The drifters and profiling float used in the coastal SLS are described in detail in Poulain (2020). Only a summary is provided below. Most drifters were Coastal Ocean Dynamics Experiment (CODE; Davis, 1985), Consortium for Advanced Research on Transport of Hydrocarbon in the Environment (CARTHE; Novelli et al., 2017) and Palo Alto Research Centre (PARC; Waterston et al., 2019; Cocker et al., 2022) drifters using GlobalStar or Iridium satellite telemetry systems. Global Positioning System (GPS) positions were measured every 5 to 20 min. They measured surface currents within 1 m of the sea surface. The effects of wind and waves on the motion of CODE and CARTHE drifters are comparable (Poulain et al., 2022). The main error is a wind-induced slip of about 0.1 % of the wind speed (Poulain and Gerin, 2019). The wind- and wave-induced slip of the PARC drifters has not yet been studied. A total of 50 CODE, 20 CARTHE (Berta et al., 2021) and 30 PARC drifters were deployed.

Additional Lagrangian instruments included (1) the RIVER drifter, a CODE-like drifter equipped with a down-looking acoustic Doppler current profiler (ADCP) to measure relative current profiles between 2 and 20 m depth with a vertical resolution of 1 m; (2) the Surface Velocity Program (SVP) drifter (Niiler, 2001) with a drogue centered at 15 m nominal depth; (3) the directional wave spectra (DWS) drifter (Centurioni et al., 2017) to measure the directional statistical properties of the surface wave; and (4) the Arvor-C float (André et al., 2010) to measure temperature and salinity profiles with a pumped conductivity, temperature and depth (CTD) sensor between the surface and ∼ 120 m depth with 1 m vertical resolution. Five SVP, two RIVER and three DWS drifters were operated.

The GPS position data of the drifters were quality controlled and interpolated at 0.5 h intervals using a kriging technique (Menna et al., 2017, and references therein). Velocities were calculated by finite differencing the interpolated positions (central difference with hourly interval).

MODIS satellite images of the chlorophyll concentration of the study area were used to describe the spatial structure and temporal evolution of the surface circulation assuming that chlorophyll is a passive tracer advected by the surface horizontal currents. As previously shown in Poulain et al. (2020), chlorophyll concentration images were preferred over sea surface temperature images as they better represent circulation features. Since we are in a coastal area where a river drains nutrient-rich water, there is a sharp contrast between coastal and offshore water, with the former being richer (higher chlorophyll) and more turbid. Daily images have a horizontal resolution of 1 km.

Atmospheric data (wind speed and direction 10 m a.s.l.) and surface wave data (significant wave height, main wave period and direction, Stokes drift) of the fifth-generation European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis (ERA5) for the global climate and weather were downloaded from the Copernicus Climate Data Store for October 2020 in the SLS. They are provided with a horizontal resolution of 0.25∘ (wind) and 0.5∘ (waves). CMEMS reanalysis products at 1/24th degree (∼ 4 km) horizontal resolution were also downloaded. Simulated hourly mean currents at the sea surface were used.

In situ data were collected as part of the Drifter Demonstration and Research 2020 (DDR20) experiment (Poulain, 2020), which took place off the coast of Italy on 8–10 October 2020. DDR20 was a rapid environment assessment (REA) exercise whose general objective was the 3D characterization of the oceanographic and acoustic environment using a network of compact and low-cost, freely drifting instruments over a few days. A total of 110 drifters and 1 float were quickly deployed in a 6 × 6 km2 array in the coastal LSL (Fig. 1) using two ships between 08:09 and 12:28 UTC on 8 October 2020. The minimum distance between drifters at release was 0.5 km if drifters deployed at the same time and position are not considered (Poulain, 2020). One-third of these drifters and the float were successfully recovered after about 2 d, starting at 09:22 UTC on 10 October 2020.

