Physical systems often produce complex structures (patterns), aperiodic in both space and time, when they are driven out of equilibrium. Examples include lasers, heart arrhythmia, path of hurricanes, and turbulent flow patterns. Information extraction from these chaotic systems is a challenge. Rayleigh-Benard convection (RBC) is a playground where researchers apply their numerical or experimental models to understand high dimensional complex systems.
We measure experimentally convective flow patterns in RBC. By applying different characterization techniques to large sets of data obtained in experiments, we try to answer some challenging questions.
In most theoretical and numerical studies of thermal convection, Oberbeck-Boussinesq (OB) approximation that simplifies the equations governing the fluid motion is frequently used. Solutions to the OB equations exhibit symmetry in the system if one assumes that the temperature dependence of the fluid properties can be neglected, except for the temperature induced density difference in the buoyant force that drives the flow. However, real flows almost never fully commit to this approximation, and non-OB effects (asymmetry) arise in natural and laboratory conditions. In collaboration with Konstantin Mischaikow , we quantitatively measure the non-OB effects in complex experimental data using a new topological characterization technique, computational homology. These asymmetries are not observable using conventional statistical measures.
Complete characterization of a complex system can be achieved by measuring the dimension D which defines the number of degrees of freedom needed to describe the system. The techniques are well-developed to extract D in low dimensional systems. However, measurements of D in high-dimensional experimental systems is extremely difficult since it requires very precise control of initial conditions and the complete knowledge of the system. Researchers, therefore, are motivated to find methodologies that extract information similar to D in experiments. One way to obtain such information is to make use of the Karhunen-Loeve decomposition (KLD) (a.k.a principal component analysis), a characterization technique well-known in many disciplines. We apply a modified KLD algorithm (for better convergence) to experimental data to extract a KLD dimension, which has been shown to have similarities with D. In addition, as a novel technique, we use a similar dimension defined by topological states acquired from computational homology. Using both techniques, we study the effect of fluid properties and of physical boundaries in experiments on the number of degrees of freedom.