Geoscience Reference
In-Depth Information
=
Ω
1
k
2
Δ
I
σ
e
A
m
+
p
1
p
2
p
χ
=
χ
+
i
χ
(86)
h
σ
e
A
m
.
w
w
1
w
2
χ
=
χ
+
i
χ
=
(87)
Taking into account that, in principle, the identity
k
2
=−
μ
(88)
τ
holds (Brzezi nski
1994
), we can appropriately modify Eq. (
86
) and insert the result-
ing matter term expression together with the motion term (Eq.
87
) into Eq. (
84
). Thus,
we obtain the following generalized, frequency-dependent equation, describing geo-
physical excitation of polar motion and nutation under consideration of both the CW
mode and the FCN resonance
χ
w
+
a
P
χ
w
σ
e
σ
1
−
σ
σ
e
σ
2
−
σ
p
p
p
ˆ
(σ )
=
+
χ
+
a
w
χ
(89)
A
f
A
η
τ
−
μ
=
10
−
2
a
p
=
9
.
509
·
A
f
A
n
0
Ω
10
−
4
a
w
=−
=
5
.
489
·
.
The given numerical values of the constants
a
p
and
a
w
are that of Brzezi nski (
1994
)
and depend on the applied structural model of the Earth. A recent reconsideration of
Sasao andWahr's theory byKoot and de Viron (
2011
) based on advanced geophysical
models and data yielded
a
p
=
10
−
4
, which is almost
two times smaller than the value noted in the above equation. A detailed examination
of the significance of both parameters can be found in Brzezi nski (
1994
). Here, it only
shall be stressed that
a
p
and
a
w
are purely theoretical quantities and have never been
confirmed by any observation (Bizouard et al.
1998
). Note also that the numerical
values given in this section revert to the different Earth models applied by Sasao and
Wahr (
1981
) and Barnes et al. (
1983
) and thus might be slightly inconsistent (1-3%
for the various Love numbers and Earth's moments of inertia). Besides, Sect.
2.4
makes use of another set of parameter values [following Gross (
2007
)], which are
generally more up to date than those introduced here. This fact is not accounted for
by a more distinguished notation, though.
The FCN term of Eq. (
89
) is resonant at
10
−
2
and
a
w
=
9
.
200
·
2
.
628
·
σ
=
σ
2
≈
σ
f
and thus plays a pivotal role
when studying diurnal retrograde perturbations of Earth rotation. However, in view
of the numerical discrepancy between
a
p
and
a
w
of about two orders of magnitude,
one has to treat matter and motion terms as two separate driving agents. On the
contrary, the CW term, which especially reigns over slow polar motion variations,
requires an equivalent weight on
w
.
In order to improve the agreement of the formulation with the dynamical behavior
of the real Earth, Brzezi nski (
1994
) suggests to replace the theoretical frequencies
p
and
χ
χ
