  Protoplanetary discs at certain radii exhibit adverse radial entropy
gradients that can drive oscillatory convection (`convective overstability';
COS). The ensuing hydrodynamical activity may reshape the radial thermal
structure of the disc while mixing solid material radially and vertically or,
alternatively, concentrating it in vortical structures. We perform local
axisymmetric simulations of the COS using the code SNOOPY, showing first how
parasites halt the instability's exponential growth, and second, the different
saturation routes it takes subsequently. As the Reynolds and (pseudo-)
Richardson numbers increase, the system moves successively from (a) a weakly
nonlinear state characterised by relatively ordered nonlinear waves, to (b)
wave turbulence, and finally to (c) the formation of intermittent and then
persistent zonal flows. In three-dimensions, we expect the latter flows to
spawn vortices in the orbital plane. Given the very high Reynolds numbers in
protoplanetary discs, the third regime should be the most prevalent. As a
consequence, we argue that the COS is an important dynamical process in planet
formation, especially near features such as dead zone edges, ice lines, gaps,
and dust rings.
