Fundamentals of Plasma Physics

170 Chapter 5. Streaming instabilities and the Landau problem

also energy in coherent particle motion - to be discussed in Chapter 14), itis seen that the
power being lost from the wave electric field is

Pwavelost∼

d

dt

〈

ε 0 E^2 wave

2

〉

∼

d

dt

[

ε 0 |Ewave^2 |

4

exp(− 2 |ωi|t)

]

=−

|ωi|ε 0 E^2 wave

2

=

√

π

8

ωpe

2 k^3 λ^3 De

exp

(

−ω^2 /k^2 v^2 Tσ

)

ε 0 E^2 wave
(5.86)
where

〈

E^2 wave

〉

=|Ewave|^2 〈cos(kx−ωt)〉=|Ewave|^2 / 2 has been used. However, in
Sec.3.8, it was shown that the energy gained by untrapped resonant particles in a wave is

Ppartgain =

−πmω

2 k^2

(

qEwave

m

) 2 [

d

dv 0

f(v 0 )

]

v 0 =ω/k

=

−πmω

2 k^2

(

qEwave

m

) 2 [

d

dv 0

{(

m

2 πκT

) 1 / 2

n 0 exp

(

−

mv^2

2 κT

)}]

v 0 =ω/k

=

πmω

2 k^2

(

qEwave

m

) (^2) (
m
2 πκT
) 1 / 2 (m
κT
ω
k

)

n 0 exp

(

−

ω^2

k^2 v^2 Tσ

)

;

(5.87)

usingω∼ωpethis is seen to be the same as Eq.(5.86) except for a factor of two. We shall
see later that this factor of two comes from the fact that the wave electric field actually
containshalfthe energy of the electron plasma wave, with the other half in coherent particle
motion, so the true power loss rate is really twice that given in Eq.(5.86).

5.2.8 Landau damping for ion acoustic waves

Ion acoustic waves resulted from a two-fluid analysis in the regime where the wave phase
velocity was intermediate between the electron and ion thermal velocities. In this situation
the electrons behave isothermally and the ions behave adiabatically. This suggests there
might be another root ofD(p)if|αe|<< 1 and|αi|>> 1 or equivalentlyvTi<<
ω/k<<vTe.From Eqs.(5.65) and (5.75), the susceptibility for|α|<< 1 is found to be

χσ =

1

k^2 λ^2 Dσ

[1+αZ(α)]

=

1

k^2 λ^2 Dσ

{

1 − 2 α^2

(

1 −

2 α^2

3

+...

)

+iαπ^1 /^2 exp(−α^2 )

}

≃

1

k^2 λ^2 Dσ

+i

α

k^2 λ^2 Dσ

π^1 /^2 exp(−α^2 ).

(5.88)

Using Eq.(5.88) for the electron susceptibility and Eq.(5.76) for the ion susceptibility
gives

D(ω) = 1+

1

k^2 λ^2 De

+i

ω

kvTe

π^1 /^2

k^2 λ^2 De

exp(−ω^2 /k^2 v^2 Te)

−

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 ).

(5.89)