Fundamentals of Plasma Physics

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

E
k,t

(a) t
 0

k

kmin kmax

resonant

particles

initial

w ave

energy

spectrum

nonresonant

particles

f 0 
v

v

(b) t→


E
k,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)