3. Numerical results
If we introduce the dimensionless time variable into Eqs. (8) and (9) then the solution is characterized by only one dimensionless parameter which is equal to the surface value of the Hall parameter at the magnetic equator, is the initial strength of the magnetic field at and . We calculated the field decay for . To solve Eqs. (8) and (9) with the corresponding boundary and initial conditions we use the standard finite difference method. The explicit scheme was applied with the time step restricted by the Courant-Friedrichs-Lewy condition (see, e.g., Potter 1973). Unfortunately, our computational facilities do allow us to study the non-linear decay for a very high magnetization, .
The behaviour of the magnetic field for all considered values of is qualitatively similar. Already aduring the early evolutionary phase, the Hall current generates higher order modes from the original dipole one. Amplitudes of these newly generated modes reach a quasi-steady state value on a relatively short time scale associated with the Hall effect, . Both toroidal and poloidal modes are generated but the strength of the mode decreases rapidly with its number (see also Naito & Kojima 1994). Thus, among poloidal modes, only those with give an appreciable contribution to the external magnetic field (in our calculations, we follow the evolution of multipoles up to ). Since the Hall current provides coupling among the multipole moments, the energy of the original dipole field is partly converted to that of higher order modes and, due to this, the strength of the dipole field decreases faster than in the case of a linear decay. One more important point, characterizing the non-linear field decay, is a non-monotonic behaviour of individual multipole components. All modes including the original dipole one undergo oscillations with the period of the order of . These oscillations are caused by a non-dissipative exchange of energy between modes and are of the same origin as well known helicoidal oscillations of a magnetized plasma (see, e.g., Kingsep et al.1990). The oscillating behaviour is especially pronounced for the newly generated modes which even can change sign. The original dipole mode evolves non-monotonously as well but the amplitude of oscillations is small in comparison with the average strength of the dipole field, at least, during the considered evolutionary phases.
Figs. 1 and 2 show the time dependence of the toroidal component of the magnetic field for different radii and for . Symmetry of the original poloidal magnetic configuration implies that the toroidal field generated due to the Hall effect is anti-symmetric to the equatorial plane thus at . The oscillating character of the evolution is seen in particular for the toroidal field component. At the initial stage while the dipole field does not decay appreciably, the period of oscillations is approximately equal to but dissipation of the field in the course of the evolution leads to an increase of this period. The amplitude of the oscillations depends on r and , reaching its maximum value at for all considered values of the parameter . Near the surface and in the vicinity of the centre where the toroidal field tends to zero, oscillations are evidently less pronounced. As a function of , the strength of the toroidal field has its maximum at and but, as was mentioned, is oppositely directed in the upper and low hemispheres. Note that for the considered values of , the maximum strength of the generated toroidal field is of . Due to the ohmic dissipation, the strength of the toroidal component decreases slowly on a time scale of the order of .
The time dependence of the poloidal field components at the surface are plotted in Figs. 3 and 4 for and 50, respectively. We followed the evolution of multipoles with but already the components with give a negligible contribution to the total magnetic configuration in all cases considered. That is why only the multipoles with and 5 are represented in Figs. 3 and 4. Like the toroidal field, the poloidal components show oscillations as well. An oscillating behaviour is typical not only for newly generated modes but for the dipole mode as well although the amplitude of the dipole variations is relatively small ( of the non-oscillating component of the dipole field). Due to these variations, there exist some phases in the course of the evolution during which the strength of the dipole field increases. The duration of these phases is , and the amplification of the dipole moment may reach . Note that this value depends on , being higher for a larger initial Hall parameter. The modes with and undergo stronger variations than the dipole one: the oscillating fraction of the mode is of the order of its average strength, whereas the mode evolves even with sign reversals. However, the strength of both these components is substantially weaker than the dipole field. Thus, the mode contributes only of the total field for and for . The contribution of the harmonic is appreciably smaller (). The period of poloidal variations is obviously the same as for toroidal ones since an oscillating behaviour is caused by the energy transfer among modes. Of course, a non-monotonic decay is typical only for individual modes but the total magnetic energy evolves monotonously. The decay of the field caused by the ohmic dissipation decreases the amplitude of the oscillations and increases their period.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998