Geoscience Reference
In-Depth Information
Fig. 3
Perturbations of the
instantaneous rotation vector
ω
(
t
)
with respect to the state
of uniform rotation around
π
30
, the axis of figure of an
undeformed Earth
y
30
m
1
(t)
1+m
3
(t)
m
2
(t)
(t)
x
The
m
i
(
are small dimensionless quantities describing excursions of the instanta-
neous rotation vector from uniform rotation. As is evident from Fig.
3
,
m
1
(
t
)
t
)
and
m
2
(
can be interpreted as angular offsets that specify polar motion of the mantle's
rotation axis, while
m
3
(
t
)
represents changes in the rate of rotation and thus LOD
variations. The corresponding perturbations of Earth's angular momentum comprise
h
as well as time-variable increments in the inertia tensor
t
)
⎛
⎝
⎞
⎠
.
Δ
I
11
(
)Δ
I
12
(
)Δ
I
13
(
)
t
t
t
Δ
I
12
(
)Δ
I
22
(
)Δ
I
23
(
)
I
(
t
)
=
I
0
+
Δ
I
(
t
)
=
I
0
+
t
t
t
(12)
Δ
I
13
(
t
)Δ
I
23
(
t
)Δ
I
33
(
t
)
For the sake of a simplified notation, the time-dependency of all quantities will
be no more explicitly stated but borne in mind. The first time derivative shall be
abbreviated by a dot above the character. Furthermore, it is recognized that the
Earth is a nearly axisymmetric body and, hence, both
A
and
B
can be replaced by
A
=
(
2, the mean equatorial moment of inertia of the whole Earth. Substi-
tuting Eqs. (
11
) and (
12
) in the Liouville equations, and considering that
A
+
B
)/
Δ
I
ij
C
,
h
i
1, the equatorial and axial equations of motion (Eq.
8
) can be
linearized and rewritten as
Ω
C
and
m
i
m
1
σ
r
+
˙
m
2
=
ψ
2
m
2
σ
r
−
˙
m
1
=−
ψ
1
(13)
m
3
=
ψ
3
,
˙
A
)
A
(
C
−
with
being the Euler frequency, the frequency of resonance of a
rigid axisymmetric Earth, see Sect.
1.3
. While the
m
i
are quantities that may be
principally inferred from astronomical or geodetic observations, the expressions on
the right hand side of the system of first-order differential equations are geophysical
measures, called
excitation functions
after Munk and MacDonald (
1960
). Neglecting
external torques, the
σ
r
=
Ω
ψ
i
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