The experiment took place after a storm with westerly winds and waves up to 15 m s-1 and 2.5 m, respectively, on 7 October (Fig. 2). During the 2 d of drifter operations mentioned above, calm meteorological conditions prevailed with winds less than 5 m s-1 and waves less than 0.5 m significant wave height. The surface Stokes drift estimated by ERA5 was as large as 20 cm s-1 on 7 October but decreased (to a few cm s-1) on subsequent days. Note that ERA5 underestimates the significant wave height by up to 0.5 m compared to the DWS drifter measurements (Fig. 2).

Unfortunately, 28 CODE drifters experienced transmission problems and did not transmit on 8 October between 14:00 and 22:00 UTC (∼ 8 h data gap) and between 9 October 10:00 UTC and 10 October 03:00 UTC (∼ 17 h data gap). Since the winds, waves and Stokes drift were relatively weak during the experiment, all CODE, CARTHE and PARC drifters were merged to investigate the kinematics and dispersion of the near-surface currents.

The relative dispersion of a drifter cluster can be quantified using both the MSSD as a function of time after deployment, D2(t), and the FSLE versus scale (Lacorata et al., 2001; Schroeder et al., 2011, 2012; Corrado et al., 2017; Boffetta et al., 2020). Unlike the MSSD, for which pair separation is averaged at a given time, the FSLE computes the averages of separation times at a given separation distance. Thus, it has the advantage of isolating different rates of dispersion due to velocity fluctuations at a given scale. The MSSD of drifter pairs is defined as 1D2(t)=<|x(1)(t)-x(2)(t)|2>, where the superscripts denote the two drifters of the pair that are located at vector position x(t) at time t, and the brackets denote the average over all pairs with the same initial spacing. The time derivative of the MSSD is referred to as the relative diffusivity. The FSLE, λ, is inversely proportional to the average time, < τ >, for two drifters initially separated by δo to reach a prescribed separation, δf : 2λδo,δf=1τln(δf/δo). Following Schroeder et al. (2011), we chose an amplification factor δf/δo= 1.2. The average time < τ > is often called the doubling time even though the amplification factor is not necessarily equal to 2.

The FSLE can be sensitive to the temporal resolution of the drifter positions, especially at a small separation distance for which the doubling time approaches the sampling time interval. Several methods have been proposed to reduce this problem (Boffetta et al., 2000; Lumpkin and Elipot, 2010; Haza et al., 2014). In this study, we have adopted two methods. For the first one, the drifter data were linearly interpolated using small time steps of 0.01 h before estimating the doubling times and averaging them. The second one was proposed by Boffetta et al. (2000). The following equation (see Eq. A4 in the Appendix of their paper) was used with the original drifter positions sampled at 0.5 h intervals: 3λδo,δf=1τ<ln(δf(τ)/δo)>. Relative dispersion by two-dimensional geophysical turbulence has the following dispersion regimes: exponential (D2 ∼ eαt, λ= constant), Richardson (D2 ∼ t3, λ ∼ δ-2/3), ballistic (D2 ∼ t2, λ ∼ δ-1) and diffusive (D2 ∼ t, λ ∼ δ-2) (Schroeder et al., 2012; Corrado et al., 2017).

To describe the small-scale surface circulation following the cluster of drifters, their motions with respect to the center of mass of the cluster were considered, and the DKPs of the surface currents were calculated. The DKPs of a flow describe how the surface water can decrease or increase in area, how it can rotate, and how it can be stretched or sheared (Okubo, 1970; Okubo and Ebbesmeyer, 1976; Molinari and Kirwan, 1975). They are defined by a first-order Taylor expansion of the velocity field: 4u=(δ+σn)/2x+(σs-ζ)/2y,5v=(σs+ζ)/2x+(δ-σn)/2y, with the following DKPs: divergence (δ=∂u/∂x+∂v/∂y), vorticity (ζ=∂v/∂x-∂u/∂y), shearing deformation rate (σs=∂v/∂x+∂u/∂y) and stretching deformation rate (σn=∂u/∂x-∂v/∂y), where u and v are the zonal and meridional velocity components; x and y are the zonal and meridional coordinates, respectively, in the system of reference moving with the center of mass of the cluster.

