# A nonlinear Monte Carlo model of super-diffusive shock acceleration with magnetic field amplification

## TL;DR

This paper introduces a nonlinear Monte Carlo model for super-diffusive shock acceleration that captures magnetic field amplification and anisotropic particle distributions, providing insights into shock precursor dynamics and magnetic turbulence.

## Contribution

It presents a novel Monte Carlo approach that models super-diffusive particle transport and magnetic field amplification in collisionless shocks, including full angular anisotropy.

## Key findings

- Super-diffusive transport causes strong quadruple anisotropy in particle distribution.
- Pressure anisotropy leads to non-resonant mirror-type instability.
- Model offers a comprehensive description of shock structure and magnetic turbulence.

## Abstract

Fast collisionless shocks in cosmic plasmas convert their kinetic energy flow into the hot downstream thermal plasma with a substantial fraction of energy going into a broad spectrum of superthermal charged particles and magnetic fluctuations. The superthermal particles can penetrate into the shock upstream region producing an extended shock precursor. The cold upstream plasma flow is decelerated by the force provided by the superthermal particle pressure gradient. In high Mach number collisionless shocks, efficient particle acceleration is likely coupled with turbulent magnetic field amplification (MFA) generated by the anisotropic distribution of accelerated particles. This anisotropy is determined by the fast particle transport making the problem strongly nonlinear and multi-scale. Here, we present a nonlinear Monte Carlo model of collisionless shock structure with super-diffusive propagation of high-energy Fermi accelerated particles coupled to particle acceleration and MFA which affords a consistent description of strong shocks. A distinctive feature of the Monte Carlo technique is that it includes the full angular anisotropy of the particle distribution at all precursor positions. The model reveals that the super-diffusive transport of energetic particles (i.e., Levy-walk propagation) generates a strong quadruple anisotropy in the precursor particle distribution. The resultant pressure anisotropy of the high-energy particles produces a non-resonant mirror-type instability which amplifies compressible wave modes with wavelengths longer than the gyroradii of the highest energy protons produced by the shock.