MHD-Waves in Layered Media - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 193

Z ( e )
g

( k τ ) − 1 = Y ( e )

g

( k τ )= Y ( e ) ( − h ; k τ ) ,

Z ( m )
g

( k τ ) − 1 = Y ( m )

g

( k τ )= Y ( m ) (

− h ; k τ ) . (7.27)

Let the ground be approximated by a half-space with constant geoelectric
permeability ε g . Then the solution for Φ and Ψ in the ground ( z<

− h )can

be written as

Φ
Ψ

∝ exp( − iκ g z )

and electric and magnetic admittances become

1
Z ( e )

= k 0 ε g
κ g

1
Z ( m )

= κ g Y ( e )

g

( m )
g

= , =

k 0 ,

(7.28)

g g

where

κ g = k 0 ε g − k τ .

Assume also that σ g is so large that the inequality holds k 0 √ ε g

k τ . Then

κ g ≈ k 0 ε g and (7.28) yields

1
Z ( e g ≈ Z ( m g ≈ √ ε g .

1

Y ( e )
g

Y ( m )
g

= =

Spectral Admittance Matrix

In the general case, the horizontal components of total electric and magnetic
fields are connected by an admittance matrix

b τ ( z )= Y ( z ) E τ ( z ) , (7.29)

where E τ ( z )and b τ ( z ) are horizontal magnetic and electric fields. We find an
expression for Y ( z ) in the atmosphere from (7.12), (7.15), (7.16) and (7.23),
(7.24):

Y ( z )= Y 11

,

Y 12

(7.30)

Y 21 Y 22

where

Y ( m ) ( z )

Y ( e ) ( z ) ,

k x k y
k τ

Y 11 = − Y 22 = − −

k x Y ( m ) ( z )+ k y Y ( e ) ( z ) ,

1
k τ

Y 12 =

k y Y ( m ) ( z )+ k x Y ( e ) ( z ) .

1
k τ

Y 21 = −