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

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4. Numerical results

In Sect. 4.1 and 4.2, we focus on axisymmetric ([FORMULA]) magnetic configurations, which are, respectively, symmetric (usually denoted by S0) and antisymmetric (denoted by A0) with respect to the midplane, and in Sect. 4.3, we study the lowest nonaxisymmetric ([FORMULA]) modes (denoted by S1 and A1).

4.1. Even axisymmetric (S0) magnetic configurations

Our reference run (Run 1) is characterized by

  • the dynamo parameters given by Fig. 1,

  • a large-scale rotation velocity with the radial dependence plotted in Fig. 2 and no vertical dependence,

  • perfect conductor boundary conditions.

The next runs are intended to analyze the physical effects of each of the input parameters. We systematically varied each of them separately and studied the impact of its variations on the time dependence and spatial structure of the large-scale magnetic field. Table 1 provides the key results of a selected number of representative runs, all of which differ from the reference run by only one input, specified in the second column. The next columns successively give the type of temporal behavior, the growth rate of the magnetic field strength, the oscillation frequency of the magnetic cycle (for oscillatory solutions), and the maximum value of the three field components normalized to the maximum field strength (or their range in the case of a cycle) near the final time of the computation (100 Gyr).


Table 1. Descriptive parameters of a few representative runs for even axisymmetric (S0) magnetic configurations

4.1.1. Reference run

When the input parameters take on their reference values, the large-scale magnetic field grows monotonically with time at an exponential rate of 0.46 Gyr-1 after the initial transients. The toroidal field component dominates by more than one order of magnitude. As indicated by Fig. 4, [FORMULA] exhibits, on each side of the equatorial plane, two well-defined polarities peaking at [FORMULA] kpc and separated by a node at [FORMULA] kpc. The poloidal field peaks at a slightly smaller radius, and its lines of force (displayed in the bottom right panel of Fig. 4) rotate about a point located close to the maximum of [FORMULA].

[FIGURE] Fig. 4. Contour lines of the three cylindrical components and poloidal lines of force of the large-scale Galactic magnetic field in a given meridional plane, for the S0 mode in Run 1 (reference run). The three field components have independent contour levels at [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA] their respective maximum absolute value (likewise for the next plots of the same kind).

This magnetic geometry is easily understood in the light of our discussion in Sect. 3. Note, in particular, the qualitative resemblance with the first three panels of Fig. 3. This resemblance as well as the predominance of toroidal field support the conventional idea that the Galactic dynamo operates in the [FORMULA]-regime ([FORMULA] referring to the production of poloidal field through the alpha-effect, and [FORMULA] to the production of toroidal field through the large-scale differential rotation). In that respect, our results are in agreement with those of most existing calculations (e.g., Parker 1971a; Vainshtein & Ruzmaikin 1971; Ruzmaikin et al. 1988; see Beck et al. 1996 for more recent references).

4.1.2. Effects of the escape velocity

The role played by the escape velocity can be clarified by comparing the results of the reference run (Run 1; Fig. 4) to those of a model with vanishing [FORMULA] (Run 2; Fig. 5). In the absence of an escape velocity, the Galactic dynamo is found to be oscillatory and slowly decaying, contrary to the monotonically growing behavior prevailing in the reference run (see Table 1). Oscillations are accompanied by an equatorward propagation, apparent in the time sequence of [FORMULA] displayed in Fig. 6. The successive snapshots are evenly spaced in time with an interval equal to one tenth the oscillation period (i.e., 0.96 Gyr), and the whole sequence covers exactly half a dynamo cycle. A peculiarity of the magnetic oscillations is that, even though they are sinusoidal at any given location, the shape of [FORMULA] varies little for almost half a period and then abruptly reverses around the time of energy minimum. Oscillatory dynamos with the same peculiarity were obtained before by Stix (1972) and by Schmitt & Schüssler (1989).

