Timo Niroma:
Sunspots: Searching supercycles by smoothing.

PART 6.
Sunspot cycles and supercycles and their tentative causes.

  • 6. Searching supercycles by smoothing.
  • PART 7: Summary of supercycles and a hypercycle of 2289 years.
  • - Short supercycles.
    - Supercycles from 250 years to a hypercycle of 2289 years.
    - The long-range change in magnitudes.
    - Stuiver-Braziunas analysis: 9000 years?

  • PART 8: Organizing the cycles into a web.

  • PART 6. SUNSPOTS: Searching supercycles by smoothing.

    6. Searching supercycles by smoothing

    6.1. Smoothing sunspot averages in 1768-1992 by one basic cycle.

    Smoothing the sunspots from years 1762 to 1998 with the mean 133 months gives two superminima and three supermaxima.

    FIGURE 4





























    Of the two superminima the first one is by far the lowest. The Wolf number falls below 35 1795.4, and below 30 already 1796.3, but then it takes 9 years to go below 25, which happens 1805.1. Then 20 is rapidly crossed 1807.8, and the Wolf value is 15 or below from 1810.3 to 1811.5. The lowest value reached is 14.5 and occurs in 1810.5. This is most probably the lowest value since the Maunder minimum whose superminimum was about 1690 or 120 years earlier. 1810 is also the only totally spotless year since the Maunder minimum.

    The Wolf number exceeds 20 1815.1, but 25 is exceeded only 9 years later or 1824.0, but 30 is exceeded already 1825.3 and 35 1826.9.

    We can divide this superminimum into three phases, that all last 7-8 years, or together 22 years or one Hale cycle. The first phase begins in 1798 and ends in 1805. During whole this period the smoothed Wolf number is 25 or 26. Then follows the very low phase from 1806 to 1814 when the value drops from 24 to 15 (1810-1811) and slightly increases back to 19. The third phase begins in 1815 and ends in 1822. During this period the smoothed Wolf value is 20 or 21. After that begins a rapid rise.

    The value reached in 1834 is the highest value for 108 years, it is broken only in 1942.

    The second superminimum is not so dramatic as the first one, which in its turn was not so dramatic as the Maunder minimum. We can divide this superminimum into four phases. There is a precursory phase which begins in 1878 and ends in 1886. During whole this period the smoothed Wolf number is either 34 or 35. A temporary rise begins in 1886 and ends in 1897/1898. Then begins a rapid fall that reaches its lowest value of 28.2 in 1901.3. The recovery is as rapid until it reaches the value of 32 in 1903. Then begins the phase four that lasts until 1911, when the increase accelerates. During this period the smoothed Wolf number is either 33 or 34.

    During the period from 1878 to 1928 the smoothed Wolf value does not exceed 48. So the minimum appears 23 years after the start and 27 years before the end of the low values. This relation 46:54 is the same as the asymmetry during the previous low. It lasted from 1798 to 1824 using 12 years to fall and 14 years to rise. The relation 46:54 appears in so many instances in the ups and down in Sun's behavior that we cannot accept the chance hypothesis on statistical basis regardless of which criteria we use.

    Both these periods are cold periods, as was also the Maunder minimum. 1810's has been a cold decade not only in Europe but also in at least North America. There are continuous temperature data from Helsinki since 1829. For reasons unknown January seems to be the most sensitive single month to the sunspot activity, at least in Finland. From 1829 to 1893 the January mean was below minus 10 degrees C 14 times (22%). Especially two occasions are worth mentioning. 1848-1850 there were three Januaries in row in this category and the two coldest Januaries occur successively in 1861 and 1862. Between 1841 and 1877 there is a continuous fall in the smoothed Wolf number from 64 to 36. Between 1894 and 1939 there is no January this cold. This seems to indicate that a value of 30 is not in itself enough to cause a cold spell, but rather the fall is the causative agent. However 15 seems to be enough in itself. The cold Januaries in 1940-1942 (all below 10 degrees) indicate the farewell to the cold spell. The Wolf number exceeds 65 already in 1941 and 70 in 1942.

