Plan MHD flows in a hyperbolic magnetic field: implications for the problem of magnetic field line reconnexion

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Abstract
This paper examines the nature of plane, steady, incompressible MHD flow in a purely hyperbolic magnetic field. It is shown that in such a magnetic field the MHD equations can yield exact analytic solutions for the plasma flow. The flow has the following properties. In the far region where the conductivity is assumed to be sufficiently high so that the plasma is effectively ‘ frozen ’ to the magnetic field, the flow pattern is radial. The plasma motion is directed towards the neutral line in the incident ‘ sectors ’ and away from it in the outgoing ‘ sectors ’ with a consequent reversal in direction across the magnetic separatrices where the solution becomes singular. The plasma pressure and density in this flow are calculated and it is shown that the latter remains constant along a streamline. It is further shown that a uniform finite conductivity is not compatible with a stagnation point at the magnetic null point. However, for a parabolic increase of conductivity with increasing distance from that point, plasma flow with uniform density along hyperbolic streamlines is shown to be possible. The relevance of these flows to the magnetic field merging problem is discussed.