In this chapter we establish the equations that govern the evolution of a fluid that is not in thermodynamic, but in local thermodynamic equilibrium. These equations are the balance or budget equations for water, salt, momentum, angular momentum, and energy within arbitrary volumes. We formulate these equations by making the continuum hypothesis. The balance equations then become partial differential equations in space and time. They all have the same general form. The rate of change of the amount within a volume is given by the fluxes through its enclosing surface and by the sources and sinks within the volume. This balance is self-evident. Specific physics enters when the fluxes and the sources and sinks are specified. If no sources and sinks are assumed then the balance equations become conservation equations. It will turn out that six balance equations, the ones for the mass of water, the mass of salt, the three components of the momentum vector, and energy completely determine the evolution of oceanic motions, given appropriate initial and boundary conditions. The vector and tensor calculus required for this chapter is briefly reviewed in Appendix B.

Continuum hypothesis

The ocean is assumed to be in local thermodynamic equilibrium. Each fluid parcel can be described by the usual thermodynamic variables such as temperature, pressure, etc., and the usual thermodynamic relations hold among these variables, but the values of these variables depend on the fluid particle. The ocean as a whole is not in thermodynamic equilibrium.