Annu. Rev. Astron. Astrophys. 1988. 36: 539-598
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5.4. Numerical Models of Jet's Head Evolution

The nonlinearity and interaction of the physical processes invoked requires a numerical approach. The pioneering work in the field was done by Norman et al (1982), who simulated (from a PPM code) the basic features and components described above.

TWO-DIMENSIONAL SIMULATIONS    Two types of geometrical configurations have been studied in some detail: cylindrical jets and slabs. In the first case, only axisymmetric features can be studied (i.e. pinches), while slabs give a preliminary view of the development of wiggles in the flow. The main results obtained are listed in the following subsections.

Working surface and cocoon    The jet deceleration is accomplished by a strong perpendicular shock at the front interface with the external medium (Mach disk), which thermalizes its bulk kinetic energy. The advancing working surface creates a bow shock in the ambient plasma, exactly as in the radio sources. Light jets (rhoext / rhoj > 1) display an extended overpressured region inflating behind the bow shock, usually called the cocoon, while heavy jets (rhoext / rhoj < 1) appear not to have a cocoon (Norman et al 1983). The cocoon is formed initially by shocked jet material compressed at the Mach disk that feeds a backflow along the sides of the jet. Then the bow shock expands in the external medium and is ram-pressure confined; expansion stops when its pressure has decreased to match the external pressure. In the case of highly supersonic light jets, Loken et al (1992) showed that the overpressure factors can be as high as a few thousand, yielding collimation of the jet over its entire length. On the other hand, the extension of the cocoon overall is smaller than in the case of slightly supersonic jets.

Internal shocks    The backflow generated at the working surface is characterized by quasiperiodically formed vortices that pinch off the jet behind the Mach disk and excite Kelvin-Helmholtz reflected modes at the contact surface; hence, internal oblique shocks develop in the recurrent pattern already found for infinite collimated flows (Norman et al 1983).

Intermittent jets    Clarke & Burns (1991) have analyzed the effects of intermittency of injection. Reborn jets are relatively short-lived as they must propagate in the wake of previous outflows encountering a hot light plasma; consequently, they are dense jets and advance quasiballistically.

Jitters    Burns et al (1991), using a slab geometry, have experimented with the effects of jitters on the injection direction and found that lobes may be more extended than predicted by diffusion of the overpressured cocoon. In fact, nonaxisymmetric kink body modes can amplify the initial jitter and produce large vortices in the lobes. Associated hydrodynamic mechanisms can then develop a complicated web of filaments and weak shocks similar to what is observed in radio sources.

External medium    Most simulations are performed for jets in pressure equilibrium with the external medium or for overpressured jets in a homogeneous external medium. The effects of inhomogeneities in the ambient medium along the jet path have been addressed by Norman et al (1988), Gouveia Dal Pino et al (1996). For instance, they showed that, when a jet crosses a shock wave in the external medium because of a preexisting supersonic wind, an internal Mach disk can form that causes a sudden transition to a subsonic, turbulent trail with extended mixing and entrainment. The condition for disruption is Mj / Mwind < 1 upstream of the shock. The results of simulations are very similar to the morphology of wide-angle-tail (WAT) sources. Decollimation of light jets (rhoj < rhoext) is also produced by steep negative pressure gradients in the external medium: Broad relaxed cocoons are formed where the Mach disk is very weak, and no internal shocks are transmitted to the jet. On the opposite side, positive pressure gradients compress and collimate jets and produce wiggling and pinching instabilities.

Relativistic jets    Relativistic jets have been simulated by Martí et al (1994) and by Duncan & Hughes (1994) for low Mach number jets and by Martí et al (1995) for high Mach number jets. The same global phenomenology of classical nonrelativistic jets is displayed in their results. However, in contrast to classical simulations for high-velocity jets, these authors determined that relativistic jets propagate more efficiently into the external medium following approximately the analytic result

Equation 23 (23)

which yields higher advancement velocities as compared with Equation 20. The Mach disk inflates large overpressured (up to 30 times) cocoons and excites a rich pattern of internal oblique shocks. However, a word of caution must be added to these results, as the authors' grid resolution is rather poor and may not reproduce accurately the boundary layer between the jet and ambient medium.

Radiative jets    The assumption of adiabatic jets is obviously untenable from the astrophysical point of view. In addition, mapping of the radiation field is necessary to make a comparison with observations. Several simulations include thermal radiative cooling by atomic transitions. Radiative cooling develops dense cool shells at the working surface and induces typically smaller cocoons. This result produces loose collimation and decreases the number and strength of internal shocks (Blondin et al 1990). On the other hand, extragalactic radio sources emit nonthermal synchrotron radiation principally. This is somewhat more difficult to calculate, as it requires understanding the formation of the suprathermal relativistic tail of the electron distribution function, which is most likely supported by acceleration processes of shocks and turbulence (see next section). Here, we remark that, following the results already obtained for infinite jets, the effect of radiative losses towards the dynamical evolution of extended radio sources does slightly affect the overall phenomenology and simply slows down the evolution of instabilities and turbulent mixing. However, a better understanding of the interaction between the thermal and nonthermal plasma components is required before these conclusions can be accepted. In addition, as magnetic fields are necessary for synchrotron emission, MHD models are specifically required.

