Fundamentals of Plasma Physics

176 Chapter 5. Streaming instabilities and the Landau problem

the dummy variable of integrationkby−kso thatdk→−dkand the±∞limits of

integration are also interchanged).

4. Plot the real and imaginary parts of the plasma dispersion function. Plotthe real and
   imaginary parts of the susceptibilities.


5. Is it possible to have electrostatic plasma waves withkλDe>> 1. Hint, consider
   Landau damping.


6. Plot the potential versus time in units of the real period of an electron plasma wave for
   various values ofω/k

√

κTe/meshowing the onset of Landau damping.

7. Plotωi/ωrfor ion acoustic waves for various values ofTe/Tiand show that these
   waves have strong Landau damping when the ion temperature approaches the electron
   temperature.


8. Landau instability for ion acoustic waves- Plasmas withTe>Tisupport propagation
   of ion acoustic waves;these waves are Landau damped by both electrons and ions.
   However, if there is a sufficiently strong currentJflowing in the plasma giving a
   relative streaming velocityu 0 =J/nebetween the ions and electrons, the Landau
   damping can operate in reverse, and give a Landau growth. This can be seen by
   moving to the ion frame in which case the electrons appear as an offset Gaussian. If
   the offset is large enough it will be possible to have[∂fe/∂v]v∼cs> 0 , giving more
   fast than slow particles at the wave phase velocity. Now, sincefeis a Gaussian with
   its center shifted to be atu 0 ,show that ifu 0 >ω/kthe portion offeimmediately
   to the left ofu 0 will have positive slope and so lead to instability. These qualitative
   ideas can easily be made quantitative, by considering a 1-D equilibrium where the ion
   distribution is
   fi 0 =

n 0

π^1 /^2 vTi

e−v

(^2) /vTi 2
and the drifting electron distribution is
fe 0 =
n 0
π^1 /^2 vTe
e−(v−u^0 )
(^2) /v (^2) Te
.
The ion susceptibility will be the same as before, but to determine the electron sus-
ceptibility we must reconsider the linearized Vlasov equation
∂fe 1
∂t
+v
∂fe 1
x

−

qe

me

∂φ 1

∂x

∂

∂v

[

n 0

π^1 /^2 vTe

e−(v−u^0 )

(^2) /vTe 2

]

=0.

This equation can be simplified by definingv′ =v−u 0. Show that the electron

susceptibility becomes

χe=

1

k^2 λ^2 De

[1+αZ(α)]

where nowα=(ω−ku 0 )/kvTe.SupposeTe>>Tiso that the electron Landau

damping term dominates. Show that ifu 0 >ωr/kthe electron imaginary term will

reverse sign and give instability.

9. Suppose a currentIflows in a long cylindrical plasma of radiusa, densityn,ion mass
   mifor whichTe>>Ti.Write a criterion for ion acoustic instability in terms of an