Geography Reference
In-Depth Information
The second terms in (6.35a) and (6.35b) are equal and opposite (hence
the cancellation referred to above), whereas the first term in (6.35b) is generally
much smaller than the first term in (6.35a). Thus, for adiabatic flow, the omega
equation (6.34) can be expressed approximately as
∂
V
g
∂p
·∇
1
f
0
∇
f
f
0
σ
∂
2
∂p
2
f
0
σ
2
2
∇
+
ω
≈
+
(6.36)
A
Equation (6.36) involves only derivatives in space. It is, therefore, a diagnos-
tic equation for the field of ω in terms of the instantaneous field. The omega
equation, unlike the continuity equation, provides a method of estimating ω that
does not depend on observations of the ageostrophic wind. In fact, direct wind
observations are not required at all; nor does the omega equation require infor-
mation on the vorticity tendency, as required in the vorticity equation method,
or information on the temperature tendency, as required in the adiabatic method
discussed in Section 3.5.2. Only observations of at a single time are needed
to determine the ω field using (6.36). The terms in (6.36), however, do employ
higher order derivatives than are involved in the other methods of estimating ω.
Accurately estimating such terms from noisy observational data can be difficult.
The terms in (6.36) can be physically interpreted by first observing that the
differential operator on the left is very similar to the operator in term A of the ten-
dency equation (6.23) and hence tends to spread the response to a localized forcing.
Because the forcing in (6.36) tends to be a maximum in the midtroposphere, and ω
is required to vanish at the upper and lower boundaries, for qualitative discussion it
is permissible to assume that ω has sinusoidal behavior not only in the horizontal,
but also in the vertical:
ω
=
W
0
sin
(
πp/p
0
)
sin kx sin ly
(6.37)
we can then write
ω
k
2
2
ω
f
0
π
p
0
f
0
σ
∂
2
∂p
2
1
σ
2
l
2
∇
+
≈−
+
+
−
which shows that the left side of (6.36) is proportional to
ω. However, ω is
proportional to
w so ω<0 implies
upward
vertical motion, and the left-hand
side of (6.36) is proportional to the vertical velocity. Thus, upward motion is
forced where the right-hand side of (6.36) is positive and downward motion is
forced where it is negative.
The right side of (6.36) represents the advection of absolute vorticity by the
thermal wind. To understand the role of this advective term in forcing vertical
−
