The large-scale magnetic fields of galaxies are thought to be generated by a galactic dynamo due to the simultaneous action of the helicity of interstellar turbulence and differential rotation (see, e.g., Ruzmaikin et al. 1988). The kinematic stage of the galactic dynamo, i.e. the evolution of a weak magnetic field with negligible influence on the turbulent flows, seems to be clear, while the nonlinear stage of dynamo evolution is a topic of intensive discussions (for reviews, see Beck et al. 1996, Kulsrud 1999). The most contentious issue is the question of the equilibrium magnetic field strength at which dynamo action saturates.
A naive viewpoint is that the saturation level for the large-scale magnetic field is given by the equipartition between kinetic energy and the energy of the large-scale magnetic field (see, e.g., Zeldovich et al. 1983). The motivation is that the equations describing large-scale dynamo action contain the mean, but not the total, magnetic field. This naive outlook leads to models of dynamo generated magnetic fields which are in basic agreement with the available observational information.
Vainshtein and Cattaneo (1992) formulated a more sophisticated argument, suggesting that the equilibrium magnetic field should be determined by a balance between the kinetic energy and the energy of the total magnetic field. The simplest models of dynamo generation then result in the estimate , where b is the small-scale magnetic field, and the magnetic Reynolds number for the interstellar turbulence (or even much larger if a microscopic diffusivity instead of ambipolar diffusion is used; cf. Brandenburg & Zweibel, 1995). Thus the ideas of Vainshtein and Cattaneo lead to the conclusion that a dynamo generated large-scale galactic magnetic field must be negligible in comparison with that observed, and so the generation of the observed field must be connected with another mechanism. However, no other general and realistic mechanism for galactic magnetic field generation is currently available.
The arguments of Vainshtein and Cattaneo do not seem inevitable. For example, a dynamo generated magnetic field can itself produce helicity, so the nonlinear effects can even amplify rather than suppress field generation at the initial stages of nonlinear evolution (Parker 1992, Moss et al. 1999); other suggestions are discussed by, e.g., Beck et al. (1996), Kulsrud (1999), Field et al. (1999) and Blackman & Field (1999). In particular, Blackman & Field (2000) argue that the Rm-dependent quenching seen in the simulations of Cattaneo & Hughes (1996) is a consequence of helicity conservation when using closed or periodic boundaries, while simulations with open boundaries by Brandenburg & Donner (1997) (see also Brandenburg 2000) do not show this effect.
The aim of this letter is to demonstrate that with open boundaries the scenario of Vainshtein and Cattaneo results in basically the same estimate for the equilibrium magnetic field strength as is given by the naive viewpoint.
The essence of our arguments can be presented as follows. According to Vainshtein and Cattaneo, the suppression of dynamo action by the small-scale magnetic field that is generated together with the large-scale is connected with the magnetic helicity of the small-scale magnetic field. Because the total magnetic helicity is an inviscid invariant of motion, the magnetic helicity of the small-scale magnetic field can be connected with the magnetic helicity of the large-scale magnetic field. The governing equation for magnetic helicity has been proposed by Kleeorin and Ruzmaikin (1982; see the discussion by Zeldovich et al., 1983), investigated by Kleeorin et al. (1995) for stellar dynamos, and self-consistently derived by Kleeorin and Rogachevskii (1999). During nonlinear stages of the dynamo, the -effect is thought to be determined by the hydrodynamic and magnetic helicities, so a closed system of equations can be obtained for the evolution of the magnetic field and the -coefficient (see below, Sect. 2). This governing system (with helicity locally conserved) leads to magnetic field behaviour which is consistent with the prediction of Vainshtein and Cattaneo (we are grateful to M. Reshetnyak, who provided us with the relevant numerical results, which will be published elsewhere).
We stress that Eq. (4) takes into account the local helicity balance at a given point inside the galactic disc , , where are cylindrical coordinates. However, the kinematic galactic dynamo is impossible without a turbulent flux of magnetic field through the surface (see, e.g., Zeldovich et al. 1983, Ch. 11). It is more than natural to believe that this flux can transport magnetic helicity to the outside of the disc. The methods of Kleeorin and Rogachevskii (1999) allow us to introduce the corresponding term into the governing equations for the galactic dynamo. We demonstrate by numerical simulations, and to some extent analytically, that this term leads to a drastic change in the magnetic field evolution. Now the steady-state large-scale magnetic field strength is approximately in equipartition with the kinetic energy of the interstellar turbulence.
© European Southern Observatory (ESO) 2000
Online publication: September 5, 2000