Geoscience Reference
In-Depth Information
temperature, and pressure in a hydrostatic atmosphere. In practice, it is simpler
mathematically to allow for the effect of vertical changes in vapor content by
calculating and using the temperature profile of an equivalent dry atmosphere
with the same density as the actual moist atmosphere, i.e., the profile of
virtual
temperature
, see Equation (2.14) and associated text. In this way, the effect of
vertical changes in water vapor content can be allowed for by making a (usually
small) adjustment to the profile of potential temperature to give the
virtual
potential temperature
profile. The several hydrostatic profiles are discussed in the
following sections.
Hydrostatic pressure law
Figure 3.1 illustrates how pressure differs between two levels,
z
and
z
z
, in the
atmosphere. Consider a thin rectangular volume of air of cross-sectional area
A
and depth
+ δ
z
r
a
], where
r
a
is the density of
the moist air. At the bottom of this volume the pressure is
P
and at the top it is
P
δ
z
. This volume of air has a mass [
A
δ
P
will be
negative
because the air below the volume must exert
a higher upward force across the area
A
to balance the additional gravitational
force acting on the mass of the air between the two levels). The forces exerted at
the top and bottom of the volume of air are
A
.(
P
+ δ
P
. (Note that
δ
+ δ
P
) and
A. P
, respectively, and
additional downward gravitational force is
g.A.
r
a
.(
δ
z
), where
g
is the acceleration
due to gravity, i.e., 9.81 m s
−2
. Therefore:
[
]
AP A P
−+=
δ
P
gA
ρ δ
z
(3.1)
a
Hence:
δ=−ρδ
Pg
z
(3.2)
a
Mass = [
A
.
δ
z
.
r
a
]
P +
δ
P
A
P
δ
z
Figure 3.1
Basis of the
hydrostatic pressure law.
