Curvature Fields and Topologically Special Points
Lagrangian trajectories are nothing more than curves in space. In the two-dimensional case, these curves can be completely characterized by their curvature, a purely geometrical quantity. It is convenient, however, to parameterize the system by time; in this case, the curvature is given as the ratio of the magnitude of the acceleration perpendicular to the direction of motion to the square of the velocity. Previously, the probability density function of curvature has been studied (see Xu, Ouellette, and Bodenschatz, Phys. Rev. Lett. 98:050201, 2007), and its shape has been explained simply using Gaussian statistics. Instead, we have computed the Eulerian field of this Lagrangian quantity, and have found that it gives us geometric and topological information about the entire flow.
We have tracked thousands of particles at a time in both a cellular flow (a regular vortex lattice) and the Kolmogorov Flow (a flow composed of parallel shear bands), and have computed the curvature fields. We find that regions of small curvature organize into lines, while regions of high curvature are pointlike. These high curvature points correspond to the topologically special points of the flow. In a two-dimensional, incompressible flow like ours, there are two types of special points: elliptic (vortex centers) and hyperbolic (saddles). By finding the local maxima of the curvature fields, we can locate these points; we then use the Okubo-Weiss parameter to classify them into elliptic and hyperbolic. Once we have their locations, we can apply our tracking algorithms to them to study their dynamics in time. Click on the two links above to see our results, or read a preprint of our paper "Curvature Fields, Topology, and the Dynamics of Spatiotemporal Chaos" (pdf, 0.5 MB).