Inhomogeneous Ionosphere - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 272

The ULF-wave skin depth is much more than the thickness of the ionosphere.
So, the approximation of thin ionosphere is applicable. Chapter 7 presents a
description for fields in this approximation. The wave electric field E is a sum
of the Alfven wave electric field E A and the FMS-electric field electric field
E S :

E = E A + E S . (9.1)
The field E A of the Alfven beam is 2D curl-free, and of E S is the curl field.
The influence of the vortex part on the curl-free part is negligible in the
horizontally homogeneous ionosphere and can be ignored under the condition
(7.135) which is written as

X K k 0 L A 1 ,

where L A is the horizontal scale of the Alfven beam, and X K is defined in
(7.107).
Both Hall conductivity and the inhomogeneities cause an FMS-wave with
the vortex field. As for the horizontally homogeneous ionosphere the influence
of this field on the electric field of the Alfven wave can be ignored for small-
scale inhomogeneities. This scale L ⊥ must satisfy the same condition except
that, L A has to be replaced by L ⊥

X K k 0 L ⊥ 1 .

It means that the phase incursion at the scale L ⊥ is small. The horizontal
scale L F of the FMS-wave field in the ionosphere must be larger than L ⊥ .
Let us estimate L F . Let horizontal wavenumber k

k A , then (8.36) re-

duces to

2 ik + k 0 X K =0 .
Then the scale of the FMS-wave L F =1 /k is

cT
πX K .

L F = (9.2)

In the equations for the Alfven electric field (potential part), FMS electric
field (vortex part) can be omitted under the condition

Σ P + Σ H
Σ P

.

X K L
Tc

X K = 4 π
c

1 , (9.3)

The ratio of the correction E (1)
A

caused by the FMS-wave to the initial Alfven

electric field E (0)
A

is given likewise (7.139) by

Σ P + Σ H
Σ P

,

E (1)
A
E (0)
A

4 πL ⊥
Tc 2

∼ (9.4)

is small.