    This second superminimum is considerably higher, with its minimum of 28.2 Wolfs, than the previous one with the value of 14.5. The interval between the superminima is 90.8 years. The previous superminimum, the minimum of the Maunder minimum, was about 1690. This makes the interval between these superminima as 210 years.

    The supermaxima with values 80, 68, and 96 occur in 1783.3, 1834.7, and 1954.1. The intervals between the supermaxima are 51.4 and 119.4 years. Thus the interval between the two extreme supermaxima is 170.8 years. Now we have for the onset of low values a period of 210 years and for the onset of high values 170 years. This is in good agreement with what we observed when searching for the limits for the 200-year cycle. This may even give added data to the oscillation of this supercycle, if the onset of low and high periods have different intervals. It may explain why some minima are more low than other minima, for example the deep dive of the Maunder or the spotless year 1810 or why some maxima are higher than others, for example four of the maxima of 20th century exceed 150, when during the 19th century none did so.

    If we held 65 (arbitrarily) as the low limit for a high period, the first one in our data lasted from 1773 to 1792 (with a temporary break in 1778-1780), the second one was short and low lasting only from 1833 until the beginning of 1836, but the third one has been very long-lasting and high one. 65 Wolfs was exceeded already in late 1941 and was temporarily broken only in 1965. After having lasted two Jovian years, it was interrupted during the cycle 20 (which lasted one Jovian year from 1964 to 1976). There was a new onset in 1974 which has had its highest value until now in 1985.1 (91 Wolfs). The mid-year smoothed values have since been:

    1985 90
    1986 89
    1987 84
    1988 79
    1989 75
    1990 73
    1991 72
    1992 73
    1993 77
    1994 76

    The rapid fall, despite of the high values, seemed even now be more decisive factor concerning the temperature than the high level, because the all-time low (since 1829) for January in Helsinki, -14 and -16.5 were reached in 1985 and 1987, respectively (170 years after the previous supercold). But when the high standstill began in 1989, the Januaries have been record-breaking warm (plus 0.5 in 1989) from 1988 to 1998 (with the exception of 1995, the coldest one has been -3 degrees).

    6.2. Smoothing by the Hale cycle

    Using the Hale cycle (266 months or 22.2 years) as the smoothing value, we get rather similar results as with one cycle. Hale cycle consists of two consecutive cycles and is regarded by some as the basic cycle because every other 11-cycle has an opposite polarity compared with the previous one. However we only know this from the middle of the 19th century so we can't say, if it is a permanent feature or does it hold also in such instances when the Sun is very unactive as in 1810 and during the Maunder minimum's minimum in 1690's.

    We have two superminima and two supermaxima when the years 1762-1997 are smoothed to years 1774-1986/87.

    The first, deeper superminimum begins when the smoothed Wolf number goes below 35 in 1800.0 and below 30 in 1801.0. 25 is passed-by in 1802.1 and 20 is reached already in 1804.7- 1804.8 but then there is a small temporary rise. The smoothed Wolf number finally goes below 20 in 1813.8 and reaches its lowest value of 17.9 from 1816.0-1816.4. Tambora is not needed the explain the cold 1816, but of course it can have helped to make it even colder.

    The rise is also rapid: 20 is exceeded in 1818.2, 25 in 1820.0, 30 in 1824.7 and 35 in 1825.9. So we are below 20 4.4 years, below 25 17.9 years, below 30 23.7 years and below 35 25.9 years.

    The second superminimum is as restless as with the 11-years smoothing. The smoothed Wolf number goes temporarily below 37 already in 1885-1886, a second attemptoccurs in 1894-1897 when we are temporarily below 35, but then we are again in the values 37-38 until finally in 1903 the 35 is broken and the lowest value, 30.6, is reached in 1906.0. The rise to 36-37 happens rapidly, but is maintained in 1909-1913.

    So all in all we have here a 110-years low in Sun and temperature on Earth from about 1800 to 1913. Both these minima last some 26-28 years. With spectrum analysis we can get a 27-year weather cycle in the northern hemisphere since about the middle of the 19th century. The intervening 60 years between the two cold superminima is not high. There is also a 60-year pattern in the weather.

