The Solar Interior

1. The Solar Interior Axel Brandenburg (Nordita, Copenhagen  Stockholm)

2. Nordita in Stockholm Nordita in Stockholm

3. Nordita in Stockholm • Expected developments • ~6 new post-docs Nordic+non-Nordic • ~2 new assist. profs • ~5 programs/year • visiting professors (1/2-2yr) • Nordita-days next August

4. Nordita in Stockholm Close proximity to AlbaNova main building (physics, big lecture theater, cafeteria)

5. Importance of solar interior

6. Cyclic variability over centuries Spörer Min. Oort Min. Wolf Min. Hallstatt “cycle” Gleisberg cycle Maunder Minimum

7. Large scale coherence Active regions, bi-polarity systematic east-west orientation opposite in the south

8. Solar cycle • Longitudinally averaged radial field • Spatio-temporal coherence • 22 yr cycle, equatorward migration butterfly diagram Poleward branch or poleward drift?

9. Solar interior physics:Radial entropy profile unstable,  convection Brunt-Väisälä osc. gravity waves

10. Basic physical parameters Density: 0.01…0.1 g/cm3 in convection zone (35 and 150 Mm depth) Mass: 2x1030 kg Luminosity: 4x1026 W Central temp: 1.5x107 K Kramers Fick’s diffusion law

11. Surface manifestation:solar granulation Horizontal size L=1 Mm, sound speed 6 km/s Correlation time 5 min = sound travel time

12. Seeing the interior:with the 5 min oscillations Discovered in 1960 (Leighton et al. 1962) Was thought to be response of upper atmosphere to convection

13. 5 min oscillation are global Franz-Ludwig Deubner (1974) Roger Ulrich (1970)

14. Only p-modes observed

15. RefractionReflection Top: reflection when wavenlength ~ density scale height Deeper down: Sound speed large

16. Inversion: input/output Duval law Sound speed

17. Internal angular velocity

18. Internal angular velocityfrom helioseismology spoke-like at equ. dW/dr>0 at bottom ? dW/dr<0 at top

19. Cycle dependenceof W(r,q)

20. Convection, mixing length theory Momentum balance rough scaling Convective flux Density: 0.01…0.1 g/cm3 or: 10 …100 kg/m3 flux: 0.7x108 W/m2 u~100 m/s

21. Magnetic fields Maxwell eqns Vector potential Uncurled induction eqn

22. Magnetic helicity

23. Dynamos: kinetic  magnetic energy surface radiation viscous heat Ohmic heat magnetic energy thermal energy kinetic energy Nuclear fusion

24. Small scale and large scale dynamos non-helically forced turbulence helically forced turbulence Scale separation :== There is room on scales Larger than the eddy scale

25. Maybe no small scale “surface” dynamo? Small PrM=n/h: stars and discs around NSs and YSOs Here: non-helically forced turbulence Schekochihin Haugen Brandenburg et al (2005) k

26. 3-D simulations in spheres • The right dynamo regime? • Or a small scale dynamo? Brun, Miesch, & Toomre (2004, ApJ 614, 1073)

27. a-effect dynamos (large scale) New loop Differential rotation (prehelioseism: faster inside) Cyclonic convection; Buoyant flux tubes Equatorward migration  a-effect ?need meridional circulation

28. a2-dynamos

29. a2-effect calculation

30. Problems with MFT • Catastrophic quenching?? • a ~ Rm-1, ht ~ Rm-1 • Field strength vanishingly small!?! • Something wrong with simulations • so let’s ignore the problem • Possible reasons: • Suppression of lagrangian chaos? • Suffocation from small scale magnetic helicity?

31. Slow saturation Brandenburg (2001, ApJ 550, 824)

32. Periodic box, no shear: resistively limited saturation Brandenburg & Subramanian Phys. Rep. (2005, 417, 1-209) Significant field already after kinematic growth phase followed by slow resistive adjustment Blackman & Brandenburg (2002, ApJ 579, 397)

33. Revised theory for a-effect 1st aspect: replace triple correlation by quadradatic 2nd aspect: do not neglect triple correlation 3rd aspect: calculate rather than Similar in spirit to tau approx in EDQNM  (Heisenberg 1948, Vainshtein & Kitchatinov 1983, Kleeorin & Rogachevskii 1990, Blackman & Field 2002, Rädler, Kleeorin, & Rogachevskii 2003)

