Geology Reference
In-Depth Information
Tabl e 6 . 2
Dimensionless numbers governing the dynamics of the fluid outer core.
Γ=
4π
G
¯ρ
0
L
/¯g
0
=
Self-gravitation number
1.2
Adiabatic to actual temperature gradient ratio
A
R
=
(α
¯g
0
T
0
/
c
p
)(
T
/
L
)
−
1
Δ
≈
1
Density ratio
D
R
=Δ
ρ/¯ρ
0
=
0.21
T
0
=
Temperature ratio τ
R
=Δ
T
/
0.27
¯
2
10
−
2
Compressibility number
C
=
¯ρ
0
¯g
0
L
/
λ
=
¯g
0
L
/α
=
8.8
×
2
L
(¯g
0
Δ
ρ/¯ρ
0
)
−
1
10
−
3
Internal Froude to Rossby number ratio
f
R
=Ω
=
3.1
×
2
L
/¯g
0
=
10
−
4
Froude number to Rossby number ratio
F
R
=Ω
6.6
×
10
3
to 2.4
10
12
Prandtl number σ
P
=
ν/κ
=
2.4
×
×
10
−
6
Rossby number
=
U
/
Ω
L
=
4.1
×
L
2
10
−
10
to 1.4
10
−
1
Ekman number
E
=
ν/
Ω
=
1.4
×
×
10
−
13
to 8.7
10
−
4
Di
ff
usivity coe
cient δ
=
ν
Ω
/
c
p
Δ
T
=
8.7
×
×
Bulk to dynamic viscosity ratio ζ/η
≈
1
Dropping primes on these scaled quantities, mass conservation, expressed by the
equation of continuity (6.92), becomes
1
+
F
R
ρ
1
F
R
∂ρ
1
∂
t
+
D
R
v
·∇
ρ
e
+
F
R
v
·∇
ρ
1
+
+
D
R
ρ
e
∇·
v
=
0.
(6.144)
The dimensionless numbers,
F
R
,
D
R
and , governing the relative importance of
the terms in this equation, are defined in Table 6.2.
At zero velocity, the momentum equation (6.117) shows that the scaled near-
equilibrium quantities are related by
D
R
ρ
e
F
R
k
r
,
p
e
=
1
D
R
ρ
e
g
e
−
1
×
k
∇
+
+
×
(6.145)
with
k
as the unit vector along the axis of rotation. Using the near-equilibrium
relation (6.145), the full equation of motion (6.117) scales to
+
F
R
ρ
1
∂
v
g
1
1
2
k
+
D
R
ρ
e
∂
t
+
v
·∇
v
+
×
v
−
E
ζ
η
+
v
)
F
R
ρ
1
k
r
×
k
1
3
2
v
=−∇
p
1
+
ρ
1
g
e
−
×
+
∇
+
∇
(
∇·
, (6.146)
where, in view of the scaling (6.142) of the gravitational potential, the gravitational
acceleration has been written
¯g
0
g
e
(
r
)
g
1
2
L
¯g
0
+
Ω
g
=
,
(6.147)
and, again, the dimensionless numbers,
D
R
, ,
F
R
and
E
, governing the relative
importance of the various terms, are defined in Table 6.2.
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