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

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5. Numerical results

We computed the non-vanishing components of the alpha- and diffusivity tensors (Eqs. (24) and (25)) at 15 equidistant Galactic radii ranging from 5 to 12 kpc. Inside 5 kpc, our model for the interstellar gas distribution becomes too uncertain and is no longer approximately plane-parallel. Beyond 12 kpc, the dynamo parameters have dropped to less than one hundredth of their maximum value. In Fig. 8, we display the results of the computation at R  = 5, 7, 8.5, 10, and 12 kpc.

[FIGURE] Fig. 8. Dynamo parameters due to SBs (thick line) and to isolated SNs (thin line, visible only for [FORMULA]) as a function of Galactic height, at Galactic radii [FORMULA]  5 kpc (dotted line), [FORMULA]  7 kpc (dot-dashed line), [FORMULA]  kpc (solid line), [FORMULA]  10 kpc (long-dashed line), and [FORMULA]  kpc (dashed line). [FORMULA] is the effective vertical escape velocity (Eq. (26)); [FORMULA], and [FORMULA] are the effective rotational velocities associated with the azimuthal, radial, and vertical alpha-effects, respectively (Eqs. (27)-(29)); [FORMULA] and [FORMULA] are the vertical and horizontal magnetic diffusivities (Eqs. (30)-(31)).

It clearly appears that the contribution from isolated SNs (thin line, visible only for [FORMULA]) is everywhere negligible compared to that from SBs (thick line): isolated SNs give rise to less than 0.1% of the alpha-parameters and to less than 1% of the escape velocity and of the magnetic diffusivities. At first sight, this huge difference is somewhat surprising, given that isolated and clustered SNs have roughly the same Galactic frequency (see below Eq. (20)). However, as mentioned in Paper III, when N SNs are clustered together, the SB they create has a fourvolume ([FORMULA]) which exceeds that of an individual SNR by a factor much larger than N. This is particularly true in a stratified medium, because the SBs that are luminous enough to push out of the dense gas layer are able to reach very large dimensions and live for a very long time, whereas most SNRs originate and remain confined to the vicinity of the midplane (Ferrière 1995). In the case of the alpha-tensor, there exists another reason why clustering is favorable: when explosion centers are distributed along the vertical, high-Z explosions partially counteract the effect of low-Z explosions, thereby reducing the components of [FORMULA]. This reduction, which is significant for isolated SNs, becomes negligible for SBs whose scale height (see Eq. (19)) is small compared to the final size.

The curves relative to [FORMULA] are similar in shape to those obtained in Paper III, but the overall magnitudes and characteristic scale heights are both smaller, mainly because a higher interstellar pressure was adopted in the present study.

The Z -dependence of the various dynamo parameters at [FORMULA] was already examined and interpreted in Paper III. We briefly re-discuss it here from a slightly different point of view. For the purpose of the discussion, we focus on SBs and suppose that they all originate at the midplane. Accordingly, we replace z by Z in Eqs. (26)-(31) and drop the differentiation symbol d in [FORMULA], [FORMULA], [FORMULA] The escape velocity, [FORMULA] (Eq. (26)), at a given height Z is one-half the collision frequency with explosion shock waves ([FORMULA]) times the thickness of the magnetic layer pushed through that height by each explosion (Z). The azimuthal alpha-parameter, [FORMULA] (Eq. (27)), is proportional to the escape velocity times the angle by which the field lines swept by an explosion rotate under the effect of the Coriolis force ([FORMULA]). The radial alpha-parameter, [FORMULA] (Eq. (28)), is also proportional to the escape velocity, but the angle by which the swept field lines rotate has a more complicated expression, which includes contributions from the Coriolis force ([FORMULA]) and from the large-scale shear ([FORMULA]). The vertical alpha-parameter, [FORMULA] (Eq. (29)), is proportional to the azimuthally-averaged effective large-scale rotation rate ([FORMULA]). The vertical and horizontal magnetic diffusivities, [FORMULA] (Eq. (30)) and [FORMULA] (Eq. (31)), are equal to one sixth of the collision frequency ([FORMULA]) times the square of the vertical and horizontal mean-free-paths (Z and [FORMULA], respectively), as expected for three-dimensional diffusion coefficients.

