Geography Reference
In-Depth Information
Fig. 1.10
Balance of forces for hydrostatic equi-
librium. Small arrows show the upward
and downward forces exerted by air
pressure on the air mass in the shaded
block. The downward force exerted by
gravity on the air in the block is given
by ρgdz, whereas the net pressure force
given by the difference between the
upward force across the lower surface
and the downward force across the upper
surface is
−
dp. Note that dp is negative,
as pressure decreases with height. (After
Wallace and Hobbs, 1977.)
1.6.1
The Hydrostatic Equation
In the absence of atmospheric motions the gravity force must be exactly balanced
by the vertical component of the pressure gradient force. Thus, as illustrated in
Fig. 1.10,
dp/dz
=−
ρg
(1.18)
This condition of
hydrostatic balance
provides an excellent approximation for
the vertical dependence of the pressure field in the real atmosphere. Only for intense
small-scale systems such as squall lines and tornadoes is it necessary to consider
departures from hydrostatic balance. Integrating (1.18) from a height z to the top
of the atmosphere we find that
∞
p(z)
=
ρgdz
(1.19)
z
so that the pressure at any point is simply equal to the weight of the unit cross
section column of air overlying the point. Thus, mean sea level pressure p(0)
=
1013.25 hPa is simply the average weight per square meter of the total atmospheric
column.
4
It is often useful to express the hydrostatic equation in terms of the
geopotential rather than the geometric height. Noting from (1.8) that d
=
gdz
and from, (1.17) that α
=
RT /p, we can express the hydrostatic equation in the
form
RT d ln p (1.20)
Thus, the variation of geopotential with respect to pressure depends only on tem-
perature. Integration of (1.20) in the vertical yields a form of the
hypsometric
equation
:
gdz
=
d
=−
(RT /p)dp
=−
R
p
1
p
2
(z
2
)
−
(z
1
)
=
g
0
(Z
2
−
Z
1
)
=
Tdln p
(1.21)
4
For computational convenience, the mean surface pressure is often assumed to equal 1000 hPa.
