Logarithmic corrections to scaling in turbulent thermal convection

We use a simplified model of turbulent convection to compute analytically h eat transport in a horizontal layer heated from below, as a function of the Rayleigh and the Prandtl number. At low Reynolds numbers, when most of the dissipation comes from the mean flow, we recover power classical scaling r egimes of the Nusselt versus Rayleigh number, with exponent 1/3 or 1/4. At larger Reynolds number, velocity and temperature fluctuations become non-ne gligible in the dissipation. In these regimes, there is no exact power law dependence the Nusselt versus Rayleigh or Prandtl. Instead, we obtain logar ithmic corrections to the classical soft (exponent 1/3) or ultra-hard (expo nent 1/2) regimes, in a way consistent with the most accurate experimental measurements available nowadays. This sets a need for the search of new mea surable quantities that are less prone to dimensional theories.