Current-sheet Evolution near a Hyperbolic Magnetic Neutral Line in Hall Magnetohydrodynamics: An Exact Solution

TL;DR

This paper presents an exact analytical solution for the evolution of current sheets near a hyperbolic magnetic neutral line in Hall MHD, showing Hall effects prevent finite-time singularities at large times.

Contribution

It provides a self-similar, exact solution demonstrating how Hall effects influence current sheet formation and prevent singularities in hyperbolic magnetic neutral lines.

Findings

Hall effect has negligible impact at small times on current sheet formation

Hall effect quenches finite-time singularity at large times

Asymptotic behavior matches steady Hall MHD predictions

Abstract

\large{\bf Abstract-} Unsteady Hall Magnetohydrodynamics (MHD) near a hyperbolic magnetic neutral line is investigated. An exact analytical solution describing a self-similar evolution is given. This solution shows a negligible impact on the current-sheet formation process near the hyperbolic magnetic neutral line at small times by the Hall effect but, subsequently, a quenching by the Hall effect of the finite-time singularity exhibited in ideal MHD and, hence a prevention of the current density blow-up at large times. The asymptotic result given by this time-dependent solution is in full quantitative agreement with the formulation of \textit{steady} Hall MHD near a $X$-type magnetic neutral line (Shivamoggi [23]). The latter formulation showed that this asymptotic result indeed corresponds to a hyperbolic configuration of the magnetic field lines in the \textit{steady} case.