This study employs absolute dynamic topography (ADT) fields derived from the satellite altimetry product SEALEVEL_GLO_PHY_L4_MY_008_047 for eddy and eddygroup identification as well as eddy tracking. The dataset, originally developed by the French Archiving, Validation, and Interpretation of Satellite Oceanographic data (AVISO) center, is now made available through the Copernicus Marine Environment Monitoring Service (CMEMS; https://resources.marine.copernicus.eu, last access: 30 May 2026). The product is generated through the Data Unification and Altimeter Combination System (DUACS), which merges multi-mission altimetry from satellites including Sentinel-3, the Jason series, Saral/AltiKa, Topex/Poseidon, and Envisat. The dataset spans from 1 January 1993–31 December 2023, with a temporal resolution of one day and a spatial resolution of 0.125°×0.125° covering the global ocean.

The identification of eddies, eddygroups, and their associated topological structures in this study is conducted through two main steps, as shown in Fig. 5. Initially, following data input, eddies and eddygroups are first identified. Subsequently, the topological structure is constructed, yielding the final multi-level identification results for eddies and eddygroups.

The algorithmic steps for the identification of eddies and eddygroups are depicted in Fig. 6. Initially, seed points are detected, followed by the extraction of closed contours. For each contour, the quantity and indices of enclosed seed points are recorded to distinguish between cases: contours enclosing a single seed point are identified as eddies, while those containing multiple seed points are identified as eddygroups. Prior to producing the final output, the identified eddygroups are sorted in descending order based on area to facilitate subsequent construction of the eddytree.

The criteria for defining eddy boundaries follow standards similar to those of (Tian et al., 2021; Mason et al., 2014), with the primary distinction arising from the spatial resolution of the dataset. The number of pixels I is used to constrain the area of mesoscale eddies. Referring to the parameter values adopted in previous studies, I between 20 and 4000 in this study; this number increases as the grid resolution becomes finer. The shape error is introduced to constrain the morphological characteristics of eddies and is defined as the ratio of the total area difference between the closed contour and its fitted circle to the area of that circle. Thus, a smaller shape error indicates that the eddy's shape is closer to an ideal circle. In addition, the constraint on eddy amplitude is related to the contour generation interval chosen for this study, which is set as 0.5 cm; therefore, the eddy amplitude is required to exceed at least two contour intervals. The size of the seed point detection window is also related to the spatial resolution of the data: higher spatial resolution leads to a denser data grid and a larger number of pixels within the seed point detection window. Specifically, an eddy is required to meet the following conditions:

Contain a single local maximum or minimum.

The procedure for constructing eddytrees using the area-sorting method is illustrated in Fig. 7. Initially, the identified eddies and the area-sorted eddygroups are input, with the largest eddygroup designated as the root node of the initial eddytree. Subsequently, the remaining eddygroups and eddies are traversed sequentially. If a given target is not contained by any existing root node, a new eddytree is initialized for that eddygroup, or the eddy is recorded as an independent entity. Conversely, if the target is contained within an existing root node, the containment relationships are evaluated within that tree: starting from the root, the hierarchy is examined level by level until the target is no longer enclosed by any eddygroup, at which stage it is inserted as a node at that level. Upon completion of this traversal, the eddytree identification result for the given day is obtained.

In general, root nodes corresponding to larger eddygroups tend to exhibit more complex hierarchical structures, encompassing a greater number of internal and leaf nodes. The area-sorting-based eddytree construction algorithm takes full advantage of this parameter by establishing larger root nodes first in descending order of area. This strategy allows the assignment of subsequent eddygroups and eddies at shallower hierarchical levels, thereby minimizing the necessity for backtracking and redundant comparisons. For independent eddies, verification is limited to confirming their exclusion from all root nodes, obviating the need for deeper containment checks within each eddytree. This approach consequently reduces computational cost and enhances overall efficiency.

