1. Introduction
The oil spill produced by the Deepwater Horizon drilling rig explosion in May 2010 (Lubchenco et al. 2012) has motivated great interest in the Lagrangian circulation of the Gulf of Mexico (GoM). This is reflected in the execution in recent years of a number of field campaigns observing its surface Lagrangian circulation. A main reason for investigating the surface Lagrangian circulation is found in the very tangible effects it had on the evolution of the oil slick that emerged from the ocean floor (Olascoaga and Haller 2012). The main campaigns have been the Grand Lagrangian Deployment (GLAD) in July 2012 (Olascoaga et al. 2013; Poje et al. 2014; Beron-Vera and LaCasce 2016) and the Lagrangian Submesoscale Experiment (LASER) in February 2016 (Miron et al. 2017; Novelli et al. 2017). These two campaigns contributed to nearly duplicating the satellite-tracked surface drifter database existing prior to the oil spill, which consisted mainly of drifter trajectories from the National Oceanic and Atmospheric Administration (NOAA) Global Drifter Program (GDP; Lumpkin and Pazos 2007) and the Surface Current Lagrangian-Drifter Program (SCULP; Sturges et al. 2001; Ohlmann and Niiler 2005); see Miron et al. (2017) for more details.
Large amounts of oil were reported to stay submerged and to persist for months without substantial biodegradation (Camilli et al. 2010). Yet, the effects that the deep Lagrangian circulation had on the submerged oil remained elusive, which directly or indirectly motivated the execution of experiments to also observe the Lagrangian circulation at depth. From 2011 to 2015, one experiment featured an important deployment of acoustically tracked floats (Hamilton et al. 2016). This augmented the existing submerged float database, consisting mainly of profiling floats from a dedicated experiment (Weatherly et al. 2005) and routine sensing of the deep global ocean (Roemmich et al. 2009). Another experiment involved the release of a chemical tracer at depth in July 2012 near the Deepwater Horizon site and its subsequent sampling over the course of one year (Ledwell et al. 2016). An aspect of the deep Lagrangian circulation highlighted by the dedicated profiling float experiment (Weatherly et al. 2005) was the restricted communication between the eastern and western GoM basins and also a cyclonic circulation at about 900 m in the southwestern sector. Analysis of the acoustically tracked float trajectories in the western basin (Pérez-Brunius et al. 2018) from the recent experiment (Hamilton et al. 2016) further revealed the existence of a cyclonic boundary current below 900 m and a cyclonic gyre over the abyssal plain consistent with numerical studies (Oey and Lee 2002) and the analysis of hydrographic data (DeHaan and Sturges 2005) and deep-water moorings (Tenreiro et al. 2018). Direct inspection of the same acoustically tracked float trajectories (Pérez-Brunius et al. 2018), as well as rough estimates of connectivity between the eastern and western basins (Hamilton et al. 2016), suggests that the exchange between them occurs along the boundary following a cyclonic circulatory motion. In turn, the analysis of the dispersion of the chemical tracer released at depth in the eastern basin (Ledwell et al. 2016) concluded that homogenization by stirring and mixing is substantially faster in the GoM than in the open ocean. The main source of energy in the deep eastern basin is presumably provided by the Loop Current by inducing a deep flow through baroclinic instabilities, deep eddies, and topographic Rossby waves, which can transfer energy toward the western basin (Sheinbaum et al. 2016; Hamilton et al. 2016; Donohue et al. 2016).
The goal of this paper is to shed new light on the deep Lagrangian circulation in the GoM by using probabilistic tools from nonlinear dynamical systems. These are applied to the above acoustically tracked float trajectories with a focus on connectivity. Investigating connectivity with the probabilistic nonlinear dynamics tools summarizes to the analysis of the eigenvectors of a transfer operator approximated by a matrix of probabilities of transitioning between boxes of a grid, which provides a discrete representation of the Lagrangian dynamics (Froyland et al. 2014b). Markov-chain representations of this type had originally been used to approximate almost-invariant sets in nonlinear dynamical systems using short-run trajectories (Hsu 1987; Dellnitz and Junge 1999; Froyland 2005), and in the ocean context to determine the extent of Antarctic gyres in two- (Froyland et al. 2007) and three-space (Dellnitz et al. 2009) dimensions. This eigenvector method has been recently applied on drifter data to construct a geography of the surface Lagrangian circulation (Miron et al. 2017). A Lagrangian geography is composed of dynamical provinces that delineate weakly interacting basins of attraction for almost-invariant attractors, which imposes constraints on connectivity. Here we construct a geography for the deep Lagrangian circulation, providing firm support to earlier inferences from the direct inspection of float trajectories and, furthermore, revealing a number of aspects transparent to traditional Lagrangian data analysis.
2. Data
The main dataset analyzed in this paper is composed of trajectories produced by a total of 152 quasi-isobaric acoustically tracked RAFOS (Sound Fixing And Ranging or SOFAR, spelled backward) floats (Rossby et al. 1986) deployed in the GoM (Hamilton et al. 2016). Starting in 2011, 121 floats ballasted for 1500 m and 31 floats for a lower depth of 2500 m were deployed in the following 2 years. Each float recorded position fixes three times daily, with record lengths varying between 7 days and 1.5 years. Given that the estimated position uncertainty is on the order of 5 km, this sampling is more frequent than required, so we here consider daily interpolated trajectories. The float deployment during the first 2 years of a 4-yr-long program was performed by several U.S. (Woods Hole Oceanographic Institution, Leidos Corporation, University of Colorado) and Mexican (Centro de Investigación Científica y de Educación Superior de Ensenada) teams sponsored by the U.S. Bureau of Ocean Energy Management (BOEM). The records of the last floats deployed ended in summer 2015. The recorded trajectories of all floats are shown in Fig. 1 (deployment locations and final positions are indicated in blue and red, respectively). The trajectories cover a region bounded for the most part by the 1750-m isobath (dashed lines in Fig. 1 indicate, from outside to inside, the 1500-, 1750-, and 2500-m isobaths). Note that while the floats are too deep to escape the GoM through the Straits of Florida (where the maximum depth is roughly 700 m), they are capable of escaping through the Yucatan Channel (where the maximum depth is about 2000 m). Mooring measurements suggest that the latter is indeed possible as they have revealed a countercurrent between 500 and 1750 m on the western and eastern sides of the Yucatan Channel (Sheinbaum et al. 2002). However, no float is seen to travel into the Caribbean Sea. Confinement of the 1500- and 1750-m floats within the region bounded by the 1750-m isobath suggests predominantly columnar motion. This is confirmed by the analysis presented below, which ignores the depth of the floats to maximize the number of available trajectories.
