Boundary Conditions for the 2d Linearized Pes of the Ocean in the Absence of Viscosity

The linearized Primitive Equations with vanishing viscosity are considered. Some new boundary conditions (of transparent type) are introduced in the context of a modal expansion of the solution which consist of an infinite sequence of integral equations. Applying the linear semi-group theory, existence and uniqueness of solutions is established. The case with nonhomogeneous boundary values, encountered in numerical simulations in limited domains, is also discussed. Introduction. The Primitive Equations of the ocean and the atmosphere are fundamental equations of geophysical fluid mechanics ([14],[20],[24]). In the presence of viscosity, it has been shown, in various contexts, that these equations are wellposed (see e.g. [9],[10], and the review article [23]).The viscosity appearing in [9] is the usual second order dissipation term. Other viscosity terms have also been considered as in the so-called δ-PEs proposed with different motivations in [20] and [22]. It has been shown in [15] and [22] that the mild vertical viscosity appearing in the δ-PEs is sufficient to guarantee well-posedness. It is generally accepted that the viscosity terms do not affect numerical simulations (predictions) in a limited domain, over a period of a few days, and these viscosities are generally not used, see [25]. Now, for the PEs without viscosity, and to the best of our knowledge, no result of well-posedness has ever been proven, since the negative result of Oliger and Sundström [12] showing that these equations are ill-posed for any set of local boundary conditions (see also the analysis in [22]). Whereas the analysis of the PEs with viscosity bears some similarity with that of the incompressible Navier Stokes equations (see [9, 10, 23]), it is noteworthy 2000 Mathematics Subject Classification: 35L50, 76N10, 47D06, 86A05.