[FIGURE] Fig. 5. Contour lines of the three cylindrical components and poloidal lines of force of the large-scale Galactic magnetic field in a given meridional plane, for the S0 mode in Run 2 ([FORMULA]).

[FIGURE] Fig. 6. Time sequence of the azimuthal component of the large-scale Galactic magnetic field over half a dynamo cycle, for the S0 mode in Run 2 ([FORMULA]). The time interval between two successive snapshots is equal to 0.96 Gyr. The same contour levels are used for all the snapshots.

In both Run 1 and Run 2, the large-scale magnetic field is predominantly azimuthal and its configuration is characterized by alternating polarities along the vertical. However, when [FORMULA], the magnetic pattern is more tightly packed and, accordingly, the reversals are more numerous. In particular, [FORMULA] in the northern hemiplane has two or three distinct peaks at [FORMULA] kpc (sometimes too weak to be visible on the contour plots) plus one or two stronger peaks straddling the midplane possibly as far out as 10 kpc. The poloidal field also reaches its maximum value in the immediate vicinity of the midplane, where it is nearly radial.

From the global viewpoint of a propagating dynamo wave, one can say that the strong equatorial peaks in [FORMULA] arise as the polarity hitting the equator gets further squeezed and eventually pushed sideways by the incoming wavetrain. Alternatively, one can hold a closer up reasoning along the lines of our detailed discussion in Sect. 3. During the main phase of a dynamo cycle, [FORMULA] concentrates near the midplane, where it is primarily amplified by shearing of [FORMULA]. [FORMULA], for its part, decays slowly with time, as the only mechanism potentially able to maintain it, the azimuthal alpha-effect, is relatively inefficacious in this region of low [FORMULA] (see Fig. 1). At some point, [FORMULA] becomes too weak to further amplify [FORMULA], which then starts decreasing and eventually reverses. For a short time interval around reversal, [FORMULA] is small everywhere and its shape looks very different than for the rest of the cycle. During this short time interval, the principal dynamo action occurs no longer near the equator, but in the peak region of [FORMULA], where the production of new [FORMULA] through the azimuthal alpha-effect takes an essential part in initiating the next cycle. An analogous reasoning applies to the dynamos of Stix (1972) and Schmitt & Schüssler (1989), although the suppression of the azimuthal alpha-effect during the main phase is due to alpha-quenching rather than to the spatial structure of [FORMULA].

The escape velocity appears to counteract downward propagation and to reduce the vertical squeezing of wave packets, in accordance with our physical expectation. By varying the overall amplitude of the escape velocity while keeping the other input parameters unchanged, we find that the dynamo loses its oscillatory character at [FORMULA] (corresponding to [FORMULA] km s-1). This exercise also reveals that the growth rate increases with increasing escape velocity from -0.026 Gyr-1 at [FORMULA] up to a maximum [FORMULA] Gyr-1 at [FORMULA], and falls off at higher [FORMULA], becoming negative again at [FORMULA]. Physically, a moderate escape velocity favors field growth by preventing magnetic features from travelling downward away from the regions where they are easily amplified, by compressing field lines in a height interval (above the peak in [FORMULA]) wherein the alpha-effect works efficiently, and by limiting the amount of magnetic dissipation taking place near field reversals. For higher escape velocities, these favorable effects are overbalanced by the rapid escape of newly generated magnetic flux, so that the growth rate drops off and eventually becomes negative.

The implications of an escape velocity for the oscillatory character and for the growth rate of the Galactic dynamo are qualitatively similar to those of an outward galactic wind as predicted by the numerical simulations of Brandenburg et al. (1993). Their dynamo model differs from ours in two important respects: first, their alpha-tensor contains a quenching factor which accounts for the dynamical feedback of the generated magnetic field on the alpha-effect; second, contrary to our escape velocity, their wind has a significant radial component and its speed increases outward. By covering a wide range of wind speeds, Brandenburg et al. (1993) came to the somewhat surprising conclusion that the dynamo becomes oscillatory again for very strong winds (a few hundred km s-1). As a possible explanation, they suggested that the kind of wind considered in their study tends to produce a field geometry which is closer to a sphere (favorable to an oscillatory dipole) than to a disk (favorable to a steady quadrupole) (the magnetic field parity is not prescribed in their nonlinear calculations).

