Environmental Engineering Reference
In-Depth Information
case the magnetic lines of force are frozen in the plasma [31], and a change in
the plasma current causes a change in the magnetic lines of force, which acts in
opposition to this current and creates oscillations of the plasma together with the
magnetic field [32-34]. Such oscillations are called magnetohydrodynamic waves.
For magnetohydrodynamic waves with wavelengths less than the radius of cur-
vature of the magnetic lines of force, we have
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
H
r
1
k
,
(5.65)
H
where
H
is the magnetic field strength. Then one can consider the magnetic lines
of force to be straight lines. We construct a simple model of oscillations of a high-
conductivity plasma, where the magnetic lines of force are frozen in the plasma.
The displacement of the magnetic lines of force causes a plasma displacement,
and because of the plasma elasticity, these motions are oscillations. The velocity of
propa
gation
of this oscillation is, according to dispersion relation (5.52), given by
c
D
p
@
MN
is the plasma density, and
M
is the
ion mass. Because the pressure of a cold plasma is equal to the magnetic pressure
p
p
/
@
,where
p
is the pressure,
D
H
2
/(8
D
π
), we have
r
H
@
H
/
@
N
c
D
M
for the velocity of wave propagation. Since the magnetic lines of force are frozen in
the plasma,
4
π
@
H
/
@
N
D
H
/
N
. This gives [32-34]
H
c
D
c
A
D
p
4
(5.66)
π
MN
for the velocity of these waves.
c
A
is called the Alfvén speed.
The oscillations being examined may be of two types depending on the direction
of wave propagation, as shown in Figure 5.9 [35]. If the wave propagates along
the magnetic lines of force, it is called an Alfvén wave or magnetohydrodynamic
wave. This wave is analogous to a wave propagating along an elastic string. The
other wave type propagates perpendicular to the magnetic lines of force. Then the
vibration of one magnetic line of force causes the vibration of a neighboring line.
Such waves are called magnetic sound. The dispersion relation for both types of
oscillations has the form
ω
D
c
A
k
.
The oscillations in a magnetic field are of importance for the Sun's plasma [36].
There are various types of waves in a plasma in a magnetic field [37].
5.2.5
Propagation of Electromagnetic Waves in a Plasma
We shall now derive the dispersion relation for an electromagnetic wave propagat-
ing in a plasma. Plasma motion due to an electromagnetic field influences the wave
