4 Conclusions

In this paper we have re-examined the recent suggestion by VC2001 that large scale dynamo action can result from velocity correlations involving higher derivatives. We found this effect to be present both in global accretion disc simulations as well as in models of forced turbulence with no rotation and just shear. Nevertheless, it does not seem to produce large scale dynamo action in the parameter regime considered here. In particular, we find no signs of any net vertical flux of magnetic helicity through the domain. This was thought to be an important property of the model suggested by VC2001.

Of course, the range of parameters considered in the present work is limited, and the degree of stratification is relatively weak. Nevertheless, the anticipated velocity correlations are strongly present and yet there are no signs of large scale dynamo action. As the magnetic Reynolds number is increased, the anticipated velocity correlations that are necessary to drive non-helical large scale dynamo action become smaller, making this new mechanism an unlikely candidate for explaining the field found in accretion disc simulations. Although we cannot exclude the possibility of different behaviour at larger magnetic Reynolds number or in more realistic representations of accretion discs, it is clear that the anticipated effect will not be easily detectable. This is quite important given the fact that large scale dynamo action owing to the helicity effect is so much stronger than nonhelical dynamo action. Thus, if one is to find the effect anticipated by VC2001 it will be quite important to isolate it from the much stronger helicity-driven dynamo effect.

Some concluding speculations regarding the viability of helicity-driven dynamo action in astrophysical settings are now in order. At large magnetic Reynolds numbers dynamo action will always generate strong magnetic fields within a short period of time. What the helicity constraint does is to prevent the formation of large scale helical patterns in a time less than a certain fraction of the magnetic diffusion time. It does not exclude however the formation of large scale patterns where the magnetic helicity cancels to zero. This would however require an exchange of magnetic helicity between various sub-domains. This does unfortunately not come automatically, as the simulations of BD2001 have shown. However, for the sun the relevant resistive time scales are estimated to be around or less than 10⁶ years. On the one hand this is long enough for large scale dynamo to be well established at the present time. On the other hand, it would suggest that the time scale for the solar cycle must essentially be controlled by non-resistive effects. One idea that deserved further attention is the possibility that the dynamo wave corresponds to actual fluid motions within the solar convection zone, such that the magnetic helicity within a Lagrangian fluid patch remains to be conserved. Since this would correspond to a systematic flux of magnetic helicity, this mechanism would be similar to that of VC2001. Here, however, the magnetic helicity flux would not be self-driven, but driven externally, e.g. by the meridional circulation. A number of recent investigations have shown that meridional circulation would be capable of reversing the sense of the dynamo wave driven by the $\alpha\Omega$-dynamo (Durney 1995; Choudhuri et al. 1995; Küker et al. 2001). This is similar to the possibility discussed above where the dynamo wave itself drives the meridional circulation.

As far as discs is concerned, the long resistive time scale is perhaps not a problem, because the possibility of strong outflows always shortens the saturation time scales, albeit at the expense of lowering the final saturation field strengths (see BD2001). The final solution to the problem may require more realistic global simulations with explicit resistivities, combined with suitable analytic approaches to enable one to extrapolate to astrophysical conditions.

Acknowledgements

We thank Wolfgang Dobler, Kandu Subramanian and Ethan Vishniac for interesting discussions and comments on the manuscript. Use of the PPARC supported supercomputers in St Andrews and Leicester (UKAFF) is acknowledged. We thank the John v. Neumann-Institut for Computing at the Forschungszentrum Jülich, Germany, for using the T90 computer. R.A. acknowledges the kind support by the Deutsche Forschungsgemeinschaft and the hospitality of Nordita, where much of this work has been carried out.