Optimal Prandtl number for heat transfer in rotating Rayleigh-B\'enard convection

The heat transfer in Rayleigh-B\'enard convection (RBC) is determined by the Rayleigh number $Ra$ and the Prandtl number $Pr$ \footnote{Ahlers et al. {\emph{Rev. Mod. Phys.}} {\bf{81}}, 503 (2009)}. In case of rotation about the vertical axis the third dimensionless control parameter is the Rossby number $Ro$. Here we present numerical data for the heat transfer in rotating RBC for $Ra = 10^8$ as a function of $Pr$ and $Ro$. When $Ro$ is fixed the heat transfer enhancement with respect to the non- rotating value as function of $Pr$ shows a maximum. This maximum is due to the reduced efficiency of Ekman pumping when $Pr$ becomes too small or too large. When $Pr$ becomes too small the heat that is carried by the vertical vortices spreads out in the middle of the cell, i.e. it makes Ekman pumping less efficient, due to the larger thermal diffusivity \footnote{Zhong et al. {\emph{Phys. Rev. Lett.}} {\bf{102}}, 044502 (2009); Stevens et al. {\emph{Phys. Rev. Lett.}} {\bf{103}}, 024503 (2009)}. For higher $Pr$ the thermal boundary layers (BLs) are much thinner than the kinetic BLs and therefore the Ekman vortices do not reach the thermal BL. This means that the fluid that is sucked into the vertical vortices is colder than for lower $Pr$ and this limits the efficiency of Ekman pumping at high $Pr$.