The subseismic equation and boundary conditions - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 421

u can be expressed entirely in terms of the generalised
potential χ. Taking the scalar product of

The scalar product g 0 ·

k with the equation of motion (7.1), scaled

2 ,wefind

by 4 Ω

2 k

g 0 g 0

1 β

2 k

2 k

− σ · u =− ·∇ χ + · · u . (7.5)

4 Ω 4 Ω

2

α

Taking the scalar product with g 0 gives

k

u

2
0

βg

1

2 g 0

− σ · u + i σ g 0 · × =−

2 g 0

·∇ χ +

2 g 0

· u . (7.6)

2

4 Ω 4 Ω α

Using the identity

k

u

k

g 0 g 0 · × =− × · u , (7.7)

for the triple scalar product, (7.6) becomes

⎝

2 ⎠

2
0

i σ k g 0

βg

1

−

2 + σ

g 0 · u − × · u =−

2 g 0

·∇ χ. (7.8)

4 Ω

2

α 4 Ω

Finally, taking the scalar product of k g 0 with (7.1), scaled by 4

2 , yields

× Ω

2 k

g 0 i σ k g 0

k

u

2 k

g 0

1

− σ × · u + × · × =− × ·∇ χ, (7.9)

Ω

4

which can be transformed with the identity
k

g 0

k

u

k

g 0 k × · × = g 0 · u − · · u (7.10)

to

2 k

g 0 − i σ k g 0 k

2 k

g 0

1

− σ × · u + i σ g 0 · u · · u =− × ·∇ χ. (7.11)

4 Ω

Equations (7.5), (7.8) and (7.11) form the system of linear equations
⎝

− β k g 0 /4

⎠ ⎝ ⎠

k

2 2 2

− σ · Ω α 0 · u

0 /4 2 2 2

0 − βg Ω α − σ − i σ g 0 · u

i σ k g 0

k

g 0 ·

2

− · i σ − σ × u

⎝ ⎠

k

·∇ χ

1

=− g 0 ·∇ χ . (7.12)

2

k

g 0 ·∇ χ

4 Ω

×

Multiplying the first equation through by k

2 , the third by i σ

· g 0 , the second by − σ

and adding all three gives

4

g 0 2 g 0 ·

2 σ

1

k

2
β σ

2

β Ω α 1

2 2 2

0

− + σ g − · u =−

2 C

·∇ χ, (7.13)

4 Ω

2

α

2

4 Ω