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In this paper, we propose a general method for perturtative solutions toBlandford-Znajek mechanism. Instead of solving the nonlinear Grad-Shafranovequation directly, we introduce an alternative way to determine relevantphysical quantities based on the horizon boundary condition and the convergencerequirement. Both the angular velocity $\Omega$ of magnetic field lines, thetoroidal magnetic field $B^\phi$ and the total electric current $I$ areself-consistently specified according to our method. As an example, stationaryaxisymmetric and force-free jet models around rotating black holes areself-consistently constructed according to the method we proposed. This jetsolution distinguishes itself from prior known analytic solutions in that it ishighly collimated and asymptotically approaches a magnetic cylinder. This jetsolution is helically twisted, since toroidal magnetic field is generated whenthe black hole spin is taken into account. For a given magnetic flux threadingthe black hole, the jet power and energy extraction rate of the collimated jetare compared with previous solutions. We find that our new solution agreesbetter with current state-of-the-art numerical simulation results. Someinteresting properties of the collimated jet and effects of field line rotationon the jet stability are also briefly discussed.

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