1. Sofia/Turekian Forum, Yale 28/03/081 A semi-empirical model for TSI variations Paul Charbonneau (+ Ashley Crouch), Département de Physique, Université de Montréal 1.Introduction: TSI and solar activity 2.A TSI model based on active region emergence and decay 3.Evidence for a secular trend in TSI 4.Evidence for additional cycle-related modulation 5.Outlook

2. Sofia/Turekian Forum, Yale 28/03/082 AshleyDanahéPaulGeneviève Submitted to Astrophys. J., 17 March 2008 Astrophys. J., 677, in press [ 10 April 2008 ]

3. Sofia/Turekian Forum, Yale 28/03/083 Observed TSI variations Foukal, Fr ö hlich, Spruit & Wigley 2006, Nature, 443, 161-166

4. Sofia/Turekian Forum, Yale 28/03/084 Cycle-related TSI variations The solar magnetic activity cycle shows strong amplitude fluctuation and/or intermittency What does TSI do in times of secular rise in activity levels? In Maunder Minimum epoch of suppressed activity?

5. Sofia/Turekian Forum, Yale 28/03/085 Origin of TSI variations 1.Solar irradiance variations reflect the changes in the photospheric coverage of various magnetic structures having different radiative emissivities; 2. On long timescales (> a few yr), TSI variations reflect a deep-seated magnetically-mediated modulation of convective energy transport. Two classes of explanations (NOT mutually exclusive!):

6. Sofia/Turekian Forum, Yale 28/03/086 A TSI model based on the emergence and decay of active regions AIM: Produce a model for TSI reconstruction based on simple physical mechanism, rather than statistical correlations; WHY Oh WHY …? WHY: Belief in the universality of physical laws suggests that such models can be extrapolated more safely outside of the parameter regime in which they were calibrated

7. Sofia/Turekian Forum, Yale 28/03/087 A TSI model based on the emergence and decay of active regions 1. Observational underpinnings 2. Model design 3. Parameter fitting 4. Results for 1978-2007 6. Long-term trends in quiet-sun irradiance 5. Reconstruction from 1874

8. Sofia/Turekian Forum, Yale 28/03/088 Evolutionary link 1: fragmentation

9. Sofia/Turekian Forum, Yale 28/03/089 Evolutionary link 2: boundary erosion

10. Sofia/Turekian Forum, Yale 28/03/0810 Emergence+fragmentation+erosion A FRAGMENTATION MODEL: 1.Sunspots of area A injected on « solar disk » (data from Royal Greenwich Observatory data) 2. Backside emergences introduced stochastically 3. Spots fragment stochastically, and erode at boundaries 4.Fragmentation/erosion process eventually produces flux tubes, which then disappear with fixed probability 5.This result in a time-evolving area distribution N(A;t), which can be convolved with the contrast curve, including center-to-limb variations, to produce a TSI time series.

11. Sofia/Turekian Forum, Yale 28/03/0811 Observational support The probability distribution function of observed sunspot areas has a lognormal form over more than two orders of magnitude in observed areas; successive fragmentation is known to yield such a distribution Bogdan et al. 1988, ApJ 327, 451 (Necessary but not sufficient!)

12. Sofia/Turekian Forum, Yale 28/03/0812 Magnetic flux transport All magnetic structures are carried in the EW direction by (differential) rotation: …and poleward by meridional circulation: Bright, small-scale magnetic elements produced by successive fragmentation and erosion accumulate as a « cloud » surrounding each decaying spot; this is the model’s equivalent to facules (Komm et al. 1993) (Charbonneau et al. 1999)

13. Sofia/Turekian Forum, Yale 28/03/0813 From sunspot and facular areas to TSI Basic 3-component model: quiet photosphere, spots, faculae: Irradiance deficit associated with « spots »: Irradiance excess associated with « faculae » (Chapman & Meyer 1986) (Lean et al. 1998; Brandt et al. 1994)

