The balance equations describe changes of extensive quantities, the amounts of water, salt, momentum, and internal energy within a (infinitesimal) volume. They represent basic physical principles. Since the internal energy is not a directly measured quantity and since sea water can for many purposes be regarded as a nearly incompressible fluid it is convenient to introduce pressure and temperature as prognostic variables, instead of the density and specific internal energy. The basic equations for oceanic motions then consist of prognostic equations for the:

• pressure;

• velocity vector;

• temperature; and

• salinity.

There are six equations. They contain:

• “external” fields and parameters like the gravitational potential and earth's rotation rate, which need to be specified (or calculated);

• molecular diffusion coefficients, which need to be specified;

• thermodynamic coefficients, which need to be specified or are given by the equilibrium thermodynamic relations of Chapter 2. Most importantly, the density or equation of state must be specified.

These equations have to be augmented by appropriate boundary conditions. The equations, together with the boundary conditions, then determine the time evolution of any initial state. Nothing else is needed. It can, of course, be elucidating to study the evolution of other quantities, such as the circulation and vorticity. The evolution of such quantities is governed by theorems that are consequences of the basic equations. One important theorem is Ertel's potential vorticity theorem.