Geology Reference
In-Depth Information
inner boundary of region
➂
is
r
0
=
a
i
(1
+
2
f
i
/3). Replacing
m
(
r
0
) and ¯ρ(
r
0
)bytheir
alternative expressions
2
r
0
GM
i
=
Ω
3
M
i
4π
r
0
,
=
m
(
r
0
)
and
¯ρ(
r
0
)
(5.234)
for
n
=
0, expressions (5.158) and (5.159) combine to give
4π
G
d
r
0
GM
i
r
0
−
1
3
Ω
2
r
0
,
U
i
=−
r
0
ρ(
r
0
)
dr
0
−
(5.235)
while for
n
=
2, expression (5.160) gives
⎩
2
r
0
⎭
,
r
0
d
1
4π
G
GM
i
r
0
5
2
Ω
r
0
ρ(
r
0
)
f
(
r
0
)
dr
0
r
0
f
i
)
=
(2
f
i
+
−
(5.236)
both correct to first order in the flattening. Replacing the integrals in expression
(5.233) by those given by the uniform method of Wavre in (5.235) and (5.236), we
find the contribution of region
➂
to the gravitational potential
V
0
to be
G
ρ
(
r
)
|
r
−
r
|
V
−
d
(5.237)
15
r
2
P
2
(cosθ)
⎩
2
⎭
,
GM
i
r
0
+
1
3
Ω
2
GM
i
r
0
5
2
Ω
r
0
f
i
)
2
r
0
+
≈
U
i
+
(2
f
i
+
−
with
r
0
the mean equivolumetric radius of the inner boundary of the region.
It remains only to add the centrifugal potential,
1
3
Ω
1
3
Ω
2
r
2
2
r
2
P
2
(cosθ),
W
=−
+
(5.238)
to the gravitational potential contributions (5.226) and (5.237) to obtain the total
geopotential
U
t
entering the torque integrals (5.219). For evaluation of the torque
integrals, we require expressions for
r
r
ρ(
r
), correct to first order.
We choose to perform the evaluations of torque integrals in the inner core frame
of reference, with the
x
3
-axis aligned with the axis of symmetry of the inner core.
For emphasis, we use double primes for variables in this frame. The surface integral
requires the cross product
×
ν
and
∇×
φ
r
sinγ,
r
×
ν
=−
(5.239)
where
r
is the radius vector,
ν
is the outward normal vector and
φ
is the unit
vector in the direction of increasing longitude. γ is the angle between
r
and
ν
.
As illustrated in Figure 5.6, the small angle γ is given by the ratio
dR
d
θ
δθ/
R
δθ
(5.240)
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