408 Chapter 14. Wave-particle nonlinearities
while the non-resonant portion can be written as a principle-part integral.Thus, Eq.(14.54)
becomes
∂WP
∂t=−ω^2 p∫
dk 2 E(k,t)[
P
∫
dvvωi(k)
(ωr−kv)^2∂f ̄ 0
∂v+πωr
k^2(
∂f ̄ 0
∂v)
v=ωr/k]
.
(14.56)
A relationship between the principle part and resonant terms in this expression can
be constructed by similarly decomposing Eq.(14.42) into a resonant portion and anon-
resonant or principle part. The principle part is Taylor expanded as a function ofωr+iωi
whereωiis assumed to be much smaller thanωr.This procedure is essentially the scheme
discussed in the development of Eq.(5.81), but to emphasize the details the expansion of
the principle part is written explicitly here. Thus, the linear dispersion relation Eq.(14.42)
can be expanded as
0 = 1−
ω^2 p
k^2∫
dv∂f ̄ 0 /∂vv−ωr
k−
iωi
k= 1−
ω^2 p
k^2
P
∫
dv
∂f ̄ 0 /∂vv−ωr
k−
iωi
k+iπ(
∂f ̄ 0
∂v)
v=ωr/k
= 1−
ω^2 p
k^2
P
∫
dv∂f ̄ 0 /∂v
v−ωr
k+
iωi
k∂
∂(
ω
k)
P
∫
dv∂f ̄ 0 /∂v
v−ω
k+iπ(
∂f ̄ 0
∂v)
v=ωr/k
= 1−
ω^2 p
k^2
P
∫
dv∂f ̄ 0 /∂v
v−ωr
k+
iωi
kP
∫
dv∂f ̄ 0 /∂v
(
v−
ω
k) 2 +iπ(
∂f ̄ 0
∂v)
v=ωr/k
.
(14.57)
The imaginary part of the last line must vanish and so
ωi
kP
∫
dv
∂f ̄ 0 /∂v
(v−ωr/k)^2+π(
∂f ̄ 0
∂v)
v=ωr/k=0 (14.58)
which leads to the usual expression for Landau damping.
If it is assumed thatωr/k>>vTthen the principle part integral in Eq. (3.238) can be
approximated as
P
∫
dv
v
(ωr−kv)^2∂f ̄ 0
∂v≃
1
ω^2 r∫
dvv
∂f ̄ 0
∂v=−
1
ω^2 r∫
dvf ̄ 0 =−1
ω^2 r