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Astron. Astrophys. 358, 125-143 (2000)

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6. Discussion

The dynamo parameters used in this paper account only for the turbulent motions associated with the expansion phase of SNRs and SBs; they neglect the return motions as well as all other kind of turbulence. As argued by Ferrière (1993a), the return motions are much less correlated than the systematic diverging motions directly driven by an explosion; as a result, they generate little alpha-effect and give but a small contribution to the alpha-parameters, [FORMULA], [FORMULA], and [FORMULA]. Furthermore, in the widely accepted picture of the Galactic fountain, return motions take place in the form of individual clouds falling back toward the midplane (Shapiro & Field 1976; Norman & Ikeuchi 1989). This means that the "downdraughts" are disconnected from each other, so that they locally stretch magnetic field lines without effectively advecting the large-scale magnetic field back down (Brandenburg et al. 1995). Hence they do not affect the escape velocity, [FORMULA], to a significant extent. On the other hand, by taking part in the random mixing of field lines, return motions augment the turbulent diffusivities, [FORMULA] and [FORMULA] (Ferrière 1993b). Their contribution added to that from all other turbulent sources is likely to exceed our background diffusivity of [FORMULA] cm2 s-1. If so, the actual dynamo growth rate is less (or possibly greater in the case when the escape velocity vanishes; see Sect. 4.1.4) than predicted by our computations.

The fact that most runs, including the reference run, yield positive growth rates lends credence to the notion of a SN-driven Galactic dynamo, while leaving some room for additional sources of magnetic diffusion. Besides, the rather low values obtained ([FORMULA] Gyr-1) suggest that the large-scale magnetic field is presently close to a state of marginal growth. This would be consistent with the observation that magnetic fields in the Milky Way as well as in external spirals have attained, and even passed, equipartition with the interstellar gas (e.g., Boulares & Cox 1990; Heiles 1995). A nearly marginal growth would also be consistent with Ko & Parker's (1989) scenario, according to which the Galactic dynamo is now in a dormant state between two bursts of star formation.

It should be emphasized that our low growth rates in no way conflict with the detection of µG-level magnetic fields in very young galaxies (Kronberg & Perry 1982; Wolfe 1988; Kronberg 1994). It is indeed likely that magnetic field amplification proceeded much faster at early stages of Galactic evolution, either because the dynamo parameters were substantially larger (see Sect. 5), or because other, more efficacious, amplification mechanisms were at work, such as the Balbus-Hawley instability (Balbus & Hawley 1991; Hawley & Balbus 1991), the Parker instability (Hanasz & Lesch 1997; 1998), the small-scale fluctuation dynamo (Tajima et al. 1992, Kulsrud et al. 1997), a buoyancy-driven alpha (Brandenburg & Schmitt 1998; Moss et al. 1999b; Schmitt 2000)[FORMULA]

The negative growth rates obtained for [FORMULA] indicate that bisymmetric magnetic fields in our Galaxy should have largely decayed away by now, leaving behind a nearly axisymmetric magnetic pattern. As it turns out, this result is not unambiguously supported by current observations. Faraday rotation measures of pulsars and extragalactic radio sources reveal the existence of at least two field reversals toward the Galactic center (Vallée et al. 1988; Rand & Lyne 1994) and possibly additional reversals outside the Solar circle (Lyne & Smith 1989; Clegg et al. 1992). These reversals have often been regarded as evidence for a bisymmetric configuration. Our study definitely disproves this interpretation, as radial reversals naturally emerge from axisymmetric runs with Z-dependent [FORMULA] (Run 7; Fig. 8). According to another axisymmetric dynamo model, described by Poezd et al. (1993), the observed reversals would be inherited from a fluctuating seed field and constitute transient nonlinear structures. It is still unclear whether the best fit to the rotation measure data is obtained with a bisymmetric Galactic magnetic field (Simard-Normandin & Kronberg 1980; Sofue & Fujimoto 1983; Han & Qiao 1994; Han et al. 1999) or with an axisymmetric one (Rand & Kulkarni 1989; Vallée 1996).

Observations of external galaxies do not help settle the question: claims have been laid to the detection of [FORMULA], [FORMULA], and mixed magnetic spiral structure, with a probably greater occurrence of [FORMULA] (or, more generally, even m) field components (Beck et al. 1996). The evidence for predominantly bisymmetric fields is still controversial (except in the case of M81; Krause et al. 1989), but it is interesting that the three best-candidate galaxies appear to be in tidal interaction with a companion, which would provide a natural explanation for their nonaxisymmetric magnetic structure. As already mentioned in Sect. 4.3, an alternative or complementary explanation could be the pre-existence of material spiral arms.

