Partially Ionized Plasma - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 19

The concentration of neutrals decreases rapidly with altitude and there is no
need to take into account the ion-neutral collisions in the upper ionosphere
and in the magnetosphere. Thus, it is possible to omit ν i in (1.102) and write

iω c 2
c 2 A

σ xx ≈− .

Using (1.58) we find

ε xx = c 2 /c 2 A . (1.103)

At the end of this section we consider the longitudinal conductivity. It is
convenient to define the electron inertial length as

λ e = c/ω pe . (1.104)

Then, for the longitudinal resistivity, (1.98) becomes

= 4 π
c 2

σ − 1 iω ) λ e . ( ν e − (1.105)

If the collisions can be neglected, the inverse longitudinal dielectric perme-
ability is

ε − 1 =1+ k 0 λ e , (1.106)

where k 0 = ω/c is the wavenumber in the free space. Note that (1.105)-
(1.106) give a correct estimate only in a very limited altitude range in the
upper ionosphere. The collisions can be neglected in comparison with the
inertial term only in the upper part of the ionospheric F -layer. On the other
hand, at high altitudes the effects of ion and electron temperatures play a key
role. The ion Larmour radius ρ i becomes a typical spatial scale related to the
longitudinal conductivity instead of the electron inertial length.

1.5 Dispersion Equation

We have found the equations for the tensor of the dielectric permeability.
Note once again that the waves in the frequency range considered are charac-
terized by tremendous spatial scales comparable with the Earth's radius and
sometimes essentially exceeding it. What are the properties of the ULF-waves
and their propagation velocities in the ionosphere, in the magnetosphere and
along the ground surface? How are their properties related to the dielectric
permeabilities and conductivities found? These questions can be answered by
studying the dispersion equations. Certainly, we need to keep in mind that for
the majority of cases the typical scales of plasma inhomogeneities are com-
parable with those of ULF-wave disturbances and, thus, the analysis of full