Partially Ionized Plasma - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 22

This is the dispersion equation for the isotropic conductor. The electromag-
netic field is described by the diffusion equation with the diffusion coecient
c 2 / 4 πσ 0 .
Equation (1.120) gives

k =(1+ i ) d − s , (1.121)

where d s = c (2 πωσ 0 ) − 1 / 2

∝

exp( i kr ) is called a skin depth. The plain wave

decays at e times at the distance d s propagating along r .
In the E -layer the polarization conductivity σ pol is small in comparison
with Pedersen conductivity σ P , and the longitudinal conductivity σ ≈ σ 0 is

large in comparison with the Pedersen conductivity:

m e
m i

1 / 2

ω pi
ω pe ∼

σ P
σ 0 ≈

ν e
ν i

σ ⊥ ≈ σ P , σ ≈− σ H and 1 .

With these conditions for not very large k ⊥ we get,

σ P
σ 0

k 2
2

4 πωσ H
c 2

,

The terms with the longitudinal conductivity can be omitted in (1.119) and

k 4
4

1 / 2

k 2
2 ±

2 i
d P −

4
d 4 H

k 2

= + , (1.122)

where

c
√ 2 πσ P ω ,

H = c

√ 2 πσ H ω .
The sign '+', in (1.122) corresponds to two normal modes; one propagating to
the positive and the other to the negative direction of the z -axes. Attenuation
of these modes is determined by Pedersen conductivity σ P . The sign '

d P =

− ', in

(1.122), corresponds to two non-propagating normal modes.
Consider in detail quasi-longitudinal propagation. Let the transversal
wavenumbers meet the condition

k ⊥ d H 1 , (1.123)

then expanding the radical expression in (1.122) into the power series over
k ⊥ d H , we obtain for the propagating normal mode
k d H 2
2

k ⊥ d H
2

2

+ i σ P =1 −

σ H .

(1.124)

Thus, a weakly decaying wave mode exists in a partly ionized plasma with
Pedersen and Hall conductivities at σ P /σ H 1[9].