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We test the ability of a low-dimensional turbulence model to predict howdynamics of large-scale coherent structures such as convection rolls change indifferent cell geometries. We performed Rayleigh-B\'enard convectionexperiments in a cubic container, in which there is a single convection rollknown as the large-scale circulation (LSC). The model describes the motion ofthe orientation $\theta_0$ of the LSC as diffusion in a potential which ispredicted as a function of the shape of the cell from an approximate solutionsof the Navier-Stokes equations. The model predicts advected oscillation modes,driven by a restoring force created by the non-circular cell cross-section. Weobserve the predicted lowest-wavenumber mode in which the LSC orientation$\theta_0$ oscillates around a corner, and a slosh angle $\alpha$ rocks backand forth, which is distinct from the higher-wavenumber advected twisting andsloshing oscillations found in cylindrical cells. The potential has quadraticminima near each corner with the same curvature in both the LSC orientation$\theta_0$ and slosh angle $\alpha$, as predicted. The new oscillation modearound corners is found above a critical Ra $=4\times10^8$, which appears inthe model as a crossing of an underdamped-overdamped transition. The naturalfrequency of the potential, oscillation period, power spectrum, and critical Rafor oscillations are all within a factor of 3 of model predictions for theRayleigh number range $8\times10^7 \le Ra \le 3\times 10^9$. However, theseuncertainties in model parameters are too large to correctly predict whetherthe system is in the underdamped or overdamped state at a given Ra. The successof the model at predicting the potential and flow modes for a cubic cellsuggests that such a modeling approach could be applied more generally todifferent cell geometries that support a single convection roll.

Low-dimensional model of the large-scale circulation of turbulent Rayleigh-B{\'e}nard convection in a cubic container