Markov State Modelling of discrete flow states in turbulent Rayleigh-B\'enard convection in a cubic container

The turbulent convection flow in closed cells, which is driven by rising and falling thermal plumes, are known to form large scale circulations (LSC). The LSC structures depend on the aspect ratio and the geometrical shape of the cell. Studies of LSC are of relevance as they can significantly affect the heat transport in the system. We carried out our investigations in a closed cubic container with no-slip boundary conditions at all boundaries, using an open source code nek5000 based on the spectral-element method. The simulations were performed for a fixed value of Rayleigh number Ra = 10⁶ for incompressible fluids of three different Prandtl numbers Pr = 0.1, 0.7, and 10, which allowed us to vary the effective Reynolds numbers while maintaining the Nusselt number Nu nearly constant. In accordance with previous studies, we identified four stable LSC states (aligned along the diagonals) and an unstable LSC state (aligned parallel to the edges). Additionally, we also found a decoherent ``null state'', where the system does not have any well-defined LSC. A long-term single trajectory analysis for 10⁵ free-fall times confirmed that the LSC state re-orients from one stable to another state via the four possible unstable and the null states. We confirmed that all stable states have equal probability of occurrence. We then performed an alternative approach based on ensemble averaging of short-term simulations. The initial conditions for these short-term simulations were randomly chosen from the single long-term trajectory run. The ensemble averaging generated a transition probability matrix and thereafter we were able to probe if the Markov property of the transition between the different large-scale flow states is effective.