The results are depending on the formulation of the problem. At first, we consider the simple case of an uniform field which initially exists in the galaxy. This part of the paper follows Ruzmaikin et al. (1988) and Otmianowska-Mazur & Chiba (1995) but with the basically modified turbulence theory described above (` -quenching').
4.1. The initial-value problem
If an initial field is given subject to induction and dissipation, the main questions concern the lifetime of the magnetic energy and the geometry of the magnetic field in its temporal dependence. The initial field is considered as being nonaxisymmetric with equatorial symmetry, i.e. as a field of type S1. It fulfills the pseudo-vacuum boundary condition. The main result of Otmianowska-Mazur & Chiba (1995) concerning the resulting magnetic energy of a similar configuration is a lifetime of 200 Myr just in correspondence to the estimate of the decay time given by Parker (1979). It is thus very doubtful whether a `predominantly nonaxisymmetric seed field may still give a significantly nonaxisymmetric field after times of order 1010 years' (Parker 1992; Moss et al. 1993; Camenzind & Lesch 1994). In our computations we have to check whether the nonlinear -effect will change this situation. If the answer is No, then there remains only little hope to explain the existence of galactic magnetic fields without a dynamo mechanism.
In the Figs. 2 - 4 the temporal evolution of the magnetic field geometry is given for various time steps. The free value is varied from Fig. 2 to Fig. 4 by two orders of magnitude. The (middle) value of 0.3 is the often used standard case (Fig. 3), but we are also working with values differing by one order of magnitude.
In early epochs we indeed find the magnetic fields concentrated between the gaseous spirals. The effect is very close to the observations. Rather fast, however, the magnetic arms are wound up by the differential rotation. In consequence, the magnetic pitch angles become smaller and smaller so that in the final state of Fig. 3 the (still existing) magnetic spirals change their form. In a certain radius they are even passing the gaseous spirals remaining between them only in the outer part of the `galaxy'. As there is no special mechanism opposing the upwinding tendency of the differential rotation (as it exists for the gaseous spirals), the initially nonaxisymmetric configuration develops towards ring-like structures (Moss & Brandenburg 1992). These are formed by magnetic fields of opposite polarity hence the magnetic dissipation is permanently reducing the field. First the central part of the galaxy becomes field-free. Fig. 5 demonstrates the decay of the magnetic energy. There is no amplification at all, not even by the factor of 10 and not even at early times (Weiss 1966; Moffatt 1978; Spencer 1994). The differences for weak and strong seed fields are surprisingly small. The maximal decay time (for very small diffusivity, ) is about 300 Myr, for the standard diffusivity value () it is 100 Myr and for one obtains 25 Myr. There are no decay times exceeding 1 Gyr. Any influence of the ambipolar diffusion was not observed in the computed results.
After a complete nonlinear analysis of the dissipating action of interstellar turbulence and its effect on winding up an initial nonaxisymmetric field in galaxies, the following can be stated in order to summarize:
For the speculative case that there is no dynamo mechanism we find the highly restricted existence of grand design magnetic patterns. The field decay is fast and even faster is the `decay' of the pitch angles.
The magnetic decay rate is much faster than the dynamo growth time. In Elstner et al. (1996) we found a dynamo growth time of about 1 Gyr in correspondence to the existence of magnetic fields in galaxies aged about 1 Gyr (Kim et al. 1990; Wolfe et al. 1992).
Also spiral arms with a lifetime of maximal 1 Gyr are short-lived phenomena (cf. v. Linden 1994). If there is indeed a causal relation between gaseous and magnetic spirals then our result of the short lifetime of magnetic spirals suggests the observed gaseous spirals as very young.
4.2. The boundary-value problem
4.2.1. Amplification and geometry
Interesting results also appear for the case that the galaxy is embedded in a long living uniform intergalactic magnetic field parallel to the galactic plane. Its amplification and/or decay could then be the most interesting phenomenon. In this case it makes only sense to apply a `weak' external intergalactic magnetic field to the galactic flow pattern. A strong (i.e. super-equipartition) external magnetic field would only be interesting for comparison and/or from mathematical reasons. The field is not believed to be of cosmological origin. Such fields are subject to various amplification mechanisms such as compression, shearing or even small-scale dynamos, so that finally a magnetic field with a strength of order G can be considered as a seed field for mean-field processes (Beck et al. 1996).
