A statistical examination of ion dynamics at 1-Hz whistler waves in the Earth's foreshock

The 1-Hz whistler wave precursor attached to shock-like structures are often observed in foreshock. Using observations from the Magnetospheric Multiscale mission, we investigate the interactions between 1-Hz waves and ions. Incoming solar wind ions do not gyro-resonate with the wave, since typically the wave is right-handed in their frame. We demonstrate that solar wind ions commonly exhibit 180 gyro-phase bunching from the wave magnetic field, understanding it with a reconciled linear picture for non-resonant ions and non-linear trapping theory of anomalous resonance. Along the longitudinal direction, solar wind ions experience Landau resonance, exhibiting either modulations at small wave potentials or trapping in phase-space holes at large potentials. The results also improve our understanding of foreshock structure evolution and 1-Hz wave excitation. Shock-like structures start with having incoming solar wind and remotely-reflected ions from further downstream. The ion-scale 1-Hz waves can already appear during this stage. The excitation may be due to shock-like dispersive radiation or kinetic instabilities resonant with these remotely-reflected ions. Ions reflected by local shock-like structures occur later, so they are not always necessary for generating 1-Hz waves. The wave leads to ion reflection further upstream, which may cause reformation. In one event, locally-reflected ions exhibit anomalous resonance in the early stage, and later approach to the gyro-resonant condition with gyro-phases ~270 . The latter is possibly due to nonlinear trapping in regions with an upstream-pointing magnetic field gradient, linked to reformation. Some additional special features like frequency dispersions are observed, requiring better explanations in the future.