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The hydrostatic equation

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.gif 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:

  1. 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;
  2. 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
  3. the surfaces of constant density (isopycnal surfaces) must be level in order that the isobaric surfaces by level.
The last statement can be proved as follows. Equation (63), Art. 2.06, stipulates that can be expressed as the gradient of the geopotential . Consequently, the hydrostatic equation can be written in the form

If we let represent any differential change of position, then the scalar products and are equivalent to the total space differentials dp and .gif 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.




next up previous contents
Next: Units of pressure Up: Fluid Density and Hydrostatic Previous: Fluid density defined

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