Earth’s rotation: observations and theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 296

To the same level of approximation, the three components of equation (4.23)
governing rotational motion about the centre of mass are

˙

Γ 1 = A Ω m 1 + r 13 Ω+ c 13 Ω+ 1

(4.30)

2 m 2 − 2 m 2 − 2 2

+ C Ω A Ω r 23 Ω − c 23 Ω − 2 Ω ,

˙

Γ 2 = A Ω m 2 + r 23 Ω+ c 23 Ω+ 2

(4.31)

2 m 1 − 2 m 1 + 2 2

+ A Ω C Ω r 13 Ω + c 13 Ω + 1 Ω ,

˙

Γ 3 = C Ω m 3 + r 33 Ω+ c 33 Ω+ 3 . (4.32)

Multiplying equation (4.31) by the unit imaginary number i and adding it to equa-
tion (4.30) produces

˜

m

2 m

r

2

˜

il

c

2

A Ω − i ( C − A ) Ω + Ω+ ir Ω = Γ− − Ω− Ω− ic Ω , (4.33)

˜

with complex phasors defined by m = m 1 + im 2 , r = r 13 + ir 23 ,

Γ=Γ 1 + i Γ 2 ,

˜

= 1 + i 2 ,and c = c 13 + ic 23 .

4.3.1 Response of the Earth to changes in the centrifugal force

When the deformation can be regarded as linear in the forcing function, a very
simple description can be given. For example, if U n is a forcing potential that is
a solid harmonic of degree n , then the disturbance in the gravity potential at the
surface of the Earth is given by

V n = k n U n , (4.34)

where k n is a Love number , in general complex, but real for a purely elastic deform-
ation.
This description applies rather well to the crust and mantle, which are elastic
solids with quite high Q factors (300 and greater). It does not apply to the oceans
and outer core where basic fluid dynamical behaviour cannot be ignored, including
possible non-linear response.
The centrifugal force per unit mass,

2 r

− ω × ( ω × r ) = ω − ( ω · r ) ω , (4.35)

is derivable by taking the gradient of the potential

2 ω

r ) 2 .

1

2 r 2

U = − ( ω · (4.36)