Geology Reference
In-Depth Information
2
1
2
0
before
2
on the container surface, given by ξ
=
ξ
0
. Multiplying by σ
−
ξ
/ξ
letting ξ
→
ξ
0
,wefindthat
ξ
1
2
∂χ
2
0
m
ξ
−
ξ
∂ξ
=
σ
χ
(6.80)
on the surface ξ
=
ξ
0
.
For modes of degree
n
, azimuthal number
m
, the secular equation for the eigen-
frequencies, σ,isthen
1
0
dP
n
(
ξ
0
)
m
ξ
0
σ
2
P
n
(ξ
0
),
−
ξ
=
(6.81)
d
ξ
0
with
1
e
2
−
ξ
0
=
σ
2
.
(6.82)
1
−
e
2
σ
Replacing the derivative of the Legendre function using the recurrence relation
(B.12), the secular equation becomes
1)
m
ξ
0
σ
m
)
P
n
−
1
(ξ
0
)
1)
P
n
+
1
(ξ
0
)
P
n
(ξ
0
). (6.83)
(
n
+
1)(
n
+
−
n
(
n
−
m
+
=
(2
n
+
Applied to the Earth's core, we might ignore the inner body and take the con-
tainer surface to be the core-mantle boundary. The flattening of the boundary is
defined as
a
−
b
a
,
=
f
(6.84)
which can be related to the eccentricity,
e
, of the spheroidal container. We find that
e
2
f
)
2
=
1
−
(1
−
.
(6.85)
The value of ξ
0
, appearing in the secular equation, is given by (6.82). The pro-
gramme POINCARE calculates the residual of the secular equation through the
function subprogramme ERR(Z,M,N) for a range of Coriolis frequencies, and iter-
ates to find a specific root, given two starting periods. The function subprogramme
ERR(Z,M,N) calls the function subprogramme PMN(Z,M,N) of Section B.2 for
the evaluation of Legendre functions of the first kind.
C
PROGRAMME POINCARE.FOR
C
C POINCARE.FOR is an interactive programme to calculate the
C eigenperiods of the Poincare inertial wave equation for an
C ellipsoidal container. The programme allows a range of values of
C the Coriolis frequency to be searched at specified intervals,
C or an iteration to be performed, starting with two initial
C trial periods.
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