Logarithmic temperature profiles in turbulent Rayleigh-B\'enard convection

We report results for the temperature profiles of turbulent Rayleigh-B'enard convection (RBC) in the interior of a cylindrical sample of aspect ratio $\mathrm{\Gamma }\equiv D\mathrm{/}L=0.50$ ($D$ and $L$ are the diameter and height respectively). Results from experiment over the Rayleigh number range $4\times 10^{12} \alt Ra \alt 10^{15}$foraPrandtlnumber$\Pra \simeq 0.8$ and from direct numerical simulation (DNS) at $Ra=2\times {10}^{12}$ for $\text{Pra}=0.7$ are presented. We find that the temperature varies as $A\ast ln(z\mathrm{/}L)+B$ where $z$ is the distance from the bottom or top plate. This is the case in the classical as well as in the ultimate state of RBC. From DNS we find that $A$ in the classical state decreases in the radial direction as the distance from the side wall increases and becomes small near the sample center.