Other metrics like the strain (ρ= [σs2+σn2]1/2), Okubo–Weiss parameter (OW =ρ2-ζ2) and instantaneous rate of separation (IROS =δ+ρ) were also estimated. The OW measures the relative importance of strain and vorticity: elliptic regions (OW < 0) are dominated by rotation, whereas hyperbolic regions (OW > 0) are dominated by strain and deformation (Provenzale, 1999; D'Ovidio et al., 2009). The IROS is the zero-order Lagrangian rate of separation at the initial time (Schaeffer et al., 2017; Lorente et al., 2021) and is therefore related to the dispersion statistics defined above, particularly the initial exponential spreading.

There are two approaches to estimating the DKPs of horizontal currents. In the first method, small clusters of n drifters (with n > = 3) are used to solve Eqs. (4) and (5) using least squares (Molinari and Kirwan, 1975; Essink et al., 2019; Tarry et al., 2021). In the second method, the drifter velocities are interpolated on a uniform regular grid to directly calculate the horizontal derivatives of velocities and the DKPs (Lodise et al., 2020). In this work, we chose the second method and used the Data Interpolating Variational Analysis (DIVA, Troupin et al., 2012) to interpolate drifter velocities on a regular horizontal grid with a cell size of 0.1 km and zonal and meridional correlation scales of 1 km. This particular interpolation method was preferred because it provides a better estimate of the error field. In practice, interpolated values were not considered if the relative error exceeded 50 %. Gradients were estimated by central finite differences of the interpolated velocity field.

Uncertainties in the abovementioned statistics are due to the drifter position error, the drifter slippage and the finite number of samples. The drifter GPS positioning error can be approximated as white-noise variability with an isotropic standard deviation σx=σy ∼ 5 m (Rypina et al., 2021), uncorrelated from one drifter to another. Using a simple back-of-the-envelope calculation, the corresponding standard deviation of the squared separation distance is equal to 4σx2 ∼ 100 m2= 10-4 km2. Velocities estimated by finite central differencing the GPS positions interpolated or measured at dt= 0.5 h intervals have an error of σu= σv= 21/2σx/2dt ∼ 0.2 cm s-1. As mentioned above, under low-wind conditions (wind speed less than 5 m s-1), the drifter velocity error due to wind and waves is less than 0.5 cm s-1. Hence the instrumental error of the drifter velocity is roughly 1 cm s-1. The standard error of the statistics due to the finite numbers of observations can be estimated by the bootstrapping method. The 95 % confidence intervals of the MSSD and FSLE were estimated using the bootstrapping estimates of the means included in their definition (squared brackets in Eqs. 1 and 2).

Since the order of magnitude of the drifter velocities is 10 cm s-1 with an error of ∼ 1 cm s-1, we used a signal-to-noise ratio of 10 in the DIVA spatial interpolation. The estimation of the DKP errors estimated from the DIVA-interpolated maps is beyond the scope of this study.

The surface drifters were released near the southern edge of a filament of coastal water extending tens of kilometers offshore. Satellite imagery (Fig. 3) shows the development and morphology of the filament whose extremities form a mushroom-like feature with anticyclonic and cyclonic eddies expanding to the north and south, respectively. The bulk of the drifters ended up in the southern cyclonic eddy. After an initial mean southward-converging drift until 9 October 00:00 UTC, they turned eastward and then northward as they diverged (Fig. 4). There is only a very qualitative agreement between the drifter velocities and the surface currents simulated by CMEMS (Fig. 4). The modeled coastal currents are essentially southeastward. At the drifter locations, they can turn toward the southwest and west, with significant differences with respect to the drifter motion. Hence, the southeast motion of the drifter cluster is more or less simulated well, but there is no signature of a cyclonic circulation in the model. In particular, on 10 October at 12:00 UTC, several drifters moved to the north, on the eastern edge of the eddy, while collocated CMEMS currents remained southeastward.

