Geoscience Reference
In-Depth Information
I
13
+
ΩΔ
I
23
+
Ω
h
1
+
h
2
2
ψ
1
=
Ω
Δ
Ω
2
(
C
−
A
)
2
I
23
−
ΩΔ
I
13
+
Ω
h
2
−
h
1
ψ
2
=
Ω
Δ
(14)
2
A
)
Ω
(
C
−
ψ
3
=
−
ΩΔ
I
33
−
h
3
.
Ω
C
These expressions indicate that if the relative angular momentum components
h
i
and
the changes of the inertia tensor
I
ij
as well as their respective time derivatives are
introduced as known quantities frommodels or observations, such as meteorological
data in case of the atmosphere, the equation system (
13
) can be solved for
m
i
and
thus
Δ
. Apparently, the system's two equatorial components are coupled differential
equations, a
nd
it is therefore convenient to define complex quantities according to
(with i
ω
≡
√
−
1)
i
m
2
,
h
i
h
2
,Δ
I
m
ˆ
=
m
1
+
=
h
1
+
=
Δ
I
13
+
i
Δ
I
23
ΩΔ
I
i
h
Δ
I
+
Ω
h
2
ψ
2
=
Ω
−
i
−
ψ
=
ψ
1
+
i
.
(15)
2
A
)
Ω
(
C
−
Thus, the bare analytical solutions of the Liouville equations for both equatorial and
axial components are
e
i
σ
r
t
σ
r
t
ψ(τ)
e
−
i
σ
r
τ
d
m
ˆ
(
t
)
=
m
ˆ
(
0
)
−
i
τ
(16)
0
m
3
=
ψ
3
+
const
(17)
and describe the motion of the instantaneous rotation pole associated with a rigid
axisymmetric body reacting to a small excitation. In the absence of any perturbing
force, the only departure from uniform rotation would be the pole's free circular
motion of period
2
σ
r
=
304 sd (sidereal days) and amplitude
ˆ
m
(
0
)
(Moritz and
Müller
1987
).
2.3 Angular Momentum Functions
Numerous authors have reformulated and modified the excitation functions in order
to estimate the geophysical contributions to polar motion and changes in LOD, most
notable Wahr (
1982
), Barnes et al. (
1983
) or Eubanks (
1993
), all highlighting a
specific deficiency of the formulation given in Eq. (
14
): if the respective scaling
factors in the denominators are disregarded, the axial excitation function
ψ
3
can be
interpreted as an angular momentum quantity, whereas the equatorial component
ψ
rather characterizes torque and contains time derivatives of inertia increments
Δ
I
