Rotating turbulent Rayleigh-B\'enard convection: Effect of weak rotation on boundary layers and heat transfer

The Grossmann-Lohse (GL) theory\footnote{Grossmann and Lohse, J. Fluid Mech. 407, 27 (2000)} for the heat transfer in turbulent Rayleigh-B\'enard (RB) convection heavily builds on the Prandtl-Blasius laminar boundary layer (BL) theory, according to which the thermal BL thickness $\lambda_\theta$ scales as $L Pr^{-1/2}$ in the low $Pr$ regime and with $L Pr^{- 1/3}$ in the high $Pr$ regime. In an attempt to extend the GL theory to the rotating RB case, we study the influence of rotation on the thermal BL thickness in flow above an infinite rotating disk. We show that with rotation $\lambda_\theta \propto Pr^{-1}$ in the low $Pr$ regime, whereas in the high $Pr$ regime the scaling remains unchanged. Furthermore, we obtain an analytic expression for the Nusselt number in the Ekman case (fluid at infinity rotates at almost the same speed as the disk). We moreover introduce a model to explain the experimentally\footnote{Rossby, {\emph{J. Fluid Mech.}} {\bf {36}}, 309 (1969)} and numerically\footnote{Oresta et al {\emph {Eur. J. Mech.}} {\bf{26}}, 1 (2007); Kunnen et al {\emph{Phys. Rev. E}} {\bf{74}}, 056306 (2006).} observed increased heat transfer (as compared to RB without rotation) at weak rotation.