Static deformations and dislocation theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 499

equation (9.30). For n ≥ 1, the equations governing deformation of the liquid outer

core degenerate to

1
g 0 y 5 ,

y 1 = (9.39)

y 2 = 0, (9.40)

y 4 = 0, (9.41)

d y 5
dr =

4 π G ρ 0
g 0 y 5

+ y 6 , (9.42)

1 4 π G ρ 0

g 0 +

2

d y 6
dr =

16 π G ρ 0
g 0 r

r Ω n ( n + 1) 2

r

− + + y 5 − y 6 . (9.43)

r 2 2g 0

Traditionally, it had been assumed that the variable y 1 was continuous at the
boundaries of the liquid core, as is the case for dynamical problems. This led
Je reys and Vincente (1966) to conclude that a solution may be impossible without

the assumptions that the Adams-Williamson condition holds and that the density
profile in the liquid outer core is perfectly adiabatic. The solution to the dilemma
of Je

ff

reys and Vincente was given by Smylie and Mansinha (1971), who pointed
out that in the static case the solid boundaries of the liquid core penetrate the equi-
potential, isobaric and equal-density surfaces (just as a loaded ship does, with the
degree of penetration being an indication of load). Since the equipotential and its
normal derivative are continuous, as shown by Israel et al. (1973), equation (9.29)
requires a compensating discontinuity iny 6 to make d y 5 / dr continuous. Of course,
an extra hydrostatic pressure is exerted on the penetrating solid boundaries of the
liquid core. Taking the radius of the core-mantle boundary to be r

ff

b , with b + the
radius just outside the core-mantle boundary and b − the radius just inside, we have
for spheroidal deformations of degree n

=

≥ 1,

1
g 0 y 5 ,

y 1 ( b − ) = (9.44)

y 2 ( b − ) = 0, (9.45)

=− ρ 0 ( b − )g 0 y 5

g 0 − y 1 ( b + )

y 2 ( b + )

, (9.46)

y 6 ( b + ) = y 6 ( b − ) 4π G ρ 0 ( b − ) + Δ y 1 , (9.47)

where

= y 1 ( b − ) − y 1 ( b + ) Δ y 1 (9.48)

is the discontinuity of y 1 . Similar conditions prevail at the inner core.