A beam pulsed amplifier mechanism responsible for effective amplification of short very low frequency (VLF) electromagnetic pulses is proposed. Effective amplification near the magnetic equator outside the plasmasphere is considered. A conditional growth rate of short electromagnetic pulses is calculated. Obtained results can explain some important features of the oblique electromagnetic chorus emissions without hiss-like radiation background.

Very low frequency (VLF) chorus emissions are very intense electromagnetic plasma waves that are naturally excited near the magnetic equatorial plane outside the plasmasphere (Burtis and Helliwell, 1969; Burton and Holzer, 1974; Tsurutani and Smith, 1974). Impressive experimental results of the chorus emission study were obtained within the framework of the CLUSTER project. These results have been presented in detail in many papers (e.g., Santolik, 2009). It is very important that a chorus is a succession of discrete emissions.

For an electromagnetic chorus with the wave vectors predominantly along the magnetic field, significant theoretical results were obtained. The usual kinetic cyclotron instability (Bespalov and Trakhtengerts, 1986) is sometimes possible for plasma parameters in the excitation region, but this instability typically does not have a sufficient growth rate to explain the rate of the chorus emission modification. The theory of chorus excitation based on the so-called backward wave oscillator (BWO) mechanism is now well known (see, e.g., Trakhtengerts, 1995; Trakhtengerts et al., 2007; Nunn et al., 2009). The implementation of this mechanism is closely related to the hiss-like radiation background and its dynamics. However, the chorus emissions are often recorded without the hiss-like radiation background. In Omura et al. (2008) and Fu et al. (2014) the generation process of the chorus emissions is analyzed by both theory and simulation assuming that the initial cyclotron wave growth is driven by the strong temperature anisotropy of energetic electrons. To explain the chorus spectrogram, the authors take into account inhomogeneity of the magnetic field and nonlinear wave decay, or the non-monotonic energy spectrum of particles. At present, there are significant achievements in the theoretical study of generation electrostatic chorus with the wave vectors near the resonance cone (see, e.g., Li et al., 2016a). There are extensive data on the wave normal angle measurements onboard THEMIS (Li et al., 2013) and a Van Allen probe (Li et al., 2016b).

Some problems connected with the theoretical analysis of chorus formation remain unsolved; for example, the excitation mechanism of an oblique electromagnetic chorus has not been studied. In this paper, we introduce a possible mechanism of oblique electromagnetic chorus excitation without hiss-like radiation background. This mechanism is related to the effective amplification of short electromagnetic pulses from the noise level even in a stable plasma. The amplification takes place in a suitable frequency band near the magnetic equator.

We consider small-scale wave processes near the local minimum of the magnetic
field, which typically is close to the magnetic equator where the plasma is
almost homogeneous. We use a linearized Vlasov equation for the disturbed
distribution function of energetic non-relativistic electrons

Let a short VLF electromagnetic pulse propagate in a homogeneous plasma along
the

Now we point out the condition favorable to the existence of a short
electromagnetic pulse that propagates along the magnetic field at a constant
velocity without the additional phase modulation and smallest dispersion
distortion of the pulse front. The solution of this problem is known.
According to electrodynamics (see, e.g., Sommerfeld, 1914; Jacson, 1962) this
takes place under condition

If the ion motion in a relatively dense background plasma is not important,
then the dispersion equation of electromagnetic waves for the frequency band

Note the first key point of our analysis. It is known (Helliwell, 1995) that
according to the dispersion Eq. (7) the conditions (6) are fulfilled independently of

On the plane

The resonance beam of energetic electrons (stars) and the short electromagnetic pulse (sine) move together in the domain along the corridor between the start and finish lines.

Effective wave-particle interaction in a homogeneous plasma in the magnetic
field takes place at the resonance conditions

We expect the following two simplifications to be fulfilled: the radiation
power of an individual energetic electron corresponds to the so-called dipole
approximation; the specific kinetic effects like the Landau damping are not
important. Both of the mentioned simplifications take place for energetic
electrons with a sufficiently small dispersion over the transverse and
longitudinal velocities (Ginzburg, 1970):

Note the second key point for a qualitative calculation of short
electromagnetic pulse amplification. According to the previous comments, the
electromagnetic field with suitable frequency in the short pulse changes its
value as the electromagnetic wave field in a homogeneous plasma with an
electron beam described by the effective distribution function:

We explain additionally the expression for the effective distribution
function (Eq. 11). Assume that a short non-spreading
pulse propagates at a constant velocity (

We expect the quasineutral plasma to consist of three fractions: unmoved
protons; cool electrons; and a weak electron beam along the magnetic field
without thermal dispersion, described by the effective distribution function
(Eq. 11). So, the complicated geophysical problem is
reduced to a typical problem of plasma physics (e.g., Akhiezer et al., 1975).
We now consider the permittivity tensor

The dispersion Eq. (14) is a cubic equation for the frequency

Numerical solution of the dispersion Eq. (14) for

According to Eq. (16), the conditional growth rate of the short
electromagnetic pulse instability is maximum for

Note that the resonance beam density

The proposed beam pulsed amplifier (BPA) mechanism produces effective
amplification of short electromagnetic pulses with frequency close to

The threshold of the BPA mechanism is mainly determined by the kinetic
Čerenkov damping (

By the further analysis it is possible to explain the gap between the lower
and upper chorus frequency bands which are well known experimentally (e.g.,
Bell et al., 2009). Actually, the short electromagnetic pulse excitation
takes place for frequency bands both below and above

Note that electromagnetic signals with a smooth envelope are not amplified due to BPA mechanisms during their propagation through the region near the magnetic equator. Therefore, it is possible to explain the excitation of chorus emissions without a hiss-like background.

BPA mechanism is not connected directly with the energetic electron
precipitation into the ionosphere because this mechanism is responsible for
the distribution function modification in the velocity space only far enough
from the loss cone. On the other hand, exactly after the amplification, a
strong electromagnetic pulse during its propagation to the ionosphere can
interact with more energetic electrons due to the cyclotron resonance
(

The paper is theoretical and no new experimental data are used. All figure data are obtained from numerical calculation in MATLAB codes. Corresponding parameters are listed in the text.

PB proposed and analyzed the BPA mechanism, and wrote the paper. OS analyzed the BPA mechanism, and wrote the paper.

The authors declare that they have no conflict of interest.

Peter Bespalov was supported by RSF, project no. 16-12-10528 (Sect. 2), the Russian Ministry of Science and Education, project no. 14.Z50.31.0007 (Sect. 3), and Fundamental Research Program no. 28 (numerical calculations). The topical editor, Matina Gkioulidou, thanks one anonymous referee for help in evaluating this paper.