Ordering of trajectories reveals hierarchical finite-time coherent sets in Lagrangian particle data: detecting Agulhas rings in the South Atlantic Ocean

The detection of finite-time coherent particle sets in Lagrangian trajectory data using data clustering techniques is an active research field at the moment. Yet, the clustering methods mostly employed so far have been based on graph partitioning, which assigns each trajectory to a cluster, i.e. there is no concept of noisy, incoherent trajectories. This is problematic for applications to the ocean, where many small coherent eddies are present in a large fluid domain. In addition, to our knowledge none of the existing methods to detect finite-time coherent sets has an intrinsic notion of coherence hierarchy, i.e. the detection of finite-time coherent sets at different spatial scales. Such coherence hierarchies are present in the ocean, where basin scale coherence coexists with smaller coherent structures such as jets and mesoscale eddies. Here, for the first time in this context, we use the density-based clustering algorithm OPTICS (Ankerst et al., 1999) to detect finite-time coherent particle sets in Lagrangian trajectory data. Different from partition based clustering methods, OPTICS does not require to fix the number of clusters beforehand. Derived clustering results contain a concept of noise, such that not every trajectory needs to be part of a cluster. OPTICS also has a major advantage compared to the previously used DBSCAN method, as it can detect clusters of varying density. Further, clusters can also be detected based on density changes instead of absolute density. Finally, OPTICS based clusters have an intrinsically hierarchical structure, which allows to detect coherent trajectory sets at different spatial scales at once. We apply OPTICS directly to Lagrangian trajectory data in the Bickley jet model flow and successfully detect the expected vortices and the jet. The resulting clustering separates the vortices and the jet from background noise, with an imprint of the hierarchical clustering structure of coherent, small scale vortices in a coherent, large-scale, background flow. We then apply our method to a set of virtual trajectories released in the eastern South Atlantic Ocean in an eddying ocean model and successfully detect Agulhas rings. At larger scale, our method also separates the eastward and westward moving parts of the subtropical gyre. We illustrate the difference between our approach and partition based k-Means clustering using a 2-dimensional embedding of the trajectories derived from classical multidimensional scaling. We also show how OPTICS can be applied to the spectral embedding of a trajectory based network to overcome the problems of k-Means spectral clustering in detecting Agulhas rings.