The ocean and the atmosphere: An applied mathematician's view

In this survey article, we provide a mathematical description of oceanic and atmospheric flows, based on the incompressible Navier–Stokes equation (for the ocean), and the compressible version with an equation of state and the first law of thermodynamics for the atmosphere. We show that, in both cases, the only fundamental assumption that we need to make is that of a thin shell on a (nearly) spherical Earth, so that the main elements of spherical geometry are included, with all other attributes of the fluid motion retained at leading order. (The small geometrical correction that is needed to represent the Earth's geoid as an oblate spheroid is briefly described.) We argue that this is the only reliable theoretical approach to these types of fluid problem. A generic formulation is presented for the ocean, and for the steady and unsteady atmosphere, these latter two differing slightly in the details. Based on these governing equations, a number of examples are presented (in outline only), some of which provide new insights into familiar flows. The examples include the Ekman flow and large gyres in the ocean; and in the atmosphere: Ekman flow, geostrophic balance, Brunt–Väisälä frequency, Hadley–Ferrel–polar cells, harmonic waves, equatorially trapped waves.