In contrast to existing “identify-while-building” algorithms (Tian et al., 2024), the present study separates the construction of the eddytree from the identification phase. The identification procedure involves only a linear traversal during which essential information is recorded. Subsequently, multi-level eddy structures are generated in a relatively straightforward manner based on area sorting. This separation strategy reduces the overall time complexity from O(n2) to O(nlog⁡n), thereby decreasing the average daily runtime from 480–43 min and enhancing efficiency by approximately a factor of eleven. The computer running the algorithm is configured with an Intel^® Core™ i5-12400F CPU and compiled and implemented in Python 3.8 (Anaconda 3).

The processes of eddy splitting and merging are tracked using a Lagrangian particle-drift approach, as outlined in the flowchart presented in Fig. 8. This procedure consists of three main steps: (1) establishing segments between consecutive days through particle drifting; (2) linking segments into branches; and (3) connecting branches to form eddygraphs.

In this study, the meridional and zonal components of the absolute geostrophic velocity are used to advect all eddy centroids and associated seed points from day i forward, employing a one-day integration step to predict their positions on day i+1. The Lagrangian particle-drift approach provides a realistic representation of particle motion in the ocean (Haller, 2015; Van Sebille et al., 2018). Due to the substantial number of particles tracked on a daily scale, only the centroid of each eddy is advected as its representative particle (Jones-Kellett and Follows, 2024).

The process of constructing segments based on particle-drift predictions is illustrated in Fig. 9. Segment classification depends on the quantity and distribution of predicted particle positions that fall within eddies or multicore eddies on day i+1: If the predicted position of an eddy centroid or a seed point on day i falls within an eddy on day i+1, the state is classified as a live segment; If the predicted positions of two objects (either eddy centroids or seed points) on day i fall within the same eddy on day i+1, the state is classified as a merged segment; If the seed points from a multicore eddy on day i are advected into distinct eddies on day i+1, the state is classified as a split segment; Finally, if the predicted position of a particle from day i does not fall within any eddy on day i+1, the corresponding eddy or seed point is designated as a dead segment.

A branch is defined as a single survival trajectory formed by temporally consecutive live segments that also exhibit spatial overlap, connected in chronological order. To construct branches, the set of segments from day i to day i+1 is first examined to extract live segments. Since each segment records the survival relationship of an eddy between two consecutive days, the surviving target identified in the segment on day i can be used to link the live segments from day i to day i+1. Following the same rule, the live segments between day i+1 and day i+2 are then connected, and this procedure continues sequentially, linking live segments from consecutive days into a continuous trajectory until no corresponding live segment can be found on the following day. The resulting branches may originate from three cases: a newly generated eddy, a child node eddy produced by splitting, or an eddy formed by merging. The termination points of branches likewise correspond to three cases: disappearance on the subsequent day, splitting, or merging (Fig. 10a). Finally, by connecting all qualifying live segments, continuous branches are obtained, as illustrated in Fig. 10b.

Branches are connected into a complete eddygraph through splitting and merging relationships at their endpoints. For example, if a branch terminates with a merging event, there must exist other branches that merge into the same target eddy at the same time. Conversely, if a branch terminates with a splitting event, multiple new eddies are generated on the next day, which are then used to establish connections with the corresponding branches on that day. As illustrated in Fig. 11, the branches depicted in the left panel are linked by their segment relationships to form a single eddygraph.

Figure 12 presents representative eddygraphs of cyclonic and anticyclonic splitting and merging events. Panels 12a and b illustrate trajectories of typical anticyclonic and cyclonic cases, respectively. For each event, the upper subpanel presents all involved branches, while the lower subpanel shows the connections among these branches through splitting and merging, thereby constructing a complete eddygraph. In Fig. 12a, the easternmost anticyclonic eddy originated on 2 January 2023, merged with another eddy on 24 January. It subsequently split into two eddies on 15 February. Of these, the southern eddy died on 26 June, while the northern one underwent another merging event on 25 June and ultimately died on 11 September, with a total lifetime of 253 d. In Fig. 12b, the eastern eddy formed on 1 January 2022, experienced a splitting event on 5 May. Among the two resultant eddies, one died on 24 May, while the other merged on 17 June with an eddy formed on 22 March. This merged eddy finally died on 8 September, with a total lifetime of 251 d. These examples demonstrate that by incorporating splitting and merging relationships, the tracking method links branches that would otherwise remain disconnected. Compared to conventional tracking approaches that consider only eddy survival and death, this method yields extended lifetimes and more coherent trajectories.

