Nearly Recurrent Components in 3D Piecewise Constant Vector Fields

Abstract

We present an algorithm for computing nearly recurrent components, that represent areas of close to circulating or stagnant flow, for 3D piecewise constant (PC) vector fields defined on regular grids. Using a number of analytical and simulated data sets, we demonstrate that nearly recurrent components can provide interesting insight into the topological structure of 3D vector fields. Our approach is based on prior work on Morse decompositions for PC vector fields on surfaces and extends concepts previously developed with this goal in mind to the case of 3D vector fields defined on regular grids. Our contributions include a description of trajectories of 3D piecewise constant vector fields and an extension of the transition graph, a finite directed graph that represents all trajectories, to the 3D case. Nearly recurrent components are defined by strongly connected components of the transition graph.