  Context. Supernova remnants are known as sources of galactic cosmic rays for
their non-thermal emission of radio waves, X-rays, and gamma-rays. However, the
observed soft broken power-law spectra are hard to reproduce within standard
acceleration theory based on the assumption of Bohm diffusion and steady-state
calculations. Aims. We point out that a time-dependent treatment of the
acceleration process together with a self-consistent treatment of the
scattering turbulence amplification is necessary. Methods. We numerically solve
the coupled system of transport equations for cosmic rays and isotropic
Alfvenic turbulence. The equations are coupled through the growth rate of
turbulence determined by the cosmic-ray gradient and the spatial diffusion
coefficient of cosmic rays determined by the energy density of the turbulence.
The system is solved on a co-moving expanding grid extending upstream for
dozens of shock radii, allowing for the self-consistent study of cosmic-ray
diffusion in the vicinity of their acceleration site. The transport equation
for cosmic rays is solved in a test-particle approach. Results. We demonstrate
that the system is typically not in a steady state. In fact, even after several
thousand years of evolution, no equilibrium situation is reached. The resulting
time-dependent particle spectra strongly differ from those derived assuming a
steady state and Bohm diffusion. Our results indicate that proper accounting
for the evolution of the scattering turbulence and hence the particle diffusion
coefficient is crucial for the formation of the observed soft spectra. In any
case, the need to continuously develop magnetic turbulence upstream of the
shock introduces non-linearity in addition to that imposed by cosmic-ray
feedback.
