Shear flows and their suppression at large aspect ratio: Two-dimensional simulations of a growing convection zone

We investigate the onset and evolution of zonal flows in a growing convective layer when a stably-stratified fluid with a composition gradient is cooled from above. This configuration allows the study of zonal flows for a wide range of values of the Rayleigh number, $Ra$, and aspect ratio of the convection zone within a given simulation. We perform a series of 2D simulations using the Boussinesq approximation, with aspect ratio of the computational domain between $1$ and $5$, and Prandtl number $Pr = 0.1$, 0.5, 1, and $7$. We find that for square domains zonal flows appear when the aspect ratio of the convective layer is smaller than two, and the evolution of the system depends on the Prandtl number. For $Pr\leq 1$, the fluid experiences bursts of convective transport with negligible convective transport between bursts. The magnitude and frequency of the bursts are smaller at low $Pr$, which suggests that the bursting regime is stronger in a narrow range around $Pr=1$, as observed in previous studies of thermal convection. For $Pr=7$, the structure of the flow consists of tilted convective plumes, and the convective transport is sustained at all times. In wider domains, the aspect ratio of the convective zone is always much larger than two and zonal flows do not appear. These results confirm and extend to fluids with stable composition gradients previous findings on thermal convection. The fact that zonal flows can be avoided by using computational domains with large aspect ratios opens up the possibility of 2D studies of convective overshoot, layer formation and transport properties across diffusive interfaces.