Nonlinear Dynamics of Acoustic Instability in a Cosmic Ray Shock Precursor and Its Impact on Particle Acceleration

An acoustic instability of a shock precursor driven by the pressure gradient of accelerated particles is studied in the nonlinear regime. The nonlinearity steepens unstable acoustic waves and turns them into shocks. The shocks form a “shocktrain” but they may merge into each other. Traveling wave solutions are obtained analytically in two different cases. In the first case, only acoustic instability is included and the characteristic scale (distance between the shocks) is limited only by the system size (while shocks merge). In the second case, the instability develops out of cyclotron unstable seed magnetohydrodynamic (MHD) waves. The spatial distance between the MHD wave packets sets the scale of the acoustic shocktrain. The internal structure of the individual shocks is presumably determined by the ion skin depth c/ωpi and by the relaxation length of slightly superthermal particle distributions near these shocks. The shocks are assumed to be arbitrarily thin compared with the distance between them which is ensured by a small viscous term. Both types of solutions are dynamically verified by numerical calculations. The hydromagnetic flow in the shock precursor emerging from the acoustic instability is crucial for two recently suggested phenomena in diffusive shock acceleration. One phenomenon is the enhancement of the acceleration rate well above its standard (Bohm) value due to the narrowing of the shock precursor. The second phenomenon is the amplification of the long-scale magnetic field by an inverse cascade of Alfvén waves generated by accelerated particles and scattered in k-space on the acoustic perturbations.