Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 51

arrive at this generalization for liquids we use the condition
of invariant density via the assumption of incompressibility.
For gases we assume that compressibility changes as a
function of height alone; so for any given height above
surface, density is similar. In both cases surfaces of equal
pressure, isobars, parallel the horizontal fluid upper sur-
face. This is known as the barotropic condition . However,
density is free to vary independently of depth or height in
Nature; we stressed in Section 2.3.3 that although density
is a material property it depends on stated conditions of
variables such as temperature or salinity. Because of such
causes of density variations, pressure at similar heights in
the atmosphere or depths in the ocean may differ laterally.
This is known as the baroclinic condition .
Horizontal gradients in hydrostatic pressure act to cause
fluid flow down the gradient from high to low pressure. In
the oceans, lateral pressure gradients often arise due to
slope of the ocean surface (Sections 4.1 and 6.4); in these
cases the horizontal pressure gradient exists despite the
fact that the barotropic condition exists, it is the slope of
the isobars with respect to horizontal that matters. Such
slopes may be caused by wind shear or variations in atmos-
pheric pressure (Sections 6.2 and 6.4). Lateral gradients
may also be due to vertical changes in the temperature
gradient and therefore water density (e.g. Fig. 8.12).
However, salinity contrasts also occur; these may either
reinforce or diminish any temperature-driven density
contrasts (see Section 6.4).
Horizontal pressure differences are commonly thermally
driven in the atmosphere; adjacent parts may be differen-
tially heated by variations in longwave radiation given off
from the ground after solar heating or by differential
shortwave radiative heating aloft. On a global scale this
may be seen in the meridional contrast in surface tempera-
ture from equator to Pole, the lateral gradient giving rise
to what is known as the thermal wind and ultimately
responsible for the jet streams (Section 6.1). On a smaller
scale, horizontal pressure gradients may be due to density
differences arising from diurnal contrasts in reradiated heat
flux from land and water surfaces, giving rise to the
phenomena of land and sea breezes.
Horizontal lithostatic pressure gradients also exist in
the outer 50 km or so of solid Earth, despite the tendency
for pressures below that to be approximately equal
(the concept of isostasy, see Section 3.6). This is because of
lateral differences in rock composition and density
above the “compensation” level. It is thought by
some that the gradients are sufficient to cause slow lower
crustal and upper mantle flow along the gradients,
especially when the rock is weakened by water or elevated
temperatures.

(Surface atmospheric pressure = p 0 at y = 0)

p 0

Surface, y = 0

da

Still fluid
of

Cylindrical
volume,
immersed
in liquid,
open at
both ends

dy

density
r

-F W

da

Depth, y = y

F H

Equal and opposite forces balanced at any point
Weight force = Hydrostatic pressure force
- F W = F H

-( r g ) d y d a = p d a

or
d p /d y = - rg and by integration p = - rgy + p 0

Fig. 3.22 Pressure gradient in stationary fluid.

ends, is immersed vertically within a stationary fluid of
constant density. There is a weight force due to gravity act-
ing positively downward on the base of the cylinder. We use
Newton's Third Law to insist that an equal and opposite
force, negative upward, must exist to balance this weight
force. This is the hydrostatic force . When we balance the
two forces and let the cylinder diminish to an infinitesimal
point, we get an expression for the pressure gradient of
universal significance. It reveals that the gravitational
weight force at a point (remember this is equal in all
directions) is given by the vertical spatial gradient of the
hydrostatic pressure (Cookie 6). For incompressible
liquids and solids at rest on and in the Earth, the effect of
surface atmospheric pressure is commonly neglected. For
the compressible atmosphere, a modification of the basic
formula is necessary (Cookie 7).

3.5.3 Horizontal gradients in hydrostatic pressure

Thus far we have considered that in any fluid the pressure
at all similar depths or heights must be equal. In order to