Geology Reference
In-Depth Information
A further transformation can be made for the viscous terms. Expanding, we find that
∂v
j
∂
x
i
+
2
∂v
i
∂
x
j
−
2
3
∇·
i
j
v
δ
∂v
j
i
j
∂v
j
i
j
∂v
j
∂
x
i
∂
x
i
+
∂v
i
2
3
∇·
+
∂v
i
∂
x
j
∂
x
i
+
∂v
i
2
3
∇·
=
∂
x
j
−
v
δ
∂
x
j
−
v
δ
j
∂v
j
2
3
∇·
2
3
∇·
j
∂v
i
4
9
(
i
i
v
)
2
i
i
−
v
δ
∂
x
i
−
v
δ
∂
x
j
+
∇·
δ
j
δ
j
.
(6.131)
Since
j
∂v
j
j
∂v
i
i
i
i
i
j
δ
∂
x
i
=
δ
∂
x
j
=∇·
v
, and δ
j
δ
=
3,
(6.132)
on the interchange of dummy summation indices, we have the transformation
identity
∂v
j
i
j
∂v
j
i
j
2
∂v
i
∂
x
j
∂
x
i
+
∂v
i
2
3
∇·
v
δ
1
2
∂
x
i
+
∂v
i
2
3
∇·
v
δ
∂
x
j
−
=
∂
x
j
−
.
(6.133)
On substitution from (6.129), (6.130) and (6.133), the equation of energy conser-
vation (6.126) takes the final form
∂
p
∂
t
+
p
α
T
ρ
c
p
∂
T
∂
t
+
v
·∇
T
−
v
·∇
⎣
∇·
v
)
2
⎦
. (6.134)
∂v
j
∂
x
i
+
j
2
1
ρ
c
p
1
2
η
∂v
i
∂
x
j
−
2
3
∇·
i
=
(
k
∇
T
)
+
H
+
v
δ
+
ζ (
∇·
In addition to the conservation equations for mass (6.92), momentum (6.117) and
energy (6.134), the dynamics of the fluid outer core are governed by gravitation,
expressed by the gravitational acceleration
g
=−∇
V
,
(6.135)
where
V
(
r
) is the gravitational potential (5.114), related to the mass density through
the Poisson equation (5.4),
2
V
∇
=
4π
G
ρ.
(6.136)
6.3 Scaling of the core equations
At the periods of several hours and longer that are considered here, the Cori-
olis acceleration is taken to be the dominant dynamical term. In a rotating fluid
exact static equilibrium is impossible, as demonstrated by the Von Zeipel style of
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