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Astron. Astrophys. 319, 340-359 (1997)

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5. Asymptotic behaviour of self-similar jets

5.1. The fate of self-similar, non-relativistic jets

When integrated up to "infinity", all our solutions display the same behaviour: after an initial widening, the magnetic surfaces reach a maximum radius [FORMULA] 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 ([FORMULA]), 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 [FORMULA] (Figs. 6 and 7). We are then faced to the following obvious questions: Why do our solutions stop ? and why do they always recollimate ?

[FIGURE] Fig. 6. Poloidal magnetic field lines for [FORMULA] and [FORMULA], for [FORMULA], 0.04, 0.03, 0.02, 0.012, 0.01, 0.009, 0.007 and 0.005 (the maximum radius increases with decreasing ejection index). The thick line connects the position of the Alfvén point for each solution. Note that in the range of allowed ejection indices for [FORMULA], the appearance of jets remains quite variable: for an anchoring radius [FORMULA] AU from a young star, jets can propagate from 1 to [FORMULA] AU of the central source, with a maximum radius ranging from 1 to approximately 300 AU. Note the logarithmic scales: small [FORMULA] jets recollimate with angles smaller than one degree (Fig. 7).
[FIGURE] Fig. 7. Jet opening angle ([FORMULA], left pannel) and pitch angle ([FORMULA], right pannel) in degrees, for [FORMULA], [FORMULA] and [FORMULA] (solid line), 0.01 (dotted line), 0.02 (short-dashed line) and 0.05 (long-dashed line). The jet opening angle is positive when the jet widens, negative when it undergoes a recollimation. While jets with high ejection indices refocuse with an angle of order [FORMULA], the others are almost cylindrical, with an angle smaller than [FORMULA] (but negative).
[FIGURE] Fig. 8. Ratio [FORMULA] of the Poynting flux to the kinetic energy flux, logarithm of the Alfvénic Mach number [FORMULA], total current I flowing within the magnetic surface (normalized to the current [FORMULA] provided at the jet basis), and logarithm of the jet density along any magnetic surface, for various ejection indices. These curves were obtained for the same parameters as in Fig. 7. The jet density is normalized to the disc midplane density [FORMULA], at the radius [FORMULA]. The jet acceleration depends on how much a magnetic surface widens ([FORMULA]), which is possible only if the current decreases. Here, this acceleration is very efficient, the magnetic structure feeding plasma with almost all its power ([FORMULA]).
[FIGURE] Fig. 9. Components of the jet poloidal velocity [FORMULA] and logarithm of the ratio of the poloidal to the azimuthal velocity, measured along a magnetic surface for various ejection indices (see Fig. 7). For these typical solutions, the jet always reaches its maximum velocity (see text), mainly as a vertical component. For small ejection indices, a full relativistic treatment should be used for jets around compact objects. Inside the disc, matter is being accreted with a velocity of order [FORMULA] the Keplerian velocity. The turning point ([FORMULA]) occurs roughly at the disc surface ([FORMULA], see FP95).

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 [FORMULA], where [FORMULA] at those altitudes (FP95). In the limit of super-Alfvénic speeds ([FORMULA]), this Mach number can be written as

[EQUATION]

where [FORMULA] is the jet opening angle (Fig. 7). In the other extreme limit ([FORMULA]), one has [FORMULA]. Thus, as the jet refocuses towards the axis (but with [FORMULA]), its opening angle grows and [FORMULA] is unavoidable.

[FIGURE] Fig. 10. Characteristic velocities in units of the disc sound speed [FORMULA] for [FORMULA], when [FORMULA], [FORMULA]. The jet becomes super-SM slightly above the disc, meets the Alfvén surface at [FORMULA], where [FORMULA] is the field line anchoring radius on the disc, and finally stops when it meets the FM critical surface. See FP95 for the expressions of these critical velocities; in particular, [FORMULA] is not the jet poloidal velocity.

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 [FORMULA]. The last critical point ([FORMULA]) would in principle constrain another parameter of the MAES (e.g. the ejection index [FORMULA]), thus leaving free (but severily bracketed) mostly one parameter, the disc aspect ratio [FORMULA] ([FORMULA] 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 ([FORMULA]) independently from the others ([FORMULA] and [FORMULA]) 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 [FORMULA] as a measure of the jet widening. At the turning point ([FORMULA]), Bernoulli equation writes

[EQUATION]

which, in the limit [FORMULA], provides the cubic

[EQUATION]

In that limit, one has [FORMULA] (PP92). Solving the cubic and using this last expression, allows us to show that jets will recollimate when they reach

[EQUATION]

In the other limiting case ([FORMULA]), Eq. (53) provides directly the condition

[EQUATION]

Both criteria, similar to those found by BP82 and PP92, are indeed verified numerically for extreme cases (all solutions have [FORMULA]). 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

[EQUATION]

where [FORMULA] provides the gradient of a quantity perpendicular to a magnetic surface ([FORMULA] for a quantity Q decreasing with increasing magnetic flux) and [FORMULA], defined by

[EQUATION]

is the local curvature radius of a particular magnetic surface (Appl & Camenzind 1993a). When [FORMULA], the surface is bent outwardly while for [FORMULA], 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).

[FIGURE] Fig. 11. Forces perpendicular to a given magnetic surface, along it, in units of [FORMULA], for [FORMULA], [FORMULA] and [FORMULA] (left), [FORMULA] (right). The solid line is the sum of gravitational, centrifugal and hoop-stress, the dotted line is the total pressure gradient and the dashed line is the magnetic structure response (namely, the term [FORMULA]). Confining forces are negative. In the high- [FORMULA] limit (left), it is the hoop-stress that is responsible for recollimation (Log [FORMULA], Log [FORMULA]). In the low- [FORMULA] limit (right), it is the pressure gradient associated to the poloidal field (Log [FORMULA], Log [FORMULA]).

