Theoretical model of steady-state magnetic reconnection in collisionless incompressible plasma based on the Grad–Shafranov equation solution

The problem of steady-state magnetic reconnection in an infinite current layer in collisionless, incompressible, nonresistive plasma, except of the electron diffusion region, is examined analytically using the electron Hall magnetohydrodynamics approach. It is found that this approach allows reducing the problem to the magnetic field potential finding, while last one has to satisfy the Grad–Shafranov equation. The obtained solution demonstrates all essential Hall reconnection features, namely proton acceleration up to Alfvén velocities, the forming of Hall current systems and the magnetic field structure expected. It turns out that the necessary condition of steady-state reconnection to exist is an electric field potential jump across the electron diffusion region and the separatrices. Besides, the powerful mechanism of electron acceleration in X-line direction is required. It must accelerate electrons up to the electron Alfvén velocity inside the diffusion region and on the separatrixes. This is a necessary condition for steady-state reconnection as well.