Waves, Instabilities, and Structures in Excited and Ionized Gases - Fundamentals of Ionized Gases - page 312

magnetic field (along the unit vector h )andthe xz plane contains the wave vector.
The x and y components of this equation are

2
p

2
p

i ωω i ωω

C k 2 k x j y k 2 j x j x D 0, j y . (5.77)

H c 2 H c 2

ω ω

The determinant of this system of equations must be zero, which leads to the dis-
persion relation

H c 2 kk z
ω

H c 2 k 2 cos

ω D ω D ω

#

. (5.78)

p

ω

p

Here is the angle between the direction of wave propagation and the external

magnetic field.
Usually whistlers are considered as electromagnetic waves excited by atmospher-
ic lightning and propagating in a magnetized plasma of the atmosphere with fre-
quencies below the electron plasma frequency and electron cyclotron frequency. In
this form whistlers were observed in the Earth's atmosphere long ago [49, 50], and
their nature was understood in the early stage of plasma physics [51]. As is seen, the
whistler frequency is considerably higher than the frequency of Alfvén waves and
magnetic sound. In particular, if the whistler propagates along the magnetic field,
dispersion relation (5.78) gives

#

2
A /

ω D ω ω i H ,where ω A is the frequency of the

Alfvén wave and ω i H is the ion cyclotron frequency. Since we assume ω ω i H ,

this implies that

ω ω ω i H . (5.79)

A

In addition, because ω ω H , the condition ω kc leads to the inequalities

ω kc ω p . (5.80)

Whistlers are determined entirely by the motion of electrons. To examine the
nature of these waves, we note first that the electron motion and the resultant cur-
rent in the magnetized plasma give rise to an electric field according to (4.157).
This electric field, in turn, leads to an electron current according to (5.75). In
the end, the whistler oscillations are generated. Note that because the dispersion
relation has the d ep endence k 2 , the group velocity of these oscillations, ω

p ω

, increases with the wave frequency. This leads to an identi-
fying characteristic of the received signal as a short-time signal with a wide band
of frequencies. Decrease of the tone of such a signal with time is the reason for
the name “whistler”. This character of whistlers as electromagnetic signals allowed
these waves to be detected long ago [49, 50] as a result of atmospheric excitation by
lightning and allows them to be separated from other types of waves. This gave the
possibility to ascertain reliably the nature of whistlers [51].
The polarization of a whistler propagating along the magnetic field can be found
from dispersion relation (5.77) together with dispersion relation (5.78). The result
is

D @ ω / @ k v g

j y D ij x , j x D ij y . (5.81)