2.1 Cycle parameters as precursors and the Waldmeier effect

The simplest weather forecast method is saying that “tomorrow the weather will be just like today” (works in about 2/3 of the cases). Similarly, a simple approach of sunspot cycle prediction is correlating the amplitudes of consecutive cycles. There is indeed a marginal correlation, but the correlation coefficient is quite low (0.35). The existence of the correlation is related to secular variations in solar activity, while its weakness is due to the significant cycle-to-cycle variations.

A significantly better correlation exists between the minimum activity level and the amplitude of the next maximum (Figure 6View Image). The relation is linear (Brown, 1976Jump To The Next Citation Point), with a correlation coefficient of 0.72 (if the anomalous cycle 19 is ignored – Brajša et al., 2009Jump To The Next Citation Point; see also Pishkalo, 2008). The best fit is

Rmax = 67.5 + 6.91Rmin . (10 )
Using the observed value 1.7 for the SSN in the recent minimum, the next maximum is predicted by this “minimax” method to reach values around 80, with a 1 σ error of about ± 25.
View Image

Figure 6: Monthly smoothed sunspot number R at cycle maximum plotted against the values of R at the previous minimum (left) and 2.5 years before the minimum (right). Cycles are labeled with their numbers. The blue solid line is a linear regression to the data; corresponding correlation coefficients are shown. In the left hand panel, cycle 19 was considered an outlier.

Cameron and Schüssler (2007Jump To The Next Citation Point) point out that the activity level three years before the minimum is an even better predictor of the next maximum. Indeed, playing with the value of time shift we find that the best correlation coefficient corresponds to a time shift of 2.5 years, as shown in the right hand panel of Figure 6View Image (but this may depend on the particular time period considered, so we will refer to this method in Table 1 as “minimax3” for brevity). The linear regression is

Rmax = 41.9 + 1.68R (tmin − 2.5). (11 )
For cycle 24 the value of the predictor is 16.3, so this indicates an amplitude of 69, suggesting that the upcoming cycle may be comparable in strength to those during the Gleissberg minimum at the turn of the 19th and 20th centuries.

As the epoch of the minimum of R cannot be known with certainty until about a year after the minimum, the practical use of these methods is rather limited: a prediction will only become available 2 – 3 years before the maximum, and even then with the rather low reliability reflected in the correlation coefficients quoted above. In addition, as convincingly demonstrated by Cameron and Schüssler (2007Jump To The Next Citation Point) in a Monte Carlo simulation, these methods do not constitute real cycle-to-cycle prediction in the physical sense: instead, they are due to a combination of the overlap of solar cycles with the Waldmeier effect. As stronger cycles are characterized by a steeper rise phase, the minimum before such cycles will take place earlier, when the activity from the previous cycle has not yet reached very low levels.

The same overlap readily explains the Rmax –tcycle,n correlations discussed in Section 1.3.3. These relationships may also be used for solar cycle prediction purposes (e.g., Kane, 2008) but they lack robustness. For cycle 24 the Rmax –tcycle,−1 correlation, as formulated by Hathaway (2010bJump To The Next Citation Point) predicts R = 80 max while the methods used by Solanki et al. (2002) yield values ranging from 86 to about 110, depending on the relative weights of tcycle,−1 and tcycle,− 3. The forecast is not only sensitive to the value of n used but also to the data set (relative or group sunspot numbers) (Vaquero and Trigo, 2008).

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