Hydromagnetic Resonators - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 108

determining the longitudinal wavenumber

k n = k n ( ω )

at a fixed oscillation frequency ω .
For perfectly reflecting ionospheres, i.e. Σ P ,Σ P →∞

,X ± →∞ and

| z =0; l z =0 , frequency ω is not included in the boundary conditions. Then
both problems in fact coincide. In this case, wavenumber k n = k n is indepen-
dent of ω and the frequency of the n -th harmonic ω An is given by
ω An ( x )= k n c A ( x ) .
If the integral conductivity at least one of the conjugate ionospheres is fi-
nite, i.e. Σ P

Q

=0or Σ P

= 0, then the wave energy dissipates in the ionospheres
and FLR-frequencies become complex. Denote the frequency of the FLR- n -th
harmonic by

iγ n ( x ) ,
where γ n ( x ) is the decrement. ω n ( x ) can be found from the problem obtained
by replacing the parameter k 2 by ω 2 /c 2 A ( x ) in (5.9). The resulting boundary
problem is the so-called generalized boundary problem on eigenfrequency ω .
It can be shown that the oscillation frequency spectrum

ω n ( x )= ω An ( x ) −

{ ω n } will be discrete

just as for the ordinary problem.
The FLR-frequencies will be found as follows. From the boundary

problem (5.9)-(5.10) we find the dependence of an n -th wavenumber k n on
frequency ω . At the known function k n ( ω ) , resonance oscillation frequencies
of the x -th field-lines can be found from

{ ω n }

ω = ± k n ( ω ) c A ( x ) , (5.18)

following from the definition of k n ( ω ). Substituting into (5.13)
q = q n ( ω )= k n l z
π

= ± ω,

where ω = ω/ω A ,ω A = πc A /l z , after a simple algebra we get
exp( − 2 iπ ω )= R S R N , (5.19)

where

Σ P
1+ Σ P

Σ P
1+ Σ P

R S = 1

−

R N = 1

−

and ,

where subscript ' ' and '+' refer to the 'Southern' and 'Northern' ionospheres,

respectively. Note, that the expressions for R S and R N yield reflection coe-
cients of the Alfven wave from the N- and S-ionospheres defined as the ratio
of the electric component in the incident wave to the same in the reflected
one. According to (5.1), Σ P is the ratio of the corresponding integral Pedersen
conductivity Σ P to the wave magnetospheric conductivity Σ A . R S,N decrease
from 1 to

−

. Two types of resonance modes
are possible determined by the sign of R S R N :

− 1 with Σ P changing from 0 to + ∞