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By utilizing diffusion maps embedding and transition matrix analysis we investigate sparse temperature measurement time-series data from Rayleigh--B\'enard convection experiments in a cylindrical container of aspect ratio $\Gamma=D/L=0.5$ between its diameter ($D$) and height ($L$). We consider the two cases of a cylinder at rest and rotating around its cylinder axis. We find that the relative amplitude of the large-scale circulation (LSC) and its orientation inside the container at different points in time are associated to prominent geometric features in the embedding space spanned by the two dominant diffusion-maps eigenvectors. From this two-dimensional embedding we can measure azimuthal drift and diffusion rates, as well as coherence times of the LSC. In addition, we can distinguish from the data clearly the single roll state (SRS), when a single roll extends through the whole cell, from the double roll state (DRS), when two counter-rotating rolls are on top of each other. Based on this embedding we also build a transition matrix (a discrete transfer operator), whose eigenvectors and eigenvalues reveal typical time scales for the stability of the SRS and DRS as well as for the azimuthal drift velocity of the flow structures inside the cylinder. Thus, the combination of nonlinear dimension reduction and dynamical systems tools enables to gain insight into turbulent flows without relying on model assumptions.

Diffusion maps embedding and transition matrix analysis of the large-scale flow structure in turbulent Rayleigh–Bénard convection