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 proportional to R(nu), where R is the cylindrical radius. We also assume that the azimuthal velocity in the disc is constant: v(phi) = Mc, where M < 1 is a constant. For each choice of the parameters nu 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 nu >= 1, the Lorentz factor of the outflow increases along a field line as gamma approximate to M(z/R(fp))((2-nu)/2) approximate to 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 nu < 1, the Lorentz factor follows the same scaling for z/R(fp) < M(-1/(1-nu)), but at larger distances it grows more slowly: gamma (z/R(fp))(nu/2). For either regime of nu, the dependence of gamma 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) approximate to (R/R(fp))(2/(2-nu)) 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.