On the Cylindrical Grad-shafranov Equation

An activity of many compact objects – Active Galactic Nuclei (AGNs), Young Stellar Objects (YSOs), microquasars – is associated with the highly collimated jets. These jets are thought to be a natural outlet of an excess angular momentum of a central object and accreting matter [1]. The latest observations indicating the jet rotation in AGNs [2] and YSOs [3] support this idea. The most attractive model for such outflows is the MHD one [1, 4, 5]. Of course, the main question within this model is the collimation itself [4–8]. We assume here that collimation is due to a finite external gas and/or magnetic pressure [9–11]. Indeed, proposing that it is the external magnetic field Bext ∼ 10 −6 G that plays the main role in the collimation, we obtain rjet ∼ Rin (Bin/Bext) 1/2 . Here r is the distance from the rotational axis, and the subscripts ’in’ correspond to the values in the vicinity of the central object. The similar evaluation can be obtained for external pressure pext ∼ B 2 ext/8π. E.g., for YSOs (Bin ∼ 10 G, Rin ∼ R⊙) we obtain rjet ∼ 10 15 cm, in agreement with observational data. Accordingly, for AGNs (Bin ∼ 10 4 G, Rin ∼ 10 13 cm) we have rjet ∼ 1 pc. It means that the external media may indeed play important role in the collimation process. The internal structure of cylindrical jets was considered both for non-relativistic [12, 13] and relativistic [9, 14–19] flows. In particular, it was shown that for constant angular velocity of plasma ΩF it is impossible to obtain reasonable solution with total zero electric current [9], but it can be constructed if the angular velocity vanishes at the jet boundary and if the external pressure is not equal to zero [11, 17]. Another result was obtained for relativistic and non-relativistic cylindrical flows [13, 15, 16, 20] is that the poloidal magnetic fieldBp has a jet-like form