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# Relation between isobaric slopes and isopycnal slopes in a baroclinic fluid

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.

Next: The equation of state Up: Fluid Density and Hydrostatic Previous: Slopes of the isobaric

Steve Baum
Mon Dec 1 08:50:29 CST 1997