Steepening of Cosmic-Ray Spectra in Shocks with Varying Magnetic Field Direction

Cosmic-ray (CR) spectra, both measured upon their arrival at the Earth's atmosphere and inferred from the emission in supernova remnants (SNRs), appear to be significantly steeper than the standard diffusive shock acceleration (DSA) theory predicts. Although the reconstruction of the primary spectra introduces an additional steepening due to propagation effects, there is a growing consensus in the CR community that these corrections fall short to explain the newest high-precision data. Using 2D hybrid simulations, we investigate a new mechanism that may steepen the spectrum during the acceleration in SNR shocks. Most of the DSA treatments are limited to homogeneous shock environments. To investigate whether inhomogeneity effects can produce the necessary extra steepening, we assume that the magnetic field changes its angle along the shock front. The rationale behind this approach is the strong dependence of the DSA efficiency upon the field angle, theta(Bn). Our results show that the variation of shock obliquity along its face results in a noticeable steepening of the DSA spectrum. Compared to simulations of quasi-parallel shocks, we observe an increase of the spectral index by Delta q=0.1-0.15. Possible extrapolation of the limited simulation results to more realistic SNR conditions are briefly considered.