Geology Reference
In-Depth Information
Tabl e 6 . 3
Numerical values assigned to outer core parameters
in the scaling of the dynamical equations.
10
−
5
rad s
−
1
Characteristic mean density ¯ρ
0
=
Ω=
×
Rotation rate
7.29
10
4
kgm
−
3
1.1
×
Characteristic mean temperature
T
0
=
4400 K
Characteristic mean gravity ¯g
0
=
8ms
−
2
10
6
m
Characteristic length scale
L
=
10
3
kgm
−
3
Density stratification
Δ
ρ
=
2.3
×
Temperature stratification
Δ
T
=
1200 K
10
−
4
ms
−
1
Mean adiabatic bulk modulus
¯
Flow speed
U
=
3
×
10
12
Nm
−
2
λ
=
¯
91 km
2
s
−
2
2
Square of P wave velocity α
=
λ/¯ρ
0
=
10
−
5
K
−
1
cient of thermal expansion α
=
Coe
1
×
10
2
Jkg
−
1
K
−
1
Specific heat at constant pressure
c
p
=
7
×
10
−
11
Nm
2
kg
−
2
Universal constant of gravitation
G
=
6.67
×
10
−
2
to 10
7
m
2
s
−
1
Kinematic viscosity ν
=
η/¯ρ
0
=
10
−
6
m
2
s
−
1
Thermal di
ff
usivity κ
=
k
/(ρ
0
c
p
)
=
5.1
×
In the absence of flow, the energy conservation equation (6.134) shows that
the near-equilibrium temperature
T
e
(
r
) obeys (see also (5.106))
∇·
(
k
∇
T
e
)
=−
H
.
(6.148)
The full equation of energy conservation (6.134), on ignoring the spatial variation
of the thermal conductivity, takes the scaled form
1
+
F
R
ρ
1
f
R
∂
T
1
T
1
+
D
R
ρ
e
∂
t
+
v
·∇
T
e
+
f
R
v
·∇
+
τ
R
T
e
+
τ
R
f
R
T
1
F
R
∂
p
1
·∇
p
1
−
A
R
1
∂
t
+
v
·∇
p
e
+
F
R
v
⎣
v
)
2
⎦
,
∂v
j
∂
x
i
+
j
2
Ef
R
σ
P
∇
1
2
∂v
i
∂
x
j
−
2
3
∇·
ζ
η
2
T
1
i
=
+
δ
v
δ
+
(
∇·
(6.149)
where, once again, the dimensionless numbers,
D
R
, ,
F
R
,
f
R
,
A
R
, τ
R
,
E
, σ
P
and δ,
indicating the relative importance of terms, are defined in Table 6.2.
The values of the dimensionless numbers indicated in Table 6.2 have been
computed using the numerical values assigned to outer core parameters listed in
Table 6.3.
In the unperturbed, near-equilibrium case, the Poisson equation (6.136) for the
gravitational potential scales to
2
V
e
∇
=Γ
(1
+
D
R
ρ
e
),
(6.150)
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