from geostrophic surface velocity fields (Isern-Fontanet et al. 2003; Morrow et al. 2004; Chelton et al. 2007; Chaigneau et al. 2008). Parameter evaluates the relative amplitude between the local deformation and the local rotation, where σn = (∂u/∂x) − (∂υ/∂y), σs = (∂υ/∂x) + (∂u/∂y), and ζ = (∂υ/∂x) − (∂u/∂y) are the shearing deformation rate, the straining deformation rate, and the vertical component of vorticity, respectively. The vortex interior is dominated by vorticity; thus, negative values of the OW parameter are expected in the vortex core. Nevertheless, these methods based on a dynamical parameter that quantifies the eddy intensity are quite sensitive to the threshold value used to identify and characterize the vortex boundary (cf. section 3a). On one hand, weak vortices could be excluded, while on the other hand intense eddies could lead to multiple contours. On average, the use of the OW parameter may induce a tendency for false positive eddy detection (Sadarjoen and Post 2000; Chaigneau et al. 2008). Other approaches (Sadarjoen and Post 2000; Nencioli et al. 2010) used only the geometrical properties of the streamlines to identify coherent vortices regardless of their intensity. The main assumption is that a coherent eddy could be characterized by closed or spiral streamlines. The fact that this type of automatic eddy detection is sensitive to both weak and intense eddies implies that several additional tests should be done to exclude unrealistic patterns from the detection. The price to pay to get an accurate eddy detection is the fine-tuning of specific geometric parameters and more computational time.

More recently, hybrid methods that combine both a physical parameter and geometrical properties (closed streamlines for instance) were used. A few studies (Viikmäe and Torsvik 2013; Halo et al. 2014; Yi et al. 2014) used the OW parameter to detect the possible eddy centers, while Mkhinini et al. (2014) introduced a new dynamical parameter: the local normalized angular momentum (LNAM). This parameter gives the normalized value of the angular momentum computed on a local area and centered on each grid point. Such local value will reach a maximum at the eddy center and the normalization will give +1 (−1) for a cyclonic (anticyclonic) solid core rotation. Hence, this parameter does not depend on the eddy intensity.

Another methodology directly used the sea surface altimetric maps for the identification of the eddy centers and their characteristic contours (Fang and Morrow 2003; Wang et al. 2003; Chaigneau and Pizarro 2005; Chelton et al. 2011). The main advantage of these methods is that they use only the sea surface height (SSH) field. Therefore, it significantly reduces the deleterious noise resulting from the second-order derivatives computed for the vorticity or the OW parameter. However, when a threshold on the sea surface deviation is required to delimit the eddy boundary (Fang and Morrow 2003; Wang et al. 2003; Chaigneau and Pizarro 2005), it will induce the same drawbacks as the eddy detection algorithms using only the vorticity or the OW parameter. Small variations in the threshold’s values will have a significant impact on the number and the location of the identified structures. Later on, Chaigneau et al. (2009) and Chelton et al. (2011) proposed threshold-free identification methods, based on the detection of the outermost closed contours of sea surface height that surround a single local extremum. Such a method allows for performing a detection over the World Ocean, and global eddy statistics was provided by Chelton et al. (2011). However, several problems still remain and large-scale eddies with more than one local extremum in their core could be identified as a cluster of small-scale structures.

All the eddy detection algorithms developed so far have specific advantages and drawbacks; therefore, it is not trivial to determine which method, if any, is the most relevant or accurate. These various methods lead to significant differences in the eddy statistics for a given area (Chaigneau et al. 2008; Souza et al. 2011). Some studies apply a combination of several algorithms in order to obtain statistical results that are more robust than those given by a single method (Souza et al. 2011; Escudier et al. 2013, 2016). However, in order to achieve the future challenges in oceanic eddies dynamics, we consider that the next generation of eddy detection and tracking algorithms should satisfy the following properties:

  1. To identify coherent eddies in model simulations and to compare them to remote sensing measurements, we need an algorithm with as few as possible tunable parameters, and the algorithm should be robust to the grid resolution. Indeed, the spatial resolutions of numerical oceanic models are generally higher than the gridded altimetric products. The main dynamical features (size, intensity, and shape) of mesoscale eddies that are resolved in a coarse-resolution dataset should not be affected if the resolution increases. In that sense, the eddy detection must be robust to the grid scale, applicable to various datasets, and weakly sensitive to any threshold values of the algorithm.
  2. The validation of the oceanic eddy detection algorithm is a challenging task. Very few independent measurements could be used to validate the accuracy of identification methods based on gridded maps (velocity or sea surface height) derived from the altimetric tracks of satellites. The direct comparison with in situ velocity measurements appears to be the best way to provide a quantitative evaluation of automatic eddy detection algorithms. Mkhinini et al. (2014) suggested using the vortex Rossby number for such comparisons; it is defined as Ro = Vmax/(f × Rmax), where f is the Coriolis parameter, Vmax is the maximal azimuthal velocity, and Rmax is the corresponding radius. Indeed, these two quantities are quite easy to extract from standard in situ measurements, such as acoustic Doppler current profiler (ADCP) transects or drifters trajectories, when a few of them remained trapped within the eddy core (Sangrà et al. 2005; Sutyrin et al. 2009; Mkhinini et al. 2014). Therefore, the eddy detection algorithms that provide Rmax (also called the speed-based radius) and Vmax facilitate the comparison with in situ measurements. We should note that several algorithms (Nencioli et al. 2010; Chelton et al. 2011; Mkhinini et al. 2014; Mason et al. 2014) already provide these features’ values.
  3. To follow accurately the dynamical evolution of eddies and the long-term transport of water masses, it is necessary to identify the merging and splitting events. Such events may indeed strongly impact the reconstruction of the eddy trajectory and its estimated lifetime if the tracking procedure does not account for them. It is only recently that automated eddy detection and tracking algorithms tend to tackle the problem of eddy–eddy interactions (Yi et al. 2014; Li et al. 2014; Du et al. 2014). It appears that a precise and robust determination of the outer eddy boundary is needed for correct identification of merging and splitting events, which becomes a new challenge for eddy detection and tracking algorithms.
In this article, we present and evaluate a new eddy detection algorithm called the angular momentum eddy detection and tracking algorithm (AMEDA; Fig. 1), which aims to satisfy the three abovementioned properties. We chose to develop and improve a hybrid algorithm (Mkhinini et al. 2014) that combines a physical parameter, the LNAM, and geometrical features of the streamlines for the determination of the eddy centers and their dynamical features. In addition to the many numerical improvements performed on the initial algorithm of 2014, we add a new eddy-tracking procedure and implement the detection of merging and splitting events. All the dynamical parameters that were needed to quantify the velocity profile of the detected eddies are now computed automatically. This type of eddy detection could be applied to a wide variety of velocity fields, such as gridded Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO) products, numerical simulations, high-frequency (HF) radar measurements, and laboratory experiments. Unlike algorithms based on SSH, this method based on the horizontal velocity field, may in the near future, be extended in numerical and allow models to detect eddies in the vertical direction (Williams et al. 2011; Petersen et al. 2013); or to account for the cyclogeostrophic correction of intense surface vortices derived from altimetric maps (Penven et al. 2014; Tuel et al. 2016).
Fig. 1.
Fig. 1.

Block diagram of AMEDA showing the main computation (gray boxes) and the conditional calculations (light gray diamond shapes). Notice that some input or output variables (white boxes) can be used in more than one sequence. User parameters (white circles) are defined in Table 1.

