Meandering of the large-scale circulation of turbulent convection in a cylindrical cell

The large-scale circulation (LSC) in cylindrical cells of aspect ratio $\Gamma \equiv D/L = 1$ ($D =$ diameter, $L = $ height) filled with water at a mean temperature of 40$^\circ$C and heated from below was studied for Rayleigh numbers $R$ in the range $10^9$ to $10^{11}$. We measured the temperatures of the cell side-wall as a function of time $t$ at eight azimuthal locations on the horizontal mid- plane and from them deduced the azimuthal orientation $\theta(t)$ of the LSC. We found that $\theta(t)$ varied irregularly in time. Although it had a preferred value, on average there was a long-term continuous rotation of the LSC. From the data for $\theta(t)$ we derived $\dot\theta \equiv \Delta\theta/\Delta t$ ($\Delta t$ is the time interval between measurements). The time averages of $\dot \theta(\theta)$ gave a deterministic force $-\partial V/\partial \theta$ corresponding to a potential of the form $V = V_0 [-\cos(\theta - \theta_0) + v_1 \theta]$, and its probability distribution-function $P_ {\dot \theta}(\dot\theta)$ yielded a Langevin force $f(t)$. Integrations of the corresponding stochastic model equation $\partial \theta /\partial t = - \partial V / \partial \theta + f(t)$ produced time series $\theta(t)$ and distribution functions $P_{\theta}(\theta)$ remarkably similar to the experimental data. We attribute $f(t)$ to the action of the turbulent background fluctuations on the LSC, and found that its intensity depended on $R$.