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
where
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.