Trajectories of RAFOS floats ballasted at (left) 1500 and (right) 2500 m in the Gulf of Mexico over 2011–15. Indicated are initial (blue dots) and final (red dots) positions of the floats. The dashed lines indicate, from outside to inside, the 1500-, 1750-, and 2500-m isobaths.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
We also analyze trajectories recorded by all available (60) profiling floats in the GoM from the Argo Program (Roemmich et al. 2009). Unlike RAFOS float positions, Argo float positions are recorded every 10 days after the float descends down to a parking depth of 1000 m, where it drifts for 9 days, and further to 2000 m to begin profiling temperature and salinity in its ascent back to the surface. The trajectories of the Argo floats roughly sample the same area as the RAFOS floats, albeit much less densely. However, the way that the Argo floats sample the deep Lagrangian circulation differs from that of the RAFOS floats, which remain at all times parked at a fixed depth. Despite this difference, we show that Argo floats replicate some important aspects of the deep circulation inferred using the RAFOS floats.
A third set of independent data considered is composed of concentrations of chemical tracer from a release experiment (Ledwell et al. 2016). In the experiment, a 25-km-long streak of CF3SF5 was injected on an isopycnal surface about 1100 m deep and 150 m above the bottom, along the continental slope of the northern GoM, about 100 km southwest of the Deepwater Horizon oil well, where oil was detected at depth after its explosion. The tracer was sampled between 5 and 12 days after release, and again 4 and 12 months after release.
3. Theory
a. Transfer operator and transition matrix
Note that the only time dependence is the duration of time T. In particular, we do not model variation of the advection–diffusion dynamics as a function of initial time. This is appropriate for a probabilistic description of the dynamics, as done in statistically stationary turbulence (Orszag 1977), yet it is also a consequence of the nature of the dataset considered here. In either case the significance of the time homogeneity assumption can only be assessed a posteriori, as we do here.
We note that the discrete evolution described by 
b. Ergodicity, mixing, attracting sets, residence time, and retention time
Because the transition matrix 
We call 
Suppose that 
c. Lagrangian geography from almost-invariant decomposition
Revealing those regions in which trajectories tend to stay for a long time before entering another region is key to assessing connectivity in a flow. Such forward time-asymptotic almost-invariant sets and their corresponding backward-time basins of attraction can be framed (Froyland et al. 2014b) by inspecting eigenvectors of 
The magnitude of the eigenvalues quantifies the geometric rates at which eigenvectors decay. Those left eigenvectors with λ closest to 1 are the slowest to decay and thus represent the most long-lived transient modes (Froyland 1997; Pikovsky and Popovych 2003). For a given 
The multiple backward-time basins of attraction are identified by boxes where the corresponding right eigenvectors take approximately constant values [see Koltai (2011) for the simpler single basin case]. Decomposition of the ocean flow into weakly disjoint basins of attraction for time-asymptotic almost-invariant attracting sets using the above eigenvector method has been shown (Froyland et al. 2014b; Miron et al. 2017) to form the basis of a Lagrangian geography of the ocean, where the boundaries between basins are determined from the Lagrangian circulation itself, rather than from arbitrary geographical divisions.
We note that the eigenvector method differs from the flow network approach (Rossi et al. 2014; Ser-Giacomi et al. 2015). The eigenvector method analyzes time-asymptotic aspects of the dynamics through spectral information from the generating Markov chain, while the flow network approach computes various graph-based quantities for finite-time durations to study flow dynamics.
4. Results
a. Building a Markov-chain model
To discretize the deep-ocean Lagrangian dynamics in the GoM, we laid down on the region X spanned by the trajectories of the RAFOS floats in Fig. 1 a grid with 
For daily interpolated trajectories, number of RAFOS floats per box in the grid into which the domain visited has been discretized, independent of time over (left) the entire 2011–15 period and (right) within each season in this period. The dashed line, here and in all the subsequent figures, represents the 1750-m isobath.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
Before getting into the specifics of the computation of the transition matrix 
Using formula (6), we computed the 
b. Assessing communication within the Markov chain
A Markov chain can be seen as a directed graph with vertices corresponding to states in the chain, and directed arcs corresponding to one-step transitions of positive probability. This allows one to apply Tarjan’s algorithm (Tarjan 1972) to assess communication within a chain. Specifically, the Tarjan algorithm takes such a graph as input and produces a partition of the graph’s vertices into the graph’s strongly connected components. A directed graph is strongly connected if there is a path between all pairs of vertices. A strongly connected component of a directed graph is a maximal strongly connected subgraph and by definition also a maximal communicating class of the underlying Markov chain.
