Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 230

d y 5
dr =

+ y 6 ,

4π G ρ 0 y 1 (3.106)

d y 6
dr =−

4π G ρ 0 n 1

n 1
r 2 y 5

2
r y 6 , r y 3

+ − (3.107)

using the shorthand notations

1
λ +

n 1 = n ( n + 1), β =

2μ ,

(3.108)

δ = 2μ(3λ + 2μ)β, = 4 n 1 μ(λ + μ)β − 2μ. (3.109)

=

For n 0, the spheroidal system degenerates to fourth order, as expressed by equa-

tions (3.59) through (3.62), becoming

d y 1 2 λ r y 1

dr =−

+ βy 2 ,

(3.110)

r 2

2g 0 +

2

d y 2
dr =

2 ρ 0
r

2 δ 4 μ r y 2 − ρ 0 y 6 −

u F 0 , − r Ω + y 1 − (3.111)

d y 5
dr =

4π G ρ 0 y 1 + y 6 , (3.112)

d y 6
dr =−

2
r y 6 .

(3.113)

In the fluid outer core, for n 1, the governing spheroidal system, expressed by

equations (3.63) through (3.67), becomes

≥

d y 1
dr =−

2
r y 1

1
λ y 2

n 1
r y 3 ,

+ + (3.114)

2g 0 +

2 y 1 +

d y 2
dr =−

2 ρ 0
r

n 1
r ρ 0 g 0 y 3 − ρ 0 y 6 −

u Fn , r Ω (3.115)

ρ 0 g 0
r y 1

1
r y 2

ρ 0
r y 5

m

= − − − v

0

Fn ,

(3.116)

d y 5
dr =

4π G ρ 0 y 1 + y 6 , (3.117)

d y 6
dr =−

4π G ρ 0 n 1

n 1
r 2 y 5

2
r y 6 .

r y 3

+ − (3.118)

The torsional deformation, for which n ≥ 1, described by equations (3.57) and

(3.58), is governed by the second-order system

dz 1
dr =

1
r z 1 +

1
μ

z 2 , (3.119)

dz 2
dr =

2] r 2 −

3
r z 2

t Fn . [ n 1 − − (3.120)