The time history of BL Lacertae has shown clear evidence of changes in jet orientation both in the plane of the sky and in the angle to the line of sight (see Figure 1). Models based on transverse shocks in a relativistic flow quantitatively fit the polarization and flux density data well and permit one to determine parameters of the flow such as the bulk Lorentz factor and the angle of the flow to the line of sight (Hughes, Aller and Aller 1989). The orientation of the jet flow to the line of sight changed by approximately 6° between the early 1980 bursts and one in 1991. There have been comparable changes in the orientation of the jet on the plane of the sky. Such changes in jet orientation may be due to a helical flow pattern arising from precession or instability.

The superluminal knots of VLBI-sources are widely believed to be shocks travelling along relativistic jets. So far only very simple quasi-one dimensional models have been used to study this phenomenon. Recent progress in numerical methods for relativistic fluid dynamics (Falle & Komissarov 1995) make it possible to carry out much more realistic simulations of such flows.

Motivated by the recent determination of proper motions of knots in the M87 jet by Biretta, Zhou, & Owen (1995), we have investigated the relationship between the dynamics of the jet and the kpc-scale radio and optical structure. We interpret the knots as internal shocks in the jet, and find that relativistic effects play an important role in their appearance. In particular, knot A, which appears to be almost transverse to the flow and seems to be viewed nearly edge-on, is in fact a highly oblique shock. The direction of its normal with respect to the flow lies within about 10° of the Mach angle, i.e., the angle of obliquity beyond which no shock is possible. In a frame comoving with the shock front, the incident streamlines make an angle of about 40° – 60° with respect to the shock normal. Knot A has a proper motion corresponding to only 0.51c, and the relativistic effects are modest. Its edge-on appearance is largely fortuitous. We have found, however, that relativistic effects would create a strong bias for shocks with apparent superluminal speeds to appear nearly edge-on.

Extragalactic jets have always had two characteristics : the presence of knots and the requirement for particle acceleration. Shock fronts provide an explanation for both. However, the knotty appearance is less obvious at kp-scale on very high resolution observations from the VLA and the HST. The evidence for shock fronts is therefore weakened. At the pc-scale (or VLBI scale), a lot of these blobs are moving superluminally and they have been interpreted and modelled as shock fronts in a relativistic jet.

The cosmic nozzle is a channel with flowing hot gas in a system containing the following components: stars, a hydrostatic gaseous corona and a cavity with hot gas in the central part of the system. In the quasi one-dimension approximation, the following solution of the cosmic nozzle problem about cosmic nozzle was obtained Here (ν,ω, T, σ) are the dimensionless density, hydro dynamical velocity, temperature and cross section of the channel (η0, ω0, T0 σ0 are these values at ζ = 1), φ(ζ), πg(ζ) are the gravitational potential and the pressure in the gaseous corona. For a polytropic state equation of the gas in the gaseous corona

Superluminal motion in extragalactic jets is successfully explained assuming a relativistic flow of material in a direction close to the observer's line of sight. In rectilinear jets, the apparent velocity measured by the observer (βapp ) is related with the intrinsic velocity of the moving component (β) and the angle between the jet direction and the observer's line of sight (θ) through the equation (eg. Rees 1966, Nature 211, p.468):

Since the pioneering work by Norman et al. (1982), many numerical studies have been devoted to the analysis of the propagation of a supersonic jet shot into an ambient medium (see Massaglia, Bodo & Ferrari 1995, hereinafter Paper I, and references therein). In spite of these strong efforts many aspects of this problem are still not well understood. This is due to the complexity of the jet-cocoon structure: in fact, the cocoon excites perturbations to the jet flow, which in turn can be amplified by the Kelvin–Helmholtz mechanism and induce a strong activity of the jet's head that affects the cocoon structure. Thus a complex feedback loop mechanism establishes between jet and cocoon which make the dynamics of the interaction very complex.