    The first supermaximum occurs in 1777-1785, when the smoothed Wolf number exceeds 66. The highest years 1779 and 1780 with 74.0 reached in 1779.9. With Hale smoothing the rapid rise to today's highs begins in 1946-1947 when in succession 70, 75 and 80 are exceeded in one and a half years. The maximum number one with a value of 84.0 is reached in 1949 and after a temporary low of 6 Wolfs reached again during the whole latter part of 1957. Then it drops to 74-75 in 1962-1966 and has a fall from 1967 to 1970 when the value drops to 69-70 until 1977 when a new rise begins. The value in mid-1987 is already 75.

    The onset between these supermaxima is 170 years.

    6.3. Smoothing by the Gleissberg cycle

    If we smooth by the Gleissberg cycle (using a value of 930 months or 77.5 years) we get from 1762-1998 the years 1801-1959.

    FIGURE 5. Wolf sunspot numbers smoothed by one Gleissberg cycle.




























    FIGURE 6. Wolf sunspot values smoothed by half a Gleissberg cycle.





























    First we notice that the whole period from the beginning (1801) until 1917 is very even and below 53 Wolfs during the whole period. Two times the value goes below 45. The first period is from 1817 to 1847 that has its low 39.7 from 1830.0 to 1830.9. The period from 1834 to 1845 has a stable value of 43- 44. A temporary onset to a little higher value begins slowly after that. The relation between the fall and rise is again 46:54 if we compare the same level.

    Second time the smoothed Wolf value goes below 45 in 1876 and rises above only in 1917. The value first oscillates between about 43.5 and 45 from 1876 until 1884, when a slow decrease in the value begins from 45 until the bottom value of 41.8-41.9 is reached. And this value is steadily maintained from 1892 until 1898. Then a slow rise begins, 43 is exceeded in 1908, 44 in 1910 and 45 in 1917.

    A slow but accelerating rise begins. 53 is reached in 1928, 55 in 1930. After that the smoothed Wolf number is 55-56 throughout till 1939/40, when there begins a new rise. The value climbs to about 61.5 in 1943 and the value is maintained between 61.5 and 62.5 until 1950, when the third rise begins. This time high is reached 1954.5 and the value reached, 69.1, is maintained until 1955.0. The last year we can achieve at the moment with this smoothing is 1959, which shows of value of little over 66.

    If the sunspots are the main reason for the socalled greenhouse warming, the Gleissberg cycle shows the connection most clearly: below 53 from at least 1801 until 1927 and above 61 from 1943 as long as we can measure. The warmest Julies since 1829 in Helsinki were in the 1930's, during the rise from low to high. The winters were warm in the beginning of 1930's and record- cold in the beginning of 1940's. The warmth record of July has not been broken, but the cold winters lost their record in 1985 and 1987, and the warm winters in 1990's.

    6.4. Double smoothing by the sunspot cycle and one Jovian year

    FIGURE 7

















    If we smooth the data both by one sunspot cycle (133 months) and one Jovian year (142 months) we get again two superminima, a short and deep from 1786 to 1838 and a long but shallow from 1838 to 1939. The smoothed Wolf value is mostly below 60 during these 52 and 101 year periods. The minima occur in 1813 and 1904 or 91 years apart. The former minimum reaches 20.9 Wolfs in 1812-1814 and the latter one 33.4 Wolfs in 1904-1905.

    The earlier maxima occur in 1785-1786 and 1838, the former reaching a value of 72 Wolfs and the latter one 63 Wolfs. The on-going supermaximum began in 1940, when the smoothed Wolf value exceeded 60 and has since been above it. There was a peak of 87 Wolfs in 1957 and after a low, reaching its bottom in 1970 (63 Wolfs), the smoothed value has been since then in a constant rise reaching 81 Wolfs in 1983-1984, lowering to 80 in 1985-1986 and 79 in 1987.

    During the whole period from 1774 to 1987 the Wolf value has exceeded 70 only in 1785-1788, 1946-1966 and since 1976.

