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# Plasmaspheric hiss waves generate a reversed energy spectrum of radiation belt electrons

## Abstract

Highly energetic electrons are trapped in the magnetic field of Earth’s radiation belts. The physical mechanisms driving the dynamics of the Van Allen belts can be understood from the electron’s energy spectrum, which is believed to be steeply falling with increasing energy. This view has been prevalent for the past 60 years since the energy spectra were first measured. Here, we report the observation of a reversed energy spectrum with abundant high-energy and fewer low-energy electrons spanning from hundreds of keV to around two MeV in electron energy in data collected with NASA’s Van Allen Probes. We find that this spectrum dominates inside the plasmasphere—a dense cold plasma region co-rotating with the Earth. Using two-dimensional Fokker–Planck diffusion simulations with a time-dependent, data-driven model of hiss waves in the plasmasphere, we demonstrate that the formation of the reversed spectrum is explained by the scattering of hiss waves. The results have important implications for understanding the distributions of charged particles and wave–particle interactions in magnetized plasmas throughout the solar system and beyond.

## Main

The Van Allen radiation belts are torus-shaped regions in which energetic electrons and protons are geomagnetically trapped, azimuthally drifting around the Earth. The energy spectra of these charged particles provide deep insight into the physical mechanisms driving the dynamics of the radiation belts. Since the discovery of the radiation belts 60 years ago, radiation belt electron energy spectra have been typically regarded as having fluxes steeply falling with increasing energies1,2,3. However, previous studies of radiation belt electron energy spectra have suffered from sparse measurements and limited in situ data quality. Since the launch of NASA’s Van Allen Probes, accurate radiation belt electron measurements with wide energy coverage and fine energy resolution have enabled new insights into radiation belt electron dynamics4.

An intense geomagnetic storm occurred on 17 March 2015, driven by a coronal mass ejection5. During this storm, the Magnetic Electron Ion Spectrometer (MagEIS)6 and Relativistic Electron Proton Telescope (REPT)7 on the Van Allen Probes observed significant electron flux enhancements with energies from hundreds of keV to several MeV in the outer radiation belt8 (Fig. 1) (detailed information of data used in this study is provided in the Supplementary Information). On 20 March 2015, the electron energy spectra at L = 2.8–4.2 (where L can be approximated as the geocentric distance in Earth radii of the location where a magnetic field line crosses the geomagnetic equator) exhibited a nearly exponential decrease of flux with increasing energy, consistent with previous observations1,2,3. As the storm recovered and the plasmasphere expanded, hundreds of keV electron fluxes gradually decreased in the outer belt, while > MeV electrons were slowly pushed inwards to lower L-shells, leading to a notable feature of ‘reversed’ energy spectra (Fig. 1c–e). Key characteristics of this remarkable type of energy spectra include a flux maximum between 1 and 2 MeV, a flux minimum at approximately hundreds of keV, and over two orders of magnitude difference between them. We call this a ‘bump-on-tail (BOT)’ energy spectrum, the electron velocity distribution of which is similar to that associated with the classic bump-on-tail instability in space plasma9,10 but is unlikely to cause plasma instability because of the very low energy density of the relativistic electron population compared to the background plasma. The radiation belt electron energy spectra at L = 2.8–4.2 during three intervals of this storm (Fig. 1f–h) show that the BOT energy spectra deepened gradually over time, and the electron energy of the flux minimum decreased with increasing L-shell. Note that the energy spectra of electrons with different equatorial pitch angles as well as electron phase space density spectrum also showed the BOT feature during this time period (the detailed information is described in the Supplementary Information).

Statistically, BOT energy spectra are observed to be the most prevalent energy spectra inside the plasmasphere at L > ~2.6 (Fig. 2b) (the detailed method of identifying BOT energy spectra is described in the Supplementary Information). The electron energy of the BOT flux maximum, ranging from hundreds of keV to several MeV, and the energy of the BOT flux minimum, commonly around a few hundreds of keV, generally increase as the L-shell decreases (Fig. 2c,d). The ratio between the flux maximum and minimum of the BOT electron energy spectrum is usually approximately one to two orders of magnitude (Fig. 2e). Note that due to the uncertainties in the measurements, as well as the energy gap between MagEIS and REPT data used in this study, the energies of the BOT flux maximum and minimum have some uncertainties; these, however, do not alter our conclusion that the BOT energy spectrum is the most prevalent energy spectrum inside the plasmasphere at L > ~2.6 (detailed analysis and discussions are shown in the Supplementary Information).

