Geology Reference
In-Depth Information
We now examine the surface integral in (8.7). Since there is no coupling of
modes across azimuthal number (see Section 3.4), expansion of χ in spherical har-
monics gives
χ
=
e
im
φ
∞
m
χ
n
(
r
)
P
n
(cosθ).
(8.8)
n
=
Similar expansion of
u
B
·
n
at the core-mantle boundary gives
e
im
φ
∞
m
y
1,
n
(
b
)
P
n
(cosθ)
u
B
·
n
=
u
B
·
s
=
n
=
∞
1
g
0
(
b
)
e
im
φ
h
n
χ
n
(
b
−
)
P
n
(cosθ),
=
(8.9)
n
=
m
using the internal load Love number for the shell defined by (7.60), while expansion
of
u
B
·
n
at the inner core boundary gives
e
im
φ
∞
n
=
m
y
1,
n
(
a
)
P
n
(cosθ)
u
B
·
n
=−
u
B
·
s
=−
∞
1
g
0
(
a
)
e
im
φ
h
n
χ
n
(
a
+
)
P
n
(cosθ),
=−
(8.10)
n
=
m
using the internal load Love number defined by (7.61) for the inner core. Spherical
harmonics obey the orthogonality relation (1.180) under integration over a sphere.
Thus, the surface integral has the expansion
h
n
χ
∗
n
(
b
−
)χ
n
(
b
−
)
∞
4
π
b
2
g
0
(
b
)
1
(
n
+
m
)!
f
χ
∗
u
B
·
n
d
S=
2
n
+
1
(
n
−
m
)!
S
n
=
m
g
0
(
a
)
h
n
χ
∗
n
(
a
+
)χ
n
(
a
+
)
f
(
a
)
a
2
g
0
(
b
)
−
b
2
=
π
b
Ω
2
Σ
S
,
(8.11)
Σ
with
S
defined by
h
n
χ
∗
n
(
b
−
)χ
n
(
b
−
)
∞
2
Ω
+
4
b
1
2
n
+
(
n
m
)!
(
n
−
m
)!
Σ
=
S
g
0
(
b
)
1
n
=
m
g
0
(
a
)
h
n
χ
∗
n
(
a
+
)χ
n
(
a
+
)
f
(
a
)
a
2
g
0
(
b
)
−
,
(8.12)
b
2
having set the decompression factor to unity at the core-mantle boundary. The
surface integral is then seen to be real. Because
T
is Hermitian,
∇
χ
∗
·
·∇
χ
∗
=∇
χ
·
T
∗
·∇
χ
∗
=∇
χ
∗
·
T
T
·∇
χ,
(8.13)
so the volume integral in the functional (8.7) is seen to be real as well.
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