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 , in the presence of AD, but ignoring the Lorentz force
where is the advective derivative, is the magnetic field, is the current density, and is the random forcing function as specified in B2000. The nonlinear drift velocity due to AD can be written as . We use nondimensional units where . Here, is the sound speed, the smallest wavenumber of the box (so its size is ), is the mean density, and is the vacuum permeability. Since AD is the only nonlinearity in Eq. (3) we can always normalize such that .
The model presented here is similar to Run 3 of B2000, where and the forcing wavenumber is 5. With a rms velocity of around 0.3 the magnetic Reynolds number based on the forcing scale is . In Fig. 1 we show a grey scale representation of a slice of the magnetic field and the current density at . Note the presence of a large scale magnetic field that varies in the x-direction with wavenumber . In Fig. 2 we show the spectra of magnetic and kinetic energies. The peak of magnetic energy at shows the development of large scale magnetic fields. Further, the current density is concentrated into narrow filamentary structures, typical of AD (see Brandenburg & Zweibel 1994).
Unfortunately, the severity of the (empirical) diffusive timestep limit, , where , prevented us from running much longer at high resolution ( meshpoints). For meshpoints this limit is unimportant, and so we were able to run until , a time when the large scale field was much more clearly defined. In the inset of Fig. 2, we show the evolution for such a case, but with a forcing at (giving larger scale separation). Note again the peak of at and also the suppression of magnetic field at the next smaller scale, corresponding to . Both these features are similar to the case with full Lorentz force and without AD (Figs. 3 and 17 of B2000).
Our main conclusion from these results is first of all that large scale field generation works in spite of AD, contrary to earlier suggestions that AD might suppress the large scale dynamo process (Kulsrud & Anderson 1992). Secondly, AD provides a nonlinear saturation mechanism for the magnetic field at all scales, except for the scale of the box, where a force-free field develops for which vanishes. Like in the simulations of B2000 this provides a `self-cleaning' mechanism, without which the field would be dominated by contributions from small scales.
Having established the close similarity between models with AD versus full Lorentz force as nonlinearity, we now move on to discuss the nonlinear closure model of S99 with AD as a `toy' nonlinearity.
© European Southern Observatory (ESO) 2000
Online publication: October 2, 2000