Fundamentals of Plasma Physics

14.3 Echoes 417

In analogy to Eq.(14.88) the linearized density can be expressed as

n 1 (t)=

1

2 π

∫

n ̃ 1 (ω,t)dω (14.101)

where againn ̃ 1 (ω,t)is not a Fourier transform becausen ̃ 1 (ω,t)contains a ballistic term
incorporating information about initial conditions. The equation for eachfrequency com-
ponent thus is
∂n ̃ 1
∂t

+ikv 0 n ̃ 1 +ikn 0 v ̃ 1 =0 (14.102)

which may be Laplace transformed to give

(p+ikv 0 ) ̃n 1 +ikn 0 ̃v 1 =0 (14.103)

or, using Eq.(14.95),

n ̃ 1 (p,ω)=

ikn 0 eE/m ̄

(p+iω)(p+ikv 0 )^2

. (14.104)

There is now a second-order pole atp= ikv 0 and so there will be a ballistic term∼
exp(ikv 0 t)associated with the density perturbation. This ballistic term will also phase mix
away if there is a Gaussian velocity distribution of beams.
In order to consider non-linear consequences, we consider the second-order continuity
equation
∂n 2
∂t

+v 0

∂n 2

∂t

+v 1

∂n 1

∂x

+

∂

∂x

(n 1 v 1 )=0 (14.105)

which has inhomogeneous (forcing) terms such asn 1 v 1 involving products of linear quanti-
ties. Suppose that in addition to the original pulse with spatial wavenumberkat timet=0
an additional pulse is also imposed with wavenumberk′at timet=τ.This additional pulse
would introduce ballistic terms having a time dependence∼exp(ik′v 0 (t−τ)).Thus the
nonlinear productn 1 v 1 has a dependence

n 1 v 1 ∼Reexp(ikv 0 t)×Reexp(ik′v 0 (t−τ)) (14.106)

which contains terms of the formexp(ikv 0 t−k′v 0 (t−τ)).In general, this product of
ballistic terms would phase mix away if there were a Gaussian distribution of beams, just
as for the linear ballistic term. However, at the special time given by

kt−k′(t−τ)=0 (14.107)

the phase of the nonlinear ballistic term would be zero for all velocities, and so no phase
mixing would occur when the velocity contributions are summed. Thus, at the special time

t=

k′τ

k′−k

(14.108)

the non-linear product is not subject to phase-mixing and the superposition of the non-linear
ballistic terms of a Gaussian distribution of beams gives a macroscopic signal. The time
at which the phase of the non-linear ballistic term becomes stationary can greatly exceed
τand so a macroscopic nonlinear signal would appear long after both initial pulses have
gone. This ghost-like non-linear signal is called thefluid echo.