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).
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