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while at the inner core boundary it gives the constraint
D
1
x
1
(
a
)
+
D
2
x
2
(
a
)
+
D
3
x
3
(
a
)
=
0,
(7.50)
≥
=
v
i
(
a
)
+
y
5
i
(
a
). For
n
=
ff
for
n
1 with
x
i
(
a
)
0, the second solution does not a
ect
the radial displacement and the second equation of the system (7.45) gives
dV
1
,
0
dr
dV
1
,
0
dr
1
w
1
(
a
)
1
4π
G
ρ
0
(
a
+
)y
11
(
a
)
(
a
+
)
(
a
+
).
D
1
=
=
(7.51)
The radial coe
cient of the radial displacement at the inner core boundary is then
1
4π
G
ρ
0
(
a
+
)
dV
1
,
0
dr
(
a
+
).
y
1,0
(
a
)
=
D
1
y
11
(
a
)
=
(7.52)
Comparison with expression (7.47) gives
h
1
I
0
=
0and
g
0
(
a
)
4π
G
ρ
0
(
a
+
)
a
.
h
2
I
=−
(7.53)
0
Under the subseismic condition, (7.50) is replaced by
D
1
x
1
(
a
)
+
D
2
x
2
(
a
)
=
0,
(7.54)
where
y
21
(
a
)
ρ
0
(
a
+
)
0,
x
1
(
a
)
=
g
0
(
a
)y
11
(
a
)
−
x
2
(
a
)
=
0.
(7.55)
Thus,
D
1
=
0 and there is no degree-zero deformation field in the inner core, so
dV
1,0
(
a
+
)/
dr
0 in agreement with (7.52).
D
2
may be non-zero, representing the
usual freedom of choice for the reference level of a potential.
The neglect of flow pressure fluctuations involved in the subseismic condition
leads to χ
=−
=
V
1
, from (6.176), so the generalised potential is equal to the change
in gravitational potential. Further, the boundary values of χ
n
and the derivatives of
V
1,
n
are no longer independent but are linearly related. If the shell were rigid, we
would expect
V
1,
n
to be proportional to 1/
r
n
+
1
at the core-mantle boundary, while
if the inner core were rigid, we would expect
V
1,
n
to be proportional to
r
n
at the
inner core boundary. Thus, we would expect the linear relations
b
n
dV
1
,
n
dr
1)χ
n
(
b
−
)
(
b
−
)
(
n
+
−
=
0
(7.56)
a
n
dV
1
,
n
dr
n
χ
n
(
a
+
)
(
a
+
)
+
=
0,
(7.57)
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