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and relative angular momentum
h
—quantities which cannot be accurately derived
from geophysical models or observations. One possibility to overcome this difficulty
would be to simply neglect the problematic terms, as done by Wahr (
1982
) under
the severe restriction of allowing only perturbations with periods much greater than
one day. Barnes et al. (
1983
) proceeded similarly and suggested partial integration of
the equatorial excitation functions. These early studies, however, were referring the
excitation equation to the instantaneous rotation axis instead of the actually observed
axis—an adaption that could have eliminated the time derivatives in question. The
present review will introduce the necessary adaptation in Sect.
3
and below we stick
with the expressions in Eq. (
14
), albeit in a slightly rewritten form
i
σ
r
i
Ω
ˆ
χ
=
ψ
=
χ
−
m
ˆ
+
m
(18)
m
3
=
ψ
3
=−
χ
3
+
const
.
(19)
Here, Eq. (
18
) has been inferred from the first two components of the Liouville
equations by an adequate linear combination.
χ
3
are the equatorial and axial
angular momentum functions
of the perturbing fluid (Barnes et al.
1983
)
χ
and
ΩΔ
I
+
h
χ
=
χ
1
+
i
χ
2
=
(20)
Ω(
C
−
A
)
χ
3
=
ΩΔ
I
33
+
h
3
.
(21)
C
Ω
The
χ
i
now uniformly relate to angular momentum and, as already anticipated in
Sect.
2.1
, they can be split into matter and motion terms, which depend on mass
displacements
3) and relative angular momentum
h
i
, respectively.
Note the differences in terminology and notation when comparing the given deriva-
tions to the formulations of other authors, e.g. Eubanks (
1993
) or Gross (
2007
), who
designates the expressions in Eqs. (
20
) and (
21
) as excitation functions, too.
Δ
I
i
3
(
i
=
1
,
2
,
2.4 Effective Angular Momentum Functions
The presented solutions for wobble and changes in rotation rate are valid for a rigid,
oblate but geometrically axisymmetric Earth which responds to small excitations
described by the angular momentum functions in Eqs. (
20
) and (
21
). In view of
Earth's fluid components and the elastic behavior of the solid Earth, perfect rigidity
represents an inadequate assumption, though. It is thus necessary to systematically
build a more realistic model of our planet and further develop the derived formalism
for all effects which are caused by departures from rigidity. Our argumentation largely
refers to Gross (
2007
):