The effect of a galactic wind was also investigated by Elstner et al. (1995). Their model includes a complicated form of alpha-quenching, and it contains, in addition to an effective escape velocity, a vertical wind with a uniform or "spiky" radial profile, existing over a finite vertical range in which its speed increases with height. Their dynamo is found to be steady at low wind speeds and oscillatory above 10 km s-1 (for the uniform profile) or 50 km s-1 (for the spiky profile). As in Brandenburg et al.'s (1993) paper, a strong wind suppresses the steady quadrupolar mode, whereas, by pushing the generated magnetic field away from the source region, it limits alpha-quenching and allows the alpha-tensor to maintain sufficiently large values to induce oscillatory modes (see next subsection).

4.1.3. Effects of the alpha-parameters

In an [FORMULA]-dynamo, the only diagonal component of the alpha-tensor which plays an important role is [FORMULA]. To verify that [FORMULA] and [FORMULA] have a negligible influence, we computed a case where they both vanish (Run 3) and a case where they take on the values of [FORMULA] (Run 4). The results of these computations turn out to be very close to the results of Run 1, not only in the magnetic field structure (the contour plots of the three field components are almost indistinguishable from those shown in Fig. 4), but also in the growth rates (see Table 1). From this, we draw the conclusion that the exact values of [FORMULA] and [FORMULA] are indeed of little importance in our [FORMULA]-dynamo.

Having established that [FORMULA] is the deciding diagonal component of the alpha-tensor, we now inquire into the impact of its overall amplitude. Repeating Run 1 with [FORMULA] multiplied by a global scaling factor, [FORMULA], shows that the growth rate increases with increasing [FORMULA], starting from [FORMULA] Gyr-1 at [FORMULA] and passing through zero at [FORMULA]. It also emerges that the dynamo begins to oscillate when [FORMULA] reaches a critical value [FORMULA] (corresponding to [FORMULA] km s-1) and that the oscillation frequency grows steadily beyond this threshold. In the present context, the switch to an oscillatory dynamo is partly due to the diminished relative importance of the escape velocity.

The situation is a little more complex when the escape velocity vanishes. From a new series of runs with [FORMULA] and [FORMULA] scaled by the varying factor [FORMULA], we infer that (1) the growth rate rises from [FORMULA] Gyr-1 at [FORMULA] to [FORMULA] Gyr-1 at [FORMULA] (vanishing at [FORMULA]), then drops down to [FORMULA] Gyr-1 at [FORMULA], and finally rises again at larger values of [FORMULA]; (2) the dynamo becomes oscillatory at [FORMULA], i.e., much sooner than with our reference escape velocity, and the oscillation frequency in the oscillatory regime is again an increasing function of [FORMULA].

The nonmonotonous dependence of the growth rate on the alpha-parameter, albeit astonishing at first sight, can be explained by the fact that the fundamental dynamo mode, as well as the higher-order modes emerging at greater [FORMULA], remain excited only throughout a range of [FORMULA], below which the combination of alpha-effect and large-scale shear is too weak to counteract magnetic diffusion, and above which both mechanisms no longer act in concert toward constructive dynamo action. More generally, the growth rate variations can be described in terms of a dependence on the dynamo number,


where G is the large-scale shear rate, [FORMULA], [FORMULA], and h is a measure of the dynamo layer thickness. Parker (1971a; 1971b) was the first to argue that the growth rate does not increase monotonically with increasing dynamo number, and he provided a physical interpretation of this phenomenon in the framework of a one-dimensional analytical model. His assertion was subsequently confirmed by numerical simulations of dynamos in galactic disks (Elstner et al. 1992), in accretion disks (Torkelsson & Brandenburg 1994), and in tori (Deinzer et al. 1993).

