A ``horizontal" hyper--diffusion $3-D$ thermocline planetary geostrophic model: well-posedness and long time behavior

In this paper we study a three dimensional thermocline planetary geostrophic ``horizontal" hyper–diffusion model of the gyre-scale midlatitude ocean. We show the global existence and uniqueness of the weak and strong solutions to this model. Moreover, we establish the existence of a finite dimensional global attractor to this dissipative evolution system. Preliminary computational tests indicate that our hyper–diffusion model does not exhibit any of the nonphysical instabilities near the literal boundary which are observed numerically in other models.