With regard to the even/odd parity in Z, the reference run in its linear version gives a slight preference to the S0 mode, whereas Run 2 ([FORMULA]) and Run 7 ([FORMULA]) give preference to the A0 mode. Neither parity, therefore, is ruled out by our linear calculations. Moreover, the near-equality of the S0 and A0 linear growth rates computed in the reference run suggest that the present-day Galactic magnetic field consists of a superposition of even and odd modes. Nonlinear models point to the same conclusion: they show that mode interactions may cause the field parity to change on timescales of the order of the Galactic lifetime (see, e.g., Fig. 15) and that the present-day field has probably not reached its final parity yet. As remarked by Brandenburg et al. (1992), it is more likely that the Galactic magnetic field is currently in a transient state still influenced by the seed field parity.

Here too the observational situation is rather muddled. It has long been known from optical polarization of starlight that the large-scale magnetic field within a few kpc from the Sun is almost horizontal (Mathewson & Ford 1970). Rotation measures of extragalactic radio sources later confirmed this field direction and showed that it prevails in most parts of the Galactic disk (e.g., Simard-Normandin & Kronberg 1980). The rotation measure vertical distribution both for extragalactic sources and for pulsars with [FORMULA] appears to be approximately antisymmetric about the equatorial plane in the inner Galactic quadrants and roughly symmetric in the outer quadrants (see Figs. 1 and 2 of Han et al. 1997). Although the observed antisymmetry in the inner Galaxy has often been attributed to nearby anomalies like the North Polar Spur, Han et al. (1997) argued for a true property of the large-scale magnetic field away from the midplane. Rotation measures of low-latitude pulsars, however, point to a symmetric distribution at all longitudes near the midplane (see Fig. 4 of Rand & Lyne 1994).

In principle, the vertical field component can also serve as an indicator of magnetic parity, but actual measurements of [FORMULA] are rendered difficult by its weakness compared to the horizontal field components and by the presence of strong irregularities such as the alternating up and down fields produced by SBs. Despite these difficulties, Han et al. (1999) attempted to estimate [FORMULA] in the vicinity of the Sun, based on rotation measures of extragalactic sources toward the Galactic poles. As expected, their inferred [FORMULA] is much smaller than the horizontal field. Unfortunately, since they proceeded from the a priori assumption that [FORMULA] is constant across the ionized layer, their estimation doesn't say anything about magnetic parity.

The detection of several filamentary radio structures at the very center of the Milky Way provides good evidence that a strong, coherent, poloidal (and hence odd) magnetic field pervades the innermost 70 pc (Morris 1990). It is not clear whether this field constitutes just a localized feature or whether it extends much farther out. Obviously, the second possibility is at odds with our numerical results, but it is easily realized by a model with a strong source of alpha-effect near the Galactic center (Donner & Brandenburg 1990).

In sum, the general observational picture emerging for the large-scale magnetic field in our Galaxy is characterized by an even topology outside the Solar circle and at low [FORMULA] and by an odd topology in the inner halo and in the central area.

In external galaxies, it is even more difficult to discriminate between even and odd magnetic parities. Edge-on galaxies, which are the best suited for that purpose, generally possess a thick synchrotron disk with magnetic fields nearly horizontal in the inner region and horizontal or inclined to the plane in the outer regions (Dumke et al. 1995; Beck et al. 1996). A notable exception is NGC 4631 where the observed magnetic field is approximately vertical in the central rigidly-rotating parts (Golla & Hummel 1994); like in our Galaxy, a vertical field going through the plane is indicative of an odd magnetic configuration. On the other hand, evidence for an even distribution was found in IC 342 (Krause 1993) and in the extended radio halo of NGC 253 (Beck et al. 1994). Another argument in favor of even modes comes from the observation that magnetic fields in mildly inclined external galaxies are predominantly directed inward (Krause & Beck 1998).

Another important issue concerns the steady vs. oscillatory behavior of the large-scale magnetic field. According to our computations, the Galactic magnetic field is steady under the reference conditions and turns oscillatory either when the escape velocity becomes small enough or when the large-scale rotation has a sufficiently steep vertical dependence. Observationally, the extremely long timescales involved (typical periods of several Gyr) make it virtually impossible to detect potential oscillations in the Milky Way. Observations of a vast number of external galaxies could, in principle, give us a clue, the idea being to look for evidence that different galaxies are presently at different phases of a dynamo cycle. The problem, however, is that dynamo cycles like those displayed in Figs. 6 and 11 would be difficult to uncover in practice, since the magnetic structure remains qualitatively the same throughout a cycle (except for a short period around the time of field reversal in Fig. 6).