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

We chose to test the accuracy and robustness of AMEDA on few specific configurations of mesoscale eddies that are generated by the unstable evolution of a coastal current. To do so we use three velocity fields that correspond to the unstable evolution of coastal current in the Algerian Basin (Fig. 2), in an idealized numerical simulation, and in a rotating tank laboratory experiment. These various datasets, used to test the algorithm, are described in section 2. The details of the algorithm and the computation of the dynamical parameters associated with each eddy are presented in section 3. The sensitivity and the robustness of the eddy detection to the grid resolution are presented in section 4. The new methodology used to follow the dynamical evolution of eddies and to characterize the eddy–eddy interactions with characteristic streamlines is detailed in section 5. We then apply AMEDA to identify the main areas of formation, splitting, and merging of coastal eddies, especially in the Algerian Basin. Finally, we summarize, in section 6, the new tools provided by AMEDA and conclude on their potential applications for coastal eddy dynamics.

Fig. 2.
Fig. 2.

(a) Surface circulation in the western Mediterranean Sea according to Millot and Taupier-Letage (2005), showing permanent currents (black arrow) with the AC, which are known as a place of eddies and meanders formation (dotted line arrow); Study area [illustrated in (b)] is indicated by the black rectangle]. (b) Snapshot of the surface geostrophic velocity field (vectors), gridded at 1/8°, derived from the AVISO ADT (background colors) in the Algerian Basin on 19 Feb 2013. Only half of the vectors are represented on this map. Bathymetry of 500 and 2000 m (black contours).

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

2. Datasets

One of the functional specifications of this eddy detection and tracking algorithm is to be able to process a wide variety of oceanic velocity fields having distinct spatial resolution: from a high-resolution numerical model used for local or regional oceanography (Cimoli et al. 2017), HF radar observations of surface currents (Paduan and Washburn 2013), and particle image velocimetry (PIV) analysis of laboratory experiments (Pennel et al. 2012; N. Géhéniau et al. 2017, unpublished manuscript) to a coarser dataset, such as surface geostrophic velocities derived from sea surface altimetry (AVISO). We used in the present work the term high resolution when a given dataset is able to resolve submesocale features, that is, structures with typical scales smaller than the local deformation radius. On the other hand, coarse resolution will correspond to gridded data having only one or two points per deformation radius. For such a coarse-resolution velocity field, only large mesoscale vortices with a typical radius equal to few deformation radii could be detected. Hence, to quantify more precisely the spatial resolution of the various datasets available, we introduce the grid parameter Γ at a given point:
e1
where dX is the grid resolution at a given point of the velocity field and Rd is the first baroclinic deformation radius of the studied area. This characteristic dynamical scale could be deduced from previous studies (Robinson et al. 2001; Escudier et al. 2016) or computed locally from a climatological stratification. The linear eigenmodes and the corresponding deformation radius will then be obtained at each grid point from the local density profile ρ(z) according to Eq. (5.204) in Vallis (2006). The efficiency and robustness of AMEDA will be tested with three specific types of datasets (remote sensing, numerical simulations, and laboratory experiments) that describe the dynamical evolution of mesoscale eddies generated by the instability of coastal currents.

a. AVISO products gridded at 1/8°

We mainly used in the present study the geostrophic velocity fields produced by SSALTO/ Data Unification and Altimeter Combination System (DUACS) and distributed by AVISO and derived from the absolute dynamical topography (ADT). Unlike the seal level anomaly (SLA), which represents the variable part of sea surface height, the ADT is the sum of this variable part and the constant part averaged over a 20-yr reference period.

The “all sat merged” series distributed regional product for the Mediterranean Sea combines, for the years 1993–2015, up-to-date datasets with up to four satellites at a given time, using all the missions available at a given time [TOPEX/Poseidon, ERS-1 and ERS-2, Jason-1 and Jason-2, the Ka-band Altimeter (AltiKa) on the Satellite with the Argos Data Collection System (Argos) and AltiKa (SARAL), Cryosat-2 and Envisat missions]. This merged satellite product, for the Mediterranean Sea, is projected onto a 1/8° Mercator grid, in 24-h intervals. Several studies (Obaton et al. 2000; Salas et al. 2002; Puillat et al. 2002; Isern-Fontanet et al. 2004; Escudier et al. 2016) have shown that the instability of the Algerian Current (AC; Fig. 2a) is a major source of mesoscale anticyclones in the western Mediterranean Sea. Hence, in order to test the efficiency of AMEDA to detect and quantify coastal eddy formations, we will mainly use the AVISO regional products in this specific area of the Algerian Basin. A typical geostrophic velocity field derived from the ADT, along the Algerian coast, is shown in Fig. 2b. This example corresponds to 19 February 2013. It depicts a phase with two large Algerian eddies generated by the instability of the current and propagating eastward along the coast. A few months later, in April, these two anticyclones eddies will merge and detach off the coast. In this study we analyzed the entire year of 2013 to get the full process of the eddy formation, their detachment, and their propagation into the Algerian Basin. We were able to detect up to 10 500 vortex centers, which was sufficient to provide accurate statistics on the eddy detection algorithm

The spatial resolution of the regional dataset is 2 times higher than the global altimetric products at 1/4°. Nevertheless, it remains a coarse-resolution product, because the horizontal resolution of the 1/8° gridded velocity fields (dX ≃ 12 km) cannot fully resolve the internal deformation radius that is around Rd = 10–18 km in the Algerian Basin (Robinson et al. 2001; Escudier et al. 2016). The typical grid parameter will then be around Γ ≃ 0.8–1. Hence, only large mesoscale eddies, with a typical radius larger than the deformation radius, could be correctly detected with this dataset. Moreover, the accuracy of the eddy center location could be affected by the limited numbers of vectors in the vortex core and the distance of the vortex center to the closest and most recent altimetric track. In other words, nonphysical fluctuations or distortions in the 1/8° gridded AVISO products could be induced by the spatiotemporal heterogeneity of the altimetric tracks of satellites.

b. Numerical simulations of an idealized coastal channel

We used in this work the simulations computed by Cimoli et al. (2017) to study the various regimes of coastal eddy formation (Fig. 3). The numerical model corresponds to the Adaptive Grid Refinement in Fortran (AGRIF) branch (Penven et al. 2006; Debreu et al. 2012) of the Regional Ocean Modeling System (ROMS; ROMS-AGRIF; Shchepetkin and McWilliams 2003, 2005). This model solves the primitive equations with a split-explicit free surface, where short-time steps are used to advance the surface elevation and barotropic momentum equation, and a larger time step is used for temperature and baroclinic momentum. The model uses terrain-following vertical coordinates. We apply the eddy detection and tracking algorithm to the surface velocity field computed for a periodic rectangular domain—Lx = 256 km and Ly = 160 km and with a grid resolution dX = 2 km and 32 sigma vertical levels, where x is the alongshore axis and y is the cross-shore axis (positive offshore). Such a grid resolution is needed to resolve accurately mesoscale eddies having a typical radius around 20 km (or larger), and it corresponds to an equivalent resolution of 1/36° for the Mediterranean Sea. The baroclinic deformation radius of Rd = 5.9 km is resolved by three to four grid points; in other words, the grid parameter will be around Γ ≃ 3–4. This configuration reproduces a coastal channel with periodic boundary conditions for the alongshore direction and no slip boundaries at the coast and offshore. The initial state consists of a steady geostrophic surface current with a moderate Rossby number Ro = 0.25 flowing along the south coast. The typical width of the current is L = 10 km, its typical depth is h = 250 m, and the total water depth is about H = 1 km. For this range of parameters (Ro = 0.25, γ = h/H = 0.25) and a flat bottom configuration, the formation of large meanders and the detachment of mesoscale eddies results from the baroclinic instability of the coastal current. The numerical simulation of Cimoli et al. (2017) corresponds to an initial value problem with no external forcing, that is, no wind shear stresses, no heat fluxes, or freshwater fluxes. The exact three-dimensional structure of the coastal current at the initial stage can be found in Cimoli et al. (2017). During 170 days of the numerical run with one step per day, AMEDA was able to detect up to 1300 eddies centers from the surface velocity fields.

