The subseismic equation and boundary conditions - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 420

7

The subseismic equation and boundary conditions

In contrast with the inertial waves studied in the previous chapter, long-period
motions in the outer core of the real Earth take place in a deformable container with
an inner boundary, where the fluid is compressible, stratified and self-gravitating.
We first derive the governing subseismic wave equation (Smylie and Rochester,
1981). Then, to incorporate the elasto-gravitational boundary conditions, we
develop a system of internal load Love numbers for the shell (mantle and crust),
and for the inner core.

7.1 The subseismic wave equation
Expression (6.181) allows the equation of motion (6.175) to be transformed to

=−∇ χ + β
α

2 u

− ω + 2 i ω Ω × u

2 g 0 ( g 0 ·

u ) = , (7.1)

showing that the displacement field u is linearly related to the gradient of the gen-
eralised potential.
The displacement field u is given in terms of the vector by equation (6.8). With

k ,and k a unit vector, we have that

Ω =Ω

k k

.

1

2

k

u =

2 σ

1

− σ + · − i σ × (7.2)

4 Ω

2

σ

2

−

Substituting the right side of equation (7.1) for ,wefindthat

2 ( g 0 · u ) C ∗

k k ·∇ χ

1

k ×∇ χ + β
α

2

u = σ ∇ χ − + i σ , (7.3)

2 σ

1

4 Ω

2

σ

2

−

with C ∗ the complex conjugate of the vector

k

g 0 k

k

2 g 0

C =− σ + · + × g 0 .

i σ (7.4)