412 Chapter 14. Wave-particle nonlinearities
In the range of velocities where∂^2 f 0 ,non−res/∂v^2 > 0 there will be a decrease in the
number of non-resonant particles and vice versa in the range where∂^2 f 0 ,non−res/∂v^2 < 0.
An initially Maxwellian distribution (which has∂^2 f 0 ,non−res/∂v^2 < 0 for very small
velocities and vice versa for very large velocities) will develop a double-plateau shape
(high plateau at low velocities and low plateau at high velocities) with a sharp gradient
between the two plateaus. The lower plateau will merge smoothly with the plateau of the
resonant particles. The non-resonant portion of the velocity distribution will appear to
become colder as the waves damp as indicated in Fig.14.1(b).
This apparent cooling can be quantified by introducing an effective temperatureTeff
which is defined to have the time derivative
d
dt
(κTeff)=2
n 0d
dt∫
dkE(k,t). (14.76)This expression can be integrated to obtain
κTeff(t)=κT 0 +2
n 0∫
dkE(k,t) (14.77)whereT 0 is the temperature for the situation where there are no waves.
Using Eq.(14.76), Eq.(14.73) can be written as
∂
∂tf 0 ,non−res=
d
dt(
κTeff
2 m)
∂^2 f 0 ,non−res
∂v^2. (14.78)
Iff 0 ,non−resis considered a function ofκTeffinstead oft,Eq.(14.78) can be written as
∂
∂(κTeff)f 0 ,non−res=1
2 m∂^2 f 0 ,non−res
∂v^2(14.79)
which has the appropriately normalized solution
f 0 ,non−res = n 0√
m
2 πκTeff(t)
exp(
−
mv^2
2 κTeff(t))
= n 0√√
√
√
√
m/ 2 π
κT 0 +2
n 0∫
dkE(k,t)exp
−
mv^2 / 2
κT 0 +2
n 0∫
dkE(k,t)
.
(14.80)
Thus wave damping [i.e., the reduction ofE(k,t)] corresponds to an effective cooling of
the non-resonant particles. As shown in Eq.(14.64) this kinetic energy reduction is accom-
panied by an equal reduction in the electric field energy and all this energy is transferred to
the resonant particles.
14.3 Echoes x
Plasma wave echoes (Gould, O’Neil and Malmberg 1967, Malmberg, Wharton, Gould and
O’Neil 1968) are a nonlinear effect that provide some very useful insights intothe Landau