Geoscience Reference
In-Depth Information
A
−
D
A
m
+
ε
c
A
c
+
C
−
σ
cw
=
Ω,
(29)
D
with
A
m
being the mean equatorial moment of crust and mantle.
Mantle Anelasticity
Using the numerical values given in Table 1 of Gross (
2007
), the theoretical period of
the CW calculated from Eq. (
29
) turns out to be about 7.5 sd smaller than the actually
observed period of 434.3
) sd (Vicente and Wilson
1997
). According to
Smith and Dahlen (
1981
), such a discrepancy can largely be attributed to anelastic
properties of Earth's mantle. Anelastic behavior, on the one hand, requires consider-
ing a complex increment
k
an
on the load Love number
k
2
, which is thereby modified
by about 4%. On the other hand, anelasticity also affects the body tide Love number
k
2
and thus the frequency of the CW. Alas, at present there is no model that considers
the impact of anelasticity on
k
2
, so that inclusion of this effect can only be achieved
within a hybrid approach (Gross
2007
), which consists of substituting the observed
Chandler frequency
±
1.7 (1
σ
σ
cw
in the equatorial Liouville
equation (
18
). The angular momentum functions have to be adapted accordingly.
For this purpose, it is necessary to single out
σ
cw
for its theoretical value
σ
cw
in the denominator of Eq. (
27
)after
eliminating the quantity
D
by aid of Eq. (
29
).
The resulting first-order differential equation system then reads
i
i
Ω
σ
cw
ˆ
χ
ˆ
+
=
χ
−
m
m
(30)
m
3
=−
χ
3
+
const
,
(31)
where
1
π
T
cw
2
i
2
Q
cw
σ
cw
=
+
(32)
is the complex Chandler frequency characterized by its period
T
cw
and quality factor
Q
cw
. The quantities
χ
3
contain all previously stated corrections for a realistic
model of our planet and, following Barnes et al. (
1983
), are called
effective angu-
lar momentum functions
. In this review, however, we shall keep the label
angular
momentum functions
, since the initial angular momentum functions in Eqs. (
20
) and
(
21
), set up for a rigid Earth, will not be needed anymore.
χ
and
k
2
+
Δ
k
an
)Δ
I
+
h
Ω(
+
1
χ
=
(33)
A
m
+
ε
c
A
c
) σ
cw
A
+
(
C
−
k
2
+
Δ
k
an
))Δ
k
r
Ω(
1
+
α
3
(
I
33
+
h
3
χ
3
=
.
(34)
Ω
C
m