There have been hints from some early studies of radiation belts using low-energy-resolution data11,12,13 that have suggested a BOT energy spectrum, although these have been little noted in the community due to the low quality of the data, and consequently ambiguous conclusions, and/or the lack of extensive time coverage of measurements. More recent studies using Van Allen Probes data14 have identified L-shell- and energy-dependent spectral features of radiation belt electrons but did not analyse the characteristics or occurrence frequencies of those features. Our study has revealed the BOT electron energy spectrum as a prevalent energy spectral property in the outer radiation belt inside the plasmasphere. On the other hand, the plasmaspheric hiss waves, which commonly exist inside the plasmasphere, were identified as an important loss mechanism for radiation belt electrons15,16,17. Recent studies also suggested that plasmaspheric hiss waves are responsible for the L-shell- and energy-dependent electron losses in the slot region (between the inner and outer belts, where energetic electron fluxes are usually low)18,19,20. We subsequently used high-resolution wave data from the Van Allen Probes to test the hypothesis that the formation of the BOT electron energy spectra inside the plasmasphere is caused by scattering of plasmaspheric hiss waves.

During the entire course of the 17 March 2015 storm, plasmaspheric hiss was observed inside the plasmasphere by the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS)21 onboard Van Allen Probe A, presenting wave power spectral density profiles similar to those for 25 March (Fig. 3a,b). During 20 to 29 March 2015, hiss wave amplitudes varied from a few pT to ~50 pT at L = 3–4 (Fig. 3c). Using a time-varying, data-driven plasmaspheric hiss wave model (the detailed method is described in the Supplementary Information), hiss-induced electron pitch angle diffusion causes efficient scattering of sub-MeV electrons inside the plasmasphere. The resultant loss timescales can be as short as a few hours, as estimated from the inverse of the pitch angle diffusion coefficients at an equatorial pitch angle equal to the equatorial loss cone ((αL)eq, where $$\sin \left[ {\left( {\alpha _L} \right)_{\rm{eq}}} \right] = \left[ {L^5\left( {4L - 3} \right)} \right]^{ - 1/4}$$)22,23 (Fig. 3d–f). Both the energy dependence and L-shell dependence of the estimated electron lifetimes agree with the observed evolution of the BOT electron energy spectra, both suggesting that the energies of the most rapidly decaying electrons shift to lower values as the L-shell increases.

To simulate the evolution of radiation belt electron fluxes during the 10-day storm recovery phase, we solve the two-dimensional Fokker–Planck diffusion equation24,25,

$$\begin{array}{lll}\frac{{\partial f}}{{\partial t}} &=& \frac{1}{G}\frac{\partial }{{\partial \alpha _0}}G\left( {\left\langle {D_{\alpha _0\alpha _0}} \right\rangle \frac{{\partial f}}{{\partial \alpha _0}} + p\left\langle {D_{\alpha _0p}} \right\rangle \frac{{\partial f}}{{\partial p}}} \right)\\{}&{}& + \frac{1}{G}\frac{\partial }{{\partial p}}G\left( {p\left\langle {D_{\alpha _0p}} \right\rangle \frac{{\partial f}}{{\partial \alpha _0}} + p^2\left\langle {D_{pp}} \right\rangle \frac{{\partial f}}{{\partial p}}} \right)\end{array}$$
(1)

where f is the electron phase space density with $$f = j/p^2$$, j is the differential electron flux, p is the electron momentum, $$\left\langle {D_{\alpha _0\alpha _0}} \right\rangle$$, $$\left\langle {D_{pp}} \right\rangle$$ and $$\left\langle {D_{\alpha _0p}} \right\rangle$$ are bounce-averaged coefficients of electron pitch angle diffusion, momentum diffusion and cross diffusion (measured in units of s−1) by the plasmaspheric hiss waves, which vary temporally corresponding to the changes in the hiss wave activity (the detailed method is described in the Supplementary Information), α0 is the equatorial pitch angle and $$G = p^2T(\alpha _0)\sin \alpha _0\cos \alpha _0$$, with $$T(\alpha _0) = 1.3802 - 0.3198\left( {\sin \alpha _0 + \sqrt {\sin \alpha _0} } \right)$$, is a function related to the bounce period26. The simulations were performed over the energy range from 10 keV to 10 MeV with initial conditions taken from electron flux measurements obtained from the Helium, Oxygen, Proton and Electron (HOPE) mass spectrometer27, MagEIS and REPT instruments around 04:00–05:00 UT on 20 March 2015, when the Van Allen Probe A was travelling through the region of L = 3–4.

