5.5 Stochastic forcing

Another means of producing amplitude fluctuations in dynamo models is to introduce stochastic forcing in the governing equations. Sources of stochastic “noise” certainly abound in the solar interior; large-scale flows in the convective envelope, such as differential rotation and meridional circulation, are observed to fluctuate, an unavoidable consequence of dynamical forcing by the surrounding, vigorous turbulent flow. Ample observational evidence now exists that a substantial portion of the Sun’s surface magnetic flux is continuously being reprocessed on a timescale commensurate with convective motions (see Schrijver et al., 1997 ; Hagenaar et al., 2003 ). The culprit is most likely the generation of small-scale magnetic fields by these turbulent fluid motions (see, e.g., Cattaneo, 1999 ; Cattaneo et al., 2003 , and references therein). This amounts to a form of zero-mean “noise” superimposed on the slowly-evolving mean magnetic field. These mechanisms can be thought of as physical noise. In addition, the azimuthal averaging implicit in all models of the solar cycle considered above will yield dynamo coefficients showing significant deviations about their mean values, as a consequence of the spatio-temporally discrete nature of the physical events (e.g., cyclonic updrafts, sunspot emergences, flux rope destabilizations, etc.) whose collective effects add up to produce a mean electromotive force. Such time-dependent, statistical deviations about the mean can be dubbed statistical noise.

Physical noise is most readily incorporated into dynamo models by introducing, on the right hand side of the governing equations, an additional zero-mean 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 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 ). This conclusion is 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 related to the lifetime of convective eddies ( $a$ -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). 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 $a_O_$ dynamo solutions with fluctuation in the dynamo number $Ca$ ranging from $± 50%$ (blue) to $± 200%$ (red). While the correlation time amounts here to only 5% of the half-cycle 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.

Figure 22:

Stochastic fluctuations of the dynamo number in an $a_O_$ mean-field dynamo solution. The reference, unperturbed solution is the same as that plotted in Panel D of Figure 7

, except that is uses a lower value for the dynamo number, $Ca = -5$ . Panel A shows magnetic energy time series for three solutions with increasing fluctuation amplitudes, while Panel B shows a correlation plot of cycle amplitude and duration, as extracted from a time series of the toroidal field at the core-envelope interface ( $r/R = 0.7$ ) in the model. Solutions are color-coded according to the relative amplitude $dCa/Ca$ of the fluctuations in the dynamo number. Line segments in Panel B indicate the mean cycle amplitudes and durations for the three solutions. The correlation time of the noise amounts here to about 5% of the mean half-cycle period in all cases.

The effect of noise has been investigated in most detail in the context of mean-field models (see Choudhuri, 1992

; Hoyng, 1993

; Ossendrijver and Hoyng, 1996

; Ossendrijver et al., 1996

; Mininni and Gómez, 2002 , 2004

). 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 19

), 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, $a$ -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, 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 $a$ -quenching as the sole amplitude-limiting nonlinearity, they find the model solutions to be quite robust with respect to the introduction of noise, and 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. They also find that the solutions exhibit good phase locking, in that shorter-than-average cycles tend to be followed by longer-than-averaged 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).