Robustness of Solar-Cycle Empirical Rules Across Different Series Including an Updated Active-Day Fraction (ADF) Sunspot Group Series

Abstract

Empirical rules of solar-cycle evolution form important observational constraints for the solar-dynamo theory. This includes the Waldmeier rule relating the magnitude of a solar cycle to the length of its ascending phase, and the Gnevyshev–Ohl rule clustering cycles to pairs of an even-numbered cycle followed by a stronger odd-numbered cycle. These rules were established as based on the “classical” Wolf sunspot number series, which has been essentially revisited recently, with several revised sets released by the research community. Here we test the robustness of these empirical rules for different sunspot (group) series for the period 1749 – 1996, using four classical and revised international sunspot-number and group sunspot-number series. We also provide an update of the sunspot-group series based on the active-day fraction (ADF) method, using the new database of solar observations. We show that the Waldmeier rule is robust and independent of the exact sunspot (group) series: its classical and $n+1$ (relating the length of $n$th cycle to the magnitude of ($n+1$)th cycle) formulations are significant or highly significant for all series, while its simplified formulation (relating the magnitude of a cycle to its full length) is insignificant for all series. The Gnevyshev–Ohl rule was found robust for all analyzed series for Solar Cycles 8 – 21, but unstable across the Dalton minimum and before it.

Appendix: Update of the Sunspot Group Numbers Using the ADF Method

The active-day fraction (ADF) method of the sunspot (group) number reconstruction was invented by Usoskin et al. (2016) and greatly upgraded by Willamo, Usoskin, and Kovaltsov (2017). It is based on the assumption that the “quality” of a solar observer is entirely defined by their observational threshold, rather than a scaling factor used in other methods. An important advantage of the ADF method is that it calibrates the observers directly to the reference series avoiding the daisy-chain calibration intrinsic to other methods. As a drawback, the ADF method tends to strongly overestimate the low-activity and moderately underestimate the high-activity cycles (Willamo, Usoskin, and Kovaltsov, 2018).

After the publication of the ADF-based reconstruction by Willamo, Usoskin, and Kovaltsov (2017), based on the sunspot-group observation database of Hoyt and Schatten (1998), an essential update was made with the upgrade and systematization of all known sunspot records, as collected in a database of Vaquero et al. (2016). Accordingly, we have revisited the ADF-based sunspot-group number reconstructed here, following exactly the algorithm described by Willamo, Usoskin, and Kovaltsov (2017), but using the new sunspot-number database (Vaquero et al., 2016). As the reference observer, we used the Royal Greenwich Observatory (solarscience.msfc.nasa.gov/greenwch.shtml), similarly to Willamo, Usoskin, and Kovaltsov (2017).

The resultant data-series for the period of 1749 – 1996 is shown in Figure 5a and tabulated in the Supplementary Information as monthly and annual sunspot-group series. The ratio of the (annual) series obtained here to that of Willamo, Usoskin, and Kovaltsov (2017) is shown in Figure 5b. It is kept close (within 10%) of unity after 1880, with small deviations being related to the Monte-Carlo procedure of the observers’ calibration. The ratio can, however, vary by up to 40% before the Dalton minimum. On the other hand, the ratio is consistent with unity within the error bars.

Figure 5

a) Monthly sunspot-group number reconstructed here using the ADF method and the updated solar-observation database (Vaquero et al., 2016). Uncertainties ($\pm 1\sigma$) are shown in gray. Tabulated values are provided in the Supplementary Information. b) Ratio of the annual values of the SGN obtained here to those from Willamo, Usoskin, and Kovaltsov (2017). The unity value is shown by the dashed line. The ratio values are not shown for the periods around solar-cycle minima (SNG<1).

This series supersedes the previous ADF-based SGN (Usoskin et al., 2016; Willamo, Usoskin, and Kovaltsov, 2017) and is used here for further analysis.