## Abstract

We examine the relative dispersion of surface drifters deployed in groups of triplets at the boundaries of a filament in the upwelling region off Namibia for both the entire ensemble and the two main subgroups. For the drifters in the group released at the northern boundary of the filament, close to the upwelling front, we find that the mean-square pair separation 〈*s*^{2}(*t*)〉 shows the characteristic distinct dispersion regimes [nonlocal, local (Richardson), and diffusive] of an ocean surface mixed layer. We confirm the different dispersion regimes by a rescaled presentation of the moments 〈*s*^{n}(*t*)〉 and thereby also explain the anomalous slow decay of the kurtosis in the transient regime. For the drifter group released at the southern boundary, 〈*s*^{2}(*t*)〉 remains constant for a short period, followed by a steep “Richardson like” increase and an asymptotic diffusive increase. In contrast to the northern release, the corresponding moments reveal a narrow distribution of pair separations for all regimes. The analysis of finite-size Lyapunov exponents (FSLEs) reveals consistent results when applied to the two releases separately. When applied to the *entire* drifter ensemble, the two measures yield inconsistent results. We relate the breakdown of consistency to the impact of the different dynamics on the respective averages: whereas, because of separation in scale, 〈*s*^{2}(*t*)〉 is dominated by the northern release, the decay of the FSLEs for small distances reflects the drifter dynamics within the filament.

## 1. Introduction

The ocean is a turbulent system, where intensive nonlinear interactions give rise to finescale structures like eddies and filaments. These features are of critical importance for the ocean circulation and for the transport and mixing of properties within the ocean. Examples like oil spills, the distribution of fish larvae, or search and rescue operations show that the understanding of dispersion properties in the ocean is not only an important scientific task but is also of public interest.

In recent years, several observational studies explored the dynamics of the ocean surface layer by deploying surface drifters and tracking them by satellite. Whereas the analysis of single-trajectory (Lagrangian) statistics derived from these observations yields information about the advective transport that results from the largest and most energetic scales, the analysis of *pair*-separation statistics of drifters can extract information about the dominant physical mechanisms acting at the different scales of motion (e.g., chaotic advection, turbulence, and diffusion) in specific regions.

Experiments and numerical studies on Lagrangian pair dispersion have been carried out, for instance, in the North Atlantic Ocean (Lumpkin and Elipot 2010), the Nordic seas (Koszalka et al. 2009), the Gulf of Mexico (Beron-Vera and LaCasce 2016; Özgökmen et al. 2012; Poje et al. 2014; Sanson et al. 2017), the southwestern Atlantic Ocean (Berti et al. 2011), the European Mediterranean Sea (Schroeder et al. 2012), and the Southern Ocean (van Sebille et al. 2015), among many others [see LaCasce (2008) for a review].

In this article, we analyze the pair separations of surface drifters in one of the major eastern boundary upwelling systems of the World Ocean, the Benguela upwelling region (Fig. 1). Although chaotic advection has been studied in this region (Bettencourt et al. 2012), to our knowledge, no experiments using paired drifters have been performed in the area so far.

In the Benguela region, offshore Ekman transport causes thermocline waters to rise to the surface, where they supply nutrients for biological activity. Persistent alongshore winds and cyclonic wind stress curl occur throughout the year off Lüderitz, Namibia, thus supporting a rich ecosystem of high economic impact. Here, the upwelling region is characterized by temperature anomalies of up to −6°C (Shannon and Nelson 1996). Variability in upwelling strength and water mass properties is induced by both the wind field and the supply of central waters, which are either of northern origin or originate from the Cape Basin farther south (Mohrholz et al. 2014). Variations occur on seasonal scales but also as intermittent episodes, caused by local forcing. Disturbances of the upwelling front may grow into eddies or filaments, which connect the coastal upwelling system to the subtropical gyre. Filaments appear as cold-water intrusions, spreading offshore from the upwelling region, and interact with mesoscale eddies, for example, shed from the Agulhas region (Capet et al. 2008).

Filaments have been reported to effectively transport properties across the temperature front, thus providing a source of nutrients and temperature anomalies for the eastern boundary system (Brink 1983; Lutjeharms et al. 1991; Muller et al. 2013). Typical filaments cover horizontal scales from 5 to 80 km in meridional extent and from 20 to >500 km in offshore extent (e.g., Kostianoy and Zatsepin 1996). Associated temperature anomalies in filaments are up to −2.5°C and are transported offshore in westerly jets of about 30 cm s^{−1} (Hösen et al. 2016). Although the features themselves cover only the mesoscale and submesoscale, their combined effect impacts the surface temperature distribution of the Atlantic Ocean and thus reflects onto the large scale. A complete understanding of filament dynamics may thus lead to accurate parameterizations of mesoscale and submesocale processes in the area and, for example, ultimately reduce the temperature bias in climate models, which cannot be eliminated by increased resolution alone (Harlaß et al. 2015).

In our analysis, we follow two distinct approaches. First, we study the mean-square separation 〈*s*^{2}(*t*)〉 for all drifter pairs that were deployed with a small initial separation in order to identify the different dispersion regimes. These dispersion regimes serve as a reference to the underlying physical mechanisms acting on different time scales and thereby characterize the Benguela region. By subsampling the data as well as by a chance-pair analysis, we account for pairs with larger initial separations. We analyze (qualitatively) the probability density functions (PDFs) and compare the corresponding kurtosis with the theoretical forecasts. Hereby, we aim to get insight into the dispersion regimes characteristic for the region.

Second, in order to get insight into the dynamics beyond the statistical average, we explore the influence of different deployment locations characteristic to the Benguela region. To this end, we analyze the dispersion of the two main subgroups of drifters deployed within and outside the environment of a filament, where the dispersion is influenced by various mesoscale and submesoscale structures. We find that, because of the underlying mesoscale system, the dispersion statistics between these subgroups vary strongly from each other, depending on the location of release. While the drifters of the group released at the southern boundary of the filament separate more slowly, the drifters in the group released closer to the cold upwelling front at the northern boundary of the filament separate faster from each other and follow distinct paths within the complex surface currents.