In addition, the eddy boundaries depicted in the figure show that the two eddies before merging and after splitting differ in area. During merging, the branch corresponding to the larger eddy exhibits stronger continuity with the post-merger branch. Similarly, in splitting events, the pre-split eddy is more strongly connected to the branch of the larger post-split eddy. To further investigate this behavior, centroid positions were statistically analyzed across all events. Results show that in 91.8 % of merging cases, the centroid of the post-merger eddy is closer to that of the larger pre-merger eddy, while in 92.5 % of splitting cases, the centroid of the pre-split eddy is nearer to that of the larger post-split eddy. These findings indicate that the larger eddy tends to dominate the splitting and merging processes, effectively functioning as the primary entity in such events.

Figure 13 presents the annual mean spatial distribution of eddies over the 31 year period. The spatial patterns indicate that eddies are predominantly concentrated along oceanic boundaries and in mid–high latitude regions, whereas their occurrence is relatively sparse in equatorial zones.

Inspection of the eddy and eddygroup distribution (Fig. 2) reveals that in some regions, eddygroups with the same polarity tend to cluster. According to our statistics, the mean areas of the anticyclonic eddygroup south of the Kuroshio Extension, the cyclonic eddygroup north of the Kuroshio Extension, the anticyclonic eddygroup south of the Gulf Stream, the cyclonic eddygroup north of the Gulf Stream, and the anticyclonic eddygroup near the Brazil Current are approximately 8.67 millionkm2 (the anticyclonic eddygroup south of the Kuroshio Extension), 4.45 millionkm2 (the cyclonic eddygroup north of the Kuroshio Extension), 7.30 millionkm2 (the anticyclonic eddygroup south of the Gulf Stream), 3.35 millionkm2 (the cyclonic eddygroup north of the Gulf Stream), and 5.05 millionkm2 (the anticyclonic eddygroup near the Brazil Current), respectively. Among them, the largest eddygroup occurred on 8 October 1994 south of the Kuroshio Extension, with an area exceeding 19.64 millionkm2. To quantify the long-term spatial distribution, the daily polarity of the root eddygroups of eddytrees was accumulated over the period 1993–2023 on a 1°×1° grid: grid cells covered by anticyclonic eddytree roots were assigned a value of +1, and those covered by cyclonic roots were assigned a value of -1. The accumulated values were then averaged annually. The results reveal a stable regional preference in eddytree polarity across the global ocean, and the annual-mean distribution of eddytree polarity exhibits a strong correspondence with large-scale ocean circulation. Figure 14 presents the annual-mean distribution of eddytree polarity superimposed on the large-scale ocean circulation, from which it can be seen that: (1) In western boundary regions, strong currents constrained by continental margins – such as the 3 – Kuroshio, 4 – Gulf Stream, 6 – Caribbean Current, 9 – East Madagascar Current, 10 – Agulhas Current, and 11 – Brazil Current – are associated with values exceeding +300yr-1 (shown in dark red), indicating a persistent dominance of anticyclonic eddytrees; (2) in frontal zones where strong cold and warm currents meet, such as the Oyashio–Kuroshio (1–3) and Labrador–Gulf Stream Current (2–4) confluence zones, a distinct boundary between cyclonic and anticyclonic eddytrees is observed, forming a dipole-like eddytree structure. This boundary extends into the interior ocean and gradually weakens; (3) in open-ocean regions, island topography such as the Hawaiian Islands obstructs the flow, resulting in von Kármán-type vortex-street structures in the wake region of the islands (as shown in Fig. 14b); and (4) within the interior ocean, polarity boundaries are typically associated with stable large-scale currents, including the 5 – North Equatorial Current, 7 – Equatorial Counter Current, 8 – South Equatorial Current, and 12 – Antarctic Circumpolar Current.

