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and the momentum equation (1.18) with taken into account (1.21) becomes
d
v
e
d
t
m
e
E
e
−
∇
e
P
e
m
e
N
e
−
=
−
ν
ei
(
v
e
−
v
i
)
−
ν
en
(
v
e
−
v
n
)
,
(1.26)
d
v
i
d
t
m
i
E
i
−
∇
e
P
i
m
i
N
i
−
=
ν
ie
(
v
i
−
v
e
)
−
ν
in
(
v
i
−
v
n
)
,
d
v
n
d
t
−
∇
P
n
m
n
N
n
−
m
e
N
e
m
n
N
n
ν
en
(
v
n
−
m
i
N
i
m
n
N
n
ν
in
(
v
n
−
=
v
e
)
−
v
i
)
,
(1.27)
E
e,i
=
E
+
1
P
α
=
kT
α
N
α
,
c
[
v
e,i
×
B
]
,
(1.28)
where
d
v
α
d
t
=
∂
v
α
∂t
+(
v
α
∇
)
v
α
,
=
e, i, n.
m
e
and
v
e
,
m
i
and
v
i
,
m
n
and
v
n
are, respectively, the mass and the mean
velocity of electrons, ions and neutrals;
P
e
,
P
i
,
P
n
are partial pressures of
electrons, ions and neutrals. Equations (1.23)-(1.27) are supplemented with
Maxwell's equations
B
=
4
π
c
j
+
1
∂
E
∂t
,
∇
×
∇
·
E
=4
πρ,
(1.29)
c
1
c
∂
B
∂t
,
∇
×
E
=
−
∇
·
B
=0
,
(1.30)
j
=
e
(
N
i
v
i
−
N
e
v
e
)
,
ρ
=
e
(
N
i
−
N
e
)
.
(1.31)
We must also supplement the energy conservation equation or heat trans-
fer equations which depend on the electron
T
e
, ion
T
i
and neutral
T
n
tem-
peratures. Below, we shall restrict our consideration to the isothermal case
T
e
=
T
i
=
T
n
=const.
Magnetohydrodynamics
Next simplifications in the plasma description is utilization of the magneto-
hydrodynamics (MHD) equations. Let the typical spatial scale of the system
and frequency of the electromagnetic field be denoted as
L,
and
ω
, and the
particle free path and thermal velocity by
l,
and
v
T
, respectively. Also let,
ω
ci
=
eB/m
i
c
be the ion cyclotron frequency and
ν
be the collision frequency
of the ion with neutrals. Then the conditions for which the generalized 3-fluid
hydrodynamics equation reduce to the one-fluid magnetohydrodynamics can
be written
1
.l<L,
2
.ω
ν
∼
v
T
/l,
3
.
ω << ω
ci
.
(1.32)
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