Variational methods and core modes - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 446

Expression (7.16) for the displacement field may be written as

u = T ·∇ χ. (8.1)

T is a second-order tensor with Cartesian components,

C ∗ i C j
B

1

2

T ij = σ δ ij − k i k j + i σξ iq j k q − , (8.2)

2 σ

1

4 Ω

2

σ

2

−

where k 1 , k 2 , k 3 are the components of the unit vector k . The components, u i ,of
the displacement field, are then given by the contraction

u i = T ij ∂χ

∂ x j .

(8.3)

The tensor T is seen to be Hermitian. On interchange of the indices i and j ,the
alternating tensor ξ iq j is replaced by ξ jqi =− ξ iq j , and the product C ∗ i C j is replaced
by its conjugate C ∗ j C i .
Consider the functional

1

2 σ 2 2

χ ∗ ∇· F = 4 Ω σ − ( f u ) d V , (8.4)

V

where χ ∗ and u are solutions of the subseismic equations obeying all boundary
conditions of the problem, including continuity of the normal component of dis-
placement at the boundaries of the outer core. The integral is over the volume of
the outer core. Using the identity
∇· f χ ∗ u = χ ∗ ∇· ·∇ χ ∗ ( f u ) + f u (8.5)

and the divergence theorem of Gauss, the functional is transformed to

1

2 σ

2 2

f u ·∇ χ ∗ d V

F =− Ω −

4 σ

V

1

2 σ 2 2

f χ ∗ u B · + 4 Ω σ − n d S , (8.6)

S

where u B is used to denote the vector displacement fields, with continuous normal
components, on the solid sides of the two elastic boundaries of the fluid outer core.
Substituting for u from (8.1), the functional becomes

1

2 σ 2 2

∇ χ ∗ · F =− 4 Ω σ − f T ·∇ χ d V

V

1

2 σ 2 2

f χ ∗ u B + 4 Ω σ − · n d S . (8.7)

S