    6.5. Calculating supercycles by omitting the cycle minima or taking into account only the active parts of the cycle.

    I have been worried about the agreed-upon character of the definition of the cycle minimum. Especially worried I am about the minima, that contain months, whose Wolf value is very low or more specifically about months whose Wolf value is, say below 10.

    But why be concerned about these zeros and near-zeros when searching for supercycles. Let's omit them altogether! And one can go further: one can omit all low values, because R(M) seems to be nearly as good an indicator as R in the area of supercycles. By experimenting a little, I observed that by omitting the months, whose Wolf number is below 10, there appear similar supercycles as by smoothing with about one or two sunspot cycles. The supercycles can be deepened or shallowed by changing the limiting Wolf number, but there is one advantage as compared to smoothing: the lengths of the supercycles do not depend on the smoothing length chosen.

    TABLE 43. Cycles derived of their minimum part.

     1. number of cycle
     2. year of minimum
     3. length of cycle (from minimum to minimum)
     4. length of cycle (from maximum to maximum)
     5. number of zero Wolf months
     6. interval between first and last zero months             
     7. number of months below 10 Wolfs
     8. second month below 10 Wolfs
     9. second last month below 10 Wolfs
    10. difference between 9 and 8 or the length of minimum
    11. length of active cycle with the minimum omitted (end of min this cycle minus beginning of min of the next)
    
    1.     2.   3.   4.  5. 6. 7.     8.     9. 10.   11.
    
    (1 1755.2 11.3  8.2  3 1.5 19 1754.0 1756.5 2.5 10.0 inaccurate) 
     2 1766.5  9.0  8.7  -   -  7 1766.5 1766.8 0.3  7.9
     3 1775.5  9.2  9.7  1   - 12 1774.7 1776.3 1.6  7.8
     4 1784.7 13.6 17.1  -   -  9 1784.1 1785.1 1.0 11.5 
     5 1798.3 12.1 11.2  6 1.9 37 1796.6 1800.3 3.7  7.4
     6 1810.4 12.9 13.5 27 5.0 65 1807.7 1814.8 7.1  5.4
     7 1823.3 10.6  7.3 17 2.9 46 1820.2 1825.0 4.8  7.7
     8 1833.9  9.6 10.9  -   - 17 1832.7 1834.6 1.9  8.6
     9 1843.5 12.5 12.0  -   -  9 1843.2 1844.5 1.3 11.0 
    10 1856.0 11.2 10.5  2 0.7 23 1855.4 1857.1 1.7  9.6
    11 1867.2 11.7 10.5  1   - 14 1866.7 1867.7 1.0  8.3
    12 1878.9 10.7 10.2  2 0.8 35 1876.0 1879.7 3.7  7.2
    13 1889.6 12.1 12.9  -   - 37 1886.9 1890.9 4.0  7.6
    14 1901.7 11.9 10.6  4 1.0 35 1898.5 1903.0 4.5  7.9
    15 1913.6 10.0 10.8  3 1.4 45 1910.9 1914.6 3.7  7.9
    16 1923.6 10.2  9.0  -   - 19 1922.5 1924.2 1.7  8.4
    17 1933.8 10.4 10.1  -   - 23 1932.6 1934.9 2.3  8.9
    18 1944.2 10.1 10.4  -   -  8 1943.8 1944.5 0.7  9.0
    19 1954.3 10.6 11.0  -   - 17 1953.5 1955.0 1.5  9.4
    20 1964.9 11.6 11.0  -   - 10 1964.4 1965.4 1.0 10.0
    21 1976.5 10.3 11.0  -   -  9 1975.4 1976.9 1.5  9.1 
    22 1986.8  9.8  9.5  -   -  8 1986.0 1987.0 1.0  9.1 
    23 1996.6    ? 10.8  -   - 11 1996.1 1997.1 1.0  9.0 (compromise min)
    24      ?    ?    ?  ?   ?  ? 2006.1      ?   ?  
    

    For safety reasons (the subjective character of the Wolf number) the first and the last below-10 months are ignored. Similarly I have ignored months, whose Wolf number temporarily exceeds 9 during the minimum.

