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Astron. Astrophys. 335, 488-499 (1998)

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6. Discussion

In this paper, we computed the dynamo parameters, i.e., the non-vanishing components of the tensors [FORMULA] and [FORMULA], which describe the effects of turbulent motions on the large-scale Galactic magnetic field. We considered only the turbulent motions directly driven by SN explosions, ignoring those associated with the contraction phase of SNRs and SBs as well as those produced by a totally different mechanism. As argued in Papers I and II, turbulent motions other than those considered here probably have a limited impact on the [FORMULA] -parameters, [FORMULA], [FORMULA], [FORMULA], whereas they are likely to reduce the escape velocity, [FORMULA], and to enhance the magnetic diffusivities, [FORMULA] and [FORMULA].

The numerical results are displayed in the form of vertical profiles at 5 different Galactic radii (Fig. 8) and in the form of contour plots (Fig. 11). The dynamo parameters appear to be almost entirely due to clustered SNs, even though these are hardly more frequent than their isolated counterparts. They peak in a double ring located at [FORMULA] and [FORMULA]. In this double ring, the dynamo number (Eq. (35)) is sufficiently high and the ratio [FORMULA] sufficiently low to allow for magnetic field amplification.

Contrary to earlier studies which found [FORMULA] above midplane (based on a second-order correlation approximation (Rüdiger & Kichatinov 1993) or on a fully numerical simulation of individual SN explosions (Kaisig, Rüdiger, & Yorke 1993)), we obtain [FORMULA] up to [FORMULA] kpc. This new result can be attributed to the "peanut" or "pear" shape of most SBs in the strongly stratified ISM. Together with the confirmation that [FORMULA] above midplane, it may have important implications for the solutions of the dynamo equation, in particular for the existence of nonaxisymmetric solutions (see Rüdiger, Elstner, & Schultz 1992).

Our results are evidently affected by uncertainties in the input ISM parameters and by working approximations. The former are reviewed and extensively discussed in Ferrière (1995) and Ferrière (1998). The latter, which concern either the modelling of a shell's dynamic behavior (thin-shell approximation, merging criterion [FORMULA]) or the analytical formulae employed for the dynamo parameters (Eqs. (26)-(31)), are discussed in Ferrière (1995) and in Papers I and II, respectively.

Because of the many sources of error, the overall amplitude of the computed dynamo parameters could easily be off by a factor of 2 or 3. Likewise, their exact spatial dependence should not be taken too seriously, especially at high altitudes where the ISM parameters become increasingly uncertain and dynamo action arises solely from a small number of particularly powerful SBs whose characteristics are not well established. The most uncertain parameter is undoubtedly [FORMULA], which turns out to be highly dependent upon the exact shape of expanding shells, with the consequence that not only its characteristic amplitude but also its very sign are still questionable.

Furthermore, our axisymmetric study is unable to provide any information on the longitudinal structure of [FORMULA] and [FORMULA]. Since our model ISM is described by the azimuthally-averaged values of the ISM parameters, it is implicitly expected to lead to the azimuthally-averaged [FORMULA] and [FORMULA]. However, owing to the numerous nonlinearities in the problem, this is not strictly the case. On the other hand, the longitudinal variation of [FORMULA] and [FORMULA] is probably weaker than that of other ISM parameters. For instance, the explosion rate varies significantly between spiral arms and interarm regions, but an increase in the explosion rate is usually accompanied by an increase in the interstellar pressure, which entails a decrease in the size and lifetime of individual bubbles, so that ultimately [FORMULA] and [FORMULA] vary only moderately with longitude. Because of this self-regulatory effect, the values obtained numerically are, in fact, likely to be reasonably close to the true azimuthally-averaged values.

Finally, the results presented in this paper are based on the present-day structure of the ISM. If introduced as such into the dynamo equation, they make it possible to predict whether SN-driven turbulence is, at the present time, able to maintain the large-scale magnetic field, [FORMULA], but they do not permit a complete determination of the exact history of [FORMULA]. The reason is that the dynamo parameters themselves evolve in the course of time.

In most dynamo calculations, the temporal dependence of [FORMULA] and [FORMULA] is usually presumed to arise only from a dependence on [FORMULA] and it is embodied in a quenching factor of the form


which decreases with increasing magnetic field strength.

In the case at hand, an increase in [FORMULA] does, indeed, lead to a decrease in the components of [FORMULA] and [FORMULA], primarily through an enhancement of the background pressure and signal speed, which slows down the expansion of SBs and forces them to merge earlier (see Eq. (21)), and secondarily through a higher magnetic tension in the swept field lines, which opposes SB expansion perpendicular to [FORMULA]. However, the quenching effect can not be described by such a simplistic law as Eq. (36), because

(1) The final size and lifetime of individual SBs, and, hence, their contributions to [FORMULA] and [FORMULA], depend in a complicated manner on the external magnetic pressure (which appears both in the momentum equations and in the merging criterion).

(2) The quenching mechanism due to magnetic tension in the swept field lines is inherently anisotropic.

(3) The dynamo parameters at a given Galactic location depend not only on [FORMULA] at that location, but also on the whole profile of [FORMULA] throughout the domain of influence of SBs reaching the considered location.

Moreover, [FORMULA] is not the only time-dependent ISM parameter that affects [FORMULA] and [FORMULA]. The SN rate, the gas density and pressure, the characteristic interstellar scale heights, the gravitational field, the rotation rate [FORMULA] are all likely to have evolved since the Galaxy's formation, in ways not entirely independent of the magnetic field's own evolution.

A rough estimation of the temporal evolution of the large-scale magnetic field can, nevertheless, be obtained from the dynamo equation by using our dynamo parameters weighted by a factor of the form shown in Eq. (36) with, say, [FORMULA] and [FORMULA] equal to the equipartition magnetic field (e.g., Brandenburg et al. 1993).

Note that some authors have criticized the use of Eq. (36), arguing that dynamo action does, in fact, saturate long before the large-scale magnetic field builds up to equipartition (e.g., Vainshtein & Cattaneo 1992; Kulsrud & Anderson 1992). Their argument is based on the notion that the dynamo requires the formation of turbulent magnetic structures down to the very small diffusive scales at which field lines can reconnect; the field strength at these diffusive scales, they claim, exceeds the large-scale field strength by a huge factor (greater than the square root of the magnetic Reynolds number), and the process of magnetic field amplification by the dynamo saturates as soon as the magnetic field at the small diffusive scales reaches equipartition, i.e., when the large-scale field is still far below equipartition.

However, the kind of turbulence envisioned by Vainshtein & Cattaneo (1992) is very different from the SN-driven turbulence considered here. While the former gives rise to a smooth magnetic energy spectrum increasing toward large wavenumbers, the latter is more likely to produce two widely separated magnetic energy peaks with comparable amplitude, namely, a first peak at the large scales characteristic of SNR and SB shells, and a second peak at the small diffusive scales presumably generated upon merging (for instance, following collision with an interstellar cloud) and allowing the swept field lines to reconnect with the background magnetic field. The resulting spectrum would be similar to that associated with the stretch-twist-fold mechanism (Vainshtein & Zel'dovich 1972), which, Vainshtein & Cattaneo (1992) themselves recognize, avoids their general criticism. We, therefore, believe that a Galactic dynamo founded on the action of SN explosions and SBs is able to amplify the large-scale magnetic field to equipartition and subsequently maintain it at that level.

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Online publication: June 18, 1998