400 Chapter 14. Wave-particle nonlinearities
where it is implicit that the magnitude of terms with subscriptnis of orderǫnwhere
ǫ << 1. Att= 0,the termsfnwheren≥ 2 all vanish because the perturbation was
prescribed to bef 1 att=0.Other variables such as the electric field will have some kind
of nonlinear dependence on the distribution function and so, for example, the electric field
will have the form
E=E 0 +E 1 +E 2 +E 3 +...
where it is implicit that thenthterm is of orderǫn.
nonresonant
particles
f 0 v
v
Ek,t
(a) t 0
k
kmin kmax
resonant
particles
initial
w ave
energy
spectrum
nonresonant
particles
f 0 v
v
(b) t→
Ek,t
k
kmin kmax
resonant
particles
final
w ave
energy
spectrum
plateau
“colder”
vresmin /kmax vresmax/kmin
vresmin /kmax vresmax /kmin
Figure 14.1: (a) Att= 0the equilibrium distribution functionf 0 (v)is monotonically de-
creasing resulting in Landau damping of any waves and there is a wave spectrum (insert)
with wave energy in the spectral rangekmin<k<kmax.Resonant particles are shown as
shaded in distribution function and lie in velocity rangevresmin<v<vresmax.(b) Ast→∞
the resonant particles develop a plateau (corresponding to absorbing energy from the wave),
the wave spectrum goes to zero, and the non-resonant particles appear to become colder.
Since by assumption,f 0 (v,t)does not depend on position, it is convenient to define a
velocity-normalized order zero distribution function
f 0 (v,t)=n 0 f ̄ 0 (v,t) (14.4)