Fundamentals of Plasma Physics

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)