Fundamentals of Plasma Physics

6.8 Resonance cones 207

oscillating

pointsource

resonance cone

‘infinite’ potential

on thissurface

B

Figure 6.4: Resonance cone excited by oscillating point source in magnetized plasma.

Resonance cones exist in the following regions of parameter space

(i) Region 3, where they are called upper hybrid resonance cones;hereS< 0 ,P> 0 ,

(ii) Regions 7 and 8, where they are a limiting form of the whistler wave and are also called

lower hybrid resonance cones since they are affected by the lower hybrid resonance

(Bellan and Porkolab 1975). Forωci<<ω << ωpe,ωcetheP andSdielectric

tensor elements become

P≃−

ω^2 pe

ω^2

, S≃ 1 −

ω^2 pi

ω^2

+

ω^2 pe

ω^2 ce

(6.116)

so that the cone angleθcone=tan−^1 r/zis

θcone=tan−^1

√

−

S

P

=tan−^1

√

(ω^2 −ω^2 lh)

(

ω−pe^2 +ω−ce^2

)

. (6.117)

Ifω>>ωlh, the cone depends mainly on the smaller ofωpe,ωce.For example, if

ωce<<ωpethen the cone angle is simply

θcone≃tan−^1 ω/ωce (6.118)

whereas ifωce>>ωpethen

θcone≃tan−^1 ω/ωpe. (6.119)

For low density plasmas this last expression can be used as the basis for asimple,

accurate plasma density diagnostic.

(iii) Regions 10 and 13. The Alfvén resonance cones in region 13 have a cone angle

θcone = ω/

√

|ωceωci|and are associated with the electrostatic limit of inertial

Alfvén waves (Stasiewicz, Bellan, Chaston, Kletzing, Lysak, Maggs,Pokhotelov,

Seyler, Shukla, Stenflo, Streltsov and Wahlund 2000). To the best of the author’s

knowledge, cones have not been investigated in region 10 which corresponds to an

unusual mix of parameters, namelyωpeis the same order of magnitude asωci.