Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 45

2 L and

although the scalar potentials (λ + 2μ) ∇ − L F can di ff er, at most, by a constant

C ,sothat

L F + C

2 L =−

∇

2μ .

(1.279)

λ +

Comparison of expressions (1.272) and (1.274) allows the vector potential A to be
written

A =∇× r P + r T , (1.280)

while comparison of expressions (1.273) and (1.275) allows the vector potential
A F to be written

A F =∇× r P F + r T F . (1.281)

Substitution in equation (1.277) yields, for the poloidal part,

2 P

μ ∇× ∇× r ∇ =−∇× ( ∇× r P F ), (1.282)

while the toroidal or torsional part yields

2 T

μ ∇×

r ∇ =−∇× r T F ,

(1.283)

where expressions (1.189) and (1.190) for the negative of the curl, taken twice, for
the poloidal part, and expressions (1.191) and (1.192) for the negative of the curl,
taken twice, for the toroidal or torsional part, have been used. Equating the poloidal
scalars on each side of expression (1.282) gives

1
μ

2 P

∇ =− P F , (1.284)

and equating the toroidal or torsional scalars on each side of expression (1.283)
gives

1
μ

2 T

∇ =− T F . (1.285)

Particular solutions of the equations of equilibrium (1.276) and (1.277) are

2 L

(λ + 2μ) ∇ =− L F , (1.286)

− μ ∇× ( ∇× A ) =− A F . (1.287)

The potentials for the body force density are given by

∇ ·

F ( r )
R

1
4π

V , L F =− d (1.288)

from expression (1.168), and

∇ ×

F ( r )
R

1
4π

d V , A F = (1.289)