A linear system derived from the Primitive Equations (PEs) of the atmosphere and the ocean is considered. Existence and uniqueness of solutions, behavior as a small viscosity parameter tends to zero are studied for different boundary conditions ; one of the boundary condition is of Dirichlet type and it produces reflections of waves at the boundary, the other one is of transparent type. Computational issues are addressed in a companion paper [RTT04], which also contains more details on the derivation of the system that we study here.