Fundamentals of Plasma Physics

5.1 Streaming instabilities 151

The maximum growth rate is found by maximizing the right hand side of Eq.(5.10) with
the minus sign chosen. Taking the derivative with respect toλand settingdz/dλ=0to
find the maximum, gives

2 z

dz

dλ

=2λ−

4 λ

√

4 λ^2 +1

=0

orλ=

√

3 / 2 .Substituting this most unstableλback into Eq.(5.10) (with the minus sign,
since this is the potentially unstable root) gives the maximum growth rate to bey=1/ 2
wherez=x+iy.
Changing back to physical variables, it is seen that onset of instability occurs when

ku 0 <

√

2 ωpe,

and the maximum growth rate occurs when

ku 0 =

√

3

2

ωpe

in which case
ω=i

ωpe

2

.

Figure 5.1 plots the normalized instability growth rate Imzas a function ofλ;both on-
set and maximum growth rate are indicated. Since the instability has a pure imaginary
frequency it is called a purely growing mode. Because the growth rate is ofthe order of
magnitude of the plasma frequency, the instability grows extremely quickly.

ymax
0. 5

y

0

0. 5

1. 0

(^31). 0 2.^0
2
2
Figure 5.1: Normalized growth rate v. normalized wavenumber
Electron-ion streaming instability