Mesoscale eddies are key drivers of ocean dynamics, playing a central role in the transport of water masses, energy, heat, salt, and biogeochemical tracers across both horizontal and vertical scales. These ubiquitous structures, typically spanning tens to hundreds of kilometers and persisting from days to several years , also exert a strong influence on marine biological and chemical processes . Coupling eddy-resolving physical observations with ecological datasets has substantially advanced our understanding of marine animal behavior as well as the dispersion and fate of pollutants . Continuous improvements in the accuracy of satellite altimetry over the past three decades have enabled global monitoring of mesoscale eddies, providing unprecedented insights into their role in ocean circulation and ecosystem functioning.

Satellite altimetry provides near-global, weather-independent observations of mesoscale ocean variability, primarily through daily, multi-satellite gridded fields of absolute dynamic topography (ADT) and sea level anomaly (SLA). Although the spatiotemporal interpolation of nadir altimeter observations limits the effective resolution of these gridded products to spatial scales larger than ∼ 100 km, they remain the backbone of global mesoscale eddy detection. Traditionally, eddy detection methods have relied on Eulerian approaches,based on fixed, time-invariant surface fields, using either geometric diagnostics or rotational metrics . Over the past decade, Lagrangian approaches, based on time-evolving surface fields, have also been introduced . However, their application at the global scale remains computationally intensive.

Several global atlases have been developed, beginning with the pioneering work of , which was later refined through collaborations between OSU, CLS, and CNES, leading to the Mesoscale Eddy Trajectory Atlas (META; ). Successive releases (META1.0exp, META2.0, META3.1exp, META3.2) progressively improved detection and tracking by refining search schemes, handling missing eddies, and extending coverage to coastal and island regions. These efforts established META as a reference global product, freely distributed via AVISO+ (https://data.aviso.altimetry.fr/aviso-gateway/data/META3.1exp_DT/, last access: 23 September 2026). Other datasets followed similar principles, such as the global atlas by , GOMEAD (), and ToEddies (), offering complementary approaches to eddy detection and tracking.

Most global atlases, including META and others (), rely on single-trajectory tracking, where eddies are followed over time by identifying candidates within a localized search window. The META2.0 version refined this method to handle the “missing eddy” problem through short temporal extrapolations and virtual eddies. Later versions (i.e., META3.x) adopted a contour-overlap strategy, linking detections across timesteps when their intersection exceeded a given threshold (). This approach proved more robust, enabling consistent trajectories even during partial occlusion or deformation, and reducing the need for virtual eddies.

Eddies were initially viewed as isolated coherent structures driven by baroclinic instabilities or topographic forcing (). However, observational and modeling evidence has revealed that mesoscale dynamics also involve frequent eddy–eddy interactions such as merging and splitting (). These processes are now recognized as a fundamental component of oceanic energy transfer and water mass transport.

Recent algorithms have been developed to explicitly identify such interactions, treating eddies as nodes within a dynamic network rather than independent trajectories. Examples include AMEDA (), which integrates merging/splitting detection into tracking, the multicore Gaussian approach of , the graph-based framework of ToEddies (), and the topological tree approach of EddyGraph (). Together, these methods have demonstrated the potential of network-based descriptions to capture the full complexity of eddy life cycles.

Building upon this progress, we present the META4.0 atlas, the new global version of the Mesoscale Eddy Trajectory Atlas distributed by AVISO+. META4.0 introduces eddy networks into the dataset, providing a unified representation of eddy evolution, merging, and splitting events. This update represents a key advancement for the META eddy products, ensuring continuity with previous versions while adding a new dimension for studying eddy connectivity and interactions. As mentioned in Sect. , previous versions of the META atlas relied on a single-trajectory tracking approach, whereas META4.0 provides a network-based representation of eddy trajectories. It should be noted that the META4.0 eddy detection framework is identical to that used in the previous META3.2 version.

The construction of the networks follows three stages:

Daily eddy detection from DT2021 ADT fields;

Beyond the methodological improvement, the increasing complexity of network representations requires appropriate tools for data manipulation, visualization, and analysis. For this reason, the full Python source code is released under a GPL v3 license (10.5281/zenodo.7197432, ) and integrated into the open-source pyeddytracker (PET) library (). The toolbox is freely available and documented online (https://py-eddy-tracker.readthedocs.io/en/latest/python_module/index.html, last access: 23 September 2026), providing ready-to-use functionalities for detecting eddies, analyzing trajectories, building networks, and reproducing the results presented in this paper.

This paper is organized as follows. Section  introduces the META4.0 algorithm and describes the data sources used. Section  presents the toolbox and main validation diagnostics. Section  details data and code availability, and Sect.  concludes with a summary and perspectives.

