On the Flat Proton Spectra at Interplanetary Shocks

Spacecraft observations of interplanetary shocks have revealed signi(cid:28)cant deviations in energetic particle spectra from the di(cid:27)usive shock acceleration (DSA) theory predictions. Within almost two decades of particle energy, spanning about seven e-folds upstream, the particle (cid:29)ux is almost energy independent. Although at and behind the shock, it falls o(cid:27) as ϵ − 1 (as predicted by DSA for reasonably strong shocks), the (cid:29)ux decreases with the coordinate close to the shock upstream progressively steeper at lower energies, which leads to a (cid:29)at energy distribution. Within a standard DSA solution under a (cid:28)xed turbulence spectrum, pre-existing or self-excited by accelerated particles, a (cid:29)at particle spectrum over an extended upstream area means that the particle di(cid:27)usivity 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 di(cid:27)usivity increases with energy as ∝ ϵ 3 / 2 , but also decays with the wave energy as 1 /E w , which compensate for the ϵ 3 / 2 rise. The main di(cid:27)erence 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 ∼ B 0 turbulence saturation amplitude). A steep, energy-dependent (cid:28)nal drop in the particle (cid:29)ux far ahead of the shock to its background level in the solar wind is likely due to a quick particle escape upstream