The simulation results of the evolution of the equatorially mirroring electron energy spectra under the effect of plasmaspheric hiss at L = 3.0, 3.5 and 4.0 during 20 to 29 March 2015 (Fig. 4d–f) (the sensitivity of simulation results is shown in the Supplementary Information) systematically reproduce the main features of observations from the MagEIS and REPT instruments (Fig. 4a–c). Remarkable quantitative agreements exist between the observations and the numerical results in the timing of formation of the BOT electron energy spectra, the profiles of the flux magnitude and energy spectrum, and the L-shell dependence, especially for electrons with energies in the range ~100 keV–2 MeV. These numerical results, in combination with the statistical results (Fig. 2), demonstrate that the BOT energy spectrum, a newly unveiled type of electron energy spectra prevalent inside the plasmasphere, is caused by resonant wave–particle interactions of electrons with plasmaspheric hiss waves. The flux evolution of greater than 2 MeV electrons is not as well captured by the simulations because radial diffusion28,29, which could have accounted for the slowly elevating fluxes of greater than MeV electrons at lower L-shells, was not included in the simulation. Such a feature of the formation and evolution of a BOT energy spectrum also occurs for electrons at pitch angles away from 90° (detailed information is shown in the Supplementary Information). Our results demonstrate the ubiquity of BOT electron energy spectra inside the plasmasphere, overthrowing the traditional picture of steeply falling spectra. Since the wave–particle interaction between whistler-mode emissions and energetic electrons is a universal physical process, the results presented here also have important implications for understanding the complexity of charged particle distributions and associated wave–particle interactions30 at magnetized planets throughout the solar system and beyond. These peculiar energy spectra can result where wave–particle interactions cause energy-dependent losses peaking at a specific energy, and consequently affect the interpretation of particle populations in such situations.

### Code availability

The computer code to simulate the temporal evolution of outer radiation belt electrons under the impact of plasmaspheric hiss is available upon request to the corresponding authors.

## Data availability

The particle data analysed during the current study are available from the ECT Science Operations and Data Center (http://www.rbsp-ect.lanl.gov). The field data are available from the EMFISIS Data Center (https://emfisis.physics.uiowa.edu/). Solar wind data and geomagnetic indices are available from OMNIWeb (http://omniweb.gsfc.nasa.gov/).

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## Acknowledgements

This work was supported by RBSP-ECT funding through JHU/APL contract 967399 (under prime NASA contract NAS5-01072), by the NSFC grants 41674163, 41474141, 41574160 and 41204120, by the Chinese Thousand Youth Talents Program, and by the Hubei Province Natural Science Excellent Youth Foundation (2016CFA044). We thank the Van Allen Probes REPT, MagEIS, HOPE and EMFISIS Science Teams for providing the particle and wave data. We also thank the NSSDC OMNIWeb for the use of solar wind and geomagnetic index data.

## Author information

### Author notes

1. These authors contributed equally: H. Zhao, B. Ni.

### Affiliations

1. #### Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO, USA

• H. Zhao
• , X. Li
• , D. N. Baker
•  & Z. Xiang

• B. Ni
• , W. Zhang
• , Z. Xiang
•  & X. Gu
3. #### Air Force Research Laboratory, Kirtland Air Force Base, Albuquerque, NM, USA

• W. R. Johnston
4. #### Department of Physics and Astronomy, University of Iowa, Iowa City, IA, USA

• A. N. Jaynes
5. #### NASA Goddard Space Flight Center, Greenbelt, MD, USA

• S. G. Kanekal
6. #### Space Sciences Department, Aerospace Corporation, Los Angeles, CA, USA

• J. B. Blake
•  & S. G. Claudepierre
7. #### Space Sciences Laboratory, University of California, Berkeley, CA, USA

• M. A. Temerin
8. #### Los Alamos National Laboratory, Los Alamos, NM, USA

• H. O. Funsten
•  & G. D. Reeves
9. #### New Mexico Consortium, Los Alamos, NM, USA

• G. D. Reeves
•  & A. J. Boyd

### Contributions

H.Z. led the study, performed the data analysis of the bump-on-tail energy spectrum and wrote the manuscript. B.N. initialized the concept of hiss wave scattering, led the simulations and their quantitative comparisons with observations, and contributed to writing of the manuscript through reviews and edits. X.L. and D.N.B. supervised the study and contributed to writing of the manuscript through reviews and edits. W.R.J. initialized the concept of bump-on-tail energy spectrum, helped with data analysis, and contributed to writing of the manuscript through reviews and edits. W.Z. conducted the Fokker–Planck simulation runs to investigate the hiss-induced electron dynamics and produced the majority of Figs. 3 and 4. Z.X. provided the wave information to establish the data-driven, time-dependent hiss wave model and helped produce Fig. 3. X.G. helped analyse the simulation results and compare them with observations. A.N.J., S.G.K., J.B.B., S.G.C., M.A.T., H.O.F., G.D.R. and A.J.B. contributed to writing of the manuscript through reviews and edits.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to H. Zhao or B. Ni.

## Supplementary information

1. ### Supplementary Information

Supplementary methods and supplementary figures