Using Persistent Homology to Describe Kolmogorov Flow and Rayleigh-B\'enard Convection*

We employ a new technique for describing the dynamics of spatially extended systems evolving in time. In particular, we study two canonical fluid flows: Kolmogorov flow and Rayleigh-B\'enard convection. The technique used, known as persistent homology, provides a powerful mathematical tool in which the instantaneous topological characteristics of the system are encoded in a so-called persistence diagram, which is independent of the global symmetries of the system. By applying a metric to measure the distances across multiple persistence diagrams, we can quantify the symmetry-independent similarities between states, providing an opportunity for unique physical insights into the time evolution of a dynamical system. The two systems studied are particularly interesting, as each display a wide range of dynamical behavior and possess their own symmetries. We perform our analysis using flow field patterns from numerical simulations of these systems; however, we emphasize that our analysis can be conducted with patterns measured in experiment. Our results show that persistent homology is a powerful way to gain new physical insights into the complex dynamics of large spatially extended systems that are driven far-from-equilibrium.