Rigorous derivation of the primitive equations with full viscosity and full diffusion by scaled Boussinesq equations

The primitive equations of large-scale ocean dynamics form the fundamental model in geophysical flows. It is well-known that the primitive equations can be formally derived by hydrostatic balance. On the other hand, the mathematically rigorous derivation of the primitive equations without coupling with the temperature is also known. In this paper, we generalize the above result from the mathematical point of view. More precisely, we prove that the scaled Boussinesq equations strongly converge to the primitive equations with full viscosity and full diffusion as the aspect ration parameter goes to zero, and the rate of convergence is of the same order as the aspect ratio parameter. The convergence result mathematically implies the high accuracy of hydrostatic balance.