Variational methods and core modes - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 457

with

1

y 1, n ( r ) = (2 n + 1) ( n

− m )!

u · n P n ( z ) dz

( n + m )!

0

n

·

s

−

1) ( n

m )!
( n + m )!

=

2 σ

1 r 0 (2 n

+

2 2

4 Ω σ −

1

z 2 m /2 m σ + z V P n ( z ) dz .

1

z 2 ,σ

2

z 2 , z 3 × − − − (8.53)

0

With a four-vector defined as

1) ( n

− m )!

e n , ( r ,σ) = (2 n +

( n + m )!

⎝ ⎠

z 2 )η ( z ) z ( z 2 1)η ( z )

m (σ +

+ −

1

1

− z 2 m /2

− z 2 ) r η ( z )
m (σ + z 2 )ψ ( z )

2

(σ

P n ( z ) dz ,

×

+ z ( z 2 1)ψ −

0

2

− z 2 ) r ψ ( z ) (σ

(8.54)

substitution for V in (8.53) from (8.41) gives, for the core-mantle boundary,

N − 1

e n , j (1,σ) x M , j +

e n , j + 1 (1,σ) x M , j + 1 ,

1

y 1, n ( b ) =

2 σ

1 b

(8.55)

4 Ω

2

σ

2

−

j = 1

while substitution for V in (8.53) from (8.43) gives, for the inner core boundary,

N − 1

e n , j ( a / b ,σ) x 1, j +

e n , j + 1 ( a / b ,σ) x 1, j + 1 . (8.56)

1

y 1, n ( a ) =

2 σ

1 a 4 Ω

2

σ

2

−

j = 1

Using the internal load Love number for the shell (7.60), y 1, n ( b ) may be related
to χ n ( b ), which in turn is expressed by (8.48). Similarly, using the internal load
Love number for the inner core (7.61), y 1, n ( a ) may be related to χ n ( a ), which in
turn is expressed by (8.49). Continuity of normal displacement at the boundaries
of the outer core then yields two linear, homogeneous constraint equations for each
harmonic degree n :

N − 1

c n , j (1,σ) x M , j +

c n , j + 1 (1,σ) x M , j + 1 = 0, (8.57)

j = 1

for the core-mantle boundary, and

N − 1

c n , j ( a / b ,σ) x 1, j +

c n , j + 1 ( a / b ,σ) x 1, j + 1 = 0, (8.58)

j = 1