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

We report experimental results for the vertical profiles of the mean temperature $\langle T \rangle$ and the rms temperature fluctuation $\sigma$ for turbulent Rayleigh-B\'enard convection in the interior of a cylindrical sample of aspect ratio $\Gamma \equiv D/L = 0.50$ ($D = 112$ cm and $L = 224$ cm are the diameter and height respectively) over the Rayleigh number range $4 \times 10^{12} \leq Ra \leq 10^{15}$ for a Prandtl number $Pr \simeq 0.8$. We found that $\langle T \rangle$ and $\sigma$ vary linearly with $ln(z/L)$ where $z$ is the distance from the bottom or top plate. Such a dependence had been predicted\footnote{S. Grossmann and D. Lohse, Phys. Fluids {\bf 23}, 045108 (2011).} for the ultimate state ($Ra > 5\times 10^{14}$), but was unexpected for the classical state ($Ra < 10^{13}$). The results for $\langle T \rangle$ and $\sigma$ suggest similarities to the logarithmic profiles found for the velocity in shear flows.\footnote{I. Marusic {\it et al.}, Phys. Fluids {\bf 22}, 065103 (2010).}$^,$\footnote{M. Hultmark {\it et al.}, Phys. Rev. Lett {\bf 108}, 094501 (2012).}