Geology Reference
In-Depth Information
On the interchange of the order of integration in the second integral, the combina-
tion of the two integrals in expression (5.215) becomes
G
ρ(
r
)ρ(
r
)(
r
−
r
×
1
2
−
r
×
r
)
V
d
−
d
V
=
0.
(5.216)
3
r
|
|
r
−
V
i
V
i
Thus, in calculating the gravitational contribution to the torque on the inner core,
we need only include the gravitational potential,
V
0
, arising from the mass con-
tained in the volume
V
0
,or
G
ρ(
r
)
V
.
V
0
=−
d
(5.217)
|
r
−
r
|
V
0
In the rotating Earth frame, the calculation of the complete gravity torque requires
the inclusion of the centrifugal potential
W
, as expressed by (5.2), in the total geo-
potential. Then, the geopotential to be used is
U
t
=
V
0
+
W
,
(5.218)
with the gravitational potential
V
0
being that contributed by the mass outside the
inner core. The total gravity torque acting on the inner core is then
V
i
r
U
t
(
r
)
ρ(
r
)
d
Γ
=−
×∇
V
V
i
∇×
r
ρ(
r
)
U
t
(
r
)
d
U
t
(
r
)
r
×∇
ρ(
r
)
d
=
V+
V
V
i
U
t
(
r
)
∇×
r
ρ(
r
)
d
=−
U
t
(
r
)ρ(
r
)
r
×
ν
d
S−
V
,
(5.219)
S
i
V
i
where integral theorem (A.19) has been used to convert one of the volume integrals
into a surface integral with
ν
the outward unit normal vector. For evaluation of the
gravity torque, it remains to calculate the gravitational potential
V
0
arising from
the mass in the volume
0
lying outside the inner core.
As shown in Figure 5.5, we divide the volume
V
0
outside the inner core into
three regions. Region
➀
is bounded on the inside by the inner core and on the
outside by a sphere with radius equal to the equatorial radius
a
i
of the inner core.
The polar radius of the inner core is
c
i
. Region
➁
lies between region
➀
and the
equipotential with polar radius
a
i
and thus equatorial radius
a
i
(1
V
f
i
), where
f
i
is the flattening of the inner core. Region
➂
is exterior to region
➀
. The surface
flattening of the inner core is then
+
a
i
−
c
i
a
i
.
=
f
i
(5.220)
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