  A sequence of two and three-dimensional simulations is conducted for the
double diffusive convection (DDC) flows in the diffusive regime subjected to an
imposed shear. The flow is confined between two horizontal plates which are
maintained at different constant temperature, salinity, and different velocity,
thus setting up a shear across the flow. The lower plate is fixed at higher
temperature and salinity, while the overall (unperturbed) density gradient is
statically stable. For a wide range of control parameters, and for sufficiently
strong perturbation of the conductive initial state, we find that
staircase-like structures spontaneously develop, with relatively well-mixed
layers separated by sharp interfaces of enhanced scalar gradient. Such
staircases appear to be robust even in the presence of strong shear over very
long times, although we typically observe early time coarsening of the number
of observed layers. For the same set of control parameters, different
asymptotic layered states, with markedly different vertical scalar fluxes, can
arise for different initial perturbation structures. The imposed shear does
significantly spatio-temporally modify the vertical transport of the various
scalars. The flux ratio (i.e., the ratio between the density fluxes due to the
total (convective and diffusive) salt flux and the total heat flux) is found,
at steady state, to be essentially equal to the square root of the ratio of the
salt diffusivity to the thermal diffusivity, consistently with the physical
model originally proposed by Linden and Shirtcliffe (1978) and the variational
arguments presented by Stern (1982) for unsheared double diffusive convection.
