Variational methods and core modes - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 459

( m may be a non-zero constant for all z , and the 'flow-through'

modes, which involve translational motion either along the axis of rotation or in
the equatorial plane. The 'flow-through' modes are the analogue of the transla-
tional modes in realistic Earth models, where the inner core and shell have oppos-
ing translational motions. The 'flow-through' mode along the axis of rotation is
also axisymmetric but is odd in the equatorial plane. For this mode

= 0), where Φ

Φ r = χ r is lin-

ear in z at the geocentre and Φ rz is constant there. The 'flow-through' modes in the

equatorial plane have azimuthal number m Φ r is a non-zero constant at the

geocentre. All of these special constraints can be cast in the linear form (8.63) and
implemented by the matrix modification described by (8.65).

= 1and

8.5 Numerical implementation and results
Construction of the coe cient matrix requires evaluation of the volume integral

(8.40) for each finite element, the boundary integral (8.47) required for the applic-
ation of the elasto-gravitational boundary conditions, and (8.54) for ensuring the
continuity of the normal component of displacement. With spline interpolation of
the decompression factor, the integrands are polynomials. We use 12-point Gaus-
sian integration in both r and z , giving complete accuracy for integrands expressed
by polynomials of degree up to and including 23. The coe cient matrix is a poly-

nomial of degree eight in the dimensionless angular frequency σ, while the matrix
constraining continuity of normal displacement at the boundaries of the outer core
is a quadratic function of σ. Thus, ten matrices are required to be calculated for
each di erent value of σ, in addition to the matrix independent of σ.

The programme MAT.FOR computes the eleven required matrices and puts them
in the output file matres.dat. Input to the programme MAT.FOR is the file
decomp.dat containing the decompression factor f (Section 6.4) and love.dat con-
taining the required internal Love numbers for the inner core and shell (Section 7.2).
The final eigenvalue-eigenvector problem then takes the form

ff

A (σ i ) x i = 0, (8.66)

for eigenfrequency σ i and eigenvector x i . The matrix A (σ) is unusual in that it is a
λ-matrix (Lancaster, 1966) or polynomial matrix of degree eight, expressible as

8 7

A (σ) = A 8 σ + A 7 σ +···+ A 1 σ + A 0 . (8.67)

A specially adapted form of inverse iteration is employed to solve the eigenvalue-
eigenvector problem (8.66). An initial estimate of the eigenfrequency σ r is found
from the vanishing of the determinant of A (σ r ). The derivative of A (σ) is easy to
calculate because A is a polynomial in σ,soforσ close to σ r we have

− σ) A (σ) A (σ r ) = A (σ) + (σ r +··· , (8.68)