  Several lines of evidence suggest that the velocity amplitude in global
simulations of solar convection, U, may be systematically over-estimated.
Motivated by these recent results, we explore the factors that determine U and
we consider how these might scale to solar parameter regimes. To this end, we
decrease the thermal diffusivity $\kappa$ along two paths in parameter space.
If the kinematic viscosity $\nu$ is decreased proportionally with $\kappa$
(fixing the Prandtl number $P_r = \nu/\kappa$), we find that U increases but
asymptotes toward a constant value, as found by Featherstone & Hindman (2016).
However, if $\nu$ is held fixed while decreasing $\kappa$ (increasing $P_r$),
we find that U systematically decreases. We attribute this to an enhancement of
the thermal content of downflow plumes, which allows them to carry the solar
luminosity with slower flow speeds. We contrast this with the case of
Rayleigh-Benard convection which is not subject to this luminosity constraint.
This dramatic difference in behavior for the two paths in parameter space
(fixed $P_r$ or fixed $\nu$) persists whether the heat transport by unresolved,
near-surface convection is modeled as a thermal conduction or as a fixed flux.
The results suggest that if solar convection can operate in a high-$P_r$
regime, then this might effectively limit the velocity amplitude. Small-scale
magnetism is a possible source of enhanced viscosity that may serve to achieve
this high-$P_r$ regime.
