MHD-Waves in Layered Media - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 194

Atmospheric Potentials and Admittances

Let us denote the solution of (7.19) by ζ 1 ( z )and ∂ζ 1 ( z ) /∂z by ik 0 ε a ( z ) ζ 2 ( z ).
They obey the ground conditions:

z = −h

1
ik 0 ε a (

∂ζ 1 ( z )
∂z

Y ( e )
g

ζ 1 ( − h )= − , ζ 2 ( − h )=

=1 . (7.31)

− h )

Denote also the solution of (7.20) by ζ 3 ( z )and ∂ζ 3 ( z ) /∂z by ik 0 ζ 4 ( z ) defined
by the boundary conditions

z = −h

1
ik 0

∂ζ 3 ( z )
∂z

Y ( m )
g

ζ 3 ( − h )=1 , ζ 4 ( z )= = − . (7.32)

Here

Y ( e )
g

= Y ( e ) ( ( m )

g

= Y ( m ) ( h ) .

Express solutions of (7.19), (7.20) and their derivatives in terms of ζ i ( z )

− h ) , −

Φ 0 ζ 1 ( z )
Y ( e )

∂Φ ( z )
∂z

ik 0 ε a ( z )
Y ( e )

Φ ( z )= − , = − Φ 0 ζ 2 ( z ) , (7.33)

g g

∂Ψ ( z )
∂z

Ψ ( z )= Ψ 0 ζ 3 ( z ) , = ik 0 Ψ 0 ζ 4 ( z ) . (7.34)

These formulae are general and are also applicable to an inhomogeneous at-
mosphere. Substituting (7.27) and (7.34) into (7.25) and (7.26), we obtain

ζ 1 ( z )
ζ 2 ( z ) ,

ζ 4 ( z )
ζ 3 ( z ) .

Y ( e ) ( z )= Y ( m ) ( z )= − − (7.35)

Then components of the admittance matrix under the ionosphere are given
by

ζ 1
ζ 2 −

,

Y 11 = k x k y
k τ

ζ 4
ζ 3

Y 22 = − (7.36)

k x ,

1
k τ

ζ 4
ζ 3

ζ 1
ζ 2

+ k y

Y 12 = −

(7.37)

k y .

1
k τ

ζ 4 ζ 1

ζ 2 ζ 3 + k x

Y 21 = (7.38)

Let

0) . (7.39)
Substituting (7.33) and (7.34) into (7.15) and (7.16), we obtain the horizontal
components b ( e ) , ( m )

ζ i = ζ i ( −

=( b ( e ) , ( m ) ,b ( e ) , ( m )

) in terms of ζ i :

τ x y

k y
−

,

k 0 Φ 0 ζ 1 ( z )
Y ( e )

b ( e )
τ

( z )= −

k x

g

k 0 Ψ 0 ζ 4 ( z ) k x

.

b ( m )
τ

( z )= − (7.40)

k y