The eddy splitting event examined here was observed northwest of the island of Mallorca in the Balearic Sea, in the Northwestern Mediterranean Sea (Fig. , bottom panel). The area is characterized by intense mesoscale and submesoscale eddy activity, fed by instabilities of the Northern Boundary Current and the Balearic Front . The mixed layer depth at the time was relatively shallow, approximately 20 m, as deduced from the Brunt-Vaisala frequency calculated from in situ hydrography (Fig. S1; see also ). The eddy could be identified in satellite chlorophyll concentration maps as early as 17 February 2022 (Animation S1) and was targeted for intense sampling starting 23 February, shortly before it began elongating (Fig. , left panel). The wind in the area was approximately from the southwest, roughly parallel to the eddy's major axis, with an intensification lasting about one day (Fig. ). Subsurface hydrography for the area shows an elliptic cyclone with doming isopycnals, characterized by subsurface vorticity reaching values of the order of the inertial frequency f , indicative of a Rossby number R of order 1. The smoothed divergence field, averaged over the top 60 m and 37 h, exhibits a quadrupole pattern , consistent with cyclogeostrophic effects that in highly curved flows can alter the geostrophic balance. In essence, as the flow approaches the narrow end of an elliptical eddy, it accelerates due to centrifugal forces creating convergence, while it diverges after passing the tip.

Over the next few days, the elongation is observed to increase, as the eddy strains (Fig.  top, central panel. Refer also to the Animation S1). Idealized modeling, suggests that the convergence areas of the quadrupole trigger a positive feedback that intensifies positive vorticity in two distinct patches, leading to the splitting . By 26 February, the center of the ridge collapses, while the smaller eddies intensify and become fully separated by 28 February (Fig.  top, right panel). Meanwhile, the wind, which was weak throughout 24 February, strengthens swiftly, likely due to a Tramontane event, starting on 25 February and lasting until 27 February, with prevalent direction from the northwest (Fig. ).

Thus, the eddy experiences both internal instabilities triggered by the eddy shape and its high intensity (high Rossby number) and significant wind forcing, especially during the period 25–27 February. These processes interact nonlinearly. The vertical vorticity ζ in the eddy modifies the `effective' Coriolis parameter feff=f+ζ/2 and thereby influences Ekman transport, potentially leading to nonlinear Ekman pumping . This, in turn, can impact the pressure field and therefore the eddy structure. The vorticity gradients associated with the edges of eddies can also modify the local nonlinear inertial oscillations (NIO), leading to surface convergence and divergence in the surface layer with associated pumping effects . Moreover, Ekman and NIO effects can interact during periods of transient forcing, through energy transfer from high to low frequencies .

Ekman transport requires a period of at least two to three days under quasi-steady forcing to be fully established, although the time scale depends on several parameters including the strength of the wind forcing and eddy viscosity . On shorter time scales, only a fraction of the full Ekman transport is found in the surface layer. The event investigated here is characterized by high variability in the wind forcing (Fig. ). The evolution and splitting of the eddy occurs over five days, while the wind variability has a scale of one to three days. This complicates the analysis. We anticipate that the interaction between eddy and wind effects is less straightforward than in the classical quasi-steady picture .

In order to identify the wind effects, we propose the following strategy. The densely distributed drifter data, from repeated launches in the first meter and at 15 m depth, are used to compute the kinematic properties (KPs), namely vorticity, strain rate, and divergence. Where drifters at the two depths are co-located, it is also possible to estimate the vertical velocity w . This allows us to describe the surface layer evolution for 23–28 February, with a temporal resolution of 1 d. Wind effects are expected to be particularly evident in the divergence field, since it is directly influenced by wind , while vorticity and strain rate are mostly dominated by mesoscale and submesoscale dynamics . While the high temporal variability in the wind inhibits the establishment of a fully developed Ekman transport and associated pumping, the tendency toward that state is expected to influence the surface layer. We therefore estimate the theoretical Ekman transport and pumping during the main wind episodes, following the theory briefly reviewed in Sect. , and relate these estimates to the observed divergence.

