## 4. ResultsThe 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 problemIf 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
10 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.
Always in the Figs. 2 - 4 the field geometry is of BSS-type. It naturally fulfills the condition , i.e. inwards fields are trailing and outwards fields are leading. In early epochs we indeed find the magnetic fields
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: - There is no amplification of the initial magnetic energy.
- After at most 1 Gyr the initial magnetic field is completely disappeared.
- In young spirals the magnetic fields are highly concentrated between the gaseous arms, the magnetic fields are grand designed.
- In old spirals the magnetic pitch angle develops to zero, the magnetic grand design does not survive.
- The influence of -quenching is strong for very small .
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 geometryInteresting 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 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 effectsDiscussing 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
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
The overview of the drift velocities given in Tables 1 and 2 demonstrate the described behavior of the magnetic field in more detail.
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 |