Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 229

After replacing − m with m , the radial coe cient of the torsional part of the body

force per unit volume becomes

2 m ω Ω ρ 0
n ( n

t Fn = ω

2

ρ 0 t n −

1) t n

+

( n

u n + 1

2 i ω Ω ρ 0
n ( n

+ 1)( n − m ) n ( n + m + 1)

u n − 1 − −

1 )

+ 2 n − 1 2 n + 3

( n

2 i ω Ω ρ 0
n ( n

− 1)( n + 1)( n − m ) n ( n + 2)( n + m + 1)

m m

n + 1

+

n − 1 +

v v .

+ 1) 2 n − 1 2 n + 3

(3.99)

Fn and
(3.99) for t Fn , into equations (3.52), (3.54) and (3.58), respectively, gives a coupled
eighth-order di

Substitution of the body force expressions (3.83) for u Fn , (3.93) for v

erential system comprising equations (3.51) through (3.58), for
each harmonic degree n . The magnitude of the Coriolis acceleration compared
with the magnitude of the local acceleration is 2ω Ω /ω

ff

2

Ω /ω, or the ratio of the
period to 12 sidereal hours. For periods shorter than an hour, in the range of con-
ventional free oscillations, the e

=

2

ect of Coriolis coupling can be handled by first-
or second-order perturbation theory. For long-period core oscillations the coupling
is much stronger and two infinite coupled chains emerge,

ff

S m ⇔ T m + 1 ⇔ S m + 2 ⇔ T m + 3 ⇔··· , the spheroidal chain (3.100)

and

T m ⇔ S m + 1 ⇔ T m + 2 ⇔ S m + 3 ⇔··· . the torsional chain (3.101)

At periods of several hours and longer (Johnson and Smylie, 1977), the conver-
gence of the coupled chains is very slow.
In summary, the sixth-order spheroidal governing system, for n ≥ 1, expressed

by equations (3.51) through (3.56), becomes

d y 1
dr =−

2 λ r y 1 n 1 λ r y 3 ,

+ βy 2 + (3.102)

r 2

n 1 ρ 0 g 0

r 2

2g 0

2

d y 2
dr =

2 ρ 0
r

2 δ 4 μ r y 2

− + r Ω + y 1 − +

r −

y 3

n 1
r y 4 − ρ 0 y 6 −

u Fn , + (3.103)

d y 3
dr =−

1
r y 1 +

1
r y 3 +

1
μ y 4 ,

(3.104)

ρ 0 g 0
r −

r 2 d y 4

dr =

λ r y 2 r 2 y 3 3

r y 4

ρ 0
r y 5

m

y 1 − + − − − v

Fn ,

(3.105)