Nonlinear Interactions Between Radiation Belt Electrons and Chorus Waves: Dependence on Wave Amplitude Modulation

by Longzhi Gan
Center for Space Physics, Boston University, Boston, MA, USA.


Whistler mode chorus waves are strong and coherent electromagnetic waves which are important for energetic electron dynamics in the Earth’s outer radiation belts. One of the unique features of chorus waves is their discrete elements, the frequency of which rises or falls with time. Cyclotron resonances between energetic electrons and relatively small amplitude chorus waves lead to diffusion processes, which are described by the quasilinear theory. Diffusion processes result in zero average electron pitch angle and energy variations, and nonzero root-mean-square of the variations. Resonances between the electrons and larger amplitude chorus waves not only cause larger electron pitch angle and energy variation, but can also lead to two types of nonlinear processes, once the wave amplitude exceeds a certain threshold. One of the nonlinear processes is phase bunching, which also leads to the scattering of electron pitch angle and energy, much like the diffusion process, but with an addition of nonzero average pitch angle and energy shift. Another type of nonlinear process is phase trapping, in which the electrons are trapped in the resonance with chorus waves and continuously accelerated until they ‘detrap’.

Many studies have demonstrated the importance of these nonlinear processes, yet further quantitative analyses are needed. In addition, quantifications of the nonlinear effects, which consider realistic chorus wave amplitude and frequency features are important subjects of research, that potentially leads to a more comprehensive understanding of nonlinear interactions between electrons and chorus waves.

Figure 1. Frequency spectrum and waveform of the observed chorus wave element and three test particle wave models. (a) Observed chorus wave magnetic spectral density and a component of chorus wave magnetic field perpendicular to the ambient magnetic field based on the waveform data from THEMIS; (b) Same format as panel (a) but for the constructed Model 1 with the constant wave frequency and amplitude, (c) Model 2 with the constant wave frequency and monotonically increasing wave amplitude, and (d) Model 3 with the constant wave frequency and realistic wave amplitude modulation.


In this study, we focus on quantifying the dependence of nonlinear interactions on the chorus wave amplitude modulations based on the THEMIS wave observation using a numerical method of test particle simulations at L = 6.

For the chorus wave input models, we use the method described in Tao et al. (2012) to reconstruct the spatiotemporal distribution of chorus waves based on the measurements from THEMIS C on 23 October 2008, and incorporate it into our test particle simulations. Figure 1a shows the rising-tone chorus wave element observed by THEMIS C. We then construct three additional wave models, including a monochromatic wave with constant frequency and wave amplitude (Figure 1b, Model 1), repeating wave elements with linearly increasing wave amplitude and constant wave frequency (Figure 1c, Model 2), repeating wave elements with realistic wave amplitude (observed wave amplitude multiplied by 5) and constant wave frequency (Figure 1d, Model 3).

Figure 2 shows the quantitative assessment of the phase trapping and phase bunching of electrons. From Model 1 to 3, the averaged pitch angle variations decrease, and the region for maximum variations in pitch angle gradually shift to smaller pitch angles. Meanwhile, the phase trapping probability becomes larger at regions near the minimum resonance boundary, as shifting from Model 1 to Model 3. For phase bunching electrons, Figures 2c, 2g, and 2k show the increase in pitch angle at small pitch angles. At higher pitch angles, phase bunching leads to negative pitch angle variations. The phase bunching features of Model 2 follow the same pattern as those of Model 1, except that the positive phase bunching region is expanded. For Model 3, the average pitch angle and energy change follows the similar pattern to Model 1 and 2, but with smaller values. Figures 2d show the pitch angle scattering rates for the phase bunching electrons in Model 1, which present the highest values at small pitch angles and energies. The pitch angle scattering rates of phase bunching electrons in Models 2 and 3 follow the similar pattern, but are in general stronger. This indicates that for more realistic wave amplitude modulation, the electron motions tend to be more ‘diffusive’, rather than more 'advective', leading to stronger pitch angle and energy scattering.

Figure 2. Test particle simulation results of phase trapping and phase bunching electrons interacting with whistler mode waves with 3 different wave amplitude structures. (a) Pitch angle change of phase trapping electrons, and (b) probability of phase trapping for electrons interacting with the Model 1 waves. (c) Averaged pitch angle change, and (d) standard deviation of pitch angle variation of phase bunching electrons interacting with Model 1. (e-h) Same as (a-d) but for Model 2. (i-l) Same as (a-d) but for Model 3. The black or white line in each panel represents the minimum resonance boundary.


Using test particle simulations, we quantified effects of wave amplitude modulation on the nonlinear interactions between energetic electrons and whistler mode chorus waves based on the THEMIS wave observation (Gan et al., 2020). The main findings of the present study are summarized below.

  1. Amplitude modulation of the chorus waves reduces the electron acceleration caused by phase trapping compared to a single frequency wave with constant wave amplitude. However, the acceleration caused by this ‘reduced’ phase trapping is still noticeable. Our results imply that the acceleration caused by phase trapping is strongly affected by the ‘shape’ of chorus wave amplitude modulation.
  2. At small pitch angles, phase bunching leads to increases in electron pitch angle and energy which is different from the quantitative prediction by conventional Hamiltonian studies (Albert et al., 2013; Artemyev et al., 2018). This feature may be important for evaluating electron precipitation loss due to nonlinear interactions with chorus waves.
  3. The realistic wave packets tend to increase the scattering of the phase bunching electrons (become more diffusive), while reducing the advection (average net changes) in pitch angle and energy compared to a single frequency wave with constant wave amplitude.
  4. The distribution of the regions, where nonlinear effects are most effective, varies significantly by the amplitude structures of the chorus wave packet.

Our simulation results clearly demonstrate the importance of wave amplitude modulation in nonlinear interactions between chorus waves and radiation belt electrons in a broad range of energy and pitch angle, and suggests that they need to be properly incorporated into future theoretical and numerical studies.


Albert, J. M., Tao, X. and Bortnik, J. (2013). Aspects of Nonlinear Wave‐Particle Interactions. In Dynamics of the Earth's Radiation Belts and Inner Magnetosphere (eds D. Summers, I. R. Mann, D. N. Baker and M. Schulz). doi:10.1029/2012GM001324

Artemyev, A.V., Neishtadt, A.I., Vainchtein, D.L., Vasiliev, A.A., Vasko, I.Y. and Zelenyi, L.M., 2018. Trapping (capture) into resonance and scattering on resonance: Summary of results for space plasma systems. Communications in Nonlinear Science and Numerical Simulation, 65, pp.111-160

Gan, L., Li, W., Ma, Q., Albert, J. M., Artemyev, A. V., Bortnik, J. (2020). Nonlinear interactions between radiation belt electrons and chorus waves: Dependence on wave amplitude modulation. Geophysical Research Letters, 47, e2019GL085987. https://doi.org/10.1029/2019GL085987

Tao, X., Bortnik, J., Thorne, R. M., Albert, J. M., and Li, W. (2012), Effects of amplitude modulation on nonlinear interactions between electrons and chorus waves, Geophys. Res. Lett., 39, L06102, doi:10.1029/2012GL051202.

Biographical Note

Longzhi Gan is a PhD student in the Astronomy Department at Boston University. His research focuses on the interactions between the radiation belt electrons and various plasma waves.

Please send comments/suggestions to
Emmanuel Masongsong / emasongsong @ igpp.ucla.edu