We apply our method to a model system that has been used frequently in studies to detect finite-time coherent sets . The velocity field of the quasi-periodically perturbed Bickley jet is defined by a stream function ψ(x,y,t), i.e. x˙=-∂ψ∂y and y˙=∂ψ∂x, with ψ(x,y,t)=ψ0(y)+ψ1(x,y,t) consisting of a stationary eastward background flow as follows: 1ψ0(y)=-ULtanh⁡(y/L), and a time-dependent perturbation, as follows: 2ψ1(x,y,t)=ULsech2(y/L)Re∑n=13fn(t)exp⁡(iknx), where Re(z) denotes the real part of the complex number z. We use the same parameter values as , with U=62.66 m/s the characteristic velocity of the zonal background flow, and L=1770 km. The parameters in Eq. () are given by kn=2n/r0 and fn(t)=ϵnexp⁡(-ikncnt), with ϵ1=0.075, ϵ2=0.4, ϵ3=0.3, c1=0.1446U, c2=0.205U and c3=0.461U. The domain of interest is Ω=[0,πr0]×[-3000km, 3000km], where r0=6371 km is the radius of the Earth, and the left and right edges of Ω are identified, i.e. the flow is periodic in the x direction with period πr0. Similar to , we seed the domain with an initial number of 12 000 particles on a uniform 200×60 grid. For this choice, the initial particle spacing is slightly above 100 km in both directions. We compute the trajectories for 40 d with a time step of 1 s using the SciPy integrate package. We output the trajectories every day, i.e. we have T=41 data points in time for each trajectory.

To test the OPTICS algorithm with a more realistic ocean flow, we simulate surface particle trajectories in a strongly eddying ocean model. Surface velocities are derived from a Nucleus for European Modelling of the Ocean (NEMO) ORCA-N006 run , which has a horizontal resolution of 1/12∘ and velocity output for every 5 d. The model is forced by reanalysis and the observed data of wind, heat and freshwater fluxes , i.e. the currents do not only contain the geostrophic component, as is the case in altimetry-derived currents . For the advection of virtual particles, we use version 1.11 of the open source Parcels framework see http://oceanparcels.org/, last access: 2 January 2021. The 2D surface current velocity is interpolated in space and time with the C-grid interpolation scheme of , using a fourth-order Runge–Kutta method with a time step of 10 min. We initially distribute particles uniformly in the ocean on the vertices of a 0.2∘×0.2∘ grid in the domain (30∘ W, 20∘ E) × (40∘ S, 20 ∘ S), which corresponds to a total number of 23 821 particles. At 30∘ S, a spacing of 0.2∘ corresponds to roughly 20 km. The particles start on 5 January 2000 and are advected for 2 years. We output the trajectories with a time interval of 5 d. We only use the first 100 d as data to detect the finite-time coherent sets, i.e. we have T=21 data points for each trajectory, but also look at later times to see how long the rings need to disperse. We provide the used trajectory data for the Agulhas flow as a NumPy file on Zenodo .

For N trajectories of dimension D and length T, the trajectory information can be stored in a data matrix X∈RN×DT, where each row results from a particle trajectory by concatenating the different spatial dimensions. The analysis of the trajectory data to detect the finite-time coherent sets of trajectories can be split into the following two essential steps:  

Step 1.  Embedding of the trajectories in an abstract (metric) space, i.e. X→X¯∈RN×M, where M≤DT. If one uses a dimensionality reduction method, then M<DT.

The embedding is necessary to represent the trajectories as points in a metric space. Different options for embedding the trajectories exist, e.g. a direct embedding of the data points along the trajectories or embeddings based on the eigenvectors derived from networks that are defined by physically motivated trajectory similarities . Once an embedding of each trajectory as a point in a metric (typically Euclidean) space is established, one can apply a clustering algorithm. Roughly speaking, clustering algorithms try to identify groups of points that are close to each other as a cluster. Partition-based clustering methods divide the entire data set into a (typically fixed) number of K clusters, such that each data point belongs to a cluster. The most popular method in this category is the k-means algorithm, which tries to find a given number of K clusters such that the sum of the pairwise squared distances of points within a cluster is minimized. Other clustering algorithms contain a concept of noisy data, i.e. data points that do not belong to any cluster or belong to a cluster only with a certain probability. Examples of the former case are density-based spatial clustering of applications with noise DBSCAN;, as discussed by in the fluid dynamics context, and the OPTICS algorithm presented here. For the latter case, the most popular method is fuzzy-c-means clustering, as discussed by in the context of finite-time coherent sets.

