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We present numerical simulations of axisymmetric, magnetically driven outflows that reproduce the inferred properties of ultrarelativistic gamma‐ray burst (GRB) jets. These results extend our previous simulations of outflows accelerated to moderately relativistic speeds, which are applicable to jets of active galactic nuclei. In contrast to several recent investigations, which have employed the magnetodynamics approximation, our numerical scheme solves the full set of equations of special relativistic, ideal magnetohydrodynamics, which enables us to explicitly calculate the jet velocity and magnetic‐to‐kinetic energy conversion efficiency – key parameters of interest for astrophysical applications. We confirm that the magnetic acceleration scheme remains robust into the ultrarelativistic regime, as previously indicated by semi‐analytic self‐similar solutions. We find that all current‐carrying outflows exhibit self‐collimation and consequent acceleration near the rotation axis, but that unconfined outflows lose causal connectivity across the jet and therefore do not collimate or accelerate efficiently in their outer regions. We show that magnetically accelerated jets confined by an external pressure that varies as z−α (0 < α≤ 2) assume a paraboloidal shape z∝ra (where r, z are cylindrical coordinates and a > 1), and we obtain analytic expressions for the one‐to‐one correspondence between the pressure distribution and the asymptotic jet shape. We demonstrate that the acceleration efficiency of jets with paraboloidal streamlines is ≳50 per cent, with the numerical value being higher the lower the initial magnetization. We derive asymptotic analytic expressions for the acceleration of initially cold outflows along paraboloidal streamlines and verify that they provide good descriptions of the simulated flows. Our modelled jets (corresponding to 3/2 < a < 3) attain Lorentz factors Γ≳ 102 on scales ∼ 1010–1012 cm, consistent with the possibility that long/soft GRB jets are accelerated within envelopes of collapsing massive stars, and Γ≳ 30 on scales ∼9 × 108–3 × 1010 cm, consistent with the possibility that short/hard GRB jets are accelerated on scales where they can be confined by moderately relativistic winds from accretion discs. We also find that Γθv∼ 1 for outflows that undergo an efficient magnetic‐to‐kinetic energy conversion, where θv is the opening half‐angle of the poloidal streamlines. This relation implies that the γ‐ray emitting components of GRB outflows accelerated in this way are very narrow, with θv≲ 1° in regions where Γ≳ 100, and that the afterglow light curves of these components would either exhibit a very early jet break or show no jet break at all.

We present numerical simulations of axisymmetric, magnetically driven relativistic jets. Our special‐relativistic, ideal‐magnetohydrodynamics numerical scheme is specifically designed to optimize accuracy and resolution and to minimize numerical dissipation. In addition, we implement a grid‐extension method that reduces the computation time by up to three orders of magnitude and makes it possible to follow the flow up to six decades in spatial scale. To eliminate the dissipative effects induced by a free boundary with an ambient medium we assume that the flow is confined by a rigid wall of a prescribed shape, which we take to be z∝ra (in cylindrical coordinates, with a ranging from 1 to 3). We also prescribe, through the rotation profile at the inlet boundary, the injected poloidal current distribution: we explore cases where the return current flows either within the volume of the jet or on the outer boundary. The outflows are initially cold, sub‐Alfvénic and Poynting flux‐dominated, with a total‐to‐rest‐mass energy flux ratio μ∼ 15. We find that in all cases they converge to a steady state characterized by a spatially extended acceleration region. The acceleration process is very efficient: on the outermost scale of the simulation as much as ∼ 77  per cent of the Poynting flux has been converted into kinetic energy flux, and the terminal Lorentz factor approaches its maximum possible value (Γ∞≃μ). We also find a high collimation efficiency: all our simulated jets (including the limiting case of an unconfined flow) develop a cylindrical core. We argue that this could be the rule for current‐carrying outflows that start with a low initial Lorentz factor (Γ0∼ 1). Our conclusions on the high acceleration and collimation efficiencies are not sensitive to the particular shape of the confining boundary or to the details of the injected current distribution, and they are qualitatively consistent with the semi‐analytic self‐similar solutions derived by Vlahakis and Königl. We apply our results to the interpretation of relativistic jets in active galactic nuclei: we argue that they naturally account for the spatially extended accelerations inferred in these sources (Γ∞≳ 10 attained on radial scales R≳ 1017 cm) and are consistent with the transition to the matter‐dominated regime occurring already at R≳ 1016 cm.