Inferred jet velocities close to the speed of light suggest that jets are formed within a few gravitational radii of the event horizon of the black hole. Moreover, very-long-baseline interferometric (VLBI) radio observations reveal that jets are already collimated at subparsec scales [133, 178]. Current theoretical models assume that accretion disks are the source of the bipolar outflows which are further collimated and accelerated via MHD processes [41, 48, 190]. There is a large number of parameters which are potentially important for jet powering: the black hole mass and spin, the accretion rate and the type of accretion disk, the properties of the magnetic field and of the environment [193, 189].
At parsec scales, the jets, observed via their synchrotron and inverse Compton emission at radio frequencies with VLBI imaging, appear to be highly collimated with a bright spot (the core) at one end of the jet and a series of components which separate from the core, sometimes at superluminal speeds . In the standard model , these speeds are interpreted as a consequence of relativistic bulk motions in jets propagating at small angles to the line of sight with Lorentz factors up to 20 or more. Moving components in these jets, usually preceded by outbursts in emission at radio wavelengths, are interpreted in terms of traveling shock waves .
Finally, the morphology and dynamics of jets at kiloparsec scales are dominated by the interaction of the jet with the surrounding extragalactic medium, the jet power being responsible for dichotomic morphologies  (the so called Fanaroff–Riley I and II classes , FR I and FR II, respectively). While current models [22, 152] interpret FR I morphologies as the result of a smooth deceleration from relativistic to non-relativistic, transonic speeds on kiloparsec scales due to a slower shear layer, flux asymmetries between jets and counter-jets in the most powerful radio galaxies (FR II) and quasars indicate that relativistic motion extends up to kiloparsec scales in these sources, although with smaller values of the overall bulk speeds . The detection of strong X-ray emission from jets at large scales (0.1–1 Mpc; e.g., PKS0637–752 ) by the Chandra satellite, interpreted as scattered CMB radiation , bears additional support to the hypothesis of relativistic bulk speeds on these scales.
Although MHD and general relativistic effects seem to be crucial for a successful launch of the jet, purely hydrodynamic, special relativistic simulations are adequate to study the morphology and dynamics of relativistic jets at distances sufficiently far from the central compact object (i.e., at parsec scales and beyond). The development of relativistic hydrodynamic codes based on HRSC techniques (see Sections 3 and 4) has triggered the numerical simulation of relativistic jets at parsec and kiloparsec scales.
At kiloparsec scales, the implications of relativistic flow speeds and/or relativistic internal energies for the morphology and dynamics of jets have been the subject of a number of papers in recent years [184, 78, 182, 183, 148]. Beams with large internal energies show little internal structure and relatively smooth cocoons allowing the terminal shock (the hot spot in the radio maps) to remain well defined during the evolution. Their morphologies resemble those observed in naked quasar jets like 3C273 . Figure 16 shows several snapshots of the time evolution of a light, relativistic jet with large internal energy. The dependence of the beam’s internal structure on the flow speed suggests that relativistic effects may be relevant for the understanding of the difference between slower, knotty BL Lac jets and faster, smoother quasar jets .
Highly supersonic models, in which kinematic relativistic effects due to high beam Lorentz factors dominate, have extended over-pressured cocoons. These over-pressured cocoons can help to confine the jets during the early stages of their evolution , and even cause their deflection when propagating through non-homogeneous environments . The cocoon overpressure causes the formation of a series of oblique shocks within the beam in which the synchrotron emission is enhanced. In long term simulations (Figure 17), the evolution is dominated by a strong deceleration phase during which large lobes of jet material (like the ones observed in many FR IIs, e.g., Cyg A ) start to inflate around the jet’s head. These simulations reproduce some properties observed in powerful extragalactic radio jets (lobe inflation, hot spot advance speeds and pressures, deceleration of the beam flow along the jet) and can help to constrain the values of basic parameters (such as the particle density and the flow speed) in the jets of real sources.
The problem of jet composition remains open for more than two decades. Measurements of circular polarization in jets  favour e– / e+ jets. However, X-ray observations of blazars associated with OVV quasars impose strong constraints on the e– / e+ pair content of jets . On the other hand, the composition of jets is tightly related to their formation mechanisms [48, 267] and can be on the basis of the FR I/FR II dichotomy . In Scheck et al.  the problem of the jet composition (e / p versus e– / e+) has been approached in the context of long-term relativistic simulations ( 6 × 106 yr) searching for signatures of the composition in the extended morphology of radio jets. Both the morphology and the dynamic behaviour are almost independent of the composition assumed for the jets in their 2D simulations (see Figure 18 and the movie in Figure 19).
