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It was shown by Oliger and Sundström, in 1978, that the initial boundary value problems for the hydrostatic primitive equations of meteorology and oceanography are ill posed if the boundaries are open and fixed in space. In this article it is shown, with theory and computation, that the same problems are well posed for a suitable set of local (pointwise applied) boundary conditions, if a mild vertical viscosity is added to the hydrostatic equation. Some indications on the behavior of thedoi:10.1175/1520-0469(2003)060<2647:obcftp>2.0.co;2 fatcat:pkse5drwpvbh3oucmh753ewedm