The Sun’s internal rotation Michael Thompson University of Sheffield

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

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

Radial Stratification

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

Helioseismic determination of location of base of convection zone at / 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.

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 submitted; C-D 2004)

Mean internal rotation

Rotation rate inferred from MDI data using two different analysis techniques

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

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

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.

The Tachocline

Artificial dataSolar data Charbonneau et al. (1999)

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.

Near-surface shear layer

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

Mean shear gradient as function of latitude Logarithmic derivative

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.

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.

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

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

Temporal variations of the Sun’s internal rotation

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

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.

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.

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

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)

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

Flows inferred with ring-diagram analysis (10 Mm deep) Haber et al

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

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

SSW and Active Complex Mm 16 Mm

Flows near and beneath active region Apr 2001

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

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

Summary: Variability in the convection zone Komm et al Angular momentum variations Aug 2002May 1996 Evolving of zonal flows Vorontsov et al 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?

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

Boberg et al Solar mean magnetic field 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

Toroidal magnetic field strengthAngular velocity variations Time (yrs) Latitude Equator Time (yrs) 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):

Summary: variability in and near tachocline Howe et al 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 ?

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?