When physical systems are driven far from equilibrium they often produce complex structures (patterns) that can be aperiodic in both space and time. Examples of pattern-forming systems include lasers, cardiac tissues, and planetary atmospheres. Today, it is possible to measure the dynamical behavior of these systems with high spatial and temporal resolution. However, characterizing the resulting data sets still poses significant challenges. Historically, our group has used a variety of pattern characterization techniques to analyze spatiotemporally chaotic data from Rayleigh-Bénard convection (RBC) experiments.
Rayleigh-Bénard convection is a paradigm in the study of pattern-forming systems. Our experimental implementation consists of a pressure cell containing a thin layer of compressed gas, which is cooled from above and heated from below (see the left image below). When the temperature difference ΔT across the gas exceeds a critical value, the fluid begins to flow. This is visualized using shadowgraphy (see the right image below), which exploits the temperature dependence of the index of refraction of the gas. When the temperature difference is small, the system forms a series of parallel convection rolls. As the temperature difference is increased, driving the system further from the equilibrium, the convection patterns become progressively more complex. Eventually, the system reaches a state known as spiral-defect chaos, shown in the video at the bottom of the page.
The resulting images can be analyzed to answer a variety of questions about convective flows. These data sets can be enormous, sometimes containing millions of images, so they must be analyzed using large computer clusters, such as Georgia Tech’s PACE cluster, and parallel algorithms. In the past, we have used traditional techniques such Karhunen–Loève decomposition (principle component analysis) and Fourier analysis to characterize convection patterns. More recently, we have added algebraic topology (homology) to this toolbox and used it to show how asymmetries arise between cold and hot flows as a result of the temperature dependence of the fluid parameters. In our current work, we are exploring persistent homology as a general tool for performing dimension reduction and gaining dynamical insight.