Likewise, the change from steady to oscillatory behavior occurs when the growth rate of the dominating steady mode is overtaken by the growth rate of the next-order mode, which is either already oscillatory or becomes oscillatory upon merging with the initially dominating mode. Bifurcations of this kind were revealed by the torus dynamo computations of Deinzer et al. (1993), which systematically spanned a whole range of values of D. Another manifestation of these bifurcations is that galactic dynamos are found to be steady in the thin-disk approximation (e.g., Ruzmaikin et al. 1988) and tend to oscillate in the case of a thick disk (e.g., Stepinski & Levy 1988). Brandenburg et al. (1993) also noted that their even solution turns oscillatory when the disk thickness and the large-scale shear become large enough.

4.1.4. Effects of the diffusivity tensor

We started by making two computations with isotropic magnetic diffusivity: in Run 5, we took [FORMULA], which roughly amounts to enhancing the horizontal diffusivity by a factor of 5.6, and in Run 6, we lowered the vertical diffusivity to [FORMULA]. In the first case, the growth rate drops by almost a factor of 2 (see Table 1) and the magnetic pattern spreads out horizontally (see Fig. 7a). In the second case, the growth rate goes up by a factor [FORMULA], while the magnetic field exhibits more spatial structure and is vertically confined to a thinner layer (see Fig. 7b); in the absence of an upward escape velocity, the field would crowd against the equatorial plane, but here it tends to concentrate right above the peak in [FORMULA], in a region into which the generated field lines are effectively convected and where they undergo further compression.

[FIGURE] Fig. 7a and b. Contour lines of the azimuthal component of the large-scale Galactic magnetic field in a given meridional plane, for the S0 mode a in Run 5 ([FORMULA]) and b in Run 6 ([FORMULA]).

As a general rule, an increase in one of the magnetic diffusivities entails both a decrease in the growth rate and a spreading of the magnetized area in the concerned direction, in accordance with the usual effect of diffusion. Furthermore, if [FORMULA] and [FORMULA] are multiplied by a common scaling factor, [FORMULA], the dynamo is found to switch from oscillatory to steady as [FORMULA] rises above [FORMULA] (i.e., [FORMULA] cm2 s-1). This change in temporal behavior is closely akin to the inverse change associated with the scaling factor of [FORMULA], [FORMULA] (see Sect. 4.1.3), both being directly related to a bifurcation at the corresponding value of the dynamo number (Eq. (13)).

When [FORMULA], the growth rate does not decrease monotonically with increasing [FORMULA]: the trend is reversed in the range [FORMULA], in the same manner as the increase of the growth rate with [FORMULA] is reversed in some intermediate range (specified in Sect. 4.1.3). Besides, the switch to a steady dynamo is now delayed until [FORMULA].

4.1.5. Effects of a Z-dependence in [FORMULA]

In the next set of runs, the large-scale rotation velocity was assigned a gaussian vertical profile with varying scale height. The solution obtained for a scale height of 5 kpc (Run 7) is plotted in Fig. 8 and its characteristic parameters are listed in Table 1. Clearly, the Z-dependence of [FORMULA] has drastic implications for the magnetic field properties. Firstly, the dynamo is no longer monotonically growing, but instead oscillatory and decaying. Secondly, the alternating magnetic polarities are now stacked along an oblique axis, which can be verified as roughly parallel to lines of constant [FORMULA]. As an immediate consequence, there is room for a greater number of distinct polarities, typically 3 or 4 for [FORMULA]. Moreover, the maximum field strength occurs at a larger radius, [FORMULA] kpc, which corresponds to the outer boundary, rather than the peak, of the amplification region.

