Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 218

approximation, the elastic properties are taken to depend on radius alone. The
displacement field u can be broken into the sum of lamellar, poloidal and torsional
vectors (1.174), dependent on lamellar, poloidal and torsional scalars, which, in
turn, can be expanded in spherical harmonics, as in (1.184), for Earth models with
spherical symmetry. The divergence of the lamellar vector, and curls of the poloidal
and torsional vectors, lead to new scalars and radial coe cients, as described by

expression (1.194).
Using the methods of Section 1.3, the equilibrium equation (3.19) can be reduced
to a system of radial equations for its radial spheroidal, transverse spheroidal and
torsional parts. The elasto-gravitational system, in addition to the equilibrium equa-
tion, includes the gravitational relation (3.20). The decrease in gravitational poten-
tial, V 1 , is expanded in spherical harmonics as

m =− n φ

∞ n

n ( r , t ) P n (cosθ) e im φ .

V 1 = (3.32)

n = 0

Although the reduction of terms in the equilibrium equation is generally straight-
forward, the term ×∇ μ) requires special consideration. By comparison with

expressions (1.195) and (1.187), the vector u

∇× ( u

×∇ μ is easily shown to be made up

t n d μ/ dr and a torsional part of a transverse spheroidal part with radial coe cient −

n d μ/ dr . The transverse spheroidal part, in turn, by (1.186),
has a poloidal part such that

with radial coe cient v

1

r ∂ ∂ r rp n =− t n d dr .

(3.33)

Since the curl of the lamellar part vanishes, the curl of the spheroidal vector field
depends only on the curl of its poloidal part. The curl of a torsional vector is pol-
oidal and the negative of the curl of a poloidal vector is torsional with scalar given
by (1.189). Thus, the curl of the vector u ×∇ μ is made up of a poloidal part with

n d μ/ dr , and a torsional part, by relations (1.189) and (1.198),

radial coe cient v

with radial coe cient

r d dr t n

1

r ∂

. (3.34)

∂ r

n and t n , the radial spheroidal, trans-
verse spheroidal and torsional parts of the equilibrium equation can be identified
in terms of their radial coe

With the definitions (1.194) of l m , p m

n

cients. The radial spheroidal part of the equilibrium
equation gives the radial equation