Geoscience Reference
In-Depth Information
d
H
d
t
,
L
=
(3)
relating the torque
L
acting upon the body to the temporal change of its angular
momentum
H
. This is the basic relation for the development of (Newtonian) preces-
sion/nutation theories for astronomical Earth rotation. In order to characterize polar
motion (or geophysical Earth rotation if referring to the traditional separation) and
changes of LOD, Eq. (
3
) has to be rewritten in the rotating reference system that has
been attached to the body (Munk and MacDonald
1960
)
d
H
d
t
+
ω
×
L
=
H
.
(4)
The time-dependent quantity
signifies the angular velocity vector of the body-fixed
reference frame with respect to the space-fixed reference frame. Hence,
ω
ω
is also the
angular velocity vector of the Earth with respect to inertial space (Gross
2007
).
Equation (
4
) is one form of Euler's dynamical equation for rigid body rotation. In
case of rigidity, the angular momentum
H
can be basically expressed as the product
of the inertia tensor
I
and
.
I
is represented by a symmetric matrix containing the
moments of inertia and the products of inertia of the rotating body, which character-
ize the internal mass distribution. In addition, the inertia tensor would be invariant
for a rigid body, since single particles do not move with respect to the attached
reference system. If deformations and fluid elements are allowed for,
I
becomes
time-variable, the mass elements move with respect to the body-frame, introducing
relative angular momentum
h
. In this case, the angular momentum
H
of the entire,
rotating, deformable body is composed of
ω
H
=
I
ω
+
h
(5)
I
ω
=
Earth
ρ
x
×
(
ω
×
x
)
d
V
(6)
h
=
Earth
ρ
x
×
v
d
V
,
(7)
where both constituents
I
and
h
contain contributions of the solid Earth as well as
the hydrosphere and the atmospheric effects of density and wind variations. The
quantity
x
denotes the position of a mass element with material density
and
v
is its
velocity relative to the terrestrial system. The first summand in Eq. (
5
) is generally
referred to as
mass
or
matter term
, the second summand is labeled
motion term
.
Moreover, Eq. (
5
) perfectly illustrates the idea of the angular momentum approach:
mass displacements and fluxes in the different components of the Earth generate both
variations of the inertia tensor and relative angular momentum. Since
H
remains
constant in the absence of external torques (
L
will be set to zero in the following
section as we do not consider astronomically-induced variations of Earth rotation),
changes in the angular velocity vector balance the equation by altering its direction
(polar motion) and magnitude (changes in LOD). Combination of Eqs. (
4
) and (
5
)
ρ
