The subseismic equation and boundary conditions - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 423

since g 0 =−∇ U 0 and ∇× g 0 =−∇×∇ U 0 ≡ 0. Further, from (5.5),

4π G ρ 0

2 . 2 U 0

∇· g 0 =−∇ =− − 2 Ω (7.22)

Finally, substituting (7.20), (7.21) and (7.22) in (7.19) yields

2 4π G ρ 0 − 2

k

g 0 . C ∗ = σ

k

∇· 2 Ω + ·∇ · (7.23)

Substituting this expression in (7.18), we arrive at

C

k 2

A C

·∇ B − ·∇ χ

B

2 2

C ∗ ·∇ σ ∇ χ − ·∇ χ − = 0, (7.24)

with the scalar A defined by

2 4π G ρ 0 2

k

g 0

2 σ

1 .

2
β σ

4 Ω

k

2

A = σ − 2 Ω + ·∇ · + − (7.25)

7.2 Deformation of the shell and inner core

At long periods of several hours and more, for which the subseismic approxim-
ation applies, the elastic forces resisting deformation dominate the acceleration
terms. There is one exception, in the case of a pure translation of the inner core,
where there is little elastic deformation. Although, in principle, a compensating
translation of the shell (mantle and crust) will accompany translation of the inner
core, the motion is small due to the much larger mass of the shell compared to the
inner core.
Neglecting shear stresses exerted by the fluid outer core, deformations of the
shell and inner core are governed by the sixth-order spheroidal system of di er-

ential equations developed in Section 3.3, for each spherical harmonic constituent
of degree n

ff

0, the governing spheroidal system degenerates to
fourth order, the terms representing transverse displacement and shear stress no
longer appearing. In each case, the frequency dependence is included through the
self-coupling body force expressions (3.125) and (3.126).
For n

≥ 1. For degree n =

1, in the shell, there are six linearly independent solutions, and the gen-
eral solution can be written as their linear combination

≥

y = C 1 y 1 + C 2 y 2 + C 3 y 3 + C 4 y 4 + C 5 y 5 + C 6 y 6 , (7.26)