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by the surface
S
a
.Thus,
j
3
r
∈S
a
=
j
·
e
3
|
r
∈S
a
=0
.
We get from this equality and from Ampere's law
e
3
=(4
π/c
)
j
3
(
∇
×
b
)
·
that
∂b
2
∂x
1
−
∂b
1
∂x
2
= 0
(6.53)
r
∈S
a
both for the toroidal mode
b
2
|
r
∈S
a
= 0, and for the poloidal mode
b
1
|
r
∈S
a
=0.
Let, for instance,
x
3
d/
2 correspond to the Southern ionosphere and
x
3
=
d/
2 to the Northern ionosphere. Then by considering (6.45) and (6.48)
the boundary conditions for (6.45), (6.46) reduce to
=
−
∂E
2
∂x
3
d
2
.
=0 at x
3
=
±
and the same for one equation of the second order (6.50) or (6.52). For the
system (6.48)-(6.49) (or for one equation (6.50) (or (6.52))), the conditions
are given by
∂E
1
∂x
3
d
2
.
=0 at
x
3
=
±
(6.54)
Equation (6.51) and (6.52) explicitly demonstrate, that the eigenfrequen-
cies of Alfven oscillations of the geomagnetic shell are somewhat different
for modes with different polarizations. This polarization spectrum splitting is
caused by a different influence of the magnetic field curvature on the spec-
trum of oscillations with the plasma displaced along or across the shell. More
precisely, it is related to the difference between the principal curvatures of
equipotential surfaces or, in other words, to the difference in the rates of
the field-lines convergence/divergence in two orthogonal surfaces containing a
given field-line. If the principal curvatures coincide at the intersection points
of the equipotential surfaces with the given field-line,
κ
1
=
κ
2
(the field-lines
converge
/
diverge at the same rate, i.e. the cross-sectional shape of a flux tube
does not change along its length), then there is no polarization splitting of
the spectrum.
It is convenient to rewrite (6.47) and (6.50) in a form similar to the equa-
tion for the Alfven waves propagating in a uniform external magnetic field
[44]. The replacement d
ξ
=
h
1
h
3
h
−
1
2
dx
3
reduces (6.47) to
∂
2
2
E
1
=0
,
∂ξ
2
+
ω
(6.55)
c
A
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