Geoscience Reference
In-Depth Information
1
Ω
˙
m
1
=
x
p
−
y
p
(76)
1
Ω
˙
m
2
=−
y
p
−
x
p
(77)
or in complex notation
i
Ω
ˆ
m
ˆ
=ˆ
p
−
p
,
(78)
where
i
y
p
characterizes terrestrial motion of the CIP in a tangential
polar plane. Equation (
78
) represents the sought-for kinematical relation between
the polar motion of the instantaneous rotation axis and that of the CIP. Considering
the arguments brought forth in the first paragraph of Sect.
3.1
, it is legitimate to
combine this result with the dynamical theory of Earth rotation. Thus, the equatorial
component of geophysical excitation in Eq. (
30
) referred to the actually observed
axis (towards the time-variable CIP) reads
p
ˆ
=
x
p
−
i
i
Ω
σ
cw
ˆ
χ.
m
ˆ
+
m
=
χ
−
p
i
Ω
i
σ
cw
d
d
t
i
Ω
i
Ω
ˆ
ˆ
χ
⇒ˆ
p
−
p
+
p
ˆ
−
=
χ
−
p
i
σ
cw
i
Ω
d
d
t
i
σ
cw
i
Ω
ˆ
ˆ
χ.
p
ˆ
+
p
−
p
ˆ
+
=
χ
−
By comparing terms on the left and right side of the latter equation, we arrive at the
probably most vital relationship within Earth rotation excitation studies
i
σ
cw
ˆ
p
ˆ
+
p
=
χ
(79)
e
i
σ
cw
t
σ
cw
t
e
−
i
σ
cw
τ
p
ˆ
(
t
)
=
p
ˆ
(
0
)
−
i
0
χ(τ)
d
τ
,
(80)
represented either as differential equation (Gross
1992
) or in integral form. Obvi-
ously, there are two main ways of comparing the reported values of polar motion
)
and the associated geophysical excitation mechanisms in the context of the angular
momentum approach. The first, more frequently applied method is a deconvolu-
tion problem (Brzezi nski
1992
) and consists of differentiating the complex series of
observed polar motion (Eq.
79
), see e.g. the study of fortnightly ocean tides by Gross
et al. (
1996
). The second approach, on the contrary, requires convolution of the equa-
torial angular momentum function with the free Chandler mode (right-hand side of
Eq.
80
) in order to get an estimate of those polar motion variations that are induced
by mass redistributions and relative motion in the fluid considered. This method,
critically assessed by Chao (
1985
), is specifically sensitive to the initial conditions
imposed on
p
ˆ
(
t
χ(
t
)
before integration.
