Shape Coherence and Finite-Time Curvature Evolution

Abstract

We introduce a definition of finite-time curvature evolution along with our recent study on shape coherence in nonautonomous dynamical systems. Comparing to slow evolving curvature preserving the shape, large curvature growth points reveal the dramatic change on shape such as the folding behaviors in a system. Closed trough curves of low finite-time curvature (FTC) evolution field indicate the existence of shape coherent sets, and troughs in the field indicate most significant shape coherence. Here we will demonstrate these properties of the FTC, as well as contrast to the popular Finite-Time Lyapunov Exponent (FTLE) computation, often used to indicated hyperbolic material curves as Lagrangian Coherent Structures (LCS). We show that often the FTC troughs are in close proximity to the FTLE ridges, but in other scenarios the FTC indicates entirely different regions.Coherence has clearly become a central concept of interest in nonautonomous dynamical systems, particularly in the study of turbulent flows, with many recent papers designed toward describing, quantifying and constructing such sets. [1][2][3][4][5][6][7]. There have been a wide range of notions of coherence, from spectral, [8], to set oriented, [9] and through transfer operators [3,5] as well as variational principles [10], and even topological methods, [11,12]. Traditionally there has been an emphasis on vorticity [13], but generally an understanding that, coherent motions have a role in maintenance (production and dissipation) of turbulence in a boundary layer, [14]. A number of theories have been developed to model and analyze the dynamics in the Lagrangian perspective (moving frame), such as the geodesic transport barriers [2] and transfer operators method [3]. These have included analysis of coherence in important problems such as how regions of fluids are isolated from each other [11] including in prediction of oceanic structures [15] and atmospheric forecasting [16,17], especially for the understanding of movement of pollution including such as oil spills, [18][19][20]. Whatever the perspectives taken, we generally interpretively summarize that coherent structures can be taken as a region of simplicity, within the observed time scale and stated spatial scale, perhaps embedded within an otherwise possibly turbulent flow, [1][2][3]5].In particular, the ridges from Finite Time Lyapunov Exponents (FTLE) fields have been widely used [7,[21][22][23]] to indicate hyperbolic material curves, often called Lagranian coherent structures (LCS). We contrast here the fundamental nonlinear notions of "stretching" encapsulated in the FTLE concept to "folding" which is a complementary concepts of a nonlinear dynamical system which must be present if a material curve can stretch indefinitely within a compact domain. As we will show that exploring the much-overlooked folding concepts leads to developing curvature changes of material curves yielding an elegant description of coherence that we call shape coherence [4]. We introduce here a method of visualizing * boll...

Pinning of reaction fronts by burning invariant manifolds in extended flows

“…For passive mixing (no reaction) systems, several techniques have been proposed during the past few years to predict persistent mixing structures, including finite-time Lyapunov exponent (FTLE) fields, 33,35 hypergraph and mesohyperbolicity/mesoellipticity techniques, 36 variational approaches, 37 ergodic partition, 38 and finite-time curvature fields. 39 Alternately, there are techniques for defining invariant manifolds for chaotic sets for time-dependent flows. [40][41][42] These various techniques use the 2D equations of motion for a particle trajectory dx/dt = u x and d y/dt = u y for their analyses.…”

A critical comparison of Lagrangian methods for coherent structure detection

“…Comparisons of a limited number of methods on specific structures in individual examples have already appeared [17][18][19] . Here the objective is to perform a systematic comparison on a variety of challenging flow fields in which a ground truth can nevertheless be reasonably established.…”

Using braids to quantify interface growth and coherence in a rotor-oscillator flow

“…There are several different definitions of coherent structures used for characterizing patches of passive tracer that do not disperse under Lagrangian transport [31,44,87]. In the context of braid theory, Allshouse and Thiffeault [49] define coherent sets as a set of trajectories surrounded by an initially "simple" topological loop that does not grow, or grows subexponentially, over the duration of the braid.…”

“…The detection of coherent structures offers especially significant applications to geophysical flows, where these techniques have been used to understand climate change and plan responses to ecological catastrophes [17,[38][39][40][41]. The most commonly used tools to detect coherent structures are based on material deformation [31,[42][43][44] and on probabilistic [32,45] properties of the flow. A comparison of the wide range of approaches for detecting coherent structures can be found in Ref.…”