Supplementary to our fixed-time analysis of pair separations, we calculate the finite-size Lyapunov exponents (FSLE) *λ*(*δ*). This fixed-scale analysis of first-passage times is performed for both the subgroups and the entire drifter ensemble. For all cases, we examine quantitatively the consistency between the fixed-time and fixed-scale analyses.

## 2. Data

The data used in this study come from 37 surface drifters, which were deployed in the eastern South Atlantic in November–December 2016 by R/V *Meteor* during cruise M132 from Walvis Bay, Namibia, to Cape Town, South Africa. The trajectories of the drifters from their time of deployment until April of 2017 are shown in Fig. 1. The drifters were of type SVP-I-XDGS from MetOcean Telematics, Inc., mounted in a 16-in. (~41 cm) hull and featuring iridium telemetry, a drogue centered at 15 m, and a sea surface temperature (SST) sensor. Data transmission was set to 30-min intervals.

Drifters were found to deliver data as planned, but one device failed to obtain position fixes after 15 December, possibly because of technical defects or extraneous causes. Data from this drifter were excluded from our analysis. The data of another device, which showed a data gap of 11 days in spring 2017, were included in our analysis up to the time gap (i.e., for 93 days). All other drifters provided data continuously in high temporal resolution. On 6 April 2017, the operation of the devices was transferred to the Global Drifter Program.

The drifter deployment strategy was developed to cover small spatial scales, thus resolving the small-scale dispersion regime. While crossing a temperature front, deployment points were chosen in distances of 5 km (Fig. 2). Each point was occupied by three devices forming a triplet in a triangular shape. Ideally, two sides of this triangle would have had a length of 200 m and one side of 100 m, representing the initial pair distances of the three pairs in one triplet. As, on average, 15 min passed between deployment and first registration of position and, furthermore, weather conditions during the deployments sometimes prevented an accurate positioning, the measured initial separations of the drifters were about 100–600 m. In total, four deployment locations were occupied, at which four, four, two, and two triplets were released, respectively. The last deployment included one additional drifter. The first and third deployments were realized at the southern boundary of an upwelling filament (at about 26°40′S, 12°3′E), and the second and fourth deployments were placed at the northern boundary of the same filament, close to the upwelling front (near 26°3′S, 12°58′E). The dropout of this drifter reduced the number of pairs, which can be generated by the combination of drifters from the same triplet deployment, to 37 pairs.

Figure 3 shows snapshots of SST and absolute dynamic topography (ADT) for the working region during the cruise and a zoomed-in view of the trajectories for deployments 1 and 2 for the first 20 days. An upwelling filament can be identified at about 26.5°S, spreading from the coastal cold-water pool westward into the South Atlantic. The filament appears to be trapped between eddies indicated in the ADT field, which exhibits local maxima at 24.6°S/10.3°E and 26.8°S/9.9°E and a local minimum at 25.5°S/12.0°E. The filament was persistent during the cruise and determined the starting positions of the drifters. The two main drifter groups were deployed with a time lag of 1 day at the southern boundary of the filament (blue in Fig. 3b) and at the northern boundary, close to the upwelling front (red in Fig. 3b), respectively. In section 4a we will present the dispersion analysis for the entire drifter group (Fig. 1), and in section 4b we will discuss the dispersion of the two deployments shown in Fig. 3 separately.

## 3. Methods

### a. Mean-square pair separation: Different dispersion regimes

The mean-square pair separation, also referred to as relative dispersion, is defined by

Here, *N*_{p} is the number of drifter pairs and the sum is over all pairs of drifters *i* and *j*, with *x*_{i,j}(*t*) and *y*_{i,j}(*t*) being their coordinates at time *t*.

The study of relative dispersion traces back to Richardson’s seminal work on pair dispersion in turbulent diffusion in the atmosphere (Richardson 1926), which is based on his observation that a pair of particles, within certain scales of separations, moves in a correlated way because of the common background velocity field. This is demonstrated in Fig. 4, which shows the trajectories of a drifter triplet released off Namibia’s coast. Although the three drifters stay together for three days following the same underlying flow structure, one drifter abruptly separates and moves far from the others. After six days, the remaining drifter pair separates but in a less abrupt way. Apparently, in the initial phase, the motions of the drifters are strongly correlated, but eventually they follow uncorrelated paths. In contrast to Brownian motion, the pair separation is not a progressive process but is formed of quiet periods and sudden bursts. According to Richardson’s empirical finding, for turbulent flows, the mean-square separation 〈*s*^{2}(*t*)〉 grows cubically in time. Later, Obukhov and Batchelor (1950) derived the *t*^{3}-dispersion law by applying scaling arguments of Kolmogorov (1941) and obtained

for the inertial range of times from *t*_{1} to *t*_{2}. Here, *ε* is the averaged energy transfer rate. The dimensionless factor *C* is the Obukhov–Richardson constant, which is determined by the specific form of the corresponding PDF *P*(*s*|*t*) of separations *s* for fixed time *t* and, therefore, in principle can serve as an indicator for the underlying PDF. By measuring the energy transfer rate *ε* independently in their two-dimensional turbulent laboratory experiment, Jullien et al. (1999) obtained an approximate estimate for *C* of 0.5. In contrast, Boffetta and Sokolov (2002), using direct numerical simulations of two-dimensional inverse cascade turbulence, found the Obukhov–Richardson constant to be C ≈ 3.8. Equation (2) describes the so-called local regime, because it rules the temporal evolution of the relative separation *s* when it is of similar size as the mesoscale eddies in the system.

For shorter times (*t* < *t*_{1}), the pair separations are smaller than the eddy sizes and one can assume that the velocity field between nearby drifters varies smoothly. In this so-called nonlocal regime, the separations grow exponentially in time:

where the unfolding-time *τ* is related to the strain rate (Badin et al. 2011). Nonlocal dispersion [Eq. (3)] is characterized by a sensitivity to initial conditions and leads to chaotic advection (e.g., Mukiibi et al. 2016). With increasing relative separation *s*, more and more pairs “break up” (see Fig. 4) and the drifters separate from each other in a highly irregular (intermittent) way that is characterized by sudden large relative separation jumps, which is the hallmark of the local regime [Eq. (2)].