As mentioned above, NIOs can also impact the divergence field in the vicinity of the eddy. The daily averaging will dampen their signal but cannot fully eliminate it, in part because the inertial period (approximately 19 h) differs from the 24 h averaging time, but more fundamentally because of the nature of Lagrangian sampling, which is by definition not fixed in space. The quickly evolving eddy field necessarily makes the interpretation of the NIO effects on divergence more qualitative than is possible in the context of a quasi-steady jet . We have to rely on the detection of smaller scale patterns and deviations with respect to the Ekman tendency.

According to linear Ekman theory , non-zero wind stress curl induces positive or negative Ekman pumping, referring to the upwelling and downwelling responses, respectively, of the Ekman layer. Assuming an eddy viscosity of 0.05 m² s⁻¹, a typical Ekman depth at the relevant latitudes is roughly 30 m. Recall that for weak surface ocean currents, the wind stress exerted on the ocean surface is given by 1τwind=ρaCDuwind|uwind| where CD is a drag coefficient, ρa is the air density, |uwind| is the wind speed (at 10 m). Non-zero wind stress induces Ekman transport Mwind=〈Uwind,Vwind〉, integrated over the depth of the surface Ekman layer, that is directed perpendicular to the stress and rotated to the right in the northern hemisphere, with magnitude 2|Mwind|=|τwind|ρ0f where ρ0 is the mean water density in the Ekman layer. Non-zero divergence in the Ekman transport gives rise to Ekman pumping, with a vertical velocity proportional to the curl of the wind stress: 3wwind=∇⋅Mwind=∇⋅(τwind×k)ρ0f=k⋅∇×τwindρ0f. Here k is the upward unit vector.

Nonlinear effects modify these formulas . When ocean currents u=〈u,v〉 have significant magnitude relative to the wind, the stress τ acting on the ocean surface depends on the relative velocity urel=uwind-u: 4τ=ρaCDurel|urel| and the associated Ekman transport and pumping velocity are modified by the vorticity ζ in the Ekman layer: 5|MEknl|=|τ|ρ0(f+ζ),6wEknl=∇⋅τ×kρ0(f+ζ).

The total Ekman pumping velocity wEknl can be decomposed into two parts, one depending only on the vorticity and the other also involving the vorticity gradient : 7wEknl=wζ+w∇ζ=k⋅∇×τρ0(f+ζ)+k⋅(τ×∇ζ)ρ0(f+ζ)2=1ρ0(f+ζ)∂τy∂x-∂τx∂y+1ρ0(f+ζ)2τx∂ζ∂y-τy∂ζ∂x, where τ=〈τx,τy〉 and we have assumed an f-plane approximation. In order to concentrate on the contribution to wEknl due to ocean currents , and in agreement with the observations, we assume a clear scale separation between the wind velocity, that provides a smooth large scale background, and the ocean currents, characterized by mesoscale and submesoscale eddies with significant local vorticity ζ, while neglecting the nonlinear dependence of the stress on the relative velocity.

The two terms in wEknl describe two different mechanisms. The first one, wζ, results from the curl of the relative surface stress τ that, according to the assumptions stated above, is dominated by the wind velocity, scaled by the `effective' Coriolis parameter f+ζ, which is modified by the local surface vorticity ζ. Within a mesoscale eddy, the resulting Ekman pumping is expected to have opposite sign with respect to the local vorticity, resulting in upwelling in an anticyclonic eddy and downwelling in a cyclonic one. evaluated wζ for an isolated idealized cyclone, finding downwelling in the core, with elongation in the direction of the wind.

The second term, w∇ζ, results from the vorticity gradients, divided by the square of the “effective” Coriolis parameter. Conceptually, within our assumptions, this term is due to the fact that Ekman transport MEknl (Eq. ) decreases with increasing magnitude of positive ζ, inducing downwelling (and upwelling for decreasing positive ζ), while the opposite is true for negative ζ. This mechanism leads to the formation of dipoles within the core of an isolated eddy , as shown in the cartoon in Fig. . For a cyclone, downwelling is expected to occur on the left side of the eddy core with respect to the wind (in the northern hemisphere) and upwelling on the right side. This is because the transport MEknl is directed to the right of the wind, and therefore enters the eddy from its left side with respect to the wind, leading to downwelling in order to adjust for increasing ζ and decreasing transport. The opposite occurs on the other side of the eddy, where ζ decreases and transport increases. For anticyclones the dipole structure is inverted with respect to the wind. Surface wind forcing has also been found to induce a vertical tilt in the cyclone . Eddy tilt is also associated with stronger vertical velocities . Unfortunately, our dataset is insufficient to study such a tilt conclusively for the present eddy.

