Fundamentals of Plasma Physics

418 Chapter 14. Wave-particle nonlinearities

14.3.4The self-consistent non-linear Vlasov-Poisson problem

These ideas carry over into the Vlasov-Poisson analysis of plasma waves, but several ad-
ditional issues occur which complicate and obscure matters. First, the problem takes place
in phase-space so instead of having a pair of coupled equations for velocity and density,
there is only the Vlasov equation. The Vlasov equation not only contains a convective
term analogous to thefluid convective terms but also incorporates an acceleration term
which describes how the velocity distribution function is modified when particles undergo
acceleration and change their velocity. As in thefluid equations there is a coupling with
Poisson’s equation. This coupling not only provides the self-consistent interaction giv-
ing plasma waves, but it also provides for Landau damping. Landau damping is essen-
tially a phase mixing of the Fourier-like driven terms, each of which scales as

∫

dv(ω−
kv)−^1 exp(−iωt)∂f 0 /∂v. However, linear ballistic terms must also be excited in order to
satisfy initial conditions. These ballistic terms scale as

∫

dv(ω−kv)−^1 exp(−ikvt)∂f 0 /∂v
and also phase mix away because of theexp(−ikvt)factor.
If there are two successive pulses, then the non-linear ballistic termwill again have a sta-
tionary phase (i.e., phase independent of velocity) at the special time given byEq.(14.108).
This time could be arranged to be long after the linear plasma responsesto the two pulses
have Landau damped away so that the echo would seem to appear from nowhere. Thus,
the Vlasov-Poisson analysis contains essentially similar echo physics, but in addition has a
self-consistent treatment of the plasma waves and their associated Landau damping.
To proceed with the Vlasov-Poisson analysis, we begin by considering Eq.(14.7) again,
which is rewritten below for convenience

∂

∂t

(f 0 +f 1 +f 2 +...)+v

∂

∂x

(f 0 +f 1 +f 2 +...)

−

e

m

(E 0 +E 1 +E 2 +..)

∂

∂v

(f 0 +f 1 +f 2 +...) = 0. (14.109)

Equilibrium is defined as the solution obtained by balancing all the zeroth orderterms,
i.e.,
∂f 0
∂t

+v

∂f 0

∂x

−

e

m

E 0

∂f 0

∂v

=0. (14.110)

This is trivially satisfied by havingE 0 =0,∂f 0 /∂t=0,∂f 0 /∂x=0andf 0 (v)arbitrary;
we will make these assumptions here. This equilibrium solution is then subtracted from
Eq.(14.109) leaving

∂

∂t

(f 1 +f 2 +..)+v

∂

∂x

(f 1 +f 2 +..)−

e

m

E 1

∂

∂v

(f 0 +f 1 +f 2 +..)

−

e

m

(E 2 +..)

∂

∂v

(f 0 +f 1 +..) = 0.

(14.111)

The first-order solution is defined as the solution to

∂

∂t

f 1 +v

∂f 1

∂x

−

e

m

E 1

∂f 0

∂v

=0, (14.112)

the equation obtained by retaining all the first-order terms. The first-order solution is then
subtracted from Eq.(14.111) and what remains are second and higher order terms. Drop-