Nonlinear wave-particle interaction upstream from the Earth's bow shock

Well-defined ring-like backstreaming ion distributions have been recently r eported from observations made by the 3DP/PESA-High analyzer onboard the WI ND spacecraft in the Earth's foreshock at large distances from the bow shoc k, which suggests a local production mechanism. The maximum phase space den sity for these distributions remains localized at a nearly constant pitch-a ngle value for a large number of gyroperiods while the shape of the distrib ution remains very steady. These distributions are also observed in associa tion with quasimonochromatic low frequency (similar to 50 mHz) waves with s ubstantial amplitude (delta B/B>0.2). The analysis of the magnetic field da ta has shown that the waves are propagating parallel to the background fiel d in the right-hand mode. Parallel ion beams are also often observed in the same region before the observation of both the ring-like distributions and the waves. The waves appear in cyclotron resonance with the ion parallel b eams. We investigate first the possibility that the ion beams could provide the free energy source for driving an ion/ion instability responsible for the ULF wave occurrence. For that, we solve the wave dispersion relation wi th the observed parameters. Second, we show that the ring-like distribution s could then be produced by a coherent nonlinear wave-particle interaction. It tends to trap the ions into narrow cells in velocity space centered on a well-defined Ditch-angle, directly related to the saturation wave amplitu de in the analytical theory. The theoretical predictions are in good quanti tative agreement with the observations.