Active Cloud Releases and MHD-Emission - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 386

⎧
⎨

√ ε m u + ρ, t

,

1 z

c A

− z 0 ,

E ρ =

ρ, t + , (14.29)

⎩

z
c A

1
√ ε m u −

− z 0 .

Substitution (14.28) and (14.29) into (14.26) yields

√ ε m
X c +2 √ ε m

u + ( ρ, t )= − q ( ρ, t ) ,

u + ( ρ, t ) .
The magnetic and electric fields for z> 0 are given by

u − ( ρ, t )= −

√ ε m
2 √ ε m + X c ( ρ, t

b ϕ = −

z/c a ) q ( ρ, t

− z/c a ) ,

−

E ρ = b ϕ / √ ε m ,

(14.30)

and the longitudinal current density by

c
4 π (

c
4 π

1
ρ

∂
∂ρ ( ρb ϕ ) .

j = ∇× B ) = (14.31)

Let us now consider the influence of the conductive ionospheric layer on the
generation of the Alfven pulse; here, as before, c A = const. According to
(14.28)-(14.29, the magnetic and electric fields are

⎧
⎨

u 1 ρ, t

,

z
c A

− z 0 ,

b ϕ =

u 2 ρ, t

+ u 3 ρ, t + ,

⎩

z +2 h
c A

z
c A

− z 0 ,

⎧
⎨

√ ε m u 1 ρ, t

,

1 z

c A

− z 0 ,

E ρ = (14.32)

√ ε m u 2 ρ, t

u 3 ρ, t +

,

⎩

1 z +2 h

c A

z
c A

− − z 0 .

Then, from (14.24) and (14.26), it follows that

u 2 ρ, t

= R AA u 3 ρ, t ,

h
c A

h
c A

− −

u 2 ρ, t

2 h
c A

− − u 3 ( ρ, t )= u 1 ( ρ, t ) ,

1+ u 1 ( ρ, t )

u 2 ρ, t

X c
√ ε m

2 h
c A

− − − u 3 ( ρ, t )= − q ( ρ, t ) , (14.33)