The effect of electron inertia in Hall-driven magnetic field penetration in electron-magnetohydrodynamics

Magnetic field penetration in electron-magnetohydrodynamics (EMHD) can be driven by density gradients through the Hall term [Kingsep et al., Sov. J. Plasma Phys. 10, 495 (1984)]. Particle-in-cell simulations have shown that a magnetic front can go unstable and break into vortices in the Hall-driven EMHD regime. In order to understand these results, a new fluid model had been derived from the L y / L n ≪ 1 limit of EMHD, where Ly is the length scale along the front and Ln is the density gradient length scale. This model is periodic in the direction along the magnetic front, which allows the dynamics of the front to be studied independently of electrode boundary effects that could otherwise dominate the dynamics. Numerical solutions of this fluid model are presented that show for the first time the relation between Hall-driven EMHD, electron inertia, the Kelvin-Helmholtz (KH) instability, and the formation of magnetic vortices. These solutions show that a propagating magnetic front is unstable to the same KH mode predicted for a uniform plasma. This instability causes the electron flow to break up into vortices that are then driven into the plasma with a speed that is proportional to the Hall speed. This demonstrates that, in two-dimensional geometry with sufficiently low collisionality [collision rate ν ≲ v Hall / ( 4 δ e ) ], Hall-driven magnetic penetration occurs not as a uniform shock front but rather as vortex-dominated penetration. Once the vortices form, the penetration speed is found to be nearly a factor of two larger than the redicted speed ( v Hall / 2 ) obtained from Burgers' equation in the one-dimensional limit.