A critical comparison of Lagrangian methods for coherent structure detection

Abstract

We review and test twelve different approaches to the detection of finite-time coherent material structures in two-dimensional, temporally aperiodic flows. We consider both mathematical methods and diagnostic scalar fields, comparing their performance on three benchmark examples: the quasiperiodically forced Bickley jet, a two-dimensional turbulence simulation, and an observational wind velocity field from Jupiter's atmosphere. A close inspection of the results reveals that the various methods often produce very different predictions for coherent structures, once they are evaluated beyond heuristic visual assessment. As we find by passive advection of the coherent set candidates, false positives and negatives can be produced even by some of the mathematically justified methods due to the ineffectiveness of their underlying coherence principles in certain flow configurations. We summarize the inferred strengths and weaknesses of each method, and make general recommendations for minimal self-consistency requirements that any Lagrangian coherence detection technique should satisfy.Keywords: Lagrangian coherent structures; nonlinear dynamical systems; vortex dynamics Coherent Lagrangian (material) structures are ubiquitous in unsteady fluid flows, often observable indirectly from tracer patterns they create, for example, in the atmosphere and the ocean. Despite these observations, a direct identification of these structures from the flow velocity field (without reliance on seeding passive tracers) has remained a challenge. Several heuristic and mathematical detection methods have been developed over the years, each promising to extract materially coherent domains from arbitrary unsteady velocity fields over a finite time interval of interest. Here we review a number of these methods and compare their performance systematically on three benchmark velocity data sets. Based on this comparison, we discuss the strengths and weaknesses of each method, and recommend minimal self-consistency requirements that Lagrangian coherence detection tools should satisfy.

Kinetic Energy of Eddy‐Like Features From Sea Surface Altimetry

“…Analytical methods define eddies as coherent structures by mathematical estimations of coherence. Some of these studies include Lagrangian coherent structures identified by material rotation relative to the mean rotation of the deforming fluid volume (Haller et al, 2016;Tarshish et al, 2018), the change in location of a fluid particle induced by infinitesimal changes in its initial position (finite-time Lyapunov exponent; Beron-Vera et al, 2008;Hadjighasem et al, 2017), and geometrical analysis using transfer operators and invariant manifolds (Froyland & Padberg, 2009;Froyland et al, 2007).…”

Generalized Lagrangian coherent structures

“…4 are the computed diagnostic fields obtained from the flow from time 1 to 0, and the right panel shows extracted LCSs based on some simple choices of procedures P. We do not use the most sophisticated methods available for such extraction, because (i) we are only interested in a qualitative comparison, and (ii) more refined extraction methods will result in approximately the same entities. We moreover emphasize that our intention is not to compare or critically evaluate different LCS methods (readers are referred to papers such as [110,143] for this), but rather to illustrate that results arising from different LCS methods are not necessarily the same, and do not necessarily have a connection to GLCS sets chosen according to some specification.…”

A novel method for unsteady flow field segmentation based on stochastic similarity of direction

“…Such methods include the bottom-up approach proposed by Kuhn et al, 19 the top-down approach proposed by Guan et al, 20 and a method based on attribute-field proposed by Zhang et al 18 The Lagrangian specification is also used to discover Lagrangian coherent structures (LCS) in flows. 21 For these studies, vortices can be identified using the finite-time Lyapunov exponent (FTLE). 41 Furthermore, methods that group trajectories using network theory 42 or trajectory analysis 43 for spatial segmentation have been proposed.…”

“…For every pair of observation points (α, α ) that satisfy |α α | ≤ ε, the distance between the limiting distributions correspond to these points is calculated with Eq. (21). Consequently, the elements of the similarity matrix can be calculated from the distance as shown in Eq.…”

“…[18][19][20] Furthermore, the Lagrangian description can detect coherent structures in the flow. 21 Given that coherent structures are thought to have great physical meaning, 22 such techniques have recently become quite popular. However, these methods require very fine-grained data to calculate the trajectories of virtual particles.…”