What is Rossby and Ekman numbers?

For a flow of speed $U$, horizontal scale $L$ and depth $H$ on a planet of Coriolis parameter $f$, the Rossby number is $\mathrm{Ro} = U/(fL)$, the ratio of the acceleration, of order $U^2/L$, to the Coriolis force, of order $fU$. The Ekman number is $\mathrm{Ek} = \nu/(fH^2)$, the ratio of the viscous force, of order $\nu U/H^2$, to the Coriolis force; in a turbulent flow $\nu$ is replaced by an eddy viscosity.