  In an attempt to understand the role of the strong radial dependence of
thermal diffusivity on the properties of convection in sun-like stars, we mimic
that effect in non-Oberbeck-Boussinesq (NOB) convection in a
horizontally-extended rectangular domain (aspect ratio 16), by allowing the
thermal diffusivity $\kappa$ to increase with the temperature (as in the case
of stars). Direct numerical simulations (i.e., numerical solutions of the
governing equations by resolving up to the smallest scales without requiring
any modeling) show that, in comparison with Oberbeck-Boussinesq (OB)
simulations (two of which we perform for comparison purposes), the symmetry of
the temperature field about the mid-horizontal plane is broken, whereas the
velocity and heat flux profiles remain essentially symmetric. Our choice of
$\kappa(T)$, which resembles the variation in stars, results in the temperature
field that loses its fine structures towards the hotter part of the
computational domain, but the characteristic large scale of the turbulent
thermal `superstructures', which are structures whose size is typically larger
than the depth of the convection domain, continue to be largely independent of
the depth.
