Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 33

∂ l l /∂ r 0and l k = = 1 in (1.206), we find the lamellar radial coe cient. By choos-

l

ing ∂( rp l )/∂ r 0and p k = = 1 in (1.207), we find the poloidal radial coe cient.

l

Finally, by choosing t k = 1 in (1.209), we find the toroidal or torsional radial

l

coe cient.

It follows that we can expand an arbitrary vector field in a series of lamellar, pol-
oidal and toroidal vector spherical harmonics, verifying the Lamb-Backus decom-
position, frequently used in geomagnetism. Similarly, an arbitrary vector field may
be expanded in a series of spheroidal and torsional vector spherical harmonics, as
is commonly done in seismology.

1.4 Elasticity theory
Although the Earth is a rotating, self-gravitating body with significant radial vari-
ation in its elastic properties, and large pre-existing stress, in some cases it may
be treated, locally, as uniform, not self-gravitating, and free of pre-stress. Much
of classical elasticity theory then applies. We delay a fuller description of realistic
Earth deformations to Chapter 3.

1.4.1 Analysis of stress
The state of stress in a continuum is studied by examining the forces acting on an
imaginary surface element in the medium, as illustrated in Figure 1.2.

F

ô

-F

Figure 1.2 Small surface element in a stressed medium. Vector ν is the unit out-
ward normal vector of the surface element. The force F acts on the outside of the
element, while an equal and opposite force − F acts on the inside.