Based on the trajectory dataset, the occurrence frequency of eddy splitting and merging events exhibits distinct spatial characteristics. As shown in Fig. 15, the annual average frequency of these events is relatively high in mid- to high-latitude regions, while much lower frequencies are found in low-latitude waters, particularly within the equatorial belt. This spatial pattern is broadly consistent with the distribution of eddies. High-frequency regions are concentrated along the North Pacific margin and central basin, the central South Pacific, the southeastern waters off Australia, the western North Atlantic (especially the Gulf of Mexico and its extension), and the southern Indian Ocean. These regions are characterized by strong eddy kinetic energy, intense shear associated with boundary currents, and pronounced frontal instabilities, all of which enhance the probability of eddy–eddy interactions and thereby lead to a higher frequency of splitting and merging events.

To examine this difference in detail, the occurrences of both types of events were accumulated within each 1° latitude band (Fig. 16). The result indicates that merging events are roughly three times more frequent than splitting events. This pattern may be influenced by survivor bias, as the limited resolution of current satellite altimeters hampers the detection of smaller eddies generated through splitting. Overall, the Southern Hemisphere exhibits a higher frequency of events compared to the Northern Hemisphere, while event counts near the equator remains close to zero. This latitudinal contrast is consistent with spatial distribution of eddies: The Southern Hemisphere has a larger oceanic area and hence more eddies, whereas in equatorial waters eddies are scarce, leading to infrequent splitting or merging occurrences.

Along with extracting representative splitting and merging events, the study examined the associated ADT background fields to visualize the evolution of eddies before and after the events. The analysis reveals that splitting and merging are gradual processes rather than instantaneous transitions. In stable merging cases, two eddies progressively approach one another and develop an egg-shaped geometry. In this configuration, their sharp poles face each other, while their dull poles are oriented outward. They then finally combining into a single eddy (Fig. 17a and b). In stable splitting cases, an eddy initially evolves into a multi-core structure; after splitting, the two resultant eddies gradually separate, again exhibiting paired egg-shaped structures with their sharp poles directed toward one another (Fig. 17c and d).

To further investigate the morphological evolution associated with splitting and merging, typical events were selected from the trajectory dataset under the criteria that persisted for more than seven days both before and after the interaction, as well as the absence of interference from other eddies during the process. This selection yielded a total of 1814 merging events, including 964 cyclonic and 850 anticyclonic cases, together with 2634 splitting events, comprising 1376 cyclonic and 1258 anticyclonic cases.

The analysis of eddy structures within a rotated and normalized coordinate system provides an effective way to minimize the impact of geometric scale and orientation, thereby enabling a clearer representation of the physical processes (Wang et al., 2023). In practice, (Sun et al., 2018) employed a normalized eddy coordinate system to align Argo profiles with individual eddies, extracting representative structures and their evolution. (Brokaw et al., 2020) explicitly applied radius normalization to compare eddy characteristics and internal structure, while (Zhou et al., 2025) used elliptical normalization as the basis for composite analyses. Collectively, these studies demonstrate that normalization greatly improves the comparability of composite analyses and facilitates the identification of evolutionary characteristics of eddies.

In this study, the dipole normalization method proposed by (Long et al., 2024) is referenced to normalize pairs of same-polarity eddies before merging and after splitting. Specifically, the origin of the coordinate system is established at the midpoint of the line connecting the centroids of the two eddies, with the larger eddy on the x<0 side and the smaller eddy on the x>0 side, as shown in Fig. 18a. To eliminate the influence of orientation differences of two-eddy structures across different events, the rotation angle θ required to align the line connecting the centroids with the horizontal direction was calculated and recorded for each event at every time point. The corresponding eddy boundaries and ADT background fields were then rotated by this angle, producing the configuration shown in Fig. 18b. To remove scale differences, the mean distance d(n) between the centroids of the two eddies on day n (prior to merging or after splitting) is used to define a scaling factor S(n). This factor is subsequently applied to the eddy boundaries and ADT background field for that day, producing the normalized field shown in Fig. 18c. Based on these normalized results, the eddy boundaries and ADT background are averaged for each day n to obtain the persistent mean morphology before merging and after splitting. The normalization window is defined as x∈[-2.5°,2.5°], y∈[-2°,2°].

