Geology Reference
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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 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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