On the long-term secular increase in sunspot number

Correlated with the maximum amplitude (R _max) of the sunspot cycle are the sum (R _sum) and the mean (R _mean) of sunspot number over the duration of the cycle, having a correlation coefficient r equal to 0.925 and 0.960, respectively. Runs tests of R _max, R _sum, and R _mean for cycles 0–21 have probabilities of randomness P equal to 6.3, 1.2, and 9.2%, respectively, indicating a tendency for these solar-cycle related parameters to be nonrandomly distributed. The past record of these parameters can be described using a simple two-parameter secular fit, one parameter being an 8-cycle modulation (the so-called Gleissberg cycle or long period) and the other being a long-term general (linear) increase lasting tens of cycles. For each of the solar-cycle related parameters, the secular fit has an r equal to about 0.7–0.8, implying that about 50–60% of the variation in R _max, R _sum, and R _mean can be accounted for by the variation in the secular fit.Extrapolation of the two-parameter secular fit of R _max to cycle 22 suggests that the present cycle will have an R _max = 74.5 ± 49.0, where the error bar equals ± 2 standard errors; hence, the maximum amplitude for cycle 22 should be lower than about 125 when sunspot number is expressed as an annual average or it should be lower than about 130 when sunspot number is expressed as a smoothed (13-month running mean) average. The long-term general increase in sunspot number appears to have begun about the time of the Maunder minimum, implying that the 314-yr periodicity found in ancient varve data may not be a dominant feature of present sunspot cycles.