Partially Ionized Plasma - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 16

(1.77) reduce to

ω pe
4 π∆ ±

ω pe
4 π∆

σ ( ± )
e

= i ( ω ∓ ω ci + iν in ) , e = i ( ω + iν in ) , (1.82)

ω pi
4 π∆ ±

ω pi
4 π∆

σ ( ± )
i

= i ( ω ± ω ce + iν en ) , i = i ( ω + iν en ) . (1.83)

These equations allow considerable further simplification. ν ei ν ie = ν ei m e /m i
can be omitted in ∆ ± if ν in >> ν ei m e /m i . This inequality holds throughout
the ionosphere, except the upper part of the F -layer. Then

∆ ± =( ω ± ω ce + iν e )( ω ∓ ω ci + iν i ) ,

∆ =( ω + iν e )( ω + iν i ) ,

and (1.82)-(1.83) become

ω pe ω pe

4 π ( ω + iν e ) ,

σ ( ± )
e

= i

ω ce + iν e ) ,

e = i (1.84)

4 π ( ω

±

ω pi ω pi

4 π ( ω + iν i ) .

σ ( ± )
i

= i

ω ci + iν i ) ,

i = i (1.85)

4 π ( ω ∓

By substituting (1.84), (1.85) into (1.71), (1.72), we obtain

X e
1+ X e

,

σ xx = c Ne
B 0

X i
1+ X i

+ (1.86)

1
1+ X i −

,

σ xy = c Ne
B 0

1
1+ X e

(1.87)

1
X e +

.

σ = c Ne
B 0

1
X i

(1.88)

where

X e = ν e −

iω
ω ce

X i = ν i −

iω
ω ci

, .

The tensor of dielectric permeability ε ( r ,ω ) could be obtained easily from
(1.56).

Ultra Low Frequency Range

Consider Ultra Low Frequency (ULF) range holding the inequality

ν n ω ω ci ω ce . (1.89)

Expand the conductivities (1.86)-(1.88) into the power series over ω/ω ce and
ω/ω ci . First we present the following approximate equations