A general relation between the slopes of the isobaric surfaces and the slope of the interface for a two-layer barotropic sea will now be derived. We will drop the restrictions of constant compressibility and constant g since these conditions are unnecessary in the development which follows.
Suppose 
is the density function for the upper
layer and 
is that for the lower layer. Then the
equations governing the pressure functions 
and 
are
where and are now taken as functions of the
geopotential 
rather than z. This procedure avoids
the complication of dealing with a variable g. Integration of
Eqs. (31) yields
where 
represents the geopotential (dynamic height) of the
sea surface and 
represents the geopotential at the position
of the interface, both of which are functions of x, y, t in
general. If x be taken in the direction of the horizontal
pressure gradient (i.e., in the direction of the greatest slope of
the pressure surfaces) then the magnitude of the geopotential
gradient along an isobaric surface is given
by , with the subscript indicating
that p is held constant. These gradients for the two layers
are, from Eqs. (32)
where the differentiation of the integrals of Eqs. (32) follows
Leibnitz' rule.
A relation between and 
is obtained from Eq. (32a), i.e.
Differentiating with respect to x gives
or, in view of Eq. (33a)
Elimination of 
between this equation
and Eq. (33b) leads to
where 
and 
represent the
values of 
and 
evaluated at the
depth of the interface. The magnitudes of the geopotential
gradients along the interface and along the isobaric surfaces
in the upper and lower layers can be expressed in terms of
dynamic meters per meter. Notice the the geopotential gradients
along the isobaric surfaces in the upper layer (Eq. 33a) are
independent of depth since both and 
depend
only on x, y, t and 
is a constant.
Furthermore, the geopotential gradient along the isobaric
surfaces in the lower layer is also independent of depth. This
result is consistent with one of the basic characteristics of
barotropy, which is that the isobars are parallel in the
plane. If the sea surface itself is an isobaric
surface (i.e. if is constant) then the geopotential
gradient of the sea surface itself is a measure of
. The isobars and
interface for the two-layer system are illustrated in
Fig. 3.09-3.
A more common form of Eq. (37) is obtained by taking
where 
is the magnitude of gravity at the depth of the
interface, 
is the geometrical slope of the interface,
is the geometrical slope of the isobaric surfaces
in the upper layer at the depth of the interface, and 
is the geometrical slope of the isobaric surfaces in the lower
layer at the depth of the interface.
Thus all three slopes are evaluated at the same depth and
represent change in elevation per unit of horizontal length.
If the above relations are inserted in Eq. (37) we obtain
where it is understood that all quantities are evaluated at the depth D. This is Margulies formula for a two layer system.
For all practical purposes the values of and
are independent of depth in the barotropic,
two-layer system since g varies so slightly with depth.
Equation (39) is very useful for estimating the slope of the isobaric surfaces in the upper layer relative to the lower layer in terms of measured values of the depth of the interface at several different stations.
In the special case where the isobaric surfaces are level in thelower layer (i.e. when the lower layer is in perfect hydrostatic balance) then
An abrupt change in temperature of 10 deg. C (from 15 to 5 deg. C)
at the interface would yield a value of about 
for the coefficient 
. An
increase of salinity of 2 parts per mille would give about the
same value, and the combination would give about
or roughly 1/300. Thus if the
interface deepens by 30 m in 100 km then the isobaric slope
in the upper layer is about 
under the
above conditions of temperature and salinity change at the
interface, and assuming that the lower layer is at rest.
This slope corresponds to 10 cm per 100 kilometers.
It is apparent in this example that the isobaric slope would be exceedingly difficult if not impossible to measure directly. However, the density structure, through measurements of temperature and salinity, is relatively easy to measure. Furthermore, the exact depths of the abrupt change in density is not too critical since the slope of the later is relatively large.
The two-layer model represents only a crude approximation
of the actual distribution of density in the sea, but is of
value for pedagogical purposes.
More refined methods of evaluating the isobaric slopes
in the sea are necessary for quantitative purposes.
The baroclinic situation is therefore examined
in this light in the following sections.