Geoscience Reference
In-Depth Information
Rotational Deformation
The contribution of spin and wobble to the inertia tensor via elastic yielding of the
rotating Earth is denoted by
I r . In the absence of oceans and secular effects like
polar wander, the components of
Δ
I r can be obtained based on the assumption of an
Earth model which responds to the changing centrifugal potential just in the same
way as a non-rotating Earth would react to an external static potential of the same
type and magnitude (Smith and Dahlen 1981 , and references therein)
Δ
I 13
Δ
k 2 0 0
0 k 2 0
00 n 0 +
m 1
m 2
m 3
a 5
2
Ω
=
.
I 23
Δ
(23)
3 G
I 33
4
Δ
3 k 2
Here, k 2 signifies the conventional body tide Love number relating the external
potential of degree 2 to the evoked change of the Earth's potential. There have been
ongoing geophysical discussions whether k 2 applies on the whole Earth (Smith and
Dahlen 1981 , p. 239) or on the mantle only (Dickman 2005 ). In this review, we
shall choose the more traditional route of k 2 as a whole-Earth value. Additional
parameters in Eq. ( 23 )are a , the radius of the sphere having the same volume as
the Earth (Smith and Dahlen 1981 ), and G , Newton's gravitational constant. The
dimensionless quantity n 0 accounts for changes in Earth's mean moment of inertia
associated with purely radial deformation (Gross 2007 ).
Equilibrium Pole Tide
The present Earth model can be augmented for the oceanic rotational response to the
matter term by applying the results of Dahlen ( 1976 ) for the equilibrium pole tide.
This basically requires rewriting the diagonal components of the tensor in Eq. ( 23 )
and considering small incremental quantities
Δ
D ij
I 13
D
+ Δ
D
Δ
D 12
Δ
D 13
Δ
m 1
m 2
m 3
I 23
=
,
Δ
Δ
D 12
D
Δ
D
Δ
D 23
(24)
I 33
D
Δ
Δ
D 13
Δ
D 23
where
k ocn , w a 5
2
= k 2 + Δ
Ω
D
(25)
3 G
n 0 +
k ocn , s a 5
2
3 G .
3 k 2 + Δ
4
Ω
D
=
(26)
Due to the irregular geographical distribution of the oceans, decoupling between
polar motion and changes in LOD is not complete—a fact expressed in Eq. ( 24 )bythe
 
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