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

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 $\mathbf{V}=\text{wh}\mathsf{z}\text{btimes}\mathrm{\nabla }\phi +V_z\text{wh}\mathsf{z}$ with the magnetic field $\mathbf{B}=\mathrm{\nabla }\psi \text{btimes}\text{wh}\mathsf{z}+B_z\text{wh}\mathsf{z}$ 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 $(B_x=\mathrm{\partial }\psi \mathrm{/}\mathrm{\partial }y,B_y=-\mathrm{\partial }\psi \mathrm{/}\mathrm{\partial }x)$. 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.