Inhomogeneous Ionosphere - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 273

Now, let the external magnetic field B 0 be orthogonal to the ionosphere.
Use the Cartesian coordinate system ( x, y, z ) with z oriented along the mag-
netic field with the origin at the ionosphere current layer. The curl-free electric
field is

E ⊥ = − ∇ ⊥ Φ 0 , (9.5)

where ∇ ⊥ denotes projection of a gradient on the plane orthogonal to B 0 ,

Φ 0 ( x, y )= Φ ( x, y, z = 0) is the potential at the ionospheric level.
The continuity equation for the electric current is

∇ · j =0 . (9.6)

Considering the following two conditions:

• current cannot leak into the atmosphere and therefore its longitudinal

component on the lower ionosphere boundary is nil;

• field line in the ionosphere is the equipotential line.

Integrate (9.6) along a field line that traverses the ionosphere, we obtain

j = ∇ ⊥ · ( Σ · ∇ ⊥ Φ 0 ) , (9.7)

where Σ is the integral conductivity tensor of the ionosphere, operators ∇ ⊥ ·

and ∇ ⊥ are the divergence and gradient in the { x, y } plane. The field-aligned

current can be found from the Ampere's law

c
4 π (

j = ∇× b ) z .

Substituting b = E / ( ik 0 ) into the above and expressing the electric field

in terms of the potential (9.5), we find

∇×

c
4 πik 0 ∇ ⊥ ·

∂ E ⊥
∂z

c
4 πik 0 ∇

∂Φ
∂z ,

2
⊥

j = = − (9.8)

∂ 2
∂x 2

+ ∂ 2

2
⊥

where

∂y 2 is a transverse Laplacian. Substituting the latter
equation into (9.7), we obtain

∇ =

∇ ⊥ Φ z =0

4 π
c

1
ik 0

∂
∂z

∇ ⊥ · Σ + =0 .

If the Alfven field is a sum of the incident and reflected waves, then Φ ( x, y, z )
in the magnetosphere is given by

Φ ( x, y, z )= Φ ( i ) ( x, y )exp( ik A z )+ Φ ( r ) ( x, y )exp( ik A z ) , − (9.9)

where k A = ω/c A . Substitution of (9.9) into (9.7) with

Φ ( x, y, z =0)= Φ ( i ) ( x, y )+ Φ ( r ) ( x, y ) ,

Φ 0 ( x, y ) ≡