Turbulent difussion and the galactic magnetic field

In this Section let us summarize the work by Battaner, Lesch and Florido (1999). The similarities of the strengths in the interstellar, the extragalactic and the pregalactic media suggest a fast and efficient connection between them. In this work, it is proposed that this connection is the result of a highly turbulent magnetic diffusion in the vertical direction.

It is an observational fact that convective phenomena are very active in disks. Galaxies sometimes exhibit "boiling" disks, with NGC 253 being a good example (Sofue, Wakamatsu and Malin, 1994), where dark filaments, lanes, arcs and other micro-structures are features revealing a very complex convective region. This fact is in part explained by "fountain" models (Shapiro and Field, 1976; Breitschwert et al., 1991; Kahn, 1994; Breitschwerdt and Komossa, 1999). Of course, these turbulent motions constitute a transporting of magnetic fields, as a result of the condition of frozen-in lines. Because of this transporting, extra and intergalactic fields merge.

Suppose first that no dynamo is acting on the galactic gas. The equation of induction will tell us (e.g. Ruzmaikin, Shukurov and Sokoloff, 1988; Battaner, 1996)

	(129)
	(130)
	(131)

where B_R, B_$\varphi$ and B_z are the magnetic field strength components and $\beta$ is the coefficient of turbulent magnetic diffusion. Cylindrical symmetry has been assumed. The usual expression to calculate $\beta$ is

(132)

where l is a typical length of the larger convective cells, say l $\approx$ 1kpc, and v is a typical convection velocity corresponding to the larger scale turbulence, say 20 km s^-1. Hence $\beta$ is of the order of 2 × 10²⁷cm²s^-1 $\approx$ 6kpc²Gyr^-1. In comparison, $\beta$ is taken as being of the order of 10²⁶cm²s^-1 in the inner disk), of 5 × 10²⁷cm²s^-1 in the galactic corona and of 8 × 10²⁹cm²s^-1 in the intergalactic medium in a cluster (Ruzmaikin, Sokoloff and Shukurov, 1989; Sokoloff and Shukurov, 1990; Ruzmaikin, Shukurov and Sokoloff, 1988).

The characteristic diffusion time is calculated with l²/$\beta$, therefore having a typical value of 0.2 Gyr, very little compared with the lifetime of the galaxy. Extragalactic magnetic fields would have spatial variations at scales much larger than a galaxy. The field strength can be assumed to be constant outside the galaxy, as a boundary condition. This external steady state magnetic field could have produced an initial penetration of magnetic fields which would have been subsequently ordered by differential rotation, resulting in a predominantly toroidal field. Or rather, the disk was born out of already magnetized material, then was magnetized at birth and maintains a permanent interchange with the magnetized environment, because of the high magnetic diffusivity.

However, the magnetic field is assumed to be homogeneous outside and toroidal inside. A configuration that continuously transforms a constant into a toroidal field was proposed by Battaner and Jiménez-Vicente (1998), but here we need to adopt convenient boundary conditions taken at a large enough height.

All three components -B_x, B_y, B_z- are constant in the extragalactic medium. But not all penetrate and diffuse inwards equally. For instance, there is no difficulty for B_z to penetrate, because it is not perturbed by rotation. And if the transport is so effective we could even assume that B_z is a constant in the whole outer disk considered, equal to the extragalactic value of B_z. We then assume as a reasonable mathematical assumption that B_z=cte everywhere in the integration region.

It is more difficult for the other components to penetrate (or exit). For instance, B_R penetrates into the disk at a given time and point (R, $\varphi$); the rotation would transport the frozen-in magnetic field lines into the azimuthally opposite position (R, $\varphi$ + $\pi$) in half a rotation period. The direction of the penetrated field vector there would be opposite to the vector transported from the opposite azimuth. The two vectors would meet with the same modulus and opposite direction and would destroy one another through the reconnection of field lines. It is therefore tempting, in a first simplified model, to assume that B_R = 0, at the boundaries. We may even adopt B_R = 0 everywhere inside the disk.

With respect to B_$\varphi$, we have a similar situation. B_$\varphi$ when penetrating at (R, $\varphi$) would be frozen-in transported to (R, $\varphi$ + $\pi$) in half a rotation and then interact with the field penetrated there. Reconnection would then act and we could reasonably adopt B_$\varphi$ = 0 at the boundaries. But B_$\varphi$ is easily amplified by rotation and can be generated from B_z, which is non-vanishing; therefore we cannot assume B_$\varphi$ = 0 everywhere; rather it is B_$\varphi$(R, z) that we want to calculate. We also assume steady-state conditions, $\partial$/$\partial$t = 0. The equations then become greatly simplified which also permits a simplified interpretation of what is essential in the process, much more understandable than a lengthy numerical calculation. In the above equations, we set $\partial$/$\partial$t = 0, B_z=constant, B_R = 0 and obtain

	(133)
	(134)
	(135)

The first and third are tautologic, telling us that we could have deduced much of what was assumed (as Battaner, Lesch and Florido did), but that is unimportant. We now see that our assumptions do not lead to incoherent results.

The second equation would provide us with B_$\varphi$ if $\theta$(z) were known. Let us further assume $\partial$$\theta$/$\partial$z = 0, which is not unrealistic given the relatively low thickness of the disk. In order to find a fast solution (which is not necessary, but just didactic) let us assume that $\partial^{2}_{}$B_$\varphi$/$\partial$z² is negligible (it cannot be zero, as B_$\varphi$ must be zero at the boundary). In fact, some galaxies have a radio halo (e.g. NGC 253, Beck et al. 1994; NGC 891, Dahlem, Lisenfeld and Golla, 1995 in other spirals). The decrease of magnetic field strength with z is observed to be very slow, even in galaxies with no radio halo (Ruzmaikin et al., 1988; Wielebinski, 1993) and also in the Milky Way (Han and Qiao, 1994). Then for small | z | we simply obtain

(136)

therefore

(137)

which is precisely the critical profile. Once we see how the critical profile is supported with this mechanism, it is expected that other more realistic calculations would be able to provide sub-critical profiles, capable therefore of producing inward magnetic forces.

The symmetries of the magnetic field predicted in this simple model agree with those obtained with Faraday rotation by Han et al. (1997) in our own galaxy. This model does not need a dynamo but provides a large-scale structure with much in common with the so called AO mode. We also predict an antisymmetry of the azimuthal field in both hemispheres for | l | < 90^o. This AO dynamo mode has also been observed in other galaxies, but in view of the symmetry similarities with our predictions, these galaxies could be interpreted in terms of the mechanisms sought by Battaner, Lesch and Florido (1999).