If we focus on the initial motion of the drifters, numerous surface instruments (i.e., the CODE, CARTHE and PARC drifters deployed in the central and northeastern portions of the array) moved anticyclonically with the inertial (17.55 h) or diurnal (24 h) period (Fig. 5). In contrast, SVP drifters deployed at the same locations moved directly southward, with no anticyclonic rotation. Considering the short record duration, it is not easy to separate inertial and diurnal currents. It is doubted that diurnal tidal currents are dominant in the SLS (Poulain et al., 2018). Therefore, we can speculate that the anticyclonic rotation in the tracks is the remnant of near-inertial surface currents, which were likely generated by the storm a day earlier.

The vertical structure of thermohaline properties and currents in the area sampled by the drifters was measured by an Arvor-C profiling float and two RIVER drifters (see their initial tracks in Fig. 5). Significant shear of horizontal currents between the surface and 20 m depth was measured by the ADCP on the two RIVER drifters. The difference between the speeds at the surface and at 15 m can be up to 10 cm s-1 (Poulain, 2020) and is compatible with the different motions of the CODE and SVP drifters discussed above. Temperature and salinity values measured by the Arvor-C float near the drifters (not shown) indicate a well-mixed surface layer with a temperature of about 21 ∘C and a salinity between 37.94 and 38.00 PSU, extending down to a depth of about 40 m. At this depth, there is a sharp thermocline and a minimum salinity (Poulain, 2020). Large vertical and temporal variations of salinity associated with the offshore-flowing filament were not observed, probably because the float deployed in the center of the drifter array (Fig. 5) remained outside of the filament (see satellite image in Fig. 3).

A quick look at a velocity scatter diagram (not shown) reveals that individual drifter speeds vary in the range of 0–50 cm s-1. The center of mass of the drifter cluster moved southeastward, with speeds in the range of 10–20 cm s-1, and relative drifter speeds with respect to this motion had a standard deviation between 3 and 8 cm s-1.

All CODE, CARTHE and PARC drifters were considered together to search for pairs near the time of deployment (8 October, 12:00 UTC) with selected separation distances of 100, 500, 1000, 2500 and 5000 m. Since the data set is limited to 10 October at 07:00 UTC and since some drifters stopped transmitting before that time, the number of pairs may decrease with time. The initial (maximum) number of pairs and the minimum number of pairs (at 07:00 UTC on 10 October) are listed in Table 1. They vary between 8 and 76.

The surface MSSD versus time, starting at 12:00 UTC on 8 October, is shown in a log–log diagram in Fig. 6. Despite the relatively small number of pairs, the rate of change of the MSSD with time (also called relative diffusivity) during the first 10 h of drift appears to be significantly larger with the short initial spacing of 100 m than with the other larger initial distances. It can possibly be approximated by exponential growth. For longer times, the MSSD can be approximated by a power law, with the slope decreasing with increasing initial distance. However, comparison with theoretical dispersion regimes of geophysical turbulence is not straightforward. After ∼ 10 h, the MSSD values starting with separations of 100 and 500 m reach similar values near 0.8 km2. Between 10 and 20 h of drift, there is a local minimum in MSSD. This decrease in relative dispersion is related to the convergence of many drifters into the small cyclonic eddy (see top panels of Fig. 4). After about a day, the MSSDs with initial separations of 100, 500 and 1000 m are all near 4 km2.

