Nonlinear aspects of stochastic particle acceleration

In turbulent magnetized plasmas, charged particles can be accelerated to high energies through their interactions with the turbulent motions. As they do so, they draw energy from the turbulence, possibly up to the point where they start modifying the turbulent cascade. Stochastic acceleration then enters a nonlinear regime because turbulence damping back-reacts in turn on the acceleration process. This article develops a phenomenological model to examine this situation in detail and to explore its consequences for the particle and turbulent energy spectra. We determine a criterion that specifies the threshold of nonthermal particle energy density and the characteristic momentum beyond which back-reaction becomes effective. Once the back-reaction sets in, the turbulence cascade becomes damped below a length scale that keeps increasing in time. The accelerated particle momentum distribution develops a near power-law of the form ${\rm d}n/{\rm d}p\propto p^{-s}$ with $s\sim2$ beyond the momentum at which back-reaction first sets in. At very high energies, where the gyroradius of accelerated particles becomes comparable to the outer scale of the turbulence, the energy spectrum can display an even harder spectrum with $s\sim 1.3-1.5$ over a short segment. The low-energy part of the spectrum, below the critical momentum, is expected to be hard ($s\sim 1$ or harder), and shaped by any residual acceleration process in the damped region of the turbulence cascade. This characteristic broken power-law shape with $s\sim 2$ at high energies may find phenomenological applications in various high-energy astrophysical contexts.