The Solar Cycle

Abstract

The solar cycle is reviewed. The 11-year cycle of solar activity is characterized by the rise and fall in the numbers and surface area of sunspots. A number of other solar activity indicators also vary in association with the sunspots including; the 10.7 cm radio flux, the total solar irradiance, the magnetic field, flares and coronal mass ejections, geomagnetic activity, galactic cosmic ray fluxes, and radioisotopes in tree rings and ice cores. Individual solar cycles are characterized by their maxima and minima, cycle periods and amplitudes, cycle shape, the equatorward drift of the active latitudes, hemispheric asymmetries, and active longitudes. Cycle-to-cycle variability includes the Maunder Minimum, the Gleissberg Cycle, and the Gnevyshev-Ohl (even-odd) Rule. Short-term variability includes the 154-day periodicity, quasi-biennial variations, and double-peaked maxima. We conclude with an examination of prediction techniques for the solar cycle and a closer look at cycles 23 and 24.

Electronic supplementary material: Supplementary material is available for this article at 10.1007/lrsp-2015-4.

Keywords: Solar activity; Solar cycle; Solar cycle prediction; Sunspots.

Figure 1:

Sunspot groups observed each year from 1826 to 1843 by Heinrich Schwabe (1844). These data led Schwabe to his discovery of the sunspot cycle.

Figure 2:

Monthly averages of the daily International Sunspot Number. This illustrates the solar cycle and shows that it varies in amplitude, shape, and length. Months with observations from every day are shown in black. Months with 1–10 days of observation missing are shown in green. Months with 11–20 days of observation missing are shown in yellow. Months with more than 20 days of observation missing are shown in red. [Missing days from 1818 to the present were obtained from the International daily sunspot numbers. Missing days from 1750 to 1818 were obtained from the Group Sunspot Numbers and probably represent an over estimate.]

Figure 3:

Boulder Sunspot Number vs. the International Sunspot Number at monthly intervals from 1981 to 2014. The average ratio of the two is 1.55 and is represented by the solid line through the data points. The Boulder Sunspot Numbers can be brought into line with the International Sunspot Numbers by using a correction factor k = 0.65 for Boulder.

Figure 4:

American Sunspot Number vs. the International Sunspot Number at monthly intervals from 1944 to 2014. The average ratio of the two is 0.97 and is represented by the solid line through the data points.

Figure 5:

Group Sunspot Number vs. the International Sunspot Number at monthly intervals from 1874 to 1995. The average ratio of the two is 1.01 and is represented by the solid line through the data points.

Figure 6:

Sunspot number revisions. Yearly sunspot numbers as reported by Wolf (1861) (red line), Wolf (1877b) (blue), Hoyt and Schatten (1998) (green), and by SILSO in 2013 (black). These sunspot numbers have disagreements as late as 1900.

Figure 7:

Smoothed RGO Sunspot Area vs. the International Sunspot Number at monthly intervals from May 1874 to December 1976. The two quantities are correlated at the 99.4% level with a proportionality constant of about 16.7.

Figure 8:

Smoothed USAF/NOAA Sunspot Area vs. the International Sunspot Number at monthly intervals from January 1977 to August 2014. The two quantities are correlated at the 99.1% level with a proportionality constant of about 11.2. These sunspot areas have to be multiplied by a factor of 1.49 to bring them into line with the RGO sunspot areas. Data obtained prior to cycle 23 are shown with filled dots, while data obtained after 1997 are shown with open circles.

Figure 9:

Sunspot area as a function of latitude and time. The average daily sunspot area for each solar rotation since May 1874 is plotted as a function of time in the lower panel. The relative area in equal area latitude strips is illustrated with a color code in the upper panel. Sunspots form in two bands, one in each hemisphere, which start at about 25 from the equator at the start of a cycle and migrate toward the equator as the cycle progresses.

