**Journal of Geophysical Research: Space Physics**

# The nonlinear gyroresonance interaction between energetic electrons and coherent VLF waves propagating at an arbitrary angle with respect to the Earth's magnetic field

## Abstract

A theory is presented of the nonlinear gyroresonance interaction that takes place in the magnetosphere between energetic electrons and coherent VLF waves propagating in the whistler mode at an arbitrary angle *ψ* with respect to the earth's magnetic field **B**_{0}. In particular, we examine the phase trapping (PT) mechanism believed by some to be responsible for the generation of VLF emissions. The phase considered in this study is the phase γ between the right-hand circularly polarized component of the wave magnetic field perpendicular to **B**_{0} and the component of the energetic particle velocity vector perpendicular to **B**_{0}. This model is an extension of one developed in earlier work [*Bell*, 1965; *Dysthe*, 1971; *Nunn*, 1974] involving the special case where *ψ* = 0. The extended theory predicts that for any finite value of *ψ* there is a range of resonant particle pitch angle α for which γ is not bounded within the range 0 ≤ |γ| < π, and thus PT in the usual sense is not possible. However, PT in an average sense can still exist and long term energy transfer between the gyroresonant electrons and the wave can still take place. It is found that for given values of *ψ* and wave frequency, the trapping frequency *ω*_{τ} has a countably infinite set of zeros when plotted as a function of α. Regions between zeros alternate between regions of normal PT and regions of anomalous PT. The average phase angle of anomalously phase trapped electrons differs by approximately 180° from the average phase of the normal phase trapped electrons. However, the average stimulated radiation from the two groups of particles tends to add in phase. Resonant particles for which *ω*_{τ}⇒0 are not phase trapped. Near the geomagnetic equatorial plane, the threshold value of wave amplitude necessary to produce PT is directly proportional to the gradient of *ψ* along **B**_{0}. A simple model predicts threshold values greater than 10 *m*γ for *ψ* > 37°. For interactions far from the magnetic equator, *ψ* variations appear to be less important than gradients in **B**_{0} and in this case we calculate the wave power increase necessary to produce the same PT efficiency for typical nonducted waves as that appropriate to ducted waves. The component of the wave electric field parallel to **B**_{0} generally enhances the strength of the PT process and tends to dominate the process for moderate to high *ψ* at midfrequency. It is concluded that near the magnetic equatorial plane gradients of *ψ* may play a very important part in the PT process for nonducted waves. Predictions of a higher threshold value for PT for nonducted waves generally agree with experimental data concerning VLF emission triggering by nonducted waves.