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

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4. Trans-Alfvénic, global disc-jet solutions

4.1. The self-similar ansatz

In order to construct global solutions from the disc equatorial plane up to a jet asymptotic regime, the full set of non-linear, partial differential equations (1) to (5) along with Eq. (7) must be solved. Since MAES are intrinsically 2-D, this requires either a numerical approach (not yet available) or some method to reduce this set to a set of ordinary differential equations (but enforcing some symmetry to solutions). This is what allows self-similarity, which is the reason why it has been used by many authors (see Tsinganos et al. (1996) for a review).

We look then for self-similar solutions so that any quantity Q can be written


where [FORMULA] is the external radius of the disc (a standard viscous disc is probably established from [FORMULA] to larger radii). The self-similar variable is chosen to be [FORMULA], consistent with the gravity field at the disc neighbourhood, but also everywhere since [FORMULA]. Such a form, [FORMULA], is expected to arise in a Keplerian disc pervaded by a large scale magnetic field (FP93a, FP95). The magnetic flux function is chosen to be


where [FORMULA] and the pressure prescription such that [FORMULA], with [FORMULA], the polytropic index [FORMULA] remaining a free parameter. Since we focus on cold jets, its value has no importance and we choose [FORMULA] for convenience (see discussion after Eq. (7)).

The set of PDEs can be separated into two sets, the first consisting of algebraic equations between the indices [FORMULA] and [FORMULA], the second of non-linear ODEs on the functions [FORMULA], [FORMULA] and their derivatives (see Appendix B in FP95). Within such a prescription, all the dynamical terms can be included, allowing henceforth a complete study of the physics of ejection from Keplerian discs. However, only non-relativistic jets can be studied within the same self-similarity prescription. Indeed, relativistic speeds enforce a scaling with the speed of light (Li et al. 1992), which is inconsistent with disc physics.

Since the boundary values [FORMULA] at the disc midplane are known, we just have to integrate the set of ODEs from [FORMULA] to infinity. Once we have a global solution as a function of [FORMULA], it is straightforward to obtain the variation of any quantity along any magnetic surface [FORMULA] with


where [FORMULA] (note the use of the notation [FORMULA] in this case). The main difficulties come from the presence of three critical points, whenever the plasma velocity [FORMULA], where


equals the three usual MHD phase speeds: slow-magnetosonic, Alfvén and fast-magnetosonic, in the direction [FORMULA]. Self-similarity not only modifies the definitions of these phase speeds, but constrains also the direction of propagation where critical points appear (see FP95 for more precisions). Hence, the critical velocity V is roughly [FORMULA] at the disc neighbourhood where the flow becomes super-SM, and [FORMULA] much farther away.

4.2. MAES parameter space

4.2.1. The slow-magnetosonic point

We obtain trans-SM solutions by adjusting the strength of the field for a given set of parameters (FP95). Thus, for µ bigger than a critical value [FORMULA], the magnetic compression is too strong and leads to a vanishing density (hence infinite vertical acceleration). When [FORMULA], the magnetic compression is insufficient and too much mass is expelled off, which eventually falls down again.

Thus, the smooth crossing of the first critical point is directly related to the disc vertical balance. Following Li (1995), we can express that a necessary (but not sufficient) condition for stationarity is


where the subscript "+" refers to the disc surface. In terms of MAES parameters, this condition for cold jets becomes


where the function [FORMULA] is of order unity (see Appendix B). This implies [FORMULA], which shows that jets cannot be produced from magnetically dominated discs. Since plasma pressure plays such an important role in sustaining the disc, only structures close to equipartition are steady (FP95).

The minimum ejection index [FORMULA] is found by requiring that all solutions become super-SM. We found [FORMULA] for an isothermal disc with [FORMULA] and [FORMULA]. If the disc aspect ratio [FORMULA] (and also the turbulence level [FORMULA], see below) decreases, the influence of advection in Eq. (30) also decreases and the radial current density [FORMULA] is smaller, hence providing a smaller value of the toroidal field at the disc surface ([FORMULA], see Appendix C). As an example, we obtain [FORMULA] for [FORMULA] respectively. The magnetic compression decreasing, the range of allowed ejection efficiencies is shifted to higher values (see Fig. 2). Moreover, by using an isothermal prescription for the disc temperature, we have underestimated the "lifting" efficiency of the plasma pressure gradient. Thus, [FORMULA] should be seen as the lower limit for cold jets from Keplerian discs. However, since [FORMULA] is a general constraint, one can expect to find that below a certain value of [FORMULA] (and [FORMULA]), no steady-state solutions can be found anymore.

