  Rotating Rayleigh-B\'enard convection is investigated numerically with the
use of an asymptotic model that captures the rapidly rotating, small Ekman
number limit, $Ek \rightarrow 0$. The Prandtl number ($Pr$) and the
asymptotically scaled Rayleigh number ($\widetilde{Ra} = Ra Ek^{4/3}$, where
$Ra$ is the typical Rayleigh number) are varied systematically. For
sufficiently vigorous convection, an inverse kinetic energy cascade leads to
the formation of a depth-invariant large-scale vortex (LSV). With respect to
the kinetic energy, we find a transition from convection dominated states to
LSV dominated states at an asymptotically reduced (small-scale) Reynolds number
of $\widetilde{Re} \approx 6$ for all investigated values of $Pr$. The ratio of
the depth-averaged kinetic energy to the kinetic energy of the convection
reaches a maximum at $\widetilde{Re} \approx 24$, then decreases as
$\widetilde{Ra}$ is increased. This decrease in the relative kinetic energy of
the LSV is associated with a decrease in the convective correlations with
increasing Rayleigh number. The scaling behavior of the convective flow speeds
is studied; although a linear scaling of the form $\widetilde{Re} \sim
\widetilde{Ra}/Pr$ is observed over a limited range in Rayleigh number and
Prandtl number, a clear departure from this scaling is observed at the highest
accessible values of $\widetilde{Ra}$. Calculation of the forces present in the
governing equations shows that the ratio of the viscous force to the buoyancy
force is an increasing function of $\widetilde{Ra}$, that approaches unity over
the investigated range of parameters.
