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 ξCR c, where ξCR 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.