[FIGURE] Fig. 8. Contour lines of the three cylindrical components and poloidal lines of force of the large-scale Galactic magnetic field in a given meridional plane, for the S0 mode in Run 7 [FORMULA].

A time sequence of [FORMULA] over half a dynamo cycle is shown in Fig. 9. Through this figure, Run 7 provides a nice illustration of dynamo wave propagation along constant-[FORMULA] surfaces (see Sect. 3). It is, incidentally, because these surfaces are no longer vertical but oblique that propagation becomes possible and that wave packets migrate away from the amplification regions. This explains both the oscillatory character and the negative growth rate obtained.

[FIGURE] Fig. 9. Time sequence of the azimuthal component of the large-scale Galactic magnetic field over half a dynamo cycle, for the S0 mode in Run 7 [FORMULA]. The time interval between two successive snapshots is equal to 1.20 Gyr. The same contour levels are used for all the snapshots.

An abrupt change in the decay rate occurs at [FORMULA] Gyr. At that time, the magnetic field has basically diffused out of the induction torus, and solely the low-diffusivity external regions remain magnetized. The dynamo then enters a regime of pure dissipation, where the field monotonically decays at the slow rate dictated by the value of the background diffusivity.

It should be noted that the results of this subsection are at odds with the conclusions of Brandenburg et al.'s (1993) investigation, in which a vertical gradient in [FORMULA] was found to have but a small effect. There exist many differences between their model and ours, in particular with regard to the vertical distribution of the magnetic diffusivities and to the adopted rotation curve, that might be responsible for this difference in behavior.

4.1.6. Influence of the boundary conditions

We tested our simulations for sensitivity to the boundary conditions by running a model with vacuum (zero conductivity) outside the computation domain (Run 8). The solution remains monotonically growing, though at a slightly faster rate, and the magnetic field structure looks very similar to that obtained in Run 1, with this small difference that the azimuthal field component falls off somewhat more rapidly at high [FORMULA]. Both the larger growth rate and the steeper falloff can be attributed to the free diffusive escape of magnetic field lines across the top surface, which makes it a little easier to get rid of reversed fields at high altitude (see Parker 1971a).

In view of our poor knowledge of the physical properties of the medium surrounding the Galaxy, it is reassuring that the interior solution is only weakly affected by the adopted boundary conditions. As already mentioned in Sect. 2, this relative insensitivity is achieved by placing the outer surface far away from the source region.

4.1.7. Effects of the radial dependence of the dynamo parameters

The first original feature of our work is the highly inhomogeneous nature of the [FORMULA] and [FORMULA] functions. To assess the importance of this distinctive feature, we made several comparison runs with each component of [FORMULA] and [FORMULA] individually smoothed out in the radial direction, i.e., replaced by its horizontally-averaged value.

In all the considered cases, we approximately recovered the magnetic structure of the corresponding unsmoothed calculation. The only notable difference is that the magnetic field in Run 7 is now excited at larger R.

The dependence of the growth rate and oscillation frequency on the dynamo number remains qualitatively the same, but the critical dynamo numbers for field amplification and for oscillatory behavior are somewhat smaller. For instance, Run 2 now has a positive growth rate.

To some extent, the results of these experiments support the robustness of our previous results.

4.2. Odd axisymmetric (A0) magnetic configurations

We repeated all the runs performed in Sect. 4.1 for magnetic configurations which are antisymmetric, i.e., odd, in Z, and we listed the key results of the most instructive runs in Table 2. The odd solutions turn out to share many features of their even counterparts and, for the most part, they can be interpreted physically in an analogous fashion. Therefore, the emphasis of this section will be placed on pointing out and explaining the main differences between both types of solutions.


