Propagation of MHD-Beams - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 255

where

α = sin 2 I
X

Y 2
X∆ (0)
S

γ (0) = , .

Let | κ a h | 1. Then

1+ κ a h √ ε g
ik 0 ε a

− √ ε g , ζ (0)

1

ζ 1 ≈ = ζ 2 ≈ ,

ik 0 h √ ε g , − √ ε g , ζ (0)

3

ζ (0)
4

ζ 3 ≈ =1 − 4 ≈ =

X + ζ (0) k 0 √ ε m , − √ ε m −

κ (0)
S

∆ (0)
S

4
ζ (0)
3

≈ ∆ S ≈ = .

The approximate expression for κ S is obtained under the assumption that

k (0) / ( k 0 ε m ) 1 , where k (0) is determined by (8.39). This inequality holds

at T< 10 4 s. Function

k 2

k (0) .

k 0 ε a |

sin I

|

i∆ (0)
S

∆ ( k )= ∆ SK ( k ) ∆ A ( k ) ≈− −

k (0) h

near k = k (0)
For the contribution of the electric mode to the ground field, we get

α 2 k 0 ε a

k (0) h∆ (0)

G 0 =

(1 − γ (0) ) 2

sin I

S

⎛ ⎞

√ ε g Y 2
ζ (0 3 ∆ (0)

√ ε g Y
ζ (0)
3

−

⎝ ⎠

sin I

S

× . (8.55)

γ (0) ) γ (0)

α

Y (1 − 1 −

∆ (0)
S

− sin I

α

If the Hall conductivity is nil ( Y =4 πΣ H /c = 0), then the wave propagates
in the atmospheric waveguide as a TEM-mode ( E z =0 , z =0).Inthe

general case ( Y = 0), we call it a TEM-type wave, keeping in mind, of course,

that the wave has a longitudinal component b x , as well.
The matrix (8.55) is suciently bulky, therefore, let us take into account
that for a wide range of frequencies and conductivities the ground conductiv-
ity is much more than the ionospheric conductivity, i.e. | ε g | >X ∼ Y and

| k 0 hX | 1 . Then (8.55) becomes (see (8.39))

⎛ ⎞

ik 0 h Y 2
X

Y
X

−

sin I

G 0 exp ik (0) x =

⎝ ⎠

exp ik (0) x .

− ik (0) (8.56)

Y
sin I

− ik 0 h 1