Robust FEM-Based Extraction of Finite-Time Coherent Sets Using Scattered, Sparse, and Incomplete Trajectories

Abstract

Transport and mixing properties of aperiodic flows are crucial to a dynamical analysis of the flow, and often have to be carried out with limited information. Finite-time coherent sets are regions of the flow that minimally mix with the remainder of the flow domain over the finite period of time considered. In the purely advective setting this is equivalent to identifying sets whose boundary interfaces remain small throughout their finite-time evolution. Finite-time coherent sets thus provide a skeleton of distinct regions around which more turbulent flow occurs. They manifest in geophysical systems in the forms of e.g. ocean eddies, ocean gyres, and atmospheric vortices. In real-world settings, often observational data is scattered and sparse, which makes the difficult problem of coherent set identification and tracking even more challenging. We develop three FEM-based numerical methods to efficiently approximate the dynamic Laplace operator, and introduce a new dynamic isoperimetric problem using Dirichlet boundary conditions. Using these FEM-based methods we rapidly and reliably extract finite-time coherent sets from models or scattered, possibly sparse, and possibly incomplete observed data.

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

“…It is important to emphasize that this approach is effective for identifying structures associated with individual flow trajectories, but is not intended to identify the full set of coherent structures in a flow, as is a common goal with other clustering and spectral graph theory methods 6,8,11 . Application of the method described here to the full set of Lagrangian trajectories could potentially be used to identify the full set of coherent structures associated with those trajectories.…”

“…This unique definition leads to corresponding changes to the conventional adjacency matrix and its analysis, as described in this paper. The more versatile definition of coherence renders the problem of identifying all of the coherent structures in the flow as ill-posed, making the present method distinct from, and complementary to, other coherent structure identification algorithms [6][7][8]10,11 . We show that it is generally more useful to consider an individual Lagrangian particle trajectory, and to identify the set of other trajectories that are coherent with the reference trajectory of interest.…”

Fast and robust computation of coherent Lagrangian vortices on very large two-dimensional domains

“…These velocities are derived from satellite altimetry data. We choose a time window (starting November 26, 2016) for which a small spatial piece has been used repeatedly as a data set "benchmark" case in the literature; cf., for instance, [24,14], and earlier studies on slightly larger domains [23,29]. The domain studied in the first references is highlighted by a small white rectangle in Figs.…”

Higher-order finite element approximation of the dynamic Laplacian

“…To this end, a scheme based on radial basis functions had been proposed [13], which showed high order of convergence, but suffered from a number of drawbacks like high sensitivity with respect to the radius parameter, a non-real spectrum and non-sparseness of the discretized operator. In [14], two finite element schemes were proposed (the "Cauchy-Green" (CG) and the "Transfer Operator" (TO) approach), which eliminated each of these drawbacks.…”

“…Eigenfunctions of this operator are used to identify and track finite-time coherent sets, which physically manifest in fluid flows as jets, vortices, and more complicated structures. Two robust and efficient finite-element discretisation schemes for numerically computing the dynamic Laplacian were proposed in [14]. In this work we consider higher-order versions of these two numerical schemes and analyse them experimentally.…”