Fundamentals of Plasma Physics

204 Chapter 6. Cold plasma waves in a magnetized plasma

verse parts

El = ˆnnˆ·E

Et = E−El (6.95)

whereˆn=kˆ=n/nis a unit vector in the direction ofn.The cold plasma wave equation,
Eq. (6.17) can thus be written as

nn·

(

Et+El

)

−n^2

(

Et+El

)

+

←→

K·

(

Et+El

)

=0. (6.96)

Sincen·Et=0andnn·El=n^2 Elthis expression can be recast as

(←→

K−n^2

←→

I

)

·Et+

←→

K·El=0 (6.97)

where

←→

I is the unit tensor. If the resonance is such that

n^2 >>Kij (6.98)

whereKijare the elements of the dielectric tensor, then Eq.(6.97) may be approximated
as
−n^2 Et+

←→

K·El≃ 0. (6.99)

This shows that the transverse electric field is

Et=

1

n^2

←→

K·El (6.100)

which is much smaller in magnitude than the longitudinal electric field by virtue of Eq.(6.98).
An easy way to obtain the dispersion relation (determinant of this system) is to dot
Eq.(6.100) withnto obtain

n·

←→

K·n=n^2 (Ssin^2 θ+Pcos^2 θ)≃ 0 (6.101)

which is just the first case discussed above. This argument is self-consistent because for
the first case (i.e.,A→ 0 ) the quantitiesS,P,Dremain finite so the condition given by
Eq.(6.98) is satisfied.
The second case,B→∞,occurs at the cyclotron resonances whereSandDdiverge
so the condition given by Eq.(6.98) is not satisfied. Thus, for the second case the electric
field is not quasi-electrostatic.

6.8 Resonance cones

The situationA→ 0 corresponds to Eq.(6.101) which is a dispersion relation having the
surprising property of depending onθ,butnoton the magnitude ofn.This limiting form
of dispersion has some bizarre aspects which will now be examined.
The group velocity in this situation can be evaluated by writing Eq.(6.101)as

k^2 xS+k^2 zP=0 (6.102)