  We present a numerical study of double-diffusive convection characterized by
a stratification unstable to thermal convection while at the same time a mean
molecular weight (or solute concentration) difference between top and bottom
counteracts this instability. Convective zones can form in this case either by
the stratification being locally unstable to the combined action of both
temperature and solute gradients or by another process, the oscillatory
double-diffusive convective instability, which is triggered by the faster
molecular diffusivity of heat in comparison with that one of the solute. We
discuss successive layer formation for this problem in the case of an
instantaneously heated bottom (plate) which forms a first layer with an
interface that becomes temporarily unstable and triggers the formation of
further, secondary layers. We consider both the case of a Prandtl number
typical for water (oceanographic scenario) and of a low Prandtl number (giant
planet scenario). We discuss the impact of a Couette like shear on the flow and
in particular on layer formation for different shear rates. Additional layers
form due to the oscillatory double-diffusive convective instability, as is
observed for some cases. We also test the physical model underlying our
numerical experiments by recovering experimental results of layer formation
obtained in laboratory setups.
