Rigorous derivation of the full primitive equations by the scaled Boussinesq equations with rotation

The primitive equations of large-scale oceanic dynamics form a fundamental model in geophysical flows. It is well-known that the can be formally derived by hydrostatic approximation. In this paper, we rigorously derive full from mathematical point view. More precisely, prove scaled Boussinesq with rotation converge to strong sense, globally time, convergence rate $$O(\varepsilon )$$ , as aspect ratio $$\varepsilon $$ goes zero.