  Supernovae remnant shock waves could be at the origin of cosmic rays up to
energies in excess of the knee ($E\simeq3\cdot 10^{15} $eV) if the magnetic
field is efficiently amplified by the streaming of accelerated particles in the
shock precursor. This paper follows up on a previous paper \citep{pell05} which
derived the properties of the MHD turbulence so generated, in particular its
anisotropic character, its amplitude and its spectrum. In the present paper, we
calculate the diffusion coefficients, also accounting for compression through
the shock, and show that the predicted three-dimensional turbulence spectrum
$k_\perp S_{3\rm d}(k_\parallel,k_\perp)\propto
k_\parallel^{-1}k_\perp^{-\alpha}$ (with $k_\parallel$ and $k_\perp$ the
wavenumber components along and perpendicular to the shock normal) generally
leads to Bohm diffusion in the parallel direction. However, if the anisotropy
is constrained by a relation of the form $k_\parallel \propto k_\perp^{2/3}$,
which arises when the turbulent energy cascade occurs at a constant rate
independent of scale, then the diffusion coefficient loses its Bohm scaling and
scales as in isotropic Kolmogorov turbulence. We show that these diffusion
coefficients allow to account for X-ray observations of supernova remnants.
This paper also calculates the modification of the Fermi cycle due to the
energy lost by cosmic rays in generating upstream turbulence and the
concomittant steepening of the energy spectrum. Finally we confirm that cosmic
rays can produced an amplified turbulence in young SNr during their free
expansion phase such that the maximal energy is close to the knee and the
spectral index is close to 2.3 in the warm phase of the interstellar medium
