We study, in this article, the well-posedness of a planetary geostrophic (PG) model of the ocean. This model is derived from the PG studied in [17] by adding a (non-linear) advection term in the horizontal momentum equation. We address in particular the question of existence and uniqueness of global (in time) weak solution to the model. We also discuss the asymptotic behavior of the weak solutions and the stability of the stationary solutions. We prove that any weak solution to the model converges exponentially with time to a weak solution to the primitive equations of the ocean provided that the data are small.
