Geoscience Reference
In-Depth Information
φ
λ
Here,
denote geocentric latitude and longitude,
r
is the geocentric radius of
the mass element and the velocity components
and
(
,
,
)
u
v
w
are given in the direction of
the unit vectors
e
k
,
k
∈ {
East
,
North
,
Ve r t
}
. After some modifications, we find
χ
=
−
1
.
100
r
4
sin
cos
2
e
i
λ
d
ρ
φ
φ
λ
d
φ
d
r
A
C
−
−
1
.
608
r
3
e
i
λ
d
+
ρ
(
u
sin
φ
+
i
v
)
cos
φ
λ
d
φ
d
r
A
)
Ω(
C
−
p
w
=
χ
+
χ
(41)
0
748
C
m
.
r
4
cos
3
χ
3
=
ρ
φ
d
λ
d
φ
d
r
0
998
Ω
.
r
3
u
cos
2
+
ρ
φ
d
λ
d
φ
d
r
C
m
p
3
w
=
χ
+
χ
3
.
(42)
Equations (
41
) and (
42
) represent the routine expressions for evaluating angular
momentum functions. Pressure and wind terms are three-dimensional integrals accu-
mulating the density distribution and velocity field over the volume of the perturbing
fluid. Depending on the data available, though, it might be useful to employ slightly
altered angular momentum functions formulae. We shall now give a brief compila-
tion of the various calculation methods, with particular focus on atmospheric angular
momentum (AAM)
H
(
a
)
and
H
(
a
)
3
H
(
a
)
1
i
H
(
a
)
2
H
(
a
)
=
=
ΩΔ
I
+
h
+
H
(
a
)
3
=
ΩΔ
I
33
+
h
3
.
Integration Over Radial Increments
This computation variant fully corresponds to Eqs. (
41
) and (
42
), except for the
scaling factors and geodetic parameters, which are not needed when considering raw
angular momentum. The vertical integral extends from Earth's surface
r
=
r
s
(
φ,λ
)
=∞
up to
r
, strictly speaking. In any practical application, the top border will
be represented by a distinct layer (for instance the pressure level of 0.1hPa), from
which upwards density values
have a negligibly small
contribution to AAM. Equatorial and axial matter and motion terms read
ρ
and wind velocities
(
u
,
v
)
∞
2
−
2
2
π
Δ
I
r
4
sin
cos
2
e
i
λ
d
=−
ρ
φ
φ
λ
d
φ
d
r
(43)
r
s
0
∞
2
2
π
h
r
3
e
i
λ
d
=−
ρ
(
u
sin
φ
+
i
v
)
cos
φ
λ
d
φ
d
r
(44)
−
2
r
s
0