For asymptotic large times (*t* ≫ *t*_{2}), the pair separations become larger than the largest eddies and the particles are moved independently from each other by different flow structures. In this asymptotic *diffusive* regime, the motions are uncorrelated, and the mean-square separation 〈*s*^{2}(*t*)〉 grows linearly in time (as for Brownian motion):

The relative diffusivity *κ* in Eq. (4) is expected to be 2 times the single-particle diffusivity.

To what extent the distinct relative dispersion regimes of Eqs. (2)–(4) can be identified experimentally in particular depends on the initial pair separation *s*_{0} with respect to the forcing injection scale *r*_{I}. In the ocean, if the turbulence is controlled by baroclinic instabilities, *r*_{I} is the Rossby radius of deformation, which determines the scale at which energy is injected to mesoscale eddies by these instabilities.

### b. Probability density function and moments

In the local regime, which is described by Eq. (2), the highly fluctuating nature of the underlying (inertial) dynamics results in an anomalous distribution *P*(*s*|*t*) of pair separations *s* at fixed time *t*. According to Richardson (1926), the distribution is a stretched Gaussian:

with *β* = 2/3 and *b* ~ *ε*^{−1/3}. Note that in Eq. (5) we wrote the PDF in terms of the scalar distance *s* instead of the directed variable **s**, which leads to the normalization condition . In contrast to Richardson, Batchelor (1950) proposed a Gaussian distribution (*β* = 2), while Kraichnan (1966) suggested a behavior like Richardson for small separations but with decaying tails as *β* = 2. The distribution *P*(*s*|*t*) therefore plays a central role in the theory of the process. In laboratory experiments (Jullien et al. 1999), the distribution *P*(*s*|*t*) has been measured by computing virtual trajectories from the integration of 2D velocity fields, which in turn have been determined by applying image velocimetry techniques to clusters of particles. Because of the large number of up to 6 × 10^{5} trajectories, the functional form in Eq. (5) could be confirmed while the theoretical functional forms of Batchelor and Kraichnan could be ruled out by the experiment. The analysis of *P*(*s*|*t*) focused on the asymptotic regime *s* ≫ 〈*s*^{2}(*t*)〉^{1/2}. The stated fit yields a reduced total probability of 0.36 [see also Beron-Vera and LaCasce (2016)], which points to additional contributions to the PDF in the regime of small separations *s*.

Although various Lagrangian drifter programs released a large number of drifters, the accuracy of the laboratory experiments is not yet available in ocean science, and the derivation of the analytic form of the PDF remains a challenge. One therefore often determines indirect measures, for example, the kurtosis

where the moments are given by . From the PDF in Eq. (5), the kurtosis can be derived directly for the local regime. Since the Richardson PDF is self-similar (in contrast to the PDF of the nonlocal regime), the kurtosis does not depend on time but yields the constant value ku ≈ 5.6.

The PDF in the diffusive regime is given by the Rayleigh distribution

which is also a self-similar function but yields a much smaller value of ku = 2.

We stress that, to receive a time-independent kurtosis for a certain time interval, according to Eq. (6) it is necessary that for the considered time span both moments 〈*s*^{4}(*t*)〉 and 〈*s*^{2}(*t*)〉 lie in the same (local or diffusive respectively) dispersion regime.

This can be proved by deriving the moments from the theoretical forecast for *P*(*s*|*t*) for the local regime [Eq. (5)] and the diffusive regime [Eq. (7)], which leads to

Here,

where Γ(*x*) is the gamma function and with *b* defined in Eq. (5). Because the transition time *t*_{x}(*n*) has to fulfill

it is given by

with *γ*_{n} = *k*_{n}/*i*_{n}. Equations (8) and (9) indicate that the transition time *t*_{x}(*n*) from the local regime to the diffusive dispersion regime differs for the different moments. The underlying reason of this property is that, for fixed time, higher moments focus on the tails of the PDF while lower moments focus on comparatively smaller separations. Accordingly, the higher moments are subject to the influence of the diffusive regime at shorter times than lower moments. In particular, according to Eqs. (8) and (9) for times *t* between the two crossover times *t*_{x}(4) and *t*_{x}(2), the two moments 〈*s*^{4}(*t*)〉 and 〈*s*^{2}(*t*)〉 increase differently in time and therefore the kurtosis in Eq. (6) should depend on time.

### c. Finite-size Lyapunov exponents

An alternative approach, which enables a systematic investigation of the effects of different scales of motion, consists in the fixed-scale analysis of first-passage times by calculating the FSLEs *λ*(*δ*), introduced by Artale et al. (1997):

Here, 〈*τ*(*δ*)〉 is the averaged (first passage) time for two particles, initially separated by distance *δ*, to reach a separation *αδ* with *α* > 0. The different dispersion regimes show up in the FSLE by the exponent *β* in the scaling of *λ*(*δ*) ~ *δ*^{β}: in the nonlocal regime, *λ*(*δ*) is constant (i.e., *β* = 0); in the local (Richardson) regime, *β* = −2/3; and in the diffusive regime, *β* = −2 (Artale et al. 1997).

In contrast to the relative separation analysis, where, for each fixed time, averages over a large scale of pair separations were considered, the FSLEs focus on the distinct scales. Therefore, we expect that the FSLE can serve as a robust measure, especially for the intermediate (local) regime, which for the fixed-time analysis is affected by the two neighboring regimes (nonlocal and diffusive).

## 4. Results

### a. Mean-square pair separation and PDF for the entire dataset

The root-mean-square (RMS) pair separation for the entire group of drifters (Fig. 1) with mean initial separation *s*_{0} ≈ 300 m calculated from Eq. (1) is plotted versus time *t* in Fig. 5a as a blue line.

