Astronomy and Astrophysics Large Scale Dynamos with Ambipolar Diffusion Nonlinearity

It is shown that ambipolar diffusion as a toy nonlinearity leads to very similar behaviour of large scale turbulent dynamos as full MHD. This is demonstrated using both direct simulations in a periodic box and a closure model for the magnetic correlation functions applicable to infinite space. Large scale fields develop via a nonlocal inverse cascade as described by the alpha-effect. However, because magnetic helicity can only change on a resistive timescale, the time it takes to organize the field into large scales increases with magnetic Reynolds number. 1. Ambipolar diffusion as a toy nonlinearity In this Letter we test and exploit the idea that the exact type of nonlinearity in the MHD equations is unessential as far as the nature of large scale field generation is concerned. At first glance this may seem rather surprising, especially if one pictures large scale field generation as the result of an inverse cascade process (Frisch et al. 1975, Pouquet et al. 1976). Like the direct cascade in Kolmogorov turbulence, the inverse cascade is accomplished by nonlinear interactions, suggesting that nonlinearity is important. However, a special type of inverse cascade is the strongly nonlocal inverse cascade process, which is usually referred to as the ‘alpha-effect’; see Moffatt (1978) and Krause & Rädler (1980). This effect exists already in linear (kinematic) theory. Until recently it was unclear which, if any, of the two effects (inverse cascade in the local sense or the αeffect) played the dominant role in large scale field generation as seen in simulations (e.g. Glatzmaier & Roberts 1995, Brandenburg et al. 1995, Ziegler & Rüdiger 2000) or in astrophysical bodies (stars, galaxies, accretion discs). A strong indication that it is actually the α-effect (i.e. the strongly nonlocal inverse cascade) that is responsible for large scale field generation, comes from detailed analysis of recent three-dimensional simulations of forced isotropic non-mirror symmetric turbulence (Brandenburg 2000, hereafter B2000). In those simulations a strong and nearly force-free magnetic field was produced, and most of the energy supply to this field was found to come from the forcing scale of the turbulence. In the absence of nonlinearity, however, the field seen in the simulations of B2000 became quickly swamped by magnetic fields at smaller scales. In that sense a purely kinematic large scale turbulent dynamo is impossible! Any hope for analytic progress is therefore slim. However, the model of Subramanian (1997, 1999) is an exception. Subramanian (1997; hereafter S97) extended the kinematic models of of Kazantsev (1968) and Vainshtein & Kitchatinov (1986) by including ambipolar diffusion (in the strong coupling approximation) as a nonlinearity. Under the common assumption that the velocity is delta-correlated in time, S97 derived a nonlinear equation for the evolution of the correlation functions of magnetic field and magnetic helicity. Although the models of Kazantsev (1968) and Novikov et al. (1983) are usually known to describe small-scale field generation, Subramanian (1999; hereafter S99) found that in the presence of fluid helicity there is the possibility of tunnelling of bound-states corresponding to small scales to unbounded states corresponding to large scale fields, which are force-free. In this Letter we present numerical solutions to the closure model of S99. We stress that we do not advocate ambipolar diffusion (AD) as being dominant over the usual feedback from the Lorentz force in the momentum equation. Instead, our motivation is to establish a useful toy model to study effects of nonlinearity in dynamos. Our numerical solutions may provide guidance for further analytic treatment of these equations in parameter regimes otherwise inaccessible. We begin however by considering first solutions of the fully three-dimensional MHD equations in a periodic box using AD as the only nonlinearity. 2. Box simulations for a finite system In this section we adopt the MHD equations for an isothermal compressible gas, driven by a given body force f , in the presence of AD, but ignoring the Lorentz force