Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 253

concept of a propagator matrix (Gilbert and Backus, 1966), Y ( r ,ρ), which is the
solution of the di ff erential system

d Y ( r ,ρ )
dr

= A ( r ) · Y ( r ,ρ) (3.299)

that propagates the fundamental solutions from radius ρ to radius r with initial
condition Y (ρ,ρ) I , where I is the unit matrix. Thus, each column vector of Y is

a linearly independent solution as guaranteed by the initial condition.
In the inner core, the y-variables are replaced by the z -variables defined by
(3.187). The sixth-order spheroidal system, represented by equations (3.102)
through (3.107), in terms of the z -variables becomes

=

dz 1
dr +

α + 1 2 λ r · z 2

r +

n 1 λ r ·

=− + β ·

z 1 z 1 z 3 , (3.300)

r

r 2

2g 0

2

dz 2
dr +

α r z 2 2 ρ 0

r

2 δ 4 μ r ·

= − + r Ω + · rz 1 − z 2

n 1 ρ 0 g 0

r 2 n 1

r ·

2

+

r −

· rz 3 + z 4 − ρ 0 · rz 6 − ω ρ 0 · rz 1 (3.301)

+ 2 m ω Ω ρ 0 · rz 3 ,

dz 3 α + 1 1

r ·

1
r ·

1
μ ·

z 4
r ,

dr +

=− + +

z 3 z 1 z 3 (3.302)

r

ρ 0 g 0
r −

r 2 dz 4

dr +

α r z 4 λ r · r 2 · 3

r ·

= · rz 1 − z 2 + rz 3 − z 4

ρ 0
r ·

2 m ω Ω ρ 0
n 1

r 2 z 5

2

− − ω ρ 0 · rz 3 + · r ( z 1 + z 3 ), (3.303)

dz 5
dr +

α + 2 z 1

r +

z 6
r ,

= ·

z 5 4π G ρ 0 (3.304)

r

α + dz 6

dr +

1

4π G ρ 0 n 1

n 1
r 2 ·

2
r ·

z 6 =−

r ·

z 3 + rz 5 − z 6 . (3.305)

r

This system can be converted to the vector di ff erential equation

d z ( r )

dr = A ( r )

− z c ,

· z ( r ) (3.306)

[ z 1 ( r ), z 2 ( r ), z 3 ( r ), z 4 ( r ), z 5 ( r ), z 6 ( r )] T

where z ( r ) = is the new solution vector and

z c is a correction to the derivative vector given by

α +

z 6 T

1

z 1 , α r z 2 , α +

1

z 3 , α r z 4 , α +

2

z 5 , α +

1

z c = . (3.307)

r r r r