  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.
