MHD-Waves in Layered Media - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 218

integration of (7.136) and (7.138) be carried out twice for each harmonic.
Converting the boundary conditions from to the upper surface of the

ionosphere, we find the reflection coecients and the field over the ionosphere.
Using the determined fields as the initial data, by numerical integration of
(7.136), (7.137), we determine the fields on the ground for fixed k 1 .
If

−∞

T> 20 −

30 s (7.140)
it is possible to avoid mentioning the tedious computations. We will use the
thin ionosphere approximation. For (7.136) this presents no diculties with
the single limitation (7.140) on the periods of oscillations because the coe-
cients of (7.136) do not depend on the horizontal wavenumber k 1 . It immedi-
ately follows from (7.136) that the electric field is conserved along a field-line.
In the computation of Alfven waves this makes it possible to replace the whole
ionosphere by a thin layer with the integral Pedersen conductivity Σ P and
use the boundary conditions for such a layer in the form

b A = − X E A | x =+0 . (7.141)

For arbitrary T , it is necessary to take into account the dependence of
B 0 and plasma density on altitude. For example, let c A in the F-layer be
300 km / s and, therefore, the wavelength of the T 1 s oscillations becomes

comparable with the thickness of the F-layer. However, for the low-frequency
range features of the height distribution of c A within the ionosphere are of
little importance.
Henceforth, we will not take into account changes in c A in the upper
ionosphere and magnetosphere and return to the model of the homogeneous
half-space. Omitting intermediate computations, we obtain for R AA from
(7.136) the same relation as (7.129):

≈

− √ ε m |

AA = X

sin I |

R (0)

R AA ≈

X + √ ε m |

. (7.142)

sin I |

Now we shall return to (7.138) which describes FMS-waves. The main
difference between FMS-waves and Alfven waves is the independence of the
phase velocity value on the angle between the direction of wave propagation
and B 0 , that is, the B 0 does not form a specific direction. The inclined system
connected with B 0 has no advantage, therefore, we return to a Cartesian
coordinate system.
We express the components of the field of FMS-waves through the potential
Ψ . Let us present (7.16) in the form

b τ = i k dΨ

E τ = − k 0 Ψ k × z ,

dz ,

where

dΨ
dz

dΨ
dx 3 −

= ik 1 cot IΨ,