Geology Reference
In-Depth Information
In the liquid outer core, the conditions of hydrostatic equilibrium prevail, even
in the deformed state. Thus, the gradient of the pressure is related to gravity by
∇
p
=
ρ
g
,
(9.31)
where
p
is the pressure, ρ is the density and
g
is the force of gravity per unit mass.
On taking the curl, we find
g
×∇
ρ
=
0.
(9.32)
Hence, equipotential surfaces, isobaric surfaces and surfaces of equal density (iso-
pycnic surfaces) remain parallel after deformation. An individual fluid particle is
able to move about force free on such surfaces in the absence of viscosity. There-
fore, y
3
becomes indeterminate and we can no longer identify individual fluid
particles. Eliminating the term iny
3
between equations (9.26) and (9.27), we obtain
g
0
+
2
y
1
−
ρ
0
g
0
d
y
2
dr
=
ρ
0
g
0
d
y
1
2
ρ
0
r
dr
−
r
Ω
λ
y
2
−
ρ
0
y
6
.
(9.33)
Substituting for y
6
from equation (9.29) and using relation (3.47), we find that
d
y
2
dr
=
ρ
0
d
dr
g
0
y
1
−
ρ
0
g
0
λ
y
2
−
ρ
0
d
y
5
dr
.
(9.34)
Multiplying equation (9.28) by
r
/ρ
0
and di
ff
erentiating gives
y
2
ρ
0
dr
g
0
y
1
ρ
0
d
=
ρ
0
d
dr
+
ρ
0
d
y
5
dr
,
(9.35)
which, on substitution in (9.34), finally yields
d
dr
lnρ
0
ρ
0
g
0
λ
+
y
2
=
0.
(9.36)
Thus, y
2
=
0, unless
dr
lnρ
0
d
=−
ρ
0
g
0
λ
,
(9.37)
which is known as the Adams-Williamson condition and becomes
2
+
λ
ρ
d
ρ
0
dr
=
+
α
d
ρ
0
dr
=
1
1
0.
(9.38)
0
g
0
ρ
0
g
0
This is just the condition for the density profile to be perfectly adiabatic (6.166) or
neutrally stratified, a condition unlikely to be met everywhere in the liquid outer
core. We conclude that y
2
=
y
5
/g
0
and that the left side of equation (9.27) vanishes, allowing substitution for y
3
in
=
0. From equation (9.28), this implies that y
1
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