Self-similar force-free wind from an accretion disc

We consider a self-similar force-free wind flowing out of an infinitely thin disc located in the equatorial plane. On the disc plane, we assume that the magnetic stream function P scales as P α R ν , where R is the cylindrical radius. We also assume that the azimuthal velocity in the disc is constant: ν Φ = Mc, where M < 1 is a constant. For each choice of the parameters v and M, we find an infinite number of solutions that are physically well-behaved and have fluid velocity ≤ c throughout the domain of interest. Among these solutions, we show via physical arguments and time-dependent numerical simulations that the minimum-torque solution, i.e. the solution with the smallest amount of toroidal field, is the one picked by a real system. For ν ≥ 1, the Lorentz factor of the outflow increases along a field line as y ≈ M(z/R fp ) (2-ν)/2 ≈ R/R A , where R fp is the radius of the foot-point of the field line on the disc and R A = R fp /M is the cylindrical radius at which the field line crosses the Alfven surface or the light cylinder. For v < 1, the Lorentz factor follows the same scaling for z/R fp < M- 1/(1-ν) , but at larger distances it grows more slowly: y ≈(z/R fp ) ν/2 . For either regime of v, the dependence of y on M shows that the rotation of the disc plays a strong role in jet acceleration. On the other hand, the poloidal shape of a field line is given by z/R fp ≈ (R/R fp ) 2/(2-ν) and is independent of M. Thus rotation has neither a collimating nor a decollimating effect on field lines, suggesting that relativistic astrophysical jets are not collimated by the rotational winding up of the magnetic field.