  A modification of the classical primitive equations of the atmosphere is
considered in order to take into account important phase transition phenomena
due to air saturation and condensation. We provide a mathematical formulation
of the problem that appears to be new in this setting, by making use of
differential inclusions and variational inequalities, and which allows to
develop a rather complete theory for the solutions to what turns out to be a
nonlinearly coupled system of non-smooth partial differential equations.
Specifically we prove global existence of quasi-strong and strong solutions,
along with uniqueness results and maximum principles of physical interest.