Applying the Tarjan algorithm to the directed graph associated with the Markov chain derived using the float trajectory data, we found a total of four maximal communicating classes. Each one of these classes is indicated with a different color in Fig. 3. One class, denoted 
Grouping of the states of the Markov chain associated with the matrix 
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
c. Forward evolution of the probability density of a tracer
Figure 4 shows selected snapshots of the forward evolution of the probability density of a tracer initially uniformly distributed within the layer between 1500 and 2500 m in the GoM under the discrete action of the underlying flow map. At the coarse-grained level given by the grid defined above, this is defined by (7) with 
Selected snapshots of the evolution of an initially uniformly distributed tracer probability density under the action of the underlying flow map, whose discrete representation is provided by the float-data-derived transition matrix 
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
The regions where the density distribution of Fig. 4 locally maximizes represent vertical “outwelling” sites. Likewise, there is vertical “inwelling” in the regions where the density locally minimizes. Volume conservation in either case implies vertical motion. While the direction of this motion cannot be determined from the analysis of the float trajectories on a single layer, some sense may be made of its magnitude by comparing area change estimates obtained using the deep floats and the satellite-tracked surface drifter data employed in Miron et al. (2017).
Let 
Probability density function estimates of relative area change computed over 7 days using the RAFOS deep float data (red) and satellite-track surface drifter data (blue).
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
d. Analysis of the Markov chain’s eigenspectrum
We now proceed to determine the level of connectivity within the horizontal domain in the layer visited by the deep floats by applying the eigenvector method on the matrix 
A portion of the discrete eigenspectrum of the transition matrix 
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
As expected from the assessment of communication within the Markov chain associated with 
(left) Left and (right) right eigenvectors associated with the largest eigenvalue of 
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
The top left and right panels of Fig. 8 respectively show the left and right eigenvectors associated with the second-largest eigenvalue of 
As in Fig. 7, but for (top) 
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
Inspection of the left eigenvector associated with the 3rd largest eigenvalue of 
Additional almost-invariant attracting sets (with shorter invariance time scales) and corresponding basins of attraction are revealed by the left–right eigenvector pairs associated with the fourth to sixth eigenvalues of 
Lagrangian geography of dynamically weakly interacting provinces formed by the domains of attraction associated with the most persistent attractors. Geographic partitions based on the analyses of the (left) second right eigenvector, (center) second through fourth right eigenvectors, and (right) second through sixth right eigenvectors are shown.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
e. Lagrangian geography
Rather than thresholding right eigenvectors as in prior applications (Froyland et al. 2014b; Miron et al. 2017), the various provinces in each Lagrangian geography constructed here were automatically obtained by applying a k-means clustering algorithm (Kaufman and Rousseeuw 1990) that minimizes squared Euclidean distance as outlined in Algorithm 1 of Froyland (2005), but with the weighted fuzzy clustering replaced with k-means clustering. The main geographic partition in the left panel of Fig. 9 was obtained by seeking 
In the two-eigenvector geography, two large provinces, one western (WE) and another one eastern (ES), split the domain nearly in half. The four-eigenvector geography incorporates two provinces in WE: a small northern subprovince (WN) and another southern subprovince (WS) even smaller. The six-eigenvector geography incorporates to WE these same small subprovinces and another, much larger, central subprovince (WC). Province ES is not modified by the four-eigenvector geography, while the six-eigenvector geography alters it by the addition of a small northern subprovince (EN).
As constructed, the provinces of the above Lagrangian geographies only weakly dynamically interact. This imposes constraints on connectivity within the 1500–2500-m layer in the GoM. More specifically, the communication between any two provinces is constrained locally by the level of invariance of the attractors contained within each of them and remotely by that of any attractors outside of the provinces but sufficiently close to them.
The level of communication among provinces can be assessed by the computation of forward-time conditional transition probabilities between pair of provinces. Over 7 days, the mean interprovince probability percentages are 98.5%, 97.5%, and 94.3% for the two-, four-, and six-eigenvector geographic partitions, respectively. Note that these are high, indicating weak dynamical interaction among provinces. Note also that the percentages decrease as the number of provinces in the geography increases. This reflects in part that communication within large provinces is less constrained than across their boundaries. The resulting transition matrices restricted to the various geographies are not symmetric, revealing the asymmetric nature of the Lagrangian dynamics in time. The asymmetry grows with the number of provinces in the partition, as can be expected.
From left to right, Fig. 10 shows residence time estimates according to formula (8) within each of the provinces in the two- to six-eigenvector Lagrangian geographies. While the basins of attraction are not necessarily highly invariant under forward-time dynamics, we record relatively high residence times. Note that 
Estimates of residence time within provinces in the (left) two-, (center) four-, and (right) six-eigenvector Lagrangian geographies. Note the scale range differences.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
The residence time calculation shows a west–east asymmetry in the two-eigenvector geography, with the WE province having longer residence times than the ES province. Specifically, the mean residence time in WE (ES) province is of about 4.53 (3.38) years. It must be mentioned that the mean here is taken with respect to a uniform probability distribution, that is, it is computed as an average according to Lebesgue (area) measure. As noted earlier in the paper, mean residence times computed using (8) coincide with retention times in (10) based on the likelihood of a trajectory to survive in a given set if the mean in the former is taken according to the probability distribution given by the leading left eigenvector of 
The provinces in the four- and six-eigenvector geographies have shorter residence times. For instance, on average within the WE, WS, WC, WW, ES, and EN provinces in the six-eigenvector partition these are about 0.74, 0.90, 0.48, 0.64, 1.18, and 0.38 years, respectively. Shorter residence times in sets covering smaller areas are expected. But note the short residence time of the WC province despite its large coverage.
