Flat Spectra of Energetic Particles in Interplanetary Shock Precursors

Abstract: The observed energy spectra of accelerated particles at interplanetary shocks often do not match the diffusive shock acceleration (DSA) theory predictions. In some cases, the particle flux forms a plateau over a wide range of energies, extending upstream of the shock for up to seven flux's e-folds before submerging into the background spectrum. Remarkably, at and behind the shock that we have studied in detail, the flux falls off in energy as $\epsilon<sup>{-1}$, consistent with the DSA prediction for a strong shock. The upstream plateau suggests a different particle transport mechanism than those traditionally employed in DSA models. We show that a standard (linear) DSA solution based on a widely accepted diffusive particle transport with an underlying resonant wave-particle interaction is inconsistent with the plateau in the particle flux. To resolve this contradiction, we modify the DSA theory in two ways. First, we include a dependence of the particle diffusivity $\kappa$ on the particle flux $F$ (nonlinear particle transport). Second, we invoke short-scale magnetic perturbations that are self-consistently generated by, but not resonant with, accelerated particles. They lead to the particle diffusivity increasing with the particle energy as $\propto\epsilon<sup>{3/2}$ that simultaneously decreases with the particle flux as $1/F$. The combination of these two trends results in the flat spectrum upstream.