Because *s*_{0} is much smaller than the ambient flow structures, the initial part of the dispersion curve for times smaller than *t*_{1} ≈ 5 days (where, on average, separations up to *s*_{1} ≈ 20 km are reached) is described very well by an exponential function and therefore supports the nonlocal regime in Eq. (3). This becomes especially evident by plotting the data half-logarithmically (Fig. 5b), where the data follow a straight line, from which we derived an unfolding time of *τ* ≈ 0.6 ± 0.1 days.

For times larger than 5 days, the data are consistent with the theoretical forecast for the local (Richardson) regime [Eq. (2)] indicated in the double-logarithmic plot by the dashed black line with slope 3/2, followed by a diffusive regime [Eq. (4)] (solid black line with slope 1/2). In the inset of Fig. 5b, the compensated evolution 〈*s*^{2}(*t*)〉/*t* according to (4) is shown, where diffusive transport would show up as a plateau. The height of the plateau corresponds to the same value for 2*κ* as is inferred for Fig. 5a (solid black line).

To get a more clear picture of the dispersion regimes for intermediate and larger times, we first focus on the asymptotic (diffusive) regime and compute dispersion curves by analyzing drifter pairs with larger initial separation in order to achieve better statistics. To this end, we apply two different methods. Since the drifter triplets were deployed in 5-km distances from each other, we first account for all combinations of drifters *between* neighboring drifter triplets. Thus, we receive 67 drifter pairs with an average initial separation of (light-blue lines in Fig. 5). Second, we analyze so-called chance pairs; that is, we average over all drifter pairs that at any time are in the prescribed distance of 6 km from each other. We take the instant time of first passage of the prescribed distance as the starting time for the considered pair. The average therefore contains drifter pairs, which start within the same triplet and are taken into account as soon as they are separated by a distance of 6 km. It also contains pairs from triplets deployed in large distances that occasionally approach each other. The corresponding relative dispersion curve, where the averages have been taken over 124 pairs, is represented by the pink lines in Fig. 5.

In contrast to the dispersion curve with smaller initial separation (see inlay of Fig. 5b), the two dispersion curves with show a clear transition toward a diffusive behavior [Eq. (4)], which for both procedures is eventually reached after roughly 80 days. Whereas the averaging for drifter pairs of neighboring triplets yields a relative diffusivity of *κ* ≈ 3.7 × 10^{4} m^{2} s^{−1}, the chance-pair analysis yields the slightly smaller value of *κ* ≈ 3.2 × 10^{4} m^{2} s^{−1}. The relative diffusivity of the original drifter pairs (blue line) seems to be smaller, but its derivation is less reliable because stronger fluctuations meant that a clear plateau could not be achieved here.

Because is relatively close to the pair separations *s*_{1} reached at the transition time *t*_{1} from the nonlocal regime to the local dispersion regime (see Fig. 5a), the nonlocal regime cannot be resolved within this data record. Furthermore, the intermediate (inertial) regime is strongly affected by the large initial separation and as a result shows a flatter increase, as expected for the Richardson regime.

Figure 5a also shows the corresponding RMS pair separations from the 37 individual pairs color coded according to their deployments (see Fig. 1). The double-logarithmic plot shows that the averaged RMS is controlled by a few extreme separations while most of the other separations lay below the average.

To explore the local dispersion for intermediate separations between the nonlocal and the diffusive regimes in more detail, in the following, we study other measures, which go beyond the averaged mean-square separation and can serve as indicators for the Richardson dynamics. We start by analyzing the fluctuations of the pair separation by calculating histograms of the PDF of pair separation for different times.

In Fig. 6, histograms of pair separations for different times derived from the field experiment are presented. To compare the fluctuations of the pair separations as an indicator for the different dispersion regimes, the histograms are plotted in a rescaled way: that is, versus *s*/〈*s*(*t*)〉.

The histograms illustrate how the PDF changes with increasing time. Figure 6 shows that for time *t* = 7 days, which according to Fig. 5 lies at the beginning of the local regime, the PDF exhibits extreme events as some separations attain values up to nearly 10 × 〈*s*(*t*)〉 while most of the separations are smaller than 〈*s*(*t*)〉. This is consistent with the variable Richardson dynamic with extreme large “sudden” jumps. With increasing time, the (rescaled) separations move closer together. Eventually, for *t* = 70 days, which corresponds roughly to the time scale at which the diffusive regime begins (see inlay of Fig. 5b), the PDF tends to cluster around its mean.

The qualitative differences of the PDF for different times shown in Fig. 6 are reflected quantitatively in Fig. 7, where the kurtosis in Eq. (6) is plotted versus time. After a smooth initial increase, the kurtosis strongly fluctuates and shows enhanced values up to 15. This value is much larger than the theoretical value ku ≈ 5.6 for the local regime and therefore seems to contradict the findings from the dispersion curve (Fig. 5a). As shown in the next section, the enhanced kurtosis reflects a situation in which different dynamics are mixed. This leads to an artificially broad (composite) distribution, in the sense that the PDF actually consists of two different distributions. Note that already after approximately 9 days the kurtosis starts to decrease slowly toward the theoretical forecast of ku = 2 for the diffusive regime.

### b. Different deployment scenarios

Of particular interest for our experiment were filaments, which often occur at the upwelling front off Namibia. In the context of Lagrangian dispersion, the question arises to what extent the filaments influence the averaged dispersion curve (Fig. 5). More precisely, the question is whether the location of deployment influences the dispersion of drifter groups only for short times or possibly has a long-term influence. In the latter case, the contributions of the different drifter groups to the average dispersion and the different dispersion regimes are analyzed in a more detailed way in order to derive the unfolding-time *τ*, the energy transfer rate *ε*, and the diffusivity *κ*.