A detailed solution    In a recent analysis based on a high-resolution 2-D hydrocode of the PPM type, Massaglia et al (1996a) have calculated in much detail the dynamics of the interaction with an extensive exploration of the parameter space especially toward high Mach numbers and for light jets, following the system evolution for long time scales through a specific renormalization algorithm that allows all parts of the cocoon to be kept inside the computational domain. The integration was performed over the full set of adiabatic, inviscid fluid (Euler) equations. In order to follow the mixing and entrainment effects between jet and IGM, an additional advection equation is solved for a scalar field, f, initially set equal to 1 inside the jet and phi outside. A collimated flow is injected from one side of the domain boundary into a medium at rest and in pressure equilibrium. The velocity and density profiles transverse to the jet are constant, rapidly changing to external values across the interface. Free outflow boundary conditions are used everywhere apart from the orifice where the jet enters.

Five different regions appear: (a) the jet; (b) the shocked jet material still flowing in the forward direction; (c) the shocked jet material reflected backwards at the contact discontinuity and flowing back at the jet side; (d) the shocked external material; (e) the unshocked external material. The shocked backflow and external material form the cocoon. The high-pressure cocoon squeezes the jet and drives shock waves into it, which reflect on the axis and assume the characteristic biconical shape. The aspect of the interaction depends on Mj : A stronger interaction between biconical shocks and jet head is obtained for high Mj (Figure 8). The dependence on nu is much weaker (for light jets). The jet thrust is modulated by the biconical shocks impinging on its head, and this can produce a periodic increase in the advance velocity of the head, leading to a strong change in the cocoon morphology.

Figure 8a
Figure 8b

Figure 8. Evolution of jet's head: (a) high Mach number jet, (b) low Mach number jet (Massaglia et al 1996a). The quantity plotted is the longitudinal momentum flux rho v2z.

The jet's head advances, generally following Begelman & Cioffi's (1989) solution over the following time scales:

Equation 24 (24)

but several significant differences appear. Two classes of dynamical evolution are found: (a) Jets with high Mj and nu have faster vh and show recurrent acceleration phases such as those due to strong thrust by the biconical shocks; (b) jets with low Mj and nu have slower vh, as the shock thrust is weak. The critical parameter is the inclination angle of the biconical shocks that determine the thrust behind the head. Oblique shocks must have a small inclination angle on the axis in order to produce a strong acceleration effect.

The cocoon's evolution follows the head's, but the influence of the density contrast is stronger. Again, two classes are found (Figure 9): (a) fat cocoons, which are more extended laterally; (b) spear-headed cocoons, bearing the sign of the recurrent acceleration of jets with high Mj. The system dynamics can be understood by following the behavior of the tracer f, which provides snapshots of the spatial distribution of the jet particles. Slow jets and fast jets with high density ratios behave differently from fast jets with low density ratios (Figure 9).

Figure 9

Figure 9. Cocoon morphology: (a) high Mach number, high density contrast, (b) high Mach numbers, low density contrast (Massaglia et al 1996a).

For high Mach numbers, Mj = 100, the shock that surrounds the cocoon engulfs matter of the external ambient medium. If this shocked region becomes the site of particle acceleration, we can say that the form of the lobe resembles the density distribution of Figure 9a, with an elongated structure that has the front part protruding from the lobe. Similar morphologies can be found in the sample of high luminosity radio sources by Leahy & Perley (1991); representative examples are 3C42, 3C184.1, 3C223, 3C441, 3C349, and 3C390.3. In the case of slower jets, shocks form only in the frontal part of the cocoon; therefore, the actual lobe has to have a morphology similar to that given by the tracer in Figure 9a,b. Examples of this second type of morphology can be found in the same sample: 3C296, 3C296, and 3C173.1 are good examples of this class; the remaining sources of the sample are more irregular.

THREE-DIMENSIONAL SIMULATIONS    The morphology of radio lobes is obviously dominated by 3-D effects, which, however, require powerful massive parallel supercomputers for their simulations. For this reason, a full exploration of parameter space in three dimensions has not been completed yet. Historically, the first 3-D simulation applied to astrophysical jets was conducted by Arnold & Arnett (1986) at very low spatial resolution. At present, the most complete simulations in fully 3-D geometry are by Clarke (1996), Norman (1996), both using PPM-type codes (ZEUS-3D and CMHOG).

