Extensive Scaling of Computational Homology and Karhunen-Lo\`{e}ve Decomposition in Rayleigh-B\'{e}nard Convection Experiments

We apply two different pattern characterization techniques to large data sets of spatiotemporally chaotic flows in Rayleigh-B\'{e}nard convection (RBC) experiments. Both Computational homology (CH) and a modified Karhunen-Lo\`{e}ve decomposition (KLD) are used to analyze the data. The KLD dimension $D_{KLD}$, the number of eigenmodes required to capture a given fraction of the eigenvalue spectrum, is computed for different subsystem sizes. A similar quantity $D_{CH}$ for the same experimental data is acquired by the probability distribution of topological states constructed from the outputs of CH. We show that both $D_{CH}$ and $D_{KLD}$ scale over a large range of subsystem sizes for the state of SDC; moreover, we find the presence of boundaries leads to deviations from extensive scaling that are similar for both methodologies.