Fig. 3.
Fig. 3.

The surface velocity (vectors) and relative vorticity (ζ/f, colors) fields illustrating the unstable evolution at (a) t = 30 days and (b) t = 50 days of an idealized coastal current computed by Cimoli et al. (2017).

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

c. PIV from laboratory experiments

We also evaluate the efficiency of AMEDA on the experimental dataset obtained by N. Géhéniau et al. (2017, unpublished manuscript). A two-layer salt stratification was used to generate unstable coastal jets and fronts in a cylindrical rotating tank. Standard PIV was used to measure the horizontal velocity field in the surface layer (Fig. 4). Small buoyant particles were illuminated by a horizontal laser sheet of wavelength 670 nm, located a few millimeters below the upper free surface. The particle motions were recorded by a Lumenera LW11059 charge-coupled device (CCD) camera fixed on the top of the rotating tank. This camera and its optical lens provide a high-resolution image (3304 × 2832 pixels) of the tank, corresponding to a resolution of 30–40 pixels per centimeter. The particle velocities were then estimated using LaVision PIV software with successive cross-correlation boxes, yielding a final 350 × 340 vector field (dX = 0.22 cm). The typical baroclinic deformation radius was around Rd = 2.3 cm for these experiments and therefore the grid parameter of the horizontal velocity field was about Γ ≃ 10. Hence, the velocity fields correspond to an equivalent resolution of 1/120° for the Mediterranean Sea.

Fig. 4.
Fig. 4.

Surface velocity (vectors) derived from PIV in the laboratory experiments of N. Géhéniau et al. (2017, unpublished manuscript). Velocities and vorticity (ζ/f, colors) exhibit unstable evolution at (a) t = 8 days and (b) t = 26 days of a density front in the cylindrical tank.

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

After a rapid stage of geostrophic adjustment, a circular current with a typical width of L ≃ 3.5 cm ≃ 1.5Rd is established in the tank with a typical Rossby number around Ro = 0.6. The initial depth of the light water in the surface layer is h = 3.6 cm, while the total water depth is H = 18 cm. As for the numerical simulation, this baroclinic surface current = h/H = 0.2) is unstable and leads to, within few rotation periods (i.e., a few “laboratory days”), the formation of large meanders and mesoscale eddies along the coast (i.e., the side of the tank). Then, these coherent eddies propagate toward the center of the basin and at a later stage smaller submesoscale eddies may appear along the coast as a result of the viscous interaction of mesoscale structures with the tank boundary. During the entire duration of the data acquisition (32 rotation periods) with four analyses per equivalent day (i.e., per rotation period), AMEDA was able to detect more than 2000 eddy centers from the postprocessed velocity fields.

3. Eddy centers, characteristic contours, and velocity profiles

We can categorize the wide variety of detection methods based on the velocity field into three types: those based on dynamical quantities, such as the vorticity or the OW parameter (McWilliams 1990; Isern-Fontanet et al. 2003; Morrow et al. 2004; Chelton et al. 2007; Doglioli et al. 2007; Chaigneau et al. 2008); those based on the geometrical features of the flow field (Sadarjoen and Post 2000; Nencioli et al. 2010); and hybrids (Mkhinini et al. 2014; Halo et al. 2014), which combine both dynamical and geometrical quantities. After a description of the limitations of the classical eddy detection based on the OW parameter, we describe the method used by AMEDA and the numerical steps to detect the eddy centers and their characteristic contours as well as dynamical features associated with the eddies. The full processing sequence of detection and tracking of AMEDA is summarized in a simplified block diagram (Fig. 1). There are four independent sequences: 1) center detection, 2) contour computation, 3) tracking procedure, and 4) merging–splitting event computation. The detection of centers and the computation of contours must be completed for all the time steps in order to move on to the tracking procedure. In the same way, the merging and splitting events can be determined only if all the eddies have been tracked during their whole lives. The tunable parameters associated with the algorithm and their optimal value are described in Table 1.

Table 1.

List of parameters tunable by users (Lb, K, Rlim, C, and Tc) for the various types of input data (AVISO, ROMS, PIV). There is no default value set for Rlim in numerical models and PIV input in this version of AMEDA.

Table 1.

a. The limits of the OW parameter

The pioneering work of Isern-Fontanet et al. (2003, 2004, 2006) suggests using the specific contours corresponding to the OW threshold OW0 = −0.2σω to identify the vortex cores, where σω is the standard deviation of the OW distribution among the domain. Another study (Chaigneau et al. 2008) suggests that the best compromise is a value of OW0 in the range −0.3σω ≤ OW0 ≤ −0.2σω. Other studies, such as Chelton et al. (2007), suggested using a fixed value OW0 = 2 × 10−12 s−2 for oceanic eddy detection, which is independent of the eddy spectrum in the studied domain. To illustrate the method, we plot in Fig. 5 an example of the OW field for a specific AVISO geostrophic velocity field focused on the Algerian Basin. The method first selects the closed contour lines corresponding to a given threshold OW0. The corresponding eddy centers could then be defined as the centroid of the selected contours. The choice of the threshold value OW0 = −0.2σω (Fig. 5a) or OW0 = 2 × 10−12 s−2 (Fig. 5b) has a significant impact on the numbers, the shapes, and the sizes of eddy contours detected by this method. Moreover, we can see in this example that the geometry of the OW contours could strongly differ from the geometry of the velocity vector field. For example, the velocity field around the cyclones C1 and C2 appears roughly axisymmetric, while the OW contours describing them are strongly elliptic (Fig. 5a). Moreover, the curvature of the flow field as a result of current meanders, for instance, may induce a strong and localized value of OW that does not correspond to any coherent vortex with closed streamlines. For instance, in Fig. 5a a closed OW contour is visible below the cyclone C2 whereas, the local velocities correspond to a only curved coastal current

Fig. 5.
Fig. 5.

Map of the OW parameter (gray scale) computed from the AVISO ADT geostrophic velocity field (black vectors) on 19 Feb 2013. Thresholds (left) OW0 = −0.2σω and (right) −2 × 10−12 s−2 are used to assign the closed contours of OW = OW0 to eddies (black solid lines).

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

b. LNAM for detection of eddy center

We follow in this study the method proposed by Mkhinini et al. (2014) to detect eddy centers, and instead of the OW parameter we use another dynamical parameter that quantifies the local angular momentum. The underlying idea is to compute an integral value of the angular momentum in a restricted area. This value will be maximized at the center of the eddy, that is, the center of swirling motion Gi. For each grid point Gi, we compute the LNAM according to the following formula:
e2
where Xj and Vj are the position and velocity vector, respectively, on a gridpoint neighbor of Gi. The sum is made over the velocity neighbor’s points inside a square domain of side Lb = 2b × dX centered on Gi, with b as the number of grid pixels (). This center is not necessarily attached to the grid velocity Vj, and we could then use a thinner grid to compute the LNAM field. In the present study, we generally increase (using cubic interpolation) the grid resolution up to a factor of 3 to compute the values of LNAM with a resolution close to three pixels per deformation radius. This dynamical parameter is an integral quantity proportional to the local angular momentum at Gi, and it is renormalized by , which is an upper bound for the angular momentum (i.e., −BLiLi ≤ BLi). We add to the renormalization term. The sum of the scalar products Si will then reach a large value for hyperbolic points and be equal to zero for elliptical points. For an axisymmetric vortex, if Gi is the vortex center, Si = 0 and the LNAM parameter will reach an extremal value of 1 (−1) for a cyclonic (anticyclonic) eddy. Hence, this parameter does not depend on the vortex intensity and is built to make a net distinction between hyperbolic and elliptical points (Mkhinini et al. 2014). We plot in Fig. 6a the LNAM field associated with the same velocity field used in Fig. 5. A first set of eddy centers, black point in Fig. 6a, are selected from the extremum of the LNAM field. We apply here a mask to exclude the grid points corresponding to positive OW values integrated inside the domain of side Lb, called the local Okubo–Weiss (LOW) parameter. We first identify the closed contours corresponding to the threshold values |LNAM|(LOW < 0)| = K = 0.7 (see Fig. 6a) and then we select the extrema of LNAM inside these contours (black dots in Fig. 6a). In other words we select the extremum only if their values exceed max[|LNAM|(LOW < 0)|] ≥ K = 0.7.
Fig. 6.
Fig. 6.