One of the key processes governing the structure and evolution of astrophysical jets is their interaction with the surrounding medium. A jet can deposit momentum and energy in the ambient medium, and entrain external material. The main physical process responsible for mixing between a jet flow and the ambient medium is the Kelvin-Helmholtz (KH) instability. We have previously analysed the 2D evolution of the axisymmetric modes of a cylindrical jet (Bodo et al 1994) and of the antisymmetric modes of a planar slab jet (Bodo et al 1995). These last are thought to give indications of the 3D evolution of the helical modes of a cylinder, since the linear behavior is very similar. In this contribution we present some preliminary results of fully 3D simulations comparing them with the mentioned 2D results.

The MHD modelling of jets in axisymmetric geometry requires the treatment of the Bernoulli and the transfield equations, that can be treated following a self-similar approach. This technique is based on two main assumptions: i) the physical variables are factorized; ii) a suitable scaling law in one direction is prescribed. Solutions self-similar in the r direction (in a spherical frame of reference) have been studied to model collimated winds from disks (Blandford and Payne 1982). Here we present solutions self-similar in the θ direction, suitable to study the collimated wind around the polar axis of a rotating object (Tsinganos and Trussoni 1991, Sauty and Tsinganos 1994). Our basic assumptions are:

• – The magnetic flux function, that describes the poloidal components of velocity and magnetic field, is expressed as A(r, θ) ∝ f(r)sin2θ.

• – The density and the pressure of the plasma are assumed to scale linearly with A: ρ(r, θ) ∝ 1 + δA and P(r, θ) ∝ Po(r) (1 + KfA). Accordingly, the surfaces with equal poloidal Alfvén number M are spherical.

From recent high angular resolution (≃ 0.1 arcsec) observations of some extragalactic jets (in particular we refer to the radio, optical, and ultraviolet images of the M 87 jet, of 3C 66B, and 3C 264) it can be inferred that the synchrotron emission from the jet comes mainly from an external, quasi-cylindrical surface on which bright helical filamentary structures can be clearly recognized.

We present a self-consistent, self-similar model for classical radio double sources (FRIIs). This model depends only on quantities which can in principle be measured.

The radio morphology of extragalactic radio sources, in particular the FR I objects, has long been recognized to be suggestive of an outflow which is undergoing a strong interaction with the surrounding medium. Two essential elements must be noted in considering this interaction. First, it is inevitable. Second, the interaction will always provide mass and energy transfer from the jet to the ambient medium, and thus it is a loss term in the overall energy balance of the radio source. As will be seen, these losses can be comparable to the overall radio luminosity and hence are a nontrivial constraint on the energy requirements placed on the central engine.

Several different properties of extragalactic radio sources have been attributed to the effects of turbulence. The morphological appearance of FRI sources has been often interpreted as the result of turbulent entrainment in subsonic or transonic flows (Bicknell 1984, 1986). Moreover, particle acceleration by MHD turbulence via a second order Fermi process is one of the possible ways for accelerating the synchrotron emitting relativistic particles (see Ferrari, Trussoni & Zaninetti 1979). Turbulence appears therefore as an important ingredient in the theoretical modelling of extragalactic radio sources; however, we do not have, unfortunately, any direct evidence of it.

With the outcoming of high angular resolution imaging and 3D spectrographic facilities (HST, TIGER-OASIS at the CFH Telescope…), the need of spatially resolved theoretical models of the kinematic interaction of radio material with the ISM (or the IGM for radio-galaxies) is increasing dramatically. In order to fill the gap between pure hydro dynamical models (3D approach, but very poor atomic physic) and photoionisation/shocks ones (planar description, full account for atomic physic processes), we have started to develop 2D or 3D hydro dynamical models including a good description of atomic physic processes (using photoionisation routines from the code MAPPINGS).

We have simulated the passage of an extragalactic jet through a medium containing an ensemble of cool, dense clouds. The hydrodynamic code uses the second-order Godunov method of Falle (Falle 1991, van Leer 1979) in three-dimensional, cartesian coordinates. We have estimated the synchrotron emissivity and used this to produce synthetic radio maps. The results are reminiscent of structures seen in many extragalactic radio sources.

