## 3. A simple exampleIn 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 where is the vector potential. The poloidal and azimuthal components of the dynamo equation, Eq. (1), are then given by Remember that the equations integrated numerically are the explicit equations for the three field components, , , and , 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, grows linearly with time according to , and since is uniform, the contour lines of coincide with those of (given in Fig. 1). The corresponding and , deduced from Eqs. (11) and (12), are plotted in the upper row of Fig. 3.
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 and plotted them in
the middle and bottom rows of Fig. 3, using our expressions for
and
(see Fig. 1) and for
and The changes in tend to produce
similar changes in by way of the
azimuthal alpha-effect (first term on the right-hand side of Eq. (9)).
As a result, a reverse polarity in
tends to appear at high 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
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 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 © European Southern Observatory (ESO) 2000 Online publication: June 26, 2000 |