1. High Altitude Observatory (HAO) – National Center for Atmospheric Research (NCAR) The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Research under sponsorship of the National Science Foundation. An Equal Opportunity/Affirmative Action Employer. Dynamics of the solar convection zone Matthias Rempel (HAO/NCAR)

2. Outline  Observations –Large scale magnetic field –Solar cycle –Large scale flows: differential rotation, meridional flow  Differential rotation –Structure of convection –Origin of differential rotation  Solar dynamo –Basic ingredients of a dynamo –Formation of sunspots

3. Measurement of magnetic field  Zeeman effect –Splitting of spectral lines –Linear+Circular polarization  Thermal and turbulent broadening of spectral lines –Splitting not observable except for strongest field (sunspots)  Most field diagnostics are based on polarization signal –Gives strength and orientation of field

4. Sunspots on solar disc  Regions of strong magnetic field (3000 Gauss)  About 20000km diameter  Lifetime of a few weeks PSPT (blue) PSPT (CaK)

5. Changing X-ray activity over 11 years Yohkoh X-ray images

6. Butterfly diagram + sunspot area over time

7. Hale’s law

8. Joy’s law

9. Solar cycle properties  Butterfly diagram –Equatorward propagation of activity starting from 35 degrees latitude over 11 years (individual lifetimes of sunspots ~ a few weeks)  Hale’s polarity law –Opposite polarity of bipolar groups in north and south hemisphere –Polarity in individual hemisphere changes every 11 years  Joy’s law –Bipolar groups are tilted to east-west direction –Leading polarity closer to equator –Tilt angle increases with latitude

10. Evolution of radial surface field

11. Everything together D. Hathaway NASA (MSFC)

12. Longterm variations

13. Variability over the past 10000 years  Cosmogenic isotopes – 14 C and 10 Be produced by energetic cosmic rays –Cosmic rays modulated by magnetic field in heliosphere –Longterm record in ice cores ( 14 C and 10 Be ) and treerings ( 14 C)  Normal activity interrupted by grant minima ~100 years duration  Persistent 11 year cycle Usokin et al. (2007)

14. Large scale flows R. Howe (NSO)  Differential rotation in convection zone, uniform rotation in radiation zone (shear layer in between: Tachocline)  Cycle variation of DR (torsional oscillations, 1% amplitdude)

15. Differential rotation and meridional flow changes through solar cycle Surface Doppler measurement R. Ulrich (2005) Changes is DR Meridional flow Butterfly diagram Radial field

16. Internal dynamics of convection zone  What drives large scale mean flows (differential rotation + meridional flow)? –Answer: small scale flows:  Reynolds stresses (correlations of turbulent motions) can drive large scale flows  Relevant for angular momentum transport:

17. How to model the solar convection zone  3D numerical simulations –Solve the full set of equations (including small and large scale flows) on a big enough computer –Problem: Computers not big enough –Only possible to simulate ingredients  Meanfield models –Solve equations for mean flows only –Problem: need good model for correlations of small scale flows (not always available) –Can address the full problem, but not from first principles

18. Correlations caused by Coriolis force North-South motions: negative (poleward) East-West motions: positive (equatorward) Latitudinal transport: Average: zero unless East-West dominates

19. Structure of convection close to surface 3D simulation (M. Miesch)

20. Structure of convection in lower convection zone 3D simulation (M. Miesch)

21. Coriolis-force causes large scale convection rolls in deep convection zone Balance between pressure and Coriolis force –Cyclonic rolls: lower pressure –Anti-cyclonic rolls: higher pressure

22. Angular momentum transport  Positive –Faster rotating equator –  -component of momentum equation  What determines radial profile of DR? –Force balance between Coriolis, pressure and buoyancy forces –r-  -component of momentum equation

23. Profile of differential rotation  Latitudinal variation of entropy essential for solar like rotation profile  Possible causes –Anisotropic convective energy transport (influence of rotation on convection –Tachocline  About 10K temperature difference between pole and equator (T~10 6 K at base of CZ)

24. Results from 3D simulations 3D simulation (M. Miesch)

25. Summary: differential rotation  Turbulent angular momentum transport –Correlations between meridional (north south) and longitudinal (east west) motions caused by Coriolis force –Anisotropic convection (“banana cells”)  Radial profile of differential rotation –Determined through force balance in meridional plane –Thermal effects important (about 10K latitudinal temperature variation needed)  Boundary layer (tachocline) important

26. The MHD induction equation  Basic laws (Ohm’s law, non-relativistic field transformation, Ampere’s law:  Combination of the three:

