5.2 The Landau problem 171On applying the Taylor expansion technique discussed in conjunction with Eqs.(5.82) and
(5.83) we find thatωris the root of
Dr(ωr)=1+1
k^2 λ^2 De−
ω^2 pi
ω^2 r(
1+3
k^2
ω^2κTi
mi)
=0, (5.90)
i.e.,
ω^2 r=k^2 c^2 s
1+k^2 λ^2 De(
1+3
k^2
ω^2 rκTi
mi)
≃
k^2 c^2 s
1+k^2 λ^2 De+3k^2κTi
mi. (5.91)
Here, as in the two-fluid analysis of ion acoustic waves,c^2 s=ω^2 piλ^2 De=κTe/mihas been
defined. The imaginary part of the frequency is found to be
ωi ≃−
Di(ωr)
dDr/dω= −
ωπ^1 /^2
k^3
1
λ^2 DevTeexp(−ω^2 /k^2 v^2 Te)+1
λ^2 DivTiexp(−ω^2 /k^2 vTi^2 )2 ω^2 pi/ω^3
= −
ω^4
k^3 c^3 s√
π
8[√
me
mi+
(
Te
Ti) 3 / 2
exp(−ω^2 /k^2 v^2 Ti)]
= −
|ωr|
(
1+k^2 λ^2 De) 3 / 2
√
π
8[√
me
mi+
(
Te
Ti) 3 / 2
exp(
−
Te/ 2 Ti
1+k^2 λ^2 De−
3
2
)]
.
(5.92)
The dominant Landau damping comes from the ions, since the electron Landau damping
term has the small factor
√
me/mi.IfTe>>Tithe ion term also becomes small because
x^3 /^2 exp(−x)→ 0 asxbecomes large. Hence, strong ion Landau damping occurs when
TiapproachesTeand so ion acoustic waves can only propagate without extreme attenu-
ation if the plasma hasTe>>Ti.Landau damping of ion acoustic waves was observed
experimentally by Wong, Motley and D’Angelo (1964).
5.2.9 The Plemelj formula
The Landau method showed that the correct way to analyze problems that lead to ill-defined
integrals such as Eq.(5.31) is to pose the problem as an initial value problemrather than as
a steady-state situation. The essential result of the Landau method can besummarized by
the Plemelj formula
lim
ε→ 01
ξ−a∓i|ε|=P
1
ξ−a
±iπδ(ξ−a) (5.93)which is a prescription showing how to deal with singular integrands of the form appearing
in the plasma dispersion function. From now on, instead of repeating the lengthy Laplace
transform analysis, we instead will use the less cumbersome, but formally incorrect Fourier
method and then invoke Eq.(5.93) as a ‘patch’ to resolve any ambiguities regarding inte-
gration contours.