Geography Reference
In-Depth Information
Combining this equation with the continuity equation (12.3) and requiring that
ρ
0
w
∗
→
0asz
→∞
it follows that
∞
∂
∂y
1
f
0
ρ
o
w
∗
=−
ρ
0
Gdz
(12.8)
z
According to (12.8), w
∗
is zero in the regions,
above
a localized forcing region,
whereas it is constant for regions directly below it, hence the term “downward
control” is sometimes used to refer to this steady-state situation.
Substitution of (12.8) into the first law of thermodynamics (12.2) and neglecting
the time-dependent term yield an expression that explicitly shows the dependence
on the zonal force distribution of the departure of the time and zonally averaged
temperature from radiative equilibrium:
∞
T
T
r
=
N
2
H
α
r
ρ
0
R
∂
∂y
1
f
0
ρ
0
Gdz
−
(12.9)
z
Thus, in steady state the departure of temperature from radiative equilibrium at a
given level depends on the meridional gradient of the zonal force distribution in
the column above that level.
For mathematical simplicity the variation of the Coriolis parameter with lati-
tude and other effects of spherical geometry have been neglected in deriving the
equations of this section. It is straightforward to extend this model to spherical
coordinates. Figure 12.7 shows streamlines of the meridional mass circulation in
response to isolated extratropical forcing for three cases corresponding to the dif-
fering frequency regimes discussed earlier, but with spherical geometry retained.
A comparison of Figs. 12.2 and 12.4 shows that the largest departures from
radiative equilibrium occur in the summer and winter mesosphere and in the polar
winter stratosphere. According to (12.9), these are the locations and seasons when
zonal forcing must be strongest. The zonal force in the mesosphere is believed to
be caused primarily by vertically propagating internal gravity waves. These trans-
fer momentum from the troposphere into the mesosphere, where wave breaking
produces strong zonal forcing. The zonal force in the winter stratosphere is due pri-
marily to stationary planetary Rossby waves. These, as discussed in Section 12.3,
can propagate vertically provided that the mean zonal wind is westerly and less
than a critical value that depends strongly on the wavelength of the w
av
es. Hence,
in the extratropical stratosphere we expect a strong annual cycle in δT , with large
values (i.e., strong departure from radiative equilibrium) in winter and small in
summer. This is indeed observed (see Figs. 12.2 and 12.4). Furthermore, because