Table 2. Descriptive parameters of a few representative runs for odd axisymmetric (A0) magnetic configurations

In the reference run (Run 1), defined in the same way as in Sect. 4.1, the large-scale magnetic field has a temporal dependence (steady exponential growth at a rate of 0.44 Gyr-1) and a spatial structure (displayed in Fig. 10) which are very close to those calculated in the even case (see Table 1 and Fig. 4). The basic reason lies in the smallness of the vertical magnetic diffusivity at low altitude (see Fig. 1), combined with the presence of an escape velocity which prevents the formation of steep gradients near the midplane. The ensuing magnetic diffusion across the equator remains weak, so that both hemiplanes stay largely decoupled. The particularity of odd configurations is that field lines of opposite polarity diffuse (however slowly) toward the equator, where they reconnect and dissipate away; for purely growing solutions, this results in a slight decrease of the growth rate.

[FIGURE] Fig. 10. Contour lines of the three cylindrical components and poloidal lines of force of the large-scale Galactic magnetic field in a given meridional plane, for the A0 mode in Run 1 (reference run).

The situation is completely different when the escape velocity vanishes (Run 2): as explained in Sect. 4.1.2, the magnetic field pattern migrates toward the equator, where the northern and southern wavetrains collide. In the even case (illustrated in Fig. 6), the colliding polarities have the same sign and they tend to merge into one strong peak. In contrast, in the odd case (illustrated in Fig. 11), they are of opposite sign, so they tend to annihilate each other through magnetic reconnection. The latter process clearly facilitates migration and, consequently, increases the oscillation frequency. In addition, the facilitated migration diminishes the vertical squeezing of the wave pattern away from the midplane; in the present case, this side effect is sufficient to reduce the global amount of magnetic dissipation and to bring the growth rate above zero (see Table 2).

[FIGURE] Fig. 11. Time sequence of the azimuthal component of the large-scale Galactic magnetic field over half a dynamo cycle, for the A0 mode in Run 2 ([FORMULA]). The time interval between two successive snapshots is equal to 0.24 Gyr. The same contour levels are used for all the snapshots.

The influence of the equatorial boundary condition is also manifest in the results of a series of runs with variable amplitude of the escape velocity. It appears that the odd parity condition raises the threshold below which the dynamo is oscillatory to [FORMULA] and lowers the maximum growth rate to about the value of 0.44 Gyr-1 reached at [FORMULA]. Up to [FORMULA], in a regime where both even and odd solutions are oscillatory, the odd mode has a larger growth rate. For greater escape velocities, the even mode grows faster, although the difference in growth rates asymptotically goes over to zero.

Regarding the diagonal components of the alpha-tensor, we find, as expected, that [FORMULA] and [FORMULA] have but a small impact on the odd solution. Increasing the overall amplitude of [FORMULA], as measured by [FORMULA], increases the growth rate, not only when the escape velocity takes on its reference value (as in the even case), but also in the absence of an escape velocity (contrary to the nonmonotonous dependence obtained in the even case; see Sect. 4.1.3). This result is in agreement with the torus dynamo simulations of Deinzer et al. (1993). The threshold for magnetic field amplification is higher than for the even solution: it is given by [FORMULA] when [FORMULA] and by [FORMULA] when [FORMULA]. However, there exists a parameter domain (approximately [FORMULA], [FORMULA] or [FORMULA], [FORMULA]) wherein the odd mode grows faster than the even mode. Finally, the odd solution begins to oscillate at [FORMULA] when [FORMULA] and at [FORMULA] when [FORMULA], i.e., each time sooner than the even solution. This is a further indication that odd parity tends to favor oscillatory behaviors.

The diffusivity tensor plays basically the same role as in the even case, and its effect on the odd mode is qualitatively the same as that described in Sect. 4.1.4, except that the growth rate now decreases monotonically with increasing [FORMULA] even when [FORMULA]. Here too it can be verified that multiplying [FORMULA] and [FORMULA] by [FORMULA] is rougly equivalent to dividing [FORMULA] by [FORMULA].

