Fundamentals of Plasma Physics

8.3 Bernstein waves 237

(a) (b)

1

2

3

4

c

p^2


c^2
p^2 
c^2

vgr  0 vgr
0

uh


c

1

2

3

4

c

Figure 8.1: Electron Bernstein wave dispersion relations

The Bernstein wave dispersion relation, Eq. (8.37), provides a transcendental relation
betweenωandk⊥and for any givenk⊥has an infinite number of rootsω.This is because

asωis varied, the factor

(

ω^2 −n^2 ω^2 cσ

)− 1

in each term of the summation takes on all values
from−∞to∞so that there are an infinite number of ways for the right hand side to equal
the left hand side. In particular, it should be noted that the right hand side becomes large
and positive wheneverωis just slightly larger thannωcσfor anyn.
We now consider electron Bernstein waves and so drop the ion terms from thesumma-
tion (the analysis for ion Bernstein waves is similar). The subscriptσwill now be dropped
and it will be understood that all quantities refer to electrons. On keepingthe electron
terms only, Eq.(8.37) reduces to

1 = 2

e−λ

λ

∑∞

n=1

n^2 ω^2 p

ω^2 −n^2 ω^2 c

In(λ) (8.40)

which has the following limiting forms whenω^2 p<<ω^2 cand vice versa.

1. ω^2 p<<ω^2 ccase:
   Here the term 2 n^2 ω^2 p/(ω^2 −n^2 ω^2 c)is negligible compared to unity except whenω^2 ∼
   O(n^2 ω^2 ce).Thus, for a givenωonly one term in the summation is near resonance and
   the dispersion relation is satisfied by this one term on the right hand side ofEq.(8.40)
   balancing the left hand side. This results in a spectrum waves that are slightly above
   the cyclotron harmonics, namely

ω^2 =n^2 ω^2 c+ 2

e−λ

λ

n^2 ω^2 pIn(λ), forn= 1, 2 ,...,∞. (8.41)