Analysis of Rayleigh-Bénard convection using latent Dirichlet allocation

We apply a probabilistic clustering method, latent Dirichlet allocation (LDA), to characterize the large-scale dynamics of Rayleigh-Bénard convection. The method, introduced by Frihat et al. [J. Fluid Mech. 920, A27 (2021)], is applied to a collection of snapshots in the vertical midplanes of a cubic cell for Rayleigh numbers in the range $[{10}^6,{10}^8]$. For the convective heat flux, temperature, and kinetic energy, the decomposition identifies latent factors, called motifs, which consist of connex regions of fluid. Each snapshot is modeled with a sparse combination of motifs, the coefficients of which are called the weights. The spatial extent of the motifs varies across the cell and with the Rayleigh number. We show that the method is able to provide a compact representation of the heat flux and displays good generative properties. At all Rayleigh numbers the dominant heat flux motifs consist of elongated structures located mostly within the vertical boundary layers, at a quarter of the cavity height. Their weights depend on the orientation of the large-scale circulation. A simple model relating the conditionally averaged weight of the motifs to the relative strength of the corner rolls and of the large-scale circulation is found to predict well the average large-scale circulation reorientation rate. Application of LDA to the temperature fluctuations shows that temperature motifs are well correlated with heat flux motifs in space as well as in time, and to some lesser extent with kinetic energy motifs. The abrupt decrease of the reorientation rate observed at ${10}^8$ is associated with a strong concentration of plumes impinging onto the corners of the cell, which decrease the temperature difference within the corner structures. It is also associated with a reinforcement of the longitudinal wind through formation and entrainment of new plumes.