For current-carrying jets (high [FORMULA], [FORMULA]), 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

[EQUATION]

where

[EQUATION]

describes how strongly the current varies across the magnetic surfaces at ([FORMULA], [FORMULA]). In the expression of the turning radius, the other term scales as [FORMULA], implying that for smaller ejection indices, the jet must open more in order to reach [FORMULA]. However, Fig. 6 shows that [FORMULA] increases with [FORMULA] much quicker than [FORMULA]. Thus, [FORMULA] plays an important role and decreases very rapidly with [FORMULA]: we found numerically that for [FORMULA], [FORMULA], we get [FORMULA], [FORMULA], [FORMULA] for [FORMULA], 0.01, 0.005 respectively. In fact, [FORMULA] is more generally a function of the fastness parameter [FORMULA] (which increases for decreasing [FORMULA]): 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 [FORMULA] is a measure of [FORMULA].

[FIGURE] Fig. 12. Ratio of the maximum radius achieved [FORMULA] to the Alfvén radius, as a function of the fastness parameter [FORMULA], for [FORMULA] and [FORMULA] (solid line), [FORMULA] (dotted line), [FORMULA] (short-dashed line), [FORMULA] (long-dashed line). Regularity conditions at SM and Alfvén points fix the range of allowed [FORMULA]. The jet is extremely sensitive to this parameter: jets with [FORMULA] reach only a few times the Alfvén radius before undergoing recollimation.

For current-free jets (low [FORMULA], [FORMULA]), [FORMULA] and the flow is only slightly super-Alfvénic. Here, the ratio of the toroidal magnetic force to the centrifugal one writes

[EQUATION]

with [FORMULA] of order unity but [FORMULA]. 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 [FORMULA] 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 [FORMULA] (for high [FORMULA] jets). There is therefore no way to slow down the poloidal motion ([FORMULA] remains too small) and, despite the decrease in the jet radius (leading to a decrease in [FORMULA]), the poloidal velocity remains roughly constant. Matter carries away the field with it, strongly decreasing the jet pitch angle. Thus, the total current I (or equivalently [FORMULA]) goes to zero. Eventually, the dominant magnetic pressure effect in Eq. (57) becomes the one due to the poloidal field. Because of recollimation, the pressure gradient associated with the poloidal field changes its sign and pinches the jet too (some sort of depression). As a result, the jet is forced to bend towards the axis, with a curvature radius [FORMULA] becoming infinitely negative. This behaviour was also obtained by Contopoulos & Lovelace (1994) and lead these authors to propose that the assumption of steady-state should break down at these distances. More generally, any steady-state jet with zero poloidal current would be asymptotically parabolic (Heyvaerts & Norman 1989). Such a situation is very far from the actual recollimating regime. Thus, it seems reasonable to guess that stationarity breaks down at the jet "end" (where [FORMULA] is met but not crossed) and that a shock is formed there.

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 [FORMULA]) and oscillating (for [FORMULA]) solutions. But, as already said, those were obtained with parameters that do not correspond to disc-driven jets and achieved regimes with [FORMULA], 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 ?

[FIGURE] Fig. 13. Isocontours of poloidal current density ([FORMULA], dotted lines) and poloidal magnetic field lines ([FORMULA], solid lines) for a cold jet with [FORMULA], launched from a disc of [FORMULA], [FORMULA]. The current circuit displays a butterfly-like shape, characteristic of tenuous ejections ([FORMULA]). The current flows down the jet axis, enters the disc at its inner edge [FORMULA] and returns along the jet itself. Force-free solutions would have [FORMULA], which is not quite achieved here, even for smaller values of [FORMULA]. The jet carries its own current, building up a global electric circuit as it propagates through the interstellar medium.

To address this question, it is worthwhile to look at the Grad-Shafranov equation (20), written in the following form

[EQUATION]

It expresses that the toroidal current is generated by two main sources, namely

[EQUATION]

that depends mostly on how the magnetic lever arm behaves from one magnetic surface to another, and

[EQUATION]

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 ([FORMULA]) with [FORMULA]. Since the toroidal current at this point,

[EQUATION]

is positive, recollimation is allowed only if the currents [FORMULA] or [FORMULA] are negative. At this turning point, one source writes

[EQUATION]

for all solutions. On the contrary, the other current source strongly depends on g (thus on [FORMULA]): for [FORMULA] it writes

[EQUATION]

and for [FORMULA]

[EQUATION]

Self-similar solutions necessarily imply that ratios like [FORMULA] and [FORMULA] are constant through all the jet. Their logarithmic derivatives are therefore zero and both current sources are indeed negative at the turning point ([FORMULA], FP93a). Thus, our systematic behaviour arises from both our MAES parameter space and the fulfilment of this condition. On the contrary, any jet with [FORMULA] (high [FORMULA]) but satisfying

[EQUATION]

would never allow [FORMULA] and therefore, would not undergo recollimation. This sufficient condition expresses that high- [FORMULA] 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 [FORMULA] (their [FORMULA] 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 [FORMULA] in Fig. 6). However, his initial split-monopole geometry for the magnetic configuration forbids the development of a jet with a constant [FORMULA] (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- [FORMULA] jets ([FORMULA]), non-recollimating solutions could in principle be obtained if

[EQUATION]

is verified. This necessary condition is much more stringent that in the case of high- [FORMULA] 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 ([FORMULA]), one of the two sources of toroidal current remains positive for any [FORMULA], namely

[EQUATION]

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 [FORMULA] decreases, from [FORMULA] to [FORMULA] with the corresponding changes in [FORMULA] (from infinity to a finite value) and [FORMULA] (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.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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