5 Discussion and comparison with observations

   
5.1 The magnetic field in the shock region

We show in Fig. 12 the variation of B² along two cuts through the shock front region for the model illustrated in Fig. 8, illustrating the variation in the magnetic field strength between the post- and pre-shock regions. Also shown is the observed profile of polarized synchrotron intensity along similarly chosen cuts in NGC 1097. In the case of constant energy density of cosmic-ray electrons, the polarized intensity depends on $B_\perp^2$, where $B_\perp$ is the magnetic field component perpendicular to the line of sight.

Figure 10: As in Fig. 5, but without any dynamo action: $R_\alpha =0$, $R_\omega =72$, $f_\alpha =0$ and $f_\eta =5$. In this case the magnetic field decays on a time scale of about $10^8\,{\rm yr}$.

Both the model magnetic field and the observed polarized intensity reveal several peaks in B²on each side of the shock front and a pronounced minimum at the shock front itself. The model and observations differ in detail, which is no surprise as inter alia the velocity field and gas distribution used in our models was not designed specifically to fit NGC 1097. We find that the best correspondence between the B² variations from the models and the observed polarized intensity profiles is obtained in models with $\widetilde\alpha \propto \omega$ (Sect. 4.3), with when $R_\omega =72$. Larger values of |D| give weaker peaks near the bar, but taking too small a value of |D| gives rather low field strengths.

Due to the inclination of NGC 1097 ($45^\circ$), the difference between $B_\perp^2$ and B² is typically a factor of 2. Projection effects reduce the polarized intensity upstream of the shock from where magnetic field has significant component parallel to the line of sight, but not in the dust lane where the value of $B_\perp$ is closer to that of B. This will suppress the amplitude of the secondary peak in polarized intensity in the pre-shock region. With allowance for the unaccounted projection effect, our model shows remarkably good agreement with observations in the number and width of the magnetic peaks in the shock region.

Our models reproduce a smooth turn of the magnetic field vector upstream of the dust lanes observed in NGC 1097 (and also NGC 1365). This turn may be difficult to see in the figures presented here because we have shown magnetic vectors only at a fraction of mesh points. It can be seen from Fig. 7 that the alignment between the magnetic and velocity fields is reduced 1-2kpc upstream of the shock fronts (dust lanes). Several effects apparently contribute to smear and displace the turn in the magnetic field with respect to that in the velocity field. We argue in Sect. 4.2 that magnetic field can be advected to the pre-shock region from smaller radii (the elongated shape of the streamlines responsible for that can be clearly seen in Fig. 2b of A92, but not that easily in Fig. 2 here). The admixture of magnetic field generated elsewhere must smear any sharp structures produced locally. Furthermore, magnetic field will tend to be aligned with the principal axis of the rate of strain tensor rather than the velocity field itself, and so its turn can start long before it becomes visible in the velocity vectors.

Figure 11: Snapshot of typical vectors of the horizontal magnetic field for the positive dynamo number model with $R_\alpha =-6$, $R_\omega =36$, $f_\alpha =0$, $f_\eta =5$, in a format similar to that of Fig. 5.

Other effects can be envisaged that might, in principle, result in a smooth turn of magnetic field upstream of the shock fronts. For example, this can be a manifestation of the dynamical influence of the regular magnetic field on the velocity field. However, this is hardly the case in our models where the regular magnetic field does not reach equipartition with even the turbulent kinetic energy in the regions concerned (see Fig. 9). Another possibility is that the turn in the observed field is smoothed by the presence of a magnetic halo with a regular magnetic field smoother than in the disc; however, there is so far no evidence for a polarized halo in NGC 1097. A further option is that the magnetic field is anchored in a warm component of the interstellar medium that has different kinematics than the cold gas (i.e., higher speed of sound); our preliminary analysis of this possibility has given no evidence for this, but we shall return to it elsewhere.

5.2 Radial profiles of magnetic field

We show in Fig. 13 the dependence of the azimuthally averaged value of B² on radius for the model shown in Fig. 8 and a similar model with larger $f_\eta$, and compare it with the observed dependence of polarized intensity on radius. The model reproduces the peak observed close to the centre and has a realistic radial scale. The slower decrease of polarized intensity with galactocentric radius in NGC 1097 can be explained by a steeper rotation curve in the A92 model, which has the turnover radius at $r/r_{\rm c}\approx0.25$ whereas it occurs at about $r/r_{\rm c}=0.4$ in NGC 1097 (Ondrechen & van der Hulst 1989). The model magnetic field has a pronounced subsidiary peak at $r/r_{\rm c}\approx0.7$; this peak is related to the strong magnetic arms visible in Fig. 9. A similar peak in the observed radial profile is less pronounced and occurs at a smaller fractional radius.

