Geography Reference
In-Depth Information
Recalling that symmetric instability requires that the slopes of the θ surfaces
exceed those of the M surfaces, the necessary condition for instability of geostrophic
flow parallel to the x axis becomes
δz
δy
δz
δy
f
f
Ri
f
2
F
2
N
s
S
4
∂u
g
∂y
θ
=
−
=
< 1
(9.26)
M
where the mean-flow
Richardson number
Ri is defined as
∂u
g
∂z
g
θ
00
2
∂θ
∂z
Ri
≡
Thus, if the relative vorticity of the mean flow vanishes (∂u
g
/∂y
=
0),Ri< 1is
required for instability.
The condition (9.26) can be related to (9.20) by observing that (9.26) requires
that F
2
N
s
S
4
< 0 for symmetric instability. As is to be shown in Problem 9.1,
−
=
ρf g
θ
00
P
F
2
N
s
S
4
−
(9.27)
Because the large-scale potential vorticity, P , is normally positive in the North-
ern Hemisphere and negative in the Southern Hemisphere, (9.27) is ordinarily
positive in both hemispheres; thus, the condition for symmetric instability is
rarely satisfied. If the atmosphere is saturated, however, the relevant static stability
condition involves the lapse rate of the equivalent potential temperature, and neutral
conditions with respect to symmetric instability may easily occur (see Section 9.5).
Finally, it is worth noting that the condition for stability with respect to symmet-
ric displacements, F
2
N
s
S
4
> 0, is identical to the condition that the Sawyer-
Eliassen equation (9.15) be an elliptic boundary value problem. Thus, when the
flow is stable with respect to symmetric baroclinic perturbations, a nonzero
forced
transverse circulation governed by (9.15) will exist when there is nonzero forcing,
Q
2
[see (9.12)].
Free
transverse oscillations, however, may occur in the absence
of forcing. These require including the horizontal acceleration term in the y com-
ponent of the momentum equation. The resulting equation for the transverse cir-
culation has the form (see Appendix F)
−
∂
2
ψ
M
∂z
2
∂
2
∂t
2
∂
2
ψ
M
∂y
2
F
2
∂
2
ψ
M
∂z
2
2S
2
∂
2
ψ
M
N
s
+
+
+
∂y∂z
=
0
(9.28)
which should be compared with (9.15). When F
2
N
s
S
4
> 0 solutions of (9.28)
−
correspond to stable oscillations, while for F
2
N
s
S
4
< 0 solutions are expo-
nentially growing, corresponding to symmetric baroclinic instability.
−
