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Astron. Astrophys. 363, L13-L16 (2000)

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3. Results

We calibrated our model so that near marginal excitation the cycle period was about 22 years. This determined [FORMULA], corresponding to [FORMULA] cm2 sec-1, given the known values of [FORMULA] and R. The first solutions to be excited in the linear theory are limit cycles with odd (dipolar) parity with respect to the equator, with marginal dynamo number [FORMULA]. The even parity (quadrupolar) solutions are also excited at a similar marginal dynamo number of [FORMULA]. It is plausible that the turbulent Prandtl number be of order unity, and we set [FORMULA]. For the parameter range that we investigated, the even parity solutions are nonlinearly stable. Given that the Sun is observed to be close to an odd (dipolar) parity state, and that previous experience shows that small changes in the physical model can cause a change between odd and even parities in the stable nonlinear solution, we chose to impose dipolar parity on our solutions.

With these parameter values, we found that this model, with the underlying zero order angular velocity chosen to be consistent with the recent (MDI) helioseismic data, is capable of producing butterfly diagrams which are in qualitative agreement with the observations. An example is shown in Fig. 1. (The polar branch is a little too strong, but this feature can be weakened by adjusting the latitudinal dependence of [FORMULA] (see also Covas et al. 2000).) The model can also successfully produce torsional oscillations (see Fig. 2) that penetrate into the convection zone, similar to those deduced from recent helioseismic data (Howe et al. 2000a) and studied in Covas et al. (2000). We note, however, that an additional interesting feature of the present model is that the torsional oscillations have larger and more realistic amplitudes near the surface, of the order of 1 nHz, much larger than was found previously using the boundary condition [FORMULA] at the surface.

[FIGURE] Fig. 1. Butterfly diagram of the toroidal component of the magnetic field [FORMULA] at fractional radius [FORMULA]. Dark and light shades correspond to positive and negative values of [FORMULA] respectively. Parameter values are [FORMULA], [FORMULA] and [FORMULA].

[FIGURE] Fig. 2. Variation of rotation rate with latitude and time from which a temporal average has been subtracted to reveal the migrating banded zonal flows, taken at fractional radius [FORMULA]. Darker and lighter regions represent positive and negative deviations from the time averaged background rotation rate. Parameter values are as in Fig. 1.

We found that the model is also capable of producing spatiotemporal fragmentation, near the base of the convection zone, hence resulting in oscillations in the differential rotation with, for example, half the basic period. To demonstrate this, we have plotted in Figs. 3-5 the radial contours of the angular velocity residuals [FORMULA] as a function of time for a cut at latitude [FORMULA], for several values of [FORMULA]. As can be seen, for smaller values of [FORMULA] (Fig. 3), we find torsional oscillations with the same period at the top and the bottom of the convection zone. As [FORMULA] is increased (Fig. 4 and Fig. 5), a spatiotemporal fragmentation occurs near the base of the convection zone, resulting in oscillations in the differential rotation with half the period of the oscillations near the top. For still higher values of [FORMULA], the temporal variations in the differential rotation at the base of the convection zone start to become non-periodic, which might be of relevance if the failure of Antia & Basu (2000) to find shorter period oscillations near the bottom of the convection zone should turn out to be correct. We have also checked that the butterfly diagrams do not fragment and keep the same period independently of the depth and [FORMULA] value, continuing to resemble Fig. 1.

[FIGURE] Fig. 3. Radial contours of the angular velocity residuals [FORMULA] as a function of time for a cut at latitude [FORMULA]. Parameter values are [FORMULA], [FORMULA], [FORMULA]. Note how the torsional oscillations are very coherent from top to the base of the dynamo region showing that only one period is present.

[FIGURE] Fig. 4. Radial contours of the angular velocity residuals [FORMULA] as in Fig. 3 for [FORMULA]. Note the emergence of spatiotemporal fragmentation towards the bottom of the convective zone, resulting in different periodicities there.

[FIGURE] Fig. 5. Radial contours of the angular velocity residuals [FORMULA] as in Fig. 3 for [FORMULA].

This fragmentation is made more transparent in Fig. 6 which shows the temporal oscillations in the angular velocity residuals [FORMULA] at a fixed point, as [FORMULA] is increased, illustrating the presence of period halving.

[FIGURE] Fig. 6. `Period halving' at [FORMULA] and latitude [FORMULA]. The panels correspond, from left to right, to [FORMULA] values -3.0 and -7.0 respectively, and display increasing relative amplitudes of the secondary oscillations. Remaining parameters values are as in Fig. 3.

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000