Geology Reference
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for the inner core boundary, with
2
σ
1
bh
n
c
n
,
(1,σ)
e
n
,
(1,σ)
2
2
g
0
(
b
)
d
n
,
=
−
4
Ω
σ
−
(8.59)
and
2
σ
1
ah
n
c
n
,
(
a
/
b
,σ)
e
n
,
(
a
/
b
,σ)
2
2
g
0
(
a
)
d
n
,
.
=
−
4
Ω
σ
−
(8.60)
The integrals (8.47) and (8.54) for
d
n
,
and
e
n
,
(
r
,σ), respectively, take into
account that they are non-vanishing only for integrands that are even across the
equatorial plane. For modes even across the equatorial plane,
n
in the constraint
equations (8.57) and (8.58) ranges over
n
=
m
,
m
+
2,
m
+
4,....While for modes
odd across the equatorial plane
n
ranges over
n
5,....For
L
spherical harmonics constrained to satisfy the boundary conditions,
n
ranges over
the values
=
m
+
1,
m
+
3,
m
+
n
−
m
−P=
0, 2, 4,..., 2(
L
−
1),
(8.61)
where the binary number
is zero for even modes and unity for odd modes.
With the application of the range and summation conventions, the functional
multiplied by σ
P
2
σ
1
is the symmetric bilinear form
2
−
2
σ
1
2
σ
−
F
=
x
i
a
ij
x
j
,
a
ij
=
a
ji
,
(8.62)
R
the
a
ij
being the elements of a matrix that is an eighth-degree polynomial inσ.The
variables
x
i
can be divided into those that are constrained,
x
p
, and those that are
free to be chosen to make the functional stationary,
x
k
.The
p
th constraint equation
can then be solved for
x
p
to give
x
p
=
c
pk
x
k
.
(8.63)
This allows the symmetric bilinear form (8.62) to be expressed entirely in terms of
unconstrained variables as
x
k
a
k
x
+
x
k
a
kp
c
p
x
+
x
k
a
p
c
pk
x
+
x
k
c
pk
a
pq
c
q
x
,
(8.64)
using the symmetry of the elements
a
ij
. Stationarity of this form leads to a modified
yet symmetric coe
cient matrix with
k
element given by
a
k
+
a
kp
c
p
+
a
p
c
pk
+
c
pk
a
pq
c
q
.
(8.65)
The new matrix is easily generated from the unconstrained matrix by sequences of
row and column operations.
In the case of the Poincare inertial wave equation,
Φ
and its derivatives
Φ
r
,
Φ
z
and
Φ
rz
can be shown to vanish at the geocentre, except for the axisymmetric modes
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