  A toy model of the post-shock region of core-collapse supernovae is used to
study the non-linear development of turbulent motions driven by convection in
the presence of advection. Our numerical simulations indicate that buoyant
perturbations of density are able to trigger self-sustained convection only
when the instability is not linearly stabilized by advection. Large amplitude
perturbations produced by strong shock oscillations or combustion
inhomogeneities before the collapse of the progenitor are efficiently shredded
through phase mixing and generate a turbulent cascade. Our model enables us to
investigate several physical arguments that had been proposed to explain the
impact of the dimensionality on the onset of explosions in global simulations
of core-collapse supernovae. Three-dimensional (3D) simulations are found to
lead to higher entropy values than two-dimensional (2D) ones. We attribute this
to greater turbulent mixing and dissipation of the kinetic energy into heat in
3D. Our results show that the increase of entropy is enhanced with finer
numerical resolution and larger perturbation amplitude.
