Multiple scaling in the ultimate regime of thermal convection

Very different types of scaling of the Nusselt number Nu with the Rayleigh number Ra have experimentally been found in the very large Ra regime beyond 10(11). We understand and interpret these results by extending the unifying theory of thermal convection [Grossmann and Lohse, Phys. Rev. Lett. 86, 3316 (2001)] to the very large Ra regime where the kinetic boundary-layer is turbulent. The central idea is that the spatial extension of this turbulent boundary-layer with a logarithmic velocity profile is comparable to the size of the cell. Depending on whether the thermal transport is plume dominated, dominated by the background thermal fluctuations, or whether also the thermal boundary-layer is fully turbulent (leading to a logarithmic temperature profile), we obtain effective scaling laws of about Nu proportional to Ra-0.14, Nu proportional to Ra-0.22, and Nu proportional to Ra-0.38, respectively. Depending on the initial conditions or random fluctuations, one or the other of these states may be realized. Since the theory is for both the heat flux Nu and the velocity amplitude Re, we can also give the scaling of the latter, namely, Re proportional to Ra-0.42, Re proportional to Ra-0.45, and Re proportional to Ra-0.50 in the respective ranges. (C) 2011 American Institute of Physics. [doi:10.1063/1.3582362]