  Convection has been discussed in the field of accretion discs for several
decades, both as a means of angular momentum transport and also because of its
role in controlling discs' vertical structure via heat transport. If the gas is
sufficiently ionized and threaded by a weak magnetic field, convection might
interact in non-trivial ways with the magnetorotational instability (MRI).
Recently, vertically stratified local simulations of the MRI have reported
considerable variation in the angular momentum transport, as measured by the
stress to thermal pressure ratio $\alpha$, when convection is thought to be
present. Although MRI turbulence can act as a heat source for convection, it is
not clear how the instabilities will interact dynamically. Here we aim to
investigate the interplay between the two instabilities in controlled numerical
experiments, and thus isolate the generic features of their interaction. We
perform vertically stratified, 3D MHD shearing box simulations with a perfect
gas equation of state with the conservative, finite-volume code PLUTO. We find
two characteristic outcomes of the interaction between the two instabilities:
straight MRI and MRI/convective cycles, with the latter exhibiting alternating
phases of convection-dominated flow (during which the turbulent transport is
weak) and MRI-dominated flow. During the latter phase we find that $\alpha$ is
enhanced by nearly an order of magnitude, reaching peak values of $\sim 0.08$.
In addition, we find that convection in the non-linear phase takes the form of
large-scale and oscillatory convective cells. Convection can also help the MRI
persist to lower Rm than it would otherwise do. Finally we discuss how our
results help interpret simulations of Dwarf Novae.
