It was already noted that in solar cycle models based on the Babcock–Leighton mechanism of poloidal field generation, meridional circulation effectively sets – and even regulates – the cycle period (cf. Section 4.8.2; see also Dikpati and Charbonneau, 1999; Charbonneau and Dikpati, 2000; Muñoz-Jaramillo et al., 2009). In doing so, it also introduces a long time delay in the dynamo mechanism, “long” in the sense of being comparable to the cycle period. This delay originates with the time required for circulation to advect the surface poloidal field down to the core-envelope interface, where the toroidal component is produced (AC in Figure 16). In contrast, the production of poloidal field from the deep-seated toroidal field (CD), is a “fast” process, growth rates and buoyant rise times for sunspot-forming toroidal flux ropes being of the order of a few months (see Moreno-Insertis, 1986; Fan et al., 1993; Caligari et al., 1995, and references therein). The first, long time delay turns out to have important dynamical consequences.

The long time delay inherent in B-L models of the solar cycle allows a formulation of cycle-to-cycle amplitude variations in terms of a simple one-dimensional iterative map (Durney, 2000; Charbonneau, 2001). Working in the kinematic regime, neglecting resistive dissipation, and in view of the conveyor belt argument of Section 4.8, the toroidal field strength at cycle is assumed to be linearly proportional to the poloidal field strength of cycle , i.e.,

Now, because flux eruption is a fast process, the strength of the poloidal field at cycle is (nonlinearly) proportional to the toroidal field strength of the current cycle: Here the “Babcock–Leighton” function measures the efficiency of surface poloidal field production from the deep-seated toroidal field. Substitution of Equation (38) into Equation (39) leads immediately to a one-dimensional iterative map, where the ’s are normalized amplitudes, and the normalization constants as well as the constant in Equation (38) have been absorbed into the definition of the map’s parameter , here operationally equivalent to a dynamo number (see Charbonneau, 2001). We consider here the following nonlinear function, with , , , and . This catches an essential feature of the B-L mechanism, namely the fact that it can only operate in a finite range of toroidal field strength.A bifurcation diagram for the resulting iterative map is presented in Panel A of Figure 24. For a given value of the map parameter , the diagram gives the locus of the amplitude iterate for successive values. The “critical dynamo number” above which dynamo action becomes possible, is here ( for smaller values). For , the iterate is stable at some finite value of , which increases gradually with . This corresponds to a constant amplitude cycle. As reaches , period doubling occurs, with the iterate alternating between high and low values (e.g., and at ). Further period doubling occurs at , then at , then again at , and ever faster until a point is reached beyond which the amplitude iterate seems to vary without any obvious pattern (although within a bounded range); this is in fact a chaotic regime.

As in any other dynamo model where the source regions for the poloidal and toroidal magnetic field components are spatially segregated, the type of time delay considered here is unavoidable. The B-L model is just a particularly clear-cut example of such a situation. One is then led to anticipate that the map’s rich dynamical behavior should find its counterpart in the original, arguably more realistic spatially-extended, diffusive axisymmetric model that inspired the map formulation. Remarkably, this is indeed the case.

Panel B of Figure 24 shows a bifurcation diagram, conceptually equivalent to that shown in Panel A, but now constructed from a sequence of numerical solutions of the Babcock–Leighton model of Charbonneau et al. (2005), for increasing values of the dynamo number. Time series of magnetic energy were calculated from the numerical solutions, and successive peaks found and plotted for each individual solution. The sequence of period doubling, eventually leading to a chaotic regime, is strikingly similar to the bifurcation diagram constructed from the corresponding iterative map, down to the narrow multiperiodic windows interspersed in the chaotic domain. This demonstrates that time delay effects are a robust feature, and represent a very powerful source of cycle amplitude fluctuation in Babcock–Leighton models, even in the kinematic regime (for further discussion see Charbonneau, 2001; Charbonneau et al., 2005; Wilmot-Smith et al., 2006).

Living Rev. Solar Phys. 7, (2010), 3
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