If all SBs were cylindrical and merged in one go (i.e., [FORMULA] and [FORMULA] independent of Z), [FORMULA], [FORMULA], and [FORMULA] would increase linearly with Z, [FORMULA] would vanish, [FORMULA] would increase quadratically, and [FORMULA] would remain constant, up to the final height reached by the least powerful SBs, and above that height all functions would increase less rapidly due to the decreasing number of contributing SBs. In reality, only low- to average- luminosity SBs are approximately cylindrical (see, for example, SB 1 in Fig. 9a); high-luminosity SBs assume a more complicated shape as they tend to balloon out to large distances in the rarefied gas away from the midplane (see, for example, SB 2 in Fig. 9a). Likewise, SB lifetime usually increases upward, because our merging criterion (Eq. (21)), which is roughly equivalent to the condition [FORMULA], 1 implies that SBs generally merge first at low Z, where the ambient interstellar pressure is strongest, and then at progressively higher Z. In consequence, the dynamo parameters in Fig. 8 (with the exception of [FORMULA]) grow faster and turn over at a higher altitude than predicted for cylindrical SBs.

[FIGURE] Fig. 9. Shape of four representative SBs at the end of their expansion phase, the asterisk giving the position of the explosion center. SB 1 is produced by [FORMULA]  30 SNs and originates at the Galactic midplane ([FORMULA]  pc); SB 2 has N  = 300 and [FORMULA]  pc; SB 3 has N  = 300 and [FORMULA]  pc; and SB 4 has N  = 1000 and [FORMULA]  pc. SBs in a are drawn at Galactic radii [FORMULA]  5 kpc (dotted line), [FORMULA]  kpc (solid line), and [FORMULA]  12 kpc (dashed line), while the SB in b is drawn at [FORMULA]  kpc (dot-dashed line) and [FORMULA] (solid line). SB 1 merges in the time interval 11.0-11.9 Myr at 5 kpc, 13.0-13.9 Myr at [FORMULA], and 15.6-16.9 Myr at 12 kpc. SB 2 merges in the time interval 12.3-12.7 Myr at 5 kpc, 17.2-17.4 Myr at [FORMULA], and 23.8-31.4 Myr at 12 kpc. SB 3 merges in the time interval 10.9-12.8 Myr at 5 kpc, 15.6-30.2 Myr at [FORMULA] and 23.1-43.0 Myr at 12 kpc. SB 4 merges in the time interval 13.7-26.0 Myr at 7 kpc and 17.3-22.8 Myr at [FORMULA]. Note that the same scaling is used along the horizontal and vertical axes.

The behavior of the vertical alpha-parameter is a little more subtle. As explained in Paper III, the sign of [FORMULA] is closely related to the shape of SBs. Consider, for instance, the SB displayed in Fig. 10, and suppose that the mean magnetic field, [FORMULA], is uniform and oriented in the positive Z -direction. Then, according to Eqs. (23) and (24), [FORMULA] is simply the Z -component of the turbulent electromotive force, [FORMULA], divided by the mean magnetic field's strength. The turbulent velocity responsible for the alpha-effect arises as the interstellar gas set into diverging motion by the explosion counterrotates with respect to the effective large-scale rotation rate, [FORMULA], in an attempt to conserve its angular momentum, so it is essentially clockwise about the Z -axis. The magnetic field swept by the explosion follows the bubble's surface, and, therefore, points away from the Z -axis up to the height of maximum horizontal cross-section and toward the Z -axis thereabove. As a result, the considered SB gives a positive (negative) contribution to the Z -component of [FORMULA], and, hence, to [FORMULA], in regions where its horizontal cross-section increases (decreases) upward. Since most SBs tend to be pinched by the dense interstellar gas and elevated pressure near the midplane (see Fig. 9), [FORMULA] is found positive at low [FORMULA] and negative high above the plane.

[FIGURE] Fig. 10. Schematics explaining the sign of the vertical alpha-parameter, [FORMULA]. The mean magnetic field, [FORMULA], is in the positive Z -direction. A SB originating somewhat above midplane (at the position indicated by the asterisk) produces an asymmetric elongated shell (thick line). The turbulent velocity responsible for the alpha-effect, [FORMULA], is opposite to the effective large-scale rotation rate, [FORMULA]. Magnetic field lines swept by the SB are deformed along the shell. Below (above) the height of maximum horizontal cross-section, the turbulent electromotive force, [FORMULA], has a positive (negative) Z -component, so that [FORMULA] is positive (negative).

