Particle acceleration at a turbulent shock

Cosmic rays can be accelerated so efficiently at shock waves by the first-order Fermi mechanism that their pressure can be comparable to or exceed that of the thermal gas. Consequently, it was recognized that CRs could modify the structure of the shock itself, typically broadening it by the CR diffusive length scale. Besides modifying the accelerated particle spectrum, the CR pressure gradient across the shock can destabilize short-wavelength compressive fluctuations (such as acoustic waves), leading to a highly turbulent shock structure. In this paper, we investigate particle acceleration at a turbulence modified shock. We show that the presence of the turbulence can be treated as an effective diffusion coefficient, an effective flow speed and an effective second order Fermi diffusion.