The hydrostatic approximation of the Boussinesq equations with rotation in a thin domain

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.