Geography Reference
In-Depth Information
10 m s
−
1
U
∼
horizontal velocity scale
1cms
−
1
W
∼
vertical velocity scale
10
6
m
L
∼
length scale [
∼
1/(2π) wavelength]
10
4
m
H
∼
depth scale
10
3
m
2
s
−
2
δ
P
/ρ
∼
horizontal pressure fluctuation scale
10
5
s
L/U
∼
time scale
Horizontal pressure fluctuation δP is normalized by the density ρ in order to
produce a scale estimate that is valid at all heights in the troposphere, despite
the approximate exponential decrease with height of both δP and ρ. Note that
δP /ρ has units of a geopotential. Referring back to (1.25) we see that indeed the
magnitude of the fluctuation of δP /ρ on a surface of constant height must equal the
magnitude of the fluctuation of the geopotential on an isobaric surface. The time
scale here is an advective time scale, which is appropriate for pressure systems
that move at approximately the speed of the horizontal wind, as is observed for
synoptic scale motions. Thus, L/U is the time required to travel a distance L at a
speed U , and the substantial differential operator scales as D/Dt
∼
U/L for such
motions.
It should be pointed out here that the synoptic scale vertical velocity is not a
directly measurable quantity. However, as shown in Chapter 3, the magnitude of
w can be deduced from knowledge of the horizontal, velocity field.
We can now estimate the magnitude of each term in (2.19) and (2.20) for synoptic
scale motions at a given latitude. It is convenient to consider a disturbance centered
at latitude φ
0
=
45
◦
and introduce the notation
2 cos φ
0
=
10
−
4
s
−
1
f
0
=
2 sin φ
0
=
Table 2.1 shows the characteristic magnitude of each term in (2.19) and (2.20)
based on the scaling considerations given above. The molecular friction term is so
small that it may be neglected for all motions except the smallest scale turbulent
motions near the ground, where vertical wind shears can become very large and
the molecular friction term must be retained, as discussed in Chapter 5.
Table 2.1
Scale Analysis of the Horizontal Momentum Equations
A
B
C
D
E
F
G
uv tan φ
a
∂p
∂x
D
Dt
u
a
ρ
x
−
Eq.
−
2v sin φ
+
2w cos φ
+
−
=−
+
F
rx
u
2
tan φ
a
∂p
∂y
D
Dt
v
a
ρ
y
−
Eq.
+
2u sin φ
+
+
=−
+
F
ry
U
2
a
U
2
/L
U
a
δ
ρL
νU
H
2
Scales
f
0
U
f
0
W
(m s
−
2
)
−
4
10
−
3
10
−
6
10
−
8
10
−
5
10
−
3
10
−
12