In our models, the regular magnetic field vanishes on the rotation axis because of its symmetry properties. However, the observed polarized intensity remains significant at r=0. We discuss implications of this in the next section.

Figure 12: Profiles of the square of the horizontal magnetic field along cuts perpendicular to the shock front of the model shown in Fig. 8 at a) 1.5kpc and b) 3kpc galactocentric distance. The total length of each cut is 10 kpc, approximately symmetrical about the position of the shock front, with distance increasing in the upstream direction. The corresponding cuts (with scaled lengths of 20kpc; $20\hbox {$^{\prime \prime }$ }$ corresponds to $r/r_{\rm c}\approx 0.14$) through the map of $\lambda 6$ cm polarized intensity (represented by the vector length of Fig. 1) are shown in c) and d) for galactocentric distances of 3 and 6kpc. The cuts are shown for the same values of $r/r_{\rm c}$ in the model and NGC 1097 and extend over the same linear length.

   
5.3 The azimuthal structure of the global magnetic field

The azimuthal structure of the regular magnetic field is a useful diagnostic of both the overall structure of the gas velocity and the field generation mechanisms. We have fitted the polarization angles observed in NGC 1097 at $\lambda\lambda3.5$ and 6.2cm, for three radial rings between 2.9 and $5.3\,{\rm kpc}$ with a model where the magnetic field is expanded into azimuthal Fourier modes. The model and the fitting procedure are discussed by Berkhuijsen et al. (1997) and Fletcher et al. (2001); Table 1 gives details of the fits. The polarization angles have been obtained from Stokes parameters averaged over $20^\circ$ sectors in the $0.8\,{\rm kpc}$ wide rings. The results are presented in terms of ${\cal R}_m=0.81\,B_m{n_{\rm e}}h$, p_m and $\beta _m$, where B_m(in $\mu$G) is the amplitude of the magnetic mode with azimuthal wave number m, ${n_{\rm e}}$ is the thermal electron density in cm^-3, h is the scale height of the thermal ionized layer in pc, p_m is the pitch angle of the mode m (i.e. the angle between the field direction and the circumference; p>0 indicates a trailing spiral) and $\beta _m$ is the azimuthal angle, measured counter-clockwise from the northern end of the major axis of the galaxy, where the mode m($m\neq0$) has maximum strength. ${\cal R}_m$ is related to the Faraday rotation measure; ${\cal R}_m=100\,{\rm rad\,m^{-2}}$ corresponds to a field strength of about $3\,\mu{\rm G}$ for ${n_{\rm e}}=0.1\,{\rm cm^{-3}}$ and h=400pc. Note that the limited azimuthal resolution of the fits $(m\leq2)$ does not allow the fitting of the sharp deflection of the $\vec B$ vectors in the shock wave region (Fig. 1).

Figure 13: The radial dependence of the azimuthally averaged square of the horizontal magnetic field for the models with $R_\alpha=1.5,\ R_\omega= 72,\ \widetilde{\alpha}\propto\omega$, but with $f_\eta =1$ (dashed) and $f_\eta =3$ (dotted). For comparison the radial profile of the azimuthally averaged polarized intensity at $\lambda 6\,{\rm cm}$ in NGC 1097 is shown with solid line. Note the logarithmic scale on the vertical axes; $\langle B^2\rangle=0$ at r=0 in our models.

The amplitudes of the three lowest azimuthal Fourier modes in the observed regular field of NGC 1097 have ratios 1:1:0.5. The corresponding values for the models shown in Figs. 5a-c and 8 are all 1:0:0.5. The axisymmetric mode is dominant in both the observed field and in all these models. The relative amplitude of the m=2 mode for these models is also in agreement with the observations, having about half the amplitude of the m=0 mode.

The magnetic fields in our models do not contain the m=1 magnetic mode present in the observed field. This is due to the strict symmetry of the model velocity field (it does not contain any modes with odd m) and the inability of the dynamo to maintain the m=1mode on its own. The only mode actively maintained by the dynamo in our models is the axisymmetric one, m=0, and the m=2 mode arises via distortion of the basic field by a flow with a strong m=2component. In turn, other magnetic modes with m even are also maintained. In real galaxies such a high degree of symmetry of the velocities cannot be expected, and thus a wider range of azimuthal modes will be present in the magnetic field. We note that NGC 1097 is perturbed by a companion, so the real velocity field will certainly be more complicated than that used in the dynamo simulations. Any m=1component of velocity will then generate a corresponding component of magnetic field from the m=0 field by the $\vec{u}\times\vec{B}$ term in the induction equation (e.g. Moss 1995), although whether the amplitude of the resulting m=1 mode is large enough to be consistent with the observed value can only be determined by a detailed calculation. Naive numerical experimentation suggests that the amplitude of the m=1 velocity component would have to be comparable with that of the m=2 component if a m=1magnetic field of the strength implied by Table 1 is to be produced solely by the $\vec{u}\times\vec{B}$ interaction. Anisotropy in the $\alpha$-effect, neglected here, can also excite the m=1 magnetic mode via dynamo action (Rohde & Elstner 1998). Neglecting the m=1mode, the implication is that all the models presented in Figs. 5 and 8 reproduce the global azimuthal structure sufficiently well.

