FIGURE 3.5. A vertical column of air of density p, horizontal cross-sectional area SA, height Sz, and mass M = pSASz. The pressure on the lower surface is p, the pressure on the upper surface is p + Sp.

where Sp is the change in pressure moving from z to z + Sz. Assuming Sz to be small, dp

Now, the mass of the cylinder is

If the cylinder of air is not accelerating, it must be subjected to zero net force. The vertical forces (upward being positive) are:

i) gravitational force

Fg = -gM = -gpSASz, ii) pressure force acting at the top face, Ft = - (p + Sp) SA, and iii) pressure force acting at the bottom face, Fb = pSA.

Setting the net force Fg + FT + FB to zero gives Sp + gpSz = 0, and using Eq. 3-2 we obtain

Eq. 3-3 is the equation of hydrostatic balance. It describes how pressure decreases with height in proportion to the weight of the overlying atmosphere. Note that since p must vanish as z ^ x—the atmosphere fades away2—we can integrate Eq. 3-3 from z to x to give the pressure at any height p(z) = g dz.

Here p dz is just the mass per unit area of the atmospheric column above z. The surface pressure is then related to the total mass of the atmosphere above: Eq. 3-4

impliesthat ps = „„rfa^ of Earth . Thus, from measurements of surface pressure, one can deduce the mass of the atmosphere (see Table 1.3).

The only important assumption made in the derivation of Eq. 3-3 was the neglect of any vertical acceleration of the cylinder (in which case, the net force need not be zero). This is an excellent approximation under almost all circumstances in both the atmosphere and ocean. It can become suspect, however, in very vigorous small-scale systems in both the atmosphere and ocean (e.g., convection; tornados; violent thunderstorms; and deep, polar convection in the ocean; see Chapters 4 and 11). We discuss hydrostatic balance in the context of the equations of motion of a fluid in Section 6.2.

Note that Eq. 3-3 does not tell us what p(z) is, since we do not know a priori what p(z) is. In order to determine p(z) we must invoke an equation of state to tell us the

Blaise Pascal (1623-1662), a physicist and mathematician of prodigious talents and accomplishments, was also intensely interested in the variation of atmospheric pressure with height and its application to the measurement of mountain heights. In 1648 he observed that the pressure of the atmosphere decreases with height and, to his own satisfaction, deduced that a vacuum existed above the atmosphere. The unit of pressure is named after him.

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