MHD-Waves in Layered Media - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 216

Chapter 3). The appearing FMS-waves damp exponentially at the charac-
teristic scale of k − 1 . Also note here that despite R SA ≈ 1, the energy flux

transformed from the incident Alfven wave into an FMS is small because the
electric field in an FMS reflected wave is s mall and is of the order of k 0 R SA /k ,
andinanAlfven wave it is R AA / √ ε m . ≈

7.7 Small-Scale Perturbations

One of the conditions for the applicability of the thin ionosphere model, viz
smallness of horizontal wavenumbers k τ l I 1, is not always satisfied. For

instance, FLR-oscillations can have a horizontal scale of about 10 km at the
ionosphere level. In that case, the approximation of large wavenumbers is
useful. In this section we want to obtain an uncoupled equation for Alfven
waves at k 1 →∞

.
As distinct from Chapter 6, where an uncoupled equation was obtained
within the same limit for Alfven waves propagating in transversely inhomoge-
neous plasma, we shall regard the plasma as horizontally homogeneous, but
Hall conductivity and the change in plasma characteristics with altitude will
be taken into account. The obtained equations are not always valid and not
applicable for an arbitrarily small horizontal perturbation scale.
The reason is that we did not take into account the interaction of Alfven
and FMS-waves with other oscillation branches remarkable at some small
scales, let us say, less than the electron inertial scale λ e = c/ω pe . The sequen-
tial derivation of asymptotic equations for small k − 1 is omitted here because
it is very cumbersome. Here we shall turn not fully faithfully. We extract a
small parameter, writing equations in the zeroth-order approximation with
respect to this parameter. And then we shall check that the corrective terms
to the obtained equations are small. The introduced earlier designations are

σ P dz, σ H dz.

k 0 = ω X = 4 π

c

Σ P = 4 π
c

Y = 4 π
c

Σ H = 4 π
c

c ,

σ P ,σ H and Σ P, Σ H are the specific and integral Pedersen and Hall conduc-
tivities of the ionosphere, respectively. It can be shown that when

k 2 k 1 (7.134)

and the following condition is satisfied:

k 0 Y 2
k 1 X

1 (7.135)

in a zero approximation for this parameter, the full wave equation in the
ionosphere are split into equations for Alfven and FMS-oscillations. We shall

write these equations in an oblique coordinate system x 1 ,x 2 ,x 3 .