5.6 Intermittency

5.6.1 The Maunder Minimum and intermittency

The term “intermittency” was originally coined to characterize signals measured in turbulent fluids, but has now come to refer more generally to systems undergoing apparently random, rapid switching from quiescent to bursting behaviors, as measured by the magnitude of some suitable system variable (see, e.g., Platt et al., 1993). Intermittency thus requires at least two distinct dynamical states available to the system, and a means of transiting from one to the other.

In the context of solar cycle model, intermittency refers to the existence of quiescent epochs of strongly suppressed activity randomly interspersed within periods of “normal” cyclic activity. Observationally, the Maunder Minimum is usually taken as the exemplar for such quiescent epochs. It should be noted, however, that dearth of sunspots does not necessarily mean a halted cycle; as noted earlier, flux ropes of strengths inferior to ∼ 10 kG will not survive their rise through the convective envelope, and the process of flux rope formation from the dynamo-generated mean magnetic field may itself be subjected to a threshold in field strength. The same basic magnetic cycle may well have continued unabated all the way through the Maunder Minimum, but at an amplitude just below one of these thresholds. This idea finds support in the 10Be radioisotope record, which shows a clear and uninterrupted cyclic signal through the Maunder Minimum (see Panels B and C of Figure 22View Image; also Beer et al., 1998Jump To The Next Citation Point). Strictly speaking, thresholding a variable controlled by a single dynamical state subject to amplitude modulation is not intermittency, although the resulting time series for the variable may well look quite intermittent.

Much effort has already been invested in categorizing intermittency-like behavior observed in solar cycle models in terms of the various types of intermittency known to characterize dynamical systems (see Ossendrijver and Covas, 2003, and references therein). In what follows, we attempt to pin down the physical origin of intermittent behavior in the various types of solar cycle models discussed earlier.

5.6.2 Intermittency from stochastic noise

Intermittency has been shown to occur through stochastic fluctuations of the dynamo number in linear mean-field dynamo models operating at criticality (see, e.g., Hoyng, 1993). Such models also exhibit a solar-like anticorrelation between cycle amplitude and phase. However there is no strong reason to believe that the solar dynamo is running just at criticality, so that it is not clear how good an explanation this is of Maunder-type Grand Minima.

Mininni and Gómez (2004Jump To The Next Citation Point) have presented a stochastically-forced 1D (in latitude) αΩ mean-field model, including algebraic α-quenching as the amplitude-limiting nonlinearity, that exhibits a form of intermittency arising from the interaction of dynamo modes of opposite parity. The solution aperiodically produces episodes of markedly reduced cycle amplitude, and often showing strong hemispheric asymmetry. This superficially resembles the behavior associated with the nonlinear amplitude modulation discussed in Section 5.3.1 (compare the top panel in Figure 23View Image herein to Figure 7 in Mininni and Gómez, 2004). However, here it is the stochastic forcing that occasionally excites the higher-order modes that perturb the normal operation of the otherwise dominant dynamo mode. Moss et al. (2008) and Usoskin et al. (2009a) present more elaborate versions of such models, that do reproduce many salient features of observed grand activity minima.

5.6.3 Intermittency from nonlinearities

Another way to trigger intermittency in a dynamo model, deterministically this time, is to let nonlinear dynamical effects, for example a reduction of the differential rotation amplitude, push the effective dynamo number below its critical value; dynamo action then ceases during the subsequent time interval needed to reestablish differential rotation following the diffusive decay of the magnetic field; in the low Pm regime, this time interval can amount to many cycle periods, but Pm must not be too small, otherwise Grand Minima become too rare (see, e.g., Küker et al., 1999Jump To The Next Citation Point). Values Pm ∼ 10−2 seem to work best. Such intermittency is most readily produced when the dynamo is operating close to criticality. For representative models, see Tobias (1996b, 1997); Brooke et al. (1998); Küker et al. (1999); Brooke et al. (2002).

Intermittency of this type has some attractive properties as a Maunder Minimum scenario. First, the strong hemispheric asymmetry in sunspots distributions in the final decades of the Maunder Minimum (Ribes and Nesme-Ribes, 1993Jump To The Next Citation Point) can occur naturally via parity modulation (see Figure 23View Image herein). Second, because the same cycle is operating at all times, cyclic activity in indicators other than sunspots (such as radioisotopes, see Beer et al., 1998Jump To The Next Citation Point) is easier to explain; the dynamo is still operating and the solar magnetic field is still undergoing polarity reversal, but simply fails to reach the amplitude threshold above which the sunspot-forming flux ropes can be generated from the mean magnetic field, or survive their buoyant rise through the envelope.

There are also important difficulties with this explanatory scheme. Grand Minima tend to have similar durations and recur in periodic or quasi-periodic fashion, while the sunspot and radioisotope records, taken at face value, suggest a pattern far more irregular (Usoskin, 2008). Moreover, the dynamo solutions in the small Pm regime are characterized by large, non-solar angular velocity fluctuations. In such models, solar-like, low-amplitude torsional oscillations do occur, but for Pm ∼ 1. Unfortunately, in this regime the solutions then lack the separation of timescales needed for Maunder-like Grand Minima episodes. One is stuck here with two conflicting requirements, neither of which easily evaded (but do see Bushby, 2006).

