jats:titleABSTRACT</jats:title> jats:pProtoplanetary 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 secondly, the different saturation routes it takes subsequently. As the Reynolds and (pseudo-) Richardson numbers increase, the system moves successively from (i) a weakly non-linear state characterized by relatively ordered non-linear waves, to (ii) wave turbulence, and finally to (iii) 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.</jats:p>