The density spike in cosmic-ray-modified shocks: Formation, evolution, and instability

We examine the formation and evolution of the density enhancement (density spike) that appears downstream of strong, cosmic-ray-modified shocks. This feature results from temporary overcompression of the flow by the combined cosmic-ray shock precursor and gas subshock. Formation of the density spike is expected whenever shock modification by cosmic-ray pressure increases strongly. That occurrence may be anticipated for newly generated strong shocks, or for strong shocks or cosmic-ray-modified shocks encountering a region of higher external density, for example. The predicted mass density within the spike increases with the shock Mach number and with shocks more dominated by cosmic-ray pressure. For very strong shocks, the total compression compared to the upstream gas may approach D(γ_g + 1)/(γ_g - 1) during the formation period, where γ_g is the gas adiabatic index and D is the compression ratio through the precursor. As the full shock reaches equilibrium, the spike detaches, lags behind the modified shock transition, and is further compressed, so that the density can exceed the limit quoted above. We find this spike to be linearly unstable under a modified Rayleigh-Taylor instability criterion at the early stage of its formation. Our linear analysis shows that the flow is unstable when the gradients of total pressure (gas pressure + cosmic-ray pressure) and gas density have opposite signs. We confirm this numerically using two independent codes based on the two-fluid model for cosmic-ray transport. These two-dimensional simulations show that the instability grows impulsively at early stages and then slows down as the gradients of total pressure and gas density decrease. Flow within the density spike becomes disordered through the instability. It seems likely that this can significantly increase the local magnetic field beyond compressional effects. Observational discovery of this unstable density spike behind shocks, possibly through radio emission enhanced by the amplified magnetic fields, would provide evidence for the existence of strongly cosmic-ray-modified shock structures.