Highly-collimated, magnetically-dominated jets around rotating black holes

In this paper, we propose a general method for perturtative solutions to Blandford-Znajek mechanism. Instead of solving the nonlinear Grad-Shafranov equation directly, we introduce an alternative way to determine relevant physical quantities based on the horizon boundary condition and the convergence requirement. Both the angular velocity $\Omega$ of magnetic field lines, the toroidal magnetic field $B^\phi$ and the total electric current $I$ are self-consistently specified according to our method. As an example, stationary axisymmetric and force-free jet models around rotating black holes are self-consistently constructed according to the method we proposed. This jet solution distinguishes itself from prior known analytic solutions in that it is highly collimated and asymptotically approaches a magnetic cylinder. This jet solution is helically twisted, since toroidal magnetic field is generated when the black hole spin is taken into account. For a given magnetic flux threading the black hole, the jet power and energy extraction rate of the collimated jet are compared with previous solutions. We find that our new solution agrees better with current state-of-the-art numerical simulation results. Some interesting properties of the collimated jet and effects of field line rotation on the jet stability are also briefly discussed.