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Rochester, 1986). This suggests that we make the change of independent variable
to
Φ
with
1
z
2
m
/2
χ
=
−
Φ
.
(8.31)
The domain of the problem is reduced to a quarter annulus with
z
ranging over the
interval (0,1). Scaling the radius by the radius
b
of the core-mantle boundary, the
dimensionless radius
r
=
r
0
/
b
ranges over the interval (
a
/
b
,1). We can then divide
−
expression (8.30) by
4π
b
and work with the reduced functional
m
2
1
z
2
/
1
z
2
0
z
2
m
V
T
⎝
⎠
2
1
0
1
+
−
−
mz
f
1
F
=
σ
−
0
1
0
V
dr dz
R
0
1
z
2
a
/
b
−
mz
−
z
2
m
V
T
⎝
⎠
⊗
1
1
mz
−
f
1
mz
,
−
1
V
dr dz
z
,
z
2
−
−
z
−
0
a
/
b
z
2
−
1
m
σ
+
z
2
V
T
⎝
⎠
⊗
1
−
z
2
m
1
1
m
σ
+
z
V
dr dz
z
2
,σ
2
z
2
,
z
3
−
2
−
z
2
−
−
f
σ
2
ζ
1
−
z
2
0
a
/
b
z
3
−
z
/
1
z
2
1
z
2
m
V
T
⎝
⎠
1
1
2
mz
2
−
−
z
f
1
+
m
σ
−
V
dr dz
1
0
0
0
a
/
b
−
z
00
2
σ
1
2
−
σ
−
Σ
S
,
(8.32)
where
V
T
=
(
Φ
,
r
Φ
,
Φ
z
) with
Φ
=
∂
Φ
/∂
r
and
Φ
=
∂
Φ
/∂
z
. The normal compon-
r
r
z
ent of displacement (8.29) is then expressed as
s
1
z
2
m
/2
m
σ
+
z
n
·
−
z
2
,σ
2
z
2
,
z
3
u
·
n
=−
2
N
2
r
0
ζ
z
2
−
−
·
V
.
(8.33)
2
4
Ω
1
−
z
2
, appearing in the denominator of the third integrand of the
functional and in the denominator of the expression of the normal component
of displacement, requires special consideration. From the definition (8.23) of ζ
1
The factor ζ
−
1
,
we have
2
σ
1
2
σ
−
2
1
z
2
2
z
2
ζ
−
=
σ
−
−
.
(8.34)
N
2
For modes with periods neither close to one-half sidereal day (σ
=
1) nor extremely
long, and for small departures from neutral stratification,
σ
N
2
2
,
(8.35)
2
σ
−
1
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