Amplification of Whistler Waves for the Precipitation of Trapped Relativistic Electrons in the Magnetosphere

In the magnetosphere, energetic electrons in the radiation belts are trapped by the Earth's dipole magnetic field and undergo bouncing motion about the geomagnetic equator. Induced electron precipitation into the ionosphere and upper atmosphere by whistler waves has been observed¹. Whistler waves in the magnetosphere can be excited during lightning events¹ or launched into the magnetosphere by the ground² or space-based VLF transmitters. In the present work, the trajectories of these trapped energetic electrons interacting with a large amplitude whistler wave are first examined. The geomagnetic field near the equator is modeled as a uniform static magnetic field with a parabolic scalar potential superimposed upon it to account for the feature of bouncing motion, which is a key factor to cause chaos in the electron/ion-wave interaction^3-5. A surface of section technique is used to examine the chaoticity of the system graphically. It is shown that the trajectories of trapped energetic electrons in the presence of a whistler wave can become chaotic. The behavior of the system depends strongly on two quantities, the electron energy and wave amplitude. The threshold field amplitude of the wave for the commencement of chaotic behavior in the electron trajectories decreases with increasing electron energy. Once the trajectory of an electron becomes chaotic, it can wander into the loss cone and subsequently, precipitates into the ionosphere and/or the upper atmosphere. Amplification of whistler waves by bulk electrons in the energy distribution through a loss-cone negative mass instability is also studied. The numerical result shows that the instability can amplify whistler waves by as large as 30 dB. This amplification process reduces considerably the required field intensity of injected whistler wave for the purpose of precipitating those tail electrons in MeV range.