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cylindrical coordinates with the origin at the center of the cloud, we have
∂b
ϕ
∂z
1
c
A
∂E
ρ
∂t
+
4
π
c
σ
P
E
ρ
+
4
π
−
=
c
2
σ
H
v
n
B
0
,
∂E
ρ
∂z
1
c
∂b
ϕ
∂t
=
−
(14.23)
with the radiation condition at
z
→
+
∞
and the impedance condition at the
ionosphere
z
=
−h
E
ρ
b
ϕ
1
X
.
=
−
(14.24)
The initial conditions for the electric and magnetic fields are
E
ρ
(
z, t
=0)=0
,
ϕ
(
z, t
=0)=0
.
(14.25)
Integrating (14.23) over the cloud thickness, we obtain a homogeneous equa-
tion for
z
≶
0
,
and conditions on the discontinuities of
b
ϕ
and
E
ρ
for
z
=0:
{
b
ϕ
}
0
+
X
c
E
ρ
|
0
=
−
q,
{
E
ρ
}
0
=0
,
(14.26)
where
{
Q
}
=
Q
|
z
=+0
−
Q
|
z
=
−
0
denotes discontinuity of a quantity
Q
,
Σ
P,H
=
σ
P,H
dz.
Y
c
v
n
c
X
c
=
4
π
c
c
=
4
π
c
Σ
P
,
Σ
c
H
,
q
=
−
B
0
,
Using equations (14.23), boundary conditions and initial data (14.24),
(14.25), one can find the hydromagnetic pulse from the region occupied by
the injected plasma.
Consider an example from which it is possible to get an estimate of the
field-aligned current in the Alfven pulse. Let the background plasma above and
below the cloud be characterized by the dielectric permeability
ε
m
=(
c/c
A
)
2
.
Then the wave equation for
b
ϕ
(
ρ, z, t
)is
∂
2
b
ϕ
∂z
2
−
∂
2
b
ϕ
∂t
2
1
c
2
A
=0
.
(14.27)
The solution of (14.27) is
⎧
⎨
u
+
ρ, t
,
z
c
A
−
z
0
,
b
ϕ
=
(14.28)
ρ, t
+
,
⎩
z
c
A
u
−
z
0
.
Then the electric field
E
ρ
is
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