Nonresonant Scattering of Relativistic Electrons by Electromagnetic Ion Cyclotron Waves in Earth’s Radiation Belts

Electromagnetic ion cyclotron waves are expected to pitch-angle scatter and cause atmospheric precipitation of relativistic ($>1\mathrm{MeV}$) electrons under typical conditions in Earth’s radiation belts. However, it has been a long-standing mystery how relativistic electrons in the hundreds of keV range (but $<1\mathrm{MeV}$), which are not resonant with these waves, precipitate simultaneously with those $>1\mathrm{MeV}$. We demonstrate that, when the wave packets are short, nonresonant interactions enable such scattering of hundred-keV electrons by introducing a spread in wave number space. We generalize the quasilinear diffusion model to include nonresonant effects. The resultant model exhibits an exponential decay of the scattering rates extending below the minimum resonant energy depending on the shortness of the wave packets. This generalized model naturally explains observed nonresonant electron precipitation in the hundreds of keV concurrent with $>1\mathrm{MeV}$ precipitation.

Figure 1

Event overview. Panels (a) and (b) show the power spectrum and wave normal angle of EMIC waves captured by MMS#1, respectively. White and yellow dashed curves in panel (a) show fractions of the equatorial proton gyrofrequency estimated with two models of MMS projection to the equatorial plane [31, 32]. The MMS#1 L-shell and MLT are shown in panel (c). Panels (d) and (e) show the spectra of trapped and precipitating electron fluxes, respectively, measured by ELFIN-B. Panel (f) shows the precipitating-to-trapped flux ratio. The ELFIN-B L-shell, magnetic latitude (MLAT), and MLT are shown in panel (g). Panel (h) shows the spectrum of precipitating electrons at the moment of the strongest precipitation, $06:25:16\mathrm{UT}$ (in black); the average of trapped spectra over the precipitation event, $06:25:14–06:24:20\mathrm{UT}$ (in blue), taken as the background level of precipitation; and the ratio of the precipitating to the aforementioned background level (in red).

Figure 2

EMIC wave packets for the event from Fig. 1. (a) The entire interval of EMIC wave observations. (b) A subinterval [shown in blue in panel (a)] with a few wave packets. The shown $B_{\perp }$ is orthogonal to the background magnetic field and the $x$ axis of the geocentric solar magnetospheric system. As EMIC wave sources can survive for hours (e.g., Refs. [43, 44]), it is reasonable to expect that EMIC wave properties do not change significantly during the ten-minute delay between wave measurements by MMS and electron precipitation measurements by ELFIN.

Figure 3

Two regimes of resonance point ${\mathrm{\Psi }}_0$ and associated contours of phase integral in $(\mathrm{Re}\mathrm{\Psi },\mathrm{Im}\mathrm{\Psi })$ for resonant and nonresonant energies. (a) Resonant regime: The resonance point ${\mathrm{\Psi }}_0$ is on the real axis. The shown path of integration is called the Hankel contour [53]. (b) Nonresonant regime: ${\mathrm{\Psi }}_0$ is in the upper half of the complex $\mathrm{\Psi }$ plane. The path of integration extending along the real $\mathrm{\Psi }$ axis from ${\mathrm{\Psi }}_l$ to ${\mathrm{\Psi }}_u$ is deformed into the upper half of the plane.

Figure 4

Comparison of pitch angle diffusion rate as a function of energy between theory and simulations for three different packet sizes: (a) $k{L}_z=30$. (b) $k{L}_z=15$. (c) $k{L}_z=5$. The inset plots sketch the EMIC wave packets. Test-particle simulation results are shown in magenta dots. Theoretical predictions of $D_{\alpha \alpha }$ using scattering factors from Eqs. (5), (9), and (10) are shown in black, red, and blue lines, respectively. The diffusion rates from the simulations with $B_w/B_{\mathrm{eq}}={10}^{-3}$ are scaled up to $B_w/B_{\mathrm{eq}}={10}^{-2}$ (as seen in the spacecraft observations) to compare them directly with those in the strong diffusion limit, depicted by the gray lines. The lower wave amplitude used in simulations is intended to separate nonresonant from nonlinear effects and to be more consistent with the perturbation analysis.