Point source dispersion of surface drifters in the southern Gulf of Mexico

The PEMEX-CICESE drifter database was obtained during a long-term program of oceanographic observations, as described in the Introduction. The drifters were built and deployed by Horizon Marine Inc. (Far Horizon Drifters), and they consist of a cylindrical buoy with a parachute attached to it with a 45 m tether line that allows for air deployment. When reaching the surface of the ocean, the parachute serves as a drogue approximately centered at 45 m, and it is estimated that the drifters follow oceanic currents below the surface Ekman layer. The geographical positions were tracked with a GPS receiver. Additional information on the drifters performance and some pre-processing steps on the data were reported by Pérez-Brunius et al (2013).

During a 7-year period (starting on September 2007), 441 surface drifters were deployed and tracked in different geographical locations. We mainly focus on 365 drifters launched in five preferential spots, defined by a 20 km radius circle, as shown in figure 1(a). The geographical positions of the spots and the corresponding depth are presented in table 1. In addition, 85 drifters were released in different places over the region, always south of 22°N. The drifters recorded hourly positions, which were interpolated to regular 6 h intervals. The time of release and the lifetimes of each drifter are presented in figure 1(b).

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The number of drifters released every month is shown in figure 1(c). In average, there are about 63 launches per year, and between 5 and 6 per month. The lifetimes of the drifters from each spot are shown in figure 1(d). Last value indicates the longest lifetime in each subset. Mean lifetimes are shown in table 1. For the whole data set, including drifters inside and outside spots, the mean lifetime is 62 days with a standard deviation of 47 days. The longest lifetime was 218 days.

In order to estimate the dispersion of drifters that start from a common origin we consider dispersion ellipses. The ellipses are calculated with the separations of the particles from the average position or 'center of mass' defined as

$$\begin{equation} \bar X_i (t) = {1 \over N}\sum\limits_{k = 1}^N {[x_i^k (t) - x_i^k (0)]}, \end{equation} \tag{ 1 }$$

where N is the number of particles, $x_{i}^{k}$ is the position in the i-direction of particle k (i = 1, 2 corresponds to zonal and meridional directions, respectively), and t is time. The overbar indicates ensemble average. The dispersion of a cloud of particles with respect to the average position is:

$$\begin{equation} \overline {R_i^2 } (t) = {1 \over {N - 1}}\sum\limits_{k = 1}^N {[x_i^k (t) - x_i^k (0) - \overline {X_i } (t)]^2 .} \end{equation} \tag{ 2 }$$

The definition of dispersion (2) is proportional to the average squared separations over all available particle pairs, that is, to relative dispersion (LaCasce 2008).

Dispersion ellipses are equivalent to the so-called variance ellipses, or covariance error ellipses, used in several fields to indicate the dispersion of two sets of data (see e.g. Waterman and Lilly 2015). In our case, the data are the zonal and meridional positions of the drifters, $x_{1}^{k} (\tau )$ and $x_{2}^{k} (\tau )$, at a given time τ. The ellipse is located at the position of the center of mass at that time, $\bar{X}_1 (\tau )$ and $\bar{X}_2 (\tau )$. The semi-axes are proportional to the square root of the relative dispersion components, according to the following expressions:

$$\begin{equation} a(\tau ) = c\sqrt {\max \;[\overline {R_1^2 } (\tau ),\overline {\;R_2^2 } (\tau )],} \end{equation} \tag{ 3 }$$

$$\begin{equation} b(\tau ) = c\sqrt {\min \;[\overline {R_1^2 } (\tau ),\;\overline {R_2^2 } (\tau )],} \end{equation} \tag{ 4 }$$

with c a positive real value that determines the size of the ellipse (we use $c = \sqrt{2}$). Then this ellipse is rotated an angle θ, with respect to the zonal direction, given by the direction of the eigenvector with the maximum eigenvalue of the positions covariance matrix. Written in terms of a parameter α that runs from 0 to 2π, the ellipse is calculated as:

$$\begin{equation} \begin{array}{*{20}l} {x(\alpha ;\tau )} & = & {\overline X _1 (\tau ) + a(\tau )\cos \alpha \cos \theta}\\ {} & {} & { - b(\tau )\sin \alpha \sin \theta}\\ \end{array} \end{equation} \tag{ 5 }$$

$$\begin{equation} \begin{array}{*{20}l} {y(\alpha ;\tau )} & = & {\overline X _2 (\tau ) + b(\tau )\sin \alpha \cos \theta } \\ {} & {} & { + a(\tau )\cos \alpha \sin \theta .} \\ \end{array} \end{equation} \tag{ 6 }$$

The time evolution of the dispersion ellipse represents the spread of the particles with respect to the center of mass. Since the drifters were not launched simultaneously, but during a 7-year period, the description is statistical in both space and time.

