MHD-Waves in Layered Media - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 217

Alfven Waves

In a zero approximation the equations for Alfven oscillations have the form

dE A
dx 3

b A
sin I ,

db A
dx 3

4 πσ P
c

E A
sin I ,

= ik 0 = − (7.136)

where

1
sin I E 1 ,

E A = b A = b 2 ,

and k i ,E i ,b i are the covariant components in an oblique coordinate system.
Equation (7.136) are supplemented at the lower boundary of the ionosphere
by the condition of no vertical current ( j n = 0) equivalent to the requirement

b A | z = − 0 =0 . (7.137)

FMS-Waves

The FMS-waves are described by
d
dx 3 −

ik 1 cot I E S = − ik 0 b S ,

d
dx 3 −

ik 1 cot I b S = 4 πσ P
c

E S −

+ i k 2
k 0

4 πσ H
c

E A , (7.138)

where

E S = E 2 , b S = b 1 , k 2 = k 1 .
The procedure for determining the ionospheric distribution of electric and
magnetic fields produced by the incident Alfven waves is as follows. Firstly,
we define the ionospheric fields caused by the Alfven wave itself making use of
(7.136) with the boundary condition (7.137). Then we define the Hall currents
initiated by the ionospheric Alfven electric field. The term proportional to E A
in (7.138) is the Hall current produced by the electric field of an Alfven wave.
In turn, the induced Hall current generates an FMS-wave (7.138) with the
electric field E S perpendicular to E A . In the next approximation, E S gives a
correction to E A = E (0)
A

of the order of

Y 2
X E A .

k 0
k 1

E (1)
A

≈ (7.139)

The approximation can be used if this a correction is small. We shall restrict
our consideration to a zero approximation for the parameter (7.135). The
same equation is correct both in the atmosphere and within the ground, but
with a corresponding replacement of σ P either by atmospheric conductivity
σ a or by the ground conductivity σ g .
The fields at the ground surface are found after the altitude-dependence
of E and b in the ionosphere is determined. This requires that numerical