Geology Reference
In-Depth Information
and the equation governing polar motion and wobble, (4.33), becomes
m
1
2
i
C
3
m
k
2
d
5
3
GA
Ω
k
2
d
5
3
GA
Ω
−
A
A
Ω−
+
−
(4.58)
˜
˜
˜
c
A
−
A
A
A
−
i
c
A
.
=
Ω
−
Ω
−
i
From equation (4.32), changes in the spin rate and length of day are governed by
m
3
1
2
˙
4
k
2
d
5
9
GC
Ω
1
=
Γ
c
33
C
−
3
C
3
3
C
Tr(
R
)
+
+
Ω
−
.
(4.59)
C
Ω
4.3.2 Free and forced polar motion and dissipation
From equation (4.58), free polar motion is governed by
m
−
i
σ
0
m
=
0,
(4.60)
with
C
3
1
2
k
2
d
5
3
GA
Ω
k
2
d
5
3
GA
Ω
−
A
A
Ω−
σ
0
=
+
.
(4.61)
Free polar motion then takes the form
ce
i
σ
0
t
m
=
,
(4.62)
where
c
is an arbitrary constant. The rotational deformation of the Earth is not
perfectly elastic, hence σ
0
is the complex number
=
ω
0
+
σ
0
i
/τ.
(4.63)
ω
0
is the resonant angular frequency and τ is the damping time of the motion. The
free motion is then described by
ce
−
t
/τ
+
i
ω
0
t
m
=
.
(4.64)
Another description of the dissipation is through the dimensionless quality factor
Q
. By definition it is given by
E
Δ
Q
=−
2π
E
,
(4.65)
where
E
is the energy of the motion and
E
is the energy dissipated per cycle.
The energy is proportional to the square of the amplitude of the motion and decays
according to
Δ
e
−
2
t
/τ
.
(4.66)
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