We first address the question under which conditions dynamo action exists. We therefore fix the Coriolis number at a value of 60 and vary from zero to one. The critical value for the onset of dynamo action lies close to 0.02. For all values of above 0.02, the field has a S1-type geometry, i.e. the radial and azimuthal components are symmetric with respect to the equatorial plane while the z-component is antisymmetric, and all components vary with the azimuthal angle like . The field has thus an approximately dipolar geometry, but with the axis of symmetry inclined by 90o relative to the rotation axis. The field rotates with the star with a slow drift with a period of about 45 years between both rotations.
Figs. 2 and 3 show the field geometry for , . In Fig. 3, a plane parallel to the equatorial plane has been chosen that cuts the northern hemisphere at an intermediate latitude. In the southern hemisphere, the z component has reversed sign while the horizontal (s and ) components are identical with those in the northern hemisphere. In the equatorial plane, vanishes and the horizontal components look very similar to those in Fig. 3.
Above the critical value the total magnetic field energy increases linearly for increasing values of . This behavior is not surprising since for a steady field the diffusion must balance the -effect. As we always find essentially the same field structure and for , the field energy must be a linear function of , hence of .
Varying means keeping the rotation rate constant while changing the efficiency of the -effect. A more relevant case is that of constant and varying Coriolis number, which corresponds to a sample of stars of the same type but different rotation rates or a single star changing its rotation rate due to external torques. We have therefore carried out a sequence of runs with and increasing Coriolis number.
The critical value of lies between 0.7 and 0.8. At , we find a stationary field of S0 geometry, i.e. the field is symmetric with respect to the equator and axisymmetric. Fig. 4 shows the field structure for .
At , the field geometry has not changed very much, but the field is now oscillatory with an oscillation period of about two years. Its time dependence cannot be described as a rotation, but is a true cycle with the total field energy varying with time. The same behavior as for is found at . In Fig. 5, five consecutive snapshots of the magnetic field for are shown. Depending on the phase, there are two or three radial shells of alternating field polarity. Field belts emerge at the center of the star and move outward to the surface where they finally disappear. The total field energy varies by about 30%. Unlike axisymmetric fields from dynamos, the magnitudes of all three field components are of the same order of magnitude, hence the field is not dominated by the toroidal component.
For and all larger values, the field has S1-type geometry and a purely rotational time dependence with no variation of the magnetic field energy. The field geometry is shown in Figs. 7 and 6. In the s- plane, it is of almost spiral-type while a plot at constant azimuthal angle shows two radial shells. Above , the field geometry does not change that much, although the profiles at constant z-values show a more spiral-type geometry for and an almost dipolar field for .
Changes of affect both and , but in the limit of fast rotation, becomes independent of , while the magnetic diffusivity decreases as . Hence, and the magnetic field energy increases linearly with increasing Coriolis number. The linear relation between the field energy and (or the Coriolis number) is a consequence of the -quenching relation (13). In Rüdiger & Kitchatinov (1993), a quenching function was derived which decreases with in the limit of strong magnetic fields. We therefore vary for fixed Coriolis number with Now and thus the magnetic energy varies as . The symmetry of the field is not affected by the change of the -quenching function.
© European Southern Observatory (ESO) 1999
Online publication: June 17, 1999