Astron. Astrophys. 362, 756-761 (2000)

1. Introduction

There is now evidence that the accretion disk dynamo works with an -effect with negative sign in the upper disk plane and positive sign in the lower disk plane. This is important because in -dynamos the sign of the -effect directs the resulting geometry. The most easily excited mode has quadrupolar geometry for positive -effect ¹ and has dipolar geometry for negative -effect (Torkelsson & Brandenburg 1994a,b). Rüdiger et al. (1995), working with positive -effect, found only solutions with the quadrupolar symmetry dominating (Fig. 1).

Fig. 1. The magnetic geometry for accretion-disk dynamos with positive - is quadrupolar, i.e. even with respect to the equator. The vertical axis at the left gives the rotation axis. Note the poloidal field lines not supporting jets and outflows. Figure taken from Rekowski et al. (2000).

In Rekowski et al. (2000) the different geometries of the dynamo-generated magnetic fields are demonstrated. For negative dynamo-, however, a stationary dipolar structure of the magnetic field results (Fig. 2). The additional magnetic torque at the disk surface significantly changes the profile of the effective temperature to a profile which is more flat. The magnetic torque becomes of the same order as the radial viscous torque. The inclination angle of the poloidal field exceeds 30^o even for a magnetic Prandtl number of order unity, and also the criterion for poloidal collimation after Spruit et al. (1997) is fulfilled. The dynamo-generated magnetic field configuration thus supports the magnetic wind launching concept for accretion disks not only for unrealisticly high turbulent magnetic Prandtl numbers.

Fig. 2. The same as in Fig. 1 but for negative -effect. The magnetic geometry is dipolar, i.e. odd with respect to the equator. Note the poloidal field lines supporting jets and outflows. Maxima of the toroidal fields are located in the halo.

On the other hand, an accretion disk can only exist if there is an instability which transports the angular momentum outwards, or, in other words, the `viscosity-' is positive. This is not a trivial constraint as we know from several hydrodynamical simulations (Ryu & Goodman 1992; Cabot & Pollack 1992; Kley et al. 1993; Goldman & Wandel 1995; Stone & Balbus 1996, see also Balbus et al. 1996). The situation drastically changes for electrically conducting media, however, if (weak) magnetic fields are allowed to play their own role and, in particular, to feedback onto the momentum transport via the Lorentz force (Balbus & Hawley 1991; Hawley et al. 1996, Brandenburg et al. 1995; Ziegler & Rüdiger 2000). On the other hand, Brandenburg (1998) proposes an interesting argument for magnetic shear flows that for positive viscosity- the dynamo- must be negative in the upper disk plane.

There is much discussion about the existence of negative -effect which is also needed in order to reproduce the observed butterfly diagram of solar activity with an -dynamo and the helioseismologically-derived profile of internal rotation ². Within the frame of the anelastic approximation, i.e. if the mass conservation can be described with for density-stratified fluids the kinetic helicity is always negative (positive) on the northern (southern) hemisphere. As there is a minus between the -effect and the helicity, the resulting -effect is positive. Also a strong differential rotation does not change this situation (Pipin et al. 2000). The only possibility for negative -effect is given if the turbulence intensity behaves in opposition to the density stratification - as it is realized in the solar tachocline layer (Krivodubskij & Schultz 1993).

In a previous paper (Rüdiger et al. 2000) we have considered quite another turbulence model ignoring the density stratification in the continuity equation

Note that here we do not apply the anelastic approximation. All the resulting effects are thus vanishingly small for a very high speed of sound, . The turbulence may be driven by Lorentz force fluctuations due to a field of magnetic field fluctuations (`flux tubes') and density fluctuations, i.e. buoyancy is included. A quasilinear second-order correlation-approximation provides the surprising result that the famous minus between kinetic helicity and -effect disappears but nevertheless the -effect proves to be positive again (see Table 1 below). As the only possibility to find negative -effects, we must consider differential rotation, i.e. the inclusion of a shear.

Table 1. The signs of the MHD coefficients for rigid rotation and Kepler rotation.

For rigid rotation the magnetically driven turbulence model yields inward transport of angular momentum. Only for shear flows, however, we can compute the total angular momentum transport in accretion disks as only in this case does the dominating eddy viscosity appear in the expression for the angular momentum transport.

In the present paper, therefore, for a magnetically driven turbulence field subject to a large-scale shear flow the dynamo-, the two helicities and the angular momentum transport (which must be outwards!) are simultaneously derived. Drastic differences of the results for rigid rotation and Kepler rotation are found. Indeed, for a sufficiently high shear rate the dynamo- changes its sign and even takes the desired negative values for the case of Kepler rotation.

© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000
helpdesk.link@springer.de