Geology Reference
In-Depth Information
and
⎝
⎠
.
(
a
/
b
)
3
M
S
1
4
−
3
4
M
I
+
M
I
M
S
1
(
a
/
b
)
3
β
=
(8.117)
M
O
+
−
M
I
is the mass of the inner core,
M
O
is the mass of the outer core,
M
S
is the mass
of the shell and
M
I
=
4/3π
a
3
ρ
0
(
a
) is the displaced mass, whereρ
0
(
a
) is the density
at the bottom of the outer core. σ
=
ω/2
is the dimensionless Coriolis frequency
corresponding to angular frequency ω. Finally,
f
a
(σ)and
f
e
(σ) are dimensionless
functions of σ given by
Ω
8
(σ
+
1)
3/2
f
a
(σ)
1)
3/2
=
+
(σ
−
1)
5/2
(8.118)
5
(σ
+
16
1)
5/2
−
−
(σ
−
,
and
f
e
(σ)
1)
1/2
1)
3/2
=
∓
24 (
±
σ
∓
−
16 (
±
σ
∓
(8.119)
1)
5/2
5
(
16
1)
5/2
−
±
σ
−
−
(
±
σ
+
,
with the upper sign referring to the retrograde mode, for which σ is positive, and
the lower sign referring to the prograde mode, for which σ is negative. The vis-
cous splitting curves are shown in Figure 8.17, together with the observed periods.
Two independent measures of viscosity are produced, as the reduction in rotational
splitting at fixed viscosity is larger for the retrograde mode than for the prograde
equatorial mode. The retrograde equatorial mode gives 1.190
10
11
Pa s,
±
0.035
×
10
11
Pa s. A balanced
while the prograde equatorial mode gives 1.304
±
0.034
×
10
11
Pa s yields viscous periods that are only 6.5 s longer
than the observed periods. The solid viscous splitting curves in Figure 8.17 are
drawn for this value of viscosity. The dashed curves surrounding the solid viscous
splitting curves are those for a
error value of 1.247
×
20% variation in viscosity.
A very stringent test of the significance of the identification of the translational
triplet in the product spectrum, shown in Figure 8.13, derives from the viscous
splitting equation (8.113). For three candidate periods,
T
R
(retrograde),
T
A
(axial)
and
T
P
(prograde), the splitting equation (8.113) provides the corresponding values
of the dimensionless viscous splitting parameter, g
ν
R
, g
ν
A
andg
ν
P
,foragivenvalueof
T
0
. Thus, the whole frequency axis can be searched for correctly split resonances.
For a resonance centred on frequency
f
j
, its form at neighbouring frequencies
f
i
is
±
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