The FSLE for initial pair spacing between 100 m and 7 km was estimated in a similar manner, i.e., for pairs starting on 8 October at 12:00 UTC and tracked until 10 October at 07:00 UTC. The FSLE was calculated for scales divided into non-overlapping 100 m intervals using Eq. (2). The results are shown in Fig. 7 in a log–log plot, along with the number of pairs in the scale intervals. Because pairs are tracked over a limited period of 43 h, some of them do not have time to separate by the prescribed amplification factor (1.2) and do not contribute to the estimate of the average separation time, < τ >, of Eq. (1). The number of pairs whose separation increases by 120 % in less than 43 h is also shown in Fig. 7. It is generally smaller than the initial number of pairs considered, especially for large scales. The doubling time increases if the separation distance varies from 0.7 h (for small scales) to 27 h (for large scales). In general, the FSLE decreases with scale. At small scales (100–200 m), it is large (∼ 6 d-1) and fairly constant. As scales increase (200–400 m), there is a strong negative slope. At larger separation distances (1–6 km), the FSLE decreases weakly with scale, with values near 0.3–0.5 d-1. Again, comparison with theoretical relative dispersion slopes is not obvious. Note that the FSLE estimated using Eq. (3) (Boffetta et al. (2000) method, red curve in Fig. 7) does not differ significantly from that obtained with Eq. (2).

For several reasons, the confidence intervals for MSSD and FSLE displayed in Figs. 6 and 9 can be quite large. First, the number of pairs is small for small separation. Second, the distributions of squared separation distances and separation times are generally not Gaussian, and their mean values may be meaningless. Third, the drift period of 43 h is short, and the FSLE may be overestimated because a substantial fraction of pairs does not reach the 120 % amplification factor during the limited tracking period (see bottom panel of Fig. 7), and long doubling times are not considered. Nonetheless, our relative dispersion results provide some useful information, as discussed later.

We now examine the DKP maps at selected times to characterize the flow within the cluster and to monitor the shape (extent and deformation) of the area covered by the drifters. Figures 8 to 15 show the interpolated currents and the DKPs at 6 h intervals between 12:00 UTC on 8 October and 06:00 UTC on 10 October, excluding the values with more than 50 % relative error. The order of magnitude of the DKPs is equal to the local inertial frequency.

On 8 October at 12:00 UTC, almost all the drifters were deployed, but the planned square geometry of the deployment was already modified due to the prevailing southward motion of the drifters released earlier and more to the north. As a result, the initial cluster (Fig. 8) became quasi-rectangular with sides of ∼ 6 and ∼ 8 km. Inside the sampled area, the DKPs were rather patchy. Approximately 6 h after the last deployments (at 18:00 UTC, Fig. 9), the cluster had mainly expanded in the meridional direction, mainly due to the drifters near the southern edge moving rapidly southward and showing significant convergence and strain. The size of the cluster reached 10 km, in both zonal and meridional directions.

After 6 h (Fig. 10), the northern portion of the cluster had further extended zonally, and its southern edge formed a thin branch extending southward and turning cyclonically. The divergence was generally weak. Positive vorticity prevailed east of the southward-flowing limb, while strain dominated on the opposite west side. The size of the cluster increased to ∼ 12 km in the meridional direction.

By the morning of the next day (9 October at 06:00 UTC; Fig. 11), the southern limb had extended further in a cyclonic eddy. The divergence was patchy. A large positive vorticity exceeding f occurred in the inner part of the eddy. Outside the eddy, a hint of negative vorticity was evident. Strain was significant, especially just outside the eddy. The cluster had reached a typical size of 15 km.

On 9 October at 12:00 and 18:00 UTC (Figs. 12 and 13) and on 10 October at 00:00 UTC (Fig. 14), the cluster size reached a saturation value near 17 km. Some drifters moved northward and nearly closed the loop of the cyclonic eddy in its northern sector. There was still a strong signature of positive vorticity in the eddy core and significant dispersion (strain and IROS) near its external edge.

On 10 October at 06:00 UTC, a few hours before the recovery operations (Fig. 15), a few drifters completed a full cyclonic loop in the eddy, now sampled more uniformly in all sectors.

As expected, the OW in the maps of Figs. 11–15 is always negative in the eddy core, corresponding to elliptic flow, while outside the eddy, the flow is hyperbolic and dominated by strain and deformation (positive OW). The horizontal divergence is essentially zero near the center of the eddy.