Dynamics: The Forces That Drive Atmospheric and Ocean Circulations - Climate Dynamics - page 119

HRT / is the atmospheric scale height . When z  H , p  p 0 / e  p 0 /
2.718, so H is an e -folding distance, or the vertical distance over which pres-
sure decreases by a factor of about 3, in an hydrostatic, isothermal atmosphere.
Figure 2.2 shows how pressure and height are related in the earth's atmo-
sphere; this is essentially a graph of Eq. 6.39, with pressure falling off expo-
nentially with height. Note that it is a one-to-one relationship, making pressure
a viable choice as a vertical coordinate in place of height. From Figure 2.2 we
can estimate that H ≈ 9 km, since pressure decreases to about one-third of its
surface value over about 9 km. The scale height is often used to characterize
atmospheric depth because it is not possible exactly to define the top of an
atmosphere.
In the ocean, where the ideal gas law cannot be applied, the relationship
between pressure and elevation (depth) is different. For the ocean we can make
the incompressibility assumption that density is constant with depth; this is
an accurate assumption below the pycnocline, that is, below about 1000 km
depth. Integrating the hydrostatic balance equation (Eq. 6.35) from the surface
of the ocean, where p  p 0 and z  0, to some depth z , where z < 0, we obtain

where /

p

z

# #

dp =− ρ gdz =− =− ρ gzppgz ρ . (6.40)

&

0

p 0

0

So, in the hydrostatic, incompressible ocean, pressure is a linear function of
depth.
We now turn to the horizontal directions. When pressure is used as a vertical
coordinate, the horizontal pressure gradient force can be expressed as a func-
tion of geopotential (or geopotential height). According to Eq. 2.3, gdzd Φ = ,

so the hydrostatic relation (Eq. 6.35) can be written as

TT T
ρ

p

.

gz == (6.41)

Then, from Eq. 6.33,

v

2 Φ Φ

2

2

t t

F =− −

x i y j

, (6.42)

PGF

2

where the horizontal derivatives must be taken on a pressure surface (holding p
constant) instead of a z surface (constant elevation). Recall that regions of low
geopotential height are regions in which the height of a pressure level is low,
that is, where isobaric surfaces dip closer to the surface. Because pressure de-
creases with elevation in the atmosphere, regions with low geopotential height
are also regions of low pressure.
The hydrostatic relation and the idea gas law can be used to express the
relationship between geopotential height and temperature in the atmosphere.
Using the ideal gas law in Eq. 6.38, we obtain

RT dp

d Φ=− . (6.43)

p

Integration over a pressure interval from p 1 to p 2 gives