Free-Fall Estimates for Rapidly Rotating Heat and Momentum Transport*

Dimensional analysis is employed here to provide free-fall scaling estimates for the convective heat and momentum transport in the limit of rapid rotation, and to relate these to scalings for non-rotating Rayleigh-B\'enard convection (RBC) systems. Our analysis shows that the scalings for free-fall dominated heat (Nusselt number, $Nu$) and momentum transfer (Reynolds number, $Re$) of rapidly rotating convection differ from their non-rotating RBC counterparts by a factor of $Ro_{ \! f \! \! f}^2$, where $Ro_{ \! f \! \! f} = \tau_\Omega/\tau_{ \! f \! \! f}$ is the free-fall Rossby number defined as the ratio of the characteristic rotation time $\tau_\Omega$ and the buoyant free-fall time $\tau_{ \! f \! \! f}$.  Since $Ro_{ \! f \! \! f} \ll 1$ in the rapidly rotating limit, our predicted rapidly rotating, free-fall transport rates remain far below the associated rates in non-rotating systems.