168 Chapter 5. Streaming instabilities and the Landau problem
5.2.6 Landau damping of electron plasma waves
The plasma susceptibilities given by Eq.(5.65) can now be evaluated. For|α|>> 1 ,using
Eq.(5.73), and introducing the “frequency”ω=ipso thatα=ω/kvTσandαi=ωi/kvTσ
the susceptibility is seen to be
χσ =
1
k^2 λ^2 Dσ
{
1+α
[
−
1
α
(
1+
1
2 α^2
+
3
4 α^4
+...
)
+iπ^1 /^2 exp(−α^2 )
]}
=
1
k^2 λ^2 Dσ
{
−
(
1
2 α^2
+
3
4 α^4
+...
)
+iαπ^1 /^2 exp(−α^2 )
}
= −
ω^2 pσ
ω^2
(
1+3
k^2
ω^2
κTσ
mσ
+...
)
+i
ω
kvTσ
π^1 /^2
k^2 λ^2 Dσ
exp(−ω^2 /k^2 v^2 Tσ).
(5.76)
Thus, if the root is such that|α|>> 1 ,the equation for the polesD(p)=1+χi+χe=0
becomes
1 −
ω^2 pe
ω^2
(
1+3
k^2
ω^2
κTe
me
+...
)
+i
ω
kvTe
π^1 /^2
k^2 λ^2 De
exp(−ω^2 /k^2 vTe^2 )
−
ω^2 pi
ω^2
(
1+3
k^2
ω^2
κTi
mi
+...
)
+i
ω
kvTi
π^1 /^2
k^2 λ^2 Di
exp(−ω^2 /k^2 vTi^2 )=0. (5.77)
This expression is similar to the previously obtainedfluid dispersion relation, Eq. (4.31),
but contains additional imaginary terms that did not exist in thefluid dispersion. Further-
more, Eq.(5.77) is not actually a dispersion relation. Instead, it is tobe understood as the
equation for the roots ofD(p). These roots determine the poles inN(p)/D(p)producing
the least damped oscillations resulting from some prescribed initialperturbation of the dis-
tribution function. Sinceω^2 pe/ω^2 pi=mi/meand in generalvTi<<vTe,both the real and
imaginary parts of the ion terms are much smaller than the corresponding electron terms.
On dropping the ion terms, the expression becomes
1 −
ω^2 pe
ω^2
(
1+3
k^2
ω^2
κTe
me
+...
)
+i
ω
kvTe
π^1 /^2
k^2 λ^2 De
exp(−ω^2 /k^2 v^2 Te)=0. (5.78)
Recalling thatω=ipis complex, we writeω=ωr+iωiand then proceed to find the
complexωthat is the root of Eq.(5.78). Although it would not be particularly difficultto
simply substituteω=ωr+iωiinto Eq.(5.78) and then manipulate the coupled real and
imaginary parts of this equation to solve forωrandωi,it is better to take this analysis as
an opportunity to introduce a more general way for solving equations of this sort.
Equation (5.78) can be written as
D(ωr+iωi)=Dr(ωr+iωi)+iDi(ωr+iωi)=0 (5.79)
whereDris the part ofDthat does not explicitly containiandDi is the part that does
explicitly containi. Thus
Dr=1−
ω^2 pe
ω^2
(
1+3
k^2
ω^2
κTe
me
+...
)
, Di=
ω
kvTe
π^1 /^2
k^2 λ^2 De
exp(−ω^2 /k^2 vTe^2 ). (5.80)