Illustration of the method used for 19 Feb 2013. (a) First step of the detection with the LNAM parameter masked when LOW is >0 (colors). These values are computed from the velocity field (black vectors) with b = 1 (Lb = 2.4Rd). Eddy centers [black dots in (a) and (b)] correspond to the extremum of LNAM’s inside contour computed with K as |LNAM(LOW < 0)| = K = 0.7. (b) Second step of the detection with the velocity field (black vectors) and the speed magnitude (colors). The method filters eddy centers that are not enclosed by streamlines (white dots) and computes the characteristic eddy contours for the other centers, with cyclones (red lines) and anticyclones (blue lines). Eddy (solid lines) and gyre (dashed lines) types, as explained in section 3c.

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

However, the LNAM parameter alone does not guarantee that the selected eddy centers identify a core area that is able to trap water masses. Therefore, in addition to the first dynamical criterion, based on the maximization of the local angular momentum, we add a second geometrical criterion. We choose, as do other eddy tracking algorithms (Chaigneau et al. 2009; Nencioli et al. 2010; Halo et al. 2014), to keep a selected center only if we can find a closed streamline around it. The local streamfunction is then computed from the velocity field in a square domain centered on the selected extremum and the size of 10Rd. For the AVISO dataset, AMEDA can directly use the isolines of the ADT and/or the streamlines of the geostrophic velocity field, which are not strictly equivalent. In this illustrative case, the black circles in Fig. 6b show the eddy centers that are extrema of LNAM that were finally retained by the algorithm, while the white circles correspond to extrema of LNAM that were not retained because of the absence around them of closed streamlines computed from the velocity field.

An eddy detection algorithm will be robust if we do not have to continuously adjust the parameters of the algorithm to the varying properties of the velocity field. Hence, we test the sensitivity of the number of detected eddies to the variations of the two main parameters used for the detection of eddy centers: Lb, the side of the box domain used to compute the integral of the LNAM; and the threshold value K. To perform a general analysis that is not restricted to a specific area or a velocity distribution, we test this sensitivity on the various velocity fields described in section 2. We plot in Fig. 6 the impact of the dimensionless size of the integration box Lb/Rd and K on the number of detected eddies Ne. AMEDA should be able to detect the largest number of eddy centers at this stage. The algorithm will filter some of them afterward with the closed streamline condition; however, if AMEDA missed them at this initial stage, they will be irreversibly lost by the detection algorithm. Hence, we search here for the values of Lb/Rd and K that will maximize the number of detected eddies. We first noticed that for all the datasets, the various curves Ne = F(K) reach a plateau of maximal values when 0.2 ≤ K ≤ 0.7–0.8 (see the right panels of Fig. 7). Hence, within this range of values, the number of detected eddies does not depend significantly on K, but only on Lb/Rd. This is a very interesting result that shows that the detection of eddy centers is indeed robust to the threshold value K. In the operational mode of AMEDA, we will fix K = 0.7. On the other hand, according to the left panels of Fig. 7, we can see that Ne varies significantly with the relative size of the integration box, Lb/Rd We expect that if the domain of integration is too large in comparison with the eddy size, then a fraction of the surrounding eddies will then fit inside the integration box and smooth the value of the LNAM parameter. For the 1/8° gridded geostrophic velocity field provided by AVISO (left panel of Fig. 7a), there is a monotonic decay of Ne for increasing values of Lb. For this specific dataset, the optimal size of the integration box corresponds to a minimal box of 2 × 2 grid points: Lb = 2dX = 2(Rd/Γ) ≃ 2.4Rd. However, for data with a higher Γ (left panel of Figs. 7b and 7c) we could find an optimal size for the integration box. Indeed, if the domain of integration is too small (Lb < 1.2Rd) in comparison with the typical eddy radius, then the angular momentum will be estimated only in a small area of the eddy core. If the structure is slightly elliptical, then multiple maxima of LNAM could then appear inside a single closed streamline. For such cases the algorithm reduces these multiple maxima to only one maximum (the highest LNAM). Hence, for a high-resolution dataset the optimal value of Lb does not correspond to the minimal box size fixed by the grid. Taking into account both the numerical simulation and the laboratory experiment, we found a mean value of Lb < 1.2Rd to maximize the total number of eddy centers. This target value will be set in the operational mode of AMEDA, and if the grid resolution is too coarse to reach this target, we will then use the minimal box size Lb = 2dX to compute the LNAM parameter as with AVISO grid.

Fig. 7.
Fig. 7.

Sensitivity of the number of detected eddies to the relative size of Lb/Rd and K for the various surface velocity fields described in section 2: (a),(b) 3 months of daily AVISO velocity fields with a grid parameter Γ ≃ 0.8, (c),(d) the full period of ROMS numerical simulation with Γ ≃ 3, and (e),(f) the full period of daily velocity fields for the laboratory experiment with Γ ≃ 10.5. The optimal value for AMEDA for each case [i.e., K = 0.7, and box size Lb ≃ 2.4Rd (= 2dX) for AVISO and Lb ≃ 1.2Rd for ROMS and PIV] is denoted the black dots.

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

If we want to apply AMEDA on a submesoscale turbulent eddy field where the typical eddy radius is much smaller than the local deformation radius Rd, the parameter Lb should be reduced consequently. However, for the case of geostrophic and mesoscale eddies generated by the instability of coastal currents, we consider that with Lb = 1.2Rd we are close to the optimal value for eddy center detection.

c. Characteristic contours for eddies and gyres, and additional features

To find for each eddy its characteristic size and intensity, we search all the closed streamlines surrounding the selected eddy center. Then for each closed streamline, we compute its mean radius, which is the equivalent radius of a circle with the same area A as the one delineated by the closed streamline,
e3
and the mean velocity derived from the circulation along the closed streamline,
e4
where Lp is the streamline perimeter. Then, we search for the highest mean velocity Vmax = max(〈V〉) and its corresponding speed radius Rmax, such as 〈V〉(Rmax) = Vmax. The specific streamline associated with Vmax defines the characteristic contour of the eddy. Examples of these characteristic contours associated with various eddy centers that are detected in Fig. 6a are plotted with bold colored lines in Fig. 6b. However, as shown in Fig. 8 two cases may occur. In the standard case, once Vmax is reached, the mean velocity along the streamlines decays for larger radius 〈R〉 (Figs. 8a and 8b). Such cases correspond to a well-defined and coherent eddy that generally remains coherent in space and with time. We consider that the “eddy criterion” is achieved when the amplitude of the mean velocity of the last streamline is at least 3% below the maximal velocity: Vend = 〈V〉(Rend) ≤ 0.97|Vmax|. If this criterion is not fulfilled (Fig. 8c), we consider that the maximal velocity is then reached at the edge of the structure and we use “gyres” for these characteristic contours and their associated structures. This geometrical distinction between eddies and gyres is generally associated with distinct dynamical features. Unlike coherent eddies, which could survive a relatively long time without any external forcing, gyres are often associated with closed circulations maintained by the external flow: a current loop, a current meander, or the strain of neighboring eddies. In what follows we will use bold solid (dashed) lines to plot the characteristic contours of the detected eddies (gyres) and use blue (red) for anticyclones (cyclones) as in Fig. 6b.
Fig. 8.
Fig. 8.

