Geoscience Reference
In-Depth Information
resulting formulation has been applied in numerous studies, most notably Brzezi nski
et al. (
2002
), Vondrák and Ron (
2007
) or Schindelegger et al. (
2011
), with the last one
attempting to consistently account for a fully decoupled core. Refer to Gross (
1993
)
for another frequently used expression of short-periodic polar motion excitation,
incorporating the effects of the FCN. The present survey will expound upon the
derivations of Brzezi nski (
1994
) in Sect.
3.2
and will also discuss the model's deficits.
For the major part of the review, though, the focus will be on the more established
long-period formalism.
2.5 Evaluation of Angular Momentum Functions
In order to study geophysical excitation of Earth rotation, it is necessary to convert
the angular momentum functions in Eqs. (
35
) and (
36
) into a practicable analytical
form that is well-suited for numerical evaluation. In particular, this requires casting
χ
χ
3
in terms of certain parameters that represent the routine output of General
Circulation Models (GCM), such as numerical weather or ocean models. In a first
step, following Munk and MacDonald (
1960
), the components of the inertia tensor
variations
and
I
and the relative angular momentum quantity
h
have to be written as
volume integrals extended over the fluid considered. This conversion is accomplished
by an appropriate modification of Eqs. (
6
) and (
7
), yielding
Δ
ΩΔ
I
1
.
100
A
)
=
−
1
.
100
p
χ
=
ρ(
x
1
x
3
+
i
x
2
x
3
)
d
V
(37)
Ω(
C
−
C
−
A
608
h
1
.
1
.
608
w
χ
=
A
)
=
(
x
2
˙
x
3
−
x
3
˙
x
2
)
−
i
(
x
1
˙
x
3
−
x
3
˙
x
1
)
d
V
(38)
A
)
Ω(
C
−
Ω(
C
−
0
.
748
ΩΔ
I
33
0
.
748
C
m
p
3
x
1
+
x
2
)
χ
=
=
ρ(
d
V
(39)
Ω
C
m
0
.
998
h
3
Ω
0
.
998
3
χ
=
=
ρ(
x
1
˙
x
2
−
x
2
˙
x
1
)
d
V
,
(40)
C
m
Ω
C
m
where the separation into matter and motion terms is underlined by the superscripts
p
(for
pressure
) and
w
(for
wind
)—a labeling which originates from the meteorological
parameters central to matter and motion terms of atmospheric angular momentum
functions. An elegant formulation of pressure terms
p
3
w
3
can be obtained by transforming the position vector
x
into spherical coordinates and
writing
p
w
χ
,χ
χ
,χ
and wind terms
x
˙
=
v
in a suitable geographical coordinate system (e.g., Moritz and Müller
1987
)
⎛
⎞
cos
φ
cos
λ
⎝
⎠
=
cos
φ
sin
λ
x
r
sin
φ
v
=
u
e
East
+
v
e
North
+
w
e
Ve r t
.