14. Sofia/Turekian Forum, Yale 28/03/0814 Model parameters In practice, 6 to 9 free parameters need to be determined The model involves a number of parameters, some that can be fixed on the basis of observations, others that need to be fitted to the data

15. Sofia/Turekian Forum, Yale 28/03/0815 Parameter fitting We need to pick model parameter values that produce best-fits to both the TSI and sunspot area (SA) time series (multi-objective optimization) The summed squared residuals between observed and modeled TSI and SA is a statistical function of the model parameters, because of the stochastic nature of the fragmentation process, and statistical treatment of backside emergences: two model runs with the same parameter values will NOT yield the same TSI and SA time series! Possible tradeoffs between model parameters make the optimization problem multimodal (secondary extrema) We use the genetic algorithm-based optimizer PIKAIA, with enhanced elitism and metric-distance-based mutation rate adjustment

16. Sofia/Turekian Forum, Yale 28/03/0816 Genetic algorithms (1) A class of biologically-inspired, population-based evolutionary algorithms that can form the core of powerful, flexible multimodal optimization schemes Breed new population from selected members Select fittest members of the population Evaluate fitness of new population member Initialization: build population of random trial solutions; evaluate fitness Is best of current population good enough?DONE! NOYES

17. Sofia/Turekian Forum, Yale 28/03/0817 Genetic algorithms (2) ENCODE: CROSSOVER: MUTATE: DECODE: 1 2 3 4 5 6 7 8 9 0 0 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 0 0 9 8 7 6 5 4 3 2 1 0 93 4 5 6 7 8 9 0 1 28 7 6 5 4 3 2 1 3 4 5 6 7 8 9 00 9 3 4 5 618 9 0 0 93 4 5 618 9 0 1 28 7 6 5 4 3 2 1 ( 1.2345, 6.7890 ) ( 0.9876, 5.4321 ) ( 0.9345, 6.1890 ) ( 1.2876, 5.4321 )

18. Sofia/Turekian Forum, Yale 28/03/0818 Genetic algorithms (3) FITNESS is defined in terms of the product of mean-squared residuals between the modeled and observed TSI and sunspot area time series: We use 81-day running boxcar averages of the time series, to avoid large contribution to the mean-squared residuals associated with timing errors in the emergence of large active regions NOTE: No derivatives of fitness w.r.t. model parameters are required

19. Sofia/Turekian Forum, Yale 28/03/0819 Genetic algorithms (4) Different runs converge at different rates, but eventually reach similar fitness levels

20. Sofia/Turekian Forum, Yale 28/03/0820 Results: It Works !! d41_61_0702 Model run

21. Sofia/Turekian Forum, Yale 28/03/0821 TSI excess in rising phase Modelled TSI is systematically below observations during rising phases of cycles… …even though SA is very well-fitted RMS(A) = 230 microHem RMS(S) = 0.202 W/m2

22. Sofia/Turekian Forum, Yale 28/03/0822 I mean, it really does !! Bright « facular » component Dark « spot » component

23. Sofia/Turekian Forum, Yale 28/03/0823 Optimal parameter values Conversion efficiency high: Nearly all sunspot flux is converted into small scale elements Faculae lifetime high: Obs. Suggest tens of days for large facular structures Facular contrast high: Observational determinations suggest ~ 0.03-0.04 Hint: we are missing an irradiance source unrelated to decay of active regions

24. Sofia/Turekian Forum, Yale 28/03/0824 Working back to 1874

25. Sofia/Turekian Forum, Yale 28/03/0825 A TSI downward trend? Allow for linear trend in quiet Sun irradiance : Repeating best-fit procedure with additional slope parameter yields an improved fit: fitness in range 1.62-1.81 as opposed to 1.50-1.67 with constant S_Q. Statistically significant! RMS(S) : 0.202 to 0.193 W/m2

26. Sofia/Turekian Forum, Yale 28/03/0826 Repeating best-fit procedure with additional slope parameter yields an improved fit (fitness in range 1.62-1.81 as opposed to 1.50-1.67 with constant S_Q. Statistically significant!

