MHD-Wave Exposure on the Ionosphere - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 355

Alfven Wave

Consider the case of an incident Alfven wave with transverse wavenumber
k =( k x > 0 ,k y = 0) [12]. The ratio of the ground magnetic signal b ( g ) to the

x

incident wave amplitude b ( i ) is the transmission coecient:

y

T SA = b ( g )

x
b ( i )

. (12.52)

y

The field-aligned FMS-mode amplitude b ( r )

above the ionosphere is associated

with b ( i ) as

y

b ( r ) b ( i y R SA exp( iI ( z )= − − kz ) , (12.53)

where R SA is the reflection coecient. The FMS-mode near the ionosphere
has a magnetostatic character and for k y = 0 its dispersion equation is reduced
to

k x + k z ≈ 0 .

Then from ∇ · b = 0 we obtain

b ( r )

( b ( r )
x

cos I + ib ( r )
z

b ( r )
x

= − sin I )= − exp iI.

This explains the appearance of an additional phase factor exp ( iI ) in (12.53).
From (12.52) and (12.53) it follows that

b ( r )
b ( g )

R SA
T SA

= − exp( iI − kz ) . (12.54)

x

In the simple case of the high conductive ground, kd g 1 and the terms

associated with the skin effect in (7.124) and (7.125) can be omitted. Then
the surface impedance Z ( m )

= 0. For the MHD-wave with the horizontal scale

g

1 /k A it can be set ( k A − k 2 ) 1 / 2 L ⊥ ≈ ik . Then (12.38) becomes

R SA

T SA ≈−

sinh ( kh ) . (12.55)

For qualitative estimations set them so that the electron concentration
decreases rapidly with height as

exp

,

z
H

N ( z ) ∝ −

where H is the magnetospheric scale height. Then, from (12.51)-(12.55) we
have

= b ( g )

δN T
N T

1
1+ kH .

x
B 0

sinh ( kh )exp( iI ) (12.56)