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the set (6.40), we also have b 2
0. As a result, (6.43) and (6.44) yield the
closed set of equations:
∂E 2
∂x 3
+ ik 0 h 2
1
h 3
h 1 b 1 =0 ,
(6.48)
∂b 1
∂x 3 + ik 0 ε
h 1
h 2 E 2 =0 ,
1
h 3
(6.49)
which reduces to the second-order equation
+ ω 2
h 2
h 1 h 3
∂x 3
h 1
h 2 h 3
∂E 2
∂x 3
c 2 ε E 2 =0 .
(6.50)
The polarization of Alfven oscillations described by (6.48), (6.49) differs from
that of the toroidal mode: the electric field is directed along the coordinate x 2 ,
while the magnetic field perturbations and plasma displacements are directed
along the coordinate x 1 . According to geophysical terminology, we will refer
to these oscillations as poloidal mode. These oscillations are mainly excited
by localized sources with large k 2 .
Equations (6.47) and (6.50) can be rewritten in an invariant form, in which
the influence of the geometrical factor - the magnetic field curvature - on the
propagation of Alfven waves is expressed explicitly. Let us introduce the field-
aligned coordinate s equal to the distance along the line, such that ds = h 3 dx 3 .
Then, (6.47) and (6.50) reduce to
2
∂s 2 +( κ 1
E 1 =0 ,
∂s + ω 2
κ 2 )
c 2 ε
(6.51)
2
∂s 2
E 2 =0 .
∂s + ω 2
κ 2 )
( κ 1
c 2 ε
(6.52)
Here
∂s log h 1 , 2
κ 1 , 2 =
are the principal curvatures of the equipotential surface x 3 = const and char-
acterize the rate of convergence/divergence of the field-lines.
Equations (6.51) and (6.52) are supplemented with the boundary condition
of the non-current flux from the ionosphere into the atmosphere. A detailed
treatment of the interaction of Alfven waves with the ionosphere will be given
in Chapters 7, 8. Here we only note that the atmospheric conductivity is very
small. It is several orders of magnitude less than the ionospheric conductivity.
That is why the current of an Alfven wave does not penetrate into the at-
mosphere and there is no component orthogonal to the atmosphere, bounded
 
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