On the central core in MHD winds and jets

We demonstrate that the 1D cylindrical version of the Grad-Shafranov equation is more rich than classical self-similar ones, and more suitable for the astrophysical jets we observe. In particular, it allows us to describe the central (and, hence, the most energetic) part of the flow. Both relativistic and non-relativistic versions are discussed. It is shown that taking into account the finite pressure of the external media one can determine the magnetic flux within the central core. We found as well that for non-relativistic flows which are magnetically dominated near the origin the solution can be constructed only in the presense of the oblique shock near the base of a jet where the additional heating is to take place.