Applying dynamical systems techniques to real ocean drifters

19 This paper presents the first comprehensive comparison of several different dynamical-systems- 20 based measures of stirring and Lagrangian coherence, computed from real ocean drifters. Seven 21 commonly used methods (finite-time Lyapunov exponent, trajectory path length, trajectory 22 correlation dimension, trajectory encounter volume, Lagrangian-averaged vorticity deviation, 23 dilation, and spectral clustering) were applied to 135 surface drifters in the Gulf of Mexico in 24 order to map out the dominant Lagrangian coherent structures. Among the detected structures 25 were regions of hyperbolic nature resembling stable manifolds from classical examples, 26 divergent and convergent zones, and groups of drifters that moved more coherently and stayed 27 closer together than the rest of the drifters. Many methods highlighted the same structures, but 28 there were differences too. Overall, 5 out of 7 methods provided useful information about the 29 geometry of transport within the domain spanned by the drifters, whereas the path length and 30 correlation dimension methods were less useful than others. 31


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Techniques from the dynamical systems theory can be used to study transport and exchange 59 processes in oceanic flows (Haller,  not typical for oceanographic applications due to high costs of vessels and manpower. However, 84 it allowed populating the domain with drifters in a manner most suitable for the dynamical 85 systems applications. Thus, this dataset provided a unique and long-awaited opportunity to try 86 applying the dynamical systems techniques to real, rather than simulated, ocean drifters and to 87 identify the real, rather than simulated, ocean LCS. 88 In this paper, seven commonly used dynamical systems techniques were applied to the real

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We start with a brief review of the 7 dynamical systems techniques that we will use. FTLEs is straightforward, computationally inexpensive, and robust with respect to noise, which 113 makes FTLEs one of the most popular methods in oceanographic studies of transport and 114 mixing. Importantly, FTLEs are also frame-independent and thus give consistent results in any 115 translating or rotating reference frame (Haller 2005;2015). Here Δ 0, and Δ are the initial and final distance in the i th -direction between neighboring 123 trajectories. This algorithm requires dense regularly-spaced orthogonal grids of trajectories. For 124 the SPLASH dataset, we manually chose quadruplets of 4 neighboring trajectories that form a 125 near-rectangle, define the local orthogonal coordinate system most strongly aligned with the axes 126 of the near-rectangle, and then estimate FTLEs using eq. (1) for the center of mass of each 127 quadruplet ( Fig. 2 shows the quadruplets and their centers of mass locations).

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Here  deployment (but because this metric is still sensitive to differences between hyperbolic/elliptic 209 behaviors even for a small deployment, it might still be able to highlight regions with different 210 transport characteristics, so we go ahead and apply it to SPLASH drifters in the next section).  Note that would only be able to identify those rotationally-coherent Lagrangian eddies 231 that are smaller than, and lay entirely within, the domain seeded with drifters. The last method for identifying the LCSs that we will be testing using drifter data is the

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Being based on the distances between trajectories, spectral clustering is frame-independent.

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Note that methods other than LLS can also be used to compute divergence and vorticity. polygon, and the polygon aspect ratio is ≤ 6. If only the aspect ratio condition is not satisfied 292 (but the number of drifters, the distance, and the center of mass conditions are), we will still 293 compute LLS estimates but we will refer to them as less trustworthy (and mark them by colored 294 diamonds). In all other cases, we do not produce estimates of divergence and vorticity.

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We start by qualitatively separating the motion of drifters into three stages. For about a day after 298 deployment, all drifters started moving together in an anticyclonic fashion to the north and then 299 northeast towards the coast (Fig. 1) this is what we will refer to as the initial stage of motion.

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Upon approaching the shelf, the drifters halted their on-shore motion and split into two groups, a 301 smaller northern group that headed northward along the coast and a larger southern group that  Finally, during the third stage of motion, the drifters started moving off-shore to the southwest.

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As they progress further from the coast, trajectories started exhibiting more looping and the 314 drifters dispersed further apart from one another, although they still remained in an elongated 315 filament configuration (not anymore aligned with the coast) all the way until day 5, which is the 316 end time of this dataset.

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Having split the drifter movement into 3 stages, we next apply our Lagrangian methods to 318 trajectory segments from = 0 days until = 0.5, 1, and 3 days, respectively (top,  Overall, was no more useful than in identifying the LCS, and, like , had the same frame 388 dependence issues. It is interesting to compare and contrast with FTLEs, which became sort of a benchmark for 414 the LCS detection problems, being frame independent, commonly used, and easy to compute.

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There are significant differences between the distributions of the two metrics, reflecting Overall, despite some challenges with undersampling, the encounter volume proved to be an 432 interesting frame-independent diagnostic that was sensitive to both enhanced clustering, 433 hyperbolic behavior, and flow convergence, and was complementary to FTLEs. this anticyclonic eddy. We think this might be because the SPLASH release domain was too 490 small and was located entirely within this vortex structure.

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Spectral Clustering (Fig 9): At early times, the number of coherent clusters identified by the SC 492 algorithm was quite large (12), although some clusters only contained a few drifters. (Recall that 493 the optimized-parameter SC is able to autonomously identify the optimal number and optimal 494 size of the clusters, without input from the user). Among the detected clusters, the yellow cluster  It is interesting to note that the two frame-dependent methodsand , which were of drifters during the first day after release). Note that even the method, which was 612 specifically designed to identify rotationally-coherent Lagrangian eddies, was also not able to 613 highlight this anticyclone, possibly because is a wrong tool for identifying an eddy from a 614 small trajectory set located entirely within an eddy. It is also possible that this anticyclone did 615 not possess a Lagrangian core, or the core was located outside of the drifter release domain 616 and/or was not properly resolved by the SPLASH drifters.

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In the future, it would be interesting to repeat the experiment with the drifter deployment site and 618 the release pattern optimized for capturing specific LCSs whose presence could have been 619 predicted based on a model or satellite data.

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The very rapid nature of evolution at submesoscales may cause an evenly spaced array of drifters were most reliable at shorter times and still meaningful at longer times in regions with strong 644 hyperbolic-type LCS located far enough away from each other to be resolved by the deployment 645 grid. and were reliable at all times, but since they did not identify any hyperbolic, elliptic, 646 or convergence-type LCS for SPLASH drifters, they were perhaps least useful among the 7 647 methods. In contrast to FTLEs, was not reliable at short times but improved its reliability at 648 longer times. and worked well at short times when drifters were still relatively close 649 and didn't form elongated filaments, but deteriorated at longer times due to the rapid