The direct evolution of tracers with 
The west–east residence time asymmetry can be further realized by computing the time it takes on average to hit or reach a given province starting in another province. This can be done using (8) with the region A set to the complement of the target province. The result of this calculation for the six-eigenvector geographic partition is shown in Table 1. The top row shows source provinces and the left column target provinces. Consider, for example, the bottom row. The mean time to hit EN in the eastern basin starting on WC in the western side of the domain is 6.42 years. Consider now the second-to-top row. To reach WC from EN, it takes on average 1.91 years. Consistent with west–east residence time asymmetry, it takes more than 3 times as long to reach EN from WC. Clearly, the mean time to reach a given province starting in the same province is 0. Note that WS has not been included in the table as this province is never reached from outside.
Mean time (years) to reach a province of the six-eigenvector Lagrangian geography indicated in the left column starting from any province in the top row.
5. Validation
a. Chemical tracer
The deep Lagrangian geography constructed here and the surface Lagrangian geography computed by Miron et al. (2017) are globally different on the overlapping domains, suggesting that the surface Lagrangian motion is to a large extent decoupled from the deep Lagrangian motion. An important exception is the partition by a roughly meridional boundary of the surface and deep domains into two basins of attraction for almost-invariant attractors revealed by the inspection of eigenvectors of the corresponding transition matrices with the second-largest nonunity eigenvalue [cf. the left panel of Fig. 9 and the left panel of Fig. 6 of Miron et al. (2017)].
The restricted connection at depth between the eastern and western GoM was suggested by the behavior of profiling floats parked at about 900 m launched in the eastern side, which tended to stay there, and those launched in the western side, which remained there for a long period of time (Weatherly et al. 2005). Here we provide support for the significance of the eigenvector methodology applied to the deep GoM domain at a deeper level using the observed evolution of the chemical tracer injected near the Deepwater Horizon oil rig during the field experiment described by Ledwell et al. (2016).
The right panels of Fig. 11 show the distribution taken by the chemical tracer 4 (top) and 12 (bottom) months after release. The release site lies about 100 km southwest of the cross, indicating the location of the Deepwater Horizon rig. The circles are colored according to the amount of tracer found during in situ casts, integrated vertically between 1000 and 2500 m. The colored background is a smoothed interpolated map based on the station data. Note the tendency of the tracer to spread over the eastern side of the domain. We may compare this tendency with the dominant (second) left eigenvector, representing the dominant forward-time almost-invariant set structure. After 12 months from release, little tracer is seen to have traversed the zero-level set of the left eigenvector of 
Comparison between the evolution of a tracer probability density under the action of (left) the transition matrix 
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
This expectation is confirmed by the evolution of a tracer probability started from a source location near the chemical tracer release site under the action of the transition matrix 
It must be noted, however, that as the tracer probability is continually being evolved under 
b. Profiling floats
Additional independent observational support for the significance of the results obtained from the analysis of the Markov-chain model derived using the RAFOS floats is provided by the analysis of a Markov-chain model constructed using Argo profiling floats drifting at an average parking depth of 1000 m. At a shallower depth, the Argo trajectories sample a similar horizontal domain of the GoM as the RAFOS trajectories, but less densely (there are only 60 Argo floats in the database analyzed). Also, the temporal coverage of the Argo floats is not as ample as the RAFOS floats. With these differences in mind, we constructed a matrix of probabilities of the Argo floats to transitioning between the boxes of a grid similar to that used with the RAFOS floats. The transition time was set to 7 days as in RAFOS floats analysis, which required us to interpolate the original 10-day trajectories. The Markov chain associated with the resulting 
(left) Left and (right) right eigenvectors associated with the dominant nonunity eigenvalue of a transition matrix computed using Argo profiling float data.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
6. Discussion
a. Cyclonic circulation and f/H
The cyclonic circulation in the western side of the GoM domain is well described by complex eigenvectors of 
Snapshots of the forward evolution of the leading complex left eigenvector of the RAFOS-float-based transition matrix 
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
Rectification of topographic Rossby waves has been identified as a driver for the cyclonic circulation along the boundary (Hurlburt and Thompson 1980; Oey and Lee 2002; Mizuta and Hogg 2004; DeHaan and Sturges 2005). In linear, unforced, inviscid, barotropic and quasigeostrophic dynamics (Gill 1982), the vorticity changes only when there is motion across f/H contours, where f is the Coriolis parameter (twice the local vertical component of Earth’s angular velocity) and H is the fluid depth. As such, f/H provides a restoring force, supporting topographic Rossby waves, which through nonlinear interaction can be rectified to give rise to a mean flow directed mainly along f/H isolines (de Verdiere 1979). Here we test f/H conservation using the Markov-chain model and further assess its effect on the evolution of probability tracer densities in the domain.
Specifically, let 
Figure 14 shows 
For each box in the partition, absolute relative error between f/H, where f is the Coriolis parameter and H is depth, and the average of f/H according to the forward evolution over 1 year of a probability vector supported in the box. Dashed lines are the isobaths of 1750, 2500, and 3000 m.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
The expected behavior of tracer trajectories deduced from the Markov-chain model is corroborated by observed float trajectory patterns. This is shown in Fig. 15 for groups of float trajectories that have gone through selected sites along the boundary of the domain (left) and the center of the western side of the domain (right). Note, in the left panel, how the red and orange trajectories tend to run along the western boundary, while the green trajectories are not so constrained to doing so along the southern boundary. Observe too in this panel how the blue trajectories cover the eastern side of the domain, consistent with f/H being preserved in that region. Finally, note that the trajectories in the right panel loop around in a largely unrestricted manner.
