Steady linear X‐point magnetic reconnection

In the main part of this paper a model for linear reconnection is developed with a current spike around the X‐point and vortex current sheets along the separatrices, which are resolved by the effects of viscosity and magnetic diffusivity. The model contains three regions. In the external ideal region, diffusion effects are negligible, and the flow is purely radial but becomes singular both along the separatrices and at the X‐point. Near the separatrix there is a self‐similar boundary layer with strong electric current and vorticity, where resistivity and viscosity resolve the singularity and allow the flow to cross the separatrix. A composite solution is set up that matches the external and separatrix solutions. Near the origin diffusion also resolves the singularity and is described approximately by a biharmonic solution. A classification of steady two‐dimensional reconnection regimes is proposed into viscous reconnection ( M e > Re ), extra slow (linear) reconnection ( M e < R me −1 ), slow reconnection ( R me −1 < M e R me −½ ), and fast reconnection ( R me −½ < M e ), where M e is the dimensionless reconnection rate, R me the magnetic Reynolds number, and Re the Reynolds number, all based on the Alfven speed far from the reconnection point. Also, an antireconnection theorem is proved, which has profound effects on the nature of linear reconnection. It states that steady two‐dimensional MHD reconnection with plasma flow across the separatrices is impossible in a plasma which is inviscid, highly sub‐Alfvenic, and has uniform magnetic diffusivity.