## 5. Asymptotic behaviour of self-similar jets## 5.1. The fate of self-similar, non-relativistic jetsWhen integrated up to "infinity", all our solutions display the same behaviour: after an initial widening, the magnetic surfaces reach a maximum radius and then start to bend towards the jet axis (see Figs. 6 and 7). The acceleration efficiency is very high, since the magnetic structure converts almost all the MHD Poynting flux into kinetic power (), allowing therefore matter to reach its maximum velocity (see Figs. 8 and 9). They finally stop at a finite distance, with a cylindrical radius larger than the Alfvén radius (Figs. 6 and 7). We are then faced to the following obvious questions: Why do our solutions stop ? and why do they always recollimate ?
The refocusing of the magnetic surfaces towards the jet axis stops because the flow meets the fast-magnetosonic (FM) critical point (see Fig. 10). The associated Mach number is defined as , where at those altitudes (FP95). In the limit of super-Alfvénic speeds (), this Mach number can be written as where is the jet opening angle (Fig. 7). In the other extreme limit (), one has . Thus, as the jet refocuses towards the axis (but with ), its opening angle grows and is unavoidable.
The integration scheme stopped because no regularity condition was imposed. A priori, such a critical point could (and should) be smoothly crossed if, inside our parameter range, we could obtain two characteristic behaviours at its vicinity. But we never obtained "breeze"-like solutions, all of them ending like in Fig. 10, in a way very similar to Fendt et al. (1995). Since we believe that our parameter space is weakly affected by the self-similar ansatz used, it seems to us doubtful that any physical trans-FM solution could be found. One should bear in mind that, to this date, there is no semi-analytical jet solution propagating to infinity that crosses all three critical points (slow, Alfvén and fast). Providing a precise answer to whether such a solution could exist is beyond the scope of the present paper. However, we will give next section an argument against such a possibility, based on the physics that leads to this critical situation. Note that all jets displayed here achieve a super fast-magnetosonic poloidal speed, namely an usual fast-magnetosonic Mach number . The last critical point () would in principle constrain another parameter of the MAES (e.g. the ejection index ), thus leaving free (but severily bracketed) mostly one parameter, the disc aspect ratio ( is supposed to be provided by calculations of MHD turbulence inside the disc). Recollimation of magnetic surfaces is an old result: all BP82's solutions displayed such a "turning radius", while PP92 found that such a behaviour could be generic for cold jets (they did not use any self-similar ansatz). Contopoulos & Lovelace (1994) however, using also a self-similar ansatz, obtained jets with different asymptotic behaviours: recollimating, ever-widening and oscillating. But these solutions were obtained by varying the magnetic configuration parameter () independently from the others ( and ) and so, cannot describe jets from Keplerian discs. These different behaviours reflect the richness of the equations governing MHD jets. Following PP92, let us define as a measure of the jet widening. At the turning point (), Bernoulli equation writes which, in the limit , provides the cubic In that limit, one has (PP92). Solving the cubic and using this last expression, allows us to show that jets will recollimate when they reach In the other limiting case (), Eq. (53) provides directly the condition Both criteria, similar to those found by BP82 and PP92, are indeed verified numerically for extreme cases (all solutions have ). However, we used only Bernoulli equation to derive them: it is the jet transverse equilibrium that will tell us whether or not such a possibility is indeed verified. Thus, we have not provided yet a satisfactory answer to why do jets recollimate. ## 5.2. Why do self-similar jets from Keplerian discs always recollimate ?In order to understand what forces the jet to bend towards the axis, one can fruitfully look at another form of the jet transverse equilibrium equation (20), namely where provides the gradient of a quantity
perpendicular to a magnetic surface ( for a
quantity is the local curvature radius of a particular magnetic surface (Appl & Camenzind 1993a). When , the surface is bent outwardly while for , it bends inwardly. The first term in Eq. (57) describes the reaction to the other forces of both magnetic tension due to the magnetic surface (with the sign of the curvature radius) and inertia of matter flowing along it (hence with opposite sign). The other forces are the total pressure gradient, gravity (which acts to close the surfaces and deccelerate the flow, but whose effect is already negligible at the Alfvén surface), and the centrifugal outward effect competing with the inwards hoop-stress due to the toroidal field (Fig. 11).
For current-carrying jets (high , ), recollimation occurs because of the constriction effect of the toroidal magnetic field (as first discovered by BP82). For such highly super-Alfvénic solutions, the jet transverse equilibrium depends mostly on the balance between this force and the centrifugal one. Therefore, it will undergo recollimation only if the jet widens enough and reaches the cylindrical radius where describes how strongly the current varies across the magnetic surfaces at (, ). In the expression of the turning radius, the other term scales as , implying that for smaller ejection indices, the jet must open more in order to reach . However, Fig. 6 shows that increases with much quicker than . Thus, plays an important role and decreases very rapidly with : we found numerically that for , , we get , , for , 0.01, 0.005 respectively. In fact, is more generally a function of the fastness parameter (which increases for decreasing ): the faster the "rotator" and the bigger the maximum radius reached (see Fig. 12). The centrifugal picture is therefore quite appealing here, but one should not forget that is a measure of .