Figure 10:

10.7cm radio flux vs. International Sunspot Number for the period of August 1947 to January 2014. Data obtained prior to cycle 23 are shown with filled dots while data obtained after 1997 are shown with open circles. The Holland and Vaughn formula relating the radio flux to the sunspot number is shown with the solid line. These two quantities are correlated at the 99.5% level.

Figure 11:

Daily measurements of the Total Solar Irradiance (TSI) from instruments on different satellites. The systematic offsets among measurements taken with different instruments complicate determinations of the long-term behavior.

Figure 12:

The PMOD (version d41-62-1204) composite TSI vs. International Sunspot Number. The filled circles represent smoothed monthly averages, with different colors representing data from the different instruments.

Figure 13:

The ACRIM composite TSI vs. International Sunspot Number. The filled circles represent smoothed monthly averages with different colors representing data from the different instruments.

Figure 14:

Hale’s Polarity Laws. A magnetogram from sunspot cycle 22 (1989 August 2) is shown on the left, with yellow denoting positive polarity and blue denoting negative polarity. A corresponding magnetogram from sunspot cycle 23 (2000 June 26) is shown on the right. Leading spots in one hemisphere have opposite magnetic polarity to those in the other hemisphere and the polarities flip from one cycle to the next.

Figure 15:

The Sun’s polar fields as reported by the Wilcox Solar Observatory. The smoothed field strength in their northernmost pixel is shown with the solid black line. The smoothed field strength in their southernmost pixel is shown with the dashed line. The smoothed sunspot number (scaled to fit on the figure) is shown with the red line.

Figure 16:

mpg-Movie (26984.0 KB) Still from a movie — A full-disc magnetogram from NSO/Kitt Peak used in constructing magnetic synoptic maps over the last two sunspot cycles. Yellow represents magnetic field directed outward. Blue represents magnetic field directed inward. (For video see appendix)

Figure 17:

A Magnetic Butterfly Diagram constructed from the longitudinally averaged radial magnetic field obtained from instruments on Kitt Peak and SOHO. This illustrates Hale’s Polarity Laws, Joy’s Law, polar field reversals, and the transport of higher latitude magnetic field elements toward the poles.

Figure 18:

Monthly M- and X-class flares vs. International Sunspot Number for the period of March 1976 to December 2013. These two quantities are correlated at the 95% level but show significant scatter when the sunspot number is high (greater than ∼ 100). Data obtained prior to cycle 23 are shown with filled dots, while data obtained after 1997 are shown with open circles.

Figure 19:

Monthly X-class flares and International Sunspot Number. X-class flares can occur at any phase of the sunspot cycle — including cycle minimum.

Figure 20:

Geomagnetic activity and the sunspot cycle. The geomagnetic activity index aa is plotted in red. The sunspot number (divided by five) is plotted in black.

Figure 21:

Geomagnetic activity index aa vs. Sunspot Number. As Sunspot Number increases the baseline level of geomagnetic activity increases as well.

Figure 22:

Cosmic Ray flux from the Climax Neutron Monitor and rescaled Sunspot Number. The monthly averaged neutron counts from the Climax Neutron Monitor are shown by the solid line. The monthly averaged sunspot numbers (multiplied by five and offset by 4500) are shown by the dotted line. Cosmic ray variations are anti-correlated with solar activity but with differences depending upon the Sun’s global magnetic field polarity (A+ indicates periods with positive polarity north pole, while A− indicates periods with negative polarity).

Figure 23:

Signal transmission for filters used to smooth monthly sunspot numbers. The 13-month running mean and the 12-month average pass significant fractions (as much as 20%) of signals with frequencies higher than one cycle per year. The 24-month FWHM Gaussian passes less than 0.3% of those frequencies and passes less than about 1% of the signal with frequencies of a half-cycle per year or higher.

