Coherent Structure Detection using Topological Tools and a Graph Theoretic Approach

For general aperiodic fluid flows, coherent structures help organize the dynamics, much as invariant manifolds and periodic orbits do for autonomous or periodic systems. The prevalence of such flows in nature and industry has motivated many successful techniques for defining and detecting coherent structures. However, these approaches often require very fine trajectory data to reconstruct velocity fields. Instead, we use topological techniques to help detect coherent trajectory sets in relatively sparse 2D fluid advection problems. More specifically, we use a homotopy-based algorithm, the ensemble-based topological entropy calculation (E-tec), which evolves fluid material curves forward in time as minimal length bands stretched about the moving data points. These bands are represented as the weighted edges of a triangulation, which allows us to analyze flows using graph theoretic tools. In this way, highly connected components of appropriately constructed graphs can be used to partition the fluid particles into coherent trajectory sets.