Geoscience Reference
In-Depth Information
A
A
m
(τ
−
μ)
Ω
σ
−
σ
1
+
A
f
A
m
η
Ω
σ
−
σ
2
Φ
L
p
ˆ
(σ )
=
A
A
m
h
A
Ω
σ
−
σ
1
−
A
f
A
m
n
f
Ω
Ω
σ
−
σ
2
−
Ω
.
(82)
Equation (
82
) describes polar motion at Earth-referred frequency
σ
generated by the
Φ
L
=
Φ
L
(σ )
complex-valued loading potential
and the relative angular momentum
h
i
h
2
=
h
of the perturbing fluid. The quantities
A
,
A
f
and
A
m
designate
the equatorial moments of inertia of the Earth, the fluid core and themantle, calculated
from an underlying Earth model.
=
h
1
+
(σ )
τ
,
μ
and
η
are dimensionless coefficients, while
σ
1
and
σ
2
=−
Ω
+
n
f
denote the theoretical frequencies of the CW and NDFW,
respectively. Both
σ
2
are valid for an oceanless Earth. However, it is not
necessary to list any numerical values for the two parameters, since we will replace
them by the corresponding observed values at the end of the present section.
Via MacCullagh's formula, as given for instance in Moritz and Müller (
1987
), the
potential of the loading mass can be converted to the corresponding perturbation of
the equatorial product of inertia
σ
1
and
Φ
L
=−
Δ
I
A
τ
.
(83)
Using this relation in Eq. (
82
) and separating the effects of the CW and the FCN, we
obtain
1
+
h
1
−
μ
τ
Ω
σ
−
σ
1
ΩΔ
I
p
ˆ
(σ )
=−
Ω
A
m
η
τ
ΩΔ
I
h
A
f
n
f
Ω
Ω
σ
−
σ
2
.
−
−
(84)
Ω
AA
m
Equation (
84
) can be rewritten elegantly in terms of the matter and motion portions
of the (effective) equatorial angular momentum function
w
. In particular,
it is possible to consistently incorporate the angular momentum function realization
of Barnes et al. (
1983
), even though their derivations have to be considered critically
to some extent, see e.g. Dickman (
2003
) or the brief discussion at the end of this
section. One of the parameters central to the study of Barnes et al. (
1983
)isthe
Chandler frequency of an elastic Earth comprising crust and mantle
p
χ
=
χ
+
χ
k
s
−
k
2
C
m
−
A
m
σ
e
=
Ω,
(85)
k
s
A
m
2
a
5
with
k
s
being the dimensionless 'secular' Love number of
the whole Earth (Munk and MacDonald
1960
). The equatorial angular momentum
function, which is associated with the specific Earth model yielding Eq. (
85
), is
corrected for rotational and loading deformation, see Barnes et al. (
1983
, pp. 46, 47).
Expressed in the notation of the present review, matter and motion terms read
=
3
(
C
−
A
)
G
/(Ω
)
