Geography Reference
In-Depth Information
Thus, it is quite a good approximation to let
=−
ω
ρgw
(3.38)
3.5.1
The Kinematic Method
One method of deducing the vertical velocity is based on integrating the conti-
nuity equation in the vertical. Integration of (3.5) with respect to pressure from a
reference level p
s
to any level
p
yields
p
∂u
∂x
+
∂v
∂y
ω (p)
=
ω (p
s
)
−
dp
p
s
p
p)
∂
(3.39)
∂x
+
u
∂
v
=
ω (p
s
)
+
(p
s
−
∂y
p
Here the angle brackets denote a pressure-weighted vertical average:
p
p
s
)
−
1
≡
(p
−
()dp
p
s
With the aid of (3.38), the averaged form of (3.39) can be rewritten as
∂
ρ
(
z
s
)
w
(
z
s
)
ρ(z)
p
s
−
p
ρ(z)g
∂x
+
u
∂
v
w(z)
=
−
(3.40)
∂y
where z and z
s
are the heights of pressure levels p and p
s
, respectively.
Application of (3.40) to infer the vertical velocity field requires knowledge of the
horizontal divergence. In order to determine the horizontal divergence, the partial
derivatives ∂u/∂x and ∂v/∂y are generally estimated from the fields of u and v
by using
finite difference
approximations (see Section 13.3.1). For example, to
determine the divergence of the horizontal velocity at the point x
0
, y
0
in Fig. 3.10
we write
∂u
∂x
+
∂v
∂y
≈
u (x
0
+
d)
−
u (x
0
−
d)
v (y
0
+
d)
−
v (y
0
−
d)
+
(3.41)
2d
2d
However, for synoptic-scale motions in midlatitudes, the horizontal velocity is
nearly in geostrophic equilibrium. Except for the small effect due to the variation of
the Coriolis parameter (see Problem 3.19), the geostrophic wind is nondivergent;
that is, ∂u/∂x and ∂v/∂y are nearly equal in magnitude but opposite in sign. Thus,
the horizontal divergence is due primarily to the small departures of the wind from
geostrophic balance (i.e., the ageostrophic wind). A 10% error in evaluating one of
the wind components in (3.41) can easily cause the estimated divergence to be in
error by 100%. For this reason, the continuity equation method is not recommended
for estimating the vertical motion field from observed horizontal winds.
