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be the radius of the region occupied by the ionized component and the ex-
pansion speed of this region's boundary, respectively. Then from the second
equation in (14.16) we get
dR
i
(
t
)
dt
=
α
(
t
)
v
n
(
R
i
(
t
)
,t
)
Substituting (14.1) in the last equation obtain
dR
i
(
t
)
dt
=
v
n
0
(
t
)
R
i
(
t
)
R
n
(
t
)
·
α
(
t
)
.
(14.19)
Integrating (14.19) we find
R
i
(
t
)=
R
i
(0) exp
t
0
α
(
t
)
dt
,
v
n
0
(
t
)
R
n
(
t
)
·
(14.20)
where
R
i
(0) =
R
n
(0). Note that for
α
(
t
)=1
,R
i
(
t
)=
R
n
(
t
). Since the total
number of charged particles is constant, the time dependency of
N
(
t
)canbe
found from the relation
N
(
t
)
R
i
(
t
)=
N
0
R
0
.
(14.21)
Equation (14.18) is obtained on the assumption that the cloud of the
injected gas is a cylinder extended along
B
0
.
For more realistic models of a
release, for example, spherically symmetric emission, can be found that an
equation is similar to (14.18) but for the concentration is integrated over the
cloud thickness. Resultant formulas enable us to define space-time distribution
of the charged particles and to estimate changes of the integral conductivities
in the cloud.
Induced MHD-Wave
The electric fields excited by the plasma cloud release generate electric cur-
rents which in turn produce hydromagnetic emission from the region occupied
by the injected plasma [14]. The current density
j
d
caused by the dynamo-field
E
d
is
E
d
=
1
j
d
=
σ
E
d
,
c
[
v
n
×
B
0
]
.
(14.22)
The electromagnetic field is found by plugging the external currents into
Maxwell's equations. The concepts developed in Chapter 7 allow us to estimate
the intensity of the MHD-wave emitted from the man-made plasma release.
One can use the small-scale approximation, also discussed in Chapter 7, to
uncouple the general system of MHD-equations to yield (7.136) and (7.146)
for the Alfven and FMS-waves, respectively.
To make the problem tractable, we consider an axially symmetrical per-
turbation in the polar region where the magnetic inclination is
I
=
π/
2
.
In
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