## 5.2 Governing Equations     continued...

The density variations in the ocean are of fundamental importance in determining the ocean circulation. Density is a complicated function of the temperature T, salinity S and the pressure (or equivalently depth z). It is common to take into account the pressure effects by considering to be potential density, which is a function of only the potential temperature and salinity S. The potential temperature is usually referred to the atmospheric pressure, meaning that it is the temperature attained by a fluid parcel of in-situ temperature T and salinity S brought adiabatically from depth z to the surface. The baroclinic pressure gradient terms in the momentum equations and the vertical stability of the fluid column can be evaluated accurately using the horizontal and vertical gradients of potential density , which can be regarded as a function of and S only:

This equation of state is usually expressed as a power series expansion in and S, but can be easily evaluated numerically. The most popular equations of state in general use in ocean models are those due to Fofonoff [15] and more recently from UNESCO [22]. The model has provisions to use either one.

The potential temperature and salinity S are governed by conservation relations of the form:

and

where is the vertical and the horizontal eddy diffusivities due to turbulent mixing of heat and salt in the water column. The last term in equation (38) accounts for the heating due to penetrative shortwave solar radiation. While the values of horizontal eddy diffusivities and are also influenced by numerical considerations arising from the need to damp subgrid scale computational noise, the vertical diffusivities and are principally determined from small scale turbulent mixing in the water column. Nevertheless, proper parameterization of , , and is the most important unresolved issue in ocean modeling since the model results are sensitive in varying degrees to the values of these coefficients and the theoretical underpinnings needed to confidently prescribe them are for the most part quite shaky.