    We have here three whole supercycles and the beginning of a fourth one. The whole supercycles contain 4, 6 and 12 cycles, if limiting ones are counted twice or 2, 4 and 10 cycles, if omitted. A very similar set of supercycles, especially the shorter and deeper from cycle 4 to cycle 9 and the longer and shallower from cycle 9 to cycle 20 is produced by smoothing the data with one sunspot cycle or with the Hale cycle. Notice how neatly the active length of the cycle (actually this is what we get by omitting the low months) first rapidly descends and then more slowly ascends in both cases, only the average speed of the latter cauldron is twice that of the former.

    The first supercycle lasts about 30 years, the second about 60 years, and the third about 120 years. This makes altogether 210 years. The asymmetry of the cauldrons again reflects the asymmetry so basic to all cyclicity in the sun.

    There is one curious thing about the minima, when calculated by the 10 Wolf limits. They are either short (at most 2.3 years) or long (at least 3.7 years), there are no minima with the intermediate length of about 3 years, as indicated by a table of the cycles 2-23:

    TABLE 44. The lengths of the minima defined by limits of 10 Wolfs.

    Second last row: years
    Last row: tenths of years
    
           x
           x
           x                          x
           x    x x                   x
    x   x  x  x xxx x   x             x  x    x  x                      x
    000000011111111112222222222333333333344444444445555555555666666666677
    345678901234567890123456789012345678901234567890123456789012345678901
    

    In the second and third cauldrons, the minima exceed 3.7 years whenever the unveiled cycle length is below 8 years. The minima are below 1.9 years whenever the cycle length is above 8 years, except with the cycle 17, whose minimum lasted 2.3 years and possibly the cycle 1 with a minimum of about 2.5 years. The cycles 1 and 17 have one thing in common: their minima is near the Jovian aphelion.

    There is a further curiosity about the relation between the minima and the unveiled cycle lengths except this quantum jump: the short minima are in reverse linear relationship to the cycle lengths as one should expect by definition, but the long minima are not, as shown by cycles 4-5 and 7-22:

    TABLE 45. The minimum and the active length crosstabulated.

     7.1- 7.5                               xx          
     7.6- 8.0                               x  x    x  x
     8.1- 8.5    x      x               
     8.6- 9.0 x           x   x                            
     9.1- 9.5    x    xx
     9.6-10.0    x      x 
    10.1-10.5         
    10.6-11.0       x
    11.1-11.5    x       
    years     000111111111122222222223333333333444444444  
    10ths     789012345678901234567890123456789012345678 
    

    So we can classify our data:

    TABLE 46. Supercycles

    1. group
    2. cycles
    3. number of cycles
    4. from minimum to mimimum
    5. length of group in years
    6. nearest supermaximum of the end of the group / smoothed by 11.1 years
    7. nearest supermaximum of the end of the group / smoothed by Hale cycle
    8. the limits of the cycle lengths
    9. months of zero Wolf per cycle
    10. the smoothed R(max) of the supermaxima in cols. 6/7
    
    1.    2. 3.        4.  5.   6.   7.    8.9.   10.
    
    1.  1- 4  4 1755-1784  30 1783 1780  9-13 1 80/73
    2.  4- 9  6 1784-1843  59 1835 1840  9-13 8 68/63
    3.  9-20 12 1843-1964 121 1954 1958 10-12 1 96/89
    4. 20- ?  ? 1964-   ?   ?    ?    ?     ? ?     ?
    

    In group 1 the minimum goes from the Jovian aphelion to the perihelion, in group 2 spirals around the perihelion, and in group 3 goes almost a full circle from perihelion to perihelion.

    Go to the
    beginning of the web of all cycles.

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    beginning of the summary of supercycles and the 2289-year cycle.

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    beginning of this part.

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    beginning of the 200-year supercycle.

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    beginning of the basic cycles to supercycles.

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    beginning of the minima, maxima and medians.

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    beginning of the avg. influence of Jupiter.

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    beginning of the sunspots.

    Comments should be addressed to timo.niroma@pp.inet.fi