Van Allen radiation belts consist of relativistic electrons trapped by Earth's magnetic field. Trapped electrons often drift azimuthally around Earth and display a butterfly pitch angle distribution of a minimum at 90° further out than geostationary orbit. This is usually attributed to drift shell splitting resulting from day–night asymmetry in Earth's magnetic field. However, direct observation of a butterfly distribution well inside of geostationary orbit and the origin of this phenomenon have not been provided so far. Here we report high-resolution observation that a unusual butterfly pitch angle distribution of relativistic electrons occurred within 5 Earth radii during the 28 June 2013 geomagnetic storm. Simulation results show that combined acceleration by chorus and magnetosonic waves can successfully explain the electron flux evolution both in the energy and butterfly pitch angle distribution. The current provides a great support for the mechanism of wave-driven butterfly distribution of relativistic electrons.

Explicit high-order non-canonical symplectic particle-in-cell algorithms for classical particle-field systems governed by the Vlasov-Maxwell equations are developed. The algorithm conserves a discrete non-canonical symplectic structure derived from the Lagrangian of the particle-field system, which is naturally discrete in particles. The electromagnetic field is spatially-discretized using the method of discrete exterior calculus with high-order interpolating differential forms for a cubic grid. The resulting time-domain Lagrangian assumes a non-canonical symplectic structure. It is also gauge invariant and conserves charge. The system is then solved using a structure-preserving splitting method discovered by He et al., which produces five exactly-soluble sub-systems, and high-order structure-preserving algorithms follow by combinations. The explicit, high-order, and conservative nature of the algorithms is especially suitable for long-term simulations of particle-field systems with extremely large number of degrees of freedom on massively parallel supercomputers.The algorithms have been tested and verified by the two physics problems, i.e., the nonlinear Landau damping and the electron Bernstein wave.

The recent launching of Van Allen probes provides an unprecedent opportunity to investigate variations of the radiation belt relativistic electrons. During the 17–19 March 2013 storm, the Van Allen probes simultaneously detected strong chorus waves and substantial increases in fluxes of relativistic (2 − 4.5 MeV) electrons around L = 4.5. Chorus waves occurred within the lower band 0.1–0.5fce (the electron equatorial gyrofrequency), with a peak spectral density ∼10−4 nT2/Hz. Correspondingly, relativistic electron fluxes increased by a factor of 102–103 during the recovery phase compared to the main phase levels. By means of a Gaussian fit to the observed chorus spectra, the drift and bounce‐averaged diffusion coefficients are calculated and then used to solve a 2‐D Fokker‐Planck diffusion equation. Numerical simulations demonstrate that the lower‐band chorus waves indeed produce such huge enhancements in relativistic electron fluxes within 15 h, fitting well with the observation.

Particle-in-Cell (PIC) simulation is the most important numerical tool in plasma physics. However, its long-term accuracy has not been established. To overcome this difficulty, we developed a canonical symplectic PIC method for the Vlasov-Maxwell system by discretizing its canonical Poisson bracket. A fast local algorithm to solve the symplectic implicit time advance is discovered without root searching or global matrix inversion, enabling applications of the proposed method to very large-scale plasma simulations with many, e.g., 10 9 , degrees of freedom. The long-term accuracy and fidelity of the algorithm enables us to numerically confirm Mouhot and Villani's theory and conjecture on nonlinear Landau damping over several orders of magnitude using the PIC method, and to calculate the nonlinear evolution of the reflectivity during the mode conversion process from extraordinary waves to Bernstein waves.

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