Geology Reference
In-Depth Information
In a spherical polar co-ordinate system (
r
,θ,φ), with the
x
3
-axis aligned with the
rotation vector
Ω
, the vector
Ω
×
u
has components
u
r
sinθ
,
−Ω
u
φ
sinθ,
−Ω
u
φ
cosθ,
Ω
u
θ
cosθ
+Ω
(3.78)
the components of the vector displacement field being (
u
r
,
u
θ
,
u
φ
). In turn, using
expressions (1.195) and (1.187), the components of the vector displacement field
can be expanded in spheroidal and torsional vector harmonics as
∞
n
u
n
P
n
(cosθ)
e
im
φ
,
u
r
=
m
=−
n
n
=
0
t
n
P
n
(cosθ)
e
im
φ
,
∞
n
dP
n
(cos
θ
)
d
θ
im
sinθ
m
n
u
θ
=
v
+
(3.79)
n
=
0
m
=−
n
im
sinθ
v
e
im
φ
,
∞
n
t
n
dP
n
d
θ
n
P
n
(cosθ)
u
φ
=
−
n
=
0
m
=−
n
where
u
n
, v
n
and
t
n
are, respectively, the radial coe
cients of the radial spheroidal
part, the transverse spheroidal part and the torsional part of the vector displacement
field. For
n
0and
P
n
=
=
0, we have
m
=
1. Thus, both the transverse spheroidal
part and the torsional part vanish for
n
0.
Using the orthogonality relations (1.205) and (1.209), and expression (3.78), the
radial coe
=
cients of the radial spheroidal part, the transverse spheroidal part and
the torsional part of the vector
Ω
×
u
can be found.
cient of the radial spheroidal part of the vector
Ω
×
u
is given by
The radial coe
⎣
−Ω
2π
π
∞
k
=
l
1)
m
2
n
+
1
u
−
Cn
=
k
l
P
l
P
n
sinθ
(
−
ik
v
4π
0
0
l
=
0
k
=−
l
⎦
dP
l
d
θ
t
l
P
n
sin
2
e
i
(
k
+
m
)φ
d
θ
d
φ. (3.80)
+Ω
θ
On using the recurrence relation (B.12), we find that
l
2π
0
π
∞
k
=
l
1)
m
2
n
+
1
u
−
Cn
=
k
P
l
P
l
sinθ
e
i
(
k
+
m
)φ
d
θ
d
φ
(
−
−
i
Ω
k
v
4π
0
l
=
0
k
=−
l
t
l
2π
0
π
1
+Ω
2
l
+
1
0
l
(
l
k
)
P
l
−
1
1)
P
l
+
1
−
×
−
k
+
(
l
+
1)(
l
+
P
n
sinθ
e
i
(
k
+
m
)φ
d
θ
d
φ
×
.
(3.81)
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