34. Implications of tau approximation • MTA does not a priori break down at large Rm. (Strong fluctuations of b are possible!) • Extra time derivative of emf •  hyperbolic eqn, oscillatory behavior possible! • t is not correlation time, but relaxation time with

35. Kinetic and magnetic contributions

36. Connection with a effect: writhe with internal twist as by-product a effect produces helical field W clockwise tilt (right handed)  left handed internal twist both for thermal/magnetic buoyancy

37. … the same thing mathematically Two-scale assumption Production of large scale helicity comes at the price of producing also small scale magnetic helicity

38. Revised nonlinear dynamo theory(originally due to Kleeorin & Ruzmaikin 1982) Two-scale assumption Dynamical quenching Kleeorin & Ruzmaikin (1982) ( selective decay) Steady limit  algebraic quenching:

39. General formula with current helicity flux • Advantage over magnetic helicity • <j.b> is what enters a effect • Can define helicity density Rm also in the numerator

40. Which is the right scenario • Distributed dynamo (Roberts & Stix 1972) • Positive alpha, negative shear • Overshoot dynamo (e.g. Rüdiger & Brandenburg 1995) • Negative alpha, positive shear • Interface dynamo (Markiel & Thomas 1999) • Negative alpha in CZ, positive radial shear beneath • Low magnetic diffusivity beneath CZ • Flux transport dynamo (Dikpati & Charbonneau 1999) • Positive alpha, positive shear • Migration from meridional circulation

41. Ideas leading toward the overshoot dynamo scenario • Since 1980: dynamo at bottom of CZ • Flux tube’s buoyancy neutralized • Slow motions, long time scales • Since 1984: diff rot spoke-like • dW/dr strongest at bottom of CZ • Since 1991: field must be 100 kG • To get the tilt angle right Spiegel & Weiss (1980) Golub, Rosner, Vaiana, & Weiss (1981)

42. In the days before helioseismology • Angular velocity (at 4o latitude): • very young spots: 473 nHz • oldest spots: 462 nHz • Surface plasma: 452 nHz • Conclusion back then: • Sun spins faster in deaper convection zone • Solar dynamo works with dW/dr<0: equatorward migr

43. Application to the sun: spots rooted at r/R=0.95 Benevolenskaya, Hoeksema, Kosovichev, Scherrer (1999) Pulkkinen & Tuominen (1998) • Df=tAZDW=(180/p) (1.5x107) (2p 10-8) • =360 x 0.15 = 54 degrees!

44. Application to the sun:spots rooted at r/R=0.95 Benevolenskaya, Hoeksema, Kosovichev, Scherrer (1999) • Overshoot dynamo cannot catch up • Df=tAZDW=(180/p) (1.5x107) (2p 10-8) • =360 x 0.15 = 54 degrees!

45. Flux storage Distortions weak Problems solved with meridional circulation Size of active regions Neg surface shear: equatorward migr. Max radial shear in low latitudes Youngest sunspots: 473 nHz Correct phase relation Strong pumping (Thomas et al.) Arguments against and in favor? Tachocline dynamos Distributed/near-surface dynamo in favor against • 100 kG hard to explain • Tube integrity • Single circulation cell • Too many flux belts* • Max shear at poles* • Phase relation* • 1.3 yr instead of 11 yr at bot • Rapid buoyant loss* • Strong distortions* (Hale’s polarity) • Long term stability of active regions* • No anisotropy of supergranulation Brandenburg (2005, ApJ 625, 539)

46. Is magnetic buoyancy a problem? Stratified dynamo simulation in 1990 Expected strong buoyancy losses, but no: downward pumping Tobias et al. (2001)

47. Cartesian box MHD equations Magn. Vector potential Induction Equation: Momentum and Continuity eqns Viscous force forcing function (eigenfunction of curl)

48. Forced LS dynamo with no stratification azimuthally averaged no helicity, e.g. Rogachevskii & Kleeorin (2003, 2004) geometry here relevant to the sun neg helicity (northern hem.)

49. Tendency away from filamentary field Cross-sections at different times Mean field

50. Simulating solar-like differential rotation • Still helically forced turbulence • Shear driven by a friction term • Normal field boundary condition