27. Differential rotation Axisymmetry + differential rotation Induction equation in spherical coordinates:

28. Properties of solution  Poloidal field always decaying  Toroidal field can grow significantly in the beginning –Stretching of field lines  Toroidal field is also decaying in the long run –The source of toroidal field decays with the poloidal field  What is missing? –Regeneration of poloidal field  Who can do it? –Again: small scale field and flows

29. Meanfield induction equation –Decomposition of velocity and magnetic field: –Averaging of induction equation: –Turbulent induction effects:

30. Induction effect of helical convection Negative kinetic helicity in northern hemisphere Induces a poloidal field from toroidal field parallel to the current of the toroidal field

31. Turbulent induction effects   -effect induces field parallel to electric current   t increases the effective diffusivity for meanfield (turbulent diffusivity)

32. Meanfield Dynamos  The  -effect closes the dynamo loop: regeneration of poloidal field from toroidal field 

33. Some more general properties   2 -dynamo –Stationary field –Poloidal, toroidal field similar strength   -dynamo –Periodic solutions, travelling waves –Toroidal field much stronger than poloidal field

34. So — what is the sun doing?  Strong differential rotation (observed), periodic behaviour   -dynamo  Propagation of activity belt –Dynamo wave (requires radial shear) –Advection effect (meridional flow)  Location of  -effect –Bulk of convection zone (helical convection  positive  ) –Base of convection zone (helical convection  negative , tachocline instabilities   of both signs ) –Rising flux tubes (positive  )

35. Dynamo wave  Surface shear layer –Positive  –Very short time scales –Significant flux loss  Tachocline shear layer –Negative  (in low latitudes) –Longer time scales, stable stratification allows for flux storage

36. Role of tachocline  Stable stratification, long time scales –Formation of large scale field, likely origin of field forming sunspots  Problems of a pure tachocline dynamo –Much stronger shear of opposite sign in high latitudes (strong poleward propagating activity belt) –Very short wavelength of dynamo wave (strongly overlapping cycles) Browning et al. (2006)

37. Advection  Meridional flow –Poleward at surface (observed) –Return flow not observable through helioseismology (so far) –Equatorward at base of CZ Mass conservation Theory: meanfield models + 3D simulations –Additional also turbulent advection effects (latitudinal pumping)

38. Rising magnetic flux tubes  Flux tubes ‘bundle of fieldlines’ form in tachocline  Rising field due to buoyancy  Fluid draining from apex  Coriolis force causes tilt of the top part of tube –Tilt increases with latitude as observed  Net effect: positive 

39. 3D simulation of rising flux-tube  Flux tube looses a lot of flux during rise (tube has to be twisted in the beginning)  Twist reduces tilt angle (Y. Fan)

40. Observations of ‘Surface’  -Effect and Flux Transport

41. . Schematic of flux-transport dynamo  Latitudinal shear producing toroidal field   -effect from decay of active regions  Transport of field by meridional flow

42. Flux-transport dynamo with Lorentz-force feedback on DR and meridional flow  Feedback of Lorentz-force on DR and MC included  Moderate variations of DR and MC –No significant change of dynamo  High latitude variations of DR –Poleward propagation, amplitude similar to observed

43. Summary: The essential ingredients of the solar dynamo I  The sun is a  -dynamo –Differential rotation profile (helioseismology) –Dominance of toroidal field (sunspots) –Cyclic behavior  Tachocline important for large scale organization of toroidal field (boundary layer) –Bulk of convection has too short time scales –Flux loss in convection zone due to magnetic buoyancy and pumping –Stable stratification allows for storage

44. Summary: The essential ingredients of the solar dynamo II  Advection by meridional flow –Certainly important at surface (observed) –Equatorward meridional flow in lower convection zone (theory, mass conservation) –How important compared to turbulent effects? Magnetic diffusivity Turbulent pumping (in radius and latitude)  Flux-transport dynamos are very successful models (consistent with observational constraints), but more research required

45. Summary: The essential ingredients of the solar dynamo III  Sunspot formation –Origin of field: stable stratification at base of convection zone –Strong magnetic flux tubes rising through convection zone (magnetic buoyancy) –Coriolis force leads to systematic tilt  Open questions –How to keep flux tube coherent in turbulent convection zone? Initial twist of tube required, but that also influences tilt angle –Rising tubes prefer long wave numbers (m=1,2) Sunspots are of much shorter wave number –Decoupling between emerged sunspot and its magnetic root at base of convection zone?

46. The Future  Much more computing power –Better understanding of essential ingredients in the short run –3D dynamo model in the long run  Observational constraints –Helioseismology Meridional flow Magnetic field in convection zone? –Solar-stellar connection How do cycle properties depend on rotation rate and depth of convection zone?