The meridional plot of Fig. 6 demonstrates the temporal behavior of the magnetic field in the interior of the galaxy. The winding produces a number of arms of alternating signs, i.e. an increasing number of field reversals. The pitch angle, however, becomes smaller and smaller hence again the magnetic spirals develop to rings. After 1 Gyr the interior of the galaxy is practically field-free. A stationary solution with the old field amplitude is reached after more than 1.66 Gyr.
In Fig. 7 the time evolution of the magnetic field amplitude is given for the turbulence model with = 0.03. There is a field amplification for the first 700 Myr by a factor of 10 due to the differential rotation. Then the field gradients become so strong that the dissipation is enforced. After less than 2 Gyr the original field amplitude is reached again due to the parallel component of the external magnetic field on the boundary which is not influenced by our boundary condition. But inside the galaxy the magnetic field has disappeared.
A perpendicular relic field, however, will not be affected by an axisymmetric configuration in our model, as it is to be seen even from the structure of the induction equation. Only nonlinear turbulence effects influence the vertical field component itself, essentially via diamagnetism, in the case of a nonaxisymmetric profile in density and turbulent velocity representing the galactic spirals. Fig. 8 shows that these effects are small as they only lead to a weak field amplification in comparison to the case of a parallel relic field.
4.2.2. Resonance effects
Discussing the initial-value problem we pointed out that it is possible to observe a relation between the spiral density structure and the geometry of the magnetic field (especially in young spirals; cf. Figs. 3 and 4). Furthermore we have to take into account that both the magnetic field and the spiral density structure rotate with different drift or pattern velocities (cf. Table 2). This leads to the question whether it is possible to observe resonance effects caused by the interaction of the drifting magnetic field and the density structure (cf. Moss 1996 and references therein).
Based on the model with = 0.3 we calculate spiral density structures rotating with different pattern velocities. The time evolution of the magnetic field strength is shown in Fig. 9 (solid curves). The amplification of the magnetic field strength arises a maximal value for a pattern velocity of = 25 Gyr-1 (Our definition of is half of that used by Moss). To understand this resonance-like behavior we compare these calculations with axisymmetric models without any spiral structure in density and turbulent velocity.
In Fig. 9 we present also two calculations (dotted curves) with axisymmetric density and turbulent velocity contributions set to minimal and maximal values respectively ( = 1, 5). In the case of lowest turbulent velocity the amplification and lifetime of the magnetic field reach the largest values. The opposite case leads to the smallest amplification and lifetime. The nonaxisymmetric configurations (solid curves in Fig. 9) induce magnetic fields living shorter or showing smaller amplifications than the less diffusive axisymmetric model.
In Figs. 10 and 11 we present the temporal evolution of the magnetic field geometry for an axisymmetric (with average values for density and turbulent velocity) and a nonaxisymmetric configuration with = 25 Gyr-1. The axisymmetric model shows a rapid winding of the magnetic field lines with a drift or pattern velocity that decreases strongly towards the outer region of the galaxy. In the nonaxisymmetric case the pattern velocity of the density structure can be equal to the drift velocity of the `frozen' magnetic field lines - but only at a special radius. Then it can be observed that the magnetic field drift velocity is reduced at inner radii but accelerated at outer radii. The magnetic field is less wound up.
Table 1. Axisymmetric turbulence: Pattern velocity of the maximal magnetic field; is the interstellar gas rotation; all angular velocities in Gyr-1, .
Table 2. Nonaxisymmetric turbulence: Pattern velocities for density profile and maximal magnetic field; is the interstellar gas rotation; all angular velocities in Gyr-1, .
Finally we discuss an explanation for the resonant behaviour observed in the time evolution of the magnetic field strength. If the drift velocities of the magnetic field and the density structure have become very similar, the decay of the magnetic field may simply be delayed. Our computations show (cf. Fig. 11) that in case of resonance the maximal field stays in the interarm region with low turbulent velocity and therefore small diffusivity. As the field destroying diffusivity is weakened in the interarm region, the field amplifying differential rotation then maximally affects the initial field in this region. This interpretation is in particular supported by our result that this amplification remains below the values of the (axisymmetric) case with lowest turbulence intensity.
© European Southern Observatory (ESO) 1998
Online publication: December 16, 1997