Nonlinear interaction between particles and ultralow frequency waves

One of the most important questions in space physics is how the energy of the solar wind is transmitted to energetic particles in the Earth's magnetosphere, part of which is in the form of ultra-low-frequency (ULF) electromagnetic waves in the mHz frequency range. More specifically, ULF waves can provide diagnostics of the magnetosphere. For example, ionospheric conductance and mass density structure can be derived; substorm onset can be timed; and geomagnetic field lines can be mapped. ULF waves can also modify the magnetosphere such as through nonlinear effects allowing Kelvin-Helmholtz surface wave energy at the magnetopause to penetrate into magnetosphere, and radial diffusion, which plays an essential role in flux enhancement and particle acceleration of the radiation belts. Hannes Alfvén first proposed the existence of transverse "electrohydrodynamic" waves in the magnetized plasma. The magnetohydrodynamic (MHD) theory was first applied to explain the observed geomagnetic pulsations in Earth's magnetosphere, which has been confirmed by spacecraft observations and ground-based magnetometers. ULF waves are usually categorized into poloidal and toroidal modes by different directions of the perturbed electromagnetic field. Magnetic line oscillations in the radial direction yields azimuthal electric fields ( E_\varphi ) and are referred to as poloidal waves, while the motion of field lines in the azimuthal direction yields electric field oscillations in the radial direction ( E_r ) which are referred to as toroidal waves. Efficient interaction between ULF waves and charged particles requires comparable periods of waves and particle's drift motion. An acceleration mechanism capable of a continuous energy exchange between ULF waves and charged particles is wave-particle drift resonance. When such resonance occurs, the azimuthal drift speed of a resonant particle matches the wave propagation speed, and the particle experiences a constant phase of the wave electric field. This process enables a sustained energy exchange between ULF waves and charged particles, which provides a major source of particle acceleration and diffusion in the Van Allen radiation belts. The conventional drift resonance theory assumes that the particle trajectories are unperturbed despite their energy gain or loss from ULF waves. This assumption is usually invalid in Earth's magnetosphere because ULF waves can have larger amplitudes and/or durations. The large wave-particle energy exchange can modify the particle trajectory and cause significant nonlinear effects. This paper mainly reviews the nonlinear drift resonance between poloidal/toroidal ULF waves and charged particles in the inner magnetosphere. The particle behavior can be described by a pendulum equation in poloidal ULF waves, with the nonlinear trapping frequency determined by the ULF wave amplitude. We further predict, based on the newly-developed theory, the observable signatures of nonlinear drift resonance such as rolled-up structures in the energy spectrum of particle fluxes. After considering how this manifests in particle data with finite energy resolution, we compare the predicted signatures with Van Allen Probes observations. Their good agreement provides the first observational identification of the nonlinear drift resonance, which highlights the importance of nonlinear effects in magnetospheric particle dynamics under ULF waves. Drift resonance between particles and toroidal ULF waves can occur even without the noon-midnight asymmetry of background magnetic field. This effect originates from the wave-carried compressional magnetic field oscillations, which turn out to play a key role in the energy exchange between toroidal ULF waves and charged particles. The resulting particle motion can be described by a modified pendulum equation with solutions depending on the wave number. These findings demonstrate that toroidal ULF waves, like their poloidal counterparts, play an important role in magnetospheric particle dynamics. This is significant because the new derivation allows a large body of existing work and understanding on the pendulum equation to be brought to bear on the problem of ULF wave-particle interactions in the inner magnetosphere. As particle detector technology improves, more nonlinear features will become observable, allowing further tests of the theory presented here.