Generation of VLF emissions with the increasing and decreasing frequency in the magnetosperic cyclotron maser in the backward wave oscillator regime

We study the mechanisms of the formation of falling tones in the dynamic spectrum of whistler-mode waves generated by energetic electrons in the Earth’s magnetosphere when the backward-wave oscillator (BWO) regime is realized in the magnetospheric cyclotron maser. As was shown earlier, this regime allows one to explain many features of ELF/VLF chorus emissions in the magnetosphere, in particular, the generation of elements with discrete frequency spectrum, characterized by a large growth rate and a fast frequency drift. On the basis of numerical simulations of a simplified system of nonlinear equations describing the magnetospheric BWO dynamics under the assumption of small efficiency of wave-particle interactions we show that the falling tones are generated in the case where the generation region is shifted from the equatorial plane (geomagnetic-field minimum) upstream with respect to the motion of energetic electrons. In this case, the resonant electrons move towards the decreasing magnetic field in the process of generation; hence, their longitudinal velocity increases, which corresponds to a decrease in the cyclotron-resonance frequency. Two mechanisms of the shift of the generation region from the equator are considered, i.e., (i) an increase in the linear instability growth rate (e. g., due to an increase in the energetic-electron density), and (ii) persistence of the phase bunching of the particles coming back to the generation region due to the bounce oscillations. We show that both of these mechanisms can result in the formation of falling tones, but the properties of the generated emissions such as the frequency drift rate and characteristic time interval between the elements are different. The conditions of preserving the phase bunching due to the bounce oscillations are discussed. Probably, this mechanism can operate in the case where the length of the generation region along the magnetic field is close to the characteristic bounce-oscillation length of energetic electrons which is realized for a sufficiently high cold-plasma density in the generation region.