Light (rhoj ~ 10-2 rhoext) and moderately supersonic (Mj leq 10) jets have been considered. The resolution adopted is still relatively poor for investigating in detail the physics of boundary layers, mass entrainment, and cocoon turbulence development. In the initial part of the jet, close to the source, the results apparently are not different from the 2-D case. An overpressured cocoon confines the collimated flow. However, a strong turbulent mixing between jet and ambient material occurs, characterized by small scale-vortices, whereas in 2-D simulations, large-scale ones were dominant; this is a well-known difference in the 2-D and 3-D turbulence cascades. The important result is that mixing does not reach the very backbone of the jet, and the cocoon ends up being overdense. The jet is more efficiently protected from disruption than 2-D results predict (or in predictions of the Kelvin-Helmholtz instability analysis of the previous section), although the usual sequence of oblique internal shocks appears.

The jet starts displaying large perturbations a few cocoon radii behind the leading working surface. Instead of the Mach disk, now a less well-defined termination shock accomplishes the transition to subsonic flow and bow shock. In addition, the jet displays a vigorous "flapping" of its head and is deflected 4-5 times before it advances by about a cocoon radius. This flapping is a 3-D process that is due to the cocoon turbulence, which causes irregular deflections of the leading part of the flow and in fact drives the "dentist drill effect" phenomenologically discussed by Scheuer (1974) as a way to increase the extremely small transverse dimensions of radio lobes in highly supersonic jets. Correspondingly, the lack of concentration in the jet head thrust slows down the working surface advancement speed to about half of the ram pressure estimates. Some recurrent reacceleration events of the type described in 2-D results occur also in 3-D scenarios. Norman & Balsara (1993) have simulated the propagation of a jet through a shocked external wind in three dimensions also. Bending and flaring of the jet are found again, showing that the interaction with the external medium can appreciably change the appearance of the physical phenomena.

EFFECTS OF MAGNETIC FIELDS    Toroidal magnetic fields, consistently generated in the jet by a longitudinal axial current, confine the plasma through a radial Jo × B force and, in the pure MHD limit, do not allow the formation of large cocoons, as the backflow and return current are small and constrained to the very surface of the jet (Clarke et al 1986, Lind et al 1989). The shocked material at the working surface accumulates in a protruding nose cone because the radial current at the Mach disk gives rise to a forward longitudinal Jr × B force. Therefore, magnetically confined jets would appear lobeless with bright noses and internal knots; this is in contrast with most radio source observations. However, laboratory experiments indicate that plasma columns are unstable to pinch and hose instabilities on dynamical scales, and this should make magnetic noses transient structures. Clarke (1994) has extended the simulation to three dimensions and has confirmed that the nose is disrupted before becoming too long; its final appearance resembles an asymmetric compact lobe with a well-ordered magnetic field and strong polarization, as in quasar jets.

Longitudinal (poloidal) magnetic fields also have been used in simulations to enhance the emission component on the jet axis where the toroidal component vanishes. Kössl et al (1990) have shown that equipartition or weaker helical fields make the flow resemble purely hydrodynamic simulations, apart from a smaller cocoon and slower head advancement. The magnetic field ends up much less ordered and confined on the edges, as in classical FR I radio galaxies.

Koide et al (1996) have addressed the simulation of relativistic MHD flows in slab geometry that are injected parallel to a preexisting homogeneous magnetic field. They used a Lax-Wendroff code with rather low spatial resolution. In the case of a weak magnetic field, a backflow forms along the jet and spreads into a cocoon, which is very similar to what is found in the pure hydrodynamic case. In the case of a strong magnetic field, no backflow and cocoon are formed at all. In both cases, however, the magnetic field is reversed at the beam surface. The overall result is that relativistic magnetized jets are well collimated and propagate without slowing down. In a subsequent paper, Nishikawa et al (1997) extended the study to 3-D relativistic MHD with a bulk Lorentz factor of 4.56 and in pressure equilibrium with the external medium. Three-dimensional (3-D) relativistic beams decelerate more efficiently than 2-D ones, are better confined, develop weak internal structures, and are associated with smaller bow shocks, thinner cocoons, and weak backflows. Koide et al (1996) have also considered jets propagating at an angle with respect to the external magnetic field and found that all jets for any Mach number are bent and, in some cases, also split. The working surface contains a compressed magnetic field and the head advancement speed is reduced. These results might explain some aspects of the phenomenology of BL Lacs.

A fully relativistic MHD code has been developed by van Putten (1996) and works at higher resolution and precision, especially at shock discontinuities. It has been applied to the standard 2-D light jets, assuming a toroidal confining magnetic field out of radial force balance at the injection point. This drives the formation of a nozzle and knots along the jet until a termination Mach disk accomplishes the deceleration. Downstream of the Mach disk, the flow bifurcates into a forward nose cone and a backflow, exactly as in the nonrelativistic case. The cocoon remains relatively small, but again, these simulations span short time scales only; perhaps they are relevant to interpreting the class of compact steep spectrum (CSS and CSO) objects.

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