The analysis here is based on the surface and near-surface observations of drifter trajectories, complemented by ancillary data from satellites and ship-board instruments. Several significant drifter deployments were carried out during the CALYPSO cruise in February/March 2022 to sample the near-surface currents in specific submesoscale features with high horizontal resolution (order of 1 km). Three types of drifters were released in or near the cyclonic eddy considered in this paper: CARTHE (Consortium for Advanced Research on the Transport of Hydrocarbon in the Environment) drifters, drogued at 0.4 m , CODE (Coastal Ocean Dynamics Experiment) drifters, drogued at 0.75 m , and SVP (Surface Velocity Program) drifters, drogued at 15 m . CARTHE and CODE drifters have been found to measure currents representative of the top meter of the water column with accuracy of a few cm s⁻¹, assuming limited wind or wave induced slippage and Stokes drift , and will be treated collectively as “surface” drifters following, e.g., with a reference depth of 0 m, referred to below as CaC drifters. The SVP drifters have been shown to follow the sub-surface currents within 1 cm s⁻¹ in winds up to 10 m s⁻¹ . GPS positions were sampled every 10 min via GlobalStar satellites for CaC drifters and every 5 min via satellite Iridium telemetry for SVP drifters. The drifter positions were processed with standard methods for quality control and interpolated to uniform 5 min intervals starting at the top of the hour. Velocities were estimated by central finite differencing of these positions. Drifter data in the area of interest on three days during the analysis is shown as cyan (CaC) and blue (SVP) curves in Fig. . The Supplemental Materials contain an hourly animation (S2) for the entire 6 da period. Of the 140 CARTHE drifters released during the cruise, 53 were in the area of interest (black box in Fig. ) during this time. Similarly, 32 of 70 released CODE drifters and 22 of 68 deployed SVP drifters entered the area and time period of the analysis.

To evaluate the wind effects on the eddy, we use the ERA5 reanalysis 10 m winds from the European Centre for Medium-Range Weather Forecasts distributed by the Copernicus Climate Change Service . The data are available hourly on a 0.25°×0.25° grid. Here we use daily averages derived from the hourly archive; see vector field in Fig. .

For local validation of the reanalysis winds, we compare wind speed and wind stress to in situ observations obtained from the standard shipboard instruments during the CALYPSO 2022 cruise. Two ships participated in the experiment, the R/V Pourquois Pas? and the R/V Pelagia. Both were equipped with anemometers recording data at a nominal frequency of 1 Hz. These were mounted at 30 and 27 m, respectively. Resulting wind measurements were converted to 10 m winds using the power law from . Wind stress magnitude is computed from the measured wind speeds using the bulk aerodynamic formula assuming negligible surface ocean currents compared to the wind |τ|=CDρa|uwind|2, where CD=0.0015 is the drag coefficient, ρa=1.2 kg m⁻³ is an average air density, and uwind is the wind velocity at 10 m. Ship tracks and time series of wind speed and stress magnitude are shown in Fig. .

Context is provided by ancillary data collected during the cruise. The eddy was first identified for intense sampling in satellite ocean color imagery, as illustrated in Fig. . The phytoplankton chlorophyll concentrations shown are a product of the Copernicus Marine Environment Monitoring Service (CMEMS) that combines data from multiple satellites (SeaWIFS, MODIS, MERIS, VIIRS-SNPP, VIIRS-JPSS1, OLCI-S3A, and OLCI-S3B) . Temporal resolution is daily, and spatial resolution is 1 km.

Hydrographic information is obtained from EcoCTD profiles taken along multiple sections during the experiment . The EcoCTD includes sensors for conductivity, temperature, and density (RBR Concerto CTD), a chlorophyll fluorometer and backscatter meter (WETLabs BB2F), and an oxygen sensor (JFE Advantech Rinko), all mounted in a weighted aluminum housing and tow-yoed manually from the stern of the R/V Pourquois Pas? with an Oceanscience UCTD winch . Typical operation of the EcoCTD was to make one profile to 250 m every 5–6 min at a vessel speed of 6 knots. Its data was calibrated against the CTD rosette. While multiple sections were sampled that intersected with the eddy, here we focus on the single temperature and salinity section for each day that is most nearly colocated with the drifter observations in time and space, shown as red lines in Fig. .

