Higher order approximation equations for the primitive equations of the ocean

In this article, we present a family of models which approximate the full primitive equations (PEs) of the ocean, with temperature and salinity, as introduced in [9]. We consider asymptotic expansions of the PEs to all orders with respect to the aspect ratio delta. At first order, we recover the well-known barotropic quasi-geostrophic (QG) equations of the ocean. At higher orders, we obtain simple linear models that share the same mathematical structure but different right-hand sides. From the computational point of view, there are two advantages. Firstly, all the higher-order expansions are linear so that they are easy to implement. Secondly, the same numerical code can be used to compute all of them. From the physical viewpoint, we expect that higher-order corrections to the first-order barotropic QG equations will capture the vertical dynamics and the thermiodynamics correctly. We will address these delicate physical issues as well as the convergence of the asymptotics in a forthcoming work.