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Astron. Astrophys. 358, 125-143 (2000)

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2. The dynamo model

The time evolution of the large-scale magnetic field, [FORMULA], is governed by the dynamo equation,

[EQUATION]

where [FORMULA] is the large-scale velocity field and [FORMULA] is the electromotive force due to turbulent motions (Steenbeck et al. 1966). In general, [FORMULA] can be expressed as a linear function of [FORMULA] and its first-order spatial derivatives:

[EQUATION]

The first term on the right-hand side of Eq. (2) embodies the "alpha-effect" (magnetic field generation in the direction perpendicular to the prevailing field) plus the effective magnetic field advection by turbulent motions, while the second term represents turbulent magnetic diffusion (e.g., Moffatt 1978).

In the case of the Galactic disk, where the ISM parameters and the sources of turbulence vary significantly faster along the vertical than horizontally, the alpha-tensor takes on the form

[EQUATION]

in a cylindrical reference frame ([FORMULA]) with origin at the Galactic center and [FORMULA] perpendicular to the Galactic disk. [FORMULA], [FORMULA], and [FORMULA] give the effective rotational velocity associated with the alpha-effect when [FORMULA] is, respectively, radial, azimuthal, and vertical, and [FORMULA] is the effective vertical velocity at which [FORMULA] is advected by turbulent motions (Ferrière 1993a). The diffusivity tensor can be written as

[EQUATION]

where [FORMULA] and [FORMULA] are the horizontal and vertical magnetic diffusivities, respectively, [FORMULA] is the unit tensor, and [FORMULA] is the three-dimensional permutation tensor (by definition, [FORMULA] = +1, -1, or 0, according to whether ijk forms an even permutation of [FORMULA], an odd permutation, or neither) (Ferrière 1993b). As explained in the introduction, we adopt, for the various components of the alpha- and diffusivity tensors, the expressions calculated numerically by Ferrière (1998b), based on the assumption that the turbulent motions responsible for dynamo action in the Galactic disk are due to SN explosions. To account for the likely existence of other sources of magnetic diffusion as well as for the physical requirement that the magnetic field be able to diffuse everywhere in the Galaxy, we add a small background diffusivity of [FORMULA] cm2 s-1 to both [FORMULA] and [FORMULA] (the exact value of the background diffusivity can be shown to have little importance). The resulting contour lines are displayed in Fig. 1.

[FIGURE] Fig. 1. Contour lines of the different dynamo parameters in a given meridional plane of our Galaxy. [FORMULA], [FORMULA], and [FORMULA] are the effective rotational velocities associated with the azimuthal, radial, and vertical alpha-effects, respectively; [FORMULA] is the effective vertical escape velocity; [FORMULA] and [FORMULA] are the vertical and horizontal magnetic diffusivities. Velocities are in km s-1 and magnetic diffusivities in units of [FORMULA] cm2 s- 1.

The large-scale flow is taken axisymmetric, like the dynamo parameters, and it is assumed to reduce to the Galactic differential rotation:

[EQUATION]

In principle, V can be determined observationally by measuring the tangent point velocities of interstellar H I inside the Solar circle and, for instance, CO velocities together with optical distances in the outer Galaxy. The tangent point method does not give reliable results within the innermost 3 kpc, because of the increased relative uncertainties in the line-of-sight positions and because of the existence of important noncircular motions (Mihalas & Binney 1981). Moreover, the data exhibit local scatter which would largely vanish upon azimuthal averaging. For this reason, we prefer to use a smooth rotation curve rather than one that tries to fit the data too closely (e.g., the curve derived by Clemens 1985). Here, we simply assume that the rotation velocity, V, increases linearly from the Galactic center to [FORMULA] kpc, varies quadratically between 3 and 5 kpc, and remains constant outside 5 kpc (see Fich et al. 1989). We further adopt the IAU standard values for the Solar galactocentric radius, [FORMULA] kpc, and rotation velocity, [FORMULA] km s-1, whereupon the rotation velocity, V, the rotation rate,

[EQUATION]

and the radial shear rate,

[EQUATION]

have the radial profiles shown in Fig. 2.

[FIGURE] Fig. 2. Parameters of the large-scale Galactic rotation curve as a function of Galactic radius. V is the linear rotation velocity, [FORMULA] the rotation rate, and G the radial shear rate.

There exists little observational information on the vertical dependence of the rotation curve. From 21-cm emission measurements of interstellar H I , the neutral gas appears to corotate up to at least 1 kpc from the Galactic plane in the radial range 3.5 kpc[FORMULA] (Lockman 1984). UV absorption line studies of interstellar high-stage ions (mainly C IV and N V ) suggest that, for [FORMULA] kpc, corotation holds up to [FORMULA] kpc (Savage et al. 1997); they also give evidence that, inside [FORMULA] kpc, departures from corotation set in at [FORMULA] kpc from the plane and bring the gas to a complete halt within the next 2 kpc (Savage et al. 1990; Sembach et al. 1991). In most of our runs, we will let the rotation law be independent of height. However, for completeness, we will devote a few runs to testing the sensitivity of the computation results on a possible Z-dependence of V.

We will integrate the three cylindrical components of the dynamo equation, Eq. (1), on a [FORMULA]-mesh by means of an explicit numerical scheme based on a method introduced by Schnack & Killeen (1980). These authors used a discretized curl-operator to compute the right-hand side of Eq. (1), so as to ensure that the initial constraint [FORMULA] is preserved during the forward integration. The method was further developed by Evans & Hawley (1988) and successfully applied to dynamo calculations by Elstner et al. (1990). Although written on a [FORMULA]-mesh, the numerical code can handle any linear nonaxisymmetric solution of given azimuthal wavenumber m (by simply making the usual substitution [FORMULA] and integrating the ensuing complex amplitudes).

Our computation domain extends radially from [FORMULA] to [FORMULA] kpc and vertically from [FORMULA] (respectively [FORMULA] kpc) to [FORMULA] kpc in the linear (respectively nonlinear) case. The restriction to the upper half-space in the linear case is justified by the fact that solutions with even and odd parity with respect to the midplane are decoupled and can be computed separately once the appropriate symmetry conditions are imposed at [FORMULA]. The outer edges at [FORMULA] kpc and [FORMULA] kpc are sufficiently remote from the induction region that the associated boundary conditions have a negligible impact on the interior solution. The external medium is assumed to be either a perfect conductor (in most runs) or a vacuum (in a few trial runs). Since exact vacuum boundary conditions are difficult to implement in practice, we simulate them by raising the magnetic diffusivity to [FORMULA] cm2 s-1 (300 times the background diffusivity) near the outer surface. At [FORMULA], we apply a set of symmetry conditions which depend on the considered azimuthal wavenumber.

The number of mesh points is usually 51 in the radial direction and 26 (respectively 51) in the vertical direction, corresponding to a grid size of 0.4 kpc in both directions. The timestep, dictated by the CFL condition for numerical stability, is on the order of one thousandth the diffusion time. Convergence is verified through a series of trial computations with finer grid and smaller timestep.

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© European Southern Observatory (ESO) 2000

Online publication: June 26, 2000
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