The subseismic equation and boundary conditions - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 422

or

2

=− α

C ·∇ B ,

g 0 · u (7.14)

β

with

2 σ

1

k

g 0 2

2
β σ

2

4 Ω α

2 2 2

0

B = − + σ g − · . (7.15)

Substitution for g 0

· u from expression (7.14) in the displacement field formula

(7.3) leads to

k k

·∇ χ

1

C ∗ C

·∇ χ
B

k

2

u =

2 σ

1

σ ∇ χ − + i σ ×∇ χ − , (7.16)

2 2

4 Ω σ −

expressing the vector displacement field u entirely in terms of the gradient of the
generalised potential χ. For this reason, χ can be referred to as the generalised
displacement potential . The displacement field must satisfy the subseismic form
(6.181) of the equation of continuity, or

k k

·∇ χ

2 σ

1 1
α

C ∗ C

·∇ χ
B

k

2 2 2

∇· σ ∇ χ − + i σ ×∇ χ − =− 4 Ω σ −

2 g 0

· u .

(7.17)

Once again using the vector identities (6.10) and (6.11), with k replacing Ω ,but
also using the identity (A.9), and substituting from (7.14), we find

C

1

k 2

2 σ

2
β σ

·∇ χ
B

C ·∇ χ

B

4 Ω

2 2

C ∗ ·∇ C ∗ +

2

σ ∇ χ − ·∇ χ − = ∇· − .

(7.18)

The divergence of C ∗ is

k

g 0 k

k

g 0 . C ∗ =− σ

2

∇· ∇· g 0 +∇· · − i σ ∇· × (7.19)

This expression can be transformed using the vector identities

k · g 0 k k · g 0 k · g 0 k · g 0

k + k ·∇ k ·∇

∇· = ∇· = (7.20)

and

k

g 0

k k

∇· × = g 0 · ∇× − · ( ∇× g 0 ) = 0, (7.21)