  This work presents numerical results on the transport of heat and chemical
species by shear-induced turbulence in strongly stratified but thermally
diffusive environments. The shear instabilities driven in this regime are
sometimes called "secular" shear instabilities, and can take place even when
the gradient Richardson number of the flow (the square of the ratio of the
buoyancy frequency to the shearing rate) is large, provided the P\'eclet number
(the ratio of the thermal diffusion timescale to the turnover timescale of the
turbulent eddies) is small. We have identified a set of simple criteria to
determine whether these instabilities can take place or not. Generally
speaking, we find that they may be relevant whenever the thermal diffusivity of
the fluid is very large (typically larger than $10^{14}$cm$^2$/s), which is the
case in the outer layers of high-mass stars ($M\ge 10 M_\odot$) for instance.
Using a simple model setup in which the shear is forced by a spatially
sinusoidal, constant-amplitude body-force, we have identified several regimes
ranging from effectively unstratified to very strongly stratified, each with
its own set of dynamical properties. Unless the system is in one of the two
extreme regimes (effectively unstratified or completely stable), however, we
find that (1) only about 10% of the input power is used towards heat transport,
while the remaining 90% is viscously dissipated; (2) that the effective
compositional mixing coefficient is well-approximated by the model of Zahn
(1992), with $D \simeq 0.02 \kappa_T /J$ where $\kappa_T$ is the thermal
diffusivity and $J$ is the gradient Richardson number. These results need to be
confirmed, however, with simulations in different model setups and at higher
effective Reynolds number.
