Variational methods and core modes - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 471

the triple scalar product u ∗ · ( Ω × u ) becomes

( u R +

i u I ) · u u ∗ = u ∗ · ( Ω × u ) = Ω × Ω · i u I ) × ( u R −

=− 2 i Ω · ( u R × u I ). (8.93)

The identity

f u ∗ ·∇ χ =∇· f u ∗ χ − χ ∇· f u ∗ (8.94)

is transformed to

f u ∗ ·∇ χ =∇· f u ∗ χ (8.95)

using the subseismic condition (6.182). The volume integral

f u ∗ ·∇ χ d V (8.96)

V

then becomes, with the use of Gauss's theorem, the surface integral over the bound-
aries of the outer core,

f χ u ∗ · n d S . (8.97)

S

With these transformations, equation (8.91) becomes

2 2 d

ω f | u | V− 4ω f Ω · ( u R × u I ) d V

V V

f χ u ∗ ·

2 2 d

= n d S+ f ω

v |

u · n | V . (8.98)

S V

The modal intensity is defined by

2 d

I = f | u | V > 0, (8.99)

V

while we might define

2 SI = f Ω · ( u R × u I ) d V (8.100)

V

as the product of I with the spin term S . The boundary deformation energy E is
represented by the surface integral, expressed by equation (8.11), for which we use
the shorthand EI . Finally, the last term on the right side represents the work done
in displacement against the non-neutrally stratified density profile and is defined as

2 2 d

FI = f ω

v |

u · s | V . (8.101)

V

Dividing through by the intensity gives a simple equation for angular eigenfre-
quency ω j ,

2
j

2
0

ω − 2ω j S = ω = E + F , (8.102)