Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 228

The radial coe cient of the torsional part of the vector Ω × u is given by

2π π ∞ k = l

2 n + 1

t − Cn = 1) m

k

l P l P n cotθ

( − Ω km v

4π n ( n + 1)

0 0

l = 0 k =− l

dP l
d θ

dP l
d θ

dP n
d θ

t l P n cosθ −Ω v

k
l

+ im Ω cosθsinθ

e i ( k + m )φ d θ d φ.

dP n
d θ

dP n
d θ

t l P l u l P l sin 2 − ik Ω cosθ −Ω θ (3.94)

Consider the third integral arising on the right,

π

dP l
d θ

dP n
d θ

cosθsinθ d θ. (3.95)

0

This is identical to the integral (3.89), which, on integrating by parts, led to (3.90),
using Legendre's equation (B.1). This combines with the first integral on the right
of (3.94), for m k , to give (3.91), which, as before, is easily evaluated using

the recurrence relations (B.11) and (B.12), along with the orthogonality relation
(1.180). Consider the fourth integral arising on the right of (3.94),

= −

k 2π
0

π

P l dP n cosθ e i ( k + m )φ d θ d φ. (3.96)

d θ 0

This is identical to the integral (3.85), which, on integrating by parts, led to (3.86).
This result combines with the second integral arising on the right of (3.94),

− k π
0

dP l
d θ

m

P n cosθ d θ, − 2π m δ (3.97)

giving, as before, expression (3.88). The remaining integral, fifth on the right of
(3.94), is easily evaluated, using the recurrence relation (B.12) and the orthogonal-
ity relation (1.180). Finally, collecting terms, we find for n

≥

1,

im Ω

t − Cn =

1) t − m

n

n ( n +

( n

n + 1

Ω
n ( n +

−

1 )( n

+

1 )( n

+

m ) n ( n

+

2 )( n

− +

1 ) m

v − m v − m − +

n − 1

1) 2 n − 1 2 n + 3

( n

1

Ω
n ( n

+ 1)( n + m ) n ( n − m + 1)

u − m u − m + − . (3.98)

n − 1 n +

+ 1) 2 n − 1 2 n + 3