  (abridged) Context: Rotation is thought to influence the size of convective
eddies and the efficiency of convective energy transport in the deep convection
zones of stars. Rotationally constrained convection has been invoked to explain
the lack of large-scale power in observations of solar flows. Aims: The main
aims are to quantify the effects of rotation on the scale of convective eddies
and velocity, the depths of convective overshoot, and the subadiabatic
Deardorff layers. Methods: Three-dimensional hydrodynamic simulations of
rotating convection in Cartesian domains were run. The results were compared
with theoretical scaling results that assume a balance between Coriolis,
inertial, and buoyancy (Archemedean) forces (CIA balance). Results: The scale
of convective eddies decreases as rotation increases, and ultimately reaches a
rotationally constrained regime consistent with the CIA balance. Using a new
measure of the rotational influence on the system, it is shown that even the
deep parts of the solar convection zone are not in the rotationally constrained
regime. The simulations capture the slowly and rapidly rotating scaling laws
predicted by theory, and the Sun appears to be in between these two regimes.
Both, the overshooting depth and the extent of the Deardorff layer, decrease as
rotation becomes more rapid. For sufficiently rapid rotation the Deardorff
layer is absent. Conclusions: Relating the simulations with the Sun suggests
that the convective scale even in the deep parts of the Sun is only mildly
affected by rotation and that some other mechanism is needed to explain the
lack of strong large-scale flows in the Sun. Taking the current results at face
value, the overshoot and Deardorff layers are estimated to span roughly five
per cent of the pressure scale height at the base of the convection zone in the
Sun.
