Regularization of the Alfvén singularity by the Hall effect

Abstract

An ideal magnetohydrodynamics (MHD) equilibrium is described by a system of rather complicated singular differential equations when a flow is included. The so-called Alfvén singularity occurs at the place where the Doppler-shifted Alfvén velocity vanishes. It is due to the vanishing of the highest-order derivative in the differential equation. The Hall effect, working as a singular perturbation to the ideal MHD system, yields a new branch of regular solutions that can smoothly connect two regions separated by the Alfvén singularity. The thickness of the transition layer is of the order of the ion skin depth, the intrinsic length scale brought about by the singular perturbation. The regularization mechanism of the nonlinear Hall effect is not as simple as that of the diffusion effect producing an entropy (viscosity) solution in a viscous fluid. The Hall effect removes the restriction binding the magnetic and flow characteristics, and creates the new branch of regularized solutions. One-dimensional analysis loses sight of this new branch of equilibria.