The comparison of the northern and southern releases (red and blue trajectories in Fig. 3) clearly shows that the location of deployment indeed can have a long-term influence. The southern drifter group follows the filament structure westward and shows a tight clustering. The pathways correspond roughly to the contour lines of ADT [for deeper insight into the evolution of these trajectories “trapped” within a filament, see Bettencourt et al. (2017)]. In contrast, drifters of the northern group separate rapidly and follow distinct paths within the complex surface currents. Here, the ADT field shows only weak gradients. Trajectories from the drifter releases 3 and 4 (Fig. 1), which were realized 2 weeks later, are mixed cases, because some of the drifters remained in the filament or were trapped later in the filament at different locations while others show a similar spreading as the first northern release. In the following, we therefore focus on the two releases depicted in Fig. 3, which can be considered as a typical inertial release for the northern position and a typical filament dispersion for the southern position.

#### 1) Northern release

We first analyze the dispersion behavior of the northern release, which, from a first glance at Fig. 3 seems to show the typical inertial (local) behavior. Figure 8 shows the RMS pair separation plotted against time. To demonstrate the different form of the PDF for different times of the northern release, we calculate the moments. The ratio between the rescaled moments 〈*s*^{n}(*t*)〉^{1/n} of *P*(*s*|*t*) expresses the broadness of the PDF, and in the double-logarithmic plot this is seen in the gaps between the lines. The diversification in the intermediate time range clearly points to an anomalous broad PDF as typical for the local (Richardson) regime, in contrast to a Rayleigh distribution [Eq. (7)] for larger times.

Figure 8 also shows that, as predicted by Eq. (9), for example, the transition from local to diffusive regime for 〈*s*^{2}(*t*)〉^{1/2} takes place at earlier times than for 〈*s*(*t*)〉. In Fig. 8, where the dashed black line shows a fit to 〈*s*^{2}(*t*)〉^{1/2} and the dashed red line a fit to 〈*s*(*t*)〉, the intersection between these dashed lines and the solid black line lies at shorter times for 〈*s*^{2}(*t*)〉^{1/2}. By means of Eqs. (8) and (9), the theoretical forecast (and thereby the Richardson PDF) can be tested by plotting the data in a rescaled way. To this end, for each curve of Fig. 8a, we first divide the time by the corresponding *n*-dependent factor *γ*_{n} in the crossover time *t*_{x}(*n*) and thereby shift the curves such that the transitions occur at the same *x* coordinate. From setting *t* = *t*_{x}(*n*) into Eq. (8), it follows that, if we further divide 〈*s*^{n}(*t*)〉^{1/n} by , we also shift the data along the *x* axis such that the data should collapse and lie on two straight lines that represent the theoretical predictions given by Eqs. (2) and (4). In the rescaled plot of the moments shown in Fig. 8a (inlay), the curves indeed follow the two lines with slope 3/2 and 1/2 and thereby support the Richardson regime and the diffusive regime.

Equations (8) and (9) also enable us to analyze the temporal evolution of the kurtosis in Eq. (6), which is shown in Fig. 9. The kurtosis shows an enhanced value for times between 1 and 10 days, which is slightly lower than the value 5.6 as predicted for the local regime, and then it decreases slowly toward the theoretical value of 2 for the diffusive regime. As can be seen also in other studies (Koszalka et al. 2009; Beron-Vera and LaCasce 2016; Sanson et al. 2017), the transition regime of ku is relatively large and on the linear scale constitutes the major part. The time dependence of the kurtosis could cast doubts on the self-similarity of the PDF in the local regime. Within our analysis, however, we show that the slow decay of the kurtosis is a generic feature for the transition between the two self-similar PDFs and originates from the *n*-dependent transition time *t*_{x}(*n*) in Eq. (9). To this end, in Fig. 9 we added the two crossover times for the fourth and the second moments. Whereas according to Eq. (8) for *t* < *t*_{x}(4) both moments lie in the local regime and for *t* > *t*_{x}(2) both are in the diffusive regime, in the time-interval *t*_{x}(4) < *t* < *t*_{x}(2) the fourth moment is already in the diffusive regime while the second moment is still in the local regime. As the figure shows, this theoretical forecast is confirmed very well by the measured kurtosis, as the two crossover times *t*_{x}(4) and *t*_{x}(2) indeed mark the time-interval [*t*_{x}(4), *t*_{x}(2)] in which the kurtosis slowly decays.

#### 2) Southern release

Figure 10 shows the RMS pair separation and the additional rescaled moments plotted against time for the drifter group released at the southern boundary (see blue trajectories in Fig. 3b). The difference to the northern release (Fig. 8) is striking. For nearly all times between deployment until the last measurement the moments stay close together, indicating a narrow PDF. As Fig. 10 shows, the separations between the drifter pairs remain constant for the first three days. The corresponding kurtosis (Fig. 11) again points to a very narrow PDF because its low value ku ≈ 1.5 is even smaller than for a Rayleigh distribution (ku = 2). Also, for about the following 10 days the kurtosis sticks to the same low value, although the corresponding 〈*s*^{2}(*t*)〉 grows rapidly and may be consistent with either a *t*^{3} law or an exponential ramp. This period is too short to set an agreement with any specific functional form. The underlying dynamics cannot be turbulence, as the small value of the kurtosis indicates. Possibly in this regime the dispersion is driven by shear forces because a linear shear with cross-jet diffusion can actually produce a *t*^{3} dispersion in the along-jet direction (LaCasce 2008). Afterward, phases in which the separation strongly increases or stagnates alternate. Separations of more similar sizes than for the northern release are achieved, and a diffusive regime (indicated by the solid black line) seems to be reached. The corresponding kurtosis, after two small peaks at days 15 and 35, indicates a behavior that is similar to that of the kurtosis for the northern release (Fig. 9) showing oscillations around ku = 2.

For completeness, we tested the theoretical approach in Eq. (8) also for the moments of the southern release. As expected, the corresponding rescaled data plot does not collapse for most of the times (Fig. 10 inlay). Nonetheless, for asymptotic large times the moments indeed collapse and lay close to the line for the northern release, which represents the diffusive regime. Therefore, the rescaled plot shows that also the southern release eventually reaches the diffusive regime with similar (relative) diffusivity. Accordingly, the diffusivities derived from the different releases and from the mean-square separation averaged over all drifter pairs (Fig. 5) only vary by roughly 20%.

