MHD-Waves in Layered Media - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 211

for which

=cosh kh tanh kh ,

ζ 1
Y ( e )

1+ ik 0 ε a
kY ( e )

− (7.111)

g g

k 0 ε a sinh kh 1

coth kh ,

ζ 2
Y ( e )

k ik 0 ε a

kY ( e )

= i − (7.112)

g g

sinh kh 1+ coth kh ,

ζ 3
Y ( m )

i k 0
k

ik
k 0 Y ( m )

= − (7.113)

g g

cosh kh 1+ tanh kh .

ζ 4
Y ( m )

ik
k 0 Y ( m )

= − (7.114)

g g

For the two-layer model

1
Z ( e )

= ik 0 ε g
κ g

1
Z ( m )

= iκ g
k 0

Y ( e )
g

Y ( m )
g

= cot κ g H, = cot κ g H, (7.115)

g g

where

κ g = k 0 ε g − k 2 = √ 2 d − 1

g

( kd g ) 2
2

i − .

Admittances Y ( e ) and Y ( m ) i , tend to the electric

and magnetic admittances of the conductive half-space.
Let us find the conditions when the ground can be considered as a perfect
conductive medium. The second term in (7.111)-(7.114) can be neglected for
high enough conductivity if the inequalities

for H

→∞

,cot κ g H

→−

g g

tan κ g H |

σ a
σ g

κ g
k

coth kh | + | tanh kh | ,

κ g tan κ g H |

k

coth kh | + | tanh kh |

hold. Then

ζ 1
Y ( e )

ζ 2
Y ( e )

k
k 0 ε a sinh kh,

= − cosh kh, = i (7.116)

g g

ζ 3
Y ( m )

i k 0
k

ζ 4
Y ( m )

= − sinh kh, = − cosh kh. (7.117)

g g

Let us consider first an influence of the conductivity on the electric mode.
The corrective term in (7.111) can be ignored if

tan κ g H tanh kh

ε a
ε g

κ g
k

1 . (7.118)

The inequality is invalid near the branch cut Im κ g = 0 (Fig.7.3) and imagi-
nary axes of the complex plane of wavenumbers k since tan κ g H and tanh kh