Trajectories of floats that pass through the indicated square boxes (a different color is assigned to the trajectories for each box). Depending on the location of the boxes, the motion of the floats is (left) constrained and (right) unconstrained by f/H conservation.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
b. Homogenization
The analysis of the chemical tracer injected at depth suggested that homogenization in the GoM is more rapid than in the open ocean (Ledwell et al. 2016). While the Markov-chain model constructed here does not predict uniform homogenization in the long run, it supports a limiting, invariant distribution which does not reveal a preferred region for accumulation but rather a multitude of different small regions where some accumulation is possible. The highly structured texture of this distribution suggests partial homogenization in the long run. This can be quite fast. For instance, for a tracer released in the eastern side of the domain, it can take as little as 1 year or so to spread over that portion of the domain (cf. Fig. 10), consistent with the good agreement between the forward evolution of a tracer probability and the observed chemical tracer spreading (recall Fig. 11). This corresponds well with the mean time required for the EN province to hit the WE province, which is of 0.86 years (cf. Table 1 and Fig. 9). A rough estimate of mean lateral eddy diffusivity is 2.4 × 104 m2 s−1, computed as the area of the eastern province (approximately N/2 = 473 times the area of the individual boxes, about 252 km2) divided by 1 year. Consistent with the idea of fast homogenization, this is quite big, on the order of values estimated using surface drifters in the vicinity of the Gulf Stream (Zhurbas and Oh 2004).
c. Ventilation
Below 1000 m the GoM is filled with oxygen-rich water which is isolated from the diffusive inflow of oxygen from the surface by the presence of a layer of oxygen-poor water (Nowlin et al. 2001). As standard deep-water formation is very unlikely due to the extreme cooling and salinity increase required for the surface layer to sink, ventilation of the deep GoM has been argued to be accomplished via horizontal transport of oxygen-rich water from the Caribbean Sea (Rivas et al. 2005).
The tendency of the Lagrangian motion as inferred from the Markov-chain model constructed using the RAFOS floats to conserve area more effectively than that using surface drifters, together with the similarities of the dominant eigenvectors of the transition matrices built using RAFOS and Argo floats, is consistent with the above observations in that Lagrangian motion within the 1500–2500-m layer is predominantly horizontal. However, the Markov-chain model does not represent exchanges through the boundary of the domain as the RAFOS floats deployed inside the domain do not escape the domain.
Yet, the above does not rule out the possibility that floats deployed outside the domain eventually enter the domain. Indeed, the Argo floats suggest that this might happen at about 1000 m. This is shown in Fig. 16 for a few Argo float trajectories that start in the Caribbean Sea. We say “might” because we do not know for certain that the Argo floats penetrate the GoM domain at their parking depth or at a shallower level in their ascent and descent. Yet, ventilation might indeed take place more effectively at a shallower level. This is suggested by hydrographic observations which situate the core of the North Atlantic Deep Water (NADW) mass inside the Caribbean Sea between 1200 and 1300 m (Hamilton et al. 2018).
Trajectories of Argo profiling floats starting inside the Caribbean Sea. Initial positions are indicated by a blue dot and final positions by a red dot.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
d. Mean circulation
We close the discussion by investigating if the results obtained using the Markov-chain model could have ever been revealed by simply considering advection–diffusion using mean velocities deduced from the RAFOS float trajectories. This is a valid question inasmuch as our assumption of time homogeneity of the statistics could be thought to be represented in this simpler way. The top panel of Fig. 17 shows ensemble-mean streamlines computed by integrating a steady velocity field resulting from averaging the float velocities in each box of the grid used to construct the Markov-chain model. While the streamlines suggest a cyclonic flow along the periphery of the domain especially on its western side, which is consistent with the transfer operator analysis and also direct inspection of float trajectories (Hamilton et al. 2016), it is difficult to find a correspondence among the many sources and sinks with the various local minima and maxima of the limiting, invariant distribution of the Markov-chain model (cf. the rightmost panel of Fig. 4 or the left panel of Fig. 7). The differences with the Markov-chain model results are most clearly evidenced when the evolution of a tracer under the corresponding flow and subjected to diffusion is compared with the evolution of a tracer probability density under the transition matrix 
(top) Ensemble mean streamlines computed using RAFOS float velocities. (middle),(bottom) Snapshots of the evolution under combined advection by the corresponding flow and diffusion of a narrow Gaussian (normalized to a probability density) initially centered about 100 km southwest of the Deepwater Horizon oil rig, indicated by a cross. The middle panels use a molecular diffusivity value appropriate for CF3SF5. The bottom panels use an eddy diffusivity estimated from the product of a characteristic length and a typical velocity (see text for details).
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
7. Summary and concluding remarks
Analyzing acoustically tracked (RAFOS) float data in the Gulf of Mexico (GoM), we have constructed a geography of its Lagrangian circulation within the deep layer between 1500 and 2500 m, revealing aspects of the circulation transparent to standard Lagrangian data examination as well as confirming, and thus providing firm support to, other aspects already noted from direct inspection of the float trajectories. The analysis was done by applying a probabilistic technique that enables the study of long-term behavior in a nonlinear dynamical system using short-run trajectories. The Lagrangian geography is inferred from the inspection of the eigenvectors of a transfer operator approximated by a transition probability matrix 
The basic geography has a single dynamical province which constitutes the backward-time basin of attraction for a time-asymptotic invariant attracting set, which is revealed by the unique left eigenvector of 
Lateral transport and mixing inside the layer scrutinized do not happen unrestrainedly. Indeed, left eigenvectors of 
The simplest nontrivial geographical partition includes two nearly equal-area western and eastern provinces. Tracers initially released within these main provinces tend to remain confined within a few years, with the western province retaining tracers for longer than the eastern province. Communication between the provinces is accomplished through a cyclonic flow confined to the periphery of the domain, which was shown to be highly constrained by conservation of f/H, where f is the Coriolis parameter and H is depth, in the western side of the domain. Smaller secondary provinces of different shapes with residence times shorter than 1 year or so were also identified, imposing further restrictions on connectivity at shorter time scales.
