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Astron. Astrophys. 331, 1130-1142 (1998)

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5. Results

We have run 9 simulations with identical initial conditions, but with different techniques for calculating the energy loss function L. None of the simulations shown here have linear growth rates which are significantly different to the adiabatic growth rates. This is in agreement with the linear results of Rossi et al. (1997), and would be expected anyway since the cooling times near equilibrium are much longer than any other time scales in the system. In addition we have run simulations, both cooled and adiabatic, to test the effect of varying the shear layers as described above. It was found, in general, that widening the shear layer, while changing the behaviour of the system for a short time at the beginning of the non-linear phase, did not significantly alter the long-term evolution of the jet.

We will begin by describing the evolution of a typical radiatively cooled simulation and comparing this with the evolution of an adiabatic simulation with identical initial conditions. We will then discuss the results of the parameter space study in terms of

  • the transfer of momentum from jet material to ambient material
  • the distribution of momentum throughout the grid
  • the strength of shocks produced on the jet axis

Each of these sections will be broken down into an analysis of the effect of the shear layer and an analysis of the effect of the different heating terms. All times are quoted in units of the sound crossing time [FORMULA]. In the simulations presented here [FORMULA] yrs.

5.1. General properties of the cooled KH instability

As already noted, the linear behaviour of the KH instability was not significantly altered by the introduction of cooling for the parameters chosen here. This would be expected since the cooling time around equilibrium is longer than any other time-scales in these simulations.

Once we enter the non-linear regime, however, cooling has a dramatic effect on the evolution of the system. This is illustrated in Fig. 3 which contains grey-scale plots of the distribution of the jet tracer variable at various times throughout an adiabatic and a cooled simulation. We can see that very little mixing occurs in the adiabatic simulation and the jet expands due to the conversion of bulk kinetic energy to internal energy by the growth of waves due to the KH instability. Differences between the adiabatic and cooled jets become noticeable at [FORMULA] and by [FORMULA] there are very significant differences in the distribution of jet material. The cooled jet material remains closer to the axis and, in addition, ambient material has been funneled onto the axis by the distortion on the surface of the jet. From this figure we would intuitively expect stronger shocks to form in the cooled jet as a result of this funneling of ambient material towards the axis. In fact this is not the case, due to the oblique nature of the shocks formed and also the damping of the body modes by cooling observed by Rossi et al. (1997) and predicted by Hardee & Stone (1997). We can clearly see the nature of the shocks formed in Fig. 4 which contains plots of the density for the same simulations at the same times as those shown in Fig. 3.


[FIGURE] Fig. 3. Grey-scale plots of the distribution of the jet tracer at time 8.5, 14.17, 19.12 and 25.51 labeled by a, b, c, and d respectively. Black indicates pure jet material, while white indicates a total absence of jet material. The scale is linear. Both simulations are for jets of initial density 100 cm-3. Note how the jet material stays closer to the axis if the jet is cooled, and also that ambient material can be found close to the jet axis. This is clearly not the case for the adiabatic simulation

[FIGURE] Fig. 4. Grey-scale plots of the distribution of the density for the same simulations, and at the same times, as those shown in Fig. 3. A square-root scale is used ranging from 4 cm-3 (white) to 700 cm-3 (black)

Fig. 5 shows the above results in a quantitative fashion. It contains plots of the proportion of material, F, lying between [FORMULA] and [FORMULA] which is jet material. We can see that in the adiabatic case, although a lot of jet material has moved away from the axis, almost all the material remaining near the axis originated in the jet. In contrast, if the jet is allowed to cool radiatively, we can see that only 70% of the material lying between [FORMULA] and [FORMULA] is jet material by the end of the simulation ([FORMULA]).

[FIGURE] Fig. 5. Plots of the proportion, F, of material which is jet material remaining within a distance R of the axis. The solid line denotes the adiabatic jet and the dotted line denotes the cooled jet. Note how little jet material there is close to the axis in the cooled, compared to the adiabatic simulation

5.2. Transfer of momentum from jet to ambient material

In this section we look at momentum transfer from material which was initially in the jet to material which was initially in the ambient medium. Note that this does not tell us about the spatial distribution of the momentum. The momentum remaining in jet material at time t is defined here as

[EQUATION]

In the plots shown in Figs. 6 and 7 this value is normalised by [FORMULA].