Based on the previously described two-eddy normalization procedure, the corresponding parent eddygroups were also normalized. First, the midpoint of the line connecting the centroids of the two eddies was chosen as the origin to establish the normalized coordinate system. Using the same origin as the two-eddy normalization avoids introducing relative displacement errors and preserves the spatial relationship between the parent eddygroup and the two-eddy structure throughout the normalization procedure. Next, the boundary of the parent eddygroup was then rotated by the same angle θ derived from the corresponding two-eddy event to ensure consistent orientation. Subsequently, the scaling parameter S(n) associated with the same event was applied to scale-normalize the boundary of the parent eddygroup, yielding normalized eddygroups for each event and each time point.

The area asymmetry of the interacting eddies results in an egg-shaped merged (or pre-splitting) eddy, defined by its sharp and dull poles. To accurately represent this geometry, an egg-shaped normalization was applied to the post-merging and pre-splitting states: for each frame, the major axis of the eddy was fitted and rotated to align with the x-axis, with the sharp pole toward the positive x-direction. The rotation angle β was recorded and then used for the rotation operation of the ADT background field. After orientation alignment, scale normalization was applied to the single-eddy structures that appear after merging and before splitting. Specifically, the scaling parameter S(n), derived from the two-eddy structure at the nearest time step within the same event, was used to adjust the single-eddy boundary and its corresponding background field. Since the size of the eddies involved in the same merging or splitting event typically remains relatively stable over a short period, adopting the scaling parameter determined during the two-eddy stage helps maintain scale consistency in the normalized results before and after the event, thereby facilitating meaningful comparisons of morphological characteristics across different stages. Based on this, the eddy boundaries and the ADT background were then averaged over all frames n to obtain the persistent mean morphology of the single eddy after merging and before splitting.

To assess the continuity of eddy morphology between the pre- and post-event stages, the difference in rotation angle was calculated from pre- to post-merging and from pre- to post-splitting. When the angular difference was less than π/4, the smaller eddy before merging corresponded to the sharp pole of the merged eddy, while the larger eddy corresponded to the dull pole; similarly, after splitting, the smaller eddy corresponded to the sharp pole of the pre-splitting eddy, and the larger eddy corresponded to the dull pole. The results show that 97.73 % of merging events and 97.15 % of splitting events satisfy these relationships. This indicates that, aside from geometric changes caused directly by the approach or separation of eddies during merging or splitting, the natural evolution of eddy morphology in the pre- and post-event stages also gives rise to egg-shaped morphologies with pointed and dull poles.

The normalized results for the splitting and merging processes of eddies with different polarities are presented in Fig. 19. During the merging process, the two eddies initially manifest as nearly circular egg-shaped structures, with their common parent eddygroup exhibiting an elliptical outline. As the interaction progresses, the eddies gradually approach each other, with the boundaries on their near side converge more rapidly. Geometrically, sharp poles gradually form on the sides that approach each other, while dull poles develop on the opposite sides, thereby accentuating the egg-shaped morphology. Concurrently, the boundary of the common parent eddygroup of the two eddies also gradually develops pointed and dull poles, taking on an egg-shaped structure, with the sharp pole containing the smaller eddy and the dull pole containing the larger one. Subsequently, the two eddies transition into seed points and eventually degenerate into a single-core eddy, completing the merging process. The resultant eddy retains an egg-shaped structure in which the originally smaller eddy corresponds to the sharp pole and the larger eddy to the dull pole. Over the following days, the contrast between sharp and dull poles diminishes, and by day seven after merging (t=7), the merged eddy approaches a more regular elliptical form. The alteration in the boundaries of the eddygroup before merging and the eddy after merging shows consistency, indicating that the boundary of the merged eddy is a degeneration of the common parent eddygroup boundary of the two pre-merging eddies.

During the splitting process, the single eddy initially exhibits an elliptical shape, which gradually develops sharp and dull poles as the event approaches, taking on an egg-shaped structure. The eddy then evolves into a multi-core structure and subsequently splits into two eddies, completing the splitting process. Notably, the two resultant eddies after splitting maintain egg-shaped morphologies, with their sharp poles oriented toward each other, while the boundary of their common parent eddygroup also exhibits an egg-shaped form. As time progresses, the egg-shaped features weaken, the separation between the two eddies increases, and the egg-shaped characteristics of the parent eddygroup gradually fade. By day seven after splitting (t=7), the two eddies tend toward a more circular shape, with the boundary of their parent eddygroup becoming elliptical. The alteration in the boundaries of the eddy before splitting and the common eddygroup after splitting shows consistency, indicating that the boundary of the pre-splitting eddy evolves into the common parent eddygroup boundary of the two post-splitting eddies.

