Using Persistent Homology to Identify Localised Defects in Rayleigh B\'enard Convection*

Complex spatiotemporal convective roll patterns are observed when a sufficiently large temperature gradient is created across a thin layer of fluid. These roll patterns are often characterized by the presence of localised defects such as centers, spirals, disclinations, grain boundaries, which play a crucial dynamical role. Our research focuses on using persistent homology (a branch of algebraic topology) to identify these defects in an experimental realization of the Rayleigh B\'enard convection in a cylindrical container. Persistent homology provides a powerful mathematical formalism in which the topological characteristics of a pattern (shadowgraph image in our case) are encoded in a so-called persistence diagram. By identifying several instants in the experiment that correspond to the appearance of a certain type of defect and computing the persistence diagrams for the corresponding shadowgraph images, we extract signatures in the persistence diagram which characterize the defect. Then, for a spatiotemporally resolved series of shadowgraph images we show that using signatures from the persistence diagrams one can automate identifying the instants when localized defects appear.