Hereafter, hydrostatic eqilibrium will be understood to imply hydrostatic equilibrium of a fluid in the presence of a field of gravity of the earth (i.e., geo-hydrostatic equilibrium).

Consider a vertical column of fluid of height *z* - *h* and
cross sectional area *A*. We will consider the density of
the fluid as defined in the previous article is endowed
with the additional property that the integral

represents the weight of the vertical column of height
*z* - *h*. Under hydrostatic conditions, the net surface

force must balance the weight *W*, thus

If this equation is differentiated with respect to *z*
we obtain

which is the scalar form of the differential equation for hydrostatic equilibrium.

The complete vector equation can be obtained by a more general approach. The property (13) can be generalized in the form of the following vector equation

where is the total force exerted upon an
arbitrary volume of fluid *V* by the field of gravity.
Equation (16) has the advantage that is is independent
of the coordinate system chosen.

THe force acting on a small element of surface area
*dA* of the volume *V* is where
is the unit outward normal to the surface at the
point in question and *p* is the fluid pressure

at that point. The total surface force is therefore given by the surface integral

where *A* is the total surface area of the
arbitrary volume *V*.

The condition of hydrostatic balance can therefore be expressed by

.

In order to put this in differential form, we will make
use of a mathematical integral tranformation known as the
*theorem of Gauss*.
This theorem states that for any continuous position function
*f*

where is the unit outward normal at any point on the
surface *A*. Actually, it can be shown that the divergence
theorem (Eq. 154, Art. 1.071, p. 65) is a special case
of the theorem of Gauss.
*in general* is that the integrand vanish. Therefore

This is the complete statement of the hydrostatic equation in vector form.

If the above theorem is applied to the pressure function
*p*, Eq. (17) takes the form

However, since *V* is arbitrary, it follows that
the only way that Eq. 19 can be satisfied *in general*
is that the integrand vanish. Therefore

This is the complete statement of the hydrostatic equation in vector form.

Equation (20) implies several things:

- the magnitude of the pressure gradient is , hence
for a given interval of
*p*the spacing of isobaric surfaces is inversely proportial to and*g*; - the gradient of
*p*is in the direction of under perfect hydrostatic conditions, therefore the isobaric surfaces are perpendicular to and are therefore level surfaces; and - the surfaces of constant density (isopycnal surfaces) must be level in order that the isobaric surfaces by level.

If we let represent any differential change of position,
then the scalar products and
are equivalent to the total space
differentials *dp* and .
Consequently, the scalar product of on Eq. (21) leads to

However, since the isobaric surfaces must be level under hydrostatic
conditions, as implied by Eq. (20) or (21), it therefore follows
that *p* is a function of only, since the latter function defines
the level surfaces. Consequently, the left hand side of
Eq. (22) is a function only of and hence must
also be a function only of . Therefore the isopycnal surfaces
are level surfaces *under perfectly hydrostatic conditions*.

In summary, the orientation of the surfaces of *p*, , and
are identical under hydrostatic conditions and would form
approximately spheroidal surfaces on the earth under this
condition.

Mon Dec 1 08:50:29 CST 1997