During the normalized morphological evolution, the processes of eddy splitting and merging exhibit a strong form of “reciprocity.” To further investigate this behavior, the egg-shaped structure indexes proposed by (Chen et al., 2021) were introduced – the ellipticity (E) and the asymmetry (A)—to provide a quantitative description of the morphological patterns associated with eddy splitting and merging. These indexes are defined as follows: 1E=2ca1+a2,2A=1-a2a1, The relevant indexes are illustrated in Fig. 20. E typically ranges between 0 and 1, where values approaching 0 correspond to shapes that are nearly circular, and values nearing 1 indicate increasingly elongated forms; in this study, E≈0.8. A varies from 0–0.8, where A=0 corresponds to a perfectly symmetric ellipse, and higher values indicate stronger asymmetry; the typical value is A≈0.1.

Building upon this framework, the study derived the morphological indexes E and A throughout the normalized processes of eddy splitting and merging. We computed the mean E and A for the two eddies (before merging or after splitting) and the values for their common eddygroup. Based on these values, time series of E and A were constructed (Fig. 21), thereby enabling a quantitative assessment and comparison of the morphological characteristics of merging and splitting events.

Figure 21a illustrates the indexes variations during the merging process. Prior to merging, the ellipticity (E, represented by dashed lines) of the two eddies increases from approximately 0.4–0.7, while their asymmetry (A, dashed lines) gradually rises from 0.1–0.2. This indicates an initial circular geometry of the eddies that progressively elongates and assumes an egg-shaped form. The common parent eddygroup shows a relatively stable E around 0.8, while A increases from 0 to about 0.15, suggesting that the parent structure begins as elliptical and gradually develops egg-shaped features. After merging, E remains stable near 0.8, whereas A decreases steadily toward zero, implying that the merged eddy maintains a relatively constant ellipticity while its shape becomes increasingly symmetric, ultimately approaching a standard ellipse. Figure 21b shows the indexes variations during the splitting process. Before splitting, the eddy maintains a nearly constant E of about 0.8, while A increases from near zero to around 0.2, indicating a transition from a symmetric elliptical shape toward an egg-shaped geometry. After splitting, the two resultant eddies exhibit a gradual decrease in A (dashed lines) from 0.2–0.1, alongside a reduction in E (dashed lines) from roughly 0.8–0.4. This suggests that the eddies retain a relatively stable degree of asymmetry but become progressively less elongated, trending toward circularity. Meanwhile, the common parent eddygroup maintains E around 0.8 while its A decreases from about 0.15 to zero, showing a transition from an egg-shaped structure to a more elliptical structure.

The variations of normalized indexes are consistent with the previously described morphological evolution. The post-merging eddy shows strong spatial and temporal continuity with the common parent eddygroup prior to merging, while the pre-splitting eddy aligns closely to the common parent eddygroup after splitting. This further underscores the crucial role of eddygroups in splitting and merging events: the boundary of the pre-merging parent eddygroup of the two eddies degenerates into the boundary of the merged eddy, and the boundary of the pre-splitting eddy evolves into the boundary of the post-splitting parent eddygroup of the two resultant eddies.

By employing the time of splitting as a symmetry axis, the indexes variations during splitting were mirrored and aligned with those of merging (Fig. 21c). The results show a high degree of consistency in E between the merging process and its inverse, the splitting process, with A also demonstrating concordance. This demonstrates that, in normalized coordinates, eddy merging and splitting can be regarded as reciprocal morphological processes. Furthermore, by treating merging and splitting as the forward and reverse phases of the same process, the overall indexes variations can be depicted as shown in Fig. 21d. And the figure illustrates that when the eddy splitting and merging events occur, the A of the eddygroup reaches its maximum.