Asymptotic studies in Dynamo Theory

1. Asymptotic studies in Dynamo Theory Kirill Kuzanyan1,2,3) 1)IZMIRAN, Moscow region, Russia 2)hosted by National Astronomical Observatory, Mitaka Japan

3. Why asymptotic? • Use analytic methods to complement numerical • Study of physical mechanism of dynamo itself. Detailed study of parametric space • Finding the role of key ingredients (specific factors) one by one

4. General line of studies(step-by-step) • 1D model kinematic dynamo • 1D nonlinear model • 2D model • 1D non-axisymmetric model (leading to 2D) • 2D model with meridional circulation

6. Large-Scale fields

7. Makarov, Sivaraman, 1983

8. Basic equations of solar magnetism Solar convection zone governed by equations of compressible MHD

9. Induction Equation

10. Mean-Field dynamo theory

11. Turbulent Electro-Motive Force

13. W a a

14. R¤ h i.e. high magnetic Reynolds number

15. not a self-adjoint operator!

16. (fast time + short waves)

17. Quantum theory analogues -U potential -E Energy levels

18. Turning points -E Location of the solution

19. Semi-classical approximation in quantum theory is usually applicable for the base level of energy (leading mode 0) as well as higher order modes. There comes only one turning point ! (see V. Maslov) now short waves! (and the maximum of the solution is not localized at the turning point!)

20. Fastest growing mode, short waves!

21. The simplest form (like Coriolis force)

22. The leading mode 0: base state is the fastest growing mode!

23. choice of root Kuzanyan, Sokoloff (1995)

24. Asymptotic solution 45o Generation Source Ù (potential) a 27o Envelope of the solution 80o Wave number (real part of)

25. Butterfly diagram(Kuzanyan and Sokoloff 1997)

27. Reversal of dynamo wave to the pole

28. (Belvedere, Kuzanyan, Sokoloff 2000)

30. maximum

31. SOHO-MDI (Schou et al. 1998) thanks to Sasha Kosovichevhttp://quake.stanford.edu/~sasha/omega.gif Internal differential rotation nHz

32. Rotation rate maximum source reversal of the wave maximum 1D solution maximum2D solution

33. Distribution of generation sources in 1D model reversal max max

34. Two waves equatorwardpoleward related to 1D solution

35. Sunspot wave and polar faculae

36. The unit of magnetic field through equipartition energy (Bassom, Kuzanyan, Soward 1999; see also Griffiths, Bassom, Soward and Kuzanyan 2001; Bassom, Kuzanyan, Sokoloff and Soward 2005)

37. Dependence of the sunspot cycle amplitude of the duration of the phase rise

38. Solar cycle amplitude versus maximum rate of rise prediction 100-114 cycle 23: 121

39. Multiple wings of butterfly diagram in active starsHale’s “number” NH~|D|1/3NH =1 NH =2 latitude latitude Cycle number Cycle number

40. Meridional circulation in dynamo (Popova and Sokoloff, 2010)

41. Meridional circulation in dynamo (Popova and Sokoloff, 2010)

42. Summary on studies of basic properties of astrophysical dynamos by WKB asymptotics • For the limit of short waves, equivalent to high magnetic Reynolds number we have used the analogue of methods or semi-classical approach in quantum mechanics. The application of the methods provided the estimates of the key trends of solar and stellar dynamos. • 　　(1) stability of the solar and stellar magnetic cycle period • 　　(2) reversal of the magnetic field dynamo wave to the pole and the equator • 　　(3) increase of magnetic activity towards the direction of the dynamo wave propagation • 　　(4) propagation of magnetic activity waves mainly along constant internal rotation rate (so-called Yoshimura-Parker law) • 　　(5) interaction of the dynamo wave branches across the solar equator • 　　(6) excitation of non-axisymmetric structures with dependence on differential rotation • 　　(7) qualitative change in dynamo waves with meridional circulation

43. The end, or the time to take a breath… Thank you спасибо! ¡Gracias! 谢谢 ! ありがとう !