Static deformations and dislocation theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 500

The generalisation of the reciprocal theorem of Betti to elasto-gravitational sys-
tems whose elastic properties are spatially varying, and which are subject to
hydrostatic pre-stress in the equilibrium reference state, yields (3.31)

( t i − ρ 0 g 0 u s ν i ) u i d F i u i d S+ V

S V

t i − ρ 0 g 0 u s ν i u i d

F i u i d = S+ V

S V

V 1 ∂ V 1

4π G ρ 0 u i − V 1 ∂ V 1 4π G ρ 0 u i

1
4π G

+

∂ x i − ∂ x i −

ν i d S , (9.49)

S

for two systems of surface tractions, body forces and decreases in gravitational
potential (primed and unprimed), t i , F i , V 1 and t i , F i , V 1 , acting on material con-
tained in a volume , producing displacement fields u i and u i ,

respectively. u s and u s are the respective components of displacement in the ortho-
metric direction (opposite to gravity). While this form of the theorem applies to
realistic Earth models, it requires generalisation to the case where the variabley 6 is
discontinuous across the boundaries of the liquid outer core. Considering first the
core-mantle boundary, the generalisation requires only that we demonstrate that
the surface integrals in (9.49) evaluated over the top surface of the deformed outer
core cancel those evaluated over the bottom surface of the mantle. A similar can-
cellation is required at the inner core boundary. y 6 , as defined by equation (3.45),
is the radial coe

V S

by a surface

cient of the outward normal component of the gravitational flux

vector ∇ V 1 − 4π G ρ 0 u . Writing

∂ V 1
∂ x i −

4π G ρ 0 u i f s = ν i (9.50)

as a shorthand for the outward normal component of the gravitational flux vector,
the physical equivalent of equation (9.47) becomes
f s ( b + )

= f s ( b − ) 4π G ρ 0 ( b − )

+ Δ u s , (9.51)

where

u s ( b − ) u s ( b + ). Δ u s = − (9.52)

Similarly, writing t s for the normal traction, the physical equivalent of equation
(9.46) becomes

t s ( b + ) =− ρ 0 ( b − ) V 1 + ρ 0 ( b − )g 0 u s ( b + ). (9.53)

Of course, t s ( b − ) 0sincey 2 ( b − )

0. The sum of the surface integrals in (9.49)
at the core-mantle boundary is then found to be

= =

S ρ 0 ( b − ) V 1

u s d

t s u s

t s u s r = b + d u s − V 1 Δ − S+ Δ S . (9.54)

S