4 Results

   
4.1 Our basic model

We first examined the simplest dynamo model with $\widetilde\alpha(\vec{r})=1$, $f_\alpha =f_\eta =0$, i.e. $\eta$ is constant, unmodified by the shear, and $\alpha$ is a function only of the magnetic field and gas density, via the $\alpha$-quenching. We experimented with several parameter combinations, and first discuss here a model with $\eta_0=2\times 10^{26}\,{\rm cm}^2\,{\rm s}^{-1}$ so $R_\omega =36$. With $R_\alpha =0.3$ (giving D=-10.8), the dynamo is slightly supercritical in the central parts of the galaxy (r<1-2kpc) and subcritical at larger radii. The eventual magnetic field configuration is steady in the corotating frame, and its structure is shown in Fig. 4, superimposed on the gas density. Here, and in the other figures, we have smoothed the magnetic field over a scale of about 0.6kpc, to approximate the resolution of the observations. For comparison, the radio observations have a beamwidth of $10\hbox{$^{\prime\prime}$ }$ ( $0.8\,{\rm kpc}$) for NGC 1097 and $15\hbox{$^{\prime\prime}$ }$ (2kpc) for NGC 1365. We see that the magnetic field is strongly concentrated to the central regions ( ) in this model with uniform diffusivity, reflecting the strong dynamo action in this region where the angular velocity and its shear are maximal. This feature remains even when the dynamo number is very substantially increased. In this model, the maximum field strength of about 0.5B₀ is reached at $r\approx0.7\,{\rm kpc}$ (corresponding to about $3\,\mu{\rm G}$); the field is negligible beyond $r\simeq2\,{\rm kpc}$. Observations of NGC 1097 suggest that the ratio of the regular field strength in the inner region ( $r/r_{\rm c}\simeq0.1$) to that in the outer bar region is about 2, and so this model is unsatisfactory. Note that this feature is little altered by the smoothing to the resolution of the observations.

Even when |D| is increased by an order of magnitude, the model magnetic field remains strongly concentrated to the central regions. Note that in all our models the field strength increases with |D|, but in order to obtain field strengths comparable with those observed near to the corotation radius, it is necessary to increase |D| to values at the margin of what is plausible, and the field in the inner regions is then significantly larger than observed. The overall field structure is relatively insensitive to the magnitude of D. An obvious problem with this basic model with $\eta=\mbox{const.}$ is that it yields an unrealistically weak magnetic field in the outer regions and perhaps a too strong contrast in B between the dust lane regions and those upstream of them. Thus we now relax the condition that $\eta$ be constant.

   
4.2 Models with turbulent diffusivity depending on velocity shear: Uniform $\widetilde\alpha$

An obvious refinement of the model is to consider the turbulent magnetic diffusivity to be modulated by the shear rate, i.e. to put $f_\eta > 0$ in Eq. (5), keeping the background value uniform, i.e. $\eta_0(\vec{r}) = \mbox{const}$. Note that, as $f_\eta$ and thus the magnetic field dissipation increase, so does the marginal dynamo number, for example $D_{{\rm cr}}\approx -27$ when $f_\eta =5$, and even D=-11 is just subcritical when $f_\eta =1$. Also, broadly speaking the effects of taking $f_\alpha > 0$ for given $R_\alpha$ and of increasing $R_\alpha$ with $f_\alpha =0$ are similar, and so we keep $f_\alpha =0$. We now set $\eta_0=10^{26}$ cm²s^-1, so that $R_\omega =72$ and adopt $R_\alpha =3$. Thus the models we discuss now are all substantially supercritical, with the local values of D (calculated from the local values of $\Omega$) of order -(10²-10³) at and about -15 near the corotation radius.

Figure 5: Vectors of the horizontal magnetic field for the models with $\widetilde\alpha(\vec{r})=1$, $f_\alpha =0$, $R_\alpha =3$, $R_\omega =72$. a) $f_\eta =1$, b) $f_\eta =3$, and c) $f_\eta =5$. The circumscribing circle has radius 8 kpc. Shades of grey show gas density, with lighter shades corresponding to higher values. The magnetic field vectors have been rescaled as $\vec{B}\to\vec{B}/B^n$ with n=0.3, to improve the visibility of the field vectors in regions with small B.

In Fig. 5 we show the field structures for calculations with $f_\eta = 1, 3, 5$. Again, the magnetic fields are steady in the rotating frame. Increasing $f_\eta$ clearly reduces the dominance of the central field. Also, increasing $\eta_0$ to $2\times 10^{26}$ cm²s^-1 ( $R_\omega =36$) produces little overall effects on the field structure, but does affects the field strength. When, for example, $R_\omega =36$, $f_\eta =1$, $R_\alpha =0.6$ (D = -22.5), the maximum field strength is about $0.7\, B_0$; this figure increases to about $6.5\, B_0$ when $R_\omega =72$, $f_\eta =1$, $R_\alpha =3$ (D = -216, Fig. 5a) Of this increase, a factor of about 4 can be attributed to enhanced induction by rotational shear which yields $B\simeq B_0\left(\vert D\vert/D_{{\rm cr}}-1\right)^{1/2}$, and the rest to the noncircular velocities.