27. Sofia/Turekian Forum, Yale 28/03/0827 Optimal parameter values, bis

28. Sofia/Turekian Forum, Yale 28/03/0828 A cyclically varying contribution to TSI? Allow for sinusoidal contribution to quiet Sun irradiance, i.e., unrelated to active region decay: Repeating fit procedure with additional sinusoidal component yields an improved fit: fitness in range 1.82-2.31 as opposed to 1.50-1.67 with constant S_Q. Statistically significant! RMS(S) : 0.202 - > 0.168 W/m2

29. Sofia/Turekian Forum, Yale 28/03/0829

30. Sofia/Turekian Forum, Yale 28/03/0830 How optimal is optimal? 100 best out of 200 GA runs Some runs remain « stuck » on secondary extrema Fittest runs have S_Q In range 0.3-0.6 W/m2 Very few runs converge With S_Q < 0.3 W/m2 Mean +/- 1 s.d. of 100 runs with fixed S_Q

31. Sofia/Turekian Forum, Yale 28/03/0831 Identifying « bad » local optima Because fitting parameters have a direct physical meaning, it is possible to assess the relative merits of globally suboptimal solutions corresponding to local extrema This solution has a reasonable facular contrast (0.045), but very low lifetime for bright small-scale « faculae » elements (9 days)

32. Sofia/Turekian Forum, Yale 28/03/0832 Optimal parameter values, coda

33. Sofia/Turekian Forum, Yale 28/03/0833 Parameter correlations 1.34 < Fitness < 1.84 0.84 < Fitness < 1.34 1.84 < Fitness < 2.34

34. Sofia/Turekian Forum, Yale 28/03/0834 Conclusions (so far…) It works !! We can simultaneously fit, and quite well, both sunspot areas and TSI over the 1978-2007 interval We find evidence for a brightness source unrelated to active regions in the rising phase of cycles 22 and 23 We find evidence for a slight downward trend in TSI, by -0.0069 +/- 0.0027 W/m^2 per year over 1978-2007 We find evidence for an cycle-phased irradiance source unrelated to active region emergence and decay, accounting for 40-50% of peak-to-peak TSI cycle variability

35. Sofia/Turekian Forum, Yale 28/03/0835 What next? Introduce and test models for long-term modulation « Differentiate » bright structures: introduce DLA model for network and faculae formation Introduce far-side emergences as inferred helioseismically Use genetic programming to « evolve » better fragmentation algorithms, contrast functions, center-to- limb contrast functions for faculae, etc. Adapt model for spectral irradiance, with special emphasis on UV range most relevant to atmospheric chemistry Use output of dynamo model to « feed » sunspot emergences into the TSI model

36. Sofia/Turekian Forum, Yale 28/03/0836 Semi-empirical models for long-term modulation of quiet-sun irradiance K. Tapping et al. 2007, Sol. Phys., 246, 309 Quiet-sun TSI based on a 2-component magnetic flux model linked to TSI via the F10.7 radio flux

37. Sofia/Turekian Forum, Yale 28/03/0837 DLA model for network formation [ Crouch, Charbonneau & Thibault, 2007, Astrophys. J., 662, 715 ] Piece of MDI quiet-sun magnetogramNumerical simulation

38. Sofia/Turekian Forum, Yale 28/03/0838 DLA model for faculae formation [ Kim Thibault, UdeM MSc thesis, 2008, in preparation ]

39. Sofia/Turekian Forum, Yale 28/03/0839 FIN

40. Sofia/Turekian Forum, Yale 28/03/0840 Parameter correlations Multiple GA-optimizarion runs allow to establish error estimates and correlations between best-fit parameters.