Geology Reference
In-Depth Information
9.3 Changes in the inertia tensor and the secular polar shift
The e
ect the displacement fields in realistic Earth models, just described, have
on the rotation of the Earth can be found from the changes they produce in the
components of the inertia tensor (Smylie and Mansinha, 1967; Smylie and Zuberi,
2009). The changes in the components of the inertia tensor,
ff
I
ij
, to first order in
the small quantities
u
r
/
r
,
u
θ
/
r
,and
u
φ
/(
r
sinθ) in the epicentral system are
Δ
2
r
u
r
1
φ
u
φ
sinθcosφsinφ
dm
,
sin
2
θcos
2
u
θ
cosθsinθcos
2
Δ
I
11
=
−
−
φ
+
2
r
u
r
1
φ
u
φ
sinθcosφsinφ
dm
,
sin
2
θsin
2
u
θ
cosθsinθsin
2
Δ
I
22
=
−
−
φ
−
2
r
u
r
sin
2
u
θ
cosθsinθ
dm
,
Δ
I
33
=
θ
+
r
2
u
r
sin
2
Δ
I
12
=−
θcosφsinφ
+
2
u
θ
cosθsinθcosφsinφ
(9.85)
u
φ
sinθ
cos
2
φ
dm
,
sin
2
+
φ
−
r
2
u
r
cosθsinθcosφ
+
u
θ
cos
2
θ
cosφ
−
u
φ
cosθsinφ
dm
,
sin
2
Δ
I
13
=−
θ
−
r
2
u
r
cosθsinθsinφ
+
u
θ
cos
2
θ
sinφ
+
u
φ
cosθcosφ
dm
.
sin
2
Δ
I
23
=−
θ
−
The integrands in (9.85) can all be cast in the form of scalar products of the dis-
placement field with particular spheroidal vectors of degrees zero and two. For
example, we may write
2
r
u
S
dm
,
Δ
I
11
=
·
(9.86)
where
S
is the spheroidal vector with radial coe
cients
2
3
,
u
0
0
=
(9.87)
1
3
,
1
6
,
u
0
2
=
0
2
v
=
(9.88)
1
12
,
1
24
,
u
2
2
=−
2
v
=−
(9.89)
u
−
2
v
−
2
=−
2,
=−
1.
(9.90)
2
2
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