Geography Reference
In-Depth Information
where P is the potential vorticity of the basic state geostrophic flow. Thus, if the
initial state potential vorticity in the Northern Hemisphere is everywhere positive,
then symmetric instability cannot develop through adiabatic motions, as potential
vorticity is conserved following the motion and will always remain positive.
To demonstrate that (9.19) is the condition for symmetric instability, we consider
the change in mean kinetic energy required for exchange of the tubes of fluid labeled
1 and 2 in Fig. 9.6. (These tubes are located at y
1
and y
2
=
δy, respectively,
and are assumed to extend infinitely along the x axis so that the problem is two
dimensional.) Because the tubes lie on the same potential temperature surface, they
have the same available potential energy. Thus a spontaneous exchange of parcels
is possible if δ(KE), the kinetic energy of the zonal flow after exchange minus that
in the initial state, is negative. Otherwise some external source of energy is required
to furnish the kinetic energy of the meridional and vertical motions required for
the exchange.
Initially the motion of the tubes is parallel to the x axis and in geostrophic
balance so that the absolute momentum for the two tubes is
y
1
+
M
1
=
fy
1
−
u
1
=
fy
1
−
u
g
(y
1
,z),
(9.21)
M
2
=
fy
2
−
u
2
=
fy
1
+
fδy
−
u
g
(y
1
+
δy, z
+
δz)
Fig. 9.6
Cross section showing isolines of absolute momentum and potential temperature for a
symmetrically unstable basic state. Motion along the isentropic path between points labeled
1 and 2 is unstable, as M decreases with latitude along the path. See text for details.