[FIGURE] Fig. 2. MAES parameter space for an isothermal structure with [FORMULA], for various disc aspect ratios: [FORMULA] (solid line), [FORMULA] (dotted line), [FORMULA] (short-dashed line) and [FORMULA] (long-dashed line). The disc magnetization µ and the magnetic Reynolds number [FORMULA] were obtained as regularity conditions, in order to get respectively trans-SM and trans-Alfvénic solutions. The other two parameters, the magnetic shear at the disc surface q and fastness parameter [FORMULA] are calculated. As [FORMULA] decreases, µ must decrease in order to balance the increase in magnetic shear q and curvature [FORMULA] (see Sect. 3.3.1). It is noteworthy that, although the disc parameters change a little, the main effect of decreasing [FORMULA] is to shift the range of allowed [FORMULA] to higher values. We confirm here that trans-Alfvénic jets require [FORMULA] (PP92, Rosso & Pelletier 1994).

The work presented here strongly differs from the work done by Wardle & Königl (1993) and Li (1995). These authors found no limiting [FORMULA] and were thus able to obtain jet solutions with an enormous lever arm [FORMULA], and an arbitrarily small mass load [FORMULA]. Here, it has been found that such a situation is impossible to achieve in steady-state. This disagreement comes from their different treatment of the highly sensitive disc vertical equilibrium. Indeed, Wardle & Königl investigated the vertical structure at a given radius and made the assumption (valid for the jet) [FORMULA] in order to deal with the mass conservation equation. Li used a self-similar approach, but did not make a smooth transition between the disc and its jets, crudely matching the disc MHS solutions to jet ones. He however found that the mass load was very sensitive to the plasma rotation rate at the disc midplane [FORMULA]: the smaller [FORMULA] the smaller [FORMULA]. This can now be understood with what was previously said. Indeed, [FORMULA] with [FORMULA]. Thus, a smaller [FORMULA] is achieved with a bigger [FORMULA], which implies a smaller ejection efficiency (hence, [FORMULA]).

4.2.2. The Alfvén point

In order to get trans-Alfvénic solutions, we adjust the magnetic Reynolds number [FORMULA], which controls the bending of the poloidal field lines at the disc surface. For [FORMULA] bigger than a critical value [FORMULA], the bending is too strong and leads to an overwhelming centrifugal effect, such that [FORMULA] (i.e., [FORMULA] becomes unphysically positive). On the contrary, if [FORMULA], the centrifugal effect cannot balance the magnetic tension, which leads to an unphysical closure of the magnetic surfaces ([FORMULA] becomes negative). At each trial for [FORMULA] one has to find another critical [FORMULA] that allows a super-SM solution. Note that fixing [FORMULA] is the same as fixing the accretion velocity at the disc midplane, for any given magnetic diffusivity. We have the freedom to do it because this accretion velocity is also controlled by the magnetic field. If we were in a situation where accretion is mainly due to the viscous stress, then [FORMULA] would also be given.

Fig. 3 shows the parameter space for cold jets in the [FORMULA] - [FORMULA] plane for [FORMULA]. Solutions with [FORMULA] do not allow super-Alfvénic jets (BP82, Wardle & Königl 1993). We found here that [FORMULA], the maximum Alfvénic Mach number reached by the flow being barely bigger than unity. For disc aspect ratios below [FORMULA], no steady-state solutions are found, [FORMULA] becoming equal to [FORMULA]. Thus, trans-Alfvénic jets are produced for discs where [FORMULA], the upper bound arising from the thin-disc approximation.

[FIGURE] Fig. 3. Parameter space for cold jets launched from an isothermal accretion disc, with [FORMULA] and for various disc aspect ratios: [FORMULA] (solid line), [FORMULA] (dotted line), [FORMULA] (short-dashed line) and [FORMULA] (long-dashed line). Due to the SM constraint, we get here a smaller parameter space as in BP82. As showed, the jet parameters do not significantly vary with any parameter but [FORMULA].

Using Eq. (16), we can express the density at the Alfvén point as a function of the disc midplane density and MAES parameters, namely


This relation is local and independent of any boundary condition that could constrain the jet behaviour. Since we approximately obtain numerically such a value for the Alfvén density, as well as a consistency with all our general requirements of Sect. 3.3.2, we believe that self-similarity doesn't affect the solutions up to this point. Therefore, our parameter space should be viewed as general for cold jets launched from Keplerian discs.

