  In our previous work dedicated to the strongly stratified Boussinesq system,
we obtained for the first time a limit system (when the froude number
$\epsilon$ goes to zero) that depends on the thermal diffusivity $\nu$ ' (other
works obtained a limit system only depending on the visosity $\nu$). To reach
those richer asymptotics we had to consider an unusual initial data which is
the sum of a function depending on the full space variable and a function only
depending on the vertical coordinate, and we studied the convergence of the
weak Leray-type solutions. In the present article we extend these results to
the strong Fujita-Kato-type solutions. We obtain far better convergence rates
(in $\epsilon$) for ill-prepared initial data with very large oscillating part
of size some negative power of the small parameter $\epsilon$. The main
difficulties come from the anisotropy induced by the presence of x3-depending
functions.
