Logarithmic Temperature Profiles in Turbulent Rayleigh-Benard Convection

We report results for the temperature profiles of turbulent Rayleigh-Benard convection (RBC) in the interior of a cylindrical sample of aspect ratio Gamma equivalent to D/L = 0.50 (D and L are the diameter and height, respectively). Both in the classical and in the ultimate state of RBC we find that the temperature varies as A X ln(z/L) + B, where z is the distance from the bottom or top plate. In the classical state, the coefficient A decreases in the radial direction as the distance from the side wall increases. For the ultimate state, the radial dependence of A has not yet been determined. These findings are based on experimental measurements over the Rayleigh-number range 4 X 10(12) less than or similar to Ra less than or similar to 10(15) for a Prandtl number Pr similar or equal to 0.8 and on direct numerical simulation at Ra = 2 X 10(12), 2 X 10(11), and 2 X 10(10), all for Pr = 0.7.