Variational methods and core modes - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 456

and

N − 1

d n , j x 1, j

d n , j + 1 x 1, j + 1 . χ n ( a ) = + (8.49)

j = 1

These lead to the squared magnitudes, given by the sums

N − 1 N − 1 1 1

2

x T M , j + q d n , j + q d n , + s x M , + s =

| χ n ( b ) | (8.50)

j = 1 = 1 q = 0 s = 0

and

N − 1 N − 1 1 1

2

x 1, j + q d n , j + q d n , + s x 1, + s .

| χ n ( a ) | = (8.51)

j = 1 = 1 q = 0 s = 0

The full contributions of Σ S to the partitioned matrix then consist of contributions

to the M , j + q row and M , + s column from the core-mantle boundary through

(8.50) and to the 1, j + q row and 1, + s column from the inner core boundary

through (8.51).

8.4 Boundary conditions and constraints
In addition to satisfying the known symmetry properties of the problem, the support
functions must also satisfy the boundary conditions at the core-mantle boundary
and at the inner core boundary. We also include the special case of the Poincare
inertial wave equation (Section 6.1), which has known analytic solutions and can
be used as a test of the accuracy of the numerical solutions. For this model of the
oscillations of a uniform, not self-gravitating fluid in a rigid container, there is no
inner body and conditions need to be imposed at the geocentre.
The remaining boundary condition to be applied to the support functions is that
of continuity of the normal component of displacement at the boundaries of the
fluid outer core. The normal component of displacement is given by expression
(8.33) in terms of the vector V . In analogy to the expansion (8.45), for small
departures from neutral stratification, the normal component of displacement has
the spherical harmonic expansion

= e im φ ∞

m y 1, n ( r ) P n (cosθ),

u · n (8.52)

n =