Aspect-ratio dependence of the large-scale circulation in Rayleigh-B\'{e}nard convection with weak rotation*

We report measurements for slowly rotating turbulent thermal convection in cylindrical samples with aspect ratios $\Gamma=1.0$ and $2.0$ for a Prandtl number $Pr = 12.3$. The results are for the large-scale circulation (LSC) strength $\delta$, Fourier-energy E$_{tot}$, and relative flow strength S, as well as for two Reynolds numbers $Re_{ret}$ and $Re_{sl}$, for the Nusselt number $Nu$, and for the vertical temperature gradient $\partial \Theta/\partial z$ at the sample center. They cover the Rayleigh-number range $3\times10^{10} \leq Ra \leq 4\times 10^{11}$ and the inverse Rossby-number range $0 \leq 1/Ro \leq 1/Ro_{c}$. $Nu$, E$_{tot}$, S, and $\partial \Theta/\partial z$ showed sharp transitions at $1/Ro_c$. The LSC underwent retrograde rotation with period $\tau_{ret}$ and showed sloshing oscillations with period $\tau_{sl} << \tau_{ret}$. At constant $Ra$ and $1/Ro$ $\delta$ grew and decayed with a period equal to $\tau_{ret}$. We found that $Re_{ret} \equiv 4 L^2 / \tau_{ret} \nu \propto Ra^{0.65}$ ($\propto Ra^{0.50}$) for $\Gamma = 1.0$ ($\Gamma = 2.0$) ($\nu$ is the kinematic viscosity and $L$ the sample height) and $Re_{sl} \equiv 4 L^2 / \tau_{sl} \nu \propto Ra^{0.50}$ ($\propto Ra^{0.42}$) for $\Gamma = 1.0$ ($\Gamma = 2.0$).