172 Chapter 5. Streaming instabilities and the Landau problem
5.3 The Penrose criterion
The analysis so far showed that electrostatic plasma waves are subject to Landau damping,
a collisionless attenuation proportional to[∂f/∂v]v=ω/k,and that this damping is con-
sistent with the calculation of power input to particles by an electrostatic wave. Since a
Maxwellian distribution function has a negative slope, its associated Landau damping is al-
ways a true wave damping. This is consistent with the physical picture developed in the
single particle analysis which showed that energy is transferred from wave to particles if
there are more slow than fast particles in the vicinity of the wavephase velocity. What
happens if there is a non-Maxwellian distribution function, in particular one where there
are more fast particles than slow particles in the vicinity of the wave phase velocity, i.e.,
[∂f/∂v]v=ω/k> 0? Becausef(v)→ 0 asv→∞,fcan only have a positive slope for
a finite range of velocity;i.e., positive slopes of the distribution function must always be
located to the left of a localized maximum inf(v).A localized maximum inf(v)corre-
sponds to a beam of fast particles superimposed on a (possible) background of particles
having a monotonically decreasingf(v). Can the Landau damping process be run in re-
verse and so provide Landau growth, i.e., wave instability? The answer isyes. We will
now discuss a criterion due to Penrose (1960) that shows how strong a beam mustbe to
give Landau instability.
The procedure used to derive Eq.(5.28) is repeated, giving
1+
q^2
k^2 mε 0∫∞
−∞k
(ω−kv)∂f 0
∂vdv=0 (5.94)which may be recast as
k^2 =Q(z) (5.95)
where
Q(z)=q^2
mε 0∫∞
−∞1
(v−z)∂f 0
∂vdv (5.96)is a complex function of the complex variablez=ω/k.The wavenumberkis assumed to
be a positive real quantity and the Plemelj formula will be used to resolvethe ambiguity
due to the singularity in the integrand.
The left hand side of Eq.(5.95) is, by assumption, always real and positive for any choice
ofk. A solution of this equation can therefore always be found ifQ(z)is simultaneously
pure real and positive. The actual magnitude ofQ(z)does not matter, since the magnitude
ofk^2 can be adjusted to match the magnitude ofQ(z).
The functionQ(z)may be interpreted as a mapping from the complexzplane to the
complexQplane. Because solutions of Eq.(5.95) giving instability are those for which
Imω> 0 , the upper half of the complexzplane corresponds to instability and the realz
axis represents the dividing line between stability and instability. Letus consider a straight-
line contourCzparallel to the realzaxis, and slightly above. As shown in Fig.5.6(a) this
contour can be prescribed asz=zr+iδwhereδis a small constant andzrranges ranges
from−∞to+∞.
The functionQ(z)→ 0 whenz→±∞and so, aszis moved along theCzcontour,
the corresponding pathCQtraced in theQplane must start at the origin and end at the
origin. Furthermore, sinceQ can be evaluated using the Plemelj formula, it is seen that
Qis finite for allzon the pathCz.Thus,CQis a continuous finite curve starting at the