Scaling of magnetic reconnection with a limited x-line extent

Magnetic reconnection is a fundamental physical process that is responsible for releasing the magnetic energy during substorms of planetary magnetotails. Previous studies of magnetic reconnection usually take the two-dimensional (2D) approach, which assumes that reconnection is uniform in the 3rd direction out of the 2D reconnection plane. However, observations suggest that reconnection can be limited in the 3rd direction, such as reconnection at Mercury's magnetotail. It turns out that reconnection can be suppressed when reconnection region is very limited in the 3rd direction. An internal x-line asymmetry along the current direction develops because of the transport of reconnected magnetic flux by electrons beneath the ion kinetic scale, resulting in a suppression region identified in Liu et al., 2019. Under the guidance of a series of 3D kinetic simulations, in this work, we incorporate the length-scale of this suppression region ~10d_i to quantitatively model the reduction of the reconnection rate and the maximum outflow speed observed in the short x-line limit. The average reconnection rate drops because of the limited active region (where the current sheet thins down to the electron inertial scale) within an x-line. The outflow speed reduction correlates with the decrease of the J×B force, that can be modeled by the phase shift between the J and B profiles, also as a consequence of the flux transport. Notably, these two quantities are most essential in defining the well-being of magnetic reconnection, which can tell us when reconnection shall be suppressed.