Geography Reference
In-Depth Information
is independent of height in a barotropic atmosphere. Thus, barotropy provides a
very strong constraint on the motions in a rotating fluid; the large-scale motion
can depend only on horizontal position and time, not on height.
An atmosphere in which density depends on both the temperature and the pres-
sure, ρ
ρ (p, T ), is referred to as a
baroclinic
atmosphere. In a baroclinic
atmosphere the geostrophic wind generally has vertical shear, and this shear is
related to the horizontal temperature gradient by the thermal wind equation (3.30).
Obviously, the baroclinic atmosphere is of primary importance in dynamic mete-
orology. However, as shown in later chapters, much can be learned by study of the
simpler barotropic atmosphere.
=
3.5
VERTICAL MOTION
As mentioned previously, for synoptic-scale motions the vertical velocity compo-
nent is typically of the order of a few centimeters per second. Routine meteoro-
logical soundings, however, only give the wind speed to an accuracy of about a
meter per second. Thus, in general the vertical velocity is not measured directly
but must be inferred from the fields that are measured directly.
Two commonly used methods for inferring the vertical motion field are the
kinematic method, based on the equation of continuity, and the adiabatic method,
based on the thermodynamic energy equation. Both methods are usually applied
using the isobaric coordinate system so that ω(p) is inferred rather than w(z).
These two measures of vertical motion can be related to each other with the aid of
the hydrostatic approximation.
Expanding Dp/Dt in the (x, y, z) coordinate system yields
w
∂p
∂z
Dp
Dt
=
∂p
∂t
+
ω
≡
V
·∇
p
+
(3.36)
Now, for synoptic-scale motions, the horizontal velocity is geostrophic to a first
approximation. Therefore, we can write
V
=
V
g
+
V
a
, where
V
a
is the
ageostrophic
(ρf )
−
1
k
wind and
|
V
a
||
V
g
|
. However,
V
g
=
×
∇
p, so that
V
g
·∇
p
=
0.
Using this result plus the hydrostatic approximation, (3.36) may be rewritten as
∂p
∂t
+
ω
=
V
a
·∇
p
−
gρ w
(3.37)
Comparing the magnitudes of the three terms on the right in (3.37), we find that
for synoptic-scale motions
10 hPa d
−
1
∂p/∂t
∼
1ms
−
1
1Pakm
−
1
1hPad
−
1
V
a
·∇
p
∼
∼
100 hPa d
−
1
gρ w
∼
