Geology Reference
In-Depth Information
In the fluid outer core, the modulus of rigidity vanishes and, neglecting viscosity,
there is no resistance to shear. From equation (3.42), the variable y
4
then vanishes
and, for
n
≥
1, the governing system (3.51) through (3.56) degenerates to
∂y
1
∂
r
=−
2
r
y
1
+
1
λ
y
2
+
n
(
n
1)
r
y
3
,
+
(3.63)
2g
0
2
y
1
∂y
2
∂
r
=−
2
ρ
0
r
n
(
n
+
1)
r
ρ
0
g
0
y
3
u
Fn
,
+
r
Ω
+
−
ρ
0
y
6
−
(3.64)
ρ
0
g
0
r
y
1
1
r
y
2
ρ
0
r
y
5
m
0
=
−
−
−
v
Fn
,
(3.65)
∂y
5
∂
r
=
4π
G
ρ
0
y
1
+
y
6
,
(3.66)
∂y
6
∂
r
=−
4π
G
ρ
0
n
(
n
1)
r
y
3
+
n
(
n
+
1)
2
r
y
6
.
+
y
5
−
(3.67)
r
2
Equation (3.53), derived from expression (3.42) for the shear stress, no longer holds
and is not included.
For
n
=
0, the governing system for the fluid outer core degenerates to
∂y
1
∂
r
=−
2
r
y
1
+
1
λ
y
2
,
(3.68)
2g
0
2
y
1
∂y
2
∂
r
=−
2
ρ
0
r
u
F
0
,
+
r
Ω
−
ρ
0
y
6
−
(3.69)
∂y
5
∂
r
=
4π
G
ρ
0
y
1
+
y
6
,
(3.70)
∂y
6
∂
r
=−
2
r
y
6
.
(3.71)
3.4 Dynamical equations
In the equation for force equilibrium (3.7), the body force
F
i
per unit volume,
over and above gravity, was left unspecified. The static equilibrium equation can
be converted to one expressing dynamic equilibrium if
F
i
is replaced by
−
ρ
a
i
,
with ρ representing the mass density and
a
i
the acceleration. This simple device is
sometimes called
d'Alembert's principle
.
Equations of motion are generally expressed with respect to an inertial or space-
fixed frame of reference. In a rotating frame of reference, such as that of the Earth,
the time rate of change of an arbitrary vector
q
must be augmented by an extra
rate of change caused by the rotation of the reference frame. In a purely kinematic
relation, the respective time derivatives are given by
d
q
dt
d
q
dt
fix
=
rot
+
ω
×
q
,
(3.72)
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