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The development of multi-dimensional relativistic hydrodynamic codes has allowed, for the first time, the simulation of parsec scale jets and superluminal radio components [110, 106, 147, 194]. The presence of emitting flows at almost the speed of light enhances the importance of relativistic effects in the appearance of these sources (relativistic Doppler boosting, light aberration, time delays). Hence, one should use models which combine hydrodynamics and synchrotron radiation transfer when comparing with observations. In these models, moving radio components are obtained from perturbations in steady relativistic jets. Where pressure mismatches exist between the jet and the surrounding atmosphere, reconfinement shocks are produced. The energy density enhancement produced downstream from these shocks can give rise to stationary radio knots as observed in many VLBI sources. Superluminal components are produced by triggering small perturbations in these steady jets which propagate at almost the jet flow speed. One example of this is shown in Figure 20 (see also [110, 106]), where a superluminal component (apparent speed 7 times the speed of light) is produced from a small variation of the beam flow Lorentz factor at the jet inlet. The dynamic interaction between the induced traveling shocks and the underlying steady jet can account for the complex behavior observed in many sources .
The linear stability analysis of relativistic flows against Kelvin–Helmholtz perturbations goes back to the seventies (see  for a review). Nowadays, the combination of hydrodynamical simulations and linear stability analysis has provided another step towards the comprehension of relativistic jets in extragalactic sources and micro-quasars. It is widely accepted that most of the features (even the large amplitude ones) observed in real jets admit an interpretation in terms of the growth of Kelvin–Helmholtz normal modes. This linear stability analysis has been succesfully applied to probe the physical conditions in the jets of several sources (e.g., S5 0836+710 , 3C273 , 3C120 ; see also the introduction of ). In [119, 252], the internal structures found in a set of relativistic axisymmetric kiloparsec jet simulations were analyzed. In the context of steady, parsec scale jets, a combination of linear stability analysis and axisymmetric hydrodynamical simulations has been used to predict the existence of fine structure appearing in the wake of superluminal components , later discovered in 3C120 . Finally, in [117, 118] the analysis is extended to the three-dimensional structures generated in steady jets by precession and focussing on the distributions of internal energy density and flow velocity.
Magneto-hydrodynamic simulations of relativistic jets have been performed in 2D [138, 136] and 3D [210, 211] to study the implications of ambient magnetic fields in the morphology and bending properties of relativistic jets. However, despite the impact of these results on specific problems like, e.g., the understanding of the misalignment of jets between parsec and kiloparsec scales, these 3D simulations have not addressed the effects on the jet structure and dynamics of the third spatial degree of freedom. This has been the aim of the work of Aloy et al.  and Hughes et al. . The latter authors have also used their three-dimensional code to study the deflection and precession of relativistic flows when impinging on an oblique density gradient.
Finally, Koide et al. [140, 141] have developed a general relativistic MHD code and applied it to the problem of jet formation from (Schwarzschild and Kerr) black hole accretion disks in the context of Blandford and Payne’s mechanism . In the case of jets from Schwarzschild black holes , jets are formed with a two-layered shell structure consisting of a fast gas pressure driven jet (Lorentz factor 2) in the inner part and a slow magnetically driven outflow in the outer part, both of which are being collimated by the global poloidal magnetic field penetrating the disk. In the case of counter-rotating disks around Kerr black holes , a new powerful magnetically driven jet is formed inside the gas pressure driven jet. This jet is accelerated by a strong magnetic field created by frame dragging in the black hole ergosphere. Through this process, the magnetic field extracts the energy from the black hole corroborating Blandford and Znajek’s mechanism . The same authors  have further explored this second mechanism for jet formation in the case of a Kerr black hole at maximal rotation immersed in a uniform, magnetically dominated corona with no disk. The magnetic field lines across the ergosphere are twisted by frame dragging. The line twist propagates outwards as a torsional Alfvén wave train carrying electromagnetic energy and leading to the generation of a Poynting flux jet. Using a 3D GRMHD code, Nishikawa et al.  have investigated the dynamics of a freely falling corona and of a Keplerian accretion disk around a Schwarzschild black hole. The disk and the corona are threaded by a uniform poloidal magnetic field. The magnetic field is tightly twisted by the rotation of the disk, and plasma in the corona is accelerated by the Lorentz force to form bipolar relativistic jets as in previous simulations assuming axisymmetry.
Finally, let us note that direct numerical simulations of the Blandford and Znajek mechanism have been undertaken by Komissarov , solving the time dependent equations of (force-free, degenerate) electrodynamics in a Kerr black hole magnetosphere. The equations are hyperbolic  and are solved by means of a Godunov type method.
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