Geoscience Reference
In-Depth Information
2.6 An alternative Modeling Approach via Torques
Alongside the classical angular momentum method, the influence of Earth's fluid
components on polar motion, nutation and changes in length of day can also be
studied by considering atmospheric and oceanic effects as external torques acting on
the mechanical system Earth (de Viron et al.
1999
). Supposing the fluid layer to be
outside the physical system contrasts with the angular momentum approach, which
retrieves information about changes in Earth rotation from the abstract balance of
angular momentum within the entire Earth-fluid system, cf. Sect.
1.5
. Nonetheless,
both methods are analytically and physically equivalent (de Viron and Dehant
1999
)
and should provide similar results when deployed for excitation studies. The rela-
tionship and numerical coherence between both approaches will be discussed in a
later paragraph. We shall now focus on the analytical expressions that are central to
the torque method.
Consider a fluid—the atmosphere in the following—surrounding and interacting
with the solid Earth. In the light of basic Newtonian physics, any variation in the total
angular momentum
H
(
s
)
of the solid body is equivalent to the total torque acting on
the solid Earth due to the atmosphere (de Viron and Dehant
1999
)
d
H
(
s
)
d
t
L
(
a
)
→
(
s
)
=
,
(55)
where d
/
d
t
is the time derivative in an inertial reference system. By recognizing that
L
(
a
)
→
(
s
)
=−
L
(
s
)
→
(
a
)
(56)
d
H
(
s
)
d
t
d
H
(
a
)
d
t
=−
(57)
holds for an isolated Earth-atmosphere system in space, we obtain the angular
momentum budget equation of the atmospheric layer in the celestial frame
d
H
(
a
)
d
t
L
(
s
)
→
(
a
)
=
.
(58)
As shown by de Viron et al. (
1999
) in a very general way, the torque exerted on the
atmosphere can be computed directly from the integral of local forces at the bottom
surface
S
of the atmosphere
L
(
s
)
→
(
a
)
=
r
×
(
−
p
s
)
n
d
S
+
r
×
p
s
n
g
d
S
+
r
×
(η
n
·∇
v
r
)
d
S
.
(59)
L
p
L
g
L
f
Herein,
r
is the position vector and
v
r
the relative velocity vector of the mass element
with respect to the terrestrial frame. The quantity
η
represents the dynamic viscosity
