Thesis Dissertation Defense

Spatiotemporal chaos, especially fluid turbulence, is ubiquitous in nature but can be difficult to characterize because analytic solutions of the strongly nonlinear partial differential equations that govern the behavior are often intractable. However, the topology of structures observed in both experiments and numerical simulations of spatiotemporally chaotic flows can provide insights into the underlying dynamics. The topological properties of spatiotemporally chaotic data can be investigated using persistent homology, a technique of topological data analysis.

In this thesis, persistent homology is used to investigate the dynamics of two different spatiotemporally chaotic fluid flows. First, in the Kuramoto-Sivashinsky equation, a popular "toy model" system that mimics the spatiotemporal chaos exhibited by fully turbulent fluid flows, persistent homology is used to detect and quantify shadowing of exact coherent structures (ECS). ECS are invariant solutions to the governing equations that structure the dynamics of spatiotemporal chaos. Persistent homology is found to be an advantageous tool for quantifying shadowing in the Kuramoto-Sivashinsky equation because it quotients out the system's continuous symmetry. Second, in Rayleigh-B\'enard convection, persistent homology is used to detect and quantify plumes, which are observable pattern features in experiments and simulations. In simulations, plumes indicate spatial regions of the convective flow in which the leading Lyapunov vector magnitude, a fundamental quantity that characterizes the dynamics of the flow, is high. A long-term goal is to use plumes to connect dynamics in simulations, where the leading Lyapunov vector can be computed, to experiments, where this quantity cannot be observed. This thesis advances research in both of these topics and demonstrates that persistent homology is a powerful tool for analyzing topological structure associated with the dynamics of spatiotemporally chaotic flow.