Self‐Consistent Theory of Triggered Whistler Emissions

Abstract

A kinetic theory of triggered VLF whistler emissions is given that is capable of predicting from a small‐scale numerical implementation the observed emission forms and frequency‐time characteristics. The present paper focuses on the theoretical developments and the explanation of the triggering process, complete with a demonstration of the threshold behavior (sometimes known as the dot‐dash anomaly) and the generation of specific falling frequency emissions whose falling rate compares quite favorably with typical observations made in the controlled experiments based in Siple Station, Antarctica. The theory that gives these results is a fully self‐consistent nonlinear treatment based on kinetic theory and valid in the asymptotic limit when several trapping periods occur within the interaction region. For a typical set of parameters (l = 4.3, n = 400 cm−3, and amplitude BT ∼ 1.6 pT) for the input wave magnetic field there are about seven trapping periods in the triggering signal, and one would expect good results from the asymptotic limit. In this limit the nonlinear dynamics can be reduced to the determination of the time τ that the resonant particles are trapped. The nonlinear currents can be expressed in terms of this function τ by simple integrals over the trapped particles’ perpendicular velocities alone. Most of the features of the emission process can be determined analytically, such as the rate of change of frequency, related to magnetospheric parameters. For more quantitative predictions, such as predictions of amplitude and frequency waveforms, a small numerical code which integrates the nonlinear wave equations is used. The theoretical picture of the triggering mechanism contains the observed threshold behavior wherein short triggering pulses of nominal amplitude BT ∼ 1 pT and duration 100 ms or less cannot generate an emission, whereas those in the range of 200 ms and longer do. Similar sensitivity is found with respect to initial frequency, where in some cases, 5.5‐kHz signals can trigger, but 5.0‐kHz signals cannot. Local saturation yielded gains in the range of 20‐30 dB with initial temporal growth rates in the range 100‐200 dB/s. The emission process requires an inverted population in the perpendicular velocity distribution function, but not necessarily linear instability. A sufficient number of high‐energy electrons are required for the driven currents to offset convection, but provided this is satisfied, a sufficiently long triggering signal will always generate a self‐sustaining emission. Interestingly, however, the self‐sustaining emission does not depend on the number of these high‐energy particles or on the details of the velocity distribution function but only on bulk magnetospheric parameters such as magnetic field, gradient scale length, and plasma density in the plasmapause. Also, the marginal signals generated just above the threshold are always fallers, as is observed. These features have not been explained by existing theories. Unresolved issues include the mechanism of termination, the generation of risers and hooks for more intense emission situations, and the detailed synchronization of the frequency‐time and amplitude waveforms. This last point requires the solution of a peculiar singular two‐way wave equation for the phase which is presently solved in a subsidiary asymptotic limit valid within the interaction region. These unexplained points could well be explained within the present theory by inclusion of the variable resonant velocity in the projection of the electron distribution function.