We now turn to the magnetic field direction. In the axisymmetric solutions, the large-scale magnetic field, [FORMULA], is roughly azimuthal, and its pitch angle, defined by


is usually negative in most places (see Figs. 4 and 5), in accordance with the preponderant effect of differential rotation. It is only when [FORMULA] falls off with height (Run 7) that the pitch angle tends to become positive in the high-[FORMULA] regions not dominated by differential rotation (see Fig. 8). The predicted roughly azimuthal direction of [FORMULA] is supported by starlight polarimetry (Mathewson & Ford 1970) and by rotation measure studies (e.g., Simard-Normandin & Kronberg 1980; Rand & Lyne 1994) in our Galaxy, and by radio polarization observations in external spirals (Beck et al. 1996). Furthermore, starlight polarimetry yields a local pitch angle [FORMULA] (Heiles 1996). The value inferred from rotation measures, which is far less reliable, varies between [FORMULA] (from extragalactic source data; Simard-Normandin & Kronberg 1980) and [FORMULA] (from pulsar data; Thomson & Nelson 1980), with the most recent studies converging toward a small negative value comprised between [FORMULA] (Rand & Lyne 1994) and [FORMULA] (Han & Qiao 1994). It should, nevertheless, be noted that a separate treatment of low- and high-latitude extragalactic sources yields a negative p in the disk and a positive p in the halo (Inoue & Tabara 1981), which would be consistent with the results of Run 7 (Fig. 8). When confronting our calculated pitch angle to observations, we should keep in mind that the actual field direction is probably affected by the underlying spiral structure, a feature not accounted for in our model.

To conclude this section, we discuss the poloidal distribution of the large-scale magnetic field. In the reference run, [FORMULA] extends up to [FORMULA] kpc from the midplane and reverses at [FORMULA] kpc (see Fig. 4). This vertical structure contrasts with that deduced from rotation measures of extragalactic sources, which suggest a scale height [FORMULA] kpc (Inoue & Tabara 1981; Han & Qiao 1994) and indicate no reversal along the vertical (as pointed out by Simard-Normandin & Kronberg 1980, the rotation measures of extragalactic sources have the same sign and are systematically larger in absolute value than the rotation measures of pulsars in a nearby direction). Synchrotron emission data yield a greater scale height [FORMULA] kpc (Ferrière 1998a), but the latter refers to the total field, not to its regular, i.e., large-scale, component. In the radial direction, the calculated B peaks at [FORMULA] kpc and decreases outward with a scale length [FORMULA] kpc. This scale length is compatible with pulsar rotation measures (Rand & Lyne 1994), whereas the absence of radial field reversals is in obvious contradiction.

When the escape velocity vanishes (Run 2), the S0 mode during most of its cycle is confined close to the plane and exhibits no significant vertical reversals (see Fig. 6), in better agreement with rotation measure data. However, it is still devoid of radial reversals, except for the short fraction of its cycle when it extends to unrealistically high altitudes (see Fig. 5). Likewise, the A0 mode has an observationally acceptable vertical distribution over part of its cycle, but it misses the expected radial reversals (see Fig. 11).

A very different situation arises when the rotation rate decreases away from the midplane (Run 7; Fig. 8). The magnetic scale height is then much greater than indicated by rotation measure analyses, and radially the magnetic field concentrates too far out (remember that the Sun is located at [FORMULA] kpc). On the other hand, Run 7 is the only model which reproduces the important observational property that [FORMULA] reverses at least a couple of times in the radial direction.

In external galaxies, the regular magnetic field generally appears to be distributed either in a circular ring or, more often, along the spiral arms (Beck et al. 1996). Radial reversals have only been observed in a few rare galaxies believed to host a bisymmetric field (e.g., in M81; Krause et al. 1989). The vertical magnetic structure is usually more difficult to establish. So far, a vertical field reversal has been detected in only one galaxy between the disk and the halo (M51; Berkhuijsen et al. 1997); apparently, this galaxy would be a good candidate for the vertical distribution obtained in the reference run (Fig. 4).

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Online publication: June 26, 2000