Magnetic-field amplification is an integral part of the process of particle acceleration at nonrelativistic shocks. It is necessary to reach the maximum energies required by observations, especially in supernova remnants, which are thought to be sources of the majority of Galactic cosmic rays. Such amplification can be caused by the acoustic instability that develops when small density perturbations interact with the cosmic-ray pressure gradient upstream of a cosmic-ray-modified shock. The vorticity induced by the nonlinear development of the instability may lead to turbulence, which amplifies the preexisting magnetic fields. To study this phenomenon, we used the PLUTO code to carry out 2D (and some 3D) magnetohydrodynamical simulations of the evolution of small density perturbations in the presence of an assigned cosmic-ray pressure gradient. Adopting more realistic values of Mach number and cosmic-ray acceleration efficiency than previously assumed in the literature, we show that the acoustic instability can transform small density perturbations into large nonlinear structures, while the fluid crosses the precursor region of a cosmic-ray-modified shock. We studied the power spectrum of turbulent magnetic fluctuations that may be important to scatter particles. We comment on the possible constructive interference between acoustic and nonresonant streaming instabilities. We discuss limitations of previous and current numerical investigations in accessing spatial scales where turbulence is expected to turn nonlinear, and we outline perspectives for future investigations.
Capanema et al. (Fri,) studied this question.