Figure shows a few possible options for trajectory embedding and clustering that have partially been explored before (see the footnotes in the figure for the combinations used in related studies). For a given trajectory data set, one can, in principle, apply an arbitrary combination of embedding and clustering methods. Only a few of the different combinations have been explored so far, and many more options for embedding and clustering (like those shown in Fig. ) exist. It is important to note that a good choice of embedding and clustering might well depend on the specific problem at hand, and there might be no combination that performs well for all possible situations.

Most of the studies that use clustering techniques to detect finite-time coherent sets have focused on developing new forms of trajectory embeddings. For example, , , and all use different forms of spectral embeddings together with k-means clustering. have developed a powerful form of embedding based on a sparse eigenbasis approximation. Here, we focus on the clustering step in Fig.  and propose the OPTICS clustering algorithm in the fluid dynamics context. We test the algorithm for the following three different kinds of embeddings: E1.

A direct embedding of the trajectory data in a high-dimensional Euclidean space, i.e. M=DT (see Sect. ).

In the following sections, we explain in detail the embeddings of E1 and E2 and the OPTICS algorithm. We introduce the network embedding of E3 together with the corresponding results in Appendix .

The detection of dense accumulations of points that are separated from each other by non-dense regions (noise) is the main goal of density-based clustering. We use the OPTICS algorithm by to detect these regions. The OPTICS algorithm can be seen as an extension of DBSCAN . As we have no prior information on the density structure of the embedded nodes, we set the generating distance of OPTICS to infinity, and our presentation here is limited to this case. The general OPTICS algorithm with finite generating distance is computationally more efficient and slightly more complicated, and we refer to for more details.

For δ∈R, the δ neighbourhood of a point p∈RM is defined as the M-dimensional ball of radius δ around p. We define Mδ(p) as the number of points that is in the δ neighbourhood of p, including p itself. OPTICS requires one parameter, i.e. an integer smin⁡ (called MinPts by ), to define the core distance of a point p as follows: 6c(p)={min⁡(δ)|Mδ(p)≥smin⁡}.

The core distance is simply the minimum radius of a ball around p, such that the ball contains smin⁡ points. Note that the generating distance that we set to infinity is a maximum cut-off distance for the computation of the core distance in Eq. (), beyond which the core distance is not defined. As we do not have an intuition for a good value of such a cut-off, we remove it by setting it to infinity.

The ordering of the points is based on the reachability distance of a point p with regards to another point q and is defined as follows: 7r(p|q)=max⁡(c(q),||p-q||), where ||p-q||, in our case, denotes the Euclidean distance between p and q. The ordering of points is then constructed with the following scheme:  

Step 1.  Pick a point p1. This is the first point in the order, and it is arbitrary.

Note that the ordering of points is achieved by constantly updating the ordered seed list (see step 3). In this way, the algorithm iterates through groups of dense points, one after the other, and it only continues with other points once a dense region has been fully explored. Note also that the entire algorithm depends on the choice of the parameter smin⁡. The value of smin⁡ should be chosen roughly as a minimum value of the expected cluster size. In the examples presented in this paper, we take values for smin⁡ that correspond to the estimated minimum size of the coherent sets.

The main result of the OPTICS algorithm is a reachability plot. This plot is the graph defined by (i,r(pi)), where r(p1)=∞ by definition. The reachability plot is a powerful presentation of the global and local distribution of a set of points at once. The valleys in this plot correspond to dense regions, which we relate to finite-time coherent sets. We show examples of reachability plots in Sect. . Given the reachability plot (i,r(pi)), we use the following two common ways to derive a clustering result:

DBSCAN clustering. Choose a cut-off parameter ϵ and define all points pi with c(pi)≤ϵ as core points. All points that are not in the ϵ neighbourhood of a core point are defined as noise. This set of noisy data points is equivalent to all points pi that are not core points and have a reachability value r(pi) with r(pi)>ϵ. A cluster of size L is then defined as a consecutive set (in the sense of the ordering) of non-noise points (pj,pj+1,…,pj+L-1), with the adjacent points of pj-1 and pj+L being noise. This is similar to the clustering result of a DBSCAN run with equal values for smin⁡ and ϵ. All possible realizations of DBSCAN clusters, with the same value for smin⁡, can therefore be derived from the reachability values, core distances and the ordering determined by OPTICS. Up to boundary points, a DBSCAN clustering result can be obtained by drawing horizontal lines in the reachability plot (see Sect. ).

