Physical noise is most readily incorporated into dynamo models by introducing, on the right hand side of the governing equations, an additional zeromean source term that varies randomly from node to node and from one time step to the next. Statistical noise can be modeled in a number of ways. Perhaps the most straightforward is to let the dynamo number fluctuate randomly in time about some preset 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). This conclusion is also supported by numerical simulations (see, e.g., OtmianowskaMazur 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 related to the lifetime of convective eddies (effectbased meanfield models), to the decay time of sunspots (BabcockLeighton models), or to the growth rate of instabilities (hydrodynamical shear or buoyant MHD instabilitybased models). Perhaps not surprisingly, the level of cycle amplitude fluctuations is found to increase with both increasing noise amplitude and longer coherence time (Choudhuri, 1992).
Figure 22 shows some representative results for three dynamo solutions with fluctuation in the dynamo number ranging from (blue) to (red). While the correlation time amounts here to only 5% of the halfcycle period, note in Panel A of Figure 22 how modulations on much longer timescales appear in the magnetic energy time series. As can be seen in Panel B of Figure 22, the fluctuations also lead to a spread in the cycle period, although here little (anti)correlation is seen with the cycle’s amplitude. Both the mean cycle period and amplitude increase with increasing fluctuation amplitude.

In the context of BabcockLeighton models, Charbonneau and Dikpati (2000) have presented a series of dynamo simulations including fluctuations in the dynamo number (statistical noise) as well as fluctuations in the meridional circulation (a form of noise both physical and statistical). Working in the supercritical regime with a form of algebraic quenching as the sole amplitudelimiting nonlinearity, they find the model solutions to be quite robust with respect to the introduction of noise, and succeed in producing a solarlike 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. They also find that the solutions exhibit good phase locking, in that shorterthanaverage cycles tend to be followed by longerthanaveraged cycles, a property they trace to the regulatory effects of meridional circulation, the primary determinant of the cycle period in these models (cf. Section 4.8 herein).
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