Geography Reference
In-Depth Information
Substituting from (8.8) into (8.5)-(8.7) and linearizing yields the perturbation
equations
∂
∂t
+
∂
2
ψ
1
∂x
2
β
∂ψ
1
∂
∂x
f
0
δp
ω
2
U
1
+
∂x
=
(8.9)
∂
∂t
+
∂
2
ψ
3
∂x
2
β
∂ψ
3
∂
∂x
f
0
δp
ω
2
U
3
+
∂x
=−
(8.10)
∂
∂t
+
ψ
1
−
ψ
3
−
∂x
ψ
1
+
ψ
3
=
∂
∂x
∂
σδp
f
0
ω
2
U
m
U
T
(8.11)
where we have linearly interpolated to express
V
2
in terms of ψ
1
and ψ
3
, and have
defined
U
3
) /2
Thus, U
m
and U
T
are, respectively, the vertically averaged mean zonal wind and
the mean thermal wind.
The dynamical properties of this system are more clearly expressed if (8.9)-
(8.11) are combined to eliminate ω
2
. We first note that (8.9) and (8.10) can be
rewritten as
U
m
≡
(U
1
+
U
3
) /2,U
T
≡
(U
1
−
∂
∂
2
ψ
1
∂x
2
β
∂ψ
1
∂t
+
U
m
+
U
T
∂
∂x
f
0
δp
ω
2
+
∂x
=
(8.12)
∂
∂
2
ψ
3
∂x
2
β
∂ψ
3
∂t
+
U
m
−
U
T
∂
∂x
f
0
δp
ω
2
+
∂x
=−
(8.13)
We now define the barotropic and baroclinic perturbations as
ψ
m
≡
ψ
1
+
ψ
3
/2
≡
ψ
1
−
ψ
3
/2
;
ψ
T
(8.14)
Adding (8.12) and (8.13) and using the definitions in (8.14) yield
∂
∂t
+
∂
2
ψ
m
∂x
2
∂
2
ψ
T
∂x
2
∂
∂x
β
∂ψ
m
∂
∂x
U
m
+
∂x
+
U
T
=
0
(8.15)
while subtracting (8.13) from (8.12) and combining with (8.11) to eliminate ω
2
yield
∂
∂t
+
∂
2
ψ
T
∂x
2
2λ
2
ψ
T
∂
2
ψ
m
∂x
2
2λ
2
ψ
m
∂
∂x
β
∂ψ
T
∂
∂x
U
m
−
+
∂x
+
U
T
+
0
(8.16)
=
where λ
2
σ(δp)
2
]. Equations (8.15) and (8.16) govern the evolution of the
barotropic (vertically averaged) and baroclinic (thermal) perturbation vorticities,
respectively.
f
0
/
≡
[
