First-Principles Fermi Acceleration in Magnetized Turbulence

This Letter provides a concrete implementation of Fermi’s model of particle acceleration in magnetohydrodynamic (MHD) turbulence, connecting the rate of energization to the gradients of the velocity of magnetic field lines, which it characterizes within a multifractal picture of turbulence intermittency. It then derives a transport equation in momentum space for the distribution function. This description is shown to be substantiated by a large-scale numerical simulation of strong MHD turbulence. The present general framework can be used to model particle acceleration in a variety of environments.

Figure 1

Statistics $|{\mathrm{\Gamma }}_l|{\mathsf{p}}_{|{\mathrm{\Gamma }}_l|}$ of the absolute values of the gradients ${\mathrm{\Gamma }}_l$ (${\mathrm{\Gamma }}_l$ expressed in units of $c/{\ell }_c$). Symbols, values recorded in a ${1024}^3$ MHD simulation at various coarse-graining scales $l$, as indicated; squares (respectively, diamonds), gradients measured along (respectively, perpendicular to) the mean magnetic field direction as coarse grained on scale $l$. The PDF reveals power-law tails at large values of $|{\mathrm{\Gamma }}_l|$ on small scales. Solid red line, adjustment of a multifractal log-normal model; solid blue line, a broken power-law approximation; dotted line, shot noise level associated with the finite sampling variance.

Figure 2

Particle spectra obtained at the final time $t=2.8{\ell }_c/c$ for various initial momenta as indicated in units of $p_c$; $p_c$ is such that $l_g(p_c)={\ell }_c$. The red and blue lines correspond to the two models shown in Fig. 1.

Figure 3

Same as Fig. 2, now showing the particle spectra at various times, for the smallest initial injection momentum, as indicated.