A feature of all these solutions is that there is a broad field minimum in the bar region, but with strong magnetic ridges at the positions of the dust lanes and field enhancement in the central part where the circumnuclear ring is observed. This agrees well with the overall distribution of polarized intensity in NGC 1097 and 1365. Nevertheless, the local magnetic energy density near the major axis (upstream of the dust lanes) exceeds the turbulent kinetic energy density. As we discuss in Sect. 5.1, both the model and observed field strengths have a well pronounced structure within the bar region in addition to the ridges elongated with the dust lanes and the central ring.

There is a relatively strong, quasi-azimuthal field upstream of the shock, especially near the ends of the bar. This feature also agrees well with the observations, cf. Fig. 1. For the dynamo parameters of the models illustrated in Fig. 5, the dynamo is locally supercritical nearly everywhere. However, the field structure is rather insensitive to the value of D. To illustrate this, we show in Fig. 6 the field vectors with only a slightly supercritical dynamo number ( $\eta_0=2\times 10^{26}\,{\rm cm}^2\,{\rm s}^{-1}$, $R_\alpha =0.6$, $R_\omega =36$). The field configuration is remarkably similar to that of Fig. 5a, with a much larger dynamo number, except perhaps near the outer boundary. This supports the idea that the field in regions upstream of the dust lanes at larger galactocentric distances is mainly produced not by local dynamo action but rather by advection over almost $180^\circ$ from regions at small radii (where the dynamo action is stronger) on the other side of the bar. We note that the half-rotation time of the gas at $r\simeq5\,{\rm kpc}$ is shorter than magnetic diffusion time $h^2/\eta_0\simeq5\times10^8\,{\rm yr}$. If the dynamo is strong enough ( for $R_\omega =72$) the advection produces a magnetic configuration where the magnetic energy density exceeds that of the interstellar turbulence in broad regions, including the region upstream of the shock front. The steep rotation curve at small radii and the large local dynamo numbers found there suggest that the dynamo is particularly efficient in the inner region. This field is then transported outwards by the noncircular velocities and further enhanced by shear.

Figure 6: As in Fig. 5a, but for $R_\alpha =0.6$, $R_\omega =36$, $f_\eta =1$, giving weak dynamo action.

Immediately downstream of the shock position, the velocity and magnetic fields are closely aligned. The alignment is fairly good throughout the galactic disc, with the mean angle $\chi$ between magnetic and velocity vectors being about $\chi\approx20^\circ$( $\langle\cos\chi\rangle\approx0.93$) in the model of Fig.  5a. As can be seen in Fig. 7, the alignment is especially close outside the bar and slightly reduced within the bar and at the corotation radius. The reduction in the bar can be plausibly attributed to enhanced magnetic diffusivity in the dust lanes and that at corotation to the small local values of $\vec u$ in the rotating frame.

   
4.3 Models with turbulent diffusivity depending on velocity shear: $\widetilde\alpha \propto \omega$

We now discuss models in which we set $\widetilde\alpha\propto \omega(r)$ (see Sect. 3.2), whilst keeping $f_\alpha =0$. Since the dynamo action now becomes more intense in the disc centre, the field strength is more strongly concentrated to the centre for given $f_\eta$, than when $\widetilde\alpha$ is uniform. As $f_\eta$ is varied, these results can be broadly summarized by saying that the field structure is very similar to that when $\widetilde\alpha(\vec{r})$ is uniform, if a larger value of $f_\eta$ is now taken. For example, we show in Fig. 8 the field structure when $R_\omega =72$, $f_\eta =3$, $R_\alpha =3$, which can be compared with that shown in Figs. 5a and b, with $\widetilde\alpha(\vec{r})=1$and $f_\eta=1, 3$ respectively. We note that the effect of setting $\widetilde\alpha\propto \omega(r)$ is the opposite of increasing $f_\eta$ - the former enhances the central field concentration, the latter decreases it. But the field structure outside of the inner 1-2kpc is altered only slightly by such changes.

For this model, the maximum field strength is about 3.9 B₀, attained at $r/r_{\rm c}\approx0.1$ (see also Fig. 13). We show in Fig. 9 a grey scale plot of B² for the model of Fig. 8 where magnetic field enhancements in the dust lanes, the spiral arms and the central region are prominent. The field is especially strong in the circumnuclear region and at the ends of the bar. Trailing magnetic arms, emanating approximately from the ends of the bar, are a common feature of the modelled and observed magnetic structures. (This feature is also seen in the model for the weakly barred galaxy IC 4214 of Moss et al. 1999a.)