Except for the main provinces, the secondary provinces identified do not resemble those of the surface Lagrangian geography recently inferred from satellite-tracked drifter trajectories. This implies disparate connectivity characteristics with possible implications for pollutant (e.g., oil) dispersal at the surface and depth.
The evolution of a chemical tracer from a release experiment as well as the analysis of a smaller set of Argo profiling floats were shown to provide independent support for the Lagrangian geography derived using the RAFOS floats. It is quite remarkable that the RAFOS and Argo floats produced similar Markov chain representations of the Lagrangian dynamics of the deep GoM given the different sampling characteristics (parking depth, temporal coverage) of these two observational platforms.
The good agreement between the results from the RAFOS and Argo float analyses suggests that the probabilistic tools employed here applied on the global Argo float array may provide important insight into the abyssal circulation of the World Ocean.
Acknowledgments
We acknowledge Amy Bower’s group at Woods Hole Oceanographic Institution for RAFOS float preparation, data acquisition, and final processing, which made the float trajectory dataset possible. We thank Alexis Lugo-Fernandez for helping us access the acoustically tracked float data, which are currently available from the National Oceanic and Atmospheric Administration (NOAA)/Atlantic Ocean and Meteorological Laboratory (AOML) subsurface float observations page (http://www.aoml.noaa.gov/phod/float_traj/index.php). The profiling float data were collected and made freely available through SEA scieNtific Open data Edition (SEANOE) at http://www.seanoe.org (doi:
APPENDIX A
Derivation of (6)
APPENDIX B
Robustness under Dataset Truncations
Evidence of the robustness of the eigenvector method results is provided in Fig. B1, which shows from left to right six-eigenvector geography partitions using transition matrices constructed by randomly excluding 10%, 25%, and 50% of RAFOS trajectories (recall that trajectories from only 152 floats are available for analysis). With a 10% reduction of the data, all regions obtained from the analysis of the complete dataset are present. The only difference is the boundary of the WE province, a transition region between the eastern and western deep GoM characterized initially by lower data density and residence time. Reducing the dataset by 25% and 50% leads to the separation of the WE province into a central region and a band around the WC province. The cyclonic circulation inferred from the analysis of the full dataset around the WC province is still represented. Using fewer trajectories eventually leads to the loss of information in some bins, for example, inside the WC province. Nonetheless, all Lagrangian provinces of the complete dataset are represented considering half of the data.
From left to right, six-eigenvector Lagrangian geographies computed using transition matrices constructed by randomly excluding 10%, 25%, and 50% of RAFOS float trajectories.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0073.1
REFERENCES
Beron-Vera, F. J., and J. H. LaCasce, 2016: Statistics of simulated and observed pair separations in the Gulf of Mexico. J. Phys. Oceanogr., 46, 2183–2199, https://doi.org/10.1175/JPO-D-15-0127.1.
Bracco, A., J. Choi, K. Joshi, H. Luo, and J. C. McWilliams, 2016: Submesoscale currents in the northern Gulf of Mexico: Deep phenomena and dispersion over the continental slope. Ocean Modell., 101, 43–58, https://doi.org/10.1016/j.ocemod.2016.03.002.
Camilli, R., and Coauthors, 2010: Tracking hydrocarbon plume transport and biodegradation at Deepwater Horizon. Science, 330, 201–204, https://doi.org/10.1126/science.1195223.
DeHaan, C. J., and W. Sturges, 2005: Deep cyclonic circulation in the Gulf of Mexico. J. Phys. Oceanogr., 35, 1801–1812, https://doi.org/10.1175/JPO2790.1.
Dellnitz, M., and O. Junge, 1999: On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal., 36, 491–515, https://doi.org/10.1137/S0036142996313002.
Dellnitz, M., G. Froyland, C. Horenkam, K. Padberg-Gehle, and A. Sen Gupta, 2009: Seasonal variability of the subpolar gyres in the southern ocean: A numerical investigation based on transfer operators. Nonlinear Processes Geophys., 16, 655–663, https://doi.org/10.5194/npg-16-655-2009.
de Verdiere, A. C., 1979: Mean flow generation by topographic Rossby waves. J. Fluid Mech., 94, 39–64, https://doi.org/10.1017/S0022112079000938.
Donohue, K., D. Watts, P. Hamilton, R. Leben, and M. Kennelly, 2016: Loop current eddy formation and baroclinic instability. Dyn. Atmos. Oceans, 76, 195–216, https://doi.org/10.1016/j.dynatmoce.2016.01.004.
Froyland, G., 1997: Computer-assisted bounds for the rate of decay of correlations. Commun. Math. Phys., 189, 237–257, https://doi.org/10.1007/s002200050198.
Froyland, G., 2005: Statistically optimal almost-invariant sets. Physica D, 200, 205–219, https://doi.org/10.1016/j.physd.2004.11.008.
Froyland, G., 2013: An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems. Physica D, 250, 1–19, https://doi.org/10.1016/j.physd.2013.01.013.
Froyland, G., K. Padberg, M. H. England, and A. M. Treguier, 2007: Detection of coherent oceanic structures via transfer operators. Phys. Rev. Lett., 98, 224503, https://doi.org/10.1103/PhysRevLett.98.224503.
Froyland, G., C. González-Tokman, and A. Quas, 2014a: Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools. J. Comput. Dyn., 1, 249–278, https://doi.org/10.3934/jcd.2014.1.249.
Froyland, G., R. M. Stuart, and E. van Sebille, 2014b: How well-connected is the surface of the global ocean? Chaos, 24, 033126, https://doi.org/10.1063/1.4892530.
Gill, A. E., 1982: Atmosphere-Ocean Dynamics. Academic Press, 662 pp.