Here we describe some essential aspects of the mesoscale circulation in the region. Although most of these features have been discussed by several authors, this brief review is necessary because one of the central points in this paper is that the surface dispersion is mainly ruled by different mesoscale processes. Our description here is based on a careful examination of sea surface height maps together with drifter trajectories.

One of the main mesoscale circulations are the anticyclonic Loop Current Eddies (LCEs) arriving from the eastern GM (Vukovich 2007). These structures, with 150 to 300 km in diameter, are shed by the Loop Current and travel westward-southwestward, mainly due to the β-effect (Cushman-Roisin et al 1990). The shedding period of LCEs is very irregular, ranging between 0.5 and 18.5 months (Lugo-Fernández and Leben 2010). The LCEs arrive to the western GM where they interact with other structures and/or collide with the western shelf (Vukovich and Waddell 1991). The collision with the continental topography might lead to several processes, such as the generation of coastal flows, the meridional translation of the eddy, the rebound of the vortex from the western boundary, and the subsequent generation of secondary eddies (Zavala Sansón et al 1998, Zavala Sansón and van Heijst 2000, Sutyrin and Grimshaw 2010).

A second, fundamental structure is the semi-permanent cyclonic gyre at the Bay of Campeche (hereafter referred to as CG), located at the southernmost part of the GM. This feature was reported by various authors (Monreal-Gómez and Salas de León 1997, Vázquez de la Cerda et al 2005), and it was described in more detail by Pérez-Brunius et al (2013).

Figure 2(a) shows an example of a large LCE arriving to the region of study on November 2013, while a CG is well-formed at the south. The CG is centered at about 20°N, 95°W, and it is clearly denoted by drifter trajectories. The LCE moves westward at 24°N approaching the western topography, in such a way that there is almost no interaction with the CG. As a result, most of the drifters in the CG remain trapped in the south.

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In figure 2(b) we present an example in which there is a strong interaction between a LCE and the CG. On January 2012 a LCE arrived to the western side of the GM at a lower latitude than in the previous case, while the CG located at the south is clearly identified by the drifter trajectories. As the LCE approaches the western slope, both structures are strongly deformed. During this interaction, the LCE traps a number of drifters originally located in the CG and pushes them northwards.

Another circulation feature relevant for the surface dispersion is the presence of an intense, along-shore current flowing parallel to the coastline at the western margin of the GM (Zavala-Hidalgo et al 2003, Dubranna et al 2011). This flow is related with the wind stress curl over the northern GM (DiMarco et al 2005). The intensity of the current varies seasonally, being stronger and more frequent during Summer. In figure 2(c) a typical case of this narrow flow is shown during August 2011. Note that this flow might exist regardless of the presence of any LCE or the CG.

There might be more processes in the region playing a role on the export or retention of drifters. Figure 2(d) presents a typical example on July 2013, when the CG is observed with a very circular shape, together with secondary cyclonic and anticyclonic structures at northern latitudes. Some drifters are advected northwards following the circulation of these eddies.

The size of most of the mesoscale features described here (vortices with 100 to 300 km in diameter) is much larger than the size of the spots where the drifters were launched (20 km). Therefore, these locations can be considered as point sources, given that we are focusing on the large-scale spreading of passive tracers released there.

Now we examine the drifter trajectories from each spot, their average position and the dispersion ellipses at different times. The ellipses show the statistical dispersion from the center of mass at a given time. Recall that the ellipses are centered at the average position of the dispersed drifters, and their orientation indicates the direction of larger dispersion.

Figure 3 presents the results at day 3. At this time we aim at detecting correlated motions near the point sources as the drifters are released (e.g. the presence of a persistent mean flow or the influence of the coastline). Consider first the western spots 1, 2 and 4, which show that the predominant flow is cyclonic. This circulation corresponds to the cyclonic gyre of the Bay of Campeche (CG). In particular, drifters from spot 1 indicate a strong southeastward flow, clearly denoted by the orientation of the dispersion ellipse, followed by zonal dispersion along the coastline from spot 4. The center of mass from the eastern spots 3 and 5 is only slightly directed northwestward, suggesting a weaker influence of the CG.

