MHD After-Effects of a Sound Impact - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 394

Ohm's law and the first equation of (15.7) lead to

σ P E x + σ H E z = j x , σ H E x + σ P E z =0 , − (15.8)

where

v z
c B 0
and E x is the electric field in the stationary coordinate system. Let us elimi-
nate E z from (15.8). The result is

E x = E x −

z = σ H j x = σ C E x ,

σ P E x

(15.9)

where the Cowling conductivity σ C is

σ C = σ P + σ 2 H

σ P .

Substituting (1.93) for Pedersen σ P and (1.94) for Hall σ H conductivities, we
obtain

ω pe
4 πν e .

σ C = σ 0 1+ β i /β e
1+ β e β i

, 0 =

( m e /m i ) 1 / 2 , we rewrite the last equation in the

Omitting the term β i /β e ∼

form

σ 0
1+ β e β i .

σ C = (15.10)

It follows from projections of the equations onto the coordinate axis, that
the system supports two uncoupled propagating modes. For the first mode
variables v z ,j x ,b y, E x ,E z

=0and

∂ 2 v z
∂z 2 −

∂ 2 v z
∂t 2 −

1
c s

1
H

∂v z
∂z

B 0
cc s N n m n

∂j x
∂t

= − ,

∂b y
∂z

4 π
c

∂E x
∂z

1
c

∂b y
∂t

= − j x , = − ,

j x = σ C E x − v z .

B 0
c

(15.11)

E z -component of this mode is

σ H
σ P σ C j x =

β e
σ 0

1 β i /β e

1+ β i /β e j x .

−

E z =

( m e /m i ) 1 / 2 , we obtain

And again, if to omit terms ∼

β e
σ 0

E z = j x .