  Double-diffusive convection, often referred to as semi-convection in
astrophysics, occurs in thermally and compositionally stratified systems which
are stable according to the Ledoux-criterion but unstable according to the
Schwarzchild criterion. This process has been given relatively little attention
so far, and its properties remain poorly constrained. In this paper, we present
and analyze a set of three-dimensional simulations of this phenomenon in a
Cartesian domain under the Boussinesq approximation. We find that in some cases
the double-diffusive convection saturates into a state of homogeneous
turbulence, but with turbulent fluxes several orders of magnitude smaller than
those expected from direct overturning convection. In other cases the system
rapidly and spontaneously develops closely-packed thermo-compositional layers,
which later successively merge until a single layer is left. We compare the
output of our simulations with an existing theory of layer formation in the
oceanographic context, and find very good agreement between the model and our
results. The thermal and compositional mixing rates increase significantly
during layer formation, and increase even further with each merger. We find
that the heat flux through the staircase is a simple function of the layer
height. We conclude by proposing a new approach to studying transport by
double-diffusive convection in astrophysics.