Figure 24:

The left panel shows cycle periods as functions of Cycle Number. Filled circles give periods determined from minima in the 13-month mean, while open circles give periods determined from the 24-month Gaussian smoothing. Both measurements give a mean period of about 132 months with a standard deviation of about 14 months. The “Wilson Gap” in the periods between 125 and 134 months from the 13-month mean is shown with dashed lines. The right panel shows histograms of cycle periods centered on the mean period with bin widths of one standard deviation. The solid lines show the distribution from the 13-month mean while the dashed lines show the distribution for the 24-month Gaussian. The periods appear normally distributed and the “Wilson Gap” is well populated with the 24-month Gaussian smoothed data.

Figure 25:

The left panel shows cycle amplitudes as functions of cycle number. The filled circles show the 13-month mean maxima with the Group Sunspot Numbers while the open circles show the maxima with the International Sunspot Numbers. The right panel shows the cycle amplitude distributions (solid lines for the Group values, dotted lines for the International values). The Group amplitudes are systematically lower than the International amplitudes for cycles prior to cycle 12 and have a nearly normal distribution. The amplitudes for the International Sunspot Number are skewed toward higher values.

Figure 26:

The average of cycles 1 to 23 (thick red line) normalized to the average amplitude and period. The average cycle is asymmetric in time with a rise to maximum over 4 years and a fall back to minimum over 7 years. The 23 individual, normalized cycles are shown with thin black lines.

Figure 27:

The average cycle (solid line) and the Hathaway et al. (1994) functional fit to it (dotted line) from Eq. (6). This fit has the average cycle starting 4 months prior to minimum, rising to maximum over the next 54 months, and continuing about 18 months into the next cycle.

Figure 28:

The Waldmeier Effect. The cycle rise time (from minimum to maximum) plotted versus cycle amplitude for International Sunspot Number data from cycles 1 to 23 (filled dots) and for 10.7 cm radio flux data from cycles 19 to 23 (open circles). This gives an inverse relationship between amplitude and rise time shown by the solid line for the Sunspot Number data and with the dashed line for the radio flux data. The radio flux maxima are systematically later than the Sunspot number data, as also seen in Table 4.

Figure 29:

The Amplitude-Period Effect. The period of a cycle (from minimum to minimum) plotted versus following cycle amplitude for International Sunspot Number data from cycles 1 to 22. This gives an inverse relationship between amplitude and period shown by the solid line with Amplitude(n+1) = 380 − 2 × Period(n).

Figure 30:

The Maximum-Minimum Effect. The maximum of a cycle plotted versus minimum preceding the cycle given by the 13-month smoothed International Sunspot Number data from cycles 1 to 23. This gives a relationship between maximum and minimum shown by the solid line with Maximum(n) = 78 + 6 × Minimum(n).

Figure 31:

Top: Latitude positions of the sunspot area centroid in each hemisphere for each Carrington Rotation as functions of time from cycle start. Three symbol sizes are used to differentiate data according to the average of the daily total sunspot area for each hemisphere and rotation. Bottom: The centroids of the centroids in 6-month intervals are shown for large amplitude cycles (red line), medium amplitude cycles (green line), and small amplitude cycles (blue line). The exponential fit to the active latitude positions [Eq. (8)] is shown with the black dashed line and 2σ error bars.

Figure 32:

Top: Latitudinal widths of the sunspot area centroid in each hemisphere for each Carrington Rotation as functions of time from cycle start. Three symbol sizes are used to differentiate data according to the daily average of the sunspot area for each hemisphere and rotation. The centroids of the centroids in 6-month intervals are shown for large amplitude cycles (red line), medium amplitude cycles (green line), and small amplitude cycles (blue line). Bottom: Latitudinal widths as functions of total sunspot area with color coded symbols for cycle strength. The black dots with 2σ error bars show the data binned in 100 µHem intervals. The black line is given by Eq. 9.

Figure 33:

Normalized north-south asymmetry in four different activity indicators for individual Carrington rotations. Sunspot area is plotted in black. The Flare Index is shown in red. The number of sunspot groups is shown in green. The Magnetic Index is plotted in blue.

