Geography Reference
In-Depth Information
based on typical observed magnitudes for synoptic-scale motions are chosen as
follows:
10 m s
−
1
U
∼
horizontal scale
1cms
−
1
W
∼
vertical scale
10
6
m
L
∼
length scale
10
4
m
H
∼
depth scale
δp
∼
10 hPa
horizontal pressure scale
1kgm
−
3
ρ
∼
mean density
10
−
2
δρ /ρ
∼
fractional density fluctuation
10
5
s
L/U
∼
time scale
10
−
4
s
−
1
f
0
∼
Coriolis parameter
10
−
11
m
−
1
s
−
1
β
∼
“beta” parameter
Again we have chosen an advective time scale because the vorticity pattern, like
the pressure pattern, tends to move at a speed comparable to the horizontal wind
speed. Using these scales to evaluate the magnitude of the terms in (4.16), we first
note that
∂v
∂x
−
∂u
∂y
U
L
∼
10
−
5
s
−
1
ζ
=
where the inequality in this expression means less than or equal to in order of
magnitude. Thus,
10
−
1
ζ/f
0
U/ (f
0
L)
≡
Ro
∼
For midlatitude synoptic-scale systems, the relative vorticity is often small (order
Rossby number) compared to the planetary vorticity. For such systems, ζ may be
neglected compared to f in the divergence term in the vorticity equation
f )
∂u
f
∂u
∂v
∂y
∂v
∂y
(ζ
+
∂x
+
≈
∂x
+
This approximation does not apply near the center of intense cyclonic storms.
In such systems
1, and the relative vorticity should be retained.
The magnitudes of the various terms in (4.16) can now be estimated as
|
ζ/f
|∼
U
2
L
2
∂ζ
∂t
,u
∂ζ
∂x
,v
∂ζ
10
−
10
s
−
2
∂y
∼
∼
w
∂ζ
WU
HL
∼
10
−
11
s
−
2
∂z
∼