Figure 7: The alignment of the regular magnetic and velocity fields in the corotating frame is illustrated with level contours of $\cos\chi=\vert\vec{u}\cdot\vec{B}\vert/uB$ for the model of Fig. 5a. The levels shown are 0.4 (dotted), 0.7 (dashed) and 0.95 (solid).

This class of models exhibits the best agreement with observations and we discuss them in detail in Sect. 5.

   
4.4 Magnetic fields in the absence of dynamo action: The case of pure advection

In this section we attempt to elucidate the roles of the two key ingredients of our model, stretching by the regular velocity and the $\alpha$-effect. In Fig. 10 we show the field structure for a computation with $R_\omega =72$, $f_\eta =5$, and $R_\alpha =0$ (no dynamo action). In the absence of dynamo action this field decays exponentially, with an e-folding time of about  $10^8\,{\rm yr}$. This decay time scale is about 5 times shorter than in normal galaxies because of the enhanced velocity shear that reduces the field scale and so facilitates its decay. Since most barred galaxies in the sample of Paper I do exhibit detectable polarized radio emission and thus possess regular magnetic fields, we conclude that the need for dynamo action in barred galaxies is even stronger than in normal galaxies.

Figure 8: As in Fig. 5a, but for $\widetilde\alpha(\vec{r})\propto\omega(r)$and $R_\alpha =1.5$.

Looking at the spatial structure of the decaying magnetic field, we see that some of the features of the dynamo maintained fields shown in Figs. 5 and 8 are present, particularly the field minimum upstream of the shock, and the trailing magnetic arms outside of the corotation radius. However there is no hint of the relatively strong fields at small radii seen in all models with dynamo action present. In addition, the relative magnetic field strength near the bar major axis is now too low.

Thus it appears that the field structure in the outer regions is relatively insensitive to the parameters of the dynamo model, but that dynamo action is essential to maintain the fields over time scales of gigayears and to produce strong magnetic fields in the region of the circumnuclear ring and upstream of the shock front.

   
4.5 Results with positive dynamo number

We made some experiments with $R_\alpha < 0$, i.e. positive dynamo number D. This is motivated by suggestions that this may be the case in accretion discs (Brandenburg et al. 1995) even though the origin of turbulence there is quite different from that in galaxies. It is a familiar result for axisymmetric galactic discs that the marginal positive dynamo numbers for dynamo excitation are much larger in magnitude than the marginal values for the usually considered case with D<0 - i.e. dynamos with D>0 are much harder to excite than those with D<0. As mentioned in Sect. 3.2, in the models with D<0 the marginal values of D for the models with the full A92 velocity field are essentially unchanged from those found by including only the axisymmetric rotational velocities. When D>0, the situation is quite different. A dynamo is excited by the full velocity field at a marginal dynamo number of magnitude considerably less than required to excite the corresponding dynamo driven by the circular motions alone. In other words, when D>0 the dynamo is driven by shear due to noncircular velocities rather than from differential rotation.

Figure 9: Grey scale map of B2 (unsmoothed) for the model shown in Fig. 8. Lighter shades indicate higher values, and the contours shown are for B2=1, so the energy density of the regular magnetic field exceeds that of turbulence inside the contours. The circumscribing circle has radius 8 kpc.

We illustrate our results by discussing a simulation with $R_\omega =72$, $f_\eta =5$, $f_\alpha =0$, $\widetilde\alpha(\vec{r})=1$and $R_\alpha =-6$, a somewhat supercritical value. The magnetic fields are now oscillatory, with a dimensionless period of about 0.48, corresponding to about $1.1\times 10^8$ yr. We see in Fig. 11 that the regions of strong magnetic field correspond to the regions of strong velocity shear. The overall field structure varies significantly during the cycle period, and at certain times the amount of structure in the magnetic field is much greater than at others. The magnetic field strength has a maximum in the inner bar region and in two elongated features parallel to the shock fronts. These structures persist through nearly all of the oscillation period. However, the field geometry within the structures changes with time as a magnetic field reversal along a line approximately parallel to the shock front propagates clockwise in Fig. 11. In the lower left-hand part of Fig. 11, weak field directed approximately radially outwards is visible. At a slightly later time, this has grown in strength and this region eats in to the adjacent area of approximately inwardly directed field, strengthening the field reversal. A little later still, the reversal has disappeared, and then the cycle repeats. We emphasize that, for this calculation, the field decays when the nonaxisymmetric velocities are removed.

It is clear that these dynamos with positive dynamo numbers generate magnetic fields that are rather different from those observed.