Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 220

The radial coe cient of the normal stress is then

∂ u n
∂ r +

2 u n −

n

2μ ∂ u n 2μ ∂ u n

n ( n + 1)v

= λ l m y 2 +

∂ r = λ

+

∂ r .

(3.41)

n

r

The shear stress arising from the spheroidal part of the deformation field is a trans-
verse spheroidal vector field with radial coe cient

∂v v

u n

n
∂ r −

n

−

y 4 = μ , (3.42)

r

while the shear stress arising from the torsional part of the deformation field is a
torsional vector field with radial coe cient

∂ t n

r t n

1

z 2 = μ

∂ r −

. (3.43)

The gravitational equation (3.20) may be expressed as

∇· ( ∇ V 1 − 4π G ρ 0 u ) = 0. (3.44)

The vector 4π G ρ 0 u may be regarded as a gravitational flux vector, with

outward normal component to an internal, spherical equipotential surface having
the radial coe

∇ V 1 −

cient

n

∂φ

4π G ρ 0 u n . y 6 =

∂ r −

(3.45)

In the approximation that the geopotential surfaces are spherical, the equation
(5.5) governing the equilibrium geopotential may be recast as

2

−∇· g 0 = 4π G ρ 0 − 2 Ω . (3.46)

Then,

r g 0

2 .

d g 0
dr =

2

4π G ρ 0 − + r Ω (3.47)

With appropriate substitutions from (1.194), (1.197), (1.198), (3.41), (3.42),
(3.43) and (3.47), equation (3.35) is reduced to

2μ) ∂ u n
∂ r

∂y 2
∂ r −

n ( n 1)

r y 4

+ 2 μ

r λ

− y 2 − (3λ +

4π G ρ

2 u n − ρ 0 g 0

r ρ 0 g 0 + r Ω

2μ) ∂ u n
∂ r

2

0

= − y 2 − (λ +

λ

n

− ρ 0 ∂φ u Fn ,

∂ r −

(3.48)