Environmental Engineering Reference
In-Depth Information
where
N
e
N
e
r
)
N
e
is the derivative at a point that moves with the plasma.
To analyze the motion of an element of plasma volume with length
d
l
and cross
section
d
s
containing
N
e
d
s
d
l
electrons, we assume at first that the vector
d
l
is par-
allel to the magnetic field
H
, so the magnetic flux through this elementary plasma
volume is
Hds
. If the plasma velocity at one end of the segment
d
l
is
w
e
,thenat
its other end the velocity is
w
e
d
dt
D
@
@
C
(
w
e
t
C
(
d
l
r
)
w
e
, so the variation of the segment length
during a small time interval
δ
t
is
δ
t
(
d
l
r
)
w
e
. Hence, the length of the segment
satisfies the equation
d
dt
(
d
l
)
D
(
d
l
r
)
w
e
,
which is identical to (4.159). From this it follows, first, that in the course of plasma
evolution the segment
d
l
has the same direction as the magnetic field and, second,
that the length of the plasma element remains proportional to the quantity
H
/
N
e
,
that is, the magnetic flux through this plasma element does not vary with time
during the plasma motion [115]. Thus, the magnetic lines of force are frozen into
the plasma [116], that is, their direction is such that the plasma electrons travel
along these lines. This takes place in the case when the plasma conductivity is high.
Interaction of charged particle flow and magnetic fields leads to specific forms of
the plasma flow which are realized in both laboratory devices and cosmic and solar
plasmas [109, 117-119].
To
find
the
steady-state
motion
of
a
high-conductivity
plasma,
we
start
with (4.157), which gives the force on each plasma electron as
1
cN
e
[
j
H
].
Inserting into the expression for the force the current density
j
D
e
c
[
w
e
F
D
e
E
D
H
]
D
(
c
/4
π
)
curl
H
,
we obtain
1
2
r
.
1
cN
e
[
j
H
]
1
1
H
2
F
D
D
N
e
[
H
curl
H
]
D
(
H
r
)
H
(4.160)
4
π
4
π
N
e
We now substitute (4.160) into the Euler equation and assume that the drift velocity
of the electrons is considerably smaller than their thermal velocity. Hence, we can
ignore the term (
w
e
r
)
w
e
compared with the term
r
p
/(
mN
e
), and obtain
p
H
2
8
(
H
r
)
H
C
D
0 .
(4.161)
r
π
4
π
The quantity
H
2
8
p
H
D
(4.162)
π
is the magnetic field pressure or magnetic pressure; it is the pressure that the
magnetic field exerts on the plasma.
