Geology Reference
In-Depth Information
The functional, given by (8.19), on integrating over the volume of the outer core,
becomes
f
m
2
1
∗
dr
0
dz
2
1
−
1
b
1
−
z
2
∂χ
∗
z
2
χχ
∗
+
r
0
∂χ
∂
r
0
∂χ
∂
z
∂χ
F =−
2πσ
+
−
∂
r
0
∂
z
a
1
b
f
r
0
z
2
∂χ
z
2
∂χ
∗
r
0
z
1
∗
∗
∂
r
0
∂χ
∂
r
0
∂χ
∂χ
∂
z
∂χ
+
2π
+
−
+
∂
r
0
∂
z
∂
r
0
−
1
a
∗
dr
0
dz
1
z
2
2
∂χ
∂
z
∂χ
+
−
∂
z
m
2
z
2
∂
r
0
χ
∗
1
b
m
σ
r
0
σ
∗
f
χ
∂χ
∂
r
0
∂χ
2
χχ
∗
+
2
+
2π
σ
−
+
2
ζ
1
−
z
2
−
1
a
z
2
∂χ
∗
r
0
σ
z
2
2
r
0
z
σ
z
2
1
∗
∗
∂
r
0
∂χ
∂χ
∂
r
0
∂χ
+
∂χ
∂
z
∂χ
2
2
+
−
−
−
−
∂
r
0
∂
z
∂
r
0
z
2
∂
z
χ
∗
∗
dr
0
dz
m
σ
z
1
∗
z
2
1
z
2
2
∂χ
χ
∂χ
∂
z
∂χ
∂
z
∂χ
−
−
+
+
−
∂
z
1
b
f
r
0
∂
r
0
χ
∗
z
∂
z
χ
∗
dr
0
dz
∗
∗
χ
∂χ
∂
r
0
+
∂χ
χ
∂χ
∂
z
+
∂χ
−
2π
m
σ
−
−
1
a
2
σ
1
2
+
4π
b
σ
−
Σ
S
.
(8.26)
Again neglecting corrections of the order of the flattening, the normal component
of the displacement is given by (7.14) as
2
=
α
C
·∇
B
.
u
·
n
(8.27)
βg
0
From (6.167), the stability factor is expressed as
2
2
N
2
α
β
=−
Ω
4
0
.
(8.28)
2
g
Substituting for β and using (8.25), the normal component of displacement to be
used at the boundaries becomes
z
1
z
2
⎣
⎦
.
σ
−
z
2
∂χ
−
1
∂χ
∂
z
+
m
σ
r
0
χ
2
u
·
n
=−
2
N
2
ζ
1
−
z
2
∂
r
0
−
(8.29)
r
0
2
Ω
4
8.2 Representation of the functional
The functional in the form (8.26) and the normal component of displacement (8.29)
are expressed in terms of equivolumetric radius
r
0
and the cosine of geographic co-
latitude
z
=
cosθ. Some general properties of the solution for χ can be deduced
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