Geography Reference
In-Depth Information
APPENDIX F
Symmetric Baroclinic
Oscillations
A variant of the derivation of the Sawyer-Eliassen equation for forced transverse
circulations in baroclinic zones (9.15) can be used to obtain an equation for free
symmetric transverse oscillations, which can be used to derive an expression for the
growth rate of unstable symmetric oscillations or the frequency of stable symmetric
oscillations.
Suppose that the flow field is zonally symmetric so that u
g
=
u
g
(y, z) and
b
b (y, z). The ageostrophic (transverse) flow is given by the streamfunction
ψ (y, z) go that v
a
=−
=
∂ψ
∂z
∂ψ
∂y. Then from (9.12) Q
2
=
0 so that
any transverse circulation is unforced and must arise from a departure from exact
geostrophic balance. This can be simply represented by adding an acceleration
term in the y-momentum equation so that (9.10) becomes
;
w
a
=
∂
2
ψ
∂z
2
∂
∂t
f
∂u
g
∂b
∂y
=
−
+
∂z
+
0
(F.1)
Then combining (9.11) and (9.13) as before and applying (F.1), we obtain
∂
∂t
∂
2
ψ
∂z
2
∂
2
ψ
∂y
2
F
2
∂
2
ψ
∂z
2
2S
2
∂
2
ψ
D
Dt
N
s
+
+
+
∂y∂z
=
0
(F.2)
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