The solar butterfly diagram: a low-dimensional model

The solar butterfly diagram: a low-dimensional model Thierry Dudok de Wit OSUC, University of Orléans With special thanks to the instrument teams (SIDC, USAF, Royal Greenwich Observatory)

Butterfly basics The solar butterfly diagram: from a low-dimensional model to new proxies of solar activity The solar butterfly diagram: from a low-dimensional model to new proxies of solar activity Sunspot area = proxy for toroidal magnetic field (flux emergence) 2

What the butterfly diagram tells us Distinguish between different dynamo types M. Schüssler and D. Schmitt: Butterfly diagram and dynamo models Dynamo model Schüssler and Schmitt, A&A (24) 4 Latitude-time diagrams Hathaway, Solar Physics (211) Large Cycles Area > 1 µhem Medium Cycles Area > 5 µhem Latitude (Degrees) 3 2 1 Small Cycles Area > 2 µhem -4-2 2 4 6 8 Time (Years from Max) 3

What the butterfly diagram tells us 3 6 6 15 Latitude [Degrees] 6 1 3 3. 2.5 2 5 4 3 6 199 6 Location of bright points and delta spots The Astrophysical 792:12 (19pp), 214 September 1 9 [McIntosh et al. ApJJournal, (214)] 1995 2 year 25 1.5 21 2. 1. McIntosh et al. (c) Merged SOHO/MDI and SDO/HMI 1 25Mm MRoI g-nodes (a) Merged SOHO/EIT 195Å and SDO/AIA 193Å EUV Brightpoints 9 9 Daily Average EUV BP Density 6 6 Latitude[Degrees] [Degrees] Latitude 2 1 log1 Mm latitude [deg] Look for anomalies or (b) Merged SOHO/MDI and SDO/HMI MRoI 9 2 long-range effects 3 3 3 3 6 6 MPS 27/1/214 3 4 9 9 9 1996 9 Daily Average EUV BP 3 1998 2 22 SOHO/MDI 24 and SDO/HMI 26 MRoI 28 (b) Merged Time [Years] 21 212 Daily Average MRoI Node Density Latitude [Degrees 4 5 4 3 2 1 5 4 3 2 1 4

The real physics is in the magnetic field We prefer the sunspot number to the magnetic field because it gives access to historical reconstructions Greenwich catalogue: 1874 older data (Arlt, Vaquero): ~161 5

Motivation Main question : what additional insight does the latitudinal distribution of the sunspot number/area give us? 6

The Sun as a dynamical system With the sunspot number only N(t) Consider the Sun as 1D system Description d N(t) =f(n,t) dt Described by an Ordinary Differential Equation (ODE) 7

The Sun as a dynamical system With the sunspot number only N(t) Consider the Sun as 1D system Description d @ N(t) =f(n,t) dt Described by an Ordinary Differential Equation (ODE) With the latitudinal distribution of the sunspot number N(t,θ) Consider the Sun as a spatially-extended system Description N(t, ) =f(n,t) @t Described by a Partial Differential Equation (PDE) Much more complex 8

How to go from a PDE to an ODE Many attempts to reduce spatially-extended systems to simpler ones: Poincaré sections, separable solutions, Our approach: look for separable solutions N(t, ) = X k S k (t) M k ( )+ (t, ) residual error amplitude (Source) latitudinal profile (Mixing coefficient) 9

Various attempts to define separable modes Principal component analysis Mininni et al., PRL (22) Spherical harmonics Stenflo & Güdel, A&A (1986) Knaack and Stenflo, A&A (25) Gokhale & Javaraiah, Solar Physics (1992) Independent component analysis Cadavid et al., Solar Physics (28) 1

A different approach All these approaches have shortcomings improper observations to adequately constrain the modes lack of realism: i.e. principal component give values < 11

A different approach All these approaches have shortcomings improper observations to adequately constrain the modes lack of realism: i.e. principal component give values < Our approach use a data-driven approach: the modes are defined from the data S k (t)m k ( )+ (t, ) N(t, ) = X k assume that the amplitudes Ak(t) are independent P(S k,s l )=P(S k ) P(S l ) assume positivity : S k M k 12

A different approach We consider a Bayesian Positive Source Separation (BPSS) technique [Moussaoui et al. IEEE (25)] This is a blind source separation problem (cocktail party): both the amplitudes and their mixing coefficients are unknown Properties uniqueness etc.: yes, under reasonable assumptions careful validation is mandatory 13

How many modes? No universal & robust criterion for this Consider how residual energy h 2 N i drops vs number of modes N N(t, ) = NX k=1 S k (t)m k ( )+ N (t, ) 1 2 residual energy < N 2 > [%] 1 1 1 5 1 15 number of modes N 1 mode 2 modes 3 modes 2 modes are a must, 3 are a bit better, 4 do not improve anymore 14

Noise properties All these methods assume that the data are stationary and that the noise is stationary too This is NOT true for the sunspot number (heteroscedasticity) 15

