  In this paper, we improve the global existence result in [9] slightly. More
precisely, the global existence of strong solutions to the primitive equations
with only horizontal viscosity and diffusivity is obtained under the assumption
of initial data $(v_0,T_0) \in H^1$ with $\partial_z v_0 \in L^4$. Moreover, we
prove that the scaled Boussinesq equations with rotation strongly converge to
the primitive equations with only horizontal viscosity and diffusivity, in the
cases of $H^1$ initial data, $H^1$ initial data with additional regularity
$\partial_z v_0 \in L^4$ and $H^2$ initial data, respectively, as the aspect
ration parameter $\lambda$ goes to zero, and the rate of convergence is of the
order $O(\lambda^{{\eta}/2})$ with
$\eta=\min\{2,\beta-2,\gamma-2\}(2<\beta,\gamma<\infty)$. The convergence
result implies a rigorous justification of the hydrostatic approximation.
