Anomalous diffusion of field lines and charged particles in Arnold-Beltrami-Childress force-free magnetic fields

Abstract

The cosmic magnetic fields in regions of low plasma pressure and large currents, such as in interstellar space and gaseous nebulae, are force-free in the sense that the Lorentz force vanishes. The three-dimensional Arnold-Beltrami-Childress (ABC) field is an example of a force-free, helical magnetic field. In fluid dynamics, ABC flows are steady state solutions of the Euler equation. The ABC magnetic field lines exhibit a complex and varied structure that is a mix of regular and chaotic trajectories in phase space. The characteristic features of field line trajectories are illustrated through the phase space distribution of finite-distance and asymptotic-distance Lyapunov exponents. In regions of chaotic trajectories, an ensemble-averaged variance of the distance between field lines reveals anomalous diffusion—in fact, superdiffusion—of the field lines. The motion of charged particles in the force-free ABC magnetic fields is different from the flow of passive scalars in ABC flows. The particles do not necessarily follow the field lines and display a variety of dynamical behavior depending on their energy, and their initial pitch-angle. There is an overlap, in space, of the regions in which the field lines and the particle orbits are chaotic. The time evolution of an ensemble of particles, in such regions, can be divided into three categories. For short times, the motion of the particles is essentially ballistic; the ensemble-averaged, mean square displacement is approximately proportional to t2, where t is the time of evolution. The intermediate time region is defined by a decay of the velocity autocorrelation function—this being a measure of the time after which the collective dynamics is independent of the initial conditions. For longer times, the particles undergo superdiffusion—the mean square displacement is proportional to tα, where α > 1, and is weakly dependent on the energy of the particles. These super-diffusive characteristics, both of magnetic field lines and of particles moving in these fields, strongly suggest that theories of transport in three-dimensional chaotic magnetic fields need a shift from the usual paradigm of quasilinear diffusion.