W.W. Hansen Experimental Physics Laboratory, Stanford University
Solar cycles are driven by a self-exciting dynamo that converts poloidal magnetic fields into azimuthal or toroidal fields that erupt as solar active regions and sunspots. Solar-cycle shapes seem to form a family of curves well characterized by a single parameter: SNMax, the maximum smoothed monthly sunspot number. Predicting the amplitude, shape, and duration of the next cycle thus concentrates on predicting SNMax for the cycle. The many empirical prediction methods that have been tried fall in two broad categories: statistical methods and precursor methods. The former assumes that the centuries-long time-series of sunspot numbers carries information about the underlying physics that can be exploited for forecasting. Precursor methods assume that some properties of the recent cycles, perhaps only part of the most recent, have predictive power for the next.
Schatten et al. suggested on assumed physical grounds (Babcock-Leighton model of the solar dynamo) that the magnetic field in the polar regions near the solar minimum would be a precursor proxy for the amount of sunspot activity in the following cycle, serving as a ‘seed’ for the dynamo when advected into the solar interior. Svalgaard et al. suggested using the average polar fields during the three-year interval preceding the solar minimum as the precursor value to regress against the amplitude of the following cycle.
Using the measurements at Wilcox Solar Observatory (WSO) of the solar axial magnetic dipole moment, DM, calculated as the difference between the north and south polar fields, we regress SNMax against the DM for Cycles 21-24, using both 3-year and 2-year averages of the DM prior to solar minima (Figure 1). Two regression fits have (likely fortuitously) very high coefficients of determination R2 of 0.99. The resulting predicted SN maximum comes to 128.
Figure 1| Smoothed monthly maximum sunspot number, SNMax, for Cycles 21-24 regressed against the (absolute) Dipole Moment averaged over three years before solar minimum (blue symbols) and over two years (violet symbols). Symbols of lighter shade are used for the more uncertain Cycle 21. Where symbols completely cover each other, they have been offset slightly for display purposes. The prediction for Cycle 25 is shown with red diamonds.
The predictions of Svalgaard et al. and Schatten were off by 6% overall. On top of that, there is uncertainty in how well the sunspot number represents actual solar activity. The SILSO data product lists a typical standard deviation of cycle-maximum of the sunspot number of 6%, for a combined ‘error’ of 8.5% or 11 sunspot units for a SN of 128. We shall round that to 10 SN-units as even the unit digit is uncertain.
That solar cycle prediction is still in its infancy is borne out by the extreme range of predictions of Cycle 25 (Ref ; see Figure 2 below) indicating that we have not made much progress since predictions were made of Cycle 24, which showed a similar spread (from half to double of actual value observed). With the widespread (from 50 to 233), someone or even several ones are bound to be ‘correct’, regardless of the possible incorrectness of the method used. The many non-overlapping error bars illustrate the folly of even assigning error bars to the predictions or, at least, to believe in them. Our prediction is shown by the yellow circle in the middle of the plot, its diameter being its error bar.
Figure 2| The 38 predictions of Solar Cycle 25 that had been registered by January 2020 (Adapted after Ref  with permission). Our prediction (128±10) is indicated by the yellow ‘sun’ in the center of the plot, near the average (123±21) of the 6 (now 7) precursor methods that seem to be preferred. The overall average is 132±47 (median 124). None of these numbers are substantially different, so one could perhaps just go with the “Wisdom of Crowds” (Aristotle, 350 BCE, “Politics”, III:xi; Ref ).
All predictions that we consider have the underlying assumption that the sun has not changed its behavior (its “spots” so to speak) on a timescale of a few centuries (the Maunder Minimum may be a possible violation of that assumption) and that there will be no such changes in the near future.
 Schatten, K. H., Scherrer, P. H., Svalgaard, L., & Wilcox, J. M. 1978, Geophys. Res. Lett., 5, 411
 Svalgaard, L., Cliver, E. W., & Kamide, Y. 2005, Geophys. Res. Lett., 32, L01104
 Schatten, K. H. 2005, Geophy. Res. Lett., 32, L21106
 Pesnell, W. D. 2020, 2020 Sun-Climate Symposium, Tucson, AZ, Jan. 2020 https://lasp.colorado.edu/media/projects/SORCE/meetings/2020/final/S6_01_Pesnell_SunClimate.pdf
 Galton, F. 1907, Nature, 75, 450