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

In this paper, the global existence of strong solutions to the primitive equations with only horizontal viscosity and diffusivity is established under the assumption of initial data $(v_0,T_0)\in H^1$ with additional regularity ${\partial }_z{v}_0\in L^4$. Moreover, we prove that the scaled Boussinesq equations with rotation strongly converge to the primitive equations with only horizontal viscosity and diffusivity, with the convergence rate $O\left({\lambda }^{min\{2,\beta -2,\gamma -2\}/2}\right)(2<\beta ,\gamma <\infty )$, in the cases of initial data $(v_0,T_0)\in H^1$ with ${\partial }_z{v}_0\in L^4$ and initial data $(v_0,T_0)\in H^2$, respectively, as the aspect ratio $\lambda$ goes to zero.