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Let us consider wave propagation, though a warm
plasma, parallel to the equilibrium magnetic field. For parallel propagation,
, and, hence, from Eq. (1065),
.
Making use of the asymptotic expansion (1069), the matrix
simplifies to

(1073) 
where, again, the only nonzero contributions are from and .
The dispersion relation can be written [see Eq. (459)]

(1074) 
where
and
. Here,
is
the species thermal velocity.
The first root of Eq. (1074) is

(1078) 
with the eigenvector .
Here, use has been made of Eq. (1024).
This root evidentially corresponds to
a longitudinal, electrostatic plasma wave. In fact, it is easily
demonstrated that Eq. (1078) is equivalent to the dispersion relation
(1034) that we found earlier for electrostatic
plasma waves, for the special case in which the distribution
functions are Maxwellians. Recall, from Sects. 6.36.5, that the
electrostatic wave described by the above expression is subject to
significant damping whenever the argument of the plasma dispersion
function becomes less than or comparable with unity: i.e., whenever
.
The second and third roots of Eq. (1074) are

(1079) 
with the eigenvector
, and

(1080) 
with the eigenvector
. The former root evidently
corresponds to a righthanded circularly polarized wave, whereas the latter
root corresponds to a lefthanded circularly polarized wave.
The above two dispersion relations are essentially the same as the corresponding
fluid dispersion relations, (538) and (539), except that they explicitly
contain collisionless damping at the cyclotron resonances. As before, the
damping is significant whenever the arguments of the plasma dispersion functions
are less than or of order unity. This corresponds to

(1081) 
for the righthanded wave, and

(1082) 
for the lefthanded wave.
The collisionless cyclotron damping mechanism is very similar to the
Landau damping mechanism for longitudinal waves discussed in Sect. 6.3.
In this case, the resonant particles are those which gyrate about the magnetic
field with approximately the same angular frequency as the wave electric field.
On average, particles which gyrate slightly faster than the wave lose energy, whereas those which gyrate slightly slower than the wave gain
energy. In a Maxwellian distribution there are less particles in the former
class than the latter, so there is a net transfer of energy from
the wave to the resonant particles. Note that in kinetic theory
the cyclotron resonances
possess a finite width in frequency space (i.e., the incident wave does
not have to oscillate at exactly the cyclotron frequency in order for there
to be an absorption of wave energy by the plasma), unlike in the cold plasma
model, where the resonances possess zero width.
Next: Perpendicular Wave Propagation
Up: Waves in Warm Plasmas
Previous: Waves in Magnetized Plasmas
Richard Fitzpatrick
20110331