Geoscience Reference
In-Depth Information
z is the unit vector of the z -axis, E τ , b τ are the horizontal components of the
E and b vectors. For the potential Ψ , from (7.138) we have
d 2 Ψ
dz 2
+ k 0 ε
k 2 Ψ =
4 πσ H
ik 1 c
E A ,
(7.143)
where ϕ =
E A sin I/ik 1 , ε = i 4 πσ P in the ionosphere. Equation (7.143)
is supplemented by the boundary conditions of the potential Ψ at z
→∞
and
the impedance condition at the ground surface
dz
Ψ z = −h
ik 0
Z ( m )
+
=0 ,
(7.144)
g
where Z ( m g is the surface impedance (7.27).
We will examine disturbances with a horizontal scale less than the wave-
lengths in the magnetosphere ( m ), ionosphere ( i ) and the atmosphere ( a ),
that is, with the wavenumbers k satisfying the conditions
1 / 2 .
k
k 0 |
ε m,i,a |
(7.145)
For the io nosphe re, it can be rewritten in terms of the Pedersen skin depth
d P = c/ 2 πωσ P as
kd P
1 .
10 1 s 1 and σ P =10 7 s 1 the skin depth d P
For frequency ω< 3
×
100 km
10 2 km 1 . Similar reasonings for
the atmosphere lead us to the conclusion that on numerous occasions the
magnetostatic approximation is adequate for the analysis of propagation of
FMS-waves in the atmosphere-ionosphere-magnetosphere system.
Thus, computations of the field with an explicit indication of the ionospheric
source can be carried out in the following way:
and the last inequality is reduced to k
d 2 Ψ
dz 2
4 πσ H
ik 1 c
k 2 Ψ =
E A ,
(7.146)
with the boundary conditions (7.144) on the ground surface. The Green's
function G ( z, z 1 ) of (7.146) is
G ( z, z 1 )= 1
R g exp [
k ( z + z 1 +2 h )]
2 k
2 k exp (
k
|
z
z 1 |
)
,
(7.147)
where
ikZ ( m )
R g = 1
/k 0
g
.
1+ ikZ ( m )
/k 0
g
Then the solution of (7.146) is given by
σ H ( z 1 ) e −k|z−z 1 |
R g e −k ( z +2 h + z 1 ) dz 1 ,
Ψ ( z )= i 2 π
c
E A
k 1 k
(7.148)
 
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