On the Viscous Boundary Layer near the Center of the Resistive Reconnection Region
Authors: Dmitri A. Uzdensky and Russell M. Kulsrud
This paper studies the behavior of the magnetic field near the center of the reconnection layer in the framework of two-dimensional incompressible resistive magnetohydrodynamics with uniform resistivity in a steady state. Priest and Cowley have presented an argument [1] showing that when the viscosity is zero, the magnetic separatrices do not cross at a finite angle but osculate at the X-point. In the present paper, it is shown that this conclusion is in fact not correct. First, some results of numerical simulations of the reconnection layer are presented. These results contradict the conclusions of Priest and Cowley. In order to explain this contradiction, an analytical theory for the neighborhood of the X-point is developed in the second part of the paper. It is found that, if the viscosity is exactly equal to zero, then one of the critical assumptions of the above mentioned argument, namely the assumption that the stream function can be Taylor-expanded near the X-point, breaks down. In the case of small but finite viscosity, a boundary layer analysis in the vicinity of the neutral point is carried out. Some of the higher derivatives of the stream function become very large near the X-point, leading to a non-zero angle between the separatrices. As viscosity goes to zero, the boundary layer shrinks and one can see the emergence of the non-analytic logarithmic terms in the expansion of the stream function in the outer region. The results of the boundary layer analysis are found to be in good agreement with the numerical simulations.