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Recent observations of supermassive black holes have brought us newinformation on their magnetospheres. In this study we attempt a theoreticalmodelling of the coupling of black holes with their jets and discs, via threeinnovations. First, we propose a semi-analytical MHD description of a steadyrelativistic inflow-outflow structure characteristic to the extraction of thehole rotational energy. The mass-loading is ensured in a thin layer, thestagnation surface, by a two-photon pair production originating to a gamma-rayemission from the surrounding disc. The double flow is described near the polaraxis by an axisymmetric meridionally self-similar MHD model. Second, the inflowand outflow solutions are crossing the MHD critical points and are matched atthe stagnation surface. Knowledge of the MHD field on the horizon give us theangular momentum and energy extracted from the black hole. Finally, weillustrate the model with three specific examples of double-flow solutions byvarying the energetic interaction between the MHD field and the rotating blackhole. When the isorotation frequency is half of the black hole one, theextracted Poynting flux is comparable to the one obtained using the force-freeassumption. In two of the presented solutions, the Penrose process dominates atlarge colatitudes, while the third is Poynting flux dominated at midcolatitudes. Mass injection rate estimations, from disk luminosity and innerradius, give an upper limit just above the values obtained for two solutions.This model is pertinent to describe the flows near the polar axis where pairproduction is more efficient.

Double flows anchored in a Kerr black hole horizon. I. Meridionally self-similar MHD models with loading terms