5.4 Time-delay dynamics

The introduction of ad hoc time-delays in dynamo models is long known to lead to pronounced cycle amplitude fluctuations (see, e.g., Yoshimura, 1978 ). Models including nonlinear backreaction on differential rotation can also exhibit what essentially amounts to time-delay dynamics in the low Prandtl number regime, with the large-scale flow perturbations lagging behind the Lorentz force because of inertial effects. Finally, time-delay effects can arise in dynamo models where the source regions for the poloidal and toroidal magnetic components are spatially segregated. This is a type of time delay we now turn to, in the context of dynamo models based on the Babcock-Leighton mechanism.

5.4.1 Time-delays in Babcock-Leighton models

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 ). 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 (A $-->$ C in Figure 17 ). In contrast, the production of poloidal field from the deep-seated toroidal field (C $-->$ D), 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.

5.4.2 Reduction to an iterative map

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 $Tn+1$ at cycle $n + 1$ is assumed to be linearly proportional to the poloidal field strength $Pn$ of cycle $n$ , i.e.,

Now, because flux eruption is a fast process, the strength of the poloidal field at cycle $n + 1$ is (nonlinearly) proportional to the toroidal field strength of the current cycle:

Here the “Babcock-Leighton” function $f (Tn+1)$ 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 $pn$ ’s are normalized amplitudes, and the normalization constants as well as the constant $a$ in Equation (38

) have been absorbed into the definition of the map’s parameter $a$ , here operationally equivalent to a dynamo number (see Charbonneau, 2001

). We consider here the following nonlinear function,

with $p1 = 0.6$ , $w1 = 0.2$ , $p2 = 1.0$ , and $w2 = 0.8$ . 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 21 . For a given value of the map parameter $a$ , the diagram gives the locus of the amplitude iterate $pn$ for successive $n$ values. The “critical dynamo number” above which dynamo action becomes possible, corresponds here to $a = 0.851$ ( $p = 0 n$ for smaller $a$ values). For $0.851 < a < 1.283$ , the iterate is stable at some finite value of $pn$ , which increases gradually with $a$ . This corresponds to a constant amplitude cycle. As $a$ reaches $1.283$ , period doubling occurs, with the iterate $pn$ alternating between high and low values (e.g., $pn = 0.93$ and $pn = 1.41$ at $a = 1.4$ ). Further period doubling occurs at $a = 1.488$ , then at $a = 1.531$ , then again at $a = 1.541$ , 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.

Figure 21:

Two bifurcation diagrams for a kinematic Babcock-Leighton model, where amplitude fluctuations are produced by time-delay feedback. The top diagram is computed using the one-dimensional iterative map given by Equations (40

, 41

), while the bottom diagram is reconstructed from numerical solutions in spherical geometry, of the type discussed in Section 4.8. The shaded area in Panel A maps the attraction basin for the cyclic solutions, with initial conditions located outside of this basin converging to the trivial solution $pn = 0$ .

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 21 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 discussed earlier in Section 4.8, for increasing values of the dynamo number in that model. 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 ).