We refer to for a more detailed discussion of the ξ-clustering method, with illustrations for example data. Note that the full ξ-clustering method presented by contains some more details related to the choice of the start and end points which we did not mention here.

The OPTICS algorithm and functions for deriving both clustering results from an OPTICS output are available in the scikit-learn library in Python. Note that the implementation in the scikit-learn library allows for a minimum cluster size that is different from smin⁡ for the ξ-clustering method (item 2c above), but we will not make use of this additional freedom to reduce the number of parameters. Note that, different from k-means, both clustering methods do not require an a priori determination of the number of clusters. For the ξ-clustering method, a larger ξ requires steeper boundaries to form a cluster, i.e. it will typically lead to a reduction in the number of resulting clusters. For DBSCAN clustering with very large ϵ, one will detect one large global cluster. Making ϵ smaller then leads to consecutive splits of this cluster, forming (up to noise) a cluster hierarchy. We will demonstrate the properties for both clustering methods in Sect.  for different situations. In the following applications, we use an estimation of the minimum number of particles per finite-time coherent set for the parameter smin⁡.

Intuitively, the two clustering methods can be understood as follows. DBSCAN detects those groups of points that have a certain minimum density defined by the minimum reachability distance ϵ. Clusters detected by DBSCAN are therefore defined by a global density criterion. This assumes no structural differences in the type of coherent sets in different regions of the fluid. Different from that, the ξ-clustering method detects clusters by finding strong changes in the density of the data points, and it is not based on absolute densities. This has an advantage in that clusters of different absolute densities can be detected. Such a situation can arise if the distribution of particles is inhomogeneous over the fluid domain or if the spatial extend of the fluid domain is very large, such that the properties of finite-time coherent sets vary significantly. It is important to note that the main result of OPTICS is the reachability plot itself. The DBSCAN- and ξ-clustering methods should be seen as useful tools for identifying the most important features of that plot.

Our method is closely related to existing methods for detecting finite-time coherent sets with clustering techniques. Most notably, also use a direct embedding of individual trajectories similar to Eq. (), together with fuzzy-c-means clustering. , , and use spectral embeddings of graphs that are defined by some form of physical intuition, or by dynamical operators, together with k-means clustering. These studies show applications of their methods to example flows where the size of almost-coherent sets is not too small compared to the fluid domain. Such examples are the Bickley jet flow, which we also study in Sect. , the five major ocean basins , a few individual eddies in an ocean or atmospheric flow . In such situations, noisy background trajectories can be detected as individual clusters by the partitioning method, as discussed by . For applications in large ocean domains, where the number of eddies is not known beforehand and where there are many more noisy trajectories than coherent trajectories, such an approach is likely to fail see also the discussion by. OPTICS does not require fixing the number of clusters beforehand, and it also contains an intrinsic concept of noisy trajectories that do not belong to any cluster, making OPTICS suitable for challenging flows in large domains.

As mentioned, OPTICS also contains an intrinsic notion of cluster hierarchy, i.e. coherent sets that are themselves part of coherent sets at larger scales. studied hierarchical coherent sets in the transfer operator framework of , in the spirit of the hierarchical clustering method proposed by . Their approach is also partition based, i.e. there is no concept of noisy trajectories. In addition, at each stage of the hierarchy, a fixed cut-off has to be chosen based on minimizing an objective function . Different from that approach, the main result of OPTICS, the reachability plot, contains such hierarchical information in a smooth and intrinsic manner.

As described in Sect. , clustering results of the DBSCAN algorithm can be derived from the reachability plot of OPTICS. DBSCAN has been used in the context of coherent sets before by , although not to identify specific clusters but to distinguish noisy from clustered trajectories. The potential of density-based clustering for applications in the ocean, and its comparison to other existing clustering methods for flow examples such as the Bickley jet (see Sect. ), has not been explored so far. Different from OPTICS, DBSCAN detects clusters with a certain fixed minimum density, although clusters with varying densities might be present in a data set . More specifically, the value for the cut-off parameter ϵ (see Sect. ) has to be set beforehand. Choosing a good value for the density parameter in DBSCAN is challenging if there is no underlying physical intuition for the density structure. As described in Sect. , OPTICS allows one to derive any DBSCAN clustering result, with the same value for the parameter smin⁡, after computing the reachability plot, i.e. after one can obtain the first insights into the clustering structure of the data set to make an appropriate choice for ϵ. Furthermore, it also allows one to use the ξ-clustering method instead of DBSCAN (see Sect. ).