Two adiabatic simulations were run each with an initial shear layer with a different width and it was found that, as expected, the results were not significantly different. In particular, the momentum transferred to the ambient medium differed by less than 6% throughout the entire duration of the simulation. Varying the width of the shear layer has a larger effect on the radiative jets at [FORMULA], as we can see in Fig. 6. The jet with the wide shear layer is initially more stable than that with the narrow shear layer. However, as noted above, the state of the two jets seems to be rather similar by [FORMULA].

[FIGURE] Fig. 6. Plots of the fraction of momentum remaining within jet material against time. The plots are labeled as follows: (1) and (2) denote the adiabatic jets with narrow and wide shear layers respectively; (3) and (4) denote the cooled jets with narrow and wide shear layers respectively. The results for the wide and narrow shear layers for both the adiabatic and cooled simulations are very similar

Now we discuss the results of simulations with the `narrow' shear layer (described in Sect. 2.3) with different techniques of maintaining initial temperature equilibrium. The cooling function itself is the same for all the simulations. We have run 9 more simulations corresponding to 3 different heating terms and 3 different densities, each with a jet to ambient density ratio of 1.

Fig. 7 contains plots of the fraction of momentum remaining in jet material against time for each of the 3 different densities and each of the 3 techniques of maintaining equilibrium. Each plot also shows the behaviour of an adiabatic jet for comparison. It is clear that in all cases the radiative jets transfer momentum to the ambient material more efficiently than the adiabatic jets. It can also be seen that, for all densities, assuming insignificant cooling below [FORMULA] causes the most efficient transfer of momentum. If a heating term proportional to the density is introduced the rate of transfer of momentum is reduced slightly. If the heating term is a constant (i.e. constant volume) then the momentum transfer is reduced even more, though it still remains more efficient than in the adiabatic case.

[FIGURE] Fig. 7. Plots of the fraction of momentum remaining within jet material against time for simulations in which initial equilibrium was maintained by a constant volume heating term, a heating term proportional to [FORMULA], and by assuming insignificant cooling below [FORMULA] (top to bottom). The plots are labeled as follows:(1) denotes the adiabatic jet result; (2), (3) and (4) denote simulations with jet densities of 20 cm-3, 100 cm-3, 300 cm-3 respectively

5.3. Distribution of momentum

Here we analyse how the momentum initially contained in the jet is distributed throughout the grid with time. We define

[EQUATION]

where [FORMULA] is the momentum at the grid point [FORMULA] at time t and [FORMULA] is the length of the grid along the axis of the jet in grid cells and [FORMULA] is chosen so that [FORMULA]. In the plots shown below we normalise these values by [FORMULA], i.e. the momentum contained within 1 jet radius of the axis at [FORMULA].

We find that in both the adiabatic and the cooled cases, widening the shear layer causes a noticeable difference in results from around [FORMULA] to [FORMULA] with the jet with the narrow shear layer distributing its momentum throughout the grid more quickly. After this time the differences in the initial widths of the shear layers has almost no effect.

[FIGURE] Fig. 8. Plots of the fraction of momentum remaining within 1 jet radius of the jet axis against time. The labels (1), (2), (3) and (4) denote the adiabatic result for narrow and wide shear layers and the cooled results for narrow and wide shear layers, respectively. Note how the cooled jets reach a quasi-steady state by [FORMULA] while the adiabatic jets continue to lose momentum

In Sect. 5.2we noted that radiative cooling caused a more efficient transfer of momentum from jet to ambient material. However, it is clear from Fig. 9 that this momentum remains closer to the jet axis if the jet is cooled.

[FIGURE] Fig. 9. Plots of the fraction of momentum remaining within R of the jet radius against time in which initial equilibrium was maintained by a constant volume heating term, a heating term proportional to [FORMULA], and by assuming insignificant cooling below [FORMULA] (top to bottom). The labels (1), (2) and (3) denote the simulations with densities of 20 cm-3, 100 cm-3 and 300 cm-3. The top two plots indicate that the loss of momentum decreases with increasing density while the reverse is the case for the bottom plot

A particularly interesting result is that when the jet has entered the non-linear regime the momentum distribution enters a quasi-steady state with between 50% and 70% of the initial jet momentum remaining within 1 jet radius of the axis. This implies that the shocks formed during the growth of the instability tend not to force longitudinal momentum `sideways' to beyond a couple of jet radii. It is also clear that increasing the density reduces the rate of loss of momentum from this region for the simulations with a heating term while the reverse is true for the simulations assuming that cooling is zero below [FORMULA].

