The solar butterfly diagram: a low-dimensional model

Transcription

1 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)

2 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

3 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) Small Cycles Area > 2 µhem Time (Years from Max) 3

4 What the butterfly diagram tells us Latitude [Degrees] Location of bright points and delta spots The Astrophysical 792:12 (19pp), 214 September 1 9 [McIntosh et al. ApJJournal, (214)] year 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 MPS 27/1/ Daily Average EUV BP SOHO/MDI 24 and SDO/HMI 26 MRoI 28 (b) Merged Time [Years] Daily Average MRoI Node Density Latitude [Degrees

5 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

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

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) 7

8 The Sun as a dynamical system With the sunspot number only N(t) Consider the Sun as 1D system Description 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, ) Described by a Partial Differential Equation (PDE) Much more complex 8

9 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

10 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

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 < 11

12 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

13 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

14 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 > [%] 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

15 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

16 Noise properties The standard deviation of the sunspot number is amplitudedependent (mix of Poisson and Gaussian noise) 35 6 month average standard devn of residuals SSN = p 2.3 SSN SSN = p SSN year SSN

17 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

18 Results 18

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

20 Temporal profiles of the modes (sources) high-latitude mode = onset of cycle low-latitude mode = decline of cycle S k (t) year 2

21 Temporal profiles of the modes (sources)

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

23 Hemispheric asymmetries Modes can also be estimated separately for each hemisphere high latitude S 1 (t) North South low latitude S 2 (t) North South

24 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

25 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

26 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

27 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

28 Phase-space plot low latitude mode SL(t) low latitude mode M 1 (t) * high latitude mode M 2 (t) high latitude mode SH(t) 28

29 Phase-space plot you are here low low latitude mode M SL(t) new cycle starts before old one has ended * high latitude mode M 2 (t) high latitude mode SH(t) crossover 29

30 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

31 Going further back : Heinrich Schwabe s sunspot data 31

32 Include data from Heinrich Schwabe Sunspot group data from H. Schwabe (provided by R. Arlt) : year average of daily sunspot area latitude [deg] year 32

33 Include data from Heinrich Schwabe Schwabe s data are substantially different : bias likely due to multiple counts of same sunspot group low latitude mode * high latitude mode 33

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

35 A phenomenological model 35

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 36

37 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

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 predator (low-latitude mode) prey (high-latitude mode) 38

39 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

40 Predator-prey model 6 x Volterra model, window = 12 year 1 3 aα b/d β/δ 5 cγ sunspots Ṡ H = S H ( S H S L ) Ṡ L = S L ( + S H S L ) 4

41 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