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 in the ocean, where many small, coherent eddies are present in a large, mostly noisy fluid flow. Here, for the first time in this context, we use the density-based clustering algorithm of OPTICS (ordering points to identify the clustering structure; Ankerst et al., 1999) to detect finite-time coherent particle sets in Lagrangian trajectory data. Different from partition-based clustering methods, 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 density-based spatial clustering of applications with noise (DBSCAN) method, as it can detect clusters of varying density. The resulting clusters have an intrinsically hierarchical structure, which allows one 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. We illustrate the difference between our approach and partition-based k-means clustering using a 2D 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.