2D simulations of hall-driven magnetic field penetration in electron-magnetohydrodynamics

Magnetic field penetration in electron-magneto-hydrodynamics (EMHD) can be driven by density gradients through the Hall term. Here we describe the effect of electron inertia on simplified one- and two-dimensional models of a magnetic front. Nonlinear effects due to inertia cause the 1D model to develop peaked solitary waves, while in 2D a shear-driven Kelvin-Helholtz (KH) like instability causes the front to break into a series of vortices which propagate into the plasma. The combination of these two effects means that in 2D, Hall driven magnetic field penetration will typically happen in the form of complex vortex-dominated penetration, rather than as a transversely-smooth shock front. Numerical solutions of the 2D KH instability are computed in the limit that the density gradient length scale is much larger than the system size. An initial shock front is found to be unstable, and the development of KH vortices is observed. The propagation speed of the vortices is found to be about a factor of two faster than the propagation speed of the initial shock front.