Recent observations of supermassive black holes have brought us new information on their magnetospheres. In this study, we attempt a theoretical modelling of the coupling of black holes with their jets and discs, via three innovations. First, we propose a semi-analytical MHD description of a steady relativistic inflow-outflow structure characteristic to the extraction of the hole rotational energy. The mass-loading is ensured in a thin layer, the stagnation surface, by a two-photon pair production originating to a gamma-ray emission from the surrounding disc. The double flow is described near the polar axis by an axisymmetric meridionally self-similar MHD model. Secondly, the inflow and outflow solutions are crossing the MHD critical points and are matched at the stagnation surface. Knowledge of the MHD field on the horizon gives us the angular momentum and energy extracted from the black hole. Finally, we illustrate the model with three specific examples of double-flow solutions by varying the energetic interaction between the MHD field and the rotating black hole. When the isorotation frequency is half of the black hole one, the extracted Poynting flux is comparable to the one obtained using the force-free assumption. In two of the presented solutions, the Penrose process dominates at large colatitudes, while the third is Poynting flux dominated at mid-colatitudes. Mass injection rate estimations, from disc luminosity and inner radius, 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 pair production is more efficient.

The dynamics of plasmas produced by low current X-pinch devices are explored. This comprehensive computational study is the first step in the preparation of an experimental campaign aiming to understand the formation of plasma jets in table-top pulsed power X-pinch devices. Two state-of-the-art magneto-hydro-dynamic codes, GORGON and PLUTO, are used to simulate the evolution of the plasma and describe its key dynamic features. GORGON and PLUTO are built on different approximation schemes and the simulation results obtained are discussed and analyzed in relation to the physics adopted by each code. Both codes manage to accurately handle the numerical demands of the X-pinch plasma evolution and provide precise details on the mechanisms of the plasma expansion, the jet-formation, and the pinch generation. Furthermore, the influence of electrical resistivity, radiation transport and optically thin losses on the dynamic behaviour of the simulated X-pinch produced plasma is studied in PLUTO. Our findings highlight the capabilities of the GORGON and PLUTO codes in simulating the wide range of plasma conditions found in X-pinch experiments, enabling a direct comparison to the scheduled experiments.