Geoscience Reference
In-Depth Information
profile in the lower troposphere for the US Standard Atmosphere shown in Fig. 1.2
illustrates that this is the case on average, the average near-surface environmental
lapse rate in this case being 6.5 K km
−1
. However, it is important to emphasize that
the behavior shown in Fig. 1.2 is a temporal and spatial average for all of the USA
and the actual environmental lapse rate on any particular day and at any particu-
lar place will differ from this average profile. The actual environmental lapse rate
and, especially, its relationship to the dry adiabatic lapse rate and the moist
adiabatic lapse rate can affect cloud and precipitation formation, as discussed
in Chapter 10.
Vertical pressure and temperature gradients
Recall that the vertical gradients of atmospheric pressure and temperature are
necessarily linked via the ideal gas law. If
R
a
is the specific gas constant for moist
air, then by combining Equation (2.5) with Equation (3.2) and rearranging, it can
be shown that the small change in pressure over a vertical distance
δ
z
is given by:
P
g
δ
=−
δ
z
(3.11)
P
R T
a
T
z
. Using this relation-
If
Γ
local
is the local environmental lapse rate, then
δ
=
Γ
local
δ
ship to substitute for
z
in Equation (3.11), it follows that the small changes in
pressure and temperature with height are related by:
δ
⎛
⎞
δ
gP T
P R T
δ
=
⎜
(3.12)
⎟
Γ
⎝
⎠
al l
If
Γ
local
is constant through a portion of the atmosphere then by taking the limit
of Equation (3.12) and integrating between two levels where the air temperatures
(in degrees K) are
T
1
and
T
2
, and air pressures
P
1
and
P
2
, it can be easily shown that
across this region:
g
⎛⎞
T
PP
T
R
Γ
al l
2
=
(3.13)
⎜
⎝⎠
2
1
1
Γ
⎛⎞
R
al l
P
TT
P
g
2
=
(3.14)
⎜
⎝⎠
2
1
1
Figure 3.2 illustrates the lapse rate and the way atmospheric pressure and air
density consequently change through the lower troposphere for the US Standard
Atmosphere for which
Γ
local
=
6.5 K km
−1
.
