Finite Element Solution of the Primitive Equations of the Ocean by the Orthogonal Sub-Scales Method

This paper introduces a method for the numerical solution of steady Primitive Equations of the Ocean. This is an adaptation of the Orthogonal Subscales – Variational Multiscale Method, using conforming finite elements. We choose this method on one hand because it is a stabilized method, thus providing a low-cost and accurate discretization. On another hand, because it also is a LES turbulence model, so no further turbulence modeling is needed. We perform a numerical analysis of stability and convergence by means of representation of stabilizing terms in spaces of bubble finite elements. In particular, we give an original proof of the inf-sup condition to estimate the surface pressure. We present some numerical experiments for 2D flows that confirm the theoretical expectations.