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The primitive equations of large-scale oceanic dynamics form a fundamentalmodel in geophysical flows. It is well-known that the primitive equations canbe formally derived by the hydrostatic approximation. On the other hand, themathematically rigorous derivation of the primitive equations without couplingwith the temperature is also known. In this paper, we generalize the aboveresult from the mathematical point of view. More precisely, we prove that thescaled Boussinesq equations with rotation converge to the full primitiveequations in a strong sense, globally in time, with the convergence rate$O(\varepsilon)$, as the aspect ratio $\varepsilon$ goes to zero.

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