Let us now consider what the balance of forces is in the vertical direction. A moment's thought indicates that there must be a pressure gradient in the vertical direction, because without one there would be nothing to hold up the fluid. A piece of fluid, be it a piece of air or a piece of seawater, is not weightless, so the force of gravity acts on it, pulling it down toward Earth. If the fluid parcel is stationary, then there must be a force in the opposite direction that balances gravity, and this is the pressure gradient force. That is to say, there is a balance between the gravitational force and the pressure gradient force, and this is known as hydrostatic balance. The balance holds exactly only when the fluid is not moving (hence the "static" in hydrostatic), but in Earth's atmosphere and ocean, it holds to a good approximation even when the fluid is moving.
To be quantitative, let us consider the vertical force balance on a thin slab of fluid of density (mass per unit volume) p, and let the slab have thickness Sz and area A. The downward force due to gravity on the slab is just equal to g (the acceleration caused by gravity) times the mass on the fluid, pASz. Thus, the total downward gravitational force per unit volume is pgASz, or pg per unit volume. The vertical pressure force on the slab is given by equation 3.6, and so we have
Downward gravitational force = gpASz, (3.7)
A BRIEF iNTRODUCTiON TO DYNAMiCS Upward pressure gradient force = —ASz^-p ■ (3.8)
If the fluid is static, then the above two forces must balance, and we obtain
This is the equation of hydrostatic balance. (Again we use a partial derivative because pressure might also be changing in other directions.) The rate of change of pressure with height is thus proportional to the density of the fluid, and because seawater is about 1,000 times more dense than air, the pressure increases rapidly indeed as we go deeper into the ocean.
In the ocean, the density of seawater is almost constant and, if we neglect the relatively small contribution of atmospheric pressure at the surface of the ocean, equation 3.9 may be integrated from z = 0 (the ocean surface) to a depth d below the ocean surface to give
where p0 is the pressure at the ocean surface, which is small compared to p itself once d exceeds a few meters. (Note that we always measure z going up (that is, increasing in the upward direction) so d = — z.) The factor pgd is just the weight of the water column per unit area from the surface to the depth d. Thus, to a good approximation, the pressure at any depth in the fluid is equal to the weight of the fluid (per unit area) above it.
This important result has many ramifications for ocean circulation.
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