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. 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 azimuthal electromotive force.

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, 1993Jump To The Next Citation Point), 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 (α-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 25View Image shows some representative results for an α Ω dynamo solutions including meridional circulation and operating in the advection-dominated regime, similar to that of Figure 11Watch/download Movie, with imposed stochastic fluctuation at the ± 100% level in Cα, 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 0.01 ≤ t∕τ ≤ 0.11 and 0.49 ≤ t∕τ ≤ 0.62 in Figure 25View ImageA). This can be traced to the buildup of strong magnetic fields in the low-diffusivity layers underlying the convective envelope.

View Image

Figure 25: Effect of stochastic fluctuations in the C α dynamo number on an advection-dominated α Ω mean-field dynamo solution including meridional circulation (see Figure 11Watch/download Movie), here with Rm = 2500, 5 C Ω = 5 × 10, Cα = 0.5, and Δη = 0.1. The fluctuation amplitude is δC α∕C α = 1, and the correlation time of the imposed fluctuations amounts to about 5% of the mean half-cycle period. Panel A shows a portion of the time series of total magnetic energy (red), used here as a proxy for cycle amplitude, and of the surface polar field strength (green), both scaled to their peak value over the full simulation run. Panel B shows a correlation plot of cycle amplitude and duration, both now normalized to their respective means over the simulation interval. Panel C snows a correlation plot of cycle amplitude versus the preceding peak value of the surface polar field.

Stochastic forcing of the dynamo number can also produce a significant spread in cycle period, although in the model run used to produce Figure 25View Image the very weak positive correlation between cycle amplitude and rise time is anti-solar (the Waldmeier rule has r = − 0.68, based on smoothed monthly SSN, cf. Figure 22View ImageD), and the positive correlation between rise time and cycle duration (r = +0.27, not shown) is comparable to solar (r = +0.4). 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, 1992Jump To The Next Citation Point; Hoyng, 1993Jump To The Next Citation Point; Ossendrijver and Hoyng, 1996Jump To The Next Citation Point; Ossendrijver et al., 1996Jump To The Next Citation Point; Mininni and Gómez, 2002, 2004Jump To The Next Citation Point; Moss et al., 2008Jump To The Next Citation Point). 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, 1993Jump To The Next Citation Point; Ossendrijver and Hoyng, 1996Jump To The Next Citation Point; 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 22View Image), 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, α-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 25View Image: 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 α-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 (2008Jump To The Next Citation Point) and Lopes and Passos (2009Jump To The Next Citation Point), 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 26View ImageA), 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 26View ImageB). They then input these variations in the kinematic axisymmetric Babcock–Leighton model of Chatterjee et al. (2004Jump To The Next Citation Point), 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 26View ImageC). 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.

View Image

Figure 26: Effect of persistent variations in meridional circulation on the amplitude of the solar cycle, as modeled by Lopes and Passos (2009Jump To The Next Citation Point). Panel A shows the signed square root of the sunspot number (gray), here used as a proxy of the solar internal magnetic field. A smoothed version of this time series (black) is fitted, one magnetic cycle at a time (green), with the equilibrium solution of the truncated dynamo model of Passos and Lopes (2008); assuming that variations in the fitting parameters are due to variations in the meridional flow speed (vp), the coarse time series of vp of panel B (in green) is obtained, scaled to the magnetic cycle 1 value and with error bars from the fitting procedure. Input of this piecewise-constant meridional flow variation (scaled down by a factor of two, in red in panel B) in the 2D Babcock–Leighton dynamo model of Chatterjee et al. (2004) yields the pseudo-SSN time series plotted in Panel C (figure produced from numerical data kindly provided by D. Passos).

  Go to previous page Go up Go to next page