Shell models for Hall effect induced magnetic turbulence

Abstract

The Hall effect occurs in strongly magnetized conductive media and results in non-dissipative currents perpendicular to the electric field. We discuss its influence on the magnetic field dynamics ignoring fluid motion and ambipolar diffusion. The magnetic field evolution can then be basically similar to that of the velocity field in hydrodynamic turbulence resulting in a magnetic turbulence. Shell models for the induction equation with Hall effect are constructed on the basis of the conservation of magnetic energy and helicity in the dissipation-free limit. Numerical simulations of these models indicate that a magnetic energy cascade does occur, but the time behaviour and spatial spectrum of the magnetic field are very different from those of the velocity in shell models of hydrodynamic turbulence.