Identification of individual coherent sets associated with flow trajectories using coherent structure coloring

Abstract

We present a method for identifying the coherent structures associated with individual Lagrangian flow trajectories even where only sparse particle trajectory data is available. The method, based on techniques in spectral graph theory, uses the Coherent Structure Coloring vector and associated eigenvectors to analyze the distance in higher-dimensional eigenspace between a selected reference trajectory and other tracer trajectories in the flow. By analyzing this distance metric in a hierarchical clustering, the coherent structure of which the reference particle is a member can be identified. This algorithm is proven successful in identifying coherent structures of varying complexities in canonical unsteady flows. Additionally, the method is able to assess the relative coherence of the associated structure in comparison to the surrounding flow. Although the method is demonstrated here in the context of fluid flow kinematics, the generality of the approach allows for its potential application to other unsupervised clustering problems in dynamical systems such as neuronal activity, gene expression, or social networks. Keywords: fluid mechanics, clustering, graph theoryIn recent years, there has been a proliferation of techniques that aim to characterize fluid flow kinematics on the basis of Lagrangian trajectories of collections of tracer particles. Most of these techniques depend on presence of tracer particles that are initially closely-spaced, in order to compute local gradients of their trajectories. In many applications, the requirement of close tracer spacing cannot be satisfied, especially when the tracers are naturally occurring and their distribution is dictated by the underlying flow. Moreover, current methods often focus on determination of the boundaries of coherent sets, whereas in practice it is often valuable to identify the complete set of trajectories that are coherent with an individual trajectory of interest. We extend the concept of Coherent Structure Coloring, an approach based on spectral graph theory, to achieve identification of the coherent set associated with individual Lagrangian trajectories. The method does not require a priori determination of the number of coherent structures in the flow, nor does it require heuristics regarding the eigenvalue spectrum corresponding to the generalized eigenvalue problem. Importantly, although the method is demonstrated here in the context of fluid flow kinematics, the generality of the approach allows for its potential application to other unsupervised clustering problems in dynamical systems such as neuronal activity, gene expression, or social networks.

Simultaneous coherent structure coloring facilitates interpretable clustering of scientific data by amplifying dissimilarity

“…On this basis, we previously hypothesized that these complex datasets can be clustered more robustly and effectively by amplifying state differences rather than state similarity [14, 15]. The rationale for this approach is that the exponential separation of dissimilar states can provide more sensitive detection of clusters than a focus on state similarity, the latter requiring longer observation to become apparent [25, 26].…”

“…Given the analogy between this approach and the problem of fuzzy graph coloring [30], wherein the connected nodes of a graph with large weights are assigned the most disparate values, we call this method coherent structure coloring (CSC) [14]. The technique has recently been demonstrated to successfully identify coherent eddies and jets associated with individual fluid particle trajectories in model geophysical flows [15].…”

“…Hence, although multiple dimensions of dissimilarity are almost always present in real data, the method cannot simultaneously distinguish between multiple types of dissimilarity in a dataset. Moreover, the method applied to individual fluid particle trajectories in a subsequent study required a subjectively defined threshold to calculate eigenspace distances [15], and it was shown to produce degenerate results for fluid particles in chaotic regions of the flow (cf. Fig 7 in Ref.…”

“…To perform sCSC, we begin with the most effective partition given by the eigenvector associated with the maximum eigenvalue, and proceed through the set of orthogonal eigenvectors in order of decreasing eigenvalue. This approach simultaneously reveals the coherent sets of the system, and eliminates the subjective user intervention required in the previous method [15].…”

“…The method of coherent structure coloring (CSC), introduced in Refs. [14] and [15], was designed to address these and other limitations of clustering algorithms for coherent structure determination based on trajectories of particle spatial coordinates. In this work, we extend CSC in the context of its own limitations, as described below.…”