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# Ultra-low-frequency wave-driven diffusion of radiation belt relativistic electrons

## Abstract

Van Allen radiation belts are typically two zones of energetic particles encircling the Earth separated by the slot region. How the outer radiation belt electrons are accelerated to relativistic energies remains an unanswered question. Recent studies have presented compelling evidence for the local acceleration by very-low-frequency (VLF) chorus waves. However, there has been a competing theory to the local acceleration, radial diffusion by ultra-low-frequency (ULF) waves, whose importance has not yet been determined definitively. Here we report a unique radiation belt event with intense ULF waves but no detectable VLF chorus waves. Our results demonstrate that the ULF waves moved the inner edge of the outer radiation belt earthward 0.3 Earth radii and enhanced the relativistic electron fluxes by up to one order of magnitude near the slot region within about 10 h, providing strong evidence for the radial diffusion of radiation belt relativistic electrons.

## Introduction

The geomagnetic field geometry allows three quasi-periodic motions of outer Van Allen radiation belt1 relativistic electrons over distinct timescales: gyration about the magnetic field line on a timescale of milliseconds; bounce along the magnetic field line between two magnetic mirror points on a timescale of seconds; drift circling the Earth on a timescale of kiloseconds. Each periodicity gives rise to an approximate constant of motion defined as the adiabatic invariant. Surrounding the highly dynamic outer radiation belt2, one outstanding question has been how electrons are accelerated to relativistic energies of several million electron volts. Two invariant-violating processes have been proposed: local acceleration by VLF (kHz) chorus waves3,4 and radial diffusion by ULF (mHz) waves5,6,7,8,9. On 30 August 2012, National Aeronautics and Space Administration launched a twin-spacecraft mission, Van Allen Probes (formerly known as the Radiation Belt Storm Probes (RBSP))10, to make the measurements for the identification of acceleration mechanisms. Recent analyses11,12 of the Van Allen Probes data have presented compelling evidence for the local acceleration in the heart of the outer radiation belt. However, the importance of radial diffusion for the radiation belt evolution has not yet been determined definitively.

Here we report a unique radiation belt event serendipitously observed by the Van Allen Probes in the plasmasphere during non-storm times. The plasmasphere is a torus-shaped region of cold and dense plasma surrounding the Earth, where the VLF chorus wave-driven local acceleration seldom occurs. By analysing the high-resolution data and performing the detailed simulation, we demonstrate that the radial diffusion by intense ULF waves was responsible for the radiation belt relativistic electron evolution in this event.

## Results

### Wave modulation of relativistic electrons

The radiation belt event of interest occurred on 15 February 2014, with the required particle/field data collected by the Relativistic Electron–Proton Telescope (REPT)23 and the Magnetic Electron Ion Spectrometer (MagEIS)24 of the Energetic particle, Composition and Thermal plasma (ECT) instrument suite25, the Electric Fields and Waves (EFW) instruments26 and the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) instrument suite27 on board the Van Allen Probes. During the event, the magnetosphere was free from magnetic storms but experienced some weak and short-duration substorms (Fig. 1a). Such magnetospheric conditions favoured the expansion of plasmasphere to at least L=6.5 (Figs 1b and 2a,c; Supplementary Table 1). The quantity L is the McIlwain’s drift-shell parameter28, equal to the equatorial radial distance (made dimensionless by Earth radii RE) of the drift orbit of the electron having the same mirror field, second invariant and energy in the dipole field. In the plasmasphere, there were no VLF chorus waves contributing to the local acceleration (Fig. 2b,d). In the time range from 13:15 to 24:00 UT, the solar wind dynamic pressure (Fig. 1a) exhibited continuous fluctuations especially around 13:15 and 22:00 UT. The solar wind variation drove the generation of ULF waves over a wide frequency range in the outer radiation belt (Fig. 1c–f). Correspondingly, the relativistic (2.0–4.5 MeV) electron fluxes oscillated periodically (Fig. 1g), which is one of the expected drift-resonance characteristics29. In contrast to the previously reported ULF modulation30,31 of electron fluxes at the relatively low energies and/or in a localized spatiotemporal region, this event presented the ULF modulation of highly relativistic electron fluxes for about 10 h throughout the outer radiation belt, serving as an observational evidence for the global and long-lasting drift resonance between ULF waves and relativistic electrons.