Noise properties The standard deviation of the sunspot number is amplitudedependent (mix of Poisson and Gaussian noise) 35 6 month average 2 3 198 standard devn of residuals 25 2 15 1 SSN = p 2.3 SSN SSN = p SSN 196 194 192 19 188 year 186 5 184 5 1 15 2 25 SSN 182 16

Noise properties To stabilise the noise, apply the Anscombe transform p If N(t) has Poisson-like noise, then N(t)+ has Gaussian-like noise We estimate the signal noise from the residual error between the observations and their a linear stationary (autoregressive) model N(t i+1 )=a N(t i )+a 1 N(t i 1 )+...+ a p N(t i p )+"(t i+1 ) Here, typically, p=5, and the noise level is p h"2 i From now on, we shall work with the square root of the sunspot number: 1879 samples (one per Carrington rotation), 5 latitude bins 17

Results 18

Latitudinal profile of the modes 1 N=2 M k ( ).5 9 45 45 9 1 N=3 2 modes The salient features are described by 2 modes only - high latitude mode - low latitude mode M k ( ).5 9 45 45 9 3 modes Additional modes merely describe near-equator dynamics 1 N=4 M k ( ).5 4 modes 9 45 45 9 latitude [deg] 19

Temporal profiles of the modes (sources) high-latitude mode = onset of cycle low-latitude mode = decline of cycle 3 2.5 2 S k (t) 1.5 1.5 186 188 19 192 194 196 198 2 22 year 2

Temporal profiles of the modes (sources) 192 194 196 198 2 22 21

Reconstruction of the butterfly diagram original data 6-month smoothing reconstruction with N=2 modes 22

Hemispheric asymmetries Modes can also be estimated separately for each hemisphere high latitude S 1 (t) 4 3 2 1 North South 186 188 19 192 194 196 198 2 22 low latitude S 2 (t) 2.5 2 1.5 1.5 North South 186 188 19 192 194 196 198 2 22 23

Main properties Provides very accurate timing for onset of new solar cycle not affected by cycle overlap Provides accurate phase lags between hemispheres no distinct pattern in phase lags, apart from slow trends confirms existing results [Muraközy and Ludmany, MNRAS (212)] 24

Gnevyshev gap Gnevyshev gap = single or multiple drops in sunspot number near solar maximum. Do not occur in phase in both hemispheres. Figure 3. An example ofgnevyshev detecting thegaps presence in N of and a Gnevyshev S hemispheres Gap for solar cycles 19 and 23 determined from sunspot Norton area et data. al. Solar The Physics total (21) sunspot area data (top), the northern hemispheric data (middle), and the southern hemisphere (bottom) are plotted with time periods 25

S(t) Gnevyshev gap All documented occurrences of Gnevyshev gaps coincide with a clear crossover from the high to low latitude mode 3 2 high latitude low latitude Northern hemishere 1 188 19 192 194 196 198 2 26

Phase-space plot We succeeded in reducing 5 observables (5 latitude bins) to 2 new pseudo-sunspot numbers a high-latitude sunspot number SH(t) a low-latitude sunspot number SL(t) By plotting SH(t) vs SL(t) we get a compact phase-space plot get rid of the temporal dimension 27

Phase-space plot 3 1982 1992 1949 low latitude mode SL(t) low latitude mode M 1 (t) 2.5 2 1.5 1.5 1929 1939 1959 1981 1884 1919 23 194 1951 214 1885 1984 1992 * 1971 1928 198 1972 1941 197 24 192 1896 199 1886 1921 193 25 1961 1993 1952 1974 18971973 1942 197 1983 1918 1927 1969 1894 195 1895 1883 196 196 213 1893 195 1938 22 1948 21 2 1968 1917 1926 1892 1898 1994 1922 191 1962 212 26 1985 1953 1882 1963 1943 1916 1975 1875 1931 1887 1888 1986 1932 1976 27 1995 194 1881 1911 1876 1923 1996 211 1899 1889 1877 19 1912 19151966 1954 1878 1933 1944 193 191 28 1964 192 1987 1977 1913 1879 29 189 1965 188 1924 1934 21 1914 1891 19251945 1955 1997 1959 1991 198 1937 1999 1936 1892 1935 1988 1998 1947 1979 199 1967 1925 1946 1978 1956 1958 1957 1989.5 1 1.5 2 2.5 3 3.5 high latitude mode M 2 (t) high latitude mode SH(t) 28

