5 Streaming instabilities and the Landau problem
5.1 Streaming instabilities
The electrostatic dispersion relation for a zero-temperature plasma is simply
1 −
∑
σω^2 pσ
ω^2=0 (5.1)
indicating that aspatially-independentoscillation at the plasma frequency
ωp=√
ω^2 pe+ω^2 pi (5.2)is a normal mode of a cold plasma. Once started, such an oscillation wouldpersist in-
definitely because no dissipative mechanism exists to quench it. On the other hand, the
oscillation would have to be initiated by some source, because no availablefree energy
exists from which the oscillation could draw to start spontaneously.
We now consider a slightly different situation where instead of being at rest in equilib-
rium, cold electrons or ions stream at some spatially-uniform initial velocity. In the special
situation where electrons and ions have the same initial velocity, the center of mass would
also move at this initial velocity and one could simply move to the centerof mass frame
where both species are stationary and so, as argued in the previous paragraph, an oscillation
would not start spontaneously. However, in the more general situation where the electrons
and ions stream atdifferentvelocities, then both species have kinetic energy in the center
of mass frame. This free energy could drive an instability.
In order to determine the conditions where such an instability could occur, thesituation
where each species has the equilibrium streaming velocityuσ 0 will now be examined. The
linearized equation of motion, the linearized continuity equation, and Poisson’s equation
become respectively
∂uσ 1
∂t+uσ 0 ·∇uσ 1 =−
qσ
mσ∇φ 1 , (5.3)∂nσ 1
∂t+uσ 0 ·∇nσ 1 =−nσ 0 ∇·uσ 1 , (5.4)