Effective Conductivity of a Cloudy Ionosphere - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 299

A numerical approach has been proposed in which all fluctuated values are
expanded as a Fourier series in spatial wavenumbers k:

+
k

σ ( r )= σ σ k exp ( i kr ) . (10.7)

=0

Here σ k is an amplitude of the harmonics of spatial wavenumber k ,and

is the mean conductivity. It has been found that in the lowest approximation
in fluctuated conductivities, the effective conductivity is

σ

1

k σ k σ − k

σ eff =

σ −

(10.8)

2

σ

and can be rewritten in the more descriptive form

1 ε 2 .

σ eff = σ − (10.9)

In order to calculate σ eff , the spatial distribution of σ ( r ) needs to be
known. Obviously, in the case of random inhomogeneities, we would first like
to theoretically determine the spectrum of inhomogeneities and only then
to calculate the resulting effective conductivity. Unfortunately in the case
of laboratory or space plasmas, we generally do not have to hand the actual
spatial correlation properties of various plasma instabilities. But, independent
of the actual form of σ ( r ) , the magnitude of σ eff for small perturbations
according to (10.9) is always less than the mean conductivity .

The situation changes drastically in magnetized media. Here, the Ped-
ersen component of the tensor σ eff

σ

ij becomes larger than the mean Pedersen

conductivity , even for small local conductivity perturbations.

Dreizin and Dychne [11] shown that the Pedersen effective conductivity

σ P

σ e P of the magnetized disorder medium is given by

( β e )= A ε
β e

µ

δσ e P σ 0 . (10.10)

Here σ 0 is the longitudinal conductivity along B 0 , A is a constant independent
of β e . For a 3D system the exponent is µ =4 / 3.
Kvyatkovsky [19] also indicated that in 2D system a specific size effect
can appear - a dependence of the conductivity on the scale-size L z along
B 0 . Moreover, in very strong magnetic fields σ e P

again becomes inversely

proportional to | B 0 | :

( β e )= ε
β e

l
L z

1
2

δσ e P · σ 0 (10.11)

where l is the size of inhomogeneities. The method of ([11], [19]) was developed
in ([5], [6]) to calculate σ eff
⊥

for an electron-ion ionospheric plasma with random

inhomogeneities.