Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 214

correct to first order in the displacements. To the same degree of approximation,
using the Poisson equation (5.4) for the gravitational potential,

2 V 1
∂ x k ∂ x k =−

∂

4π G ∂
∂ x k

4π G ρ 1 = (ρ 0 u k ), (3.11)

where V 1 is defined as the decrease in gravitational potential arising from the dens-
ity perturbation ρ 1 . Then, the perturbation in gravity is given by

∂ V 1
∂ x i

g 1 i = (3.12)

and, from (3.7),

∂σ ki
∂ x k =− ρ 0 g 0 i − ρ 0 g 1 i − ρ 1 g 0 i −

F i . (3.13)

Substitution from expressions (3.5) and (3.6) yields a modified equilibrium
equation,

∂τ ki
∂ x k =−

∂
∂ x i

(ρ 0 u k g 0 k ) − ρ 0 g 1 i − ρ 1 g 0 i − F i . (3.14)

On substitution of the generalised Hooke's law (3.3), we have

∂ u i

∂
∂ x k

λ ∂ u l

∂ x k + ∂ u k

i

∂ x l δ

+ μ

∂ x i

∂ u i

2 u l 2 u i 2 u k

∂ x k ∂ x i

= ∂λ

∂ x i ∂ u l ∂ x l + λ ∂ ∂ x i ∂ x l + ∂μ ∂ x k + ∂ u k

+ μ ∂

∂ x k + μ ∂

∂ x k ∂ x i

∂
∂ x i

=− (ρ 0 u k g 0 k ) − ρ 0 g 1 i − ρ 1 g 0 i − F i . (3.15)

culty, the equilibrium equation (3.15) can be converted to sym-
bolic vector notation. Making use of the properties of the alternating tensor ξ ijk ,
the i th component of

With some di

∇× ( ∇× u )is

2 u k
∂ x i ∂ x k −

2 u i
∂ x k ,

∇×

u ) i = ξ ijk ∂

∂ x j ξ klm ∂ u m

∂ ∂

( ∇×

∂ x l =

(3.16)

while the i th component of ∇× ( u ×∇ μ)is

∇×

×∇ μ) i = ξ ijk ∂

∂ x j ξ klm u l ∂μ

( u

∂ x m

2 2

∂ x k ∂ u i

∂μ

u i ∂

μ
∂ x k −

∂ x i ∂ u k

∂μ

u k ∂

μ
∂ x k ∂ x i .

=

∂ x k + ∂ x k −

(3.17)