Consider a vertical section of a baroclinic fluid such as the ocean oriented parallel to the x-axis. The isobars and isopycnals in this vertical plane are defined by the equations
Thus the slopes of the isobars and isopycnals are
Under quasi-static conditions, p is related to through the equation
Consequently, Eq. (42a) can be written in the form
Differentiating this equation with respect to z gives
The interchange of differentiation indicated above is permissible since p is a continuous function of x and z. Substituting from Eq. (43) gives
or taking g as constant for all practical purposes gives
The term remaining on the right can be replaced by its equivalent from Eq. (42b) to obtain
Multiplying by dz and integrating from to gives
This can also be expressed in the form
where is the mean slope of the isopycnal surfaces for the interval to . If the isopycnals are constructed for equal increments of then the value of can be estimated from the arithmetic mean of the slopes of those isopycnals lying between and .
Equation (49) can be written approximately as
where is the mean density for the layers to . If is selected as that level at which the isobaric slope is zero, then
This equation is the analogue of Eq. (40) for the case of a baroclinic fluid.
A relation identical to Eq. (50) or (51) applies for the slopes of the isobaric surfaces in the y direction. The magnitude of the combined slope is given by
and has the direction
relative to the x axis, where and are the slopes determined from the relation (51) based on the distribution of in the vertical planes oriented in the x and y directions, respectively.