Mean averaged velocity along each streamline (white dots) as a function of the corresponding radius 〈R〉 for the (a) anticyclone A, and cyclones (b) C1 and (c) C2 identified in Fig. 6b. Positive (negative) streamline velocities correspond to a cyclonic (anticyclonic) circulation. Corresponding Vmax = 〈V〉(Rmax) = max(|〈V〉|) are identified on the curves (black dot), and location of the largest closed streamline (i.e., Rend, Vend; black square) is denoted. Vortex Rossby numbers of A, C1, and C2 are equal to Ro = 0.11, 0.06, and 0.05, respectively; a typical eddy period is τe = 4, 8 and 11 days. Generic fit of the velocity profiles (solid lines) as described in section 3d with a steepness parameter α = 1.15 for (a) anticyclone A and (b) α = 1.72 for the cyclone C1.

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

From the characteristic size Rmax and the intensity Vmax of every detected eddy, we can directly compute the vortex Rossby number
e5
and the typical eddy turnover time (or eddy period τe), defined by
e6
To calculate the eddy period, we assume a solid rotation in the vortex core and fit the first points in the velocity profiles with a straight line (solid lines in Fig. 8). From the eddy period we can also get a first estimation of the relative core vorticity,
e7
where ζ(r) = ∂rV + (V/r) is the vorticity in cylindrical coordinates. For a circular vortex, the core vorticity is directly proportional to the slope of the velocity profile when the radius tends to zero: ζ(0) = (2V/R).

At this stage we interpolate the characteristic vortex contour by an ellipse and estimate an equivalent ellipticity ε = 1 − a1/a2 (also called the flattening parameter), where a1 is the semiminor axis and a2 is the semimajor axis. To ignore highly distorted contours, we also compute the local curvature along each streamline. If more than 20% of the streamline perimeter has a relative negative curvature, then the streamline is removed from the analysis and cannot be used as a characteristic contour. Moreover, in order to filter out the large gyres that may be formed at the basin scale, corresponding to the path or the loop of semipermanent regional currents, we fix a limit to the size of the characteristic contours RmaxRlim. This limit is fixed at Rlim = 59Rd ≃ 50−100 km for the AVISO dataset, while this limit is fixed at half the size of the numerical domain or the laboratory tank for the idealized numerical simulations or the laboratory experiments.

d. Steepness of the velocity profile

When there are enough closed streamlines to characterize the eddy, we can try to fit the velocity profile with a generic curve. The Gaussian vortex is often used as a first guess for isolated eddies; however, once Vmax and Rmax are fixed, there is no other parameter left to adjust the fit to the velocity profile. Hence, we chose to use a more general class of velocity profiles (see Figs. 8a–c) parameterized by the steepness parameter α as in Carton et al. (1989) and Stegner and Dritschel (2000),
e8
This equation describes a wide range of profiles having smooth (1 < α ≤ 2) or steep velocity and vorticity gradients (large α). The relative core vorticity and the vortex Rossby number are then fixed by the simple relation
e9

We should notice that the case α = 2 corresponds to a Gaussian velocity profile. For larger α the velocity decays more rapidly at the outer edge after Vmax, while for a smaller value of α the velocity signature of the eddy extends a larger distance after Rmax. The stability of such a class of vortices was studied by Carton et al. (1989) for purely two-dimensional flows and by Stegner and Dritschel (2000) for shallow-water flows. In the shallow-water framework, the barotropic stability of such class of vortices depends on their relative size. According to Stegner and Dritschel (2000), when the speed radius of mesoscale eddies matches the deformation radius (RmaxRd), stable solutions could exist if the steepness parameter is not too high (α < 2.3). However, for larger structures, when Rmax = 2.5Rd, stable solutions could be found with a parameter α slightly larger than 3. Hence, a large range in the value of the steepness parameter could occur and, indeed, the analysis of various circular eddies (ε ≤ 0.2) detected by AMEDA in the three datasets shows a wide distribution of α. The median value is around (α ≃ 1.8, while 90% of the steepness parameter values are between α = 1.1 and α = 2.7. This range of values are in correct agreement with previous studies of oceanic vortices, which suggest velocity profiles corresponding to α ≃ 1 for Gulf Stream rings (Olson 1980; Joyce and Kennelly 1985) or α = 2 as a universal structure of mesoscale eddies in the ocean (Zhang et al. 2013).

4. Sensitivity of the eddy detection to the grid resolution

Once an eddy is detected, the center is located, and its characteristic size and intensity are quantified, we wonder how accurate these values are. We could expect that the accuracy of the eddy detection will mainly depend on the relative eddy size in comparison to the grid resolution. If the typical eddy size is at the limit of resolution of the velocity grid, then it cannot be accurately quantified or even detected. However, when the eddy is larger, how many grid points per eddy radius are needed by AMEDA to quantify correctly the features of the detected eddy? To estimate the accuracy of the algorithm, we apply it to the same velocity field but interpolated on several grids of decreasing resolution. The eddy detection performed on the highest-resolution grid will be used as the reference case and the other ones will be used to test the sensitivity to the grid resolution. The numerical and experimental datasets, described in section 2, will be used for this sensitivity analysis, and their grid parameter will be decreased to the coarse value of Γ ≃ 1 of the AVISO dataset.

We first plot in Fig. 9 the distribution of the detected eddies as a function of their relative size Rmax/Rd. We can see that the three datasets have approximately the same probability distribution function for the eddy field. Indeed, the baroclinic instability of a geostrophic coastal current is the principal source of coastal eddies for these three datasets, and the vortex distribution is centered on mesoscale eddies with a peak radius around 1.5Rd for the Algerian Basin (AVISO) and the laboratory experiment (PIV), while the numerical simulation (ROMS) has a slightly larger peak radius centered around 2.2Rd. This larger value is fixed by the most unstable wavelength of the initial current, which is λ ≃ 8.85Rd ≃ 4Rmax for this specific numerical simulation (Cimoli et al. 2017). Moreover, for all the cases, the vortex distribution extends to significantly larger eddies up to 4Rd. Such large structures are often due to relatively weak and transient gyres, which are formed between intense mesoscale vortices.

Fig. 9.
Fig. 9.

(a) Probability distribution function of the number of detected eddies as a function of their relative size Rmax/Rd for the three datasets, (b) number of detected eddies for the numerical simulation (ROMS), and (c) laboratory experiment for various grid parameters. Shown in (b) and (c) are the eddy size distribution of datasets at the native resolution (solid lines) and those obtained with the artificially reduced grid resolution (gray lines and dashed lines). These eddy size distributions are computed in intervals of 0.1Rd, and the plotted curves are smoothed with a moving average of 0.5Rd.

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

Figures 9b and 9c show how the distribution of detected eddies is affected by the grid resolution. The black solid lines correspond to the original resolution of the datasets: Γ = 3 for the numerical simulation and Γ = 10.5 for the laboratory experiments. Note that we did not reduce the resolution of the AVISO dataset because it is already a coarse-resolution dataset with Γ ≃ 1. When the resolution of the velocity field is diminished—in other words, when we reduced artificially the grid parameter Γ—the number of detected eddies is first reduced at small scale, while the eddy size distribution remains almost unchanged for large structures. The interesting part of the distribution is the part that remains unchanged when the grid parameter is reduced. For both the numerical run (Fig. 9b) and the laboratory experiment (Fig. 9c), we can see that even if the grid parameter is reduced to a coarse resolution of Γ = 0.9 (the gray solid curve), the eddy size distribution above 9Rmax > 1.8 − 2Rd is almost unchanged. Hence, the number of detected eddies Ne will stay constant if their speed-based radius is larger than Rmax > Rd ≃ 2ΓdX≃ 2dX. Below this characteristic eddy radius, the probability to be detected by AMEDA starts to be significantly reduced. We also run AMEDA on a very coarse grid resolution of Γ = 0.5 (dotted line), and for such extreme cases, fewer than half of the eddies are detected.

