Geography Reference
In-Depth Information
forc
ing
has a simple harmonic dependence in time with frequency σ . Then, w
∗
and G are of the form:
w
∗
G
Re
e
iσt
w (φ,z)
G (φ,z)
ˆ
=
(12.5)
w and G are complex la
ti
tude and height dependent amplitudes. Combining
(12.1) and (12.3) to elim
in
ate v
∗
and combining the resulting equation with (12.2)
and (12.4) to eliminate T yields a partial differential equation for
where
ˆ
ˆ
w:
1
ρ
0
N
2
f
0
∂
2
∂ G
∂z
∂
ρ
0
ˆ
w
∂z
iσ
iσ
∂
∂z
w
∂y
2
ˆ
1
f
0
∂
∂y
+
=
(12.6)
+
α
r
The structure of
w in the meridional plane depends on the latitudinal and vertical
distribution of the forcing G. The differential equation (12.6) is elliptic in form.
Such a form implies that a forcing applied locally in any region will produce a
nonlocal response in the form of a large-scale circulation that acts to maintain the
zonal flow in thermal wind balance with the meridional temperature gradient.
As suggested by the coefficients of (12.6), the character of the nonlocal response
depends on the ratio of the frequency of the forcing to the Newtonian relaxation
rate. We consider three cases:
ˆ
(i) High-frequency variations σ
α
r
: The motions are approximately adia-
batic (as is true in sudden stratospheric warmings); that is
iσ
α
r
→
1
iσ
+
Away from the forcing region, the right-hand side of (12.6) vanishes so that
the two terms on the left must balance each other. For this balance to hold, a
simple scaling argument indicates that in midlatitudes the vertical scale δz
of the response must be related to the horizontal scale as δz
∼
∼
(f
0
/N ) δy
10
−
2
δy.
(ii) Low-frequency variations, such as the annual cycle: The Newtonian relax-
ation coefficient has a magnitude of order α
r
≈
1/ (20 days) in the lower
1/ (5 days) near the stratopause. Thus
α
r
>σand the coefficient multiplying the second term on the left in (12.6)
is reduced from its value in case (i) so that the vertical penetration scale is
increased.
(iii) Steady-state regime σ/α
r
stratosphere, increasing to α
r
≈
0, and rather than
using (12.6), it is simpler to note that (12.1) in this case reduces to a simple
balance between the Coriolis force and the zonal drag force:
→
0, In this limit ∂u//∂t
=
f
0
v
∗
=
−
G
(12.7)
