Extension of the Electron Dissipation Region in Collisionless Hall MHD Reconnection

This paper presents Sweet-Parker type scaling arguments in the context of hyper-resistive Hall magnetohyrdodynamics (MHD). The predicted steady state scalings are consistent with those found by Chac\'on et al. [PRL 99, 235001 (2007)], though as with that study, no prediction of electron dissipation region \emph{length} is made. Numerical experiments confirm that both cusp-like and modestly more extended geometries are realizable. However, importantly, the length of the electron dissipation region, which is taken as a parameter by several recent studies, is found to depend explicitly on the level of hyper-resistivity. Furthermore, although hyper-resistivity can produce more extended electron dissipation regions, the length of the region remains smaller than one ion skin depth for the largest values of hyper-resistivity considered here. These electron dissipation regions are significantly shorter than those seen in many recent kinetic studies. The length of the electron dissipation region is found to depend on electron inertia as well, scaling like $(m_e/m_i)^{3/8}$. However, the thickness of the region appears to scale similarly, so that the aspect ratio is at most very weakly dependent on $(m_e/m_i)$. The limitations of scaling theories which do not predict the length of the electron dissipation region are emphasized.