2. The dynamo model
As usual, is the turbulent magnetic diffusivity and represents a conventional alpha-effect. comprises the large-scale velocities (circular and non-circular), , and represents the turbulent diamagnetism (Vainshtein & Zeldovich 1972, Roberts & Soward 1975). In general we allow , . We solve eqn (1) as either a 2D or a 3D initial value problem. Initial conditions were usually that the seed field be axisymmetric and of low energy compared to the saturation value. For the 3D calculations, it is also localized near the disc plane. However the initial configurations are rapidly forgotten as the simulations proceed. The 3D version of the code allows , and to vary also with z, the coordinate perpendicular to the disc plane. Note that both the 2D code (implicitly) and 3D (explicitly) can only consider fields of even (quadrupolar) parity with respect to to disc plane . Given that the currently used velocity data is two dimensional, and that theory and observation predict that fields of quadrupolar parity are generally to be expected, we feel that this is not a fundamental restriction at this time. Nevertheless, we recognize the desirability of implementing a more general code, especially if we wish to use truly 3D velocity data. For computational convenience we put
thus separating the circular and non-circular velocities. In general we write , , , with , and being typical values of , and respectively. Thus gives a basic form of -quenching with field strengths limited at approximately . Plausibly would be the field strength in equipartition with a typical value of the kinetic energy of the turbulent velocities. Alternatively, would yield a linear calculation. A more sophisticated model might have , representing the change in the magnetic field-gas equilibrium condition as the gas density and turbulent velocity vary with position.
We make the conventional assumption that , where and are typical values of the velocity and length scale of the turbulence. In the Milky Way, near the disc plane, km s-1, pc, gives an estimate for of about cm2 s-1. However conditions in the sort of galaxies we are considering may be rather different, and the above expression for is only an order of magnitude estimate, so the value of is rather uncertain. is chosen to be equal to or slightly greater than the maximum value of , so that . We can define the corresponding magnetic Reynolds numbers , , where L is a convenient length scale, either h (a measure of the disc thickness) or R (the disc radius) - see below. We define km s-1, and choose such that the maximum value of is approximately 1-5 km s-1, and satisfies cm2 s-1. Note (Sect. 6) that the local value of only approaches the largest of these values of in the 3D calculations high in the halo. In principle, is fixed exactly by the dynamo model parameters, but in practice we explore the effects of modest changes in this parameter. We always use the nominal value of the circular velocity , and so the other conventional dynamo parameter, , is fixed by the dynamical model via eqn (2). Lengths are scaled with R, the disc radius, and units of time are , say.
2.1. The 2D model
The 2D `no-z ' approximation is based on the idea of replacing z -derivatives by terms , where h is the disc thickness or scale height (e.g. Subramanian & Mestel 1993, Moss 1995), and the code is in the form implemented in Moss (1996), solving a modified version of Eq. (1) on , , with grid points and uniform meshing.
We assume that is uniform, i.e. and, for most of the computations, also. The length scale L is taken as h. Boundary conditions are that at . Results were little affected for other plausible choices of boundary condition, provided that R was large enough for the dynamo-active region to be included within the computational domain.
2.2. The 3D model
The 3D code is described in Moss (1997), and comparisons are there made with the 2D code. Eq. (1) is solved on a grid of size covering , , . The spacing is uniform in each of the coordinate directions. We found that taking produced fields at this boundary that were small compared to their maximum values, and that results were insensitive to increasing further. Mostly we considered and only to vary with z, and we took, slightly arbitrarily,
with varying smoothly between and . It is the z -dependence of these quantities that effectively defines the disc. Thus the disc plane value of is . It is plausible that the diffusivity is larger in the halo than in the disc (eg Sokoloff & Shukurov 1990), and we thus take . We set for the scalings. We arbitrarily prescribed a decrease in the size of the non-circular gas velocities as z approached .
Our boundary conditions are that on and should be decreasing in magnitude at least as fast as for a general dipolar field. Similar `open' conditions are applied on , and the condition on the third field component is found by satisfying . As for the 2D calculations, we verified that plausible variations in these boundary conditions, such as setting some field components to zero at the boundary, did not alter significantly our results.
The restriction to uniform mesh size in the z -direction does mean that it is not possible to investigate discs of aspect ratio as small as those of real spiral galaxies, with the computing resources available. Practically, we are limited to . With the 2D velocity data this is probably not a real limitation when investigating the generation of nonaxisymmetric structure. However, we will obviously represent less well the z -structure of the fields.
© European Southern Observatory (ESO) 1998
Online publication: December 16, 1997