Fundamentals of Plasma Physics

15.5 Digging a hole in the plasma via ponderomotive force 451

Caviton instability
The instability outlined above can be described in a quantitative manner usingthe 1-D
version of Eq.(15.129), namely

i

∂χ

∂τ

+iηχ+|χ|^2 χ+

∂^2 χ

∂ξ^2

=0. (15.133)

It is assumed that a stable solutionχ 0 (x,t)exists initially and satisfies

i

∂χ 0

∂τ

+iηχ 0 +|χ 0 |^2 χ 0 +

∂^2 χ 0

∂ξ^2

=0 (15.134)

where|χ 0 (x,t)|is bounded in both time and space. Next, a slightly different solution is
considered,
χ(x,t)=χ 0 (x,t)+ ̃χ(x,t) (15.135)
where the perturbation ̃χ(x,t)is assumed to be very small compared toχ 0 (x,t).The equa-
tion forχ(x,t)is thus

i

∂

∂τ

(χ 0 + ̃χ)+iη(χ 0 + ̃χ)+|χ 0 + ̃χ|^2 (χ 0 + ̃χ)+

∂^2

∂ξ^2

(χ 0 + ̃χ)=0. (15.136)

Subtracting Eq.(15.134) from (15.136) yields

i

∂

∂τ

̃χ+iηχ ̃+|χ 0 + ̃χ|^2 (χ 0 + ̃χ)−|χ 0 |^2 χ 0 +

∂^2

∂ξ^2

̃χ=0. (15.137)

Expansion of the potential-energy-like terms while keeping only terms linearin the
perturbation gives

|χ 0 + ̃χ|^2 (χ 0 + ̃χ)−|χ 0 |^2 χ 0 ≈χ^20 χ ̃∗+2|χ 0 |^2 χ ̃ (15.138)

so Eq.(15.137) becomes

i

∂

∂τ

χ ̃+iηχ ̃+χ^20 ̃χ∗+2|χ 0 |^2 ̃χ+

∂^2

∂ξ^2

̃χ=0. (15.139)

It is now assumed that the perturbation is unstable and has the space-time dependence

̃χ∼eikξ+γt (15.140)

in which case Eq.(15.139) becomes
(
iγ+iη+2|χ 0 |^2 −k^2

)

̃χ=−χ^20 ̃χ∗ (15.141)

which has the complex conjugate
(
−iγ−iη+2|χ 0 |^2 −k^2

)

χ ̃∗ =−χ∗ 02 ̃χ. (15.142)

Combining the above two equations gives a dispersion relation for the growth rate

(γ+η)^2 =−k^4 +4k^2 |χ 0 |^2 − 3 |χ 0 |^4. (15.143)