Typical temperature, salinity, and density profiles for the mid-latitude regions of the sea are shown in Fig. 3.09-1. The situation shown represents a stable one with respect to density structure, with light water at the higher level.
As a rough approximation to the real situation one might regard the region of rapid transition of temperature and salinity with depth as a discontinuity, and imagine the fluid as composed of two barotropic layers with an interface at depth D, which corresponds roughly to the depth of maximum density gradient in the real fluid (Fig. 3.09-2). Furthermore, consider that the compressibility is such that the density in each layer is a linear function of pressure, i.e.
where K is a constant representing a mean compressibility coefficient. The values of and depend upon the mean temperature and salinity of each layer.
For quasi-static conditions
where and denote the pressure in the upper and lower layers, respectively, which are functions of the elevation z.
If we regard g as constant with respect to depth, the integration of Eqs. (27) leads to the following expressions for the functions and :
where is the atmospheric pressure at the actual sea surface located at elevation above the mean sea level reference, and is the pressure at the position of the interface . The value of can be evaluated from Eq. (28a) by setting , since at the interface the pressure must be continuous.
The mean compresibility of sea water is about decibars (The Oceans, Table 15). Consequently, the maximum value of in the ocean is about 0.04 which corresponds to the depth of the Mindanao Deep ( ). Therefore, the exponential terms in Eqs. (28) can be represented very accurately by the first three terms of the power series expansion of these functions. After carrying out this expansion and neglecting terms involving squares and higher powers of K we obtain for the final expressions
If, as a first approximation, the term modified by K in each of these equations is neglected, then we have the equations for an ideal incompressible fluid. In the ocean, the term involving K in Eq. (29b) is about 200 decibars for a depth of 10,000 meters, which represents about 2 per cent of the magnitude of the first two terms (roughly 10,000 decibars). However, for a depth of 1000 meters the last term in Eq. (29b) is only about 2 decibars or roughly 0.2 per cent of the sum of the first two terms.