Geoscience Reference
In-Depth Information
Δ
Δ
minute off-diagonal elements
D
23
(Smith and Dahlen
1981
). However,
as shown by Dahlen (
1976
), this effect is negligible, as is the coupling between both
equatorial components conveyed by
D
13
and
Δ
D
12
. The quantities of interest are the
oceanic
Love numbers
k
ocn
,
s
for the spin and wobble component in Eqs. (
25
)
and (
26
). Superimposing them on the degree 2 body tide Love number increases
D
and
D
by 16% and 10%, respectively. Hence, the pole tide correction is considerably
smaller than the effect of rotational deformation.
Δ
k
ocn
,
w
and
Δ
Surface Loading Deformation
The matter term of the excitation process is associated with changes in the mass
distribution of the geophysical fluid in question and thus represents a loading effect
on the solid parts of the Earth. The resulting surface deformations partly compensate
the influence of the direct mass effect of excitation on Earth's inertia tensor and are
analytically accounted for by multiplying the matter terms of the angular momentum
functions with a factor of
k
2
)
(
1
+
(for the equatorial component in Eq.
20
)or
(
1
+
α
3
k
2
)
(for the axial component in Eq.
21
). Herein,
k
2
denotes the load Love number
of degree 2 and
α
3
is a dimensionless coefficient that allows for axial core-decoupling
(Merriam
1980
).
The preliminary scheme for excitation studies established in Sect.
2.3
can now
be updated for the described rotational and loading response of a more accurate
Earth model by adding the components of
I
r
and
Δ
Δ
h
, given by Eqs. (
24
)-(
26
) and
(
22
), to the initial angular momentum functions in Eqs. (
20
) and (
21
). By substi-
tuting the so-corrected functions
χ
3
into Eqs. (
18
) and (
19
), relative angular
momentum of the core as well as rotational deformation of the Earth together with
the passive response of equilibrium oceans are incorporated into the linearized Liou-
ville equations. While the mathematical formulation in Eqs. (
18
) and (
19
) can be
fully maintained, both the excitation terms as well as the resonance frequency of
the (equatorial) system change as part of the necessary rearrangements. The updated
angular momentum functions become
χ
and
k
2
)Δ
I
+
h
χ
=
Ω(
1
+
(27)
A
−
(
C
−
D
)Ω
+
α
3
k
2
)Δ
k
r
Ω(
1
I
33
+
h
3
χ
3
=
.
(28)
Ω
C
m
Clearly, the matter terms have now also been corrected for surface loading defor-
mation. The leading factor
k
r
in Eq. (
28
) compactly describes the effect of axial
rotational deformation. On the left-hand side of Eq. (
18
), the Euler frequency has
to be expanded as
A
)
A
(
C
−
and rewritten in order to yield the theoretical
frequency of the CW, given also in Smith and Dahlen (
1981
) (p. 249)
σ
r
=
Ω
