Turbulence and particle acceleration in collisionless supernovae remnant shocks

I. Anisotropic spectra solutions

¹ Laboratoire d'Astrophysique de Grenoble, CNRS, Université Joseph Fourier II, BP 53, 38041 Grenoble, and Institut Universitaire de France, France e-mail: Guy.Pelletier@obs.ujf-grenoble.fr
² Institut d'Astrophysique de Paris, UMR 7095 CNRS, Université Pierre & Marie Curie, 98bis boulevard Arago, 75014 Paris, France e-mail: lemoine@iap.fr
³ Centre d'Études Spatiales et du Rayonnement, CNRS, Université Paul Sabatier, 9 avenue du Colonel Roche, 31028 Toulouse Cedex, France e-mail: Alexandre.Marcowith@cesr.fr

Received: 20 December 2005
Accepted: 16 March 2006

Abstract

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 , where  is the ratio of the cosmic-ray pressure to the shock kinetic energy. The instability determines the dependence of the spectrum with respect to  (wavenumbers along the shock normal). The transverse cascade of Alfvén waves simultaneously determines the dependence in . 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 . Downstream, anisotropy is further enhanced through the compression at shock crossing.