1. The Sun’s internal rotation Michael Thompson University of Sheffield michael.thompson@sheffield.ac.uk

2. Global-mode seismology Measure mode properties ω; A, Γ; line-shapes Eigenfunctions / spherical harmonics Frequencies ω nlm (t) depend on conditions in solar interior determining wave propagation ω nlm – degeneracy lifted by rotation and by structural asphericities and magnetic fields Inversion provides maps such as of c and ρ and rotation and wave-speed asphericities North-south averages Snapshots at successive times: typically 2-month averages Spherical harmonics

3. Data from space – in particular from three instruments on the SOHO satelite SOI-MDI GOLF VIRGO Data from ground-based networks, in particular GONG, BiSON, LOWL GONG network sites

4. Radial Stratification

5. Sound speed profile Temperature (Sound speed is in units of hundreds of km/s)

6. Helioseismic determination of location of base of convection zone at 0.713 +/- 0.003 R (Christensen-Dalsgaard et al. 1991). No temporal or latitudinal variation in this location (Basu & Antia 2001). Note that this measures the extent of the essentially adiabatically stratified region. Sun’s transition smoother than in models.

7. Christensen-Dalsgaard (2004) Sun minus model New model (Z=0.015) Old model (Z=0.020) Old ‘standard’ Model S (Christensen-Dalsgaard et al. 1996) left little room for overshoot – near-perfect match to depth of Sun’s convection zone. But newly determined abundances give a model with 1-2% shallower convection zone. (See Basu & Antia 2004; Bahcall et al. 2004 submitted; C-D 2004)

8. Mean internal rotation

9. Rotation rate inferred from MDI data using two different analysis techniques

10. Radial cuts through inferred rotation profile of the solar interior (at latitudes indicated)

11. Dashed lines are at 25 degrees to polar axis – a better match to the isorotation contours than radial lines (or Taylor columns!)

12. Rotation rate in the deeper interior (LOWL + BiSON data) Fast core strongly ruled out (unless very small). Possibly slightly slow core rotation. But inversions consistent with uniform rotation in deep interior.

13. The Tachocline

14. Artificial dataSolar data Charbonneau et al. (1999)

15. Kosovichev (1996) – fit to early data, characterizing the tachocline transition with the function 0.5{ 1 + erf [2 ( r - r 0 ) / w ] } Obtained width w = (0.09+/-0.05)R and central location r 0 = (0.692+/-0.005)R, i.e. just beneath base of convection zone. More recent analyses have typically yielded substantially smaller width, e.g. Charbonneau et al. (1999): width w = (0.039+/-0.013)R Position r0 = (0.693+/-0.002)R at equator. Prolate with location different by (0.024+/-0.004)R at 60 degrees. Extent of prolateness confirmed by Basu & Antia (2001), who further found no temporal variation in the tachocline properties.

16. Near-surface shear layer

17. Near-surface rotation rate from inversion of f-mode splittings

18. Mean shear gradient as function of latitude Logarithmic derivative

19. Logarithmic derivative Gradient using only l<250 f modes Conclusion: Rotational gradient negative (logarithmic derivative approx. -1) at low-/mid-latitudes. Vanishes just above 50 degrees latitude. Small or positive gradient in shallow layer at high latitude.

20. High-latitude jet (Schou et al. 1998) Evidence weak – could be real but time-dependent. Jet at 75 degrees apparent in RLS inversion of first 144-days of MDI data.

21. Mean Angular Velocity in 3-D Models REASONABLE CONTACT WITH HELIOSEISMICDEDUCTIONS Case AB Case AB Brun & Toomre Courtesy J. Toomre

22. Angular Momentum Balance REYNOLDS STRESSES (R) transport angular momentum TOWARD equator, unlike MERIDIONAL CIRCULATION (MC) SLOW POLES from weak MC at high latitudes R MC R MCV V total total MERIDIONALCIRCULATION Poleward flow at equator Courtesy J. Toomre

23. Temporal variations of the Sun’s internal rotation

24. 19862000 Torsional oscillations (Howard & LaBonte 1980) Evolution of surface velocity field (upper panel) and surface magnetic field (lower panel) Ulrich 2001

25. Torsional oscillations of whole convection zone Differencing rotation inversions relative to solar minimum (1996) at successive 72-day epochs reveals the torsional oscillations at low and high latitudes through much of the convection zone.

26. It is also revealing to stack fewer results, so the evolution of the whole convection zone can be seen. Here we difference rotation inversions relative to solar minimum at successive 1-year epochs.

27. Variation of rotation at fixed depth (3 different inversions) GONG RLS MDI RLS MDI OLA 0.99 R0.95 R0.90 R0.84 R

28. 0.99 R 0.95 R 0.90 R 0.84 R Equator30 o 60 o 75 o Systematic variations – consistent between analyses Howe et al. (2004)

29. Variation of rotation at fixed depth (3 different inversions) 30 o 45 o 15 o Equator GONG MDI 0.8 R 0.9 R

30. Flows inferred with ring-diagram analysis (10 Mm deep) Haber et al. 1999 1997

31. Longitudinal averages of flows from ring-diagram analysis Zonal flows at 1 Mm and 7 Mm depth (note torsional oscillation) Meridional flows Mostly poleward but with transient counter-cell in N hemisphere

32. Comparison of near-surface torsional oscillation flow from global analysis (contours) and ring analysis (coloured shading) – symmetrized.

33. SSW and Active Complex 9393 7 Mm 16 Mm

34. Flows near and beneath active region Apr 2001

35. Phase of variations in the CZ (A) Amplitude and ( B) phase of 11-year harmonic fit (C) 22-yr extrapolation as function of latitude; with residuals also shown (D) 22-yr extrapolation as function of depth; with residuals also shown

36. Comparison of phase of variations with isorotation contours (nb different colour scheme) Constant-phase surfaces tend to align with contours of constant rotation.

37. Summary: Variability in the convection zone Komm et al. 2003 Angular momentum variations Aug 2002May 1996 Evolving of zonal flows Vorontsov et al. 2002 Torsional oscillations – what is relation to magnetic field? Whole convection zone involved Strong evolution at high latitude Physical understanding of the phase propagation of perturbations through the convection zone? Role of polar open field-lines?

38. Variability in and near the tachocline Rotation residuals show a quasi-periodic 1.3-yr variation at low latitudes (Howe et al. 2000).

39. Boberg et al. 2002 Solar mean magnetic field 19752000 Wavelet analysis of the Sun’s mean photospheric magnetic field: prominent periods are the rotation period and its 2 nd harmonic, and the 1.3/1.4-yr period

40. 0 11 22 Toroidal magnetic field strengthAngular velocity variations Time (yrs) Latitude Equator Time (yrs) 0 11 22 Depth Surface CZ base The evolution of the angular velocity and the magnetic field are coupled. Results of Covas et al. (2000, 2001) model (spatio-temporal fragmentation):

41. Summary: variability in and near tachocline Howe et al. 2000 19962002 Variations in Ω ( r, θ ; t ) 1.3-yr variations in inferred rotation rate at low latitudes above and beneath tachocline Signature of dynamo field evolution? Radiative interior also involved in solar cycle? Link between tachocline and 1.3/1.4-yr variations in solar wind, aurorae, solar mean magnetic field ?

42. Some questions What further constraints on tachocline / c.z. base would be useful? Is comparison of variability in c.z. between Sun and numerical models worthwhile? What determines the phase propagation of variations in the c.z.? What is the nature of the torsional oscillation? What causes the high-latitude variability?