When the large-scale rotation rate varies with height (Run 7), even and odd modes behave similarly, namely, they migrate along lines of constant [FORMULA] and slowly decay in time. The small differences in the decay rates and in the oscillation frequencies are due (directly or indirectly) to the weak magnetic reconnection taking place at the midplane.

A computation performed with vacuum boundary conditions (Run 8) leads to the exact same conclusions as those presented in Sect. 4.1.6.

Likewise, comparison runs with the [FORMULA] and [FORMULA] functions horizontally smoothed out have the same properties as in Sect. 4.1.7, not only with regard to the magnetic structure, but also for the temporal behavior and its dependence on the dynamo number.

From the above discussion, it emerges that (1) the odd mode oscillates more readily than the even mode, except when the large-scale rotation velocity is a function of height, in which case both modes oscillate with almost equal ease; (2) the even mode is often, but not always, easier to excite, and it is only when both modes are oscillatory that the odd mode is liable to grow faster.

For comparison, early calculations of galactic dynamos systematically found a preference for the even (S0) mode compared to the odd (A0) mode (see, e.g., reviews by Wielebinski & Krause 1993and by Beck et al. 1996). Subsequent studies showed that the S0 mode is indeed preferred under typical galactic conditions, but they also demonstrated the possible dominance of the A0 mode in the following circumstances: when the galactic disk is sufficiently thick (Stepinski & Levy 1988), when the alpha-effect is active in the halo as well as in the disk and exceeds a certain threshold (Brandenburg et al. 1992), when dynamo action is dominated by the alpha-effect near the galactic center (Donner & Brandenburg 1990; Elstner et al. 1992), or when differential rotation is weak and the alpha-tensor possesses strong anisotropies (Meinel et al. 1990). In all these cases, the steady S0 mode has entered the decaying regime (see discussion on the nonmonotonous dependence of the even mode's growth rate on the dynamo number in Sect. 4.1.3), allowing the dynamo to operate in a higher-order mode.

4.3. Bisymmetric ([FORMULA]) modes

We conducted a number of nonaxisymmetric computations with azimuthal wavenumber [FORMULA]. In all cases, the magnetic field decays in time after going through the same successive stages. At the beginning, field lines are essentially sheared out by the large-scale differential rotation into an increasingly tighter bisymmetric spiral. This winding brings field lines of opposite sense closer together, thereby creating a magnetic configuration conducive to efficient dissipation. Within a few rotation periods, a magnetic void appears in the radial range [FORMULA] kpc, where the combination of large-scale shear and magnetic diffusivity is the most destructive. This void rapidly spreads to the edge of the Galactic disk. After [FORMULA] Gyr, magnetic fields have virtually vanished everywhere outside 5 kpc; inside 3 kpc, they are regenerated by an [FORMULA]-process, which is, however, insufficient to maintain them against the background diffusivity. As a result, the magnetic field evolution ends up with a long phase of exponential decay. During this phase, the whole magnetic pattern propagates in the azimuthal direction at a frequency ([FORMULA] Gyr-1 for Run 1 (reference) and Run 2 ([FORMULA])), which is intermediate between the fast rotation rate at small R (56 Gyr-1) and the slow magnetic decay rate ([FORMULA] Gyr-1 for Run 1 and Run 2).

Let us start by analyzing the reference run. To help visualize the magnetic field distribution during the final decay phase, we plotted contour lines of the three field components in two orthogonal meridional planes, both for the S1 solution (Fig. 12) and for the A1 solution (Fig. 13). In both figures, the magnetic field is confined to the vicinity of the rotation axis, in accordance with its removal from differentially rotating areas, and its three components have comparable strength, as expected for an [FORMULA]-dynamo. The differences between even and odd solutions are more pronounced than in the axisymmetric case. This is because magnetic diffusion across the equator plays a more important role now that dynamo action takes place in a region where the amplification mechanism is less efficient than diffusion and where the vertical diffusivity at midplane is no longer much smaller than the diffusivity at higher altitudes.

