  The prevalence and consequences of convection perpendicular to the plane of
accretion discs have been discussed for several decades. Recent simulations
combining convection and the magnetorotational instability have given fresh
impetus to the debate, as the interplay of the two processes can enhance
angular momentum transport, at least in the optically thick outburst stage of
dwarf novae. In this paper we seek to isolate and understand the most generic
features of disc convection, and so undertake its study in purely
hydrodynamical models. First, we investigate the linear phase of the
instability, obtaining estimates of the growth rates both semi-analytically,
using one-dimensional spectral computations, as well as analytically, using
WKBJ methods. Next we perform three-dimensional, vertically stratified,
shearing box simulations with the conservative, finite-volume code PLUTO, both
with and without explicit diffusion coefficients. We find that hydrodynamic
convection can, in general, drive outward angular momentum transport, a result
that we confirm with ATHENA, an alternative finite-volume code. Moreover, we
establish that the sign of the angular momentum flux is sensitive to the
diffusivity of the numerical scheme. Finally, in sustained convection, whereby
the system is continuously forced to an unstable state, we observe the
formation of various coherent structures, including large- scale and
oscillatory convective cells, zonal flows, and small vortices.
