The subseismic equation and boundary conditions - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 426

by y 1 , y 2 and y 3 , the most general solution in the inner core, regular at the geo-
centre, is then formed by the linear combination

y = D 1 y 1 + D 2 y 2 + D 3 y 3 , (7.38)

where, once again, each y j is a six-vector with components representing the values
of the variables (y 1 ,y 2 ,y 3 ,y 4 ,y 5 ,y 6 ) describing the spheroidal deformation field,
and each D j is a linear combination coe cient.

At the inner core boundary of radius a the shear stress vanishes and the nor-
mal displacement, normal stress, the perturbation in gravitational potential and the
gravitational flux vector are all continuous. Following the same arguments as used
at the core-mantle boundary, continuity of both the normal stress and the perturb-
ation in gravitational potential imply that

=− ρ 0 ( a + ) y 5 ( a ) + χ n ( a + ) + ρ 0 ( a + )g 0 ( a )y 1 ( a ), y 2 ( a ) (7.39)

while continuity of the gravitational flux vector implies

dV 1 , n
dr

( a + ) 4π G ρ 0 ( a + )y 1 ( a ), y 6 ( a ) = − (7.40)

a + indicating the fluid side of the inner core boundary. Application of
the conditions at the inner core boundary (again excepting continuity of normal
displacement) leads to the system of linear equations,

with r =

D 1 y 41 ( a ) + D 2 y 42 ( a ) + D 3 y 43 ( a ) = 0,

= χ n ( a + ), D 1 v 1 ( a ) + D 2 v 2 ( a ) + D 3 v 3 ( a ) (7.41)

dV 1 , n
dr

( a + ), D 1 w 1 ( a ) + D 2 w 2 ( a ) + D 3 w 3 ( a ) =

with

1
ρ 0 ( a + ) y 2 i ( a )

v i ( a ) = g 0 ( a )y 1 i ( a ) − − y 5 i ( a ), (7.42)

and

4π G ρ 0 ( a + )y 1 i ( a ) w i ( a ) = + y 6 i ( a ). (7.43)

For n 0, as shown in Section 3.5, there is only one solution in the inner core,

regular at the geocentre, which corresponds to a true deformation. This involves
only the variables y 1 , y 2 and y 5 . A second solution, regular at the geocentre, is the
trivial constant solution for which only y 5 is a non-zero constant. For this second

=