ASYMPTOTIC STRUCTURE OF HYDROMAGNETICALLY DRIVEN RELATIVISTIC WINDS

We study the asymptotic properties of fully relativistic, steady, axisymmetric, hydromagnetic winds. In agreement with Heyvaerts and Norman, who considered only the nonrelativistic limit, we find that all flux surfaces generally converge to either cylinders or paraboloids that are nested around the rotation axis. First we exclude asymptotically horizontal flux surfaces solely from consideration of energy and magnetic flux conservation. Then, by taking into account the cross-field force balance, we show that asymptotically current-carrying paraboloidal and conical flux surfaces cannot occur arbitrarily close to the rotation axis. Asymptotically cylindrical flux surfaces, while having no difficulty in filling out the polar region, cannot extend to infinity transversely unless the terminal velocity on the last flux surface vanishes, which renders it implausible for them to match onto other types of asymptotes. Asymptotically cylindrical flux surfaces with finite transverse radius, either self-confined or externally confined, are possible. We also show that asymptotically current-free paraboloidal flux surfaces may exist, filling up the entire space without difficulty. We show that the terminal Lorentz factor on any asymptotically cylindrical flux surface is directly proportional to the current enclosed within that surface, provided that the surface extends to a radius well beyond the light cylinder. For current-free paraboloidal surfaces, all of the energy in the flow goes asymptotically into kinetic energy, in sharp contrast to other flow geometries in which the Poynting flux remains important at large distances. The results presented here are potentially important for understanding the structures of relativistic jets in active galactic nuclei and of pulsar winds.