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A good summary of many results emerging from HD and MHD computations of jets was given in Ferrari (1998). In this section we concentrate on more recent simulations.

4.1. The launching of extragalactic jets

Early efforts to incorporate GR effects in simulations of accretion onto a BH indicated the likelihood of significant outflow, even in purely HD situations (Hawley et al. 1984). However, until very recently, this work and other attempts along the same lines were greatly hampered by severe numerical difficulties. These demanded the development of techniques, such as adaptive mesh refinement, that allow one to efficiently and simultaneously compute flows in the high density regions in the accreting gas and in the low density expelled gas; following the latter also requires much greater spatial scales which make the computations extremely expensive.

Pioneering work on MHD launching of jets from disks was performed by Uchida & Shibata (1985), who evolved an initially vertical magnetic field tied to a cold thin disk rotating around a point mass assuming axisymmetry. Differential rotation in the disk produces a substantially toroidal field and this magnetic tension is released through strong torsional Alfvén waves, which expel mass. This approach has recently been extended to 3-D relativistic flows around Schwarzschild BH's by Koide et al. (1998) and Nishikawa et al. (1999). They find that a shock forms in the disk and yields a gas-pressure driven jet which dominates the outflow, though a weaker MHD jet is present outside the pressure driven jet. In a truly impressive computation, this work has recently been extended to the environs of a rapidly rotating Kerr BH (Koide et al. 2000); while the results for a corotating disk do not greatly differ from those of the Schwarzschild situation, for (the relatively unlikely case of) counter-rotating disks a very powerful magnetically driven jet is formed inside the gas-pressure driven jet.

The launching of cold gas from a disk under circumstances carefully designed to emulate the BP magneto-centrifugal mechanism has recently been simulated in 3-D by Krasnopolsky et al. (1999). If the field is set up to be ``propelling'' then rapid acceleration and collimation of the flow are indeed observed. A simulation of the situation where a Keplerian disk is initially threaded by a dipolar poloidal magnetic field has been recently performed by Ustyugova et al. (2000); they find that a quasi-stationary collimated Poynting jet arises from the inner part of the disk, while a steady uncollimated hydromagnetic outflow emerges from the outer part of the disk. Although these calculations are focussed on the types of overdense cooling jets that are to be found in protostellar systems instead of AGN, it is also worth noting the sophisticated numerical techniques involved in the simulations of Stone & Hardee (2000).

4.2. The propagation and stability of extragalactic jets

Early 2-D simulations of HD jets (e.g., Norman et al. 1982) were of great importance in establishing that extragalactic jets were of very low density and of high Mach number, for the morphology of FR II radio galaxies could only be reproduced under those circumstances. The jet is preceded by a bow shock; the cocoon is comprised of shocked ambient medium, separated by a contact discontinuity from jet material that has passed through a Mach disk shock at the head of the jet, which corresponds to the hot-spot. Since then, as the largest computers have been turned to this task, the computations have greatly improved in both spatial resolution and temporal duration. Very long term 2-D simulations, which allowed the growth of axisymmetric Kelvin-Helmholtz instabilities to go non-linear (typically after the jets propagated distances corresponding to hundreds of initial radii) indicated that the lobes could become detached from the jets, but that new Mach disks could form behind them, thereby explaining some of the ``double-double'' radio source morphologies (Hooda et al. 1994). A suite of 2-D relativistic and nonrelativistic jets have recently been compared to show that the velocity field of nonrelativistic jet simulations cannot be scaled up to give the spatial distribution of Lorentz factors seen in relativistic simulations, as had been often speculated to be the case (Rosen et al. 1999); however, each relativistic jet and its nonrelativistic equivalent do have similar ages, if expressed in the appropriate dynamical time units.

Three-dimensional simulations have clearly shown that non-axisymmetric instabilities will become important if even small perturbations are applied (e.g., Hooda & Wiita 1998). Nonetheless, the HD jets can propagate to very substantial distances without completely breaking up if they have high enough Mach numbers. A careful comparison of numerical simulations and normal mode analysis for relativistic 3-D jets has shown that a wide variety of helical modes can be generated; these imply that dramatic variations in Doppler boosting are possible without much overall bending of the jet (Hardee 2000). Higher resolution simulations of relativistic jets indicate that the instabilities are greatly reduced in comparison to nonrelativistic situations (Aloy et al. 1999). Other relativistic simulations have convincingly shown that the knot structures seen in VLBI observations can be reasonably reproduced in terms of shocks within those jets (e.g., Martí et al. 1995, Mioduszewksi et al. 1997, Gómez et al. 1998).

The collision of a jet with a much denser cloud have recently been reexamined using high resolution 3-D HD simulations (e.g., Higgins et al. 1999, Wang et al. 2000). While powerful jets will destroy most obstructions and weak jets will be stalled and destabilized by them (as probably happens in many Seyfert galaxies), there is a rather small region of parameter space where jets can bend and survive; this could explain some rare ``dog-leg'' morphologies.

The instability of MHD jets, particularly focussed on the question of entrainment, has been carefully studied under various situations recently (Hardee & Rosen 1999, Rosen & Hardee 2000). By precessing the jets at the origin to excite the KH instability, results can be compared with linear stability analyses, and it is concluded that the KH instability is the primary cause for mass entrainment but that expansion of the jet reduces the rate of mass entrainment.

An interesting approach to MHD jet stability has been taken by Frank et al. (2000). The initial conditions for the jets are taken from analytical models for magneto-centrifugal launching and have a more complicated structure than most earlier work. They find new behavior including the separation of an inner jet core from a low density collar. The wavelengths and growth rates from a linear stability analysis are in good accord with 2.5 dimensional numerical simulations (Lery & Frank 2000). For a sub-class of current-driven instabilities in cold supermagnetosonic jets, 3-D MHD simulations have also found good agreement with a linear analysis (Lery et al. 2000). If the initial equilibrium structure has a pitch profile that increases with radius, an internal helical ribbon with a high current density forms, which yields localized dissipation; this might produce particle acceleration within the jet.

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