On the Flat Proton Spectra at Interplanetary Shocks

Spacecraft observations of interplanetary shocks have revealed signi cant deviations in energetic particle spectra from the diffusive shock acceleration (DSA) theory predictions. Within almost two decades of particle energy, spanning about seven e-folds upstream, the particle ux is almost energy independent. Although at and behind the shock, it falls off as ϵ⁻¹ (as predicted by DSA for reasonably strong shocks), the ux decreases with the coordinate close to the shock upstream progressively steeper at lower energies, which leads to a at energy distribution. Within a standard DSA solution under a xed turbulence spectrum, pre-existing or self-excited by accelerated particles, a at particle spectrum over an extended upstream area means that the particle diffusivity must be energy-independent, contrary to most transport models. We propose a resolution of this paradox by invoking a strongly nonlinear solution upstream under a self-driven but short-scale turbulence, in which the particle diffusivity increases with energy as ∝ ϵ³/², but also decays with the wave energy as 1/E_w, which compensate for the ϵ³/² rise. The main difference with the traditional DSA is that the wave-particle interaction is nonresonant, and the turbulence is not saturated at the Bohm level (that would require δB ∼ B₀ turbulence saturation amplitude). A steep, energy-dependent nal drop in the particle ux far ahead of the shock to its background level in the solar wind is likely due to a quick particle escape upstream caused by turbulence de ciency.