From Fig. 9 we can see that the momentum distribution attains this quasi-steady state at [FORMULA] crossing times for all the heating terms and densities investigated here. Generally, the simulations with a constant volume heating term retain a higher proportion of jet momentum in this region. The simulations which used the assumption of insignificant cooling below [FORMULA] lose most momentum from this region.

It is interesting to note that, for [FORMULA], the simulation with the assumption of insignificant cooling below [FORMULA] is very similar to an adiabatic system in terms of the momentum distribution. However, the adiabatic system continues to transport momentum beyond [FORMULA] after the cooled simulations have reached a quasi-steady state.

Overall, these results are quite surprising since the radiative jets have been found (see Sect. 5.2) to transfer more momentum to ambient material. This is a reflection of the result that significant amounts of ambient material are transported to within 1 jet radius of the jet axis (see Sect. 5.1).

5.4. Shock strengths and morphologies

We have measured the maximum shock strength on the axis of the jet against time. Since shocks are typically smeared over 3 to 4 cells in this code we measure the maximum shock strength using the equation

[EQUATION]

Thus we only look at the amplitude of the velocity discontinuity in the direction parallel to the jet axis.

Fig. 10 shows plots of the maximum shock strength against time for the adiabatic and cooled simulations with the wide and narrow shear layers. We can see that shocks form earlier in the adiabatic jet with the narrow shear layer and reach a maximum strength of about 50 km s-1. The jet with a wide shear layer takes slightly longer to develop shocks but, when they do form, they reach a higher amplitude of almost 70 km s-1.


[FIGURE] Fig. 10. Plots of the amplitude of the largest velocity discontinuity on the jet axis against time for the adiabatic and cooled jets with narrow and wide shear layers. The labels (1), (2), (3) and (4) denote the adiabatic results for narrow and wide shear layers, and the cooled results for narrow and wide shear layers respectively. See text

The same type of effect of the shear layer is seen in the cooled case. Here, though, the differences in the time taken for shocks to form is greater while the difference in the maximum shock strength attained throughout the simulations is less.

Fig. 11 shows plots of the maximum shock strength against time for the adiabatic simulation with the narrow shear layer and the 9 other simulations with different densities and heating terms. In all cases where the heating term is either constant or proportional to the density it is clear that increasing the density slows down the development of strong shocks and reduces the maximum strength of these shocks over the duration of the simulation. The same cannot be said, however, for the simulations where we assume that cooling is insignificant below [FORMULA]. Here the simulation with [FORMULA] cm-3 develops stronger shocks than either of the simulations with [FORMULA] cm-3 or 20 cm-3.


[FIGURE] Fig. 11. Plots of the amplitude of the largest velocity discontinuity on the jet axis against time for simulations in which initial equilibrium was maintained by a constant volume heating term, a heating term proportional to [FORMULA], and by assuming insignificant cooling below [FORMULA] (top to bottom). The labels (1), (2), (3) and (4) denote the adiabatic result, and simulations with densities of 20 cm-3, 100 cm-3 and 300 cm-3 respectively. See text

As might be expected from Sects. 5.2and 5.3, the heating term which leads to the slowest and weakest shock development is the constant volume one. If the heating term is set proportional to [FORMULA] the shocks are formed slightly faster and evolve to a greater strength. The simulation which simply assumes the cooling to be insignificant below [FORMULA] forms the strongest shocks.

In general, the shocks produced by the instability in radiative jets are quite flat, but with a slight bow shape pointing towards the jet source. In the adiabatic simulations these shocks tend to be more curved with the apex of the curve pointing away from the jet source.

5.5. Proper motions

The proper motions of the shocks produced in the simulations have been measured as the motion of the shock with respect to the ambient medium. The adiabatic simulations produced shocks with proper motions in the range 0.7 [FORMULA] -0.9 [FORMULA] while the shocks in the cooled simulations moved with proper motions of 0.3 [FORMULA] -0.9 [FORMULA]. No significant differences were found between the proper motions of shocks produced by different heating terms or densities. Strong shocks were found to move slower (with respect to the ambient medium) than weaker shocks, as would be expected from momentum balance arguments.

While making these measurements it was noted that, in cooled jets, the shock pattern tends to coalesce into one shock within a few crossing times of the first shocks appearing. The rate of coalescence was found to increase with the density of the jet. Shocks were not found to coalesce in the adiabatic simulations.

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© European Southern Observatory (ESO) 1998

Online publication: March 3, 1998
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