For the local regime, the drifter pairs trapped in the filament do not contribute much to the averaged pair separation 〈*s*^{2}(*t*)〉 because their separations are smaller by two orders of magnitude than those of the northern release (cf. Figs. 8 and 10). This leads to a reduced effective value of *Cε* as determined from the entire drifter group in comparison with the northern release, for which the derived proportionality constant *Cε* is 3 times as large.

The trapping of trajectories in the filament also leads to a slightly enhanced value for the unfolding-time *τ* in the nonlocal regime. The comparison of Figs. 8 and 10 shows that for the first four days the pair separations of the southern release are nearly constant. To first order, this constant contribution shows up in the averaged pair separation as an effectively enhanced *τ* in the nonlocal regime.

Last, the differences between the dispersion curves of the northern and southern releases imply that the PDF cannot be regarded as a homogeneous distribution of a specific form but actually constitutes two distinct PDFs, namely for the southern and northern releases, with very different means. The kurtosis derived from the entire ensemble then does not measure the typical inertial behavior [Eq. (5)] but shows an enhanced value because of the artificial broad distribution, which actually is the sum of two distinct distributions.

### c. FSLEs

We complement our fixed-time analysis of pair separations with a fixed-scale analysis of finite-size Lyapunov exponents *λ*(*δ*) [Eq. (10)], where, for the parameter *α* in the analysis presented here, the value *α* = 1.4 is used (Artale et al. 1997). Figures 12a–c show the FSLE for pairs of drifters from the northern release (Fig. 12a; red dots), the southern release (Fig. 12b; blue dots), and for the entire deployment (Fig. 12c; green dots). In the inlays of the figures, the corresponding drifter-pair statistics, that is, the number of pairs that could be detected (at any time) with separation *δ* and that achieved a distance *αδ* within the total time of measurement, are plotted.

The dotted line (plateau) in Fig. 12a corresponds to the nonlocal dispersion regime in terms of FSLEs, while the dashed line with a slope of −2/3 represents the local (Richardson) regime. To compare the FSLE record with the dispersion curve of pair separations more quantitatively, we refer to the mean pair separations 〈*s*(*t*)〉 (brown in Fig. 8) rather than to the RMS separations 〈*s*^{2}(*t*)〉^{1/2} (blue in the same figure), which, because of the anomalous broad PDF (stretched exponential) in the local regime, for the northern release are shifted against each other. From the points of intersection of the corresponding fits to the nonlocal, local, and diffusive regimes for 〈*s*(*t*)〉 in Fig. 12, we expect the FSLEs to decay as *δ*^{−2/3} for distances *δ* between 14 and 650 km. Despite some fluctuations, this is consistent with the FSLE record in Fig. 12a. A transition toward a diffusive regime, which is hardly detectable for the pair separations, is not detectable by means of FSLEs. The origin should be that the FSLE data (in contrast to the pair separations for fixed times) are strongly affected by the finite time of measurement, since especially those pairs that need an above-average long time to reach *αδ* after passing *δ* are missing in the statistics. According to the inlay of Fig. 12a, this finite-time effect, which shows up in the inlay as a declining number of detected pairs, starts already at distances *δ* of about 200 km. For distances smaller than about 1 km, deviations from the nonlocal forecast (plateau) are detectable for the northern release. In this regime also the corresponding dispersion curves in Fig. 8 show a deviation from nonlocal behavior.

For the southern release, the initial regime is consistent with a *δ*^{−2/3} decay for distances below 5 km, which in principle reflects the corresponding increase of 〈*s*(*t*)〉 for times between 3 and 13 days in Fig. 10. For larger *δ*, the FSLE record shows an irregular behavior that does not allow us to derive any functional form. According to the inlay of Fig. 12b, the bias due to the finite time of measurement affects the southern release at much smaller distances *δ* than for the northern release. It is nonetheless striking that, for both releases the *δ* regimes, which correspond to the *t*^{3} increase in the different dispersion curves (Figs. 8 and 10), are well described by the slope −2/3; that is, despite the probably very different underlying physics, which cause the *t*^{3} law for the northern release (turbulence) and a similar steep increase detected for the southern release (possibly shear dispersion with across-stream fluctuations), both dynamics also show up in the FSLEs in terms of a *δ*^{−2/3} decay of *λ*(*δ*), which occurs, however, in very different regimes of *δ*.

Although the results for the two methods—relative dispersion and FSLE—are consistent with each other in cases in which they are applied to the single releases, the situation is different when we compare the results of the fixed-time and fixed-scale analysis applied to the *entire* drifter ensemble. The corresponding FSLEs (Fig. 12c) do not reflect the different separation regimes detected for the pair separations (Fig. 5). Given that we found clear evidence for a nonlocal regime for the pair separations, we would expect that the FSLEs reveal a plateau *β* = 0 for the initial *δ* regime. In contrast, the FSLE record for the entire release is more consistent with a *δ*^{−2/3} decay for scales up to roughly 3 km (gray dashed line in Fig. 12c). For larger *δ*, the FSLE curve is unspecific. To understand this apparent inconsistency, one has to keep in mind that, while the mean-square pair separations 〈*s*^{2}(*t*)〉 differ by up to two orders of magnitudes for the two releases (see Figs. 10 and 8), the FSLEs are of comparable sizes. Accordingly, because of scale separation, for pair separations the northern release dominates the dispersion curve, but this is not the case for the FSLE record. For the FSLE, the small *δ* regime seems to reflect the dispersion behavior of the southern release, because the FSLEs remain roughly constant in this regime for the northern release. Note that for the smallest scales the number of contributing pairs *N*(*δ*) is larger for the southern release than for the northern release (see inlays of Figs. 12a,b), which is due to the enhanced probability of finding chance pairs with small *δ* within the filament. To test if this could contribute to the fact that the combined FSLE shows similar behavior to the southern-release FSLE, we repeated the FSLE analysis allowing only original pairs and found no difference in the effect of the southern release dominating the combined dataset for small distances (not shown). For larger *δ*, the FSLEs of both releases are of similar size. As a result, the FSLEs of the total release (Fig. 12c) reflect a mixed dynamic that cannot be described by the exponents expected from the pair-separation analysis.

