Here we present the Boussinesq approximation. It consists of two steps: the anelastic approximation and a set of additional approximations exploiting the characteristics of the oceanic density field.

In many oceanographic problems one is only interested in time scales slower than those associated with acoustic phenomena. The simplest and most straightforward way to eliminate sound waves is to regard sea water as incompressible, i.e., to consider the limit c2 → ∞ in the basic equations of oceanic motions. While this limit is adequate for many purposes compressibility (though not sound waves) plays a crucial role in the deep thermohaline circulation. For this reason one introduces the anelastic approximation, which assumes that the pressure field adjusts instantaneously. The anelastic approximation eliminates sound waves but does not remove compressibility effects. It is the lowest order in an expansion with respect to the small parameter ∈a := Ta/T where Ta is the fast time scale of acoustic waves and T the slow time scale of the motions under consideration. The main structural change in the equations brought about by the anelastic approximation is that the pressure and vertical velocity become diagnostic variables.

The density field of the ocean is characterized by the facts that:

• the density of the ocean at a point does not differ very much from a reference density; and

• the reference density does not change very much from surface to bottom.