The temporal evolution of the large-scale magnetic field in spiral galaxies is governed by the dynamo equation (Steenbeck, Krause, & Rädler 1966; Parker 1971; Vainshtein & Ruzmaikin 1971). According to this equation, the large-scale magnetic field is amplified through the combined effects of a large-scale velocity field and small-scale turbulent motions. The large-scale velocity field basically consists of the Galactic rotation and of a possible Galactic wind. The former is reasonably well established observationally, mainly thanks to HI and CO velocity measurements (e.g., Fich, Blitz, & Stark 1989). In contrast, the properties of turbulent motions are still poorly understood. It is commonly assumed that their impact on the large-scale magnetic field can be parameterized with the help of two tensors: the alpha-tensor, which represents the alpha-effect (generation of magnetic field perpendicular to the prevailing field) plus the net advection of field lines by turbulent motions, and the diffusivity tensor, which describes turbulent magnetic diffusion (e.g., Moffatt 1978).
For most calculations, each of these tensors is reduced to a single parameter, which ignores the anisotropic nature of the alpha-effect and of the turbulent magnetic diffusion as well as the net advection of field lines by turbulent motions. Furthermore, the diffusivity parameter is usually taken to be uniform throughout the Galactic disk, and while the alpha-parameter is often allowed to vary in space, its spatial dependence is chosen for mathematical convenience rather than on physical grounds. Finally, there exist only order-of-magnitude estimates for these parameters, founded, for instance, on mixing-length theory or on crude pictures of interstellar turbulence.
Our goal is to obtain realistic curves for all the components of the alpha- and diffusivity tensors throughout the Galaxy. We proceed from the assumption that turbulence in the interstellar medium (ISM) is driven by supernova (SN) explosions (McCray & Snow 1979), either isolated SNs which produce individual supernova remnants (SNRs) or clustered SNs which are responsible for the formation of superbubbles (SBs). We utilize the formal analytical expressions established by Ferrière (1993a) (hereafter Paper I) for the alpha-tensor and by Ferrière (1993b) (hereafter Paper II) for the diffusivity tensor due to an arbitrary distribution of axisymmetric explosions. These expressions are written in terms of the explosion rate, the large-scale rotation and shear rates, and the dynamical properties (lifetime, final shape ) of the shock wave driven by an explosion. We determine these properties with the help of the numerical code developed by Ferrière (1995) in order to follow the time evolution of SNRs and SBs in a prescribed ISM. The model ISM, in turn, is taken from Ferrière (1998). Note that the current paper constitutes a generalization of the study by Ferrière (1996) (hereafter Paper III) designed to compute the alpha- and diffusivity tensors in the vicinity of the Sun.
In Sect. 2, we review the main components of the model ISM. In Sect. 3, we explain how we model the expansion of SNRs and SBs. In Sect. 4, we provide formal expressions for the alpha- and diffusivity tensors. In Sect. 5, we present the numerical results and interpret them physically. In Sect. 6, we summarize our study and discuss its limitations.
© European Southern Observatory (ESO) 1998
Online publication: June 18, 1998