MHD-Waves in Layered Media - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 197

Let us relate the admittance matrix above the ionosphere with the admit-
tance matrix below the ionosphere. It follows from (7.47) that

Y I ≡ Y (+0) = Y a + G ,

where

G = −

,

Y/ sin I X

X/ sin 2 I − − Y/ sin I

X = 4 π Y = 4 π

c Σ P , c Σ H .

This results in

k x
k τ

+ X

⎛
⎝

⎞
⎠

ζ 1
ζ 2 −

ζ 3 + k y

k x k y
k τ

ζ 4
ζ 3

Y
sin I

ζ 4 ζ 1

ζ 2

− − −

k τ

Y I = .

ζ 1
ζ 2 −

k y
k τ

+ k x
k τ

ζ 4
ζ 3

ζ 1
ζ 2 −

X
sin 2 I

k x k y
k τ

ζ 4
ζ 3

Y
sin I

−

In the next sections will be found both the matrix of transformation T Σ of

the total horizontal magnetospheric fields b ( m )

(+0) above the ionosphere into

τ

the ground magnetic fields b ( g )

b ( g ) = T Σ b ( m )

(+0) (7.51)

τ

and the matrix T relating b ( g ) and the amplitude of an initial wave b ( i τ (+0)
by

b ( g ) = T b ( i τ (+0) . (7.52)

Since

(+0) = 1 + R b ( i )
τ

b ( m )
τ

(+0) ,

these two matrices are related with one another as
T = T Σ 1 + R . (7.53)

7.5 Homogeneous Magnetosphere

The Alfven and FMS-Wave Potentials

The magnetospheric electric and magnetic wave fields we present as a sum of
the Alfven ( E A ( r ) , b A ( r )) and FMS ( E S ( r ) , b S ( r )) fields:

E ( r )= E A ( r )+ E S ( r ) , b ( r )= b A ( r )+ b S ( r ) . (7.54)

Let us first obtain expressions for the fields in the coordinate system ( x ,y ,z )
connected with the external magnetic field. In the oblique coordinate system