Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 29

The lamellar, poloidal and toroidal or torsional scalars can be expanded in the
form (1.181) to give
⎩ ⎭ ⎩ ⎭

l n ( r , t )
p n ( r , t )
t n ( r , t )

L
P
T

∞ n

P n (cosθ) e im φ .

= (1.184)

m =− n

n = 0

cients l n , p n , t n , in general, can be functions of time, so we need
to use partial derivatives when di

The radial coe

erentiating them.
The resulting vector spherical harmonics can be found from expression (1.174).
In what follows, where convenient, we take the summations over m and n to be
implied.
The components, in the spherical polar co-ordinate directions, of the lamellar
vector are

ff

∂ l n

l n P n e im φ ,

l n
r

dP n

im
r sinθ

L n =

∂ r P n , d θ ,

(1.185)

while those of the poloidal vector are

n ( n

∂ r rp n P n e im φ ,

dP n

+ 1) 1

r ∂ ∂ r rp n

im
r sinθ

∂

P n = p n P n ,

d θ ,

(1.186)

r

where Legendre's equation (B.1) has been used to simplify the expression for the
radial component. The components of the toroidal or torsional vector spherical
harmonic are

0, e im φ .

t n P n , − t n dP n

im
sinθ

T n = (1.187)

d θ

The divergence of the lamellar vector L is a scalar L that, with the use of
Legendre's equation (B.1), can be shown to be given by
L =∇

2 L .

(1.188)

The negative of the curl of the poloidal vector P is a toroidal or torsional vector
with scalar P given by

P =∇

2 P .

(1.189)

The negative of the curl, taken twice, of the poloidal vector P is a poloidal vector
P given by

P =∇× ∇× r P . (1.190)

The curl of the toroidal or torsional vector T is a poloidal vector with scalar T .The
negative of the curl, taken twice, of the toroidal or torsional vector T is a toroidal
or torsional vector T given by

T =∇× r T , (1.191)