Gigahertz Peak Spectrum (GPS) and Compact Steep Spectrum (CSS) sources have attracted a large amount of attention in recent years since they occupy a surprisingly large fraction of the extragalactic radio source population. In this paper we summarise a theory which attempts to unify the optical emission line and radio properties of these sources. In outline the theory is as follows: The bow shock preceding the radio lobe driven into the ISM by a powerful radio jet ionizes the ISM producing both optical line emission and a free-free absorbing screen. The free-free absorption can explain the relationship between size and turnover frequency (Stanghellini et al., 1995) and the prediction of the line emission is in accord with the observation for a small sample of sources for which optical data are available.

Extended emission-line regions (EELRs) in radio ellipticals are generally thought to trace gas acquired externally, eg. through interaction with a gas-rich disk galaxy (Athanassoula and Bosma 1985, Barnes and Hernquist 1992, Hernquist and Mihos 1995). We examine here the dynamical evolution of gas in mergers, focussing on the conditions required for collisions between streams of gas. We find that such collisions can occur over a relatively wide range of encounter geometries, producing large-velocity-amplitude kinematic signatures characteristic of those observed in EELRs. This is relevant to the formation of shocks, which can account for the ionization properties of EELRs (Koekemoer and Bicknell, this conference).

We present evidence for the viability of “auto-ionizing” shocks as the dominant ionization mechanism in extended emission-line regions (EELRs) in two radio galaxies, PKS 0349–27 and PKS 2356–61. The application of this model, rather than the nuclear photoionization hypothesis of unified schemes (Barthel 1989), is motivated by observed EELR properties: large line-of-sight velocity widths (up to δv ≃ 500 km s–1 for nearby objects and ≳ 1000kms–1 at higher z); kinematics/excitation relationships (Baum et al. 1992); the EELR/radio axis alignment (Chambers et al. 1987, McCarthy et al. 1987); and the correspondence between the brighter EELR and the shorter radio lobe (McCarthy et al. 1991), suggestive of jet/gas interactions. We show that the flux, excitation and kinematics across the gas is self-consistently accounted for in terms of shocks as a single physical mechanism, requiring fewer unknown parameters than nuclear photoionization.

It seems now important to consider bulk velocity gradients inside jets (Blandford, 1993; Biretta et al, 1995) and recent results from tomography technique suggest the presence of different components in radio sources (Rudnick, this symposium). A few jet models take explicitly into account two components with different bulk velocities (Smith, Raine, 1985; Baker et al, 1988; Sol et al, 1989; Achatz et al, 1990). A fast beam comes from the vicinity of the black hole while a slower collimated wind is emitted by the accretion disk. When stable, the two components can survive along the jet. If a fast instability develops at a distance Dc from core, the beam is destroyed over some relaxation length. Its energy and momentum are transmitted to the wind. Apparent change in the flow regime such as slowing down, decollimation, local bending or discontinuity in surface brightness is the expected observational signature of such critical zones on radio maps. The exact appearance of the zone depends on the parameters of the two flows and on their interaction regime.

BL Lac objects often show a quite distorted radio morphology. Almost 75% of the BL Lacs for which the information is available show an apparent misalignment angle ΔPA between the VLBI jet and the large scale radio structure larger than 45 degrees. This can be explained by strong enhancement of slight bending due to projection effects, especially if BL Lacs are the most highly beamed sources. However we recently performed a statistical analysis of misalignment angle histograms for 155 extragalactic radio sources of different types and found that the intrinsic distortion is significantly more important in BL Lacs than in quasars and even CSS sources. Indeed the best fits of the δPA histograms by a simple bend model correspond to γɸ = 123° for BL Lacs, 37° for quasars and 36° for CSS sources, where ɸ and γ are the jet typical intrinsic bend and Lorentz factor within a given class of sources (Appl et al, 1995). If, as currently thought, jets in BL Lacs have smaller Lorentz factors than in quasars, high intrinsic bending and misalignment appear to be the rule in BL Lac sources.