Hamilton, P., A. Bower, H. Furey, R. Leben, and P. Pérez-Brunius, 2016: Deep circulation in the Gulf of Mexico: A Lagrangian study. OCS Study BOEM 2016-081 Tech. Rep., Bureau of Ocean Energy Management, 289 pp.
Hamilton, P., R. Leben, A. Bower, H. Furey, and P. Pérez-Brunius, 2018: Hydrography of the Gulf of Mexico using autonomous floats. J. Phys. Oceanogr., 48, 773–794, https://doi.org/10.1175/JPO-D-17-0205.1.
Horn, R. A., and C. R. Johnson, 1990: Matrix Analysis. Cambridge University Press, 561 pp.
Hsu, C. S., 1987: Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems. Applied Mathematical Sciences, Vol. 64, Springer, 354 pp.
Hurlburt, H. E., and J. D. Thompson, 1980: A numerical study of Loop Current intrusions and eddy shedding. J. Phys. Oceanogr., 10, 1611–1651, https://doi.org/10.1175/1520-0485(1980)010<1611:ANSOLC>2.0.CO;2.
Kaufman, L., and P. J. Rousseeuw, 1990: Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, 342 pp.
Khatiwala, S., M. Visbeck, and M. A. Cane, 2005: Accelerated simulation of passive tracers in ocean circulation models. Ocean Modell., 9, 51–69, https://doi.org/10.1016/j.ocemod.2004.04.002.
Koltai, P., 2011: A stochastic approach for computing the domain of attraction without trajectory simulation. Proc. Eighth AIMS Int. Conf., Dresden, Germany, American Institute of Mathematical Sciences, 854–863, https://doi.org/10.3934/proc.2011.2011.854.
LaCasce, J. H., 2008: Statistics from Lagrangian observations. Prog. Oceanogr., 77, 1–29, https://doi.org/10.1016/j.pocean.2008.02.002.
Lasota, A., and M. C. Mackey, 1994: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics. 2nd ed. Applied Mathematical Sciences, Vol. 97, Springer, 474 pp.
Ledwell, J. R., R. He, Z. Xue, S. F. DiMarco, L. J. Spencer, and P. Chapman, 2016: Dispersion of a tracer in the deep Gulf of Mexico. J. Geophys. Res. Oceans, 121, 1110–1132, https://doi.org/10.1002/2015JC011405.
Lehoucq, R. B., D. C. Sorensen, and C. Yang, 1998: ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. Society for Industrial and Applied Mathematics, 160 pp.
Lubchenco, J., M. K. McNutt, G. Dreyfus, S. A. Murawski, D. M. Kennedy, P. T. Anastas, S. Chu, and T. Hunter, 2012: Science in support of the Deepwater Horizon response. Proc. Natl. Acad. Sci. USA, 109, 20 212–21 221, https://doi.org/10.1073/pnas.1204729109.
Lumpkin, R., and M. Pazos, 2007: Measuring surface currents with Surface Velocity Program drifters: The instrument, its data and some recent results. Lagrangian Analysis and Prediction of Coastal and Ocean Dynamics, A. Griffa et al., Eds., Cambridge University Press, 39–67, https://doi.org/10.1017/CBO9780511535901.003.
Maximenko, A. N., J. Hafner, and P. Niiler, 2012: Pathways of marine debris derived from trajectories of Lagrangian drifters. Mar. Pollut. Bull., 65, 51–62, https://doi.org/10.1016/j.marpolbul.2011.04.016.
McAdam, R., and E. van Sebille, 2018: Surface connectivity and interocean exchanges from drifter-based transition matrices. J. Geophys. Res. Oceans, 123, 514–532, https://doi.org/10.1002/2017JC013363.
Miron, P., F. J. Beron-Vera, M. J. Olascoaga, J. Sheinbaum, P. Pérez-Brunius, and G. Froyland, 2017: Lagrangian dynamical geography of the Gulf of Mexico. Sci. Rep., 7, 7021, https://doi.org/10.1038/s41598-017-07177-w.
Mizuta, G., and N. G. Hogg, 2004: Structure of the circulation induced by a shoaling topographic wave. J. Phys. Oceanogr., 34, 1793–1810, https://doi.org/10.1175/1520-0485(2004)034<1793:SOTCIB>2.0.CO;2.
Norris, J., 1998: Markov Chains. Cambridge University Press, 254 pp., https://doi.org/10.1017/CBO9780511810633.
Novelli, G., C. Guigand, C. Cousin, E. H. Ryan, N. J. Laxague, H. Dai, B. K. Haus, and T. M. Özgökmen, 2017: A biodegradable surface drifter for ocean sampling on a massive scale. J. Atmos. Oceanic Technol., 34, 2509–2532, https://doi.org/10.1175/JTECH-D-17-0055.1.
Nowlin, W., A. E. Jochens, S. F. DiMarco, R. O. Reid, and M. K. Howard, 2001: Deepwater physical oceanography reanalysis and synthesis of historical data: Synthesis report. OCS Study MMS 2001-064, Minerals Management Service, 528 pp., https://www.boem.gov/ESPIS/3/3129.pdf.
Oey, L.-Y., and H.-C. Lee, 2002: Deep eddy energy and topographic Rossby waves in the Gulf of Mexico. J. Phys. Oceanogr., 32, 3499–3527, https://doi.org/10.1175/1520-0485(2002)032<3499:DEEATR>2.0.CO;2.
Ohlmann, J. C., and P. P. Niiler, 2005: A two-dimensional response to a tropical storm on the Gulf of Mexico shelf. J. Mar. Syst., 29, 87–99, https://doi.org/10.1016/S0924-7963(01)00011-2.