Figure 34:

Smoothed, normalized north-south asymmetry in sunspot area. The hemispheric asymmetry is shown by the black line while the total area scaled by 1/1000 is shown by the red line for reference.

Figure 35:

Smoothed monthly sunspot areas for northern and southern hemispheres separately. The difference between the two curves is filled in red if the north dominates or in blue if the south dominates.

Figure 36:

Active longitudes in sunspot area. The normalized sunspot area in 5° longitude bins is plotted in the upper panel (a) for the years 1878–2009. The dotted lines represent two standard errors in the normalized values. The sunspot area in several longitude bins meets or exceeds these limits. The individual cycles (12 through 23) are shown in the lower panel (b) with the normalized values offset in the vertical by the cycle number. Some active longitudes appear to persist from cycle to cycle.

Figure 37:

Active region tilt from SOHO/MDI magnetograms over cycle 23. Each black data point gives the tilt (angle between leading-polarity center of gravity and following-polarity center of gravity measured clockwise from the west to east line) for a NOAA active region on a given day as a function of its latitude. The red dots with 2σ error bars give the averages in 5° latitude bins.

Figure 38:

The Maunder Minimum. The yearly averages of the daily Group Sunspot Numbers are plotted as a function of time. The Maunder Minimum (1645–1715) is well-observed in this dataset.

Figure 39:

The Gleissberg Cycle. The best fit of cycle amplitudes to a simple sinusoidal function of cycle number is shown by the solid line (which includes the secular trend).

Figure 40:

Gnevyshev-Ohl Rule. The ratio of the odd-cycle sunspot sum to the preceding even-cycle sunspot sum is shown with the filled circles. The ratio of the odd-cycle amplitude to the preceding even-cycle amplitude is shown with the open circles.

Figure 41:

Short-term variations. The lower panel shows the daily International Sunspot Number (SSN) smoothed with a 24-rotation FWHM Gaussian. The upper panel shows the residual SSN signal smoothed with a 54-day Gaussian and sampled at 27-day intervals.

Figure 42:

Morlet wavelet transform spectrum of the bandpass-limited daily International Sunspot Number. Increasing wavelet power is represented by colors from black through blue, green, yellow, and red. The Cone-Of-Influence (outside of which the data isn’t long enough to give good measurements of wavelet power) is outlined by the white curves. Periods of 154 days are indicated by the horizontal red line.

Figure 43:

Predictions for cycles 22 and 23 using the modified McNish-Lincoln (M-M-L) auto-regression technique and the Hathaway, Wilson, and Reichmann (H-W-R) curve-fitting technique 24 months after the minima for each cycle.

Figure 44:

Ohl’s method for predicting cycle amplitudes using the minima in the smoothed aa index (panel a) as precursors for the maximum sunspot numbers of the following sunspot number maxima (panel b).

Figure 45:

A modification of Feynman’s method for separating geomagnetic activity into a sunspot number related component and an “interplanetary” component (panels a and b). The maxima in aa_I prior to minimum are well-correlated with the following sunspot number maxima (panel c).

Figure 46:

Thompson method for predicting sunspot number maxima. The number of geomagnetically disturbed days in a cycle is proportional to the sum of the maxima of that cycle and the next.

Figure 47:

Sunspot group size (area) distributions for cycles 22 and 23. The maximum sizes attained by active regions in the USAF database are used to count each active region in a size bin for its respective cycle. On the left, the bins extend from 10 to 200 µHem in 10 µHem increments. On the right, the bins extend from 100 to 2000 µHem in 100 µHem increments. The deficit of small spots (with areas of 10 and 20 µHem) in cycle 23 appears to be significant. Cycle 23 also appears to have fewer large active regions. While those deviations are well within the 1σ errors, there are consistently fewer numbers in all area bins above ∼ 700 µHem.