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Astron. Astrophys. 329, 895-905 (1998)

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5. Two dimensional calculations

Most of these simulations were performed with a grid defined by [FORMULA], but some were repeated at higher resolution to verify that these values were large enough. We adopted standard galactic parameters of [FORMULA] kpc (4 exponential scale lengths), [FORMULA] pc (equal to the gravity softening parameter in the dynamical calculations), so the disc aspect ratio is [FORMULA]. Thus with canonical values of [FORMULA] cm2 s-1, [FORMULA] cm s-1, we obtain [FORMULA].

5.1. The main calculations

We used as standard velocity data that from the first calculation described in Sect. 3(Model I), reduced by Method 1, including azimuthal Fourier components [FORMULA]. We did verify that the results were not radically changed by taking M (Sect. 4) to be 3.

We attempted firstly to complete a simulation over 10 Gyrs with [FORMULA] cm2 s-1, [FORMULA], i.e. approximately the values quoted above. We encountered several situations, at intervals of 1-2 Gyr, when the interpolated non-circular velocities, in particular the radial component, became especially large at smaller radii. This resulted in the magnetic field being swept rapidly to the centre of the disc, and the numerical scheme eventually became unstable. (The problem can be somewhat alleviated by using a finer spatial mesh, but it appears that higher than practical spatial and temporal resolution would be needed to remove it altogether.) We thus adopted two methods to circumvent this problem, which probably is connected with the spatial sparsity of the raw velocity data (see Sect. 4. Either we arbitrarily reduced the magnitude of the non-circular velocities slightly, by reducing [FORMULA] to 50 or 80% of its nominal value, or we increased the diffusion coefficient, to [FORMULA], and, for comparison, also to [FORMULA] cm s-1. Of course, we can also use a combination of these two changes. When we increase [FORMULA], we keep [FORMULA], ie we increase the value of [FORMULA], in order to ensure that a dynamo is still excited. We found that the gross features of the field evolution were very similar in all cases. Increasing the diffusion means (unsurprisingly) that the field features are somewhat broader. Both of these changes mean that the field concentration to the centre is reduced during episodes of strong radial velocities. Also, we note that, during intervals when the code does run satisfactorily with the canonical parameter values, then the results are again quite similar to those obtained with these modified values. As it may be that the local sparsity of the raw velocity data causes the interpolation process somewhat to exaggerate the radial velocities (Sect. 4), we consider either of these proceedures to be reasonable. We also verified that, when [FORMULA] was set to zero, the magnetic fields did decay; that is there is no `false' dynamo effect arising from our representation of the imposed nonaxisymmetric velocities or boundary conditions.

We now discuss in more detail a simulation with [FORMULA] cm2 s-1, [FORMULA], [FORMULA] (i.e. a reduction of [FORMULA] to 50% of its nominal value). Projections of the magnetic field vectors on to the disc plane at successive times are shown in Fig. 2. The position of the axis of the bar is shown by the radii projecting beyond the circle representing the computational boundary. Ring and spiral-like structures appear and disappear, and at certain epochs (e.g. near time 4.6 Gyr), the field is concentrated near the centre of the disc. Fig. 1 shows the relation of the streaming velocities to the bar at a typical instant, and also shows the relation between the vectors of the magnetic field and streaming velocities. This is typical of our solutions.

[FIGURE] Fig. 2. Snapshots of magnetic field for the case with [FORMULA] kpc, [FORMULA] cm2 s-1, [FORMULA]. at times 0.6, 1.2, 2.8, 4.0, 4.6, 6.0, 6.6, 8.4 Gyrs. The radii projecting beyond the circumscribing circle (the outer computational boundary at [FORMULA] kpc) indicate the current position of the bar axis.

[FIGURE] Fig. 2. (continued)

Fig. 3 shows plots of the energies in azimuthal modes [FORMULA] against time. In these plots, the [FORMULA] mode is clearly dominant. This might appear a little surprising given the nature of the field plots in Fig. 2. However, there are large parts of the disc where the field is predominantly axisymmetric, and the rather more striking nonaxisymmetric features are quite localized. This is more clearly seen in a short trial run with [FORMULA] kpc, with a slightly different velocity field to that used for the calculations described above. This excludes much of the region where the field is approximately axisymmetric, and accordingly gives larger relative energies in modes [FORMULA] - see Fig. 4. (Note that here the scale for the energies differs from that of Fig. 2.) In general, the global magnetic energy in the mode [FORMULA] is rather less than in [FORMULA], and that these contributions fluctuate quite strongly, see Fig. 3.

[FIGURE] Fig. 3. Variation of energies in modes [FORMULA] (respectively continuous, long-dashed, medium-dashed, short-dashed curves) with time for case in Fig. 2. The time unit is 0.21 Gyr.
[FIGURE] Fig. 4. As Fig. 3, but for a calculation with [FORMULA] kpc.

For comparison, in Fig. 5 we show field configurations at time 1.2 Gyr for simulations with parameters [FORMULA], [FORMULA] cm2 s-1 (Fig. 5a), and [FORMULA], [FORMULA] cm2 s-1 (Fig. 5b), both with [FORMULA]. Thus in Fig. 5a, the value of the noncircular velocities is closer to the `raw' value, to be compared with the smaller values of the standard case (Fig. 2). In Fig. 5b the velocities again are nearer to their raw values, but [FORMULA] is increased (remember that [FORMULA]).

[FIGURE] Fig. 5a and b. Snapshots of field structure at time 1.2 Gyr, [FORMULA] for a [FORMULA], [FORMULA] cm2 s-1 ; b [FORMULA], [FORMULA] cm2 s-1 ;

It is clear that the larger value of [FORMULA] adopted for Fig. 5b tends to give rather broader magnetic features and increasing the value of [FORMULA] for fixed [FORMULA] has the opposite effect. Note that, although in the case illustrated in Fig. 5b we have increased [FORMULA] in keeping [FORMULA], this has little effect on the field geometry.

5.2. Test with Method 3 velocities

We performed a limited comparison between results obtained using velocity data obtained by use of procedures 1 and 3. For this we took a time independent velocity field, corresponding to an early epoch of the simulation, and followed the evolution of the magnetic field for about 2 Gyr. The magnetic structures obtained showed strong similarities, but for the same parameter values those obtained with the Method 3 velocities were spatially narrower and generally (and not unexpectedly) exhibited rather more shearing.

5.3. Results using data from the Model II dynamical simulation

We also investigated magnetic field evolution, using data from the second simulation described in Sect. 3. We used the same procedure as for the two dimensional calculation described in Sect. 5.1. Clearly, the detailed results were different. However we found that the same general features of magnetic field morphology appeared, namely ring-like and short armed structures, with vectors of magnetic field and non-circular velocities well aligned.

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© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997