The (relative) geometrical and dynamical simplicity of the various types of dynamo models considered earlir severely restricts the manner in which such stochastic effects can be modeled. Perhaps the most straightforward is to let the dynamo number fluctuate randomly in time about some pre-set mean value. By most statistical estimates, the expected magnitude of these fluctuations is quite large, i.e., many times the mean value (Hoyng, 1988, 1993), a conclusion also supported by numerical simulations (see, e.g., Otmianowska-Mazur et al., 1997; Ossendrijver et al., 2001). One typically also introduces a coherence time during which the dynamo number retains a fixed value. At the end of this time interval, this value is randomly readjusted. Depending on the dynamo model at hand, the coherence time can be physically related to the lifetime of convective eddies (-effect-based mean-field models), to the decay time of sunspots (Babcock–Leighton models), or to the growth rate of instabilities (hydrodynamical shear or buoyant MHD instability-based models).

Figure 25 shows some representative results for an dynamo solutions including meridional circulation and operating in the advection-dominated regime, similar to that of Figure 11, with imposed stochastic fluctuation at the ± 100% level in , and coherence time amounting to 5% of the cycle period in the deterministic parent solution. The red curve is the total magnetic energy in the solution domain, used here as a measure of cycle amplitude and proxy for the sunspot number. The green curve is the absolute value of the N-hemisphere surface polar field strength. Perhaps the most striking feature of these curves is the fact that even with a coherence time much smaller than the cycle period, zero-mean stochastic forcing can induce patterns of amplitude modulation with characteristic timescales spanning many cycles (e.g., at and in Figure 25A). This can be traced to the buildup of strong magnetic fields in the low-diffusivity layers underlying the convective envelope.

Stochastic forcing of the dynamo number can also produce a significant spread in cycle period, although in the model run used to produce Figure 25 the very weak positive correlation between cycle amplitude and rise time is anti-solar (the Waldmeier rule has , based on smoothed monthly SSN, cf. Figure 22D), and the positive correlation between rise time and cycle duration (, not shown) is comparable to solar (). It must be kept in mind that these inferences are all predicated on the use of total magnetic energy as a SSN proxy; different choices can lead to varying degrees of correlation.

The effect of noise has been investigated in most detail in the context of classical mean-field models (see Choudhuri, 1992; Hoyng, 1993; Ossendrijver and Hoyng, 1996; Ossendrijver et al., 1996; Mininni and Gómez, 2002, 2004; Moss et al., 2008). A particularly interesting consequence of random variations of the dynamo number, in mean-field models at or very close to criticality, is the coupling of the cycle’s duration and amplitude (Hoyng, 1993; Ossendrijver and Hoyng, 1996; Ossendrijver et al., 1996), leading to a pronounced anticorrelation between these two quantities that is reminiscent of the Waldmeier Rule (cf. Panel D of Figure 22), and hard to produce by purely nonlinear effects (cf. Ossendrijver and Hoyng, 1996). However, this behavior does not carry over to the supercritical regime, so it is not clear whether this can indeed be accepted as a robust explanation of the observed amplitude-duration anticorrelation. In the supercritical regime, -quenched mean-field models are less sensitive to noise (Choudhuri, 1992), unless of course they happen to operate close to a bifurcation point, in which case large amplitude and/or parity fluctuations can be produced (see, e.g., Moss et al., 1992).

In the context of Babcock–Leighton models, introducing stochastic forcing of the dynamo numbers leads to amplitude fluctuation patterns qualitatively similar to those plotted in Figure 25: long timescale amplitude modulation, spread in cycle period, (non-solar) positive correlations between cycle amplitude and rise time, and (solar-like) positive correlation between duration and rise time, with the interesting addition that in some model formulations cycle-to-cycle amplitude variation patterns reminiscent of the Gnevyshev–Ohl Rule are also produced (see Charbonneau et al., 2007). Charbonneau and Dikpati (2000) have presented a series of dynamo simulations including stochastic fluctuations in the dynamo number as well as in the meridional circulation. Working in the supercritical regime with a form of algebraic -quenching as the sole amplitude-limiting nonlinearity, they succeed in producing a solar-like weak anticorrelation between cycle amplitude and duration for fluctuations in the dynamo numbers in excess of 200% of its mean value, with coherence time of one month. However, these encouraging results did not prove very robust across the model’s parameter space.

A different approach is followed by Passos and Lopes (2008) and Lopes and Passos (2009), who used a low-order dynamo model resulting from truncation of the 2D axisymmetric mean-field dynamo equations, with flux loss due to magnetic buoyancy as the amplitude-limiting nonlinearity. Fitting equilibrium solutions to their low-order model to the smoothed SSN time series, one magnetic cycle at a time (Figure 26A), they can plausibly interpret variations in their fitting parameters as being due to systematic, persistent variations of the meridional flow speed on decadal timescales (Figure 26B). They then input these variations in the kinematic axisymmetric Babcock–Leighton model of Chatterjee et al. (2004), conceptually similar to that described in Section 4.8 but replacing the nonlinearity on the poloidal source term by a threshold function for magnetic flux loss through magnetic buoyancy. The resulting SSN-proxy time series reconstructed in this manner shows some remarkable similarities to the true SSN time series, including an epoch of strongly reduced cycle amplitude in the opening decades of the nineteenth century, and secular rise of cycle amplitudes from the mid-nineteenth to the mid-twentieth century (Figure 26C). This suggests that relatively small but persistent changes in the meridional flow, at the 5 – 30% level, could account for much of the variation in amplitude and duration observed in the solar cycle, and possibly even Grand Minima of activity (see Passos and Lopes, 2009), the topic to which we now turn.

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