Fundamentals of Plasma Physics

150 Chapter 5. Streaming instabilities and the Landau problem

and

∇^2 φ 1 =−

1

ε 0

∑

σ

qσnσ 1. (5.5)

As before, all first-order dependent variables are assumed to vary asexp(ik·x−iωt).
Combining the equation of motion and the continuity equation gives

nσ 1 =nσ 0

k^2

(ω−k·uσ 0 )^2

qσ

mσ

φ 1. (5.6)

Substituting this into Eq.(5.5) gives the dispersion relation

1 −

∑

σ

ω^2 pσ

(ω−k·uσ 0 )^2

=0 (5.7)

which is just like the susceptibility for stationary cold species except that hereωis replaced
by the Doppler-shifted frequencyωDoppler=ω−k·uσ 0.
Two examples of streaming instability will now be considered: (i) equal densities of
positrons and electrons streaming past each other with equal and opposite velocities, and
(ii) electrons streaming past stationary ions.
Positron-electron streaming instability
The positron/electron example, while difficult to realize in practice, is worth analyzing
because it reveals the essential features of the instability with aminimum of mathematical
complexity. The equilibrium positron and electron densities are assumed equal so as to
have charge neutrality. Since electrons and positrons have identical mass,the positron
plasma frequencyωppis the same as the electron plasma frequencyωpe. Letu 0 be the
electron stream velocity and−u 0 be the positron stream velocity. Definingz=ω/ωpe
andλ=k·u 0 /ωpe, Eq. (5.7) reduces to

1=

1

(z−λ)^2

+

1

(z+λ)^2

, (5.8)

a quartic equation inz.Because of the symmetry, no odd powers ofzappear and Eq.(5.8)
becomes
z^4 − 2 z^2 (λ^2 +1)+λ^4 − 2 λ^2 =0 (5.9)
which may be solved forz^2 to give

z^2 =(λ^2 +1)±

√

4 λ^2 +1. (5.10)

Each choice of the±sign gives two roots forz.Ifz^2 > 0 then the two roots are real, equal
in magnitude, and opposite in sign. On the other hand, ifz^2 < 0 ,then the two roots are
pure imaginary, equal in magnitude, and opposite in sign. Recalling thatω=ωpezand
that the perturbation varies asexp(ik·x−iωt),it is seen that the positive imaginary root
z=+i|z|corresponds to instability;i.e., to a perturbation whichgrowsexponentially in
time.
Hence the condition for instability isz^2 < 0 .Only the choice of minus sign in Eq.(5.10)
allows this possibility, so choosing this sign, the condition for instability is
√
4 λ^2 +1>λ^2 +1 (5.11)

which corresponds to
0 <λ <

√

2. (5.12)