We now turn to the R -dependence. The vertical profiles of the dynamo parameters in Fig. 8 remain qualitatively the same throughout the radial range of our computation. Their typical values vary little between 5 and 8 kpc, and they fall off rapidly beyond the solar circle, though not as rapidly as the SB rate per unit area (Fig. 6b). This relatively weak radial dependence arises because the steep drop in the SB rate is counteracted by an outward decline in the interstellar pressure (Fig. 2) and column density (Fig. 1b), which allows most SBs to live longer and reach greater dimensions at larger Galactic radii (see Fig. 9a).

On the other hand, there exist a few particularly luminous off-center SBs (like SB 4 in Fig. 9b) which are so severely affected by the gas density stratification that they actually grow higher at smaller radii. After bursting through the dense gas layer, these SBs find it easier to expand upward at smaller R, where they encounter less resistance from the ambient density (see Fig. 1a). Moreover, after their equatorial region merges and lets the interior pressure equilibrate with the external ambient pressure, the net driving pressure force on their upper portions is stronger at smaller R, thereby pushing their polar cap to higher altitudes. The existence of powerful SBs with very different behaviors regarding the vertical expansion explains why the vertical profiles of the dynamo parameters cannot be characterized by a single scale height which varies monotonically with Galactic radius.

To close up this section, we provide contour plots of the dynamo parameters in Fig. 11. Because of the limited radial range of our computation, we had to extrapolate the numerical results presented above. Inside [FORMULA]  kpc, we opted to take the functions obtained at 5 kpc and simply weight them by the factor [FORMULA], where [FORMULA] is the SB rate per unit area (Fig. 6b) and the exponential factor is meant to account for the increase in interstellar pressure toward the Galactic center. Outside [FORMULA]  kpc, all functions have become so small that the exact extrapolation procedure has no impact on the contour plots.

[FIGURE] Fig. 11. Contour plots of the dynamo parameters in a given meridional plane. [FORMULA] is the effective vertical escape velocity (Eq. (26)); [FORMULA], and [FORMULA] are the effective rotational velocities associated with the azimuthal, radial, and vertical alpha-effects, respectively (Eqs. (27)-(29)); [FORMULA] and [FORMULA] are the vertical and horizontal magnetic diffusivities (Eqs. (30)-(31)). Velocities are in km s [FORMULA] and magnetic diffusivities are in units of [FORMULA]  cm2  s-1.

Broadly speaking, the dynamo parameters peak at [FORMULA]  kpc and (except for [FORMULA]) [FORMULA]  kpc. The fact that the peak in dynamo activity occurs farther out than the peak in SB rate (see Fig. 6b) is a direct consequence of the longer lifetime and larger volume reached by SBs at greater R (see Fig. 9a). The peak values are [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA], about half an order of magnitude larger than the maximum values reached at the solar circle.

The standard Galactic dynamo operates through a combination between the large-scale differential rotation and the alpha-effect. The strength of these two processes against magnetic diffusion can be measured by the dimensionless numbers




respectively, and their combined strength is given by the dynamo number,


With [FORMULA], [FORMULA], and [FORMULA] (appropriate for a disk geometry), the dynamo number works out to be [FORMULA] in the peak region. This is a comfortable value, large enough to guarantee dynamo action (when [FORMULA]) and small enough to exclude very high dynamo modes. When [FORMULA], dynamo action also requires that the ratio [FORMULA] be less than a critical value [FORMULA] (Schultz, Elstner, & Rüdiger 1994); this condition, too, is fulfilled in the peak region, where we find [FORMULA].

Two more conclusions can be drawn from the values of the ratio [FORMULA] in the peak region. First, for [FORMULA], this ratio yields the magnetic pitch angle, p, defined as the angle between the large-scale magnetic field and the azimuthal direction, via [FORMULA]. Here, we have [FORMULA], leading to a pitch angle [FORMULA], which lies at the lower end of the observational range (Beck et al. 1996). Second, for [FORMULA], the ratio [FORMULA] quantifies the relative importance of the radial alpha-effect compared to the large-scale shear in the production of large-scale azimuthal magnetic field. The relatively high value [FORMULA] obtained here indicates that the Galactic dynamo is of the [FORMULA] -type, rather than the commonly assumed [FORMULA] -type.

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998