## 5. Discussion and conclusions

We have analyzed the relative dispersion of surface drifters deployed at the boundaries of a filament during a cruise in November–December 2016 in the Benguela upwelling region by a fixed-time analysis of pair separations supplemented by the fixed-scale analysis of first-passage times (FSLEs). The drifters were released in groups of triplets, with an initial separation of 100–600 m, a distance of 5 km between the triplets, and a high-frequency sampling of 30 min. In our study, we found that the location of release has a strong influence on the corresponding dispersion curves. We therefore analyzed the groups separately and investigated their specific impact on the averages in the fixed-time and the fixed-scale study, respectively.

### a. Northern release

For the drifter pairs released at the northern boundary of the filament closer to the upwelling zone, the mean-square pair separation 〈*s*^{2}(*t*)〉 shows an exponential increase with an unfolding time of *τ* ≈ 0.5 days for the first five days at spatial scales below the Rossby radius, which for the Benguela region is on the order of 30 km (Houry et al. 1987). Subsequently, the increase of 〈*s*^{2}(*t*)〉, within the limits of having a small number of drifters, is consistent with the *t*^{3} law of Richardson (local regime) for times shorter than about 30 days followed by a diffusive regime.

To overcome restrictions due to the small degrees of freedom, we derived a scaling hypothesis for the corresponding moments 〈*s*^{n}(*t*)〉 of the theoretical PDFs *P*(*s*|*t*) for local and diffusive dispersion. The rescaled data plot confirms the theoretical forecast (local to diffusive dispersion) for the northern release. We suggest the scaling approach as a valuable tool to detect (or to rule out) the different dispersion regimes as it takes advantage of more information on the dynamics by considering a hierarchy of moments of the PDF. The scaling approach might be helpful especially when the degrees of freedom are reduced (as in our case) but also if (e.g., because of the finite lifetime of the drifters or the restricted time of measurement) the diffusive regime is hardly reached. Our approach also explains why the kurtosis ku, which is theoretically expected to have a constant value for the local dispersion regime, in (ocean) experiments instead typically shows up as a function slowly decaying in time. The origin is that, because of the anomalous broadness of the PDF, the moments of the PDF experience the transition from local to diffusive transport at very different times *t*_{x}(*n*) such that within the time-interval [*t*_{x}(4), *t*_{x}(2)] the kurtosis is influenced by both the local and the diffusive dynamics. The time dependence of ku therefore is a generic feature of the transition between the two self-similar PDFs (for the local and the diffusive regime) and in particular does not question the self-similarity of the PDF of the local regime.

We showed that the fixed-scale analysis of FSLEs *λ*(*δ*) applied to the northern release is in quantitative consistency with the fixed-time analysis of the mean pair separation 〈*s*(*t*)〉 because it shows a plateau for *δ* between 1 and roughly 10 km (which corresponds to the transition from nonlocal to local regime for 〈*s*(*t*)〉) followed by a *δ*^{−2/3} decay. A diffusive regime (*δ*^{−2}) could not be identified via FSLEs. We relate this to two different effects. On the one hand, it is related to the boundary effect due to the finite duration of measurement. On the other hand, the relevant transition time *t*_{x}(1) for the mean separation 〈*s*(*t*)〉 is larger than the corresponding time *t*_{x}(2) for the mean-square separation 〈*s*^{2}(*t*)〉 and is closer to the end in time of our measurements.

Our finding that the relative dispersion 〈*s*^{2}(*t*)〉 increases exponentially for scales smaller than the Rossby radius is consistent with several other studies, which have experimentally investigated relative dispersion for scales larger than ~1 km, namely, those of LaCasce and Ohlmann (2003) in the Gulf of Mexico, Koszalka et al. (2009) in the Nordic seas, Schroeder et al. (2011) in the Mediterranean Sea, van Sebille et al. (2015) in the Southern Ocean, Berti et al. (2011) in the southwestern Atlantic Ocean, and Sanson et al. (2017) in the southern Gulf of Mexico, who all found an exponential regime with similar unfolding times. Also, Lumpkin and Elipot (2010) detected an exponential dispersion regime referring to a nonlocal regime followed by a Richardson regime. As the spatial scale achieved at the transition time to the Richardson regime is considerably smaller than the Rossby radius, this indicates the presence of significant energy at the submesoscale.

### b. Southern release

In contrast to the drifters released at the northern boundary, the drifters released at the southern boundary remain in constant distance from each other for the first three days. Afterward, for the following 10 days, 〈*s*^{2}(*t*)〉 increases strongly in time. Roughly, this regime can be described by a *t*^{3} increase, but other functional forms (e.g., a ramped exponential) cannot be ruled out because of the limited length of this period. During this period, 〈*s*^{2}(*t*)〉 is approximately two orders of magnitude smaller than for the northern release. For longer times, 〈*s*^{2}(*t*)〉 shows a steep but irregular increase before 〈*s*^{2}(*t*)〉 eventually reaches the diffusive regime. The (effective) *t*^{3} regime of the pair separations described above appears as a *δ*^{−2/3} decay for the FSLEs in the corresponding regime for distances *δ* smaller than 4 km. Contrary to the northern release, the distribution *P*(*s*|*t*) of pair separations remains narrow with a kurtosis of ku ≈ 1.5 [i.e., even smaller than for a Rayleigh distribution (ku = 2)] until the end of the first increase mentioned above and then, after some sharp peaks, fluctuates around ku = 2.