[FIGURE] Fig. 12. Contour lines of the three cylindrical components of the large-scale Galactic magnetic field in two orthogonal meridional planes, for the S1 mode in Run 1 (reference run).

[FIGURE] Fig. 13. Contour lines of the three cylindrical components of the large-scale Galactic magnetic field in two orthogonal meridional planes, for the A1 mode in Run 1 (reference run).

When the escape velocity goes to zero, the field pattern is squeezed along the vertical and displaced toward the equator, but its structure at late times remains similar to that obtained in the reference run. Likewise, the decay rate is either reduced (for the S1 mode) or increased (for the A1 mode), but not by a large amount. Hence the escape velocity appears to have a less dramatic effect than in the axisymmetric case. The basic explanation is again that the magnetic field evolution is dominated by dissipation (which remains the same in both runs) rather than by true dynamo action (to which the escape velocity belongs).

Along the same lines, it can be verified that the alpha-parameters have but a limited influence on the solutions. Increasing their overall amplitude slightly modifies the field configuration and leads to a small decrease in the decay rate.

The two primary factors controlling the behavior of bisymmetric magnetic fields are the magnetic diffusivities and the dimensions of the rigidly-rotating region. The former are determining for the decay rate, as can be checked by varying their values. The latter, on the other hand, delimit the zone within which magnetic fields survive the transient destructive phase due to differential rotation. This magnetized zone typically extends somewhat beyond the rigidly-rotating region (see Figs. 12 and 13). In the particular case when the rotation rate falls off with height (Run 7), the whole Galaxy rotates differentially and magnetic fields are rapidly killed off everywhere through azimuthal shearing (of both [FORMULA] and [FORMULA]) followed by magnetic dissipation (along R and, even more efficient, along Z). Under these conditions, there is no prolonged [FORMULA]-dynamo phase.

Finding [FORMULA] modes more difficult to excite than [FORMULA] modes agrees with the conclusions of all existing galactic dynamo models with axisymmetric background properties (see Beck et al. 1996). According to these studies, the difference in growth rates may, however, be improved in favor of the [FORMULA] modes by factors such as the absence of a large-scale shear (Rädler 1986; Baryshnikova et al. 1987), the thinness of the disk (Ruzmaikin et al. 1988; Moss & Brandenburg 1992), anisotropies in the alpha-tensor (Rüdiger et al. 1993), azimuthal variations in [FORMULA] and [FORMULA], arising from either the underlying spiral structure (Rohde et al. 1999a; Schreiber & Schmitt 2000) or a gravitational encounter with a nearby galaxy (Moss et al. 1993; Vögler 1999), parametric resonances with spiral arms (Schmitt & Rüdiger 1992; Hanasz & Chiba 1994; Moss 1996; Rohde et al. 1999b), and gas streaming (Moss 1998; Moss et al. 1999a).

The results of this section also confirm the tendency of nonaxisymmetric magnetic fields to avoid the differentially rotating parts of a galaxy (e.g., Ruzmaikin et al. 1988). In that respect, it is noteworthy that the exact R-dependence of the rotation law may have crucial implications for the radial distribution of bisymmetric fields. For example, solid-body rotation inside 3 kpc (as in Fig. 2) or a large-scale shear approaching zero toward the axis (as in Moss & Brandenburg 1992) facilitates their survival at small R. In contrast, with the polynomial approximation of the rotation curve of M51 adopted by Baryshnikova et al. (1987), bisymmetric modes are found to be localized farther out than axisymmetric modes. Finally, if [FORMULA] increases all the way in to very small R (as in M31; Braun 1991), the region of approximately rigid rotation almost vanishes and nowhere do bisymmetric fields survive the brief destructive phase due to large-scale shearing. Since [FORMULA] is in fact ill-determined near to the Galactic center, it may not be ruled out that such a situation prevails in our own Galaxy.

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Online publication: June 26, 2000