A variety of methods exist for extracting KPs from drifter observations . Here we obtain area-averaged KP estimates from drifter triplets, following Method I from . Divergence δ, averaged over time and over the area A of a triangle defined by a drifter triplet, is computed from the change in that area over 15-minute intervals: 8δ=log⁡(A(t+Δt))-log⁡(A(t))Δt.

Vorticity ζ, normal strain rate N, and shear strain rate S are computed similarly, from rotated velocity components, following . The latter two quantities are combined in a total strain rate α=N2+S2. To eliminate those estimates that are subject to large errors due to the geometry of the drifter triplets, we filter out groups that do not satisfy the threshold Λ>0.2 on the scaled triangle aspect ratio Λ=123A/P2, where A is the area and P the perimeter of the triangle . Area-averaged KPs are known to be scale-dependent . So, only triplets with the distance between each pair of drifters (side length of the resulting triangle) between 1 and 10 km are used. To ensure independent sampling, triplets sharing more than one drifter with an already listed triplet are eliminated. The resulting estimates are associated with the corresponding triplet centers of mass, then binned in 2 km × 2 km boxes and averaged over bins and time for daily gridded maps.

These KP estimates are subject to errors due GPS position uncertainty, as well as due to representation errors arising since the advected triangles are generally not the triangles defined by their advected vertices and only partially addressed by the geometric subsampling . These errors are somewhat mitigated by the use of spatio-temporal means over each 2 km × 2 km box and each day.

GPS positioning uncertainties (order of 10 m) result in larger KP errors for small-scale triplets and/or small-time intervals due to their larger magnitude relative to the actual triangle area and/or area change. For triads of a scale of 1–16 km, these errors were found to impact 4 %–14 % of samples from an experiment in the Alboran Sea by , resulting in an error estimated to be <10 %. The dynamics here are expected to be comparable. However, the pre-processing of the drifter data (see Sect. ) is designed to mitigate position error, so that the anticipated errors are actually smaller than these theoretical results.

While choosing larger triplet scales minimizes the impact of position errors , the representation error grows with triplet size . The latter is of particular concern for triangles that are far from equilateral . By applying the threshold of Λ>0.2, we remove those estimates most likely to be effected by these errors. Model results from showed that with this threshold, more than 99 % of samples had an error less than 0.25f. Therefore, 0.25f could be considered as an upper bound for any individual estimate of δ. Since we are averaging both spatially (2 × 2 km bins) and temporally (24 h), the error associated with our results should be even less and a function of the number of samples in each spatio-temporal bin. Typically, 10 or more triplets were used for each individual bin in our results (see maps in Fig. S2), suggesting an uncertainty on our KP estimates of no more than 0.1f.

Drifter clusters can also be used to derive pointwise KP estimates by using least-squares fits . Pointwise-estimates tend to have larger extremes than averages. So, a direct one-to-one comparison with area-averaged estimates is not advisable. However, as a form of validation, we carried out a qualitative comparison between the triplet-based area-averaged and cluster-based pointwise estimates of vorticity from CaC drifters with spatial scales of 2–4 km. (This is a subset of the estimates entering the bins used in the rest of the paper). For the least-squares fits, clusters were formed consisting of 6 or more drifters, all at most 4 km apart from the centroid, with an aspect ratio equal or larger than 0.1, defined as the ratio of the smaller to larger eigenvalue of the position covariance matrix . To ensure independent sampling, redundant clusters, i.e. those with overlapping drifter membership of more than 75 %, were removed. Figure  shows the vorticity scaled by the Coriolis parameter f=9.4717×10-5 s⁻¹ derived from triplets (triangles) and from clusters (circles) for 23 February 2022. The results demonstrate consistency between the two very different methods, in line with method comparisons reported elsewhere (e.g. ).