### c. Entire dataset

The mean-square pair separation 〈*s*^{2}(*t*)〉 averaged over all 37 drifter pairs shows a similar dispersion behavior as for the northern drifter release described above with three dispersion phases (nonlocal, local, and diffusive). The separation in scales by two orders of magnitude between the different releases results in the domination of the northern release for the total average of mean-square pair separations in the relevant time regime. The slow separation of the drifters in the southern release nonetheless has an impact on 〈*s*^{2}(*t*)〉, as it effectively leads to a slightly larger unfolding-time of *τ* ≈ 0.6 days, a considerably smaller dissipation rate *Cε* ≈ 4.38 × 10^{−9} W kg^{−1}, and a reduced relative diffusivity *κ* ≈ 3.5 × 10^{4} m^{2} s^{−1}, where *κ* was confirmed by two different subsampling methods. In comparison with the northern release, the transition from nonlocal to local dispersion is shifted toward shorter distances (20 km) and thereby lies below the first baroclinic Rossby radius (approximately 30 km) when the average is taken over the entire drifter ensemble. We suggest that, depending on the distribution of deployment locations, the presence of filaments might lead in general to a bias in the set of parameters (*τ*, *Cε*, and *κ*) that, according to Corrado et al. (2017), describe relative dispersion in ocean subbasins. In particular, for the entire southern Atlantic subtropical gyre, Corrado et al. (2017) derived from their FSLE analysis a similar value for *κ* but values for *Cε* and *τ* that are smaller by factors of 2 and 3, respectively. In addition, the relatively small number of degrees of freedom and seasonal to interannual variability could affect the conclusions. We note that a change of the parameters is also accompanied by a shift of the corresponding transition times and scales of 〈*s*^{2}(*t*)〉 and *λ*(*δ*), respectively.

The different methods (mean-square separation and FSLE) reveal consistent results when applied to the two releases separately. However, consistency breaks down when the two methods are applied to the *entire* drifter ensemble. For distances *δ* from 100 m to 3 km, *λ*(*δ*) decays according to *δ*^{−2/3}. This would point to a local behavior in the submesoscale range and seems to contradict the nonlocal regime apparent for 〈*s*^{2}(*t*)〉, which would show up as a plateau for *λ*(*δ*). Similar controversial results occur in several other studies. For example, Poje et al. (2014) found for the DeSoto Canyon region that an energetic submesoscale field produces local dispersion from approximately 200 m to 50 km. Their results were based on an analysis of the time dependence of dispersion ellipses and the scale-dependent relative diffusivities from the Grand Lagrangian Deployment (GLAD) experiment. However, Beron-Vera and LaCasce (2016) reanalyzed the same data in the context of a study of synthetic trajectories and concluded that the results would be ambiguous because the dispersion curves would suggest nonlocal behavior while the structure functions are more consistent with a Richardson regime. Van Sebille et al. (2015) released 10 pairs of drifters in the western Pacific Ocean sector of the Southern Ocean, from either side of the ship (13 m apart) and reported their position every hour. In the FSLE analysis, they found two regimes with local behavior for scales smaller than approximately 3 km and a mixed behavior for larger scales up to approximately 70 km. In contrast, their analysis of relative dispersion reveals nonlocal behavior for small scales.

In our case, we argue that the origin of the apparent inconsistency between the fixed-time and the fixed-scale analysis lies in the strong influence of the area of location of deployment on the dispersion behavior, which results in situations where different kinds of dynamics are mixed. These different dynamics in turn show up differently in the fixed-time analysis, *λ*(*δ*), and the fixed-scale analysis, 〈*s*^{2}(*t*)〉, respectively: The pair separations 〈*s*^{2}(*t*)〉 reflect the typical dispersion behavior closer to the upwelling region (northern release) where the achieved pair separations are by two orders of magnitude larger then within the filament (southern release). In contrast, the FSLE analysis reveals the small-scale behavior *within* the filament, because for small distances *λ*(*δ*) is constant for the northern release, whereas for the filamental (southern) release *λ*(*δ*) ~ *δ*^{−2/3} holds for the same *δ* regime.

### d. Fluctuations/PDF

To explore the dispersion for intermediate separations between the nonlocal and the diffusive regime in more detail, we also analyzed (qualitatively) the fluctuations of the (rescaled) pair separations *s*/〈*s*(*t*)〉. The corresponding histograms reveal extreme events for times within the (local) Richardson regime with extreme large sudden (relative) jumps, which vanish with increasing times. The kurtosis reflects the different dispersion regimes despite that for the local regime it shows enhanced values up to 15. We relate this to the scale separation for the pair distances between the different releases of up to two orders of magnitudes, which leads to a nonhomogenous distribution of pair separations *P*(*s*|*t*) for fixed times. An enhanced value of ku has also been observed, for example, in Sanson et al. (2017) and Beron-Vera and LaCasce (2016). We have discussed here that the increase of ku at the beginning of the local regime in combination with the slow decay of ku toward the end of the local regime does not question the self-similarity of the underlying PDF but stems from a nonhomogeneous PDF in combination with the generic (transition) effect mentioned above.

### e. Submesoscale features

An important issue is whether our findings point to the existence of local submesoscale features, in our case in the area of the southern release within the filament. When only considering the FSLE analysis for small *δ*, our results are similar, for example, to the findings of Schroeder et al. (2012) in the coastal frontal zone in the northwestern Mediterranean Sea. Their results are consistent with a *δ*^{−2/3} decay for similar distances *δ*. The assumption that the *δ*^{−2/3} decay in our case points to the influence of submesoscale features, however, might be premature because for the southern release the corresponding PDF is very narrow, with ku ≈ 1.5. This points to the absence of intermittent dynamics as would be typical for the local (Richardson) dynamic. Our results therefore stress the importance of an accompanied study of the fluctuations in the system, for example, by the moments of the respective PDF.

## Acknowledgments

We thank the science party and the crew of cruise M132 on R/V *Meteor* for their support during our survey. Drifter data are available through the Global Drifter Program (http://www.aoml.noaa.gov/phod/dac/index.php). This paper is a contribution to the project L3 (Diagnosing and Parameterizing the Effects of Eddies) of the Collaborative Research Centre TRR 181 “Energy Transfer in Atmosphere and Ocean” funded by the German Research Foundation (DFG). Author G. Badin was also funded by Research Grants DFG 1740 and DFG BA5068/8-1. We thank Inga Koszalka and Joe LaCasce for valuable discussions.

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## Footnotes

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