Fundamentals of Plasma Physics

238 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma

The small and largeλlimits of this expression are determined using the asymptotic

values of theInBessel function, namely forn≥ 1 ,

lim

λ<< 1

In(λ) =

1

n!

(

λ

2

)n

lim

λ>> 1

In(λ) = eλ

(8.42)

to give

ω^2 =n^2 ω^2 c+ (1−λ)n^2 ω^2 p

1

n!

(

λ

2

)n− 1

forλ<< 1 ,

ω^2 =n^2 ω^2 cforλ>> 1.

(8.43)

Then= 1mode differs slightly from then> 1 modes because then= 1mode has

the smallλdispersion

ω^2 =ω^2 uh−λω^2 p (8.44)

showing that this mode is the warm plasma generalization of the upper hybridreso-

nance. The modes for whichn> 1 start atω^2 =n^2 ω^2 cwhenλ= 0, have a maximum

frequency at some finiteλ, and then revert toω^2 =n^2 ω^2 casλ→∞. To the left

of the frequency maximum the group velocity∂ω/∂k⊥is positive and to the right of

the maximum the group velocity is negative. This system of modes is sketchedin

Fig.8.1(a).

2. ω^2 p>>ω^2 c
   For largeλ,the product of the exponential factor and the modified Bessel function
   in Eq.(8.40) is unity, leaving one more factor ofλin the denominator, so again the
   dispersion reduces to cyclotron harmonics,ω^2 =n^2 ω^2 casλ→∞. For smallλthe
   situation is more involved because to lowest order, then= 1term is independent of
   λand so must always be retained when approximating Eq.(8.40). Let us suppose that
   some other term, say the(n+ 1)thterm is near resonance, i.e.,ω^2 ∼(n+ 1)^2 ω^2 c.
   Then, keeping the left hand side, then= 1term and the(n+1)thterm, expansion of
   Eq.(8.40) results in

1 ≃

ω^2 p

[

(n+ 1)^2 − 1

]

ω^2 c

+

(n+ 1)ω^2 p

[

ω^2 −(n+ 1)^2 ω^2 c

]

n!

(

λ

2

)n

(8.45)

which may be solved forωto give

ω= (n+ 1)ωc







1 −

ω^2 p

[

(n+ 1)^2 − 1

]

(

ω^2 p−

[

(n+ 1)^2 − 1

]

ω^2 c

)

λn

2 n+1(n+ 1)!







. (8.46)

Thus, for smallλthenthmode starts atω≃(n+ 1)ωcand then asλ→∞the
frequency decreases towards the asymptotic limitnωc.This dispersion is sketched in
Fig.8.1(b).
Excitation of Bernstein waves requiresk‖= 0which is a quite stringent requirement.
The antenna must be absolutely uniform in the direction along the magnetic field because
otherwise a finitek‖will be excited which will result in the waves being subject to strong
Landau damping.