When horizontally colocated divergence estimates are available at different depths, vertical velocities can be approximated from the continuity equation ux+vy+wz=0, where u, v, and w are the eastward, northward, and upward velocity components, respectively. It follows that 9w(-h)=w(href)+∫-hhrefux+vydz.

To obtain a vertical velocity estimate, both the vertical velocity at a reference depth href and a vertical profile of horizontal divergence are needed. The reference depth is typically either taken to be the surface, where vertical velocities approximately vanish , or a deep “depth of no motion” . Here, we follow and use the surface at z=0. In lieu of a continuous vertical profile of the horizontal divergence, we rely on a first-order approximation, interpolating linearly between the two known depths . Thus, if we denote the known horizontal divergences at depths -h0 and -h1 as δ0 and δ1, then 10w(-h)=2(δ0h1-δ1h0)h-(δ0-δ1)h22(h1-h0). For our sampling depths of h0=0 and h1=15 m, the vertical velocity at h=15 m is then given by 11w(-15)=15(δ0+δ1)2.

The uncertainty in the estimates for δ0 and δ1, in terms of median standard errors, are 0.08f and 0.07f, respectively. These values are typical for most bins, although for some of the bins near the edges of the computational domain, the standard errors reach 0.48f and 0.34f for δ0 and δ1, respectively (see Fig. S2). Propagating these standard errors to the w estimates yields a median standard error for w of 8 m d⁻¹.

In this section, we describe the evolution of the eddy day-by-day during the splitting period, 23–28 February, skipping 24 and 27 February when insufficient data are available for a meaningful analysis. KP values are superimposed on satellite chlorophyll isolines, when available, for context and are normalized by the local Coriolis parameter f, taken to be 9.57×10-5, computed for a latitude of 40.5° N. Subsurface drifters are more sparse and primarily add insight to the eddy splitting analysis on 23 February, when the co-location of the two drifter datasets allows direct computation of the vertical velocity w. While available, we therefore do not show SVP results for other dates. The selected ecoCTD sections provide a three-dimensional context.

Our investigation into possible wind effects considers contributions from nonlinear Ekman pumping wEknl (see Sect. ) and NIOs (see Sect. ), concentrating on the days of significant wind forcing, i.e., 23 and 25–27 February. As discussed in Sect. , even though the time scales are such that wEknl is not expected to be fully established, the tendency toward wEknl is expected to leave a signature in the observed patterns.

Recall that there are two components of nonlinear Ekman pumping (Eq. ): wζ and w∇ζ. For an isolated cyclone , wζ is expected to lead to downwelling in the core, with elongation in the direction of the wind. Here, the observations (Sect. ) clearly show the presence of a cyclonic eddy but are too sparse to delineate its complete spatial structure, let alone its surroundings. Consequently, we can only provide a qualitative description of the expected contribution of wζ. The observed pattern of isopycnal downwelling and increasing ellipticity observed during the eddy evolution (Sect. ) is consistent with a tendency toward downwelling and eddy elongation along the wind direction. Its relevance, though, depends on the value of the expected wζ. An order of magnitude estimate, using |uwind|≈ 7–10 m s⁻¹, |u|≈ 0.10–0.5 m s⁻¹, |ζ|≈ 1 – 2f yields a corresponding range for wζ of ≈ 1–1.5 m d⁻¹, which is more than an order of magnitude smaller than the w derived from the drifter divergence calculations (Fig. ) and implied by the observed isopycnal drop of more than 40 m on 26 February (Fig. ). We therefore conclude that, even though wζ may contribute to the eddy evolution and its vertical velocity, its contribution is secondary and cannot explain the observed evolution.

The second term w∇ζ depends on the vorticity gradient and on wind direction. As for wζ, we cannot provide a complete direct assessment of w∇ζ since our measurements are spatially limited. Some considerations and estimates, though, can be provided combining the information from the wind data during the two events and the observed vorticity ζ. As a first step, we again perform an order of magnitude analysis, assuming the same ranges for |uwind|≈ 7–10 and |u|≈ 0.10–0.5 m s⁻¹, and for the vorticity gradients ∇ζ≈Δζ/Δr≈f/(5–10) km, where r is the relevant spatial scale. The corresponding w∇ζ is estimated in the range 5–20 m d⁻¹, which is comparable to the order of magnitude of the measured w.

