Geography Reference
In-Depth Information
The resulting zonally averaged
2
terms are denoted by overbars as done previously
in Chapter 7:
L
−
1
L
0
()
=
()dx
where L is the wavelength of the perturbation. Thus, for the first term in (8.9) we
have, after multiplying by
−
ψ
1
, averaging and differentiating by parts:
∂
2
ψ
1
∂x
2
ψ
1
∂ψ
1
∂t
∂ψ
1
∂x
∂ψ
1
∂x
∂
∂t
∂
∂x
∂
∂x
∂
∂t
ψ
1
−
=−
+
The first term on the right-hand side vanishes because it is the integral of a perfect
differential in x over a complete cycle. The second term on the right can be rewritten
in the form
∂ψ
1
∂x
2
1
2
∂
∂t
which is just the rate of change of the perturbation kinetic energy per unit mass
averaged over a wavelength. Similarly,
ψ
1
times the advection term on the left
in (8.9) can be written after integration in x as
−
∂ψ
1
∂x
ψ
1
∂ψ
1
∂x
∂ψ
1
∂x
∂
2
ψ
1
∂x
2
∂
2
∂x
2
∂
∂x
∂
∂x
U
1
ψ
1
−
=−
U
1
+
U
1
∂ψ
1
∂x
2
U
1
2
∂
∂x
=
=
0
Thus, the advection of kinetic energy vanishes when integrated over a wave-
length. Evaluating the various terms in (8.10) and (8.11) in the same manner after
multiplying through by
ψ
3
and (ψ
1
−
ψ
3
), respectively, we obtain the following
−
set of perturbation energy equations:
∂ψ
1
∂x
2
1
2
∂
∂t
f
0
δp
ω
2
ψ
1
=−
(8.34)
∂ψ
3
∂x
2
1
2
∂
∂t
f
0
δp
ω
2
ψ
3
=+
(8.35)
∂t
ψ
1
−
ψ
3
2
U
T
ψ
1
−
ψ
3
∂x
ψ
1
+
ψ
3
+
ω
2
ψ
1
−
ψ
3
1
2
∂
∂
σδp
f
0
=
(8.36)
where as before U
T
≡
(U
1
- U
3
)/2.
2
A zonal average generally designates the average around an entire circle of latitude. However, for
a disturbance consisting of a single sinusoidal wave of wave number k
=
m/(a cos φ), where m is an
integer, the average over a wavelength is identical to a zonal average.