Phase-space plot 3 1982 you are here 1992 1949 low low latitude mode M SL(t) 1 2.5 2 1.5 1.5 new cycle starts before old one has ended 1929 1939 1959 1981 1884 1919 23 194 1951 214 197 1893 195 1938 1948 1885 1984 1992 * 22 21 1983 1971 1928 1918 198 1972 1927 1969 1941 197 1894 24 195 192 1896 199 2 1886 1895 1883 196 196 1968 1917 1921 1961 1993 1952 193 25 1974 18971973 1926 213 1942 1892 1898 1994 1922 191 1962 212 26 1985 1953 1882 1963 1943 1916 1975 1875 1931 1887 1888 1986 1932 1976 27 1995 194 1881 1911 1876 1923 1996 211 1899 1889 1877 19 1912 19151966 1954 1878 1933 1944 193 191 28 1964 192 1987 1977 1913 1879 29 189 1965 188 1924 1934 21 1914 1891 19251945 1955 1997.5 1 1.5 2 2.5 3 3.5 high latitude mode M 2 (t) 1991 198 1936 1959 1999 1892 1937 1935 1988 1998 1979 1967 high latitude mode SH(t) 1925 1947 1978 199 1946 1956 1958 1957 1989 crossover 29

Main properties 2 similar cycles = their orbits overlap ( same sunspot number) cycle 24 is similar to cycle 12, not to cycle 14/15 Look for determinism = criteria to predict solar activity there are some, but more investigation needed would be more meaningful with 22-year cycle (signed sunspot numbers?) 3

Going further back : Heinrich Schwabe s sunspot data 31

Include data from Heinrich Schwabe Sunspot group data from H. Schwabe (provided by R. Arlt) : 1825-1867 1 year average of daily sunspot area latitude [deg] 5 5 185 1875 19 1925 195 1975 2 year 32

Include data from Heinrich Schwabe Schwabe s data are substantially different : bias likely due to multiple counts of same sunspot group low latitude mode 1829 1862 183 1.6 1861 1827 1831 186 1863 1848 1859 1864 1849 1851 185 1847 1982 1852 184 1.4 1841 1839 1929 1949 1959 1865 1853 1838 1919 1981 19924 1832 198 192 1884 1991 1841 1941 1885 194 1939 1942 1897 193 1952 1971 195 1958 1961 1972 1837 1951 23 1983 197 1992 1928 22 196 1.2 1866 1921 1846 197 1918 1969 1886 1858 1893 21 25 1921993 1984 1938 2141894 1854 * 1948 199 1842 1953 1896 191 1962 1836 196 1927 1931 1943 1973 1975 19941874 1974 198 1 26 1898 1985 1883 195 1875 1845 1895 1932 1963 213 2 1887 1855 1922 1887 1937 1833 27 1867 1843 1917 1968 1911 1899 1986 1995 1976 1844 1892 1947 1957.8 1877 1876 191888 1979 1996 1857 1926 1878 1933 1954 1989 1856 1835 1912 1882 212.6 1964 28 1987 191 1916 1834 1944 194 18891923 1936 1946 1977 1966 1999 1967 1956.4 1913 1934 1965 1935 1891978 1988 211 1879 1954 1997 1924 1945 1987 1915 1881 1955 1925 1998 192 189 1914 188 193 29 21.2 1828.5 1 1.5 2 high latitude mode 33

Phase-space plot Same plot, by using principal components only [Mininni et al., Solar Physics, 28].4.2.2 No immediate interpretation because modes can be <.4.6.8.1.2.3.4.5.6 34

A phenomenological model 35

Predator-prey model High-latitude mode SH(t) = prey is fed by the polar magnetic field transfers magnetic flux to low-latitude mode Low latitude mode SL(t) = predator feeds on magnetic flux from high-latitude mode decays and eventually feeds the polar magnetic field 36

Predator-prey model High-latitude mode SH(t) = prey is fed by the polar magnetic field transfers magnetic flux to low-latitude mode Low latitude mode SL(t) = predator feeds on magnetic flux from high-latitude mode decays and eventually feeds the polar magnetic field Heuristic model Ṡ H = S H ( S H S L ) Ṡ L = S L ( + S H S L ) 37

Predator-prey model Our model Ṡ H = S H ( S H S L ) Ṡ L = S L ( + S H S L ) is a predator-prey (aka Lotka-Volterra) model predator (low-latitude mode) prey (high-latitude mode) 38

Predator-prey model Our model Ṡ H = S H ( S H S L ) Ṡ L = S L ( + S H S L ) is a predator-prey (aka Lotka-Volterra) model Interpretation α > growth rate from polar field β > equatorward motion during cycle γ > decay rate of solar cycle δ > conservation law should imply δ β Let s estimate these parameters not so easy 39

Predator-prey model 6 x Volterra model, window = 12 year 1 3 aα b/d β/δ 5 cγ sunspots 4 3 2 1 186 188 19 192 194 196 198 2 22 Ṡ H = S H ( S H S L ) Ṡ L = S L ( + S H S L ) 4

Conclusions the spatio-temporal dynamics of the butterfly diagram has been reduced to 2 modes only (high/low latitude) important to take noise statistics into account phase space representation gives new insight into the solar cycle but raises also new questions Open questions predator/prey analogy reveals centennial trends define new pseudo-sunspot proxies 41