Fundamentals of Plasma Physics

200 Chapter 6. Cold plasma waves in a magnetized plasma

suffices to keep only the leading term ofΓ.Thus, in this limit

Γ≃ 2

∣

∣

∣

∣

∣

ωce

ω

(

1 −

ω^2 pe

ω^2

)

cosθ

∣

∣

∣

∣

∣

=− 2

(

1 −

ω^2 pe

ω^2

)

∣

∣

∣

ωce

ω

cosθ

∣

∣

∣ (6.77)

sinceP=1−ω^2 pe/ω^2 is assumed to be negative. Upon substitution forΓin Eq.(6.71) and
then simplifying one obtains

n^2 +=1−

ω^2 pe/ω^2

1 −

∣

∣

∣

ωce

ω

cosθ

∣

∣

∣

, QLRmode (6.78)

and

n^2 −=1−

ω^2 pe/ω^2

1+

∣

∣

∣

ωce

ω

cosθ

∣

∣

∣

, QLLmode. (6.79)

These simplified dispersions are based on the implicit assumption that|P|is large, because
ifP→ 0 the presumption that Eq.(6.77) gives the leading term inΓwould be inappropriate.
Whenω<|ωcecosθ|the QLR mode (quasi-longitudinal, right-hand circularly polarized)
is called the whistler or helicon wave. This wave is distinguished by having a descending
whistling tone which shows up at audio frequencies on sensitive amplifiers connected to
long wire antennas. Whistlers may have been heard as early as the late 19thcentury by tele-
phone linesmen installing long telephone lines. They become a subject of some interest in
the trenches of the First World War when German scientist H. Barkhausen heard whistlers
on a sensitive audio receiver while trying to eavesdrop on British military communications;
the origin of these waves was a mystery at that time. After the war Barkhausen (1930) and
Eckersley (1935) proposed that the descending tone was due to a dispersive propagation
such that lower frequencies traveled more slowly, but did not explain the source location
or propagation trajectory. The explanation had to wait over two more decades until Storey
(1953) finally solved the mystery by showing that whistlers were caused by lightning bolts
and identified two main types of propagation. The first type, called a short whistler re-
sulted from a lightning bolt in the opposite hemisphere exciting a wave which propagated
dispersively along the Earth’s magnetic field to the observer. The second type, called a
long whistler, resulted from a lightning bolt in the vicinity of the observer exciting a wave
which propagated dispersively along field lines to the opposite hemisphere, then reflected,
and traveled back along the same path to the observer. The dispersion wouldbe greater in
this round trip situation and also there would be a correlation with a click from the local
lightning bolt. Whistlers are routinely observed by spacecraftflying through the Earth’s
magnetosphere and the magnetospheres of other planets.
The reason for the whistler’s descending tone can be seen by representingeach lightning
bolt as a delta function in time

δ(t)=

1

2 π

∫

e−iωtdω. (6.80)

A lightning bolt therefore launches a very broad frequency spectrum. Because the ionospheric
electron plasma frequency is in the range 10-30 MHz, audio frequenciesare much lower
than the electron plasma frequency, i.e.,ωpe>> ωand so|P|>> 1 .The electron cy-
clotron frequency in the ionosphere is of the order of 1 MHz soωce>>ωalso. Thus, the