Highly rotating fluids with vertical stratification for periodic data and vanishing vertical viscosity

We prove that the primitive equations without vertical diffusivity are globally well-posed (if the Rossby and Froude number are sufficiently small) in suitable Sobolev anisotropic spaces. Moreover if the Rossby and Froude number tend to zero at a comparable rate the global solutions of the primitive equations converge globally to the global solutions of a suitable limit system. The space domain considered has to belong to a class of tori which is general enough to include all non-resonant tori and many resonant tori as well.