Comparing Turbulence in Plasmas and Neutral Fluids for Understanding Solar Wind Origins

2. Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary

3. Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary

4. the solar wind Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary natural turbulence laboratory We aim to: Compare turbulence in plasmas and neutral fluids so that we can learn more about the universal aspects of turbulent flows; Specific understanding of plasma turbulence phenomenology and modeling of space weather; Learn more about the conditions in the solar corona from which the solar wind originates. High magnetic Reynolds number ~ 105

5. the solar wind Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary natural turbulence laboratory We aim to: Compare turbulence in plasmas and neutral fluids so that we can learn more about the universal aspects of turbulent flows; Specific understanding of plasma turbulence phenomenology and modeling of space weather; Learn more about the conditions in the solar corona from which the solar wind originates. High magnetic Reynolds number ~ 105

6. the solar wind Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary natural turbulence laboratory We aim to: Compare turbulence in plasmas and neutral fluids so that we can learn more about the universal aspects of turbulent flows; Specific understanding of plasma turbulence phenomenology and modeling of space weather; Learn more about the conditions in the solar corona from which the solar wind originates. High magnetic Reynolds number ~ 105

7. y t bHy bt self-similarity Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary Change scale from ttobt AND scale y to bHy

8. y t bHy bt self-similarity Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary Change scale from t to bt AND scale x to bHx If the statistics of bHy is the same as y then process is statistically self-similar Hurst exponent H

9. Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary x(t,) = y(t+) - y(t) 

10. Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary x(t,) = y(t+) - y(t)  probability density function (pdf)

11. P(x , ) x pdf collapse Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary

12. Ps(x  - H) x  - H pdf collapse Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary

13. moment scaling Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary pdf scaling implies scaling of moments AND monoscaling implies a linear increase in the scaling of the moments with moment order p (p)=Hp

14. log10Sp() moment scaling Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary pdf scaling implies scaling of moments AND monoscaling implies a linear increase in the scaling of the moments with moment order p (p)=Hp

15. S1 S2 S3 probability density Sn Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary

16. Lévy processes Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary For Levy H=1/

17. heavy-tails Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary

18. finite-size effects Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary From extreme value theory (EVT)

19. finite-size effects Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary From extreme value theory (EVT) gradient=1/

20. Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary K. Kiyani, S. C. Chapman and B. Hnat, Phys. Rev. E 74, 051122 (2006)

21. multifractals Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary non-linear Why does this happen?

22. multifractals Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary non-linear Why does this happen?

23. digression Nigel Goldenfeld. Roughness-induced criticality in a turbulent flow. Phys. Rev. Lett. 96, 044503:1-4 (2006)

24. Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary ‘real-world’ data

25. solar cycle change Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary • In situ data from solar wind monitors ACE & WIND around the Sun-Earth L1 point (R. Lepping, K. Ogilvie) • To study ~ solar max (2000) and ~ solar min (1996) • Probe in detail the scaling of B2 - magnetic field energy density

26. Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary Monofractal Lévy process Multifractal p-model

27. results Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary 2000 - Solar max 1996 - Solar min (p) (p) Moment p Moment p H = 0.44 ± 0.02 K. Kiyani, S. C. Chapman, B. Hnat and R. M. Nicol, Phys. Rev. Lett 98, 211101 (2007)

28. Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary @ 1% 1 year (2000 solar max) ACE 64 seconds calibrated magnetometer data (R. Lepping) ~ 2 hrs ~ 5 mins log10(Sp) K. Kiyani, S. C. Chapman, B. Hnat and R. M. Nicol, Phys. Rev. Lett 98, 211101 (2007)

29. pdfs Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary Functional form of pdf different at solar max and min Power law tail at solar maximum reminiscent of a Lévy process B. Hnat et. al. , Geophys. Res. Lett. 34, L15108 (2007)

30. Lévy? Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary H=1/ for a Lévy process Measured values: H=0.44 ≠ 1/=0.71

31. Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary ‘All epistemological value of the theory of probability is based on this: that large scale random phenomena in their collective action create strict, non-random regularity.’ (Gnedenko and Kolmogorov, Limit Distributions for Sums of Independent Random Variables)

32. limit theorems Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary Central Limit Theorem(De Moivre, Laplace, Lyapunov)

33. limit theorems Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary Central Limit Theorem(De Moivre, Laplace, Lyapunov) Generalized Central Limit Theorem(Lévy)

34. Joseph effect Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary Introduce memory via Mandelbrot’s notion of Joseph (memory) and Noah (large events) effects using Linear fractional stable motion (lfsm) Memory parameter d = H - 1/ tail parameter  d>0 => persistence d<0 => anti-persistence self-similarity parameter H Hurst exponent

35. lfsm Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary fft based algorithm obtained from S. Stoev and M. Taqqu (fractals 2004)

36. Hd 0.99 1.4 +0.276 0.71 1.4 ~0 0.44 1.4 -0.274 pd+1 1.52 1 0.476 finite-size scaling Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary

37. finite-size scaling Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary (p) Moment p

38. Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary 2000 - Solar max lfsm H=0.44 =1.4 (p) (p) Moment p Moment p

39. conclusions Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary • We find a unique monofractal signature in the solar wind at 1 AU in the ecliptic at solar maximum. • Use of a robust statistical estimator which operationally excludes a small percentage of poorly represented extreme events. • Solar Max -- Complex topology & larger events =>are we seeing remnants of solar activity? • To obtain a more complete picture it is necessary to take into account non-Markovian effects Possible signatures of the processes driving the turbulence? • Outlook - look to different spacecrafts to scan interplanetary solar wind to study the evolution of its turbulence. Check for detailed solar cycle dependence.

40. references Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary • Self-similar scaling of B2 • K. Kiyani, S. C. Chapman, B. Hnat and R. M. Nicol, Phys. Rev. Lett 98, 211101 (2007). • B. Hnat, S. C. Chapman, K. Kiyani, G. Rowlands and N. W. Watkins, Geophys. Res. Lett. 34, L15108 (2007). • Conditioning extreme events • K. Kiyani, S. C. Chapman and B. Hnat, Phys. Rev. E 74, 051122 (2006). • Applications to anomalous transport • G. Zimbardo, Plasma Phys. Control. Fusion 47 B755-B767 (2005). • D. F. Escande and F. Sattin, PRL 99, 185005 (2007). • Reading on stable processes • Samorodnitsky & Taqqu, Stable Non-Gaussian Random Processes.

41. Motivation Coronal signatures Scaling Introducing memory Heavy-tails Summary