At any rate, the notion of a nicely regular 11/22-year cycle does not hold long upon even cursory scrutiny, as the amplitude of successive cycles is clearly not constant, and their overall shape often differs significantly from one cycle to another (cf. cycles 14 and 15 in Panel A of Figure 19). Closer examination of Figure 19 also reveals that even the cycle’s duration is not uniform, spanning in fact a range going from (cycle 2) to nearly (cycle 4). These amplitude and duration variations are not a sunspot-specific artefact; similar variations are in fact observed in other activity proxies with extended records, most notably the radio flux (Tapping, 1987), polar faculae counts (Sheeley Jr, 1991), and the cosmogenic radioisotopes and (Beer et al., 1991; Beer, 2000).
The various incarnations of the sunspot number time series (monthly SSN, 13-month smoothed SSN, yearly SSN, etc.) are arguably the most intensely studied time series in astrophysics, as measured by the number of published research paper pages per data points. Various correlations and statistical trends have been sought in these datasets. Panel D of Figure 19 and Panel E of Figure 19 present two such classical trends. The “Waldmeier Rule”, illustrated in Panel D of Figure 19, refers to a statistically significant anticorrelation between cycle amplitude and rise time (linear correlation coefficient ). A similar anticorrelation exists between cycle amplitude and duration, but is statistically more dubious (). The “Gnevyshev-Ohl” rule, illustrated in Panel E of Figure 19, refers to a marked tendency for odd (even) numbered cycles to have amplitudes above (below) their running mean (blue line in Panel E of Figure 19), a pattern that seems to have held true without interruption between cycles 9 and 21 (see also Mursula et al., 2001).
A number of long-timescale modulations have also been extracted from these data, most notably the so-called Gleissberg cycle (period ), but the length of the sunspot number record is insufficient to firmly establish the reality of these periodicities. One must bring into the picture additional solar cycle proxies, primarily cosmogenic radioisotopes, but difficulties in establishing absolute amplitudes of production rates introduce additional uncertainties into what is already a complex endeavour (for more on these matters, see Beer, 2000; Usoskin and Mursula, 2003). Likewise, the search for chaotic modulation in the sunspot number time series has produced a massive literature (see, e.g., Feynman and Gabriel, 1990; Mundt et al., 1991; Carbonell et al., 1994; Rozelot, 1995, and references therein), but without really yielding firm, statistically convincing conclusions, again due to the insufficient lengths of the datasets.
The aim in this section is to examine in some detail the types of fluctuations that can be produced in the various dynamo models discussed in the preceding section10. After going briefly over the potential consequences of fossil fields (Section 5.2), dynamical nonlinearities are first considered (Section 5.3), followed by time-delay effects (Section 5.4). We then turn to stochastic forcing (Section 5.5), which leads naturally to the issue of intermittency (Section 5.6).
© Max Planck Society and the author(s)