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

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3. A simple example

In order to facilitate the interpretation of the computation results, we start by looking in detail at a special case where the time evolution of the large-scale magnetic field can easily be described in terms of the various processes at play (shear, alpha-effect, escape, diffusion). In this special case, the magnetic field in the induction region is initially uniform and purely azimuthal. Hence it remains axisymmetric at all times and can, therefore, be written as

[EQUATION]

where [FORMULA] is the vector potential. The poloidal and azimuthal components of the dynamo equation, Eq. (1), are then given by

[EQUATION]

and

[EQUATION]

with

[EQUATION]

and

[EQUATION]

Remember that the equations integrated numerically are the explicit equations for the three field components, [FORMULA], [FORMULA], and [FORMULA], not Eqs. (9) and (10), whose sole purpose is to serve as a support for the current discussion.

Initially, the magnetic field is purely azimuthal, so the only source term in the dynamo equation is the first term on the right-hand side of Eq. (9). This term obviously represents the production of poloidal field from azimuthal field via the azimuthal alpha-effect. Thus, at early times, [FORMULA] grows linearly with time according to [FORMULA], and since [FORMULA] is uniform, the contour lines of [FORMULA] coincide with those of [FORMULA] (given in Fig. 1). The corresponding [FORMULA] and [FORMULA], deduced from Eqs. (11) and (12), are plotted in the upper row of Fig. 3.

[FIGURE] Fig. 3. Top row: contour lines of the poloidal components of the large-scale magnetic field generated by the azimuthal alpha-effect acting on a purely azimuthal field. Middle and bottom rows: contour lines of the associated source terms for the azimuthal field (first four terms on the right-hand side of Eq. (10)). The solid lines correspond to positive values and the dotted lines to negative values.

The poloidal field produced by the azimuthal alpha-effect acts back on the azimuthal field, first through field line shearing by the large-scale differential rotation (first two terms on the right-hand side of Eq. (10)), and second through the radial and vertical alpha-effect (next two terms in Eq. (10)). We calculated the four source terms for [FORMULA] and plotted them in the middle and bottom rows of Fig. 3, using our expressions for [FORMULA] and [FORMULA] (see Fig. 1) and for [FORMULA] and V (see Fig. 2), and assuming, for the purpose of illustration, that V falls off with height at the rate [FORMULA]. It clearly appears that the radial shear [FORMULA] and the radial alpha-effect [FORMULA] together reduce the initial [FORMULA] at high Z and reinforce it at intermediate Z. Their combined effect at low Z depends on their relative importance; with our adopted values, the radial shear dominates by about one order of magnitude, so that [FORMULA] is also enhanced at low Z. Likewise, the vertical shear [FORMULA] and the vertical alpha-effect [FORMULA] together reduce [FORMULA] at small R and reinforce it beyond [FORMULA] kpc. Hence the presence of a poloidal field tends to create a reverse polarity in the azimuthal field at high Z, small R, while it amplifies the existing polarity at low Z, large R.

The changes in [FORMULA] tend to produce similar changes in [FORMULA] by way of the azimuthal alpha-effect (first term on the right-hand side of Eq. (9)). As a result, a reverse polarity in [FORMULA] tends to appear at high Z, small R, while the existing polarity moves toward low Z, large R. This behavior is consistent with the propagation of a dynamo wave toward the midplane. In the absence of poloidal alpha-effect ([FORMULA]), the dynamo wave would propagate along surfaces of constant [FORMULA], as predicted by the standard [FORMULA]-dynamo theory; in particular, if [FORMULA] is independent of Z, the propagation would be strictly vertical.

The dynamo wave cannot freely migrate downward, for, at the equatorial midplane, it encounters another dynamo wave coming from the southern hemiplane. Both wave trains are effectively stopped at the equator, and regions of alternating polarity gradually squeeze up as new polarities are generated at high [FORMULA] by the mechanism described above. The number of reversals increases until magnetic diffusion (last two terms in Eqs. (9) and (10)) prevents any further squeezing. Typically, if the rotation rate varies only with R, [FORMULA] will end up having, on each side of the midplane, two or three well-defined polarities along the vertical. If the rotation rate varies also with Z, the alternating polarities of [FORMULA] will be stacked obliquely and, therefore, likely to become more numerous. Obviously, the number of distinct polarities is a decreasing function of the magnetic diffusivities, [FORMULA] and [FORMULA], and an increasing function of both the "source factor", [FORMULA], and the thickness of the dynamo layer (region with non-negligible [FORMULA]).

How the field pattern obtained in this manner evolves past the transient phase depends on the relative importance of magnetic diffusion. If the latter is sufficiently strong that magnetic field lines reconnect and dissipate faster than they are regenerated by the combination of large-scale shear and alpha-effect, then the whole field pattern will eventually decay with time. In the opposite case, there will be exponential growth. The magnetic field growth/decay, in turn, may be either oscillatory or steady (i.e., monotonous), depending on the ability of the polarities closest to the midplane to dissipate away rapidly and give way to the next closest polarities (of opposite sign).

Other factors are expected to affect the oscillatory vs. steady character of the dynamo. In particular, the poloidal alpha-effect (bottom row in Fig. 3) helps destroy the azimuthal field component in the immediate vicinity of the midplane, thereby favoring propagation. Note that this conclusion is not in contradiction with the fact that the standard [FORMULA]-dynamo (which assumes uniform [FORMULA]) is steady, because here [FORMULA] increases upward at low Z, and it is precisely this Z-dependence that sets the sign of [FORMULA] just above the midplane. The large-scale vertical shear, [FORMULA], which tilts the direction of the wavevector away from the vertical, also favors propagation. In contrast, the existence of an escape velocity, [FORMULA], directed away from the midplane clearly opposes propagation. Finally, the influence of the equatorial boundary condition is such that propagation is generally easier for odd magnetic configurations, which allow reconnection across the equator.

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© European Southern Observatory (ESO) 2000

Online publication: June 26, 2000
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