For current-free jets (low , ), and the flow is only slightly super-Alfvénic. Here, the ratio of the toroidal magnetic force to the centrifugal one writes with of order unity but . Thus, the force responsible for recollimation can only be the magnetic pressure gradient associated with the poloidal field. Indeed, the jet transverse equilibrium (57) implies that this force is negative, hence trying to close the jet. If both centrifugal force and magnetic force associated to the toroidal field cannot overcome it, the jet cannot avoid a recollimation (Fig. 11). If is a necessary condition to obtain trans-Alfvénic jets, it is obviously not sufficient: the "rotator" must be fast enough to allow the propagation of jets much farther away from the Alfvén surface. Since this result appears at the Alfvén surface vicinity, we believe that it is real, namely independent of our self-similar modelling. We can now try to understand the fate of cold jets from Keplerian
discs, once they start to recollimate. At the turning radius, matter
has almost reached its maximum poloidal velocity, with
(for high jets). There
is therefore no way to slow down the poloidal motion
( remains too small) and, despite the decrease
in the jet radius (leading to a decrease in ),
the poloidal velocity remains roughly constant. Matter carries away
the field with it, strongly decreasing the jet pitch angle. Thus, the
total current It is interesting to note that, althought they used the same self-similarity ansatz as we did, Contopoulos & Lovelace (1994) found also non-recollimating (for ) and oscillating (for ) solutions. But, as already said, those were obtained with parameters that do not correspond to disc-driven jets and achieved regimes with , thereby allowing a different asymptotic state. On the same line of thought, any neglect of the poloidal field (like in Contopoulos 1995) would miss the magnetic depression effect due to recollimation, hence modifying also the jet asymptotic equilibrium (and possibly allowing trans-FM solutions by fine-tunning a parameter). ## 5.3. Why do self-similar jets widen so much ?For current-carrying jets (Fig. 13), it has been shown that jets recollimate through the constriction action of the magnetic field. This is possible because the jet radius keeps on opening, allowing a huge acceleration efficiency as well as a decrease of the matter rotation rate. Why do we obtain such a systematic behaviour ?
To address this question, it is worthwhile to look at the Grad-Shafranov equation (20), written in the following form It expresses that the toroidal current is generated by two main sources, namely that depends mostly on how the magnetic lever arm behaves from one magnetic surface to another, and strongly dependent on the mass load at each surface. It is clear that the behaviour of global quantities has a tremendous importance on the jet equilibrium, thus casting doubts on approaches that are not trully 2-D. In the self-similar description of disc-driven jets and within our parameter range, one always achieves a turning point () with . Since the toroidal current at this point, is positive, recollimation is allowed only if the currents or are negative. At this turning point, one source writes for all solutions. On the contrary, the other current source
strongly depends on and for Self-similar solutions necessarily imply that ratios like and are constant through all the jet. Their logarithmic derivatives are therefore zero and both current sources are indeed negative at the turning point (, FP93a). Thus, our systematic behaviour arises from both our MAES parameter space and the fulfilment of this condition. On the contrary, any jet with (high ) but satisfying would never allow and therefore, would not undergo recollimation. This sufficient condition expresses that high- jets would not widen so much as they do here, the magnetic structure keeping stored a significant amount of the power provided by the disc. Since our parameter space is only slightly dependent on our self-similar modelling, we claim that it describes realistic boundary conditions for any given magnetic surface. Therefore, we expect that any model of a magnetically-driven jet would also undergo recollimation if conditions (69) are not met. "Over-widening" and then recollimation would be general features, independent on the analytical model used, of cold jets from discs described by constant parameters. This is consistent with PP92, who did not use any self-similar assumption, but found recollimating solutions by making the approximation (their parameter) constant through the jet. Sakurai (1987) performed full numerical simulations of jets from accretion discs that crossed the three usual critical points (measured with the poloidal velocity), but without taking self-consistently into account the disc. Because of the limited computational domain, it is difficult to figure out whether or not his solutions display a recollimation (see the vertical scale involved for small in Fig. 6). However, his initial split-monopole geometry for the magnetic configuration forbids the development of a jet with a constant (see his Fig. 3). To determine if the final configuration is realistic would require a full 2-D treatment of the disc-jet interrelations. For current-free or low- jets (), non-recollimating solutions could in principle be obtained if is verified. This necessary condition is much more stringent that in the case of high- jets, and we have strong doubts that it could ever be met. As already mentionned however, we expect that the behaviour of these kind of jets is mostly independent of our modelling. It is noteworthy that differential rotation is of so great importance for both jet formation and collimation. Indeed, it has already been found that without the counter current due to the disc differential rotation, no magnetically-driven jet could be possible. Here, we find that in the absence of differential rotation (), one of the two sources of toroidal current remains positive for any , namely Thus, differential rotation of magnetic surfaces has a tremendous importance in recollimating outflows, or more generally, in their asymptotic behaviour. This is a very important effect and shows that jets from a rigidly rotating object (e.g., a star) will certainly have a different asymptotic behaviour than jets from accretion discs. In particular, they can display non-recollimating or oscillating behaviours (Tsinganos & Trussoni 1991, Sauty & Tsinganos 1994). MAES are expected to be settled in the innermost part of a larger accretion disc, where both the energy reservoir and magnetic field strength are bigger. Such a picture implies that a viscous-like transport of angular momentum acts in the outer disc until magnetic braking due to the jet overcomes it in the inner regions. One would then obtain a transition, as decreases, from to with the corresponding changes in (from infinity to a finite value) and (from zero to a finite value). Thus, the conditions (69) could be naturally fulfilled in a self-consistent picture taking into account realistic boundary conditions. Needless to say that only 2-D numerical calculations could achieve it. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |