On the self-consistent model of the axisymmetric radio pulsar   magnetosphere
  We consider a model of axisymmetric neutron star magnetosphere. In our approach, the current density in the region of open field lines is constant and the returning current flows in a narrow layer along the separatrix. In this case, the stream equation describing the magnetic field structure is linear both in the open and closed regions, the main problem lying in matching the solutions along the separatrix (Okamoto 1974; Lyubarskii 1990). We demonstrate that it is the stability condition on the separatrix that allows us to obtain a unique solution of the problem. In particular, the zero point of magnetic field is shown to locate near the light cylinder. Moreover, the hypothesis of the existence of the nonlinear Ohm's Law (Beskin, Gurevich & Istomin 1983) connecting the potential drop in the pair creation region and the longitudinal electric current flowing in the magnetosphere is confirmed. 
V.S. Beskin and L.M. Malyshkin