Precipitation of trapped relativistic electrons by amplified whistler waves in the magnetosphere

Numerical study of a loss-cone negative mass instability to amplify whistler waves by energetic electrons in the radiation belts is presented. The results show that a very low intensity whistler wave can be amplified by 50 keV electrons more than 25 dB, consistent with the Siple experimental result [Helliwell, J. Geophys. Res. 85, 3360 (1980)]. The dependencies of the amplification factor on the energetic electron density and on the initial wave intensity are evaluated. It is shown that the amplification factor decreases as the initial wave intensity increases. However, this gain can still exceed 15 dB for a 30 dB increase of the initial wave intensity, which is needed for the purpose of precipitating MeV electrons in the radiation belts. We then show that there exists a double resonance situation, by which, as an example, a wave is simultaneously in cyclotron resonance with 50 keV electrons as well as with 1.5 MeV electrons; the wave is first amplified by 50 keV electrons and then precipitates 1.5 MeV electrons. With the aid of the cyclotron resonance, the threshold field for the commencement of chaos in the electron trajectories is reduced considerably from that for a general case. Pitch angle scattering of 1.5 MeV electrons is demonstrated. The results show that a whistler wave with magnetic field amplitude of 0.08% of the background magnetic field can scatter electrons from an initial pitch angle of 86.5° to a pitch angle <50°.