I will first briefly describe the different types of oceanic circulations and their link with climate change. The shallow-water model will then be considered since it is often used as a prototype of the primitive equations and it is frequently employed as a benchmark for numerical schemes to be used in more complex oceanic models. In particular, the shallow-water model support the propagation of Kelvin, Inertia-gravity and Rossby waves implied in the El Niño phenomenon. A few results will be presented to show the difficulty of choosing an appropriate numerical method to solve the shallow-water equations. Finally, I will show that a class of Discontinuous Galerkin methods has very interesting properties to solve geophysical flows. The analytical results will be illustrated with the numerical simulation of equatorial waves and oceanic eddies.
