Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 216

where ν i is the unit outward normal vector to the surface S . Transforming the sur-
face integral to a volume integral by Gauss's theorem gives, for the work done,

∂

+ F i u i d V=

∂ x j u i + F i u i d V . ∂ x j τ ji u i

τ ji ∂ u i

∂τ ji

∂ x j +

(3.22)

V V

Substitution from the equilibrium equation (3.14) gives the combination of terms

F i u i +

∂ x i ρ 0 u j g 0 j

∂τ ji ∂

∂ x j u i +

F i u i = F i u i . t c = − − ρ 0 g 1 i − ρ 1 g 0 i − (3.23)

Using expression (3.10) for ρ 1 ,

∂ x i u j g 0 j u i + g 0 i ∂ ∂ x j ρ 0 u j u i − ρ 0 g 1 i u i .

t c =− g 0 j ∂ρ 0

∂ x i u j u i − ρ 0 ∂

(3.24)

With the exchange of the repeated subscripts, the second term on the right side
becomes − ρ 0 ∂/∂ x j ( u i g 0 i ) u j , while the third term on the right can be replaced

identically by

∂ x j ρ 0 g 0 i u j u i ∂ x j u i g 0 i .

∂

− ρ 0 u j ∂ (3.25)

The expression for the work done then takes the form

τ ji ∂ u i

∂ x j − g 0 j ∂ρ 0 ∂ x i u j u i − ρ 0 u j ∂

( u i g 0 i )

∂ x j

V

− ρ 0 g 1 i u i d

∂ x j u i g 0 i ∂ x j ρ 0 g 0 i u j u i

− ρ 0 u j ∂

∂

+ V . (3.26)

On substitution from (3.3), the first term in the integrand can be transformed as

∂ u i
∂ x j +

∂ u i

∂ u j

i ∂ u i

∂ x j ∂ u i ∂ x i ∂ u i

∂ u j
∂ x i ∂ x j + μ ∂ u j

λ ∂ u k

∂ x j = λ ∂ u i ∂ x j + μ ∂ u i

j

∂ x k δ ∂ x j + μ ∂ x i ∂ x j

∂ u j ∂ u j

∂ x i , ∂ x j ∂ u i

= λ ∂ u i
∂ x i ∂ x j + μ ∂ u i ∂ x j + μ ∂ u i

(3.27)

∂ x j

the latter form being realised on the interchange of repeated subscripts in the last
term. The second term in the integrand can be transformed by the recognition
that surfaces of equal density (isopycnic surfaces) coincide with equipotentials of