A more precise estimate of w∇ζ is possible where there is enough data coverage. The linear Ekman transport Mwind (Eq. ) is shown in Fig.  (top panels) for 23, 25, and 26 February, respectively, with the nonlinear transport |MEknl| (Eq. ) superimposed wherever vorticity ζ is available from drifters (Figs. , , ). As expected, |MEknl| is lower than the background in the cyclonic eddy (positive ζ), but higher in the anticyclonic areas (negative ζ).

A corresponding estimate of w∇ζ for 23 February (Fig. D) shows a prominent upwelling pattern, reaching values of w∇ζ≈ 20 m d⁻¹ in the transition area between positive and negative ζ (Fig. B). In this region, the transport |MEknl| increases significantly in the direction of propagation of transport (Fig. A). Conceptually, this corresponds to the upwelling on the side of the cyclone to the right of the wind (Fig. ). Some downwelling can be seen at the southern edge of the available data, where negative vorticity starts to decrease. The vertical velocity w computed from the drifters for the same day (Fig. E) shows a similar predominantly upwelling pattern in the same area. The surface divergence (Fig. D) is also consistently positive there, indicating upwelling.

A quantitative comparison between the estimated w∇ζ and the actual w from the drifters is shown in Fig. . In grid cells where both values are available, the ratio w/w∇ζ is computed (panel a). It is almost entirely positive, reflecting that both estimates consistently show upwelling in this area. The few values greater than 1 indicate that a downwelling tendency from another source is softening the wind effect. Figure B compares the histograms of the full vertical velocity and the Ekman pumping component. It shows that w reaches values of 50–60 m d⁻¹, while w∇ζ has a more peaked distribution, reaching only 20 m d⁻¹.

For 25 February, the background transport direction (northwestward, Fig. B) and the cyclone vorticity structure (Fig. B) characterized by a maximum in the core of the eddy lead us to expect upwelling to occur north of the maximum and downwelling south of the maximum, consistent with Fig. . Because of the reduced drifter coverage on this day that limits the gradient calculation and because of the orientation of the vorticity maximum, the computation of w∇ζ is limited to the downwelling side in the northwestern portion of the eddy and to the upwelling side in the southeastern portion (Fig. E), with maximum w∇ζ≈10 m d⁻¹. For this day, the drifter-based w estimates have very limited coverage, but the surface divergence δ (Fig. D) exhibits a pattern of positive and negative δ to the northwest and to the southeast of the maximum ζ, respectively, suggestive of upwelling and downwelling as in Fig. E.

Finally, for 26 February, the w∇ζ estimates (Fig. F) are mostly confined within the southernmost emerging eddy structure, since the elongated filament is too narrow to compute significant cross gradients. Due to the alternating negative and positive vorticity within the structure (Fig. B), and the transport direction (Fig. C), a pattern of positive and negative w∇ζ to the east and west, respectively, emerges (Fig. F). For this day, no w estimates are available from drifters for comparison, but the δ estimates (Fig. D) show a consistent pattern of positive and negative divergence in the same area. We note that the main convergence/downwelling area observed in the central structure between the two eddies, and consistent with the isopycnal drop (Figs. C to C), does not have an obvious signature in terms of wind effects and likely results from internal dynamics, as suggested by the subsurface analysis of .

In addition to nonlinear Ekman pumping effects, the wind is also likely to induce NIOs that can affect the divergence and w distributions. As discussed in Sect. , we are not in a position to provide a quantitative estimate of these effects since we have only a partial picture of the eddy vorticity structure, which is moreover rapidly evolving, as is the wind. On the other hand, some general observations should be considered. For instance, in the presence of a vorticity maximum, we can expect that the NIOs tend to create two bands of opposite δ on either sides of the maximum , with alternating sign in time at the inertial period (approximately 19 h). For 25 February, a maximum ζ is indeed recorded along the axis of the eddy (Fig. ), with bands of opposite δ on each side. It is therefore possible that NIOs contribute to the observed pattern, even though we cannot verify whether or not the sign and the time dependence recorded by the drifters are indeed consistent with NIOs fluctuations. Similarly, NIOs could also contribute to the observed patterns in the submesoscale eddies on 26 and 28 February (Figs. , ).