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Figure 4 presents the drifter trajectories and dispersion ellipses at day 30 (panels a to e), a time scale at which the mesoscale features play a fundamental role on the drifter dispersion. The CG is now more clearly depicted by the trajectories starting from the western spots 1, 2 and 4, and it is still visible with trajectories from spot 5 at the east. The drifters from the easternmost spot 3 do not show this behavior, which indicates that the eastern zonal extension of the CG is about 93°W, approximately. In panel (f) the trajectories and the dispersion ellipse of the drifters initially located outside the spots are plotted. The center of mass is initially located at 19.5°N, 94.5°W, approximately, because these drifters were deployed over a wide area in the southern GM (magenta dots in figure 1). The trajectory of the center of mass is directed northwestward, in agreement with the corresponding result from the eastern spots 3 and 5.

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The northwest-southeast orientation of all the ellipses reflects the northward advection of drifters, preferentially along the western side of the GM. A typical situation is that some drifters from all spots are trapped in the CG, and at some point are diverted northward. This happens more frequently with drifters from spot 2, as suggested by the more elongated shape of the dispersion ellipse. Most of such motions are associated with some of the circulation features described in previous section, specially the presence of anticyclonic LCEs arriving from the eastern GM, and wind-driven along-shore flows. The southern CG might be important in this process as well, because the cyclonic circulation accumulates drifters in the south, which are then transported northward during the arrival of a LCE. This description is clear from altimetry images, as shown in figure 2(b).

Another important observation is that the drifters hardly move towards deeper water in the central GM. Furthermore, the drifters practically never penetrate into the Yucatan shelf at the eastern side (Rodríguez Outerelo 2015).

We present now the dispersion of drifters from the spots during some particular months that describe two dominant, contrasting scenarios. The main idea is that there are two dispersion patterns that dominate the behavior of surface drifters released in the region: (i) the northward advection of drifters, and (ii) the retention of drifters at the southern GM. These scenarios were distinguished quantitatively by means of two-particle statistics in ZPS16.

The northward advection scenario is identified when 25% of drifters in a given month move beyond 24°N. The blocking scenario is determined when 75% of the drifters remain south from 22°N. Since both cases are essentially different, they can be distinguished by visual inspection of the drifter trajectories during the month. The scenarios are identified during 26 months each, presented in table 2. The selected months might or might not be consecutive through the 84-month period of study. The reason for using some particular months to illustrate the dominant dispersion scenarios, instead of using monthly or seasonal data, is because dispersion is strongly influenced by the already described mesoscale circulation events. This point is further addressed in the Discussion section.

In order to show the differences between the two scenarios, figure 5 presents the drifter trajectories in both cases during 30 days, or less for drifters with a shorter lifetime. The first 15-day segments are colored differently than the subsequent section. In the northward advection scenario several drifters move beyond 24°N and even reach the 26°N latitude. As mentioned before, the northward path is mainly along the western margin of the GM. In contrast, for the blocking scenario only a few drifters reach 24°N. Another important difference is the apparent shape and position of the GC: in the northward advection scenario the CG is more compact and located more to the west, while in the blocking case the CG is wider and centered more to the east.

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Figures 6 and 7 show the drifter trajectories and dispersion ellipses for the northward advection and blocking scenarios, respectively. The first feature to notice is that the dispersion ellipses in the northward advection scenario are indeed much more elongated in the meridional direction than those in the blocking scenario. The ellipses in the latter case are almost circular, except in spot 1. The shape of the ellipses can be interpreted as a graphical representation of the anisotropy of relative dispersion during the northward advection scenario, contrasting with the isotropy observed during the blocking scenario.

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A second observation is that the north-south elongation of the ellipses in the northward advection scenario is more pronounced for drifters starting from spot 2, in the central region. This is interpreted in terms of the presence of LCEs in the southern GM. Drifters released from this spot might be trapped during the arrival of a LCE and then strongly advected northwards by the western flank of the vortex.

Another important point is that drifters from the eastern spot 3 are dispersed northwestward on average, regardless of the dispersion scenario (the dispersion ellipses in figures 6(c) and 7(c) are very similar). This is an indication that dispersion from the eastern part of the region tends to be northwestward in general, as observed with the full set of drifters in figure 4.