## CHAPTER

and adding the heat gained by turbulent convection on the same footing as the external heat sources p Q (radiation and gain of latent heat by condensation). As noted by Lorenz (1967), not all potential and internal energy is available for conversion into kinetic energy. The sum of potential and internal energy must be measured relative to a reference state of the atmosphere. Lorenz defines the reference state as a state in which the pressure equals the average value on the isentropic surface passing through the point considered (hence the convenience of introducing the potential temperature (2.51)).

### Modelling the oceans

As mentioned in section 2.3, the state of the oceans given by the temperature T, the density p = pw (or alternatively the pressure P), the salinity S (salt fraction by mass), and possibly other variables such as oxygen and organic materials present in the water. S is necessary in order to obtain a relation defining the density that can replace the ideal gas law used in the atmosphere. An example of such an equation of state, derived on an empirical basis by Eckart (1958), is

1 - Xo p where x0 is a constant and x1 and x2 are polynomials of second order in T and first order in S."

The measured temperature and salinity distributions were shown in Figs. 2.63-2.65. In some studies the temperature is instead referred to surface pressure, thus being the potential temperature d defined according to (2.51). By applying the first law of thermodynamics, (2.52), to an adiabatic process (Q = 0) the system is carried from temperature T and pressure P to the surface pressure P0 and the corresponding temperature d = T(P0). The more complex equation of state (2.58) must be used to express d(1/p) in (2.52) in terms of dT and dP, d p where

x0 = 0.698 x 10-3 m3 kg-1, x1 = (177950+ l 125 T - 7.45 T2 - (380 + T) 1000 S) m2 s-2, x2 = (5890 + 38 T - 0.375 T2 + 3000 S) 105 N m-2, where T should be inserted in °C (Bryan, 1969).

This is then inserted into (2.52) with Q = 0, and (2.52) is integrated from (T, P) to (9, P0). Because of the minimal compressibility of water (as compared to air, for example), the difference between T and 9 is small.

### Basic equations governing the oceanic circulation

Considering first a situation where the formation of waves can be neglected, the wind stress may be used as a boundary condition for the equations of motion analogous to (2.46). The "Reynold stress" eddy transport term may, as mentioned in connection with (2.46), be parametrised as suggested by Boussinesq, so that it gets the same form as the molecular viscosity term, and - more importantly - can be expressed in terms of the averaged variables,

Here use has been made of the anticipated insignificance of the vertical velocity, w*, relative to the horizontal velocity, V*. A discussion of the validity of the Boussinesq assumption may be found in Hinze (1975). It can be seen that (2.59) represents diffusion processes, which result from introducing a stress tensor of the form (2.30) into the equations of motion (2.46). In (2.59), the diffusion coefficients have not been assumed to be isotropic, but in practice the horizontal diffusivities, kx and ky are often taken to be equal (and denoted K; Bryan, 1969; Bryan and Cox, 1967). Denoting the vertical and horizontal velocity components in the ocean waters ww and Vw, the averaged equations of motion, corresponding to (2.47) and (2.48) for the atmosphere, become dP _ 