8 Vlasov theory of warm electrostatic waves in a magnetized plasma
8.1 Uniform plasma
It has been tacitly assumed until now that the wave phase experienced by a particle is
just what would have been experienced if the particle had not deviated from its initial
positionx 0 .This means that the particle trajectory used when determining the wave phase
experienced by the particle isx=x 0 instead of the actual trajectoryx=x(t).Thus the
wave phase seen by the particle was approximated as
k·x(t)−ωt=k·x 0 −ωt. (8.1)This approximation is fine provided the deviation of the actual trajectory from the assumed
trajectory satisfies the condition
|k·(x(t)−x 0 )|<<π/ 2 (8.2)so any phase error due to the deviation is insignificant. Two situations exist where this
assumption fails:
- the wave amplitude is so large that the particle displacement due to the wave is signif-
icant compared to a wavelength, - the wave amplitude is small, but the particle has a large initial velocity so that it moves
substantially during one wave period.
The first case results in chaotic particle motion as discussed in Sec.3.7.3 while the
second case, the subject of this chapter, occurs when the particles have significant thermal
motion. If the motion is parallel to the magnetic field, significant thermal motion means
thatvTis non-negligible compared toω/k‖,a regime already discussed in Sec.5.2 for
unmagnetized plasmas. Thermal motion in the perpendicular direction becomes an issue
whenk⊥rL∼π/ 2 ,i.e., when the Larmor orbitrLbecomes comparable to the wavelength.
In this situation, the particle samples different phases of the wave as the particle traces out
its Larmor orbit. The subscript‖is used here to denote the direction along the magnetic
field. If the magnetic field is straight and given byB=Bzˆ,‖would simply be thez
direction, but in a more general situation the‖component of a vector would be obtained
by dotting the vector withB.ˆ
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