Partially Ionized Plasma - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 14

Hence

ω

ω + iν n iν in ,

i eE ( ± )
m e ∆ ±

ω

v ( ± )
e

= − ∓ ω ci + (1.67)

ω

ω + iν n iν en ,

= i eE ( ± )
m i ∆ ±

ω

v ( ± )
i

± ω ce + (1.68)

where

∆ ± =[ ω ± ω ce + i ( ν e − aν en ][ ω ∓ ω ci + i ( ν i − bν in )]

+( ν ei + bν en )( ν ie + aν in ) .

The magnetic field has no effect on the velocity and current components along
B 0 . Therefore, in order to find longitudinal velocities, one can put ω ce = ω ci =
0 , then

ω +

ω + iν n iν in ,

eE
m e ∆

ω

v e = − i (1.69)

ω +

ω + iν n iν en ,

eE
m i ∆

ω

v i = i (1.70)

where

∆ =[ ω + i ( ν e − aν en )][ ω + i ( ν i − bν in )] + ( ν ei + bν en )( ν ie + aν in ) .

The electron and ion current densities and corresponding conductivities in the
basis of circle polarizations are

e = j ez and j ( ± ) j ( ± )

e

= j ex ± ij ey , = j ix ± ij iy , i = j iz ,

i

σ ( ± )
i

σ ( ± )
e

= σ exx ∓ iσ exy , e = σ ez and = σ ixx ∓ iσ ixy , i = σ iz ,

and

σ (+)
e

, σ (+)

i

,

1
2

ixx = 1
2

+ σ ( − )
i

+ σ ( − )
e

σ exx = (1.71)

σ (+)
e

, σ (+)

i

.

i
2

i
2

σ ( − )
i

σ ( − )
e

σ exy = − ixy = − (1.72)

Here we take into account that σ xx = σ yy ,σ xy = σ yx . The transversal and

longitudinal components of the total conductivity is a sum of the electron and
ion parts:

−

+ σ ( ± )
i

σ ( ± ) = σ ( ± )
e

, σ = σ e + σ i . (1.73)

Substituting (1.67)-(1.70) into

j ( ± )
e

Nev ( ± )
e

= σ ( ± )
e

E ( ± ) ,

= − e = − Nev e = σ e E , (1.74)

j ( ± )
i

= Nev ( ± )
i

= σ ( ± )
i

E ( ± ) , j i = Nev i = σ i E . (1.75)