A more recent and powerful technique for detecting finite-time coherent sets in sparse trajectory data was presented by , based on dynamic Laplacian and transfer operators . apply their method to a trajectory data set in the western boundary current region in the North Atlantic Ocean and successfully detect many eddies by superposing individual eigenvectors. The methods presented there are based on a form of spectral embedding derived from discretized dynamical operators. Based on this embedding, clustering results have also been derived with k-means by and with individual thresholding by . also show how the low-order eigenvectors correspond to large-scale coherent features, while the individual eddies are derived by a sparse eigenbasis approximation of a number of eigenvectors. The latter approach is essentially a transformation of the embedding to represent the most reliable features, such that a superposition of the eigenvectors alone yields the information about the location and size of finite-time coherent sets (without a clustering step). This is essentially an optimized form of embedding, i.e. the second step in Fig. . Our aim here is to focus on the third step in Fig. , i.e. to demonstrate the potential of the density-based clustering algorithm OPTICS, together with a very simple embedding of Eq. ().

A downside of our method compared to other approaches is the rather ad hoc choice of embedding (see Eq. ). Different from many other methods, most notably the ones of , and , this type of embedding is not derived from a meaningful dynamical operator. It could be fruitful to explore a combination of these more meaningful embeddings together with OPTICS as a clustering algorithm in future research.

We start with the direct embedding of the Bickley jet flow trajectories (see Sect. ). The data matrix has the dimension X∈R12000×123. We apply the OPTICS algorithm to the resulting points, together with DBSCAN clustering, choosing smin⁡=80 as a minimum size of the finite-time coherent sets. In the following, all axis units are in multiples of 1000 km. Figure  shows the reachability plot, together with the DBSCAN clustering result of three different choices of ϵ. The six vortices and the jet are clearly visible as the major valleys in the reachability plot. The hierarchical structure of the DBSCAN clustering with decreasing ϵ is visible in the figures from top (large-scale coherence) to bottom (small-scale coherence). Note that for the DBSCAN clustering results, boundary points of the clusters can be above the horizontal line at y=ϵ. This is because of the definition of the DBSCAN clustering in Sect. .

To illustrate the difference between OPTICS and k-means, we use the embedded trajectories and apply classical MDS to obtain a 2D embedding. As described in Sect. , this assures the capturing of the major variance along the embedding axes. The spectrum of B in Eq. () is shown in Fig.  in the appendix, with two clearly dominant eigenvalues. The fact that there are two very dominant eigenvalues ensures that the illustration of the data in the plane captures the major variance in the data points. Figure a shows the corresponding embedding of the trajectories in the 2D Euclidean space. The star-shaped distribution of data points reflects the strong symmetries of the underlying idealized Bickley jet flow. Such symmetry is not expected to be present for more realistic flows. Figure b and c show the cluster labels for OPTICS with DBSCAN clustering at ϵ=106 km, and for a k-means clustering with K=8 clusters, respectively. K=8 corresponds to the six vortices, the jet, and one noise cluster as suggested by .

The corresponding clustering results in real space are shown in Figs.  and for OPTICS and k-means, respectively. The jet and the six vortices are clearly recognizable as dense accumulations of points in the 2D space of Fig. b (see Fig.  for the corresponding colours). The clustering result with k-means in Fig.  shows that the clusters corresponding to the vortices are much less focused. In addition, each of the eight clusters in Fig. c contains some of the noisy points of Fig. b, which shows that using one additional cluster for noise does not work in this situation. It is interesting to note that capturing the noisy data points of Fig. b with an additional cluster in k-means is geometrically impossible, simply because k-means clusters are circular. Covering all noisy points without including the centre, i.e. the jet in Fig. b, is not possible for k-means.

It should be noted here that the poor performance of k-means in Figs. c and is not representative for other methods that use k-means. For example, the method of captures the coherent structures in the Bickley jet rather well, including the jet in the middle. We emphasize again that we use classical MDS here mostly for visualization purposes as the computation of the classical MDS embedding is difficult for large particle sets. In our case, a dense 12000×12000 symmetric matrix has to be diagonalized, which already takes a significant amount of computation time.

We finally also tested the performance of our algorithm with a random subset of 2000 particles, using data for every 5 d instead of every day (see Fig.  in the Appendix). OPTICS still detects the six vortices and the jet, although the cluster boundaries are less clearly defined compared to Fig. . detect the vortices and the jet by using the data of 3000 particles only at initial and final times (t=0 and t=40 d). Our method is not able to detect the expected finite-time coherent sets by using only initial and final particle data. This is likely to be a result of the ad hoc direct embedding; see Eq. () and the discussion at the end of Sect. .