Okubo, A., 1971: Oceanic diffusion diagrams. Deep-Sea Res. Oceanogr. Abstr., 18, 789–802, https://doi.org/10.1016/0011-7471(71)90046-5.
Olascoaga, M. J., and G. Haller, 2012: Forecasting sudden changes in environmental pollution patterns. Proc. Natl. Acad. Sci. USA, 109, 4738–4743, https://doi.org/10.1073/pnas.1118574109.
Olascoaga, M. J., and Coauthors, 2013: Drifter motion in the Gulf of Mexico constrained by altimetric Lagrangian Coherent Structures. Geophys. Res. Lett., 40, 6171–6175, https://doi.org/10.1002/2013GL058624.
Orszag, S. A., 1977: Lectures on the statistical theory of turbulence. Fluid Dynamics, R. Balian and J.-L. Peube, Eds., Gordon and Breach, 235–374.
Pérez-Brunius, P., H. Furey, A. Bower, P. Hamilton, J. Candela, P. García-Carrillo, and R. Leben, 2018: Dominant circulation patterns of the deep Gulf of Mexico. J. Phys. Oceanogr., 48, 511–529, https://doi.org/10.1175/JPO-D-17-0140.1.
Pikovsky, A., and O. Popovych, 2003: Persistent patterns in deterministic mixing flows. Europhys. Lett., 61, 625–631, https://doi.org/10.1209/epl/i2003-00117-6.
Poje, A. C., and Coauthors, 2014: The nature of surface dispersion near the Deepwater Horizon oil spill. Proc. Natl. Acad. Sci. USA, 111, 12 693–12 698, https://doi.org/10.1073/pnas.1402452111.
Rivas, D., A. Badan, and J. Ochoa, 2005: The ventilation of the deep Gulf of Mexico. J. Phys. Oceanogr., 35, 1763–1781, https://doi.org/10.1175/JPO2786.1.
Roemmich, D., and Coauthors, 2009: The Argo Program: Observing the global ocean with profiling floats. Oceanography, 22, 34–43, https://doi.org/10.5670/oceanog.2009.36.
Rossby, T., D. Dorson, and J. Fontaine, 1986: The RAFOS system. J. Atmos. Oceanic Technol., 3, 672–679, https://doi.org/10.1175/1520-0426(1986)003<0672:TRS>2.0.CO;2.
Rossi, V., E. Ser-Giacomi, C. Lopez, and E. Hernandez-Garcia, 2014: Hydrodynamic provinces and oceanic connectivity from a transport network help designing marine reserves. Geophys. Res. Lett., 41, 2883–2891, https://doi.org/10.1002/2014GL059540.
Rousseeuw, P. J., 1987: Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math., 20, 53–65, https://doi.org/10.1016/0377-0427(87)90125-7.
Ser-Giacomi, E., V. Rossi, C. Lopez, and E. Hernandez-Garcia, 2015: Flow networks: A characterization of geophysical fluid transport. Chaos, 25, 036404, https://doi.org/10.1063/1.4908231.
Sheinbaum, J., J. Candela, A. Badan, and J. Ochoa, 2002: Flow structure and transport in the Yucatan Channel. Geophys. Res. Lett., 29, 1040, https://doi.org/10.1029/2001GL013990.
Sheinbaum, J., G. Athié, J. Candela, J. Ochoa, and A. Romero-Arteaga, 2016: Structure and variability of the Yucatan and loop currents along the slope and shelf break of the Yucatan channel and Campeche bank. Dyn. Atmos. Oceans, 76, 217–239, https://doi.org/10.1016/j.dynatmoce.2016.08.001.
Sturges, W., and A. Lugo-Fernandez, 2005: Circulation in the Gulf of Mexico: Observations and Models. Geophys. Monogr., Vol. 161, Amer. Geophys. Union, 347 pp., https://doi.org/10.1029/GM161.
Sturges, W., P. P. Niiler, and R. H. Weisberg, 2001: Northeastern Gulf of Mexico inner shelf circulation study. OCS Rep. MMS 2011-103, Minerals Management Service, 69 pp., https://www.boem.gov/ESPIS/3/3216.pdf.
Tarjan, R., 1972: Depth-first search and linear graph algorithms. SIAM J. Comput., 1, 146–160, https://doi.org/10.1137/0201010.
Tenreiro, M., J. Candela, E. P. Sanz, J. Sheinbaum, and J. Ochoa, 2018: Near-surface and deep circulation coupling in the western Gulf of Mexico. J. Phys. Oceanogr., 48, 145–161, https://doi.org/10.1175/JPO-D-17-0018.1.
Ulam, S., 1960: A Collection of Mathematical Problems. Interscience Tracts in Pure and Applied Mathematics, No. 8, Interscience, 150 pp.
van Sebille, E., E. H. England, and G. Froyland, 2012: Origin, dynamics and evolution of ocean garbage patches from observed surface drifters. Environ. Res. Lett., 7, 044040, https://doi.org/10.1088/1748-9326/7/4/044040.
Vardner, J., and B. Loose, 2016: Molecular diffusion of CF3SF5 in pure water and artificial seawater. Mar. Chem., 180, 51–56, https://doi.org/10.1016/j.marchem.2016.01.005.
Weatherly, G. L., N. Wienders, and A. Romanou, 2005: Intermediate-depth circulation in the Gulf of Mexico estimated from direct measurements. Circulation in the Gulf of Mexico: Observations and Models, Geophys. Monogr., Vol. 161, Amer. Geophys. Union, 315–324, https://doi.org/10.1029/161GM22.
Zhurbas, V., and I. S. Oh, 2004: Drifter-derived maps of lateral diffusivity in the Pacific and Atlantic Oceans in relation to surface circulation patterns. J. Geophys. Res., 109, C05015, https://doi.org/10.1029/2003JC002241.
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