Network construction is performed separately for anticyclonic and cyclonic eddies. While cyclonic–anticyclonic interactions do exist (), they are comparatively rare and involve complex dynamical processes. Restricting the analysis to eddies of the same polarity allows for a more robust and computationally tractable description of eddy interactions, while retaining the dominant merging and splitting dynamics observed globally.

The grouping step links daily detections into spatio-temporal networks based on their geometric overlap across consecutive days. Two eddies are connected if their intersection-over-union (IoU) overlap score O exceeds 10 %, as defined : 1O(eddyref,eddystudy)=Area(eddyref∩eddystudy)Area(eddyref∪eddystudy).

In addition, a hybrid criterion is used to retain configurations for which the area of the smaller contour is almost entirely included within the larger contour, even when the IoU overlap ratio defined above falls below the threshold.

To account for occasional missing detections, resulting from mapping limitations or temporary detection failures, the search is extended up to seven days forward. This allows eddies belonging to the same trajectory to be reconnected when intermediate detections are missing. No constraint is imposed on radius or amplitude variations, allowing the detection of rapid size changes associated with merging or splitting.

These tracking parameters were selected empirically based on extensive expert evaluation of trajectory continuity and network consistency. They build upon the methodology developed for previous versions of the META atlas , while being adjusted to improve the identification of merging and splitting events in the network framework. In particular, the temporal search window was extended from 5 d in META3.2 to 7 d in META4.0. Indeed, merging and splitting events are often associated with deformations of the eddy contours and rapid changes in their properties (see Sect. ), which can increase the occurrence of temporary detection gaps. A longer temporal window therefore improves the reconnection of eddies belonging to the same network while remaining short enough to limit spurious associations. It should however be noted that the selected overlap threshold (10 %) and temporal search window (7 d) are empirical choices and their influence on the resulting network statistics has not yet been systematically evaluated through dedicated sensitivity analyses. While the extended search window helps reduce trajectory fragmentation caused by temporary detection gaps, it may also increase the likelihood of reconnecting unrelated eddy structures, particularly in regions characterized by high eddy density, mapping uncertainties, or short-lived spurious detections. Consequently, some aspects of the reconstructed network topology, trajectory lifetimes, and merging/splitting statistics may depend on the selected parameter values.

It should be added that the complete workflow is implemented in the open-source pyeddytracker library, which allows users to modify the tracking parameters and generate customized eddy trajectories, networks, and associated statistics according to their specific applications.

The process can be summarized as follows: 1.

Preprocessing of files

Input daily eddy detections (polygons) are sorted chronologically;

Each eddy observation is thus assigned to a group, corresponding either to a simple trajectory or to a more complex network involving multiple interacting eddies. These steps are illustrated through dedicated Jupyter notebooks available in the documentation (https://py-eddy-tracker.readthedocs.io/en/latest/python_module/16_network/pet_group_anim.html#sphx-glr-python-module-16-network-pet-group-anim-py, last access: 23 September 2026).

The segmentation step decomposes each group into coherent trajectory segments bounded by an event. An event can be a birth, death, merging, or splitting, defining the start or end of one or several eddy segments. Figure illustrates an example of a network composed of three segments connected through two successive events. The first segment is composed of four observed eddies represented by the dots. It ends by merging with the second observations of segment 2 resulting in a larger eddy belonging to segment 2. The fifth observation of this segment splits and gives birth to the third segment.

The segmentation algorithm proceeds as follows: 1.

Initialization

Each observation in a group is assigned an identifier and a time index;

It should be noted that the segmentation algorithm does not impose any restriction on the number of eddies involved in merging or splitting events. Consequently, the resulting network topology is not limited to pairwise interactions: more than two eddies may merge into a single segment, and a single segment may split into multiple offspring segments. This design choice avoids introducing a priori constraints on the network structure and allows the atlas to represent all interaction patterns identified by the tracking algorithm.

The segmentation procedure is illustrated in the PET documentation notebooks (https://py-eddy-tracker.readthedocs.io/en/latest/python_module/16_network/ pet_segmentation_anim.html#sphx-glr-python-module-16-network-pet-segmentation-anim-py, last access: 23 September 2026), which provide visual examples and interactive tools for reproducing this step.

In this section, the capabilities of the pyeddytracker toolbox are illustrated through a representative case study. This local example highlights some of the graphical diagnostics available to investigate the structure and dynamics of individual eddy networks.

A single network was selected from the META4.0 anticyclonic networks atlas to demonstrate the diagnostic possibilities. Networks can be extracted using flexible selection tools:

Spatio-temporal criteria (e.g., specifying a time window and/or longitude–latitude box);

The chosen network characteristics are summarized in Table .

