5.1 The observational evidence: An overview

Panel A of Figure 19

shows a time series of the so-called Zürich sunspot numbers, starting in the mid-eighteenth century and extending to the present. The thin line is the monthly sunspot number, and the red line a 13-month running mean thereof. The 11-year sunspot cycle is the most obvious feature of this time series, although the period of the underlying magnetic cycle is in fact twice that (sunspot counts being insensitive to magnetic polarity). Since sunspots are a surface manifestation of the toroidal magnetic flux system residing in the solar interior, cycle-to-cycle variations in sunspot counts are usually taken to indicate a corresponding variation in the amplitude of the Sun’s dynamo-generated internal magnetic field. As reasonable as this may sound, it remains a working assumption; at this writing, the process via which the dynamo-generated mean magnetic field produces sunspot-forming concentrated flux ropes is not understood. One should certainly not take for granted that a difference by a factor of two in sunspot count indicates a corresponding variation by a factor of two in the strength of the internal magnetic field; it is not even entirely clear whether the two are monotonically related.

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 $9 yr$ (cycle 2) to nearly $14 yr$ (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 $10.7 cm$ radio flux (Tapping, 1987 ), polar faculae counts (Sheeley Jr, 1991 ), and the cosmogenic radioisotopes $14 C$ and $10 Be$ (Beer et al., 1991 ; Beer, 2000 ).

Figure 19:

Fluctuations of the solar cycle, as measured by the sunspot number. Panel A is a time series of the Zürich monthly sunspot number (with a 13-month running mean in red). Cycles are numbered after the convention introduced in the mid-nineteenth century by Rudolf Wolf. Note how cycles vary significantly in both amplitude and duration. Panel B is a portion of the $10Be$ time series spanning the Maunder Minimum (data courtesy of J. Beer). Panel C shows a time series of the yearly group sunspot number of Hoyt and Schatten (1998 ) (see also Hathaway et al., 2002 ) over the same time interval, together with the yearly Zürich sunspot number (purple) and auroral counts (green). Panels D and E illustrate the pronounced anticorrelation between cycle amplitude and rise time (Waldmeier Rule), and alternance of higher-than-average and lower-that-average cycle amplitudes (Gnevyshev-Ohl Rule, sometimes also referred to as the “odd-even effect”).

Equally striking is the pronounced dearth of sunspots in the interval 1645-1715 (see Panel C of Figure 19

); this is not due to lack of observational data (see Ribes and Nesme-Ribes, 1993

; Hoyt and Schatten, 1996 ), but represents instead a phase of strongly suppressed activity now known as the Maunder Minimum (Eddy, 1976 , 1983 , and references therein). Evidence from cosmogenic radioisotopes indicates that similar periods of suppressed activity have taken place in ca. 1282-1342 (Wolf Minimum) and ca. 1416-1534 (Spörer Minimum), as well as a period of enhanced activity in ca. 1100-1250 (the Medieval Maximum).

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 $r = - 0.68$ ). A similar anticorrelation exists between cycle amplitude and duration, but is statistically more dubious ( $r = - 0.37$ ). 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 $= 88 yr$ ), 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).