We next apply OPTICS to the Agulhas trajectories. As described in Sect. , we have X∈RN×63 with N=23821. We choose smin⁡=100 in the following, which corresponds initially to a square cell of 2∘×2∘, i.e. a reasonable minimum size of an Agulhas ring. Figure  shows the result of the direct embedding. The reachability plot in Fig. a is much more jagged than for the Bickley jet model flow (see Fig. a). The narrow, deep valleys and the wider valleys in the reachability plot indicate the presence of large- and small-scale coherence patterns. Figure a–c show the DBSCAN clustering result for a relatively large value of ϵ. The main separation of fluid domains is between the red and the blue particles, with a few vortices at their boundary. These two water masses are the northern and southern parts of the subtropical gyre in the South Atlantic, with the red particles moving to the west and the blue particles moving to the east. The second and third rows of Fig.  show other clustering results for the DBSCAN- and the ξ-clustering method, respectively. The valleys in Fig. g with steepest boundaries, as detected by the ξ-clustering method, mostly correspond to eddy-like structures separated by background noise. Note that not all clusters in the figure correspond to eddies. For example, the blue cluster in Fig. g–i stays approximately coherent over the considered time interval, although it is certainly not an Agulhas ring. An animation of the detected finite-time coherent sets for the full 2 years of trajectory data, based on the ξ-clustering method as in the last row of Fig. , can be found on Zenodo , showing that many of the sets stay coherent for significantly longer times than the first 100 d.

Figure shows that, for this situation, the ξ-clustering method detects more Agulhas rings than DBSCAN. While the clustering results shown in the figure all depend on the parameter values for ξ and ϵ, it is visible in the reachability plot of Fig. g that the definition of some eddies includes the entire boundary of the valleys, i.e. up to very high reachability values. At the same time, the detection of the large-scale clusters, as in Fig. a–c, is not possible with the ξ-clustering method. These findings are in fact expected; see the discussion of the two clustering methods at the end of Sect. . DBSCAN is best for detecting global density structures, i.e. when the reachability values of all points are compared to the same cut-off ϵ. Regions that are dense locally but not necessarily globally are better detected with the ξ-clustering method. Despite these differences between the two clustering methods, we again emphasize that the main result of OPTICS is the reachability plot itself. Figure  shows a colour map at the initial time of the reachability values. We clearly see Agulhas rings as the dark regions corresponding to lowest values of reachability. The regions of large reachability correspond to trajectories that are relatively noisy compared to all the other trajectories.

In order to illustrate again the difference between OPTICS and k-means for this example, we choose 12 000 random trajectories and again embed the trajectories in a 2D space with classical MDS (see Sect. ). The reduction in the particle set is necessary for simplifying the eigendecomposition of the matrix B in Eq. (), and we therefore choose smin⁡=30. The corresponding spectrum of B is shown in Fig.  in the Appendix, showing that there are again two dominant eigenvectors, i.e. visualizing the network in the plane captures the main variance of the data. Figure  shows the embedded trajectories together with OPTICS and DBSCAN clustering (Fig. b) and k-means (Fig. c) for K=40. Figures  and show the corresponding clustering results in the fluid domain. It is clear that k-means does not detect a single vortex but instead splits the fluid domain into regions of approximately similar size. OPTICS detects multiple Agulhas rings by finding the deepest valleys in the reachability plot.

It is interesting to note that the use of classical MDS in Fig.  has led to the detection of many of the vortices of Fig. d–f with DBSCAN instead of the ξ-clustering method. The transformation to the reduced 2D space has hence led to a simplification of the reachability plot, which now represents the major variations in the distances of the embedded trajectories. At the same time, the large-scale structure of Fig. a is not visible any more in Fig. . This indicates that exploring more dimensionality-reduction techniques could be useful for future research, in particular for those that are computationally more efficient than classical MDS.

Spectral embeddings derived from networks, together with partition-based clustering, have a similar problem to the one illustrated in Figs. c and . Similar to the case discussed here, OPTICS can be used to overcome the problems of k-means. We show this in Appendix  for the network proposed by for the Agulhas region, together with a brief introduction to the network and how to construct spectral embeddings. In summary, k-means again fail to detect any of the vortices, while OPTICS detects many of the coherent vortices in the spectrally embedded network. Yet, other flow features are also present that result from the physical motivation of the network definition (see the results in Appendix ).