Effective Conductivity of a Cloudy Ionosphere - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 297

The basic idea is adequately illustrated by the present elementary example
and shows how, with increasing β e , the effective Pedersen conductivity σ e P
can grow beyond proportions for small local inhomogeneities.
Since the discovery of the strong influence of small perturbations on the
effective transport properties of magnetized media, many physicists have been
intrigued by this effect (e.g., [11], [12], [13], [19]). The key questions was: how
is the integral current caused by an external electric field applied to randomly
inhomogeneous medium?
This chapter is organized as follows: in the first section we briefly review
existing theories of effective conductivity both for regular media with scalar
local conductivity and for magnetized media. Next we present an expression
for the effective conductivity of partially ionized plasma. Then, we derive
expressions for σ eff for small perturbations of local conductivity in a strong
magnetic field. Theoretical outcomes are compared with results of laboratory
experiments on silicic inhomogeneous films placed into a strong magnetic field.

10.2 Existing Theories

Assume that the ionosphere at a certain altitude is replaced by an anisotropic
sheet with a constant conductivity σ 1 containing rarefied inhomogeneities of
conductivity σ 2 . Let us consider a region of the ionospheric sheet of volume V .

Suppose also that the inhomogeneities occupy a volume V k , then p = V k /V
is the inhomogeneity concentration.
An effective conductivity σ eff defines the relation between the volume av-
erage current density

j and the electric field E :

= σ eff

j ( r ) E ( r ) . (10.4)

σ eff is actually measured in experiments and appears in averaged Maxwell's
equations. j ( r )and E ( r ) in (10.4) are the local current and electric field. It
would appear reasonable that the spatial distribution of local electron con-
centration inhomogeneities and hence, of conductivity σ ( r ) are both random.
σ eff does not in general coincide with the mean conductivity and can

greatly differ from it. A wide range of theoretical approaches has been ap-
plied to this problem. One such approach concerns the case of sparse inhomo-
geneities (i.e. p

σ

1). In this case the mutual impact of inhomogeneities may
be neglected and we may assume that only E influences the inhomogeneities

and therefore

j ( r ) = σ 1 E + p ( σ 2 − σ 1 ) E 2 ,

where E =(1 − p ) E 1 + p E 2 , and

1 1

V 2

E 1 = E ( r ) dv, E 2 = E ( r ) dv.

V − V 2

V −V 2 V 2