Scenarios for magnetic X-point collapse in 2D incompressible dissipationless extended magnetohydrodynamics

Abstract

The equations of 2D incompressible dissipationless extended magnetohydrodynamics (XMHD) extend the equations of incompressible Hall MHD (HMHD) by retaining finite-electron inertia. These XMHD equations couple the fluid velocity ${\bf V} = \wh{\sf z}\btimes\nabla\phi + V_{z}\,\wh{\sf z}$ with the magnetic field ${\bf B} = \nabla\psi\btimes\wh{\sf z} + B_{z}\,\wh{\sf z}$ in a process that is known to support dissipationless solutions that exhibit finite-time singularities associated with magnetic X-point collapse in the magnetic plane $(B_{x} = \partial\psi/\partial y, B_{y} = -\,\partial\psi/\partial x)$. Here, by adopting a 2D self-similar model for the four XMHD fields $(\phi,\psi,V_{z},B_{z})$, we obtain five coupled ordinary differential equations that are solved in terms of the Jacobi elliptic functions based on an orbital classification associated with particle motion in a quartic potential. Excellent agreement is found when these analytical solutions are compared with numerical solutions, including the precise time of a magnetic X-point collapse.