Geography Reference
In-Depth Information
v
df
10
−
10
s
−
2
dy
∼
Uβ
∼
f
∂u
∂v
∂y
f
0
U
L
10
−
9
s
−
2
∂x
+
∼
∂w
∂x
∂v
∂z
−
∂w
∂y
∂u
∂z
WU
HL
∼
10
−
11
s
−
2
∂ρ
∂x
1
ρ
2
∂p
∂y
−
∂ρ
∂y
∂p
∂x
δρ δp
ρ
2
L
2
10
−
11
s
−
2
∼
The inequality is used in the last three terms because in each case it is possible that
the two parts of the expression might partially cancel so that the actual magnitude
would be less than indicated. In fact, this must be the case for the divergence term
(the fourth in the list) because if ∂u/∂x and ∂v/∂y were not nearly equal and oppo-
site, the divergence term would be an order of magnitude greater than any other
term and the equation could not be satisfied. Therefore, scale analysis of the vor-
ticity equation indicates that synoptic-scale motions must be quasi-nondivergent.
The divergence term will be small enough to be balanced by the vorticity advection
terms only if
∂u
∂x
+
∂v
∂y
10
−
6
s
−
1
so that the horizontal divergence must be small compared to the vorticity in
synoptic-scale systems. From the aforementioned scalings and the definition of
the Rossby number, we see that
∂u
∂x
+
f
0
∂v
∂y
Ro
2
and
ζ
∂u
∂x
+
∂v
∂y
Ro
Thus the ratio of the horizontal divergence to the relative vorticity is the same
magnitude as the ratio of relative vorticity to planetary vorticity.
Retaining only the terms of order 10
−
10
s
−
2
in the vorticity equation yields the
approximate form valid for synoptic-scale motions,
f
∂u
D
h
(ζ
+
f)
∂v
∂y
=−
∂x
+
(4.22a)
Dt
