We have performed a total of 12 different simulations of the KH instability in jets. The parameters were chosen to approximate the physical conditions thought to prevail in jets from young stellar objects.
6.1. General differences introduced by cooling
We have found that introducing cooling
The first and second of these results are somewhat surprising as they contradict the results presented by Rossi et al. (1997). However, we found it possible to reproduce the results of Rossi et al. (1997) by performing the simulations in cylindrical symmetry. Therefore we can conclude that, although in the linear regime we do not expect significant differences between the 2 symmetries, the non-linear growth of the instability is significantly affected by the choice of symmetry.
We will now explain how radiative jets can transfer more momentum to the ambient medium while the momentum flux in these systems remains better collimated than in an adiabatic system. As the KH instability grows, longitudinal momentum will be converted to internal energy by the resulting compressions and shocks within the jet. If the jet is not cooled it will, as a result, become overpressured and begin to expand and so momentum will be moved away from the jet axis. This mechanism allows longitudinal momentum to be transported away from the jet axis while still remaining in jet material. If, on the other hand, the jet is cooled this internal energy is radiated away causing the jet to remain in approximate pressure equilibrium with the surrounding medium and thus momentum will not be transported away from the jet axis in this way. In fact, as the instability grows in the cooled jet the surface of the jet develops a saw-tooth profile which steepens and eventually funnels ambient material towards the jet axis, causing mixing. It is in this way that momentum is transferred from jet to ambient material in the radiatively cooled simulations. It is interesting that the shocks produced by this process do not appear to decollimate the jet.
The process by which the steepening of the disturbance on the surface of the jet occurs can be understood as follows. As the jet's surface becomes perturbed by the transverse velocity perturbation, the expanding parts of the jet begin to compress the surrounding ambient medium. This process heats both ambient and jet material near the boundary between the two. The region is then cooled and a dense filament will form at the boundary. The inertia of this filament will cause the smooth wave pattern to be converted to a saw-tooth one as the wave grows in amplitude. This process is illustrated in Fig. 12a and b . Put another way, the waves steepen and `break' earlier in the radiative than the adiabatic case because of a stronger dependence of the wave speed on its amplitude.
The shocks produced in the adiabatic jet are formed by the growth of body modes. Our results suggest that cooling of the type expected in YSO jets inhibits the development of shocks and this is in agreement with the results of Rossi et al. (1997). This is because the development of the instability takes longer due to the increased levels of cooling in high temperature regions and vice versa for low temperature regions with respect to equilibrium values (see Hardee & Stone 1997). Therefore, in a cooled jet, the time taken for shocks to develop in the jet is increased. Note however, that while this explanation is valid for these simulations, it does not imply that, in general, shocks resulting from the KH instability are weaker if cooling is introduced. We emphasize that whether this is the case in general depends on the form of the heating and cooling functions.
6.2. Effect of shear layers
Our simulations suggest that widening the shear layer does not significantly affect the long term evolution of the system.
This result is easily explained. Previous studies of the KH instability in adiabatic jets (e.g. Payne & Cohn 1985; Hardee & Norman 1988) have shown that the surface modes do not grow as fast as the body modes in high Mach number flows for . A wide shear layer tends to damp the growth of surface modes and body modes with wavelengths less than the width of the layer (for example, Bodo et al. 1994). Therefore, although widening the shear layer will damp surface modes, we know that in a high Mach number adiabatic jet these modes do not have a significant effect on the jet anyway. Since it is these modes which are expected to disrupt the jet and thus cause mixing and transfer of momentum from jet material to ambient material, we do not expect widening the shear layer to have much effect on momentum transfer in the adiabatic case. However, according to Hardee & Stone (1997) introduction of cooling can allow a new surface mode to grow which has higher growth rates than any other modes present if the cooling function is shallow enough around . In our simulations it is difficult to tell whether the cooling function is shallow enough around as it is a non-equilibrium one and we also include energy dumped into ionisation in our calculations. Since the differences in growth rates introduced by the narrowing of the shear layer are of order a few percent, it seems reasonable to say that this surface mode does not exist, or at least has a very low growth rate, for the energy loss functions studied here.
6.3. Effect of changing the heating function
Detectable differences were found between simulations which included a constant volume heating term and those which included a heating term proportional to . Generally, the inclusion of the former resulted in behaviour more similar to an adiabatic system than simulations with the latter kind of heating term, though this difference is relatively small.
Simulations with the assumption that the cooling is insignificant below were dramatically different from adiabatic simulations and were generally more unstable than the simulations which used a heating function to maintain initial equilibrium. This difference is unlikely to arise from the growth of the surface mode mentioned by Hardee & Stone (1997) because of the very low level of the cooling around for the modified cooling function described in Eq. 20. It is more likely to be due to the fact that there is no heating term in the system and this allows the instability to grow faster than in the other cooled systems.
The injection of energy by a heating term makes the system as a whole more similar to an adiabatic one. It is plausible to suggest that the lack of a heating term will therefore accentuate any differences between adiabatic and cooled systems.
6.4. Differences between slab and cylindrical symmetry
As already stated, the results of simulations of the cooled KH instability in slab and cylindrical symmetry differ in 2 respects. In slab symmetry the introduction of cooling
whereas the introduction of cooling in cylindrical symmetry does the opposite.
We can understand this as follows. The increase in mixing and momentum transfer in slab symmetry results from disturbances on the surface of the jet steepening due to cooling as discussed in Sect. 6.1. While we do not expect this steepening process to be significantly modified by the choice of symmetry we note that axisymmetric waves of a given amplitude on the surface of the jet require higher pressures (and hence higher temperatures) on the jet axis in cylindrical symmetry. This is because the amplitude of an axisymmetric wave in cylindrical symmetry goes as whereas it is independent of r in slab symmetry. Thus if the system is cooled, and the cooling function is an increasing function of T, we expect cooling to damp the waves more in cylindrical than in slab symmetry. Hence the disturbances on the surface of the jet will be lower in amplitude and take longer to steepen. This is precisely what we observe in cylindrically symmetric simulations. By the time the waves have steepened significantly the corresponding adiabatic jet will have disrupted.
Hence we conclude that, while slab symmetric simulations are adequate to approximate the linear behaviour of the axisymmetric modes of the KH instability in cooled cylindrical jets, such simulations give misleading results in the the non-linear regime. In principle one would expect the above argument to hold for studies which examine the growth of the helical modes of the KH instability using slab symmetric simulations of the sinusoidal modes (e.g. Stone et al. 1997). However, this assumes that the most significant waves are ones which pass through the body of the jet rather than along its surface.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998