The azimuthal mode number m of ULF waves is required to quantify the drift-resonance process. Wavelet transforms32 have been performed on the electron fluxes at different energy channels to identify their dominant oscillation periods (Fig. 3). During some time intervals (10:00–13:15, 15:30–16:00, 17:30–19:30, 20:20–21:00 and 23:30–24:00 UT), the regular oscillations of fluxes were so weak (as marked by the black arrows in Fig. 1g) that the wavelet transforms mainly captured the power associated with the irregular noise and/or the radial variation of fluxes. Most of the time the dominant oscillation periods of electron fluxes are found to change with energy and location but remain close to the corresponding electron drift periods, which can be considered as the result of the m=1 mode drift-resonance between relativistic electrons and broadband ULF waves29,33. It should be mentioned that recent magnetohydrodynamic simulation34 for a magnetic storm had suggested the dominance of m=1 mode ULF waves in the radiation belt region.

### Global evolution of relativistic electrons

The Van Allen Probes passed through the radiation belt six times in the time range of interest. The comparison of the L-dependent electron fluxes observed during each passage is made to identify the global evolution of radiation belt electrons (Fig. 4). Within about 10 h, the inner edge of the outer belt moved inward about 0.3 RE at 2.0 MeV and 0.8 RE at 4.5 MeV. Around the inner edge, the electron fluxes increased by an average factor of up to 10. To exclude the possible adiabatic effect associated with the compression of magnetosphere, these electron fluxes are transformed into phase space density (PSD) in the adiabatic invariant coordinate system (Fig. 5a; Supplementary Figs 1–3; Supplementary Note 1). For the fixed L* near the slot region, the variation of mapped L with time was below 0.05 throughout the event (Fig. 6). The initial PSD peaked at the large L* (that is, the external source region) and monotonically decreased with the decreasing L*. As time went on, the external source showed some fluctuations. At the centre of the outer belt (L*≈4.3), the PSD remained unchanged, once again demonstrating the absence of local acceleration. In the slot region, the inner edge of the outer belt was transported inward about 0.3 RE. Specifically, at L*≈3.6, the PSD gradually increased by one order of magnitude within 10 h. These global evolution characteristics were qualitatively consistent with the prediction of radial diffusion theory35,36. It should be mentioned that the very strong fluctuations of electron PSDs after 22:00 UT were perhaps associated with the coherent transport process33.

The radial diffusion of radiation belt electrons is simulated by solving the equation37

with the electron PSD F and the radial diffusion rate DL*L* calculated from the observed ULF waves9 (Fig. 7). The simulations (Fig. 5b; Supplementary Fig. 4; Supplementary Note 2) well reproduce the average rate and extent of the observed PSD variations. The radial diffusion process can effectively reduce the electron PSD gradients35. In the region L*>4.3, the electron PSD behaved smoothly and consequently presented minor changes. In contrast, the electron PSD possessed a very steep gradient in the region L*<4.3 and then allowed the significant enhancement.

## Discussion

There was an average enhancement of relativistic electron fluxes by up to one order of magnitude within about 10 h on 15 February 2014, which could be easily misinterpreted in the framework of the VLF chorus wave-driven local acceleration. The high-resolution data and the detailed simulation clearly show that, in the absence of VLF chorus waves, the ULF waves could radially diffuse and effectively accelerate the radiation belt electrons in this event. The ULF waves38,39,40,41,42,43,44, as well as the VLF chorus waves45,46,47,48,49,50,51,52,53,54, are commonly observed in the magnetosphere. To provide a more comprehensive picture, we additionally analyse two more radiation belt events with the concurrence of ULF and VLF chorus waves (Supplementary Figs 5–14; Supplementary Tables 2 and 3; Supplementary Note 3). During the 18 January 2013 event, there were moderate ULF waves but quite weak VLF chorus waves. The radial diffusion, dominating over the local acceleration, caused the earthward movement of the inner edge of the outer radiation belt. In contrast, both moderate ULF waves and strong VLF chorus waves occurred during the 22 September 2014 event. The local acceleration produced the relativistic electron PSD peaks, and the radial diffusion redistributed the electrons along the radial direction to reduce the PSD gradients.

