Download Two stream instability 1 Consider two beams of electrons, each with

Magnetic Confinement Fusion - Two stream instability
1
Consider two beams of electrons, each with density n0 , travelling in opposite directions
with velocity ±v0 through a background of cold (stationary) ions of density 2n0 . In
equilibrium there is no net charge, no current and no magnetic field. The beams are wide
enough that the only important direction is parallel to the beams (x). This problem tests
the stability of this situation to electrostatic perturbations by treating each electron beam
as a fluid:
∂np,m
+ vp,m · ∇np,m = −np,m ∇ · vp,m
∂t
!
me
∂vp,m
+ vp,m · ∇p,m vp,m
∂t
= −eE
where the p, m subscripts refer to the fluid with initial velocity plus and minus v0 .
1. Linearise these equations with a perturbation of the form exp (−iωt + ikx), and
show that the velocity perturbation is given by
ṽp,m = −
e2
(ñp + ñm ) / (±kv0 − ω)
kme 0
2. By substituting in expressions for ñp and ñm , show that
e2 n0
1
1
1=
2 +
me 0 (kv0 − ω)
(kv0 + ω)2
"
#
Hint: write vm = −vp (kv0 − ω) / (kv0 + ω)
3. What happens when v0 = 0?
4. Assuming ω 2 is real, find the range of k which is unstable by considering when
ω 2 = 0.