Variational methods and core modes - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 448

Now consider variations, δχ, in the generalised displacement potential χ, subject
only to essential boundary conditions , excluding continuity of the normal compon-
ent of displacement at the boundaries of the fluid outer core. The variation in the
functional F ,causedbyavariationδχ in χ,is

1

2 σ

f ·∇ δχ d V

2 2

∇ δχ ∗ · ·∇ χ +∇ χ ∗ ·

δ F =− 4 Ω σ − T T

V

1

2 σ

f δχ ∗ u B · n d

2 2

+ δχ u ∗ B · + 4 Ω σ − n S , (8.14)

S

using the symmetry of χ n and χ ∗ n in expression (8.11). The volume integral can be
transformed with the vector identity,

+∇· f δχ ∗ T ·∇ χ . ∇ δχ ∗ · ·∇ χ =− δχ ∗ ∇·

( f T

·∇ χ ) f T (8.15)

·∇ χ) ∗ =

∇ χ ∗ · T ∗ ·∇ χ ∗ = u ∗ , and a second vector

Since T is Hermitian, T = ( T

identity emerges,

·∇ δχ =− δχ ∇· f T ∗ ·∇ χ ∗ +∇· f δχ T ∗ ·∇ χ ∗ . ∇ χ ∗ · f T (8.16)

Using both vector identities, we find

1

2 σ

δχ ∗ ∇·

+ δχ ∇· f T ∗ ·∇ χ ∗ d

2 2

δ F = 4 Ω σ − ( f T ·∇ χ) V

V

1

2 σ

·∇ χ +∇· f δχ T ∗ ·∇ χ ∗ d ∇· f δχ ∗ T

2 2

− 4 Ω σ − V

V

1

2 σ

f δχ ∗ u B · n d S

2 2

n + δχ u ∗ B ·

+ 4 Ω σ −

S

1

2 σ

δχ ∗ ∇·

+ δχ ∇· f T ∗ ·∇ χ ∗ d

2 2

= 4 Ω σ − ( f T ·∇ χ) V

V

1

2 σ

f δχ ∗ T ·∇ χ + δχ T ∗ ·∇ χ ∗

2 2

− 4 Ω σ − · n d S

S

1

2 σ

f δχ ∗ u B n d

2 2

+ δχ u ∗ B ·

+ Ω − ·

n

S ,

4 σ (8.17)

S

on using the divergence theorem of Gauss to transform the second volume integral
to a surface integral. The vector displacement field in the interior of the outer core
is given by equation (8.1) as u = T ·∇ χ. Thus, the variation in the functional F ,

causedbythevariationδχ in χ, becomes

1

2 σ

( f u )δχ ∗ +∇· f u ∗ δχ d

2 2

δ F = 4 Ω σ − ∇· V

V

1

2 σ

f u n δχ d u B · n δχ ∗ + u ∗ − u ∗ B ·

2 2

− 4 Ω σ − − S . (8.18)

S