Acceleration and collimation of magnetized winds

The acceleration-collimation problem is discussed for stationary, axisymmetric, polytropic, non-relativistic MHD outflows, with causality and the current-closure condition taken into account. To elucidate the properties of physically realizable ‘quasi-conical’ winds, we consider four kinds of rather unphysical flows in contrast, namely ‘radial’, ‘asymptotic’, ‘conical’ and ‘current-free’ flows. ‘Radial’ flows are supposed to possess the radial structure from the source to infinity, thereby not fulfilling the transfield equation, though keeping causal contact with the source. ‘Asymptotic’ flows coincide in the asymptotic domain with the ‘quasi-conical’ winds, and ones extrapolated inwards from them through the subasymptotic domain to the source. Thirdly, ‘conical’ flows are supposed to satisfy the transfield equation in the subasymptotic domain; thus they are not literally conical, but are supposed to satisfy the ‘solvability condition at infinity for the conical structure’. It is, however, argued that there is one difficulty in connecting the asymptotic conical structure causally to the structure upstream. Finally, ‘current-free’ flows with no poloidal and toroidal currents everywhere in the wind zone are treated, but it is pointed out that there is no means of satisfying the current-closure condition in the wind zone. Of physical relevance are the ‘quasi-conical’ winds, for which it is shown that the condition that open field lines in the wind zone can reach infinity leads to the requirement that the Poynting flux, proportional to ζ=αρϖ2η, is not carried to infinity along these field lines, i.e., ζ→0, where α is the angular velocity of field lines, ρ the gas density, and η the mass flux per unit flux tube. While ζ decreases from a value of ζB=ζA+4πηδα near the coronal base through at the Alfvénic surface to null at infinity, the specific angular momentum of the flow increases up to and the flow energy reaches nearly at infinity, where δ is a constant of the Bernouilli integral, and ϖA is the axial distance of the Alfvénic surface. It is also argued that ‘quasi-conical’ winds with the current-closure condition fulfilled in the wind zone possess the two-componentness of outflow as one of their generic properties.