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yields the Euler-Liouville equations or simply Liouville equations
d
d t (
L
=
I
ω +
h
) + ω × (
I
ω +
h
) .
(8)
2.2 Excitation Functions
The basic task is now to decompose the Liouville equations into a mathematically
practicable formulation that can be used for studying variations in Earth rotation in
view of geophysical (and especially atmospheric) excitation. Basically, this can be
accomplished by reverting either to a linear analytical approach or to a non-linear
numerical approach. As sketched in Seitz and Schuh ( 2010 ), the numerical approach
solves the differential equation system as an initial value problem via numerical
integration. Further details on the implementation and sensitivity of this method are
outlined in Seitz ( 2004 ). This review, though, focuses entirely on the conventional
analytical approach, of which the initial derivations of Munk and MacDonald ( 1960 )
are still valid today. Yet, our formulation largely follows Gross ( 2007 ).
If we neglect secular effects like polar wander, Earth's rotation departs only
slightly from uniform rotation. It is therefore justified to linearize Eq. ( 8 ) by con-
sidering only small deviations from an initial state of motion (subscript 0) in which
the entire system Earth, including all solid and fluid portions, is rotating with con-
stant angular velocity
around the polar axis z of the body-fixed reference system.
The orientation of this frame is realized in such a manner that the inertia tensor of
the Earth becomes diagonal and z coincides with the axis of figure
Ω
π 30 so that
ω 0 = Ω
z
(9)
A 00
0 B 0
00 C
.
I 0 =
(10)
Here, A , B and C denote the equatorial and polar principal moments of inertia of the
whole Earth. We now allow for slight perturbations of the defined reference state by
mass redistribution and relative particle motion. The resulting incremental rotation
can be described if a reference frame is appropriately attached to the disturbed Earth.
Following Munk and MacDonald ( 1960 ), the body-fixed frame is defined so that h
has contributions from the atmosphere, the oceans and the core, but not from relative
motion in the deformable mantle. This is the 'Tisserand mean-mantle' frame of the
perturbed Earth, its angular velocity with respect to inertial space equivalent to the
mean rotation vector averaged through the mantle (Wahr 2005 ) being
ω 1
ω 2
ω 3
m 1 (
t
)
=
Ω.
m 2 (
)
ω (
t
) = ω 0 + Δ ω (
t
) =
t
(11)
+
m 3 (
)
1
t
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