Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 222

Written in y-notation, equation (3.45) becomes
∂y 5
∂ r =

4π G ρ 0 y 1 + y 6 . (3.55)

ff

A sixth equation is obtained on di erentiating (3.45) and substituting the result in

the gravitational relation (3.38). With further substitution from (1.197), (3.41) and
(3.55), it takes the form
∂y 6
∂ r =−

4π G ρ 0 n ( n +

1 )
r y 3

n ( n + 1 ) 2

r y 6 .

+ y 5 − (3.56)

r 2

The spheroidal deformation field is then governed by the six first-order linear dif-
ferential equations, (3.51) through (3.56).
The torsional deformation field may be described similarly by writing z 1

t n .

=

From equation (3.43), we then have
∂ z 1
∂ r =

1
r z 1

1
μ

+ z 2 . (3.57)

Substitution of this relation in the torsional part of the equilibrium equation (3.50)
produces

r 2 n ( n

2 z 1

∂ z 2
∂ r =

3
r z 2

t Fn . + 1) − − − (3.58)

The torsional part of the deformation field is then described by the variables z 1 , z 2 ,
governed by the two first-order linear di ff erential equations (3.57) and (3.58).

0and P n =

1. Thus, from (1.195), there is no trans-
verse spheroidal vector field, and, from (1.187), there is no torsional part of the
equilibrium equation. Equations (3.36) and (3.42), derived from transverse spher-
oidal parts, no longer appear, and, for n

For degree n = 0, we have m =

0, there is no torsional displacement
field. The spheroidal displacement field is entirely in the radial direction, and, from
(3.40), the shear stress vanishes. The variables y 3 and y 4 no longer appear and the
governing spheroidal system degenerates to the four first-order linear di

=

ff

erential

equations

∂y 1
∂ r =−

2λ
(λ +

1
λ + 2μ) r y 1 + 2μ y 2 ,

(3.59)

2μ) r 2

2g 0

2

∂y 2
∂ r =

2 ρ 0
r

(3λ + 2μ) 4μ

(λ +

− + r Ω + 4μ y 1 −

2μ) r y 2 (λ +

u F 0 , − ρ 0 y 6 − (3.60)

∂y 5
∂ r =

4π G ρ 0 y 1 + y 6 , (3.61)

∂y 6
∂ r =−

2
r y 6 .

(3.62)