Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 219

2μ) ∂ l m

n ( n + 1)

d dr +

d dr

l m

n
∂ r + μ

p m (λ + +

n n

r

d 2 d dr ∂ u n d 2

d dr n ( n + 1)

r v

μ 1

r

n

dr 2 u n + dr 2 ( r μ) u n

+ +

∂ r −

n

dr g 0 u n − ρ 0 ∂φ ∂ r ρ 0 g 0 u n − ρ 0 g 0 l m

∂ d ρ 0

u Fn , = −

∂ r −

(3.35)

n

with u Fn representing the radial coe

cient of the radial spheroidal part of the body
force per unit volume. The transverse spheroidal part of the equilibrium equation
gives the radial equation

r d dr v

n

d 2
dr 2 ( r μ ) v

(λ + 2μ) μ r ∂

∂ r rp n +

1

r ∂

1
r

d dr u n − 1

r

l m

n

+ +

n

r ∂ r

1 1

r ρ 0 φ r ρ 0 g 0 u n −

n m

= − v

Fn ,

(3.36)

m

with v cient of the transverse spheroidal part of the

body force per unit volume. The torsional part of the equilibrium equation gives
the radial equation

Fn representing the radial coe

r d dr t n

d 2 1

r ∂

1
r

μ t m

dr 2 ( r μ) t n =−

t Fn , + − (3.37)

n

∂ r

with t Fn representing the radial coe

cient of the torsional part of the body force
per unit volume. A fourth radial equation follows from the gravitational equa-
tion (3.20),

4π G

dr u n

2

1

r ∂ ∂ r 2 r φ

n −

n ( n + 1) d ρ 0

n

ρ 0 l m φ = + . (3.38)

n

r 2

Together with the defining relations (1.196), these four radial equations form six
equations in the six radial coe

n .
We may recast these equations in terms of more physical variables by consider-
ing the stresses associated with the deformations. The stress normal to an internal
spherical surface of radius r is

cients u n , v

n , l m

, p m , t m

and φ

n n n

2μ ∂ u r

2 L

τ rr = λ ∇ +

∂ r ,

(3.39)

while the tangential shear stress has components

∂ u θ
∂ r +

1

r ∂ u r

u θ
r

τ r θ = μ

∂θ −

,

∂ u φ
∂ r +

u φ
r

1
r sinθ

∂ u r
∂φ −

τ r φ = μ . (3.40)