Turbulence and particle acceleration in collisionless supernovae remnant shocks. I. Anisotropic spectra solutions

This paper investigates the nature of the MHD turbulence excited by the streaming of accelerated cosmic rays in a shock wave precursor. The two recognised regimes (non-resonant and resonant) of the streaming instability are taken into account. We show that the non-resonant instability is very efficient and saturates through a balance between its growth and non-linear transfer. The cosmic-ray resonant instability then takes over and is quenched by advection through the shock. The level of turbulence is determined by the non-resonant regime if the shock velocity V_sh is larger than a few times ξ_CR c, where ξ_CR is the ratio of the cosmic-ray pressure to the shock kinetic energy. The instability determines the dependence of the spectrum with respect to k_∥ (wavenumbers along the shock normal). The transverse cascade of Alfvén waves simultaneously determines the dependence in k_⊥. We also study the redistribution of turbulent energy between forward and backward waves, which occurs through the interaction of two Alfvén and one slow magneto-sonic wave. Eventually the spectra at the longest wavelengths are found almost proportional to k_∥^-1. Downstream, anisotropy is further enhanced through the compression at shock crossing.