Vertical Velocity in the Gray Zone

Atmospheric models exhibit convection-like phenomena over a wide range of horizontal resolutions (the `gray zone'), but it is unclear at what resolution convective vertical velocities w should converge, or how this convergence might differ for a hydrostatic model. Here, we argue that considerations of parcel geometry alone imply that O(100 m) resolutions are required for convergence. We also present analytical formulae which describe how w converges, and which suggest that hydrostatic models over-estimate w by a factor of 2-3 in the high-resolution limit. We validate our results with numerical simulations, and give intuition for them in terms of Davies-Jones's "effective buoyancy" and the closely related "buoyancy pressure" of Jeevanjee and Romps.