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Astron. Astrophys. 361, L33-L36 (2000)

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4. Resistively limited growth on large scales

In an unbounded system the magnetic helicity, [FORMULA], can only change if there is microscopic magnetic diffusion and finite current helicity, [FORMULA],


The closure model of S97 and S99 also satisfies this constraint. (Note that ambipolar and/or turbulent diffusion do not enter!) As explained in B2000, this constraint limits the speed at which the large scale field can grow, but not its final amplitude. One way to relax this constraint is if there is a flux of helicity through open boundaries (Blackman & Field 2000, Kleeorin et al. 2000), which may be important in astrophysical bodies with boundaries. Here, however, we consider an infinite system.

In Fig. 5 we show that, after some time [FORMULA], [FORMULA] reaches a finite value. This value increases somewhat as [FORMULA] is decreased. In all cases, however, [FORMULA] stays below [FORMULA], so that [FORMULA] remains finite; see (9). A constant [FORMULA] implies that [FORMULA] grows linearly at a rate proportional to [FORMULA]. However, since the large scale field is helical, and since most of the magnetic energy is by now (after [FORMULA]) in the large scales, the magnetic energy is proportional to [FORMULA], and can therefore only continue to grow at a resistively limited rate, see Fig. 5. We emphasize that this explanation is analogous to that given in B2000 for the full MHD case; the helicity constraint is independent of the nature of the feedback!

[FIGURE] Fig. 5. a Evolution of [FORMULA] for different values of [FORMULA]. The corresponding value of [FORMULA] is shown on the right hand side of the plot. b  Evolution of magnetic energy for the same values of [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: October 2, 2000