 

 

Table 1: The global magnetic structure in NGC 1097 represented in terms of the amplitudes of the azimuthal Fourier modes ${\cal R}_m$, their pitch angles p_m and phases $\beta _m$, with m the wavenumber. The bottom line shows azimuthally averaged magnetic field strengths obtained assuming ${n_{\rm e}}=0.1\,{\rm cm^{-3}}$ and $h=0.4\,{\rm kpc}$.

		Radial range (kpc)
		2.9-3.7	3.7-4.5	4.5-5.3
${\cal R}_0$	$\rm {rad\,m^{-2}}$	$-152\pm6$	$-141\pm13$	$-152\pm17$
p0	degrees	$18\pm1$	$27\pm2$	$22\pm2$
${\cal R}_1$	$\rm {rad\,m^{-2}}$	$-75\pm8$	$-88\pm12$	$-90\pm10$
p1	degrees	$-50\pm13$	$-71\pm13$	$-58\pm7$
$\beta_1$	degrees	$-119\pm6$	$-98\pm6$	$-122\pm8$
${\cal R}_2$	$\rm {rad\,m^{-2}}$	110+3-185	57+8-145	75+15-8
p2	degrees	0+1-130	-7+9-102	-31+8-91
$\beta_2$	degrees	$13\pm4$	$11\pm15$	$28\pm8$
$\langle B\rangle$	$\mu\rm {G}$	5.3+0.3-1	$4.9\pm0.6$	5.3+0.6-1

The absence of modes with odd m explains why the model magnetic field vanishes at the disc centre as all modes with even m have B=0 at r=0 from symmetry considerations. Models with a more realistic velocity field can have strong magnetic fields with $m=1,3,\ldots$ at the disc centre. The significant observed polarized intensity at r=0 (see Fig. 13) may be due to magnetic modes with odd m.

The estimates of azimuthally averaged magnetic field strength resulting from the fits of Table 1, shown in the bottom line of the table, have been obtained using an electron density of ${n_{\rm e}}=0.1\,{\rm cm^{-3}}$ and an ionized disc scale height of $h=0.4\,{\rm kpc}$. (The errors given do not include any uncertainties in ${n_{\rm e}}$ and h.) Both parameters are poorly known for barred galaxies. The diffuse H$_\alpha$ flux in barred galaxies is only moderately lower than in normal spirals (Rozas et al. 2000) and the H I scale height is comparable to (or somewhat larger than) in the Milky Way (Ryder et al. 1995), so electron densities can be expected to be similar in different galaxy types. The electron density at comparable fractional radii (with respect to corotation) of 4-5kpc in the Milky Way are about $0.1\,{\rm cm^{-3}}$ (Taylor & Cordes 1993). This justifies our crude estimate ${n_{\rm e}}=0.1\,{\rm cm^{-3}}$ for $r\simeq5\,{\rm kpc}$ in NGC 1097. A significantly lower value of ${n_{\rm e}}$ would imply a systematically larger magnetic field strength than estimates resulting from energy equipartition with cosmic rays.

   
5.4 The field in the central ring

In the top right of Fig. 1, the field at a galactocentric radius of about 1.5- $2\,{\rm kpc}$ can be seen to have a strongly nonaxisymmetric structure apparently dominated by the m=2 mode with a significant m=0 mode also present. The overall field structure features strong magnetic fields at those azimuthal angles where the dust lanes intersect the nuclear ring. This can be compared with the magnetic fields in the inner regions of our models (seen most clearly in Figs. 4, 5a, 8 where the field in the inner regions is emphasized). The form of these fields is remarkably similar to that shown in Fig. 1. We emphasize that the fact that the observed polarized intensity does not vanish at small galactocentric radii (see Fig. 13), as it should for modes with even m, indicates that the modes with odd m should become increasingly dominant at smaller radii. Angular momentum transfer due to the magnetic stress produced by the regular field can provide a mass inflow rate into the central region of about $1\,M_\odot\,{\rm yr}^{-1}$, which is comparable to that required to feed the active nucleus (Beck et al. 1999).