Intermittency has also been observed in strongly supercritical model including α-quenching as the sole amplitude-limiting nonlinearity. Such solutions can enter Grand Minima-like epochs of reduced activity when the dynamo-generated magnetic field completely quenches the α-effect. The dynamo cycle restarts when the magnetic field resistively decays back to the level where the α-effect becomes operational once again. The physical origin of the “long” timescale governing the length of the “typical” time interval between successive Grand Minima episodes is unclear, and the physical underpinning of intermittency harder to identify. For representative models exhibiting intermittency of this type, see Tworkowski et al. (1998).

5.6.4 Intermittency from threshold effects

Intermittency can also arise naturally in dynamo models characterized by a lower operating threshold on the magnetic field. These include models where the regeneration of the poloidal field takes place via the MHD instability of toroidal flux tubes (Sections 4.7 and 3.2.3). In such models, the transition from quiescent to active phases requires an external mechanism to push the field strength back above threshold. This can be stochastic noise (see, e.g., Schmitt et al., 1996), or a secondary dynamo process normally overpowered by the “primary” dynamo during active phases (see Ossendrijver, 2000aJump To The Next Citation Point). Figure 27View Image shows one representative solution of the latter variety, where intermittency is driven by a weak α-effect-based kinematic dynamo operating in the convective envelope, in conjunction with magnetic flux injection into the underlying region of primary dynamo action by randomly positioned downflows (see Ossendrijver, 2000a, for further details). The top panel shows a sample trace of the toroidal field, and the bottom panel a butterfly diagram constructed near the core-envelope interface in the model.

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Figure 27: Intermittency in a dynamo model based on flux tube instabilities (cf. Sections 3.2.3 and 4.7). The top panel shows a trace of the toroidal field, and the bottom panel is a butterfly diagram covering a shorter time span including a quiescent phase at 9.6 ā‰² t ā‰² 10.2, and a “failed minimum” at t ā‰ƒ 11 (figure produced from numerical data kindly provided by M. Ossendrijver).

The model does produce irregularly-spaced quiescent phases, as well as an occasional “failed minimum” (e.g., at t ā‰ƒ 11), in qualitative agreement with the solar record. Note here how the onset of a Grand Minimum is preceded by a gradual decrease in the cycle’s amplitude, while recovery to normal, cyclic behavior is quite abrupt. The fluctuating behavior of this promising class of dynamo models clearly requires further investigation.

5.6.5 Intermittency from time delays

Dynamo models exhibiting amplitude modulation through time-delay effects are also liable to show intermittency in the presence of stochastic noise. This was demonstrated in Charbonneau (2001Jump To The Next Citation Point) in the context of Babcock–Leighton models, using the iterative map formalism described in Section 5.4.2. The intermittency mechanism hinges on the fact that the map’s attractor has a finite basin of attraction (indicated by gray shading in Panel A of Figure 24View Image). Stochastic noise acting simultaneously with the map’s dynamics can then knock the solution out of this basin of attraction, which then leads to a collapse onto the trivial solution pn = 0, even if the map parameter remains supercritical. Stochastic noise eventually knocks the solution back into the attractor’s basin, which signals the onset of a new active phase (see Charbonneau, 2001, for details).

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Figure 28: Intermittency in a dynamo model based on the Babcock–Leighton mechanism (cf. Sections 3.2.4 and 4.8). The top panel shows a trace of the toroidal field sampled at (r,šœƒ) = (0.7,πāˆ•3). The bottom panel is a time-latitude diagram for the toroidal field at the core-envelope interface (numerical data from Charbonneau et al., 2004Jump To The Next Citation Point).

A corresponding behavior was subsequently found in a spatially-extended model similar to that described in Section 4.8 (see Charbonneau et al., 2004Jump To The Next Citation Point). Figure 28View Image shows one such representative solution, in the same format as Figure 27View Image. This is a dynamo solution which, in the absence of noise, operates in the singly-periodic regime. Stochastic noise is added to the vector potential A ˆeĻ• in the outermost layers, and the dynamo number is also allowed to fluctuate randomly about a pre-set mean value. The resulting solution exhibits both amplitude fluctuations and intermittency.

With its strong polar branch often characteristic of dynamo models with meridional circulation, Figure 28View Image is not a particularly good fit to the solar butterfly diagram, yet its fluctuating behavior is solar-like in a number of ways, including epochs of alternating higher-than-average and lower-than-average cycle amplitudes (the Gnevyshev–Ohl rule, cf. Panel E of Figure 22View Image), and residual pseudo-cyclic variations during quiescent phases, as suggested by 10Be data, cf. Panel B of Figure 22View Image. This later property is due at least in part to meridional circulation, which continues to advect the (decaying) magnetic field after the dynamo has fallen below threshold (see Charbonneau et al., 2004, for further discussion). Note also in Figure 28View Image how the onset of Grand Minima is quite sudden, while recovery to normal activity is more gradual, which is the opposite behavior to the Grand Minima in Figure 27View Image.

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