Variational methods and core modes - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 455

s (8.12) appearing in the functional
and representing integrals over the boundaries of the outer core. To evaluate this
sum, we require the radial coe

Finally, we consider the support for the sum Σ

cients χ n ( b − )andχ n ( a + ) of the spherical harmonic
expansions of χ on the outer core boundaries. From expressions (8.38) and (8.39)
on the core-mantle boundary, we have

N − 1

D M , j (1, z ) x M , j

D M , j + 1 (1, z ) x M , j + 1 , V = + (8.41)

j = 1

with

⎝ ⎠

η ( z ) 0ψ ( z ) 0
0 η ( z ) 0ψ ( z )
η ( z ) 0ψ ( z ) 0

D M , (1, z ) = , (8.42)

while on the inner core boundary we have

N − 1

D 1, j ( a / b , z ) x 1, j +

D 1, j + 1 ( a / b , z ) x 1, j + 1 , V = (8.43)

j = 1

with

⎝ ⎠

η ( z ) 0 ψ ( z ) 0

D 1, ( a / b , z ) = 0 a η ( z )/ b 0 a ψ ( z )/ b . (8.44)

η ( z ) ψ ( z ) 0 0

The coe cients χ n ( r ) in the spherical harmonic expansion (8.8) of χ can be

recovered, using the formula (1.182) and the relation (1.179), as

1
0 χ P n ( z ) dz .

1) ( n

− m )!

χ n ( r ) = (2 n + (8.45)

+

( n m )!

In terms of the vector V ,thecoe cients may be expressed as

1

z 2 ) m /2 1,0,0 V P n ( z ) dz . 1) ( n

− m )!

χ n ( r ) = (2 n + (1 − (8.46)

+

( n m )! 0

In terms of the four-vector with transpose

1

1

z 2 m /2 η ( z ),0,ψ ( z ),0 P n ( z ) dz ,

1) ( n

m )!
( n + m )!

−

d n , = (2 n + − (8.47)

0

with substitution from (8.41) and (8.43), the radial coe cients on the boundaries

of the outer core are found to be

N − 1

d n , j + 1 x M , j +

d n , j + 1 x M , j + 1 χ n ( b ) = (8.48)

j = 1