Azimuthal diffusion of the large-scale circulation of turbulent Rayleight-B\'enard convection*

We present measurements of the azimuthal orientation $\theta_0(t)$ of the large-scale circulation (LSC) of turbulent Rayleight-B\'enard convection. The sample was a cylinder with height and diameter equal to 1.12 m. We used compressed SF$_6$ gas at pressures up to 19 bars as the fluid. The measurements covered the Rayleigh-number range $10^{12} \leq Ra \leq 10^{14}$ at a Prandtl number $Pr \simeq 0.80$. We found that the preferred orientation of the LSC upflow was aligned to the West, consistent with Earth's Coriolis force. The LSC azimuthal dynamics was diffusive, driven by the small-scale turbulent fluctuations. For $Ra\leq 10^{13}$ the Reynolds number $Re^{\dot{\theta}}$ based on the azimuthal diffusivity had a $Ra$ dependence similar to that seen for $10^{9} \leq Ra \leq 10^{11}$ and $Pr=4.38$. The $Pr$ dependence $Re^{\dot{\theta}} \propto Pr^{\alpha}$ with $\alpha \simeq -1.2$ was the same as that found for the Reynolds number based on the root-mean-square fluctuation velocity in the interior bulk flow. For $Ra = Ra^*_1 \simeq 2\times10^{13}$ $Re^{\dot{\theta}}$ showed the ultimate-state transition and for $Ra \geq Ra^*_2 \simeq 8\times10^{13}$ it had a $Ra$ dependence with an exponent of $0.40 \pm 0.02$.