At this stage, one cannot tell with precision what will be the asymptotic state of the jet (unless of course by propagating the solution, as we will do next section). However, by using general arguments based on the work of Heyvaerts & Norman (1989), one can give clues on the degree of collimation achieved, according to the amount of current [FORMULA]


still available at the Alfvén point. Hereafter, we make a distinction between two kinds of jets, depending upon their state at the Alfvén surface (see Fig. 4).

[FIGURE] Fig. 4. Low-efficiency ([FORMULA], left pannels) and high-efficiency ([FORMULA], right pannels) trans-Alfvénic jets for [FORMULA], [FORMULA]. The upper pannels show the angular momentum transfer from the field ([FORMULA], dashed line), to the plasma ([FORMULA], solid line). At the disc surface ([FORMULA]), all the specific angular momentum [FORMULA] is carried by the field, but it is afterwards completely transferred into the plasma. Below, the corresponding function [FORMULA] is displayed for both cases. The cross labelled with "A" tells the position of the Alfvén surface. In the low-efficiency limit, the jet reaches it with [FORMULA] ([FORMULA]), most of the power being still carried by the field. In the high-efficiency case, the flow requires already almost all the power to reach Alvén speeds ([FORMULA]).

Current-carrying jets ([FORMULA]) reach the Alfvén surface while most of the angular momentum is still stored in the magnetic structure. They are powerful (corresponding here to "low" ejection indices [FORMULA]) and could reach highly super-Alfvénic speeds ([FORMULA]), with [FORMULA] and a maximum poloidal velocity


Such jets could allow a cylindrical collimation if this current does not vanish completely.

Current-free jets ([FORMULA]) become super-Alfvénic at the expense of almost all the available current. They correspond to "high" ejection efficiencies [FORMULA] (for [FORMULA], see below). According to Heyvaerts & Norman, only parabolic collimation could be asymptotically achieved. These jets could reach moderate speeds ([FORMULA]), with [FORMULA] and a maximum poloidal velocity


4.2.3. Influence of the magnetic diffusivity

In the above two sections, we restricted ourselves to discs with [FORMULA]. Fig. 5 shows that it has a profound effect on the fastness parameter [FORMULA]: at constant [FORMULA], the smaller [FORMULA] the smaller [FORMULA]. Since trans-Alfvénic jets require [FORMULA], we obtain that there is a minimum turbulence level required. Because usual dimensional arguments restrict [FORMULA] to unity, we conclude that steady state MAES require an MHD turbulence with [FORMULA].

[FIGURE] Fig. 5. Influence of the turbulence parameter [FORMULA] on other MAES parameters (see Fig. 2), for [FORMULA] and various ejection indices: [FORMULA] (solid line), 0.005 (dotted line), 0.01 (short-dashed line) and 0.02 (long-dashed line). The minimum level of MHD turbulence is limited by the value of the induced toroidal field that allows trans-Alfvénic jets ([FORMULA]). The maximum level is arbitrarily fixed to unity. Note that for fixed [FORMULA], the fastness parameter [FORMULA] grows with decreasing [FORMULA] (see text).

Eq. (35) provides, for small ejection indices (Pelletier et al. 1996),


The fastness parameter [FORMULA] strongly depends on both the shear parameter q (henceforth on [FORMULA]) and the ejection index [FORMULA]. The highest value of [FORMULA] (hence, of [FORMULA]) will be achieved with [FORMULA] and the smallest value of [FORMULA]. Since the disc vertical equilibrium limits the latter, [FORMULA] is therefore also limited ([FORMULA], see Figs. 3 and 5).

As [FORMULA] decreases, the influence of plasma on the field grows: the counter current due to the disc differential rotation increases, as well as the effect of advection (see Appendix C). Thus, the toroidal magnetic field at the disc surface decreases, decreasing accordingly the magnetic torque. This has two consequences. First, the mass load is slightly lowered (see Eq. (47)), providing a higher Alfvén speed at the Alfvén surface and a lower fastness parameter. Second, the jet is less powerful ([FORMULA] decreases) and the magnetic force ([FORMULA], see Eq. (15)) is less efficient in opening the jet: [FORMULA] decreases while [FORMULA] increases. In order to accelerate plasma up to the Alfvén surface, the magnetic structure has to provide more power (through [FORMULA] and [FORMULA]), thereby reducing the available current flowing inside a given magnetic surface ([FORMULA] decreases). Therefore, powerful jets with current still available at the Alfvén surface require a high diffusivity level ([FORMULA]).

We can now generalize our understanding of the asymptotic behaviour of super-Alfvénic jets by using [FORMULA] instead of [FORMULA]. Indeed, although the jet parameters are mostly described by [FORMULA], the jet asymptotic behaviour is drastically sensitive to [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998