In the second step of this sensitivity analysis, we will test how the estimation of the eddy center location, the speed-based radius Rmax, and the maximal velocities Vmax are affected by the value of the grid parameter. We compute in both datasets for each detected eddy the distance Δrc between its initial center location obtained with the highest grid parameter and the eddy center estimated by AMEDA with a lower grid resolution. The eddy center position obtained with the highest grid parameter will be considered as the reference center, and we then estimate the relative deviation Δrc/Rmax of the other ones. A similar calculation is performed to estimate for each individual eddy the deviation induced by a lower Γ on the speed radius and velocity, and we then compute the relative deviations: ΔRmax/Rmax and ΔVmax/Vmax. Here again, the Rmax and the Vmax obtained with the highest grid parameter will be considered as the reference radius and the reference velocity, respectively. According to Figs. 10a–c, which show the results for the ROMS experiment, these relative errors mainly depend on the relative eddy size Rmax/dX for various grid parameters Γ.

Fig. 10.
Fig. 10.

Error estimation as a result of the degradation of the resolution during the ROMS experiment. The relative errors of (a) the eddy center Δrc/Rmax, (b) the eddy radius ΔRmax/Rmax, and (c) the maximal velocity ΔVmax/Vmax are plotted as a function of the relative eddy size Rmax/dX for various grid parameters Γ = 0.5–3. Errors bars only for Γ=1 (gray circles) are represented. Each side of a bar corresponds to the root-mean-square of the relative error distribution.

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

We notice a systematic bias when Rmax is too small in comparison with the grid scale dX, which leads (on average) to an increase of the distance center (Fig. 10a), to a spread of the eddies’ size (Fig. 10b), and to an underestimation of the maximum velocity of the eddies (Fig. 10c). Besides, the dispersion of the error increases (i.e., the root-mean-square of the relative error distribution) and could exceed 15%–20% when Rmax/dX decreases below 2. Hence, all the analysis on the eddy features, obtained with AMEDA, should be taken with care for eddies having a small characteristic radius Rmax below 2dX.

5. Dynamical evolution of the detected eddies

a. Eddy-tracking procedure and trajectories

The purpose of eddy-tracking techniques is to be able to follow the temporal evolution of the detected eddies with time (i.e., their trajectories) despite the fact that the center, shape, and intensity of each eddy may vary with time. Moreover, the eddy detection scheme is sensitive to noise in the dataset or sampling errors induced by the algorithm; therefore, an eddy could be “missed” for a few time steps. For instance, the spatiotemporal heterogeneity of the altimetric tracks of satellites could induce such a deficiency in the detection when an eddy passes into the gaps between satellite ground tracks. Hence, in order to optimize the eddy-tracking procedure, AMEDA is able to search eddies for several time steps. To do so we use a global cost function that quantifies the similar neighbor assignments for all the possible eddy tracks from several time steps preceding the last one.

The local nearest neighbor (LNN) assignment was the first method to be used for eddy tracking (Isern-Fontanet et al. 2006; Chelton et al. 2007, 2011). This method consists of associating with each detected eddy ei at the last time step t the closest eddy ej of the same sign detected at the previous time step tdt in a given search area. The search area is usually based on the theoretical distance an eddy can travel during the period dt between two-time steps. In the present study, the search area first considers a minimum overlapping of the contours between two consecutive time steps and will therefore depend on the two characteristic radii of the eddies ei(t) and ej(tdt). To account for the propagation of ej, we introduce the speed parameter C, and the maximum search distance Dij is then given by
e10
where 〈Rmax〉(j) is the mean speed-based radius of ej averaged during the five preceding steps of its track, while Rmax(i) is the speed radius of ei(t). The speed parameter C is a constant value fixed by the user and should correspond to an upper bound of the propagation speed of eddies for a given dataset. By default, the value used is 6.5 km day−1, as in Mkhinini et al. (2014). If no eddies are found in the search area (i.e., when the spatial distance dij between the eddy centers is always larger than the maximal search distance Dij), then ei(t) is identified as a new vortex and a new trajectory is associated with the eddy.
Despite its comprehensiveness and quickness, this simple method presents a few biases. First, the minimal spatial distance between the eddy centers may not be the most robust criteria for tracking eddies between two-time steps. To account for the physical similarity between eddy pairs, Penven et al. (2005) and Chaigneau et al. (2008) suggest comparing the size and intensity of the eddies in addition to their spatial distance. Hence, instead of the spatial distance, they use a cost function of the following type:
e11
where Xd, XR, and XI are dimensionless values that quantify the relative distance between the eddy centers, the relative difference between the eddy radii, and the relative difference between the intensity of the initial eddy (ej) and the search eddy (ei) If several eddies are found in the search area, then ei will be associated with ej, which minimizes the cost function. In AMEDA we used a cost function with similar dimensionless terms, but instead of using constant characteristics to normalize the distance, size, and intensity, we adapt the value to the initial and search eddies. Also, we add another dimensionless term related to the time separation between ej and ei. This leads to the following cost function:
e12
where dij is the distance between ej and ei; here, we average the distance between the eddy centers and the eddy barycenters of the characteristic contours. The quantity Dij(Tc) is the distance Dij at the correlation time (Tc, defined below), which is the maximal possible distance between ei and ej. The parameters ΔR and ΔRo are the radius and the Rossby number difference between ei and ej, respectively; while 〈R〉(j) and 〈Ro〉(j) are respectively the mean radius and the mean Rossby number of the eddy ej averaged during the five preceding steps. The radius R represents the mean between the speed radius and the last radius (or the shared contour if any), whereas Ro is the average between the Rossby number from the speed radius and the characteristic shared contour if any. Finally, if two eddies ej similar in size, intensity, and distance could be associated with ei, then the last term (1/2)(dt/Tc) will give an advantage to the one detected more recently.

Another main source of error for the eddy-tracking algorithm is the occurrence of missing eddies during one or several time steps. Such a lack of detection occurs quite often with altimetric datasets where vortices may totally disappear between consecutive maps if altimetric tracks do not cross the structure for several days. For other datasets the lack of detection is mainly due to noise in the measurements (for laboratory experiments or a HF radar) or algorithmic errors (for the output of numerical models). In the latter cases, the lack of detection hardly exceed one time step, while for altimetric datasets an eddy could be missed up to 2 weeks (DUACS and AVISO 2014). Hence, to account for the temporal coherence of these various datasets we introduce Tc. This characteristic time parameter fixes the limit to the number n of time steps used to search for a preexisting eddy (a similar neighbor) backward in time. In other words, the period ndt between two-time steps of detection for the same eddy will be bounded by the correlation time: ndt < Tc. The correlation value has to be fixed by the end user according to the input dataset. We will use two-time steps for the numerical and the laboratory velocity fields, and a correlation time of 10 days for the AVISO dataset. However, when we use daily AVISO products, it means that the eddy-tracking procedure could be iterated up to 10-time steps. Such a large number of iterations may induce another important source of errors in the eddy-tracking procedure. Indeed, for the standard eddy-tracking procedure, the association of eddy pairs depends on the scanning order of the eddies, which means that once an eddy has been associated with another one, the latter is no longer available for another association, even if this new eddy pair has a smaller cost function. To avoid such a problem, we build for each t an assignment global matrix containing the cost function of all the possible associations between the current eddies detected at this time step ei(t) and the last preceding detection of the eddies during the n previous time step ej(tndt), with 1 < n < Tc/dt.

The assignment matrix [εji] contains the cost function of every association allowed (i.e., dij < Dij) or a very high value when the association is not allowed (1000 < εji < 106). The assignment closest to the perfect coupling is then resolved thanks to the Khun–Menkres algorithm (Kuhn 1955; Munkres 1957) available in various MATLAB tools (Buehren 2004). Before resolving the assignment, the algorithm imposes that a maximum of previously detected eddies will be assigned to one, and only one, current eddy and vice versa. Then, the optimal assignment of the matrix is computed by minimizing the overall cost. This methodology can ensure a unique trajectory for each detected eddy with one starting point and one ending point. The impacts of possible merging and splitting of eddies on the tracking procedure are detailed below.

b. Identification of merging and splitting events

The merging and splitting of coherent eddies are important dynamical events that strongly impact the eddy-tracking algorithm and may therefore influence the estimation of the horizontal transport induced by coherent eddies. Indeed, for the standard eddy-tracking algorithm, when two like-sign eddies merged, the smallest eddy disappears, and its trajectory is stopped even if a significant fraction of its water mass could be captured and transported by the largest one. On the other hand, the splitting of a coherent eddy into two smaller ones will lead to the detection of a new independent vortex with no connection to its origin. In addition, both splitting and merging events will induce a rapid variation of the size and the kinetic energies of the interacting eddies. Hence, to describe accurately the dynamical evolution of coherent eddies (transport of mass, energy budget, etc.), it is necessary to identify the merging and splitting events and to account for them in the tracking procedure.

We developed for AMEDA a new methodology for detecting eddy–eddy interactions. A typical merging event that was detected in the ROMS numerical simulation is described in Fig. 11. Two isolated anticyclones are getting closer (Fig. 11a), their ellipticity increases when they rotate around each other (Fig. 11b), and they finally merge and lead to a quasi-circular end state that is reached within a few days (Fig. 11c). We plot for this typical example different types of contours. The characteristic contours for each anticyclone are plotted in blue. These contours correspond to the streamlines having the maximum averaged velocity [〈V〉(Rmax) = Vmax] that encloses a single eddy center. If now we consider streamlines that enclose two eddy centers (i.e., shared contours), we can then define a characteristic shared contour as when the mean velocity along the shared contours reaches a maximum value larger than the Vmax of at least an individual eddy. Such characteristic shared contours are plotted in green in Figs. 11a–c. The streamlines associated with the velocity field of Fig. 11b are plotted in Fig. 12a, while the mean velocities 〈V〉 associated with closed streamlines (which enclose one or two eddy centers) are plotted as a function of their mean radius 〈R〉 in Fig. 12b. We clearly see in this example that the 〈V〉 calculated for the shared contours may exceed the Vmax of at least one eddy (black circles labeled contour A1 and contour A2). When such a characteristic shared contour is detected, we consider that the two eddies experience a dynamical interaction.

Fig. 11.
Fig. 11.

Merging of two anticyclonic vortices in the numerical simulation ROMS. Snapshots show the surface velocity field (black vectors) and the relative vorticity ζ/f (blue shading: anticyclonic; red shading: cyclonic) at t of (a) 44, (b) 48, and (c) 52 days. Characteristic contours of anticyclones A1 and A2 (blue with black dots at their center), and characteristic shared contour (green contour).

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

Fig. 12.
Fig. 12.

(a) Streamlines (thin black contours) corresponding to the velocity field where anticyclones A1 and A2 are merging at t = 48 days during the numerical simulation ROMS (Fig. 11b). Their respective characteristic contours (black bold line) and shared contour (gray bold line) are also reproduced. (b) Mean averaged velocity along each streamline (+ symbols) as a function of 〈R〉 for eddies A1 and A2 and their shared streamlines. Values of Vmax at Rmax for A1 and A2 (black circles) and for their shared contour (gray circle).

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

We should notice that the formation of a characteristic shared contour cannot be avoided when two eddies having a similar size and intensity merge. In such a case, the centers stay coherent until they merge (Griffiths and Hopfinger 1987; Melander et al. 1988; Verron and Valcke 1994; Dritschel 1995). The reverse is also true when an eddy splits into two smaller eddies. Hence, the formation of a characteristic shared contour, depicted in Figs. 11 and 12, could be generalized to most of the merging or splitting events between similar eddies. In the case of strongly asymmetric merging, the small eddy may wrap around the larger one (Dritschel and Waugh 1992; Yasuda and Flierl 1995) and thus its core will be destroyed before the formation of the shared contour. Moreover, if the eddy–eddy interaction is too fast, the formation of the characteristic shared contour could also be missed by the detection, especially for the AVISO dataset when a lack of altimetric measurements occurs at the location of such specific events.

However, such eddy–eddy interaction (the formation of a characteristic shared contour) is not always the signature of a merging or splitting event. To be more precise and to identify these specific events, we need to consider the whole period of interaction and look at the trajectories of the two interacting eddies. As it is shown in the schematic in Fig. 13, a merging event will be established only if one of the two trajectories ends after the interaction period; conversely, a splitting event will be considered if before the interaction period there is only one trajectory. To account for possible spatiotemporal uncertainties in the computation of the characteristic shared contours, the interaction period is extended by Tc/2 backward and forward in time, where Tc is the correlation time used in the eddy-tracking procedure described in section 5a. If the two interacting eddies preexist and survive after the interaction period, we assume a neutral interaction and do not take it into account. Moreover, the simultaneous interaction between three eddies is not considered by AMEDA.

Fig. 13.
Fig. 13.

Four cases of eddy–eddy interactions are depicted in this spatiotemporal schematic. (left to right) Splitting, merging, and two neutral interactions. Characteristic contours of individual eddies (black solid or black dashed contours), characteristic shared contours (bold gray), and individual eddy trajectory (gray dashed lines). Depicted are the first detection points (large black filled circles), and the splitting (white square) and merging points (white triangle).

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

With this procedure we are then able to make the distinction between the formation point of an eddy (large black circle in Fig. 13) and the point of first detection of a new independent vortex that results from a splitting event (green square in Fig. 13). The latter will be called a splitting point and the identification of the “parent” eddy will be recorded in order to be able to build the genealogical tree of any detected eddies. Moreover, when a merging event is detected, the end point of the merged eddy (pink triangle in Fig. 13) is labeled as a merging point and the identification of the interacting eddy is recorded. Hence, the whole tracking procedure and the identification of merging and splitting events allow for following the path of water masses from one eddy to another, from the generation point to the final location where the coherent structure breaks down.

c. Formation point of eddies

We plot in Fig. 14 the formation points (black filled circles) of cyclonic and anticyclonic eddies for the three datasets: the AVISO maps of the Algerian Basin (upper panels), the idealized numerical simulation (middle panels), and the laboratory experiment (lower panels). The size (area) of these formation points is proportional to the eddy lifetime. The last two datasets correspond to an initial value problem, where an unstable current leads to the formation of mesoscale eddies along the coast. As expected the formation points of these coastal eddies are then located around the center of the initial current (i.e., the location of the maximal jet velocity), which is represented here by a dashed line. For the idealized coastal current configuration used by Cimoli et al. (2017), the anticyclones are first formed by the meander of the current, as shown in Fig. 3a, while cyclonic structures appear later on and slightly offshore when the anticyclones detach from the coast and propagate into the center of the domain. For the rotating tank experiment (Figs. 14e and 14f), the spatial distribution of the formation points is more dispersed. Nevertheless, we find a similar pattern where the anticyclones are formed “close to the coast” between the outer wall of the tank and the initial location of the coastal current (the dashed circle), while cyclones are formed slightly offshore, closer to the center of the tank. The eddy detection and tracking procedure applied to these idealized configurations works efficiently, and the formation points of coherent eddies are correctly identified. If now we apply AMEDA to the AVISO dataset, the spatial distribution of the formation points in the Algerian Basin is more complex. The most intense current in this area is the AC, and the unstable dynamics of the latter is expected to be a major source of long-lived anticyclones that detach from the coast (Taupier-Letage and Millot 1988; Obaton et al. 2000; Puillat et al. 2002; Isern-Fontanet et al. 2004). Indeed, the longest-lived eddy that was formed in the Algerian Basin in 2013 is an Algerian eddy (AE) that detached at 8°E. The trajectory of this mesoscale anticyclone, plotted in gray in Fig. 14a, is in good agreement with the mean pathway of the AE (Isern-Fontanet et al. 2004; Escudier et al. 2016). However, several other eddies of both signs are formed in the center of the Algerian Basin. These eddies do not seem to be related to AEs, and other currents, such as the Balearic Front or the wind forcing, could be responsible for their formation. A more exhaustive analysis of the AVISO dataset over several years is probably needed here to identify the formation regions of long-lived eddies in the Algerian Basin.

Fig. 14.
Fig. 14.

Formation points (black dots), splitting (white squares), and merging points (white triangles) of (left) anticyclonic and (right) cyclonic eddies for the three datasets used in this study. Area of formation, and splitting or merging points are proportional to the eddy lifetime. Trajectory of the longest eddy for each dataset is plotted (gray, gray diamond represents the last position). (a),(b) Algerian Basin analyzed with the AVISO 1/8° gridded products with the AC (dashed line). (c),(d) Periodic coastal channel studied by Cimoli et al. (2017). Initial location of the maximal jet velocity of the unstable coastal current (horizontal dashed line). (e),(f) Rotating tank experiment performed by N. Géhéniau et al. (2017, unpublished manuscript) and initial location of the density front (dashed circle).

Citation: Journal of Atmospheric and Oceanic Technology 35, 4; 10.1175/JTECH-D-17-0010.1

d. Splitting and merging points

In addition to the formation points, the splitting points (white squares) and the merging points (white triangles) are also plotted in Fig. 14. The size (area) of the splitting points is proportional to the lifetime of the new eddy that is formed by the splitting event. For the initial value configurations, once the eddies are formed the inverse energy cascade associated with a decaying quasi-geostrophic turbulent flow (McWilliams 1989) should induce more merging events than splitting events. Indeed, for both the idealized coastal channel simulations (Figs. 14c and 14d) and the laboratory experiments (Figs. 14e and 14f), the number of merging points exceeds the splitting points if any. Merging events could occur anywhere in the domain and are not necessarily close to the formation area of eddies. The merging depicted in Fig. 11 increases the energy of the lifetime of the long-lived anticyclone (gray trajectory) plotted in Fig. 14c. Hence, long-lived eddies could experience one or several merging events during their lives, and AMEDA is able to identify these specific events that impact significantly the lifetime and size of the detected eddies. Unlike the initial value configurations, the coastal current in the Algerian Basin (i.e., the AC) is permanently forced by the entrance of Atlantic surface water through the Strait of Gibraltar and by the surface wind stress. This may explain why the relative number of splitting events plotted in Figs. 14a and 14b in comparison with the formation points is significantly higher than for the two idealized cases.

6. Summary and conclusions

A new eddy detection and tracking algorithm, based on physical parameters and geometrical properties of the velocity field, is presented. AMEDA could be applied to various velocity fields having different spatial resolutions without a specific fine-tuning of the parameters. The two main tunable parameters used for eddy identification are the relative box size Lb/Rd used to compute the local angular momentum and the dimensionless threshold K (Table 1). We have shown that for three distinct velocity datasets (gridded AVISO products of the Mediterranean Sea, outputs of numerical simulations, and laboratory experiments), eddy detection is optimal when the two parameters stay in the following range: Lb/Rd = 1–2 and K = 0.4–0.8. Hence, eddy identification is weakly sensitive to a moderate variation of these two parameters, and we suggest using Lb/Rd = 1.2 and K = 0.7 as the default values. Moreover, we have tested the sensitivity of eddy detection to the spatial grid resolution dX of the velocity field. When the detected eddy is sufficiently large—that is, when the characteristic speed radius Rmax exceeds 2dX —the accuracy of the eddy size and its intensity appears to be insensitive to the grid resolution and the root-mean-square errors stay below 15%–20%. As far as we know, this is the first time that the sensitivity and accuracy of an automatic eddy detection algorithm to the spatial resolution are tested. Our analysis shows that AMEDA is robust to the grid resolution and could be easily used on a wide variety of velocity fields. The dynamics of large mesoscale eddies in high-resolution model simulations could then be accurately compared to real observations and especially to the coarser datasets provided by altimetric measurements, such as the 1/8° AVISO geostrophic velocity fields available for the Mediterranean Sea. For this dataset AMEDA should be able to perform an automatic detection and an accurate quantification of large mesoscale eddies that have a characteristic speed radius larger than Rmax ≥ 2dX ≃ 25 km. Conversely, the quantification of the dynamical features of oceanic eddies below a radius of 20–25 km, regardless of the accuracy of the 1/8° altimetric gridded maps, should be taken with care because of the inaccuracy of the algorithm. If we apply AMEDA to the global ocean, the accuracy of the detection will drop as a result of the coarser resolution of the AVISO global products. For such a dataset, gridded at 1/4°, this specific eddy detection and tracking algorithm may not provide valid information for eddies that have a typical speed radius below 50 km.

This version of AMEDA provides quantitative information on the eddy properties such as the position of the center, the maximal azimuthal velocity Vmax, and both the speed-based radius Rmax and the typical radius of the outermost closed contour Rend. The vortex intensity is then estimated with Rossby number Ro = Vmax/(f Vmax)and the relative core vorticity ζ(0)/f = 4π/(e), where τe is the typical eddy turnover time. These two dimensionless parameters could be derived from standard in situ measurements, such as boat ADCPs or the analysis of drifters’ trajectories. Therefore, quantitative comparisons between the outputs of AMEDA (applied to the AVISO gridded product) with independent in situ measurements could be easily done. Moreover, unlike a standard algorithm, AMEDA provides additional information on the shape of the mean velocity profile and its ellipticity. Here again this quantitative information allows for detailed comparisons with in situ measurements. Such comparisons with a small set of (well documented) Mediterranean eddies will be provided in a future work.

We also developed for AMEDA a new methodology, based on the detection of a characteristic shared contour, to identify eddy–eddy interactions. This characteristic shared contour corresponds to a streamline that encloses two eddy centers with a maximum value of the mean velocity averaged along the streamline. This specific contour is formed when two eddies merge or when an eddy splits. Hence, the eddy-tracking procedure of AMEDA is able to make the distinction between the formation point of an eddy and the point of first detection of a new independent vortex that results from a splitting event. Moreover, the trajectory of eddies that merge could be prolonged into the new structure, and the origin of the water masses captured and transported by a long-lived mesoscale eddy could be identified. This is crucial for quantifying the mean transport of heat, salt, and biogeochemical species induced by coherent structures in a semi-enclosed basin such as the Mediterranean Sea. For instance, the recent analysis with AMEDA of the trajectory of a Balearic eddy, surveyed during the Prévision Océanique, Turbidité, Ecoulements, Vagues et Sédimentologie (PROTEVS) campaign in 2016, showed that the water mass trapped in its core originated from an Algerian eddy formed one year earlier. Hence, the detection of eddy–eddy interactions may provide a better description of the water mass exchanges between the Algerian Current and the Balearic Front.

Acknowledgments

This work was funded by the ANR-Astrid Project DYNED-Atlas (ANR 15 ASMA 0003 01). Briac Le Vu thanks the national support from the Direction Générale de l’Armement (DGA) in the context of the ADETOC project 2015. The altimeter products were produced by SSALTO/DUACS and distributed by AVISO, with support from CNES (http://www.aviso.altimetry.fr/duacs/). The code written in MATLAB and a tutorial are made freely available (ftp://ftp.lmd.polytechnique.fr/blevu/AMEDA/). Images of detection and tracking results for the three datasets presented in this study are also stored on this FTP site.

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