Static deformations and dislocation theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 511

u n is the excess of the radial displacement of the base of the mantle over
that of the gravitational equipotential coincident with the core-mantle boundary in
the undeformed state. Similar results are found for the inner core boundary.
A sudden redistribution of mass within the Earth, such as that accompanying
a major earthquake, produces a secular polar shift (Smylie and Mansinha, 1967;
Smylie and Zuberi, 2009),

where Δ

ΩΔ c

σ 0 A ,

(9.93)

where Ω is the angular velocity of Earth's rotation, Δ c =Δ c 13 + i Δ c 23 is the change

in the o -diagonal components of the inertia tensor in the geographical system

of reference, σ 0 is the angular frequency of the Chandler wobble and A is the
equatorial moment of inertia of the Earth. The real part of the secular polar shift
is measured towards Greenwich and the imaginary part is measured in the direc-
tion of 90 ◦ E. The changes in the o

ff

c 23 of the
inertia tensor in the geographical system of reference are found by similarity trans-
formation of the changes

ff -diagonal components Δ c 13 and Δ

J ij in the inertia tensor in the epicentral system.
A closely equal and opposite contribution to the Chandler wobble occurs, so that
only a small instantaneous shift in pole position is realised, with the pole path
embarking on a new circular arc about a new centre of rotation.
From expressions (9.91) and (9.92), the changes in the inertia tensor in the epi-
central system of reference depend on spheroidal displacement fields of zeroth and
second degree. The zeroth-degree displacement field makes an equal contribution
only to the diagonal components of the inertia tensor. This contribution will, there-
fore, be the same in all centre of mass co-ordinate systems. Since the excitation of
wobble and secular polar shift depend only on two of the o

Δ I ij +Δ

ff

-diagonal components
of the inertia tensor, the degree-zero spheroidal displacements can be ignored. In
the mantle and crust, the second-degree spheroidal displacement fields obey the
system of ordinary di

erential equations (3.102) through (3.107). Four separate
solutions of the non-homogeneous equations are involved with singular sources
proportional to

ff

5
8π r 3 μδ( r − r 0 ),

u F 2 =

(9.94)

5

u F 2 =

8π r 2 μδ ( r

− r 0 ), (9.95)

5
8π r 3 μδ( r

m
F 2

v = − r 0 ), (9.96)

5

m
F 2

8π r 2 μδ ( r

v = − r 0 ). (9.97)