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

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.

The error bars associated with the average inaccuracy of magnetic field model are overplotted every 0.1 L * -shell (calculation details are given in Supplementary Note 1). Note that Van Allen Probe B gave the results generally consistent with those from Van Allen Probe A. 14 Supplementary The electron phase space density (PSD) is a function of the adiabatic invariants whose calculation requires information about the global configuration of the geomagnetic field 2, 3 . Such global configuration cannot be obtained from the local satellite measurements and has to be described by an appropriate magnetic field model. An assumed geomagnetic field model can lead to the generation of errors on the PSD profiles 2, 3 . We have analyzed the radial profiles of relativistic electron PSDs in the four different magnetic field models ( Supplementary Fig. 1): OP77Q 4 , T89Q 3, 5 , T89D 5 and TS04D 6 . These models give a qualitative consistent picture, illustrating the earthward movement of the inner edge of the outer radiation belt. The difference between observed and TS04Dmodeled geomagnetic fields ( Supplementary Fig. 2) was relatively small (averagely 4%) in the region L * < 4.2, but became much larger (averagely 18%) in the region L * > 4.2. Such large differences were probably produced by the enhanced compression of dayside magnetosphere (Fig.   1a in main text). The uncertainties of the magnetic field amplitude translate to the uncertainties in the kinetic energy for a fixed first adiabatic invariant, and then the corresponding uncertainties in the electron flux and PSD can be calculated based on the observed energy spectrum 3 . The errors associated with the average inaccuracy of TS04D geomagnetic field model are found to be smaller than the extent of electron PSD variation in the slot region ( Supplementary Fig. 3).

Supplementary Note 2 | Relative importance of electric and magnetic perturbations
To illustrate the relative importance of electric and magnetic perturbations of ULF waves, we have performed the radial diffusion simulations driven by D E L * L * or D B L * L * alone (Supplementary Fig.   4). The electric perturbations were able to largely explain the electron PSD evolution, while the contribution of magnetic perturbations was quite limited.

Supplementary Note 3 | Radiation belt events under different magnetospheric conditions
By analyzing a radiation belt event in the plasmasphere without detectable VLF chorus waves, we have illustrated the ability of ULF waves to radially diffuse the relativistic electrons. In fact, the radial diffusion characteristics can still be found under other different magnetospheric conditions.
Here we show two more radiation belt events with the concurrence of ULF and VLF chorus waves.  Fig. 7), suggesting the occurrence of m = 1 mode drift-resonance between ULF waves and relativistic electrons. We adopt the TS04D geomagnetic field model ( Supplementary   Fig. 8) to calculate the adiabatic invariants and the relativistic electron PSDs (Supplementary Fig.   9). The relative difference between the observed and modeled magnetic fields were large (aver- mode drift-resonance driven by ULF waves was supported by the wavelet transform analysis for the electron fluxes ( Supplementary Fig. 12). The TS04D geomagnetic field model well reproduced the observed magnetic fields (with the relative difference 2% on average) throughout the event 20 ( Supplementary Fig. 13), and the obtained relativistic electron PSDs behaved quite differently from those of the events on 18 January 2013 and 15 February 2014 ( Supplementary Fig. 14).
The local acceleration by VLF chorus waves explained the formation of a significant peak in the relativistic PSD around L * = 5.2. In the region L * < 5.2 with steep gradients of PSDs, the radial diffusion by ULF waves could redistribute the electrons toward the Earth.