The temporal evolution of the network is represented as timelines in Fig. . Timelines can be ordered either by segment identifiers or by latitude, both as a function of time. Merging and splitting events are symbolized by black hexagons and stars, respectively. Such representations are useful for quickly assessing the internal complexity of a network: the number of coexisting segments, their relative durations, and the timing of interaction events.

To complement this view, a geographical view of the segments is shown in Fig. . In this map-based representation, the merging and splitting events are explicitly localized, allowing for a joint spatial and temporal understanding of network dynamics.

Timelines can also be enriched with eddy properties such as effective radius or shape error, as illustrated in Fig. . On this timeline, each dot corresponds to an observed eddy. Marker color indicates the shape error, while marker size is proportional to the effective radius. This combined representation directly links the temporal organization of eddy segments to the evolution of their physical characteristics.

For example, Fig.  reveals that the effective radius tends to increase after merging events and before splitting events, suggesting a reorganization of the eddy structure when mass and energy are redistributed. At the same time, the timeline indicates that eddies become more distorted around interaction events, corresponding to increased dynamic stress during merging or division. Together, these diagnostics provide physically interpretable signatures of merging and splitting: growth in size accompanied by shape deformation is a strong indicator of structural reorganization. Similar diagnostics could also be performed with other eddy properties such as amplitude or mean translation speed, however, they are omitted here for conciseness.

Overall, these diagnostics can be particularly informative for:

Detecting structural changes in eddy properties (e.g., increase in radius, deformation) around merging and splitting events;

These exploratory diagnostics therefore provide a practical toolkit for investigating the structure and dynamics of individual eddy networks. While they are demonstrated here on a single representative case, the tendencies observed will be assessed and statistically confirmed at the global scale in the following sections.

This section analyzes the characteristics of merging and splitting events identified with the network-based approach.

This section explores whether mesoscale eddy networks can be grouped into a limited number of recurrent types based on their main properties. The objective is to identify regions of the global ocean where networks share similar structural and dynamical characteristics, thus providing a typology of mesoscale organization. This investigation is performed using a clustering analysis based on some chosen network attributes, as follows:

The number of observations;

Different clustering algorithms and parameterizations were tested (e.g., hierarchical clustering, Gaussian Mixture Models). The k-Means algorithm () with five clusters was retained for its convergence speed, accuracy and robustness. For the sake of conciseness, this exploratory phase is not reported here.

Figure presents the mean characteristics of the five clusters, while Fig.  shows the distribution of the underlying attributes. Together, these figures provide the basis for interpreting the different categories of eddy networks. The geographical distribution of the resulting classes is presented afterwards in Fig. .

The five groups can be interpreted as distinct dynamical regimes of mesoscale organization:

Group 1: Small networks of small eddies, distributed nearly everywhere. These networks likely represent the background turbulent field, with limited impact on large-scale transport.

This typology extends the single-eddy perspective of previous atlases (e.g., META3.2) by explicitly considering the collective organization of eddies into networks. Rather than only identifying where eddies are dense, the clustering highlights where eddies evolve as isolated, long-lived structures versus where they form complex, interacting systems. From a dynamical point of view, the results distinguish between regions dominated by transient, localized structures (Groups 1–2), zones of persistent solitary eddies (Group 3), regions of intense mesoscale interactions (Group 4), and major corridors of inter-basin transport (Group 5).

Overall, the clustering highlights the heterogeneous roles of eddy networks in ocean circulation: from local stirring and storage to long-distance connectivity and redistribution of water masses.

In the next section, Lagrangian particle advection is used to complement the spatial clustering analysis by providing a dynamical assessment of water mass transport and retention within eddy networks.

The coherence of eddy segments and their associated networks is assessed using Lagrangian particle advection experiments. Lagrangian approaches based on particle trajectories and Lagrangian coherent structures have been widely used to characterize eddy transport and coherence properties (e.g. ).

Approaches based directly on Lagrangian coherent structures or particle trajectories characterize eddy coherence using explicitly Lagrangian criteria. In contrast, META4.0 relies on an Eulerian detection and tracking framework based on contour geometry and overlap criteria. The purpose of the particle-advection experiments presented here is therefore not to reproduce such Lagrangian tracking methodologies, but rather to assess whether the reconstructed network topology is consistent with the transport properties implied by the velocity field.

Here, virtual particles are advected using surface geostrophic velocities derived from the same altimetric sea level anomaly (ADT) product employed for eddy detection. An eddy (or eddy segment) is considered coherent when a significant fraction of the particles initially enclosed within its contour remain trapped within coherent eddy structures over time, whereas it is deemed incoherent when particles rapidly escape the eddy boundaries.

To perform this assessment, particles are uniformly initialized at a spatial resolution of 1/50° within the eddy speed contour at a reference date d. Particle trajectories are then integrated both forward and backward in time over a 7 d window. Particle trajectories are computed using the Lagrangian advection module of the open-source pyeddytracker package, which relies on a fourth-order Runge-Kutta (RK4) integration scheme. At d±7 d, particle positions are analyzed and assigned to the detected eddies at those dates. The coherence score is defined as the sum of the fractions of particles contained within the two eddy effective contours gathering the largest proportions of the initial particle population. This metric provides a quantitative estimate of the retentive capacity of individual segments and their associated networks, expressed as a percentage of retained particles.

Performing particle advection both forward and backward in time allows characterization of the directionality of coherence and to assess the intrinsic irreversibility of eddy interactions. Forward advection evaluates the ability of eddies to retain particles in the future, while backward advection measures the extent to which particles observed within a given eddy originate from a coherent structure in the past. Differences between forward and backward coherence therefore provide insight into the irreversible nature of eddy dynamics and particle transport.

Figure presents the raw and cumulative distributions of segment autocoherence scores for anticyclonic and cyclonic eddy networks. For a given segment, the resulting score is referred to as autocoherence because particle positions are compared with the future or past evolution of the same segment. In both atlases, the raw distributions exhibit a pronounced maximum at high autocoherence values (around 90 %), indicating that most detected segments behave as strongly Lagrangian coherent structures. In particular, more than 50 % of the segments display autocoherence scores exceeding 75 %, which supports the robustness of the segmentation procedure and highlights the persistence of coherent transport pathways within eddy networks.

Interaction events are further evaluated using the same framework, by computing the fraction of particles that remain involved in merging or splitting processes, as opposed to escaping from interacting structures.

As described in Fig. , a merging event is characterized by a minor eddy M2 merging into a dominant eddy M1, resulting in a single eddy M that follows the trajectory of M1, based on overlapping criteria. Conversely, a splitting event is modeled as an initial eddy S separating into a main eddy S1, which retains the original trajectory, and a secondary eddy S2.

Figure shows the cumulative distributions of coherence scores for merging and splitting events.

Forward advection reveals a strong coherence of merging processes, as particles initially enclosed within interacting eddies remain largely confined within the merged structure. In contrast, backward advection exhibits significantly lower coherence, reflecting the irreversible mixing of particle trajectories once the eddies have merged. More than 60 % of detected merging events display forward coherence scores above 80 % for the dominant eddy M1, whereas the secondary eddy M2 generally exhibits lower coherence, with only about 40 % exceeding the same threshold. This asymmetry likely reflects the weaker dynamical integrity of M2, which is typically smaller and more rapidly distorted during the merging process.

Conversely, Fig. b shows that splitting events display higher backward autocoherence. Backward integration highlights the convergence of particle trajectories toward a single coherent structure in the past, while forward advection results in a dispersion of particles among multiple offspring eddies. More than 60 % of detected splitting events exhibit backward coherence scores above 80 % for the main eddy S1. The secondary eddy S2 shows systematically lower coherence, with only about 35 % of events exceeding this threshold, consistent with its more recent formation and reduced ability to retain particles.

Overall, these diagnostics show that most detected interaction events are associated with coherent Lagrangian transport patterns that are consistent with the reconstructed network structure.

It should be noted, however, that the velocity fields used for particle advection are derived from the same altimetric product as that employed for eddy detection. As a result, the coherence assessment is not fully independent of the detection methodology, unlike diagnostics based on external tracers such as chlorophyll or sea surface temperature. Nevertheless, the particle-based analysis evaluates a different aspect of the dataset. While the detection and tracking procedures are based on Eulerian contour geometry, the advection experiments assess the Lagrangian transport properties implied by the same velocity field. The analysis therefore provides a consistency check of the reconstructed networks rather than an independent validation of the underlying eddy detections.

Additional limitations include the finite 7 d advection window, which may miss shorter-lived interactions or intermediate events occurring within this period. Despite these limitations, the consistency of the results across multiple diagnostics supports the robustness of the proposed framework.

Finally, tools to perform Lagrangian particle advection analyses on individual eddy networks are provided within the pyeddytracker toolbox (https://py-eddy-tracker.readthedocs.io/en/v3.6.1/python_module/16_network/ pet_follow_particle.html#sphx-glr-python-module-16-network-pet-follow-particle-py, last access: 23 September 2026). These tools allow, for instance, the advection of particles within a given network, the visualization of their trajectories, and the analysis of interaction events along their temporal evolution.