Geography Reference
In-Depth Information
Absolute momentum is conserved by the tubes so that after exchange the pertur-
bation zonal velocities are given by
M
1
=
u
1
=
M
1
,M
2
=
u
2
=
fy
1
+
fδy
−
fy
1
−
M
2
(9.22)
Eliminating M
1
and M
2
between (9.21) and (9.22) and solving for the disturbance
zonal wind, we obtain
u
1
=
u
1
,u
2
=−
fδy
+
fδy
+
u
2
The difference in zonal kinetic energy between the final state and the initial state
is given by
u
1
2
u
2
2
u
1
+
u
2
1
2
1
2
δ (KE)
=
+
−
=
fδy(u
1
−
u
2
+
fδy)
=
fδy(M
2
−
M
1
))
(9.23)
Thus, δ(KE) is negative, and unforced meridional motion may occur, provided that
f(M
2
−
M
1
)<0. This is equivalent to the condition (9.19), as the tubes lie along
the same θ surface.
To estimate the likelihood that conditions for symmetric instability may be
satisfied, it is useful to express the stability criterion in terms of a mean-flow
Richardson number
. To do this we first note that the slope of an M surface can be
estimated from noting that on an M surface
∂M
∂y
∂M
∂z
δM
=
δy
+
δz
=
0
so that the ratio of δz to δy at constant M is
δz
δy
M
=
−
∂M
∂z
f
∂u
g
∂z
∂M
∂y
∂u
g
∂y
=
−
(9.24)
Similarly, the slope of a potential temperature surface is
δz
δy
−
∂θ
∂z
f
∂u
g
∂z
g
θ
00
∂θ
∂y
∂θ
∂z
θ
=
=
(9.25)
where we have used the thermal wind relationship to express the meridional tem-
perature gradient in terms of the vertical shear of the zonal wind. The ratio of
(9.24) to (9.25) is simply
δz
δy
f
2
∂u
g
∂z
2
δz
δy
f
f
g
θ
00
F
2
N
s
S
4
∂u
g
∂y
∂θ
∂z
θ
=
−
=
M
where the notation in the last term is defined in (9.16).