Our results present evidence for the importance of ULF wave-driven radial diffusion in the outer radiation belt dynamics. Under differing magnetospheric conditions, the accurate contributions of local acceleration and radial diffusion to the radiation belt evolution should be carefully examined. The radial diffusion can play a dominant role in some radiation belt electron acceleration events (such as the one discussed here). Compared with the local acceleration in the previously reported marked events11,12,21, the radial diffusion in the present event yielded a smaller acceleration rate and a smaller relativistic electron flux enhancement. However, no matter how strong the action of local acceleration is, the additional redistribution of relativistic electrons by the radial diffusion would constitute an important part of the outer radiation belt dynamics11,33,35,55,56,57.

## Methods

### Calculation of wave power spectral density

We use the level-3 magnetic field data with the time resolution of 1 s in the Geocentric Solar Magnetospheric coordinate system to obtain the magnetic power spectral density of ULF waves. A three-step procedure is performed as follows. First, the magnetic field data are running averaged over 11 s to reduce the spin modulation of Van Allen Probes44. Then, these magnetic field data are projected on the so-called mean field aligned (MFA) coordinates58. In the MFA coordinate system, the parallel direction is determined by the 2,048-s running average of the instantaneous magnetic field, the azimuthal direction is obtained by the cross product of the parallel vector and the satellite position vector, and the radial direction completes the triad. This system is used to separate magnetic field perturbations into the toroidal (azimuthal), poloidal (radial) and compressional (parallel) components26. Finally, an overlapped (80%) 2,048-point fast Fourier transform is adopted to calculate the power spectral densities of each magnetic field components. The obtained power spectra densities have the time resolution of 6.83 min and the frequency resolution of 0.49 mHz. We use the level-3 electric field data with the time resolution of 10.9 s in the modified geocentric solar ecliptic (mGSE) coordinate system to obtain the electric power spectral density of ULF waves. The mGSE unit vectors (XmGSE, YmGSE and ZmGSE) can be generally expressed in terms of the GSE unit vectors (XGSE, YGSE and ZGSE) and the spin axis unit vector SGSE: XmGSE=SGSE, , and ZmGSE=XmGSE × YmGSE. When the spin axis points towards the sun (SGSE=XGSE), the mGSE system is exactly the same as the GSE system. The EFW instrument only measured the two electric field components in the spin plane, and the third component has to be derived from the assumption E·B=0. For this event after 22:00 UT, the magnetic field component Bx of RBSP-B in the mGSE system was nearly zero and the electric field component Ex was unconstrained by the E·B=0 condition, not allowing the calculation of electric field vectors in the MFA system. Note that the most significant ULF waves happened to be generated after 22:00 UT. Hence, we calculate the power spectral densities of the electric field components in the mGSE system (rather than in the MFA system) to ensure the continuous monitoring of ULF waves throughout this event. Specifically, an overlapped (80%) 256-point fast Fourier transform is performed on the electric field data (with the preliminary substraction of the co-rotation electric field and the Vsc × B electric field induced by the satellite motion through the Earth’s magnetic field), and the obtained the electric field power spectral densities possess the time resolution of 9.3 min and the frequency resolution of 0.36 mHz. It should be mentioned the power spectral densities exhibited unphysical intensification44 associated with the steep variation of the background magnetic field during the averaging time of 2,048 s in the region L<3 or the contamination by the large Vsc × B electric field in the region L<2.5.

### Calculation of radial diffusion rates

The radial diffusion is generally produced by both the ULF electric and magnetic perturbations. In this event, the m=1 mode symmetric drift resonance is suggested to be dominant. The corresponding radial diffusion rates are written as9

with the equatorial magnetic field magnitude at the Earth’s surface B0=0.312 G, the Lorentz factor of electrons γ, the Earth’s radii RE, and the power spectral densities of the azimuthal wave electric field PE and the compressional wave magnetic field PB at the resonant frequency ω=ωd in the MFA system. As discussed in the previous section, the azimuthal wave electric field cannot be obtained continuously. In this study, PE in the MFA system is approximated as the power spectral density of the y component electric field in the mGSE system. We calculate the radial diffusion rates in two time ranges (14:00–22:00 and 22:00–24:00 UT). In each time range, we average the wave power in three spatial regions (L*=3.4±0.4, 4.2±0.4 and 5.0±0.4) and then obtain the L*-dependent ULF wave power distribution through the linear interpolation. In the inner region L*<3.4, the wave power is assumed to equal that at L*=3.4; in the outer region L*>5.0 before 22:00 UT or L*>4.2 after 22:00 UT, the wave power is assumed to equal that at L*=5.0 or 4.2. Note that there were no ULF wave data in the region L*=5.0±0.4 after 22:00 UT. These assumptions would not significantly affect our conclusions, since the ULF wave-driven variation in electron PSDs mainly occurred in the region L*=3.4–4.2. In the frequency range 1–5 mHz, the obtained electromagnetic power spectral densities (Fig. 7a–d) were comparable to the previous statistical results38,39,40,41,43. The obtained electric diffusion rates dominated over the magnetic diffusion rates (Fig. 7e,f), consistent with the previous calculations38,40,43.

### Simulation of radiation belt evolution

The simulation code is extracted from our previously developed STEERB model37. The fully implicit finite difference method is adopted to solve the radial diffusion equation with the computational domain 2.5≤L*≤5.5. The initial condition is given by the observed electron PSD during the first passage (12:06–13:32 UT) of the outer radiation belt. The electron PSD is assumed to be fixed at the outer boundary L*=5.5 and to be zero at the inner boundary L*=2.5.

### Software availability

The software for wavelet analysis is available at http://paos.colorado.edu/research/wavelets/.

How to cite this article: Su, Z. et al. Ultra-low-frequency wave-driven diffusion of radiation belt relativistic electrons. Nat. Commun. 6:10096 doi: 10.1038/ncomms10096 (2015).

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## Acknowledgements

This work was supported by the National Natural Science Foundation of China grants 41422405, 41274169 and 41274174, the Chinese Academy of Sciences grants KZCX2-EW-QN510 and KZZD-EW-01-4, the National Key Basic Research Special Foundation of China grant 2011CB811403, the Fundamental Research Funds for the Central Universities WK2080000077. We acknowledge the University of Iowa as the source for the EMFISIS data in this study (this acknowledgement does not imply endorsement of the publication by the University of Iowa or its researchers), acknowledge J.H. King, N. Papatashvilli and CDAWeb for providing interplanetary parameters and magnetospheric indices, and acknowledge C. Torrence and G. Compo for providing the wavelet analysis software.

## Author information

Authors

### Contributions

Z.S. designed this study, performed most of the data analysis and simulation and wrote the paper. H.Z., Q.Z., X.Z. and Y.H. contributed to the ULF wave analysis. F.X. assisted with the simulations and the English improving. H.Z., Y.W. and S.W. contributed to the discussions. Z.G. and Z.H. assisted with the electron flux analysis. D.N.B., H.E.S., G.D.R. and J.B.B. provided the ECT-REPT and ECT-MagEIS data. J.R.W. provided the EFW data.

### Corresponding author

Correspondence to Zhenpeng Su.

## Ethics declarations

### Competing interests

The authors declare no competing financial interests.

## Supplementary information

### Supplementary Information

Supplementary Figures 1-14, Supplementary Tables 1-3, Supplementary Notes 1-3 and Supplementary References (PDF 16334 kb)

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Su, Z., Zhu, H., Xiao, F. et al. Ultra-low-frequency wave-driven diffusion of radiation belt relativistic electrons. Nat Commun 6, 10096 (2015). https://doi.org/10.1038/ncomms10096

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