Fundamentals of Plasma Physics

186 Chapter 6. Cold plasma waves in a magnetized plasma

and again two distinct modes appear. The first mode has as its eigenvector the condition
thatEz =0.The associated eigenvalue equation isP−n^2 =0or

ω^2 =k^2 c^2 +

∑

σ

ω^2 pσ (6.36)

which is just the dispersion for an electromagnetic plasma wave in an unmagnetized plasma.
This is in accordance with the prediction that modes involving particlemotion strictly par-
allel to the magnetic field are unaffected by the magnetic field. This mode is called the
ordinarymode because it is unaffected by the magnetic field.
The second mode involves bothExandEyand has the eigenvalue equationS(S−
n^2 )−D^2 =0which gives the dispersion relation

n^2 =

S^2 −D^2

S

=2

RL

R+L

. (6.37)

Cutoffs occur here when eitherR=0orL=0and a resonance occurs whenS=0.Since
this mode depends on the magnetic field, it is called theextraordinarymode.TheS=0
resonance is called a hybrid resonance because it depends on a hybrid ofω^2 cσandω^2 pσ
terms (note thatω^2 cσterms depend on single-particle physics whereasω^2 pσterms depend
on collective motion physics). BecauseSis quadratic inω^2 , the equationS=0has two
distinct roots and these are found by explicitly writing

S=1−

ω^2 pi

ω^2 −ω^2 ci

−

ω^2 pe

ω^2 −ω^2 ce

=0. (6.38)

A plot of this expression shows that the two roots are well separated. The large root may
be found by assuming thatω∼O(ωce)in which case the ion term becomes insignificant.
Dropping the ion term shows that the large root ofSis simply

ω^2 uh=ω^2 pe+ω^2 ce (6.39)

which is called theupper hybrid frequency. The small root may be found by assuming that
ω^2 <<ω^2 cewhich gives thelower hybrid frequency

ω^2 lh=ω^2 ci+

ω^2 pi

1+

ω^2 pe

ω^2 ce

. (6.40)

6.2.4 Very low frequency modes whereθis arbitrary

Equation (6.31) shows that forω<<ωci

S ≃ R≃L≃1+c^2 /vA^2

D ≃ 0 (6.41)

so the cold plasma dispersion simplifies to



S−n^2 z 0 nxnz

0 S−n^20

nxnz 0 P−n^2 x



·





Ex

Ey

Ez



=0. (6.42)