Geography Reference
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mean flow. Alternatively, as shown below, symmetric instability may be regarded
as isentropic inertial instability.
For convenience in deriving the conditions for symmetric instability, we use the
Boussinesq equations of the previous section and assume that the flow is inde-
pendent of the x coordinate. The mean wind is directed along the x axis and is in
thermal wind balance with the meridional temperature gradient:
f∂u
g
/∂z
=−
∂b/∂y
=−
(
g/θ
00
)
∂/∂y
(9.18)
where as before θ
00
is a constant reference value of potential temperature.
Following Sections 2.7.3 and 7.5.1, we measure the stability with respect to
vertical displacements by the distribution of total potential temperature θ
and that with respect to horizontal displacements by the distribution of the absolute
zonal momentum, M
=
θ
0
+
≡
fy
−
u
g
(where ∂M/∂y
=
f
−
∂u
g
/∂y is the zonal mean
absolute vorticity).
For a barotropic flow, potential temperature surfaces are oriented horizontally
and absolute momentum surfaces are oriented vertically in the meridional plane.
When the mean flow is westerly and increases with height, however, the potential
temperature and absolute momentum surfaces both slope upward toward the pole.
The comparative strengths of the vertical and horizontal restoring forces in the
midlatitude troposphere are given by the ratio N
s
/ (f∂M/∂y). For typical tropo-
spheric conditions this ratio is
10
4
. Thus, parcel motion in the plane orthogonal
to the mean flow will remain much closer to θ surfaces than to M surfaces. It is
thus natural to utilize isentropic coordinates to analyze parcel displacements. The
arguments of Section 7.5.1 still apply, provided that derivatives with respect to y
are taken at constant θ . The stability of such motions thus depends on the relative
slope of the θ and M surfaces. Ordinarily the M surfaces will slope more than the
θ surfaces (see Fig. 9.5), and parcel displacements are stable. However, when the
θ surfaces slope more than the M surfaces so that
∼
f(∂M/∂y)
θ
< 0
(9.19)
the flow is unstable with respect to displacements along the θ surfaces. This sit-
uation may occur in regions of a very strong horizontal temperature gradient and
weak vertical stability. The condition in (9.19) is similar to the criterion of (7.54)
for inertial instability, except that the derivative of M is here taken along a sloping
θ surface.
If (9.19) is multiplied by
g(∂θ/∂p), the criterion for symmetric instability can
be expressed in terms of the distribution of Ertel potential vorticity (4.12) in the
simple form
−
¯
P<0
f
(9.20)
