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

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Appendix A: quadrupolar topology and the production of jets

In this appendix, we briefly investigate the structure of a Keplerian accretion disc, thread by a large scale magnetic field of quadrupolar topology. Such a topology is described by an even toroidal field and a magnetic flux [FORMULA] (see Eq. (6)), being an odd function of z. The magnetic field inside the disc follows the plasma, with [FORMULA], [FORMULA] and [FORMULA] (subscripts "o" refer to quantities at the disc midplane). Since both plasma angular velocity and poloidal magnetic field increase towards the center (we don't take into account a possible boundary layer between the disc and the central object), the disc can be divided into two distinct regions (see Lovelace et al. 1987, Khanna & Camenzind 1992). The inner region is characterized by a positive vertical current density at the disc midplane ([FORMULA]), but negative at its surface ([FORMULA]). The outer region displays the opposite behaviour, with [FORMULA] and [FORMULA] (the transition between the two is of course at the radius where [FORMULA]).

One would like to build up a Keplerian disc (that is, in quasi-MHS equilibrium in both radial and vertical directions), accreting towards the center ([FORMULA]) and giving rise to a magnetically propelled jet. This last demand requires the following (necessary) conditions: (a) open field lines ([FORMULA] and [FORMULA] both positive), (b) a positive MHD Poynting flux (for [FORMULA], namely [FORMULA]) and (c) a MHD acceleration ([FORMULA], see Eq. (15)).

The inner region is unable to steadily produce MHD jets. This arises from the topology itself, which provides [FORMULA] (therefore [FORMULA]): the magnetic structure would slow down any plasma that could have been (thermally) ejected out from the underlying disc.

On the contrary, the outer region can in principle produce jets. Let us then examine its structure. Since the Lorentz force is accelerating the matter at the disc midplane, accretion is achieved only if the viscous torque is dominant ([FORMULA]). Thus, the accretion velocity is

[EQUATION]

where [FORMULA] is the anomalous viscosity (measured at the disc midplane). Both radial and vertical components of the Lorentz force are positive, thus counter-acting gravity inside the disc. If the disc is in quasi-MHS vertical equilibrium, then the deviation from Keplerian rotation law is of the order [FORMULA]. At the disc surface, the vertical Lorentz force changes its sign, keeping nevertheless [FORMULA] if [FORMULA] is satisfied (condition (c)), with [FORMULA]. Condition (a) is fulfilled at the disc surface only if the vertical scale of variation of the magnetic flux is of the order of the disc scale height. Steady-state diffusion of the poloidal field (Eq. (4)) requires then a "poloidal" magnetic diffusivity such that [FORMULA]. The toroidal field at the disc surface can be written as [FORMULA], where [FORMULA] is provided by the disc differential rotation (Eq. (5)). Since [FORMULA] must be negative (condition (b)),

[EQUATION]

where [FORMULA] is the "toroidal" diffusivity (see Sect. 2.1). With these estimates, one obtains that with the following restrictions, namely

[EQUATION]

where [FORMULA] and the plasma pressure [FORMULA] (quasi-MHS vertical equilibrium), quadrupolar topologies fulfill the disc requirements as well as the jet conditions (a) and (b) (condition (c) requires a full calculation of the vertical structure). Note that above the disc surface, the current system is similar to the one of a bipolar magnetic topology: the magnetic energy that feeds the jet arises from the disc plasma itself, extracted where [FORMULA] (in a layer around [FORMULA]). Since the torque there cannot be as strong as in the bipolar case, one gets the general result that jets from quadrupolar topologies would be weaker.

However, this configuration demands very peculiar conditions on the disc turbulence. Indeed, a situation giving rise to [FORMULA] requires [FORMULA] with [FORMULA], whereas [FORMULA] is achieved for [FORMULA], and [FORMULA] for [FORMULA]. Such a situation is against our current understanding of turbulence, where all transport coefficients should achieve a comparable level (Pouquet et al. 1976). Therefore, it is dubious that such a topology could produce MHD jets. It could nevertheless play a role in a quadrupolar "cored apple" circulation around protostellar sources (Henriksen & Valls-Gabaud 1994, Fiege & Henriksen 1996).

Appendix B: toroidal field at the disc surface

We start with Eq. (30), by making a second order Taylor expansion of all the quantities and neglecting the advection term, and we obtain

[EQUATION]

where [FORMULA] is a measure of the degree of anisotropy of the MHD turbulence inside the disc. The disc radial equilibrium provides the angular velocity [FORMULA], where the deviation from Keplerian rotation law comes mainly from the radial magnetic tension,

[EQUATION]

with [FORMULA]. The above expression shows that, as the density decreases vertically, the magnetic effect increases and the plasma rotates with a lower rate. This gives rise to an enhanced accretion velocity above the disc midplane (and a corresponding source of toroidal current, see Figs. 4, 5 and 8 in FP95). The disc vertical equilibrium provides the density profile, namely

[EQUATION]

where the first term is the tidal force, the second one is the magnetic pressure due to shear ([FORMULA]) and the last one is the magnetic pressure due to curvature ([FORMULA]). These three effects are comparable (see Fig. 2). Using that the "toroidal" magnetic resistivity decreases vertically as [FORMULA], we get

[EQUATION]

and therefore, [FORMULA], where [FORMULA] and [FORMULA]. The toroidal field inside the disc is then mainly measured by [FORMULA], with [FORMULA]. Since the magnetic diffusivity decreases on a disc scale height ([FORMULA] of order unity), the only way to allow a decrease of [FORMULA] on such a scale ([FORMULA] and of order unity) is to require [FORMULA]. This is the key factor for ejection (Ferreira & Pelletier 1993b, FP95). This, in turn, implies [FORMULA] to 1. Such an important result is independent of the diffusivity used, as long as it decreases on a disc scale height.

Appendix C: minimum ejection index

The disc is not in a perfect MHS equilibrium, but there is a slight motion towards the disc midplane. From mass conservation, we obtain that this tiny motion is

[EQUATION]

Thus, this velocity increases as [FORMULA] diminishes, describing the basic fact that the plasma pressure gradient is less effective in sustaining the disc against both tidal and magnetic compression. This then implies that [FORMULA] decreases (see Appendix B), which leads to a decrease of [FORMULA], hence an increase of [FORMULA] at the disc surface ([FORMULA] increases). Now, a look at Eq. (46) shows that a vertical equilibrium will not be achieved for [FORMULA] too high (how high being obtained only by numerical means).

Eq. (30), taking into account the advection term, can be written as

[EQUATION]

This expression shows that inside the disc, where this expansion is valid, the effect of advection is to decrease the radial current density (although this contribution, of order [FORMULA], is negligible). On the contrary, above the disc when [FORMULA], this term changes its sign and contributes to maintain [FORMULA]. Eventually, as one goes from the resistive disc to the ideal MHD jet, this term will completely balance the effect of differential rotation. What the above expression shows, is that its influence will start at lower altitudes for higher [FORMULA] and [FORMULA]